QuantumChromodynamicsandstronginteractionphysicsfjeger/QCD-lectures.pdf · The Lie-algebra (commutation rules) [Ti,Tk] = iciklTl determines the structure constants cikl, which are

Quantum Chromodynamics and strong interaction physics

F. Jegerlehner∗

Humboldt-Universitat zu Berlin, Newtonstrasse 15, D-12489 Berlin, Germany

∗Lectures given at the University of Silesia, Katowice, Poland. * * * * * * * * * * * * * * * * * * * * * * * * * ** * * * * * * * * * * * * * * * * * * * * * * * * (see: http://www-com.physik.hu-berlin.de/ fjeger/books.html )

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Contents

1 Internal symmetries, unitary groups 1

1.1 Finite dimensional representations of SU(n) . . . . . . . . . . . . . . . . . . . . . . 4

1.2 Combining representations, reduction . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Application to SU(3)flavor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.4 The spectrum of low lying hadrons: . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 One particle states: pions and nucleons . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5.1 Interpolating fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5.2 Noether Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.6 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Spontaneous symmetry breaking and Nambu–Goldstone bosons 26

2.1 The Goldstone theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2 Models of spontaneous symmetry breaking . . . . . . . . . . . . . . . . . . . . . . . 30

2.2.1 A model with spontaneous breaking of a discrete symmetry . . . . . . . . . 31

2.2.2 The Goldstone model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3 Chiral symmetry and quark flavor mixing 38

3.1 Prolog: QCD as a part of the Standard Model of fundamental interactions . . . . . 38

3.1.1 The matter fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.1.2 The gauge fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.2 Chiral transformations, chiral symmetry and the axial-vector anomaly . . . . . . . 42

3.2.1 Chiral fields and the U(1)-axial current . . . . . . . . . . . . . . . . . . . . 42

3.2.2 The chiral group U(n)V ⊗ U(n)A . . . . . . . . . . . . . . . . . . . . . . . . 45

3.2.3 The Adler Bell, Jackiw triangle anomaly . . . . . . . . . . . . . . . . . . . . 47

3.3 Yukawa interaction of quarks and quark flavor mixing . . . . . . . . . . . . . . . . 51

3.3.1 Flavor mixing pattern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4 The Chiral Structure of Low Energy Effective QCD . . . . . . . . . . . . . . . . . 61

4 Gauge principles and gauge invariance 68

4.1 Abelian gauge theory: Quantum Electrodynamics . . . . . . . . . . . . . . . . . . . 68

4.2 Non-Abelian gauge theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2.1 Global non-Abelian symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2.2 Local non-Abelian symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2.3 Minimal couplings of the matter fields . . . . . . . . . . . . . . . . . . . . . 78

4.2.4 Non-Abelian gauge field strength tensor, Yang-Mills action . . . . . . . . . 80

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4.2.5 Equations of motion and currents . . . . . . . . . . . . . . . . . . . . . . . . 82

5 Quantization of gauge theories 87

5.1 The QCD Feynman rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6 Dimensional regularization 93

6.1 Tools for the Evaluation of Feynman Integrals . . . . . . . . . . . . . . . . . . . . . 96

6.1.1 ǫ = 4− d Expansion, ǫ→ +0 . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.1.2 Bogolubov-Schwinger Parametrization . . . . . . . . . . . . . . . . . . . . . 97

6.1.3 Feynman Parametric Representation . . . . . . . . . . . . . . . . . . . . . . 97

6.1.4 Euclidean Region, Wick–Rotations . . . . . . . . . . . . . . . . . . . . . . . 98

6.2 Scalar One–Loop Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.3 Selected Properties of Scalar Integrals . . . . . . . . . . . . . . . . . . . . . . . . . 102

6.4 Tensor Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.5 Special Integrals for massless propagators . . . . . . . . . . . . . . . . . . . . . . . 105

7 One–Loop Renormalization 111

7.1 The quark self–energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.2 The gluon self–energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7.3 The Faddeev-Popov ghost self–energy . . . . . . . . . . . . . . . . . . . . . . . . . 125

7.4 The quark-gluon vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

8 The renormalization group, QCD and asymptotic freedom 138

8.1 Renormalization Group: the general solution . . . . . . . . . . . . . . . . . . . . . 138

8.2 Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

8.2.1 Short distance or UV behavior . . . . . . . . . . . . . . . . . . . . . . . . . 143

8.2.2 Long distance or IR behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 144

8.3 Universality of the asymptotic behavior . . . . . . . . . . . . . . . . . . . . . . . . 148

8.4 RG-invariance of physical observables . . . . . . . . . . . . . . . . . . . . . . . . . 149

8.5 The “running” parameters of QCD . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

8.6 αs in perturbation theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

8.7 How does the RG work for asymptotically free theories? . . . . . . . . . . . . . . . 157

8.8 Decoupling of heavy states and the decoupling theorem . . . . . . . . . . . . . . . 159

8.9 MOM scheme, automatic decoupling . . . . . . . . . . . . . . . . . . . . . . . . . . 161

8.10 Nf - flavor effective QCD and matching conditions . . . . . . . . . . . . . . . . . . 164

8.11 Renormalization Scheme Dependence . . . . . . . . . . . . . . . . . . . . . . . . . . 170

8.12 General O(α2) framework for predicting R(s) . . . . . . . . . . . . . . . . . . . . . 176

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8.12.1 Current correlators and the Kallen-Lehmann spectral representation . . . . 186

8.13 High energy hadron-production in e+e−–annihilation . . . . . . . . . . . . . . . . . 188

8.14 Non–Perturbative Effects, Operator Product Expansion . . . . . . . . . . . . . . . 195

8.14.1 Testing non–perturbative hadronic effects via the Adler function . . . . . . 197

8.15 Some considerations on hadron production at low energy . . . . . . . . . . . . . . . 200

8.15.1 Pions and Scalar QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

8.15.2 The Vector Meson Dominance Model . . . . . . . . . . . . . . . . . . . . . . 203

8.15.3 Mass and Width of Vector Mesons . . . . . . . . . . . . . . . . . . . . . . . 205

8.15.4 ρ0 − γ–Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

8.15.5 Vector–Meson Production Cross–Sections . . . . . . . . . . . . . . . . . . . 208

8.15.6 ρ0 − ω–Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

8.15.7 Gounaris-Sakurai Parametrization of Fπ(s) . . . . . . . . . . . . . . . . . . 210

8.15.8 The Theory of the Pion Form Factor in the Threshold Region . . . . . . . . 211

8.15.9 Beyond Chiral Perturbation Theory: Models with Massive Spin 1 Fields . . 217

9 Quarkonia 223

9.1 Charmonium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

9.2 Charmed Mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

9.3 The Υ resonances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

9.4 Bottom Mesons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232

9.5 Non-Relativistic Potential Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233

9.5.1 More on the heavy quark potential . . . . . . . . . . . . . . . . . . . . . . . 240

9.6 A digression: positronium in QED . . . . . . . . . . . . . . . . . . . . . . . . . . . 245

9.6.1 e+e−–annihilation into photons: e+e− → 2γ, 3γ . . . . . . . . . . . . . . . 247

9.7 Quarkonia Decays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251

9.8 Matrix elements of mesonic bound states . . . . . . . . . . . . . . . . . . . . . . . . 254

9.9 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

9.9.1 Determination of αs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

10 Jets 269

10.1 QCD corrections O(αs) to the 2–jet cross–section . . . . . . . . . . . . . . . . . . . 276

10.1.1 Virtual corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

10.1.2 Real soft gluons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

10.1.3 Virtual plus real soft gluons . . . . . . . . . . . . . . . . . . . . . . . . . . . 280

10.2 3–Jet Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

10.3 Summary and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

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A Solved Problems 289

A.1 Exercises: Section ?? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289










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1 Internal symmetries, unitary groups

Empirically, physical states are known to often show up in multiplets of symmetry groups. Fa-miliar examples are the flavor symmetries in strong interaction physics SU(2)flavor (= isospin),SU(3)flavor (= isospin plus hypercharge), etc. Internal symmetries are to be distinguished fromthe space-time symmetries, which are unified in the Poincare group. The symmetry groups G ofinterest here are the groups SU(n), defined by the set of complex n × n-matrices U which areunitary (U−1 = U+) and unimodular (detU = 1) and with matrix multiplication as the groupoperation. The requirement of unitarity ensures that the transition probabilities between statesare preserved:

|< ϕ | ψ >|2=|< ϕ′ | ψ′ >2=|< ϕ | U+U | ψ >|2

of course | ψ >→| ψ′ >= U | ψ > is a symmetry if and only if all group elements U commutewith the total Hamiltonian H of the system:

[U,H] = 0 ∀ U ∈ G.

Since any unitary matrix U can be written as a product U = Ueiϕ of a matrix U with detU = 1and a phase factor eiϕ, a unitary group U(n) is equivalent to a direct product SU(n) ⊗ U(1).Therefore we may restrict ourselves to a consideration of the simple groups SU(n). Possible U(1)factors may be discussed separately.

The groups SU(n) have r = n2− 1 real continuous parameters ωi (i = 1, . . . , r). A complex n×nmatrix has 2n2 real parameters, unitarity implies n2 conditions and detU = 1 yields one furthercondition. Therefore, SU(n) is characterized by r infinitesimal generators Ti and a generalSU(n) transformation can be written as

U = U(ω) = exp

i

n2−1∑

i=1

Tiωi

and r is called order of the group.

The generators are Hermitian Ti = T+i (which guarantees that U is unitary), traceless Tr Ti = 0

(which implies detU = 1) and may be normalized so that Tr (TiTj) = 12δij .

A convenient (non unique) basis for the matrices Ti, written conventionally as Ti = λi/2, can beconstructed as follows. For the n− 1 possible diagonal traceless Hermitian λi choose

1−1

0. . .

0

,

1√3

11−2

0. . .

0

, . . . ,

√2

n(n− 1)

11

1. . .

−(n− 1)

. (1.1)

Then form the n(n−1)2 off-diagonal matrices λi with 1 in a given off-diagonal position above the

diagonal, 1 in the transposed position and zeros elsewhere. Also form the n(n−1)2 off-diagonal

matrices λi with a −i in a given off-diagonal position above the diagonal, +i in the transposedposition and zeros elsewhere.

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The Lie-algebra (commutation rules)

[Ti, Tk] = iciklTl

determines the structure constants cikl, which are real, totally antisymmetric and satisfy theJacobi identity

cikncnlm + cyclic terms in(ikl) ≡ 0.

A Lie group G and its structure constants cikl uniquely determine each other in a neighborhoodof the identity element of G. In a Lie-algebra there is a maximum number l of simultaneouslycommuting (i.e. diagonal) elements. l is called the rank of the group. The SU(n) groups haverank l = n − 1, which is obvious in the basis given above. The states belonging to a SU(n)multiplet may be labeled, as usual, by the eigenvalues of the simultaneous eigenstates of the ldiagonal matrices which we denote by H1, . . . ,Hl. The structure of a multiplet is characterizedby a weight diagram which displays the eigenvalues of the states on a (H1, . . . ,Hl) plot.

The remaining generators may be combined into pairs of ladder operators (a raising and alowering operator) E±α(α = 1, . . . , r−l2 ) which map the different eigenstates of a multiplet intoeach other. The E±α’s are non-Hermitian matrices with 1 in a given off-diagonal position andzeros elsewhere.

For SU(2) and SU(3) we list some basic properties in the following.

a) SU(2) : Order r = 3 , rank l = 1

Structure constants: cikl = ǫikl the fully antisymmetric permutation tensor.

Generators: Ti = τi2 ; τi the Pauli matrices 1

τ1 =

0 1

1 0

, τ2 =

0 −i

i 0

, τ3 =

1 0

0 −1

Diagonal operators: H1 = τ32 = I3 : 3rd component of isospin.

Eigenvectors: 1

0

,

0

1

Eigenvalues of I3 : 12 ,−1

2Ladder operators: E±1 = 1

2 (τ1 ± iτ2)

E+1 =

0 1

0 0

, E−1 =

0 0

1 0

1Properties of the Pauli matrices:

[τi, τk] = 2iǫiklτl , τi, τk = 2δik

τ+i = τi , τ 2i = 1 , T r τi = 0

τiτk =1

2τi, τk+

1

2[τi, τk] = δik + iǫiklτl

2

• •+d u

−12 +1

20

I3

E+1

E−1

Figure 1.1: Weight diagram for the (u, d) quark doublet.

b) SU(3) : Order r = 8 , rank l = 2

Structure constants : cikl = fikl , where the non-vanishing entries are permutations of theelements

f123 = 1

f147 = f165 = f246 = f257 = f345 = f376 = 1/2

f458 = f678 =√

3/2.

Generators: Ti = λi2 ; λi the Gell-Mann matrices 2

λ1 =

0 1 01 0 00 0 0

, λ2 =

0 −i 0

i 0 0

0 0 0

, λ3 =

1 0 0

0 −1 0

0 0 0

λ4 =

0 0 1

0 0 0

1 0 0

, λ5 =

0 0 −i0 0 0

i 0 0

, λ6 =

0 0 0

0 0 1

0 1 0

λ7 =

0 0 0

0 0 −i0 i 0

, λ8 = 1√

3

1 0 0

0 1 0

0 0 −2

Diagonal operators:

H1 =λ32

= I3 3rd component of isospin

2Properties of the Gell-Mann matrices:

[λi, λk] = 2i fiklλl , λi, λk =4

3δik + 2diklλl

Tr λi = 0 , T r λiλk = 2δik

Tr λi [λk, λl] = 4i fikl , T r λi λk, λl = 4i dikl

d118 = d228 = d338 = −d888 = 1/√3 ; d448 = d558 = d668 = d778 = −1/(2

√3) ;

d146 = d157 = −d247 = d256 = d344 = d355 = −d366 = −d377 = 1/2 .

3

H2 =λ82

.=

√3

2Y , Y hypercharge

Eigenvectors:

100

,

010

,

001

Eigenvalues of (I3, Y ) :(12 ,

13

),(−1

2 ,13

),(0,−2

3

)

Ladder operators:

E±1 = T1 ± iT2 : E+1 =

0 1 00 0 00 0 0

, E−1 =

0 0 01 0 00 0 0

E±2 = T4 ± iT5 : E+2 =

0 0 10 0 00 0 0

, E−2 =

0 0 00 0 01 0 0

E±3 = T6 ± iT7 : E+3 =

0 0 00 0 10 0 0

, E−3 =

0 0 00 0 00 1 0

d u

I3

s

Y

13

−23

+ +−1

2 +12

E+1

E+3 E+2

E−1

E−3 E−2

Figure 1.2: Weight diagram for the (u, d, s) quark triplet.

1.1 Finite dimensional representations of SU(n)

Given the structure constants cikl of SU(n) any set of Hermitean traceless N × N matricesTi(i = 1, . . . n2 − 1) satisfying the Lie-algebra

[Ti, Tk

]= iciklTl

is called a representation of the SU(n) Lie-algebra. The unitary unimodular matrices U =

exp(i∑n2−1

i=1 Tiωi) then form a representation of SU(n). The smallest non-trivial irreducible rep-resentation is the fundamental representation of dimension N = n. This is the representationwhich defines SU(n). In gauge theories the fundamental spin 1/2 matter fields of quarks and lep-tons are in this representation. The Jacobi identity implies that there always exists the adjointrepresentation of dimension r with generators

(Ti)kl = −icikl

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In gauge theories the spin 1 gauge fields in any case must be in this representation, as we shallsee below.

The complex conjugate of a representation is also a representation since

(U1U2)∗ = U∗1U∗2 .

Two representations are equivalent if we can transform one into the other by a change of basis:

SD1(U)S−1 = D2(U) ∀ U ∈ G.

The conjugate representation n∗ of the fundamental representation n is not a new representationif it is equivalent to n. In fact for the fundamental representation 2 of SU(2) 2∗ is equivalent to2. In contrast, the conjugate representations 3∗ of the fundamental representation 3 of SU(3) isa new (inequivalent) representation. In QCD this crucial property of SU(3) allows to distinguishcolor triplets of quarks (which transform according to the 3 representation) from color triplets ofantiquarks (which transform according to the 3∗ representation).

A representation is called irreducible if it cannot be transformed by a change of basis to block-diagonal form:

D(U) =

D1(U) 0

0 D2(U)

= D1(U)⊕D2(U) ∀ U ∈ G.

If such a transformation exist, the representation is reducible. The irreducible representationsare the basic building blocks of any representation. Particle multiplets are classified by theirreducible representations of a symmetry group.

The possible irreducible representations can be constructed by decomposing products of thefundamental representation into irreducible blocks. In the following we briefly discuss how thiscan be done.

1.2 Combining representations, reduction

Let ψi(i = 1, . . . n) be a vector transforming under the fundamental representation n of SU(n).A tensor product ψi1 . . . ψim forms a tensor ψi1...im which transforms according to

ψi1...im → ψ′i1...im = Ui1i′1 . . . Uimi′mψi′1...i′m .

For m > 1 this product representation, denoted by n⊗ n⊗ . . .⊗ n (m factors), is reducible.

One can decompose ψi1...im into a sum of tensors of different symmetry class with respect topermutations of the indices i1 . . . im as follows:

Choose a set of positive integers n1, n2 . . . , nk with n1 ≥ n2 ≥ . . . ≥ nk which form a partitionof m : n1 + n2 + . . . + nk = m. Then group the indices i1 . . . im into k classes (i11 . . . i1n1),(i21 . . . i2n2), . . . , (ik1 . . . iknk

) and write them in form of a tableau of k stacked rows where thefirst row has n1 boxes containing the indices i11 . . . i1n1 , the second row has n2 boxes containingthe indices i21 . . . i2n2 and so on. The tableau obtained is called a Young tableau (often called

5

Young diagram).

2.antisymmetrize

in columns↓

1. symmetrize in rows→

i11 i12 · · · · . . . i1n1

i21 i22 · · . . . i2n2

...

ik1 ik2 . . . iknk

With a Young tableau we associate a tensor of a given symmetry class by the following con-struction. By convention first symmetrize ψi1...im in each group of indices appearing in therows. Afterwards anti-symmetrize in the indices appearing in each column. By this a tensor ofa given symmetry class is defined. According to the convention (first symmetrize in rows thenanti-symmetrize in columns) tableaus with indices permuted in columns represent the same ten-sor. Tableaus with indices permuted in rows represent the same tensor if and only if the indicesare not anti-symmetrized with indices in a different row.

For SU(n) a tensor index can take the values i = 1, . . . , n only. Hence, there cannot be morethan n rows for anti-symmetrization.

One easily verifies that group transformations do not mix tensors from different symmetry classes.The following central theorem holds:

a) Tensors in a given symmetry class form an invariant irreducible subspace. The grouprepresentation induced (by projection to the invariant subspace) in this subspace by thefundamental representation is irreducible.

b) The irreducible representations generated through all possible symmetry classes are exhaus-tive (i.e. there are no irreducible representations which cannot be obtained this way).

Symmetrization and anti-symmetrization obviously reduces the number of independent compo-nents of a tensor. The number of independent components of a tensor of a given symmetry classis equal to the dimension of the irreducible representation.

The irreducible representation of highest dimension is represented by the totally symmetric tensor.

i1 i2 . . . im

There is only one such representation in n⊗ . . .⊗ n (m factors)

ψ(i1...im) =1

m!

∑

permutations p

ψip(1) . . .ip(m) .

A column with n boxes represents a tensor of rank zero i.e. a singlet and corresponds to a1-dimensional trivial representation:

ϕ = ǫi1...inψi1 . . . ψin

6

Therefore, if a column with n boxes is part of a larger Young tableau it can be omitted!

A n− 1 fold antisymmetric product of ψi’s transforms as a n∗ (complex conjugate of the funda-mental representation n):

χi = ǫii1...in−1ψi1 . . . ψin−1

since

ψiχi = ϕ = ǫii1...in−1ψiψi1 . . . ψn−1

is a singlet.

We now present some specific properties of SU(2) and SU(3):

a) SU(2) :The fundamental representation is 2 . 2∗ is equivalent to 2 . Indices have 2 possible valuesi = 1, 2.

A tableau is a singlet. as a part of a larger tableau can be omitted i.e. ≡ etc.All nontrivial representations are characterized by a row:

Tableau: , , , etc.

Dimension: 2 3 4

Product representations and their reduction follow by combining corresponding tableaus inall possible ways.

Examples: SU(2) interpreted as spin

1.

2⊗ 2 = × = + = 1⊕ 3

i.e. two spin 1/2 particles can group into a singlet of spin 0 and a triplet of spin1

2.

2⊗ 2⊗ 2 = ( + )× = + +

= + +

= 2⊕ 2⊕ 4

i.e. three spin 1/2 particles can group into two doublets of spin 1/2 and a quartet ofspin 3/2.

b) SU(3) :The fundamental representation is 3 . 3∗ is inequivalent to 3. Indices have 3 possible valuesi = 1, 2, 3.

A tableau is a singlet. as part of a larger diagram can be omitted i.e. ≡ . etc.

All non-trivial representations are characterized by tableaus with one or two columns:

︸︷︷︸q︸︷︷︸p

Each corresponds to a n∗ i.e. an irreducible representation is characterized by two indices(p, q) and transforms as a tensor

ψj1...jqi1...ip

symmetrized in (i1 . . . ip) and (j1 . . . jq)

where i1 . . . ip transform under 3 and j1 . . . jq under 3∗.

7

We may write ψj1...i1...in product form

ψj1...jqi1...ip

= χj1 . . . χjqψi1 . . . ψiq

with χi = ǫiklψkψl. Together with the symmetrization it can be shown that the trace condition

3∑

j=1

ψjj2...jqji2...ip

= 0

must hold. This restricts the number of independent components of the tensor, which equals thedimension of the irreducible representation D(p, q): One finds

D(p, q) =1

2(p + 1)(q + 1)(p + q + 2).

The generators Ti of a given irreducible representation can be worked out from the transformationlaw

ψ′j1...jqi1...ip= U∗j1j′1 . . . U

∗jqj′q

Ui1i′1 . . . Ujpj′pψj′1...j

′q

i′1...i′p

for infinitesimal transformations.

The simplest irreducible representation are given in the following table:

(p, q) D(p, q) tableau tensor

(0, 0) 1 1 singlet

(1, 0) 3 ψi triplet

(0, 1) 3∗ ψi antitriplet

(2, 0) 6 ψik sextet

(0, 2) 6∗ ψik antisextet

(1, 1) adjoint 8 = 8∗ ψik octet

(3, 0) 10 ψikl decaplet

(0, 3) 10∗ ψikl antidecaplet

1.3 Application to SU(3)flavor

Low lying hadronic states may be classified in SU(3)flavor multiplets. The relevant quantumnumbers are the baryon number B, isospin I and strangeness S. We can achieve that multipletsare centered on the origin if we replace strangeness S by hypercharge Y

Y = B + S.

Empirically, the electric charge of a hadron is given by

Q = I3 +Y

2.

8

In the quark model of hadrons mesons (B = 0) are quark – antiquark states

M = (qq)

baryons (B = 1) are three quark states

B = (qqq)

where q = u, d, s. The quarks (u, d, s) are in the fundamental representation 3, the antiquarks(u, d, s) in the representation 3∗.

d u

I3

s

Y

13

−23

+ +−1

2 +12

du

I3

s

Y

−13

23

+++1

2−12

Figure 1.3: Basic building blocks of the SU(3) quark model.

Direct products of representations may be reduced (decomposed) into irreducible blocks by com-bining boxes of the corresponding Young tableaux in all possible ways with the restriction thatantisymmetric pairs must be preserved. The latter condition is nontrivial but may be satisfiedby the following construction:

In order to append to the first tableau the second one in all admissible ways which respect the(anti -) symmetrization, we place in each box of the second tableau letters (in lexicographic order)with identical letters in each given row (symmetrized). Thus we insert a’s in the first row, b’s inthe second row, etc. All boxes of the second tableau are now appended to the right-hand ends ofthe rows of the first one (which represents the upper left-hand corner of the new diagram) in allpossible ways. Thus we first append all a’s to the first tableau (in all admissible ways) with nomore than one a per column (anti-symmetrized). To the such obtained enlarged tableaux appendall b’s (in all admissible ways) with no more than one b per column, etc.

Some of the tableaux such obtained are not admissible because they do not take into accountproperly the (anti -) symmetrization of the original boxes and have to be thrown away (also inorder to avoid double counting).

Here we need a definition: a sequence of letters a, b, c, · · · is admissible if at any point of thesequence at least as many a’s have occurred as b’s, at least as many b’s have occurred as c’s etc.

Examples: a) admissible: abcd, aabcb, ....

b) not admissible: abb, acb, ...

Now consider for each tableau constructed above the full sequence of letters formed by readingfrom right to left in the first row, then in the second row etc. The tableaux which we have tothrow away are those which lead to sequences of letters which are not admissible.

The properties of the composed new tableaux may be summarized as follows:

1. Each tableau must be a Young tableau.

9

2. The number of boxes in the new tableau must be equal to the sum of the number of boxesin the original two tableaux.

3. If dealing with SU(n), no tableau has more than n rows.

4. Making a journey through the tableau starting with the top row and entering each row fromthe right, at any point the number of b’s encountered in any of the attached boxes mustnot exceed the number of previously encountered a’s and the number of c’s encountered inany of the attached boxes must not exceed the number of previously encountered b’s, etc.

5. The letters must be in anti-lexicographical order when reading across a row from left toright.

6. The letters must differ and be in lexicographic order when reading a column from top tobottom.

The first three rules should be obvious. The purposes of the three rules 4) to 6) are to assurethat states which were previously symmetrized are not anti-symmetrized in the product and viceversa, and to avoid double counting states.

Examples:

1. 3⊗ 3 = × = + = 3∗ ⊕ 6

2. 3⊗ 3∗ = × = + = 1⊕ 8

3. 3⊗ 3⊗ 3 = ( + )× = + + +

= 1⊕ 8⊕ 8⊕ 10

More than two tableaux may be combined by first combining the first two, then combining theresult with the third one and so on.

Exercise: Show that 8⊗ 8 = 1⊕ 8⊕ 8⊕ 10⊕ 10⊕ 27.

The quantum numbers of quarks are given by:

Quark spin B Q I3 S Y

u 1/2 1/3 2/3 1/2 0 1/3

d 1/2 1/3 −1/3 −1/2 0 1/3

s 1/2 1/3 −1/3 0 −1 −2/3

Exercise: Use the Young tableaux to construct the meson states in

3⊗ 3∗

and the baryon states in

3⊗ 3⊗ 3 .

Notice that the indices of the tensors ψj1...jqi1...ip

, which characterize a irreducible representation (p, q),in the quark model have two different interpretations. Each upper index has associated either

10

an antiquark or an anti-symmetrized pair of quarks. For lower indices antiquarks and quarks areinterchanged: i.e.

Upper index :

u

d

s

or

(ds)

(su)

(ud)

B = −1/3 B = 2/3

Lower index :

u

d

s

or

(ds)

(su)

(ud)

B = 1/3 B = −2/3

where (ud) etc. denote anti-symmetrized pairs. Which interpretation is to be used is uniquelyfixed if we specify the baryon number B of the state.

1.4 The spectrum of low lying hadrons:

Mesons: qq′ bound statesA qq′ with orbital angular momentum L has Parity P = (−1)L+1. For q′ = q we have a qq boundstate which is also an eigenstate of charge conjugation C with C = (−1)L+S , where S is the spin 0or 1. The L = 0 states are the pseudoscalar mesons, JP = 0−, and the vectors mesons, JP = 1−.

In the limit of exact SU(3) the pure states would read

π0 = (uu− dd)/√2

η1 = (uu + dd + ss)/√3

η8 = (uu + dd− 2ss)/√6

ρ0 = (uu− dd)/√2

ω1 = (uu + dd + ss)/√3

ω8 = (uu + dd− 2ss)/√6

In fact SU(2)flavor breaking by the quark mass difference md−mu leads to ρ−ω–mixing [mixingangle ∼] (Glashow 1961) [4]:

ρ0 = cos θ ρ′ + sin θ ω′

ω = − sin θ ρ′ + cos θ ω′

Similarly, the substantially larger SU(3)flavor breaking by the quark masses, leads to large ω−φ–mixing [mixing angle ∼ 36 close to so called ideal mixing where φ ∼ is a pure ss state] (Okubo1963) [5]:

φ = cos θ ω8 + sin θ ω1

ω = − sin θ ω8 + cos θ ω1

11

•

••

•

• •

•••π0 η η′

π+π−

K+K0

K0K−

duud

susd

dsus

I3-1 - 1

2 0 + 1

2+1

Y

-1

0

+1

•

•••

• •

•••ρ0 φ ω

ρ+ρ−

K∗+K∗0

K∗0K∗−

duud

susd

dsus

Y

-1

0

+1

Figure 1.4: The pseudoscalar-meson and vector-meson octets plus the singlets (nonets). Stateswith the same conserved quantum numbers mix, like η8, η1 → η, η′ and ρ− ω and ω − φ.

12

•

••

•

• •••Λ Σ0

uds

Σ+Σ−

pn

Ξ0Ξ−

uusdds

uududd

ussdss

I3

- 3

2 -1 - 1

2 0 + 1

2+1 + 3

2

Y

-1

0

+1

•

••

•• •

•

••

•Σ0

uds

Σ+Σ−

∆+∆0

Ξ0Ξ−

∆++∆−

Ω−

uusdds

uududd

ussdss

uuuddd

sss

Y

-2

-1

0

+1

Figure 1.5: The baryon octet and decaplet

13

1.5 One particle states: pions and nucleons

Here we briefly present some notation for one particle states and their interpolating fields. Thenormalization of the states is given by

〈α(p)|β(p′)〉 = (2π)3 2ω(p) δ(3)(~p− ~p′ ) δαβ ,

with ω(p) =√~p 2 +m2, m the mass of the state, ~p its momentum.

π–mesons:

|π+(p)〉, |π0(p)〉, |π−(p)〉 are one pion states of momentum p. Some times it is convenient to usea cartesian basis for the π’s: |πa(p)〉 a = 1, 2, 3. The charge eigenstates are then given by

|π+(p)〉 =1√2

|π1(p)〉+ i|π2(p)〉

,

|π0(p)〉 = |π3(p)〉 ,

|π−(p)〉 =1√2

|π1(p)〉 − i|π2(p)〉

,

such that

T±|π0〉 = ∓√

2 |π±〉 .

The pseudoscalar mesons may be written as a SU(2) isospin triplet |t3; p〉π with t3 = 1, 0,−1 forπ+, π0, π−.

Nucleons:

the nucleon states |p(p, s)〉, |n(p, s)〉 are in a SU(2) isospin doublet |t3; p, s〉N with t3 = 12 the

proton and t3 = −12 the neutron. s denotes the spin [= ±1

2 ]. Similar for the other hadrons.

1.5.1 Interpolating fields

For the pion in the Cartesian basis the pseudoscalar fields are real

ϕπa(x) = ϕπa(x)+

and create or annihilate a bare πa and are normalized by (free field = plane wave solution)

〈0|ϕπa (x)|πb(p)〉 = e−i px δ ba .

As a SU(2) representation one may write the multiplet field as a triplet field

φ(x) =

ϕ+

ϕ0

ϕ−

or as 2× 2 matrix field

Φ(x) =1√2

3∑

a=1

Taϕa(x) =

ϕ0/

√2 ϕ+

ϕ− −ϕ0/√

2

in the Cartesian and charged basis, respectively. In the latter representation the field is an ele-ment of the SU(2) Lie algebra.

14

Similarly, for the nucleons we have an interpolating field ψp(x) and ψn(x), which we may writeas an isodoublet field

ΨN (x) =

ψp(x)

ψn(x)

.

As usual proton field ψp(x) creates a bare p at x and/or annihilates a bare p at x and similar forthe neutron. With (ΨN )t3=+ 1

2= ψp and (ΨN )t3=− 1

2= ψn the normalization in this case reads

〈0| (ΨN )t3 (x) |t′3; p, s〉N = e−i px δt3,t′3 .

1.5.2 Noether Theorem

Symmetries of physical systems usually correspond to an invariance under a change of frames andmathematically are described by Lie Groups of transformations like the unitary groups discussedabove. The key insight about symmetries we owe to Emmy Noether 1918: the Noether theorem.It says that if a system (e.g. as defined by a certain Lagrangian) is invariant under a set ofsymmetry transformations there esists a corresponding set of conserved four–currents jµi (x) =(ρi(t, ~x),~ji(t, ~x)) satisfying the continuity equation ∂µj

µi (x) = 0. The trsnsformations are then

generated by a corresponding set of generalized charges [which represent the generators of thesymmetry group] which are given by space - integrals

Qi =

∫d3x j0i (t, ~x)

of the j0i component of a conserved current. Current conservation implies the time independenceof the “charges” Qi:

dQi

dt = 0 [“charge” conservation].

In hadron physics the flavor currents related to the flavor symmetries SU(2) (isospin), SU(3)(isospin+strangeness), · · · play an important role. This will be discussed in Sects. 3.2and 4.2.1.

15

1.6 Outlook

Nature plays all kind of games with symmetries. There are global and local space-time symmetriesand global and local internal symmetries. The latter determine strong, weak and electromagneticinteractions by the local gauge group

Gloc = SU(3)c ⊗ SU(2)L ⊗ U(1)Y .

SU(3)c is the color gauge group of the strong interactions (QCD). SU(2)L ⊗ U(1)Y is the elec-troweak gauge group (SU(2)L the left-handed weak isospin and U(1)Y the weak hypercharge)which is broken to the Abelian electromagnetic U(1) em of QED. Local space-time symmetry(general coordinate transformation invariance, general relativity) leads to classical gravity theory.Gravitational interactions break Poincare invariance, which “in practice” appears as an absoluteglobal symmetry due to the extreme weakness of the gravitational force. Global charge-like sym-metries remain the most mysterious symmetries. The only known exact global symmetries areAbelian symmetries with the associated quantum numbers being quantized.

Q electric charge U(1)Q

B baryon number U(1)B .

The quantization of these charges is barely understood, notice that U(1) invariance only impliesthe conservation of the corresponding charge not its quantization3.

Baryon number B is an additive quantum number like the electric charge Q. It derives from aglobal U(1)B invariance of all standard model interactions. Empirically it is tested most accu-rately by the stability of the proton. Proton lifetime limits are

τgeop > 1.6× 1025 years (geochemical estimate independent on decay modes)

τ labp > 1031 − 3× 1032 years (absence of “main” decay modes) .

Possible decay modes which where searched for are:

p → e+γ, e+π0, e+ρ0, e+ω, e+K0, . . . ,

νeπ+, νeρ

+, νeK+, . . . , (e→ µ)

p → π+ + π0

By convention B(p) = 1, B(e−) = 0 . All observations support the assignments B(B) = 1 andB(B) = −1 for baryons B and antibaryons B, respectively. All other particles have B = 0 .

Excursion on the Baryon Asymmetry Of The Universe: While we have no direct evidence that baryon number is notstrictly conserved we know that at some level, presumably not far from current experimental limits, baryon numbermust be violated. Otherwise the observed baryon asymmetry, the asymmetry between matter and antimatterobserved in our universe (galaxies, stars, planets,... but no anti–galaxies,anti–stars, anti–planets,...) could not be

3Note that also the lepton numbers Lℓ(ℓ = e, µ, τ ), related to approximate U(1)Lℓsymmetries, are known to be

conserved with high accuracy. They are broken at the level of the tiny neutrino masses, which must be non-zeroand non-degenerate in order to allow for the observed neutrino oscillations.

16

explained from properties of the fundamental interactions of nature. It would (and could) be just an accidentalasymmetry in the initial condition at the moment of the creation of the universe at the big–bang.

In the early very hot universe matter and antimatter was created by the highly energetic photon collisions at(almost) equal rate. In fact with a tiny asymmetry: per unit density ρB=1.000000000 of antimatter a portionρB=1.000000001 of matter must have been created. The expansion of the universe cooled down the radiation andmatter and antimatter annihilated almost completely into photons with the relict ρB − ρB=0.000000001 of matter.Thus the baryon number in units of the number of photons in the universe is NB/Nγ ∼ 10−9.

A theory which is able of explaining the origin of the baryon asymmetry must satisfy three conditions (Sakharov1967):

• It must violate B,

• it must violate CP and

• the universe must be out of thermal equilibrium.

The latter condition is satisfied since we know that the universe is expanding and not in a stationary equilibriumstate.

Within the SM B is conserved to all orders in perturbation theory and only violated by extremely tiny non–perturbative effects. The latter are due to the existence of non–trivial classical solutions of the SU(2) Yang–Mills equations, the so called instantons (Belavin et al. 1975). Quantum effects due to these four-dimensionalpseudoparticles lead to symmetry breaking via Adler-Bell-Jackiw anomalies (’t Hooft 1976).

While B is practically conserved in the SM, CP violation is naturally incorporated in the SM with three (or more)families of quarks and leptons (Kobayashi and Maskawa 1973). In fact three families are known to exist, the lastmember of the third family the top quark with a mass of about 175 GeV has been found some time ago (CDFand D0 at Fermilab 1995). More recently CP violation has been established (Babar at SLAC and Belle at KEK2001) in the B–meson system to be a large effect in accord with the SM prediction given the CP violation in theK0 − K0 system, which has been discovered as a small effect ε ≃ 2.3 × 10−3 long time ago (Christenson, Cronin,Fitch and Turlay 1964).

The SM is very unlikely able to predict the correct size of the baryon asymmetry. The latter thus is a clearindication that the SM is only part of the full story.

The lepton numbers Lℓ (ℓ = e, µ, τ) are other additive quantum numbers which seem to be strictlyconserved at first sight. By convention Lℓ(ℓ

−) = 1 . That Lµ is separately conserved follows fromthe non-observations of the decays

µ+ → e+ + γ Γ(µ→ eγ)/Γ(µ→ all) < 1.2 × 10−11

µ+ → e+ + e− + e+ Γ(µ→ 3e)/Γ(µ→ all) < 1.0× 10−12

KL → e+ µ Γ(KL → eµ)/Γ(KL → all) < 4.7 × 10−12

K+ → π+ + e+ µ Γ(K+ → π+eµ)/Γ(K+ → all) < 2.1× 10−10

µ− + (Z,A)→ e− + (Z,A) Γ(µ−Ti→ e−Ti)/Γ(µ−Ti→ all) < 4.0 × 10−12

µ− + (Z,A)→ e+ + (Z − 2, A) Γ(µ−Ti→ e+Ca)/Γ(µ−Ti→ all) < 3.6 × 10−11 .

Tests of the separate conservation of Lτ are much less stringent: The best limits are:

Γ(τ → eγ)/Γ(τ → all) < 2.7× 10−6 and Γ(τ → µγ)/Γ(τ → all) < 1.1× 10−6 .

Within the experimentally well established electroweak standard model strict lepton numberconservation is only possible if the neutrinos are strictly massless. Non-vanishing neutrino masseslead to neutrino–oscillations. Neutrino mixing searches (ν-oscillations νℓ ↔ νℓ′) have confirmedthe effect recently which implies the existence of non–vanishing neutrino masses. Present directupper limits on the neutrino masses are:

17

mνe < 3.0 eV (from 3H → 3He e− νe)

mνµ < 190 keV (from π → µ νµ)

mντ < 18.2 MeV (from τ− → 3π ντ )

Lower bounds are not yet so easy to establish at present but observed neutrino mixing phenom-ena indicate values of about two to three orders of magnitude lower than the above direct upperlimits. In any case this implies corresponding lepton numbers Lℓ (ℓ = e, µ, τ)–violations.

Another important limit is the absence of ∆Le = 2 transitions. The limit from neutrino-lessdouble beta decay (Z,A) → (Z + 2, A) + e+ + e− is t1/2 > 1.6 × 1025 years for 76Ge . Theobservation of such reactions would imply that the electron-neutrino is a massive Majorananeutrino, a self-conjugate fermion which is its own antiparticle.

Global non-Abelian symmetries are approximate (broken) only and correlated with the hierarchyof the fundamental interactions. The weaker the interaction the less symmetries it respects.Strong interaction symmetries are:

Isospin (charge independence of nucleon forces) I SU(2)I

Strangeness S

SU(3)flavor

Charm C

SU(4)flavor

...

The larger the symmetry group the stronger it is broken by growing mass differences of thestates in the multiplets. These symmetries are furthermore broken by electromagnetic and weakinteractions. The latter in addition breaks parity P maximally and CP in accordance with theCabibbo-Kobayashi-Maskawa (CKM) mixing scheme of the three quark–lepton family electroweakSM. CP violation was observed for the first time in K0 decays in 1964 as a small effect at the3 ppm level. The corresponding effect in B0 decays has been established 2001 at dedicatedB–factories.

Let us elaborate more on Quantum Chromodynamics and the chiral symmetry group: The moderntheory of strong interaction is QCD. It views hadrons like the pions, the nucleons etc. as compositeobjects made out of quarks. Mesons are quark antiquark bound states, nucleons are three quarkbound states etc. The quarks not only carry flavor quantum numbers like isospin, strangeness,charm, etc. but an additional one called color. More precisely quarks are color triplets antiquarksare color antitriplets of SU(3)c. Color is a local symmetry (much like the local U(1)em gaugeinvariance in QED) and requires the existence of eight colored gauge bosons the gluons whichglue together the quarks in the hadrons. QCD is a unbroken non-Abelian gauge theory, which hasthe property of asymptotic freedom, the strength of the interaction becomes weaker and weakeras we look at shorter and shorter distances, i.e., inside the hadrons. Complementary it becomesstronger and stronger as we go to larger and larger distances. This means that we cannot separatethe quarks in the hadrons to become free particles. If we try to separate a qq–pair the color fieldbetween them get squeezed into flux tubes which at the end form strings which execute a linearlyrising force. In this way quarks and gluons get in fact permanently confined inside the hadrons.This phenomenon is called confinement. Only objects not carrying net color can become free,these are the hadrons. They have typical sizes of about 1 fermi (=10−13 cm) and the color forcesare screened at distances beyond the size of a hadron. The remnant forces are what we observeas nuclear forces in atomic nuclei or in low energy hadron scattering. Thus in spite of the fact

18

the force carriers, the gluons, are massless the strong interaction forces are rather short ranged.Thus, interestingly, the spectrum of possible states of QCD are not the fields in the Lagrangian,the quarks and gluons, but the hadrons, and those must be color neutral which means they mustbe color singlets.

Thus from the point of view of the color SU(3)c the spectrum can be found by determining allpossible singlets which we may form from quarks in 3 or 3∗ and gluons in 8:

3⊗ 3∗ = 1⊕ 8

3⊗ 3⊗ 3 = 1⊕ 8⊕ 8⊕ 10

3∗ ⊗ 3∗ ⊗ 3∗ = 1⊕ 8⊕ 8⊕ 10∗

8⊗ 8 = 1⊕ 8⊕ 8⊕ 10⊕ 10∗ ⊕ 27

Only the singlets play a role here. They correspond to the mesons ψicα(Γ)αβψicβ baryons, an-tibaryons and the so called glueballs, respectively. The glueballs, the singlet in 8 ⊗ 8 of gluons,do not contain any valence quarks. They are expected to show up as broad resonances between1 and 2 GeV and have not yet been established by experiments.

A completely different kind of application concern the flavor symmetries, which are approximateglobal symmetries of the strong interactions and hence, in modern terminology, of the QCDLagrangian. Since some of the quarks are rather light one may look at the approximation wherequark masses are switched off. The QCD Lagrangian then has a huge global flavor symmetry,namely, the chiral group

GF = U(NF )V ⊗ U(NF )A ≃ SU(NF )V ⊗ SU(NF )A ⊗ U(1)V ⊗ U(1)A (1.2)

with NF the number of quark flavors. In the context of the SM, as we shall see later, thiscorresponds to the symmetric phase of the electroweak– or flavor– sector of the SM, i.e., beforethe local gauge symmetry SU(2)L⊗U(1)Y is broken by the Higgs mechanism to the residual exactelectromagnetic U(1)em local gauge symmetry. The unbroken phase may be understood as anasymptotic symmetry which is approached asymptotically at very high energies, when all masseswhich are small relative to a given energy scale are negligible. Since as we increase the numberof flavors from NF = 2 to 6, the above symmetry is broken more and more by increasingly heavyquark masses4

quark flavor u d s c b t

mass (MeV) ∼5 ∼9 190 1650 4750 172600

the chiral symmetry is good only for the light flavors: for NF = 2 we have with very goodaccuracy the isospin SU(2) for NF = 3 the slightly more broken SU(3) of isospin plus strangeness,symmetries which manifest themselves in the hadronic spectrum. One might ask what is theprecise sense of a broken symmetry, a symmetry which is not truly a symmetry? Associatedwith a symmetry there are currents and generalized charges, the generators of the symmetrytransformations (see next Sec.) of particle multiplets. The point is that in spite of the fact thatthe symmetry is not perfect the states may be classified or labeled in terms of corresponding

4Note the in each doublet the quarks with the larger charge magnitude like c and t have also larger mass thans and b, respectively. The lightest two quarks are an exception, the u quark is lighter than the d quark which is ofexistential importance as it makes the proton to be lighter than the neutron and the neutron to decay into protonsand not vice versa. Thus the inversion is crucial for the stability of the proton and hence for all structure in theuniverse.

19

quantum numbers like isospin and stangeness which satisfy the appropriate Lie algebra. As thesymmetry is approximate only, the corresponding charges are not strictly conserved and thus aretime-dependent to some extent.

Surprisingly the chiral symmetry group (1.2) is not just SU(NF ) which is what we observe inthe hadron spectrum. The chiral group is doubled by the axial part, in fact in the masslesslimit left–handed and right–handed fields satisfy an SU(NF ) independently and thus we obtainSU(NF )L⊗SU(NF )R which is equivalent to vector times axial-vector SU(NF )V ⊗SU(NF )A (seeSec. 3.2 for more details). The only explanation for the absence of the parity doublers in thespectrum is that in fact the SU(NF )A is spontaneously broken, i.e., the symmetry is manifestin the dynamics, represented by the massless QCD–Lagrangian, but absent in the ground state(vacuum) and hence in the space of states built up about the non–symmetric vacuum (see Sec. 2).In turn this implies the existence of a set of Nambu–Goldstone bosons, which must be massless:for SU(2) we must have 3 Goldstone bosons the three pions π±, π0, for SU(3) we must have8 Goldstone bosons the pions plus K±, K0, K0 and η. Since in reality quark masses are non-vanishing the “would be Goldstone bosons” aquire a mass and one calls them pseudo Goldstonebosons. This mechanism explains why the pseudoscalar mesons are the lightest hadrons and whythey have masses substantially lower than the other hadrons.

We finally mention that the U(1)V factor in (1.2) in fact corresponds to the baryon numberconservation, while U(1)A is not a symmetry at all. The latter is broken by quantum corrections,the famous Adler–Bell–Jackiw anomaly.

For a more detailed discussion we refer to Sec. 3.2 for the chiral group and to Sec. 2 for thespontaneous symmetry breaking and the Goldstone phenomenon.

We have seen that we are often dealing with imperfect symmetries in nature. The various possi-bilities we have encountered may be classified as follows:

• Symmetries broken by weaker interactions: What are the enhanced symmetries we get ifwe switch of gravity, the weak forces and the electromagnetic interactions?

• Symmetries broken by mass terms or other terms in the Lagrangian of dimension less thanfour. In general such breakings disappear if we go to higher energies only if they aregenerated by

• spontaneous symmetry breaking.

• What is barely considered in the literature: symmetries may show up at low energies becauseone does not see the short distance details, which means that at high enough energies whenprobing short distances symmetries observed at low energy may be violated. Typically,interaction terms of dimension larger than four (non-renormalizable terms which naturallyarise in low energy effective theories but which are suppressed if the high-energy cut-offis large enough) could violate symmetries we see at low energies. Low energy here meanspresent accelerator energies as we expect the Planck scale MPlanck ∼ 1019 GeV to be thefundamental reference scale. Such non-renormalizable terms are absent in the SM andpossibly only show up when we go much closer to the Planck scale MPlanck. Effects at the 1ppm level we may expect only at energies E/MPlanck ∼ 10−3, i.e., E ∼MGUT ∼ 1016GeV.

Appendix to Section 1:

Some useful formulae for matrix transformations

20

1.

eiABe−iA = B + i[A,B] + i2

2! [A, [A,B]] + . . .

+ in

n! [A, [A, . . . , [A, [A,B ]] . . .]︸︷︷︸n

+ . . .

holds for any two operators or matrices A and B .

Proof:

Replace A by λA and perform a Taylor expansion in λ

F (λ) = eiλABe−iλA =

∞∑

n=0

λn

n!

(∂nF

∂λn

)|λ=0

and evaluate the Taylor coefficients. Using that A commutes with eiλA we get

∂F

∂λ= eiA i [A,B]e−iλA

and by repeated differentiation

∂nF

∂λn= eiλA in [A,A, . . . [A,B ]] . . .]︸︷︷︸

n

e−iλA.

For λ = 1 the result follows.

2.

ei∑

l Tlωl Ti e−i

∑

l Tlωl = Tk(ei

∑

l tlωl)ki

i.e. Ti transforms as a vector under the adjoint representation (tl)ik =−i clik or (tl)ki = i clik.

Proof:

ei∑

l Tlωl Ti e−i

∑

l Tlωl

= Ti + i [Tlωl, Ti] +i2

2![Tl2ωl2 [Tl1ωl1 , Ti]] + . . .

+in

n![Tlnωln , [Tln−1ωln−1 , . . . , [Tl1ωl1 , Ti ] . . .]]︸︷︷︸

n

+ . . .

= Tkδki + Tk i (tlωl)ki + Tki2

2!(tlωl)

2ki + . . .+ Tk

in

n!(tlωl)

nki + . . .

= Tk

(ei

∑

l tlωl

)ki

We have used 1. and [Tl, Ti] = Tk i clik = Tk(tl)ki such that

[Tlωl, Ti] = Tk(tlωl)ki

[Tl2ωl2 , [Tl1ωl1 , Ti]] = [Tl2ωl2 , Tk1 ] (tl1ωl1)k1i

= Tk (tl2ωl2)kk1 (tl, ωl1)k1i

= Tk(tω)2ki etc.

where repeated indices have to be summed over.

21

3.

e−i∑

l Tlωl∂µ(ei

∑

l Tlωl)

= Tk

(1−e−i

∑l tlωl

∑

l tlωl

)ki∂µωi

is a linear combination of the generators and thus an element of theLie-algebra

Proof:

e−i∑

l Tlωl∂µ

(ei

∑

l Tlωl

)

= −i [Tlωl, ∂µ] +i2

2![Tl2ωl2 , [Tl1ωl1 , ∂µ]] + . . .

+(−i)n!

n

[Tω, [Tω, . . . , [Tω, ∂µ] . . .]] + . . .

= −i Tk δki (−∂µωi) +i2

2![Tlωl, Ti] (−∂µωi) + . . .

+(−i)n!

n

[Tω, [Tω, . . . , [Tω, Ti ] . . .]]︸︷︷︸n−1

(−∂µωi) + . . .

= −Tk∞∑

n=1

(−i)n!

n

(tω)n−1ki ∂µωi

= Tk

(1− e−i

∑

l tlωl

∑l tlωl

)

ki

∂µωi = TkΛki(ω) ∂µωi

We have used 1. and [Tlωl, ∂µ] = Tlωl∂µ − ∂µTlωl = −Tl(∂µωl) and then proceed as in theproof of 2.

Exercises: Section 1

① Show that 8⊗ 8 = 1⊕ 8⊕ 8⊕ 10⊕ 10∗ ⊕ 27.

② Isospin symmetry was introduced by Heisenberg in 1932 right after the discovery of the neu-tron (Chadwick 1932). It is obvious thar isospin symmetry is violated by elctromagnetism:the isodoublet members (p,n) have different charge (1, 0). The nucleon system is describedby a isospinor Dirac field

Ψ =

ψp

ψn

.

Write down the charges Q, B and Ti (i = 1, 2, 3) in terms of the nucleon field operator andshow that B commutes with isospin, but Q does not. Write down the T = 1 and T = 0 twonucleon states.

③ Discuss the isospin properties of the triplet of pions (π+, π0, π−) .

The isospin symmetry of the scattering operator S not only leads to relations betweenmatrix elements but also to selection rules: Suppose

(a) T is a generator of a symmetry transformation such that [T, S] = 0 ,

(b) | α > and | β > are eigenstates of T i.e. T | α >= tα | α >, T | β >= tβ | β >

22

What does this imply for the S-matrix elements

Sβα =< β | S | α > ?

Find a few examples.

④ Use the Young tableaux to construct the meson states in

3⊗ 3∗


3⊗ 3⊗ 3 .

The states in the pseudoscalar meson octet of flavor SU(3) are characterized by the 3rd

component of isospin and by hypercharge Y = B+S (B baryon number B = 0 for mesons,S strangeness S = 0 for pions). Display the weight diagram (I3 − Y plot) of the mesonstates. How are they composed of u, d and s quarks in the SU(3)flavor quark model ?

⑤ The structure constants cikl of a Lie-algebra [Ti, Tk] = iciklTl satisfy the Jacobi identity.

cikncnlm+ terms cyclic in (ikl) = 0

Use this to show that (Ti)kl = −icikl also satisfies the Lie-algebra (adjoint representation).

⑥ Discuss the electromagnetic decays of π0 and η0 into photons. Use the fact that the Cparity of the photon is ηγC = −1. What are the observed decays? What do they imply forthe C parities of the neutral pseudoscalar mesons π0 and η0 ?. Which decays are strictlyforbidden? Which Feynman diagram is responsible for these decays?

⑦ Which experiments (processes) allow us to determine the intrinsic parity (ηP ) of the pions?

⑧ Lepton number Le is another additive quantum number which is almost strictly conserved.Le(e

−) = 1 by convention. Determine Le for the other particles from the observed reactions:

1. Le(e+) = −1, Le(γ) = 0 :

p+ e → p+ e+ γ

γ∗ → e+ + e−

2. Le(π0) = Le(π

±) = 0 :

π0 → 2γ, γ + e+ + e−

p+ π− → n+ π0

p+ π0 → n+ π+

3. Le(νe) = −1, Le(νe) = 1 :

π− → e− + νe

π+ → e+ + νe

From the last two reactions we learn the important result νe 6= νe !

23

⑨ Baryon number conservation is responsible for the stability of the proton. By conventionB(p) = 1, B(e−) = 0 . Determine the baryon numbers of particles from the observation ofthe following reactions:

a.) Baryons and mesons:

1. B(π0) = 0 :

p+ p → p+ p+ π0

2. B(n) = B(p), B(π±) = B(π0) = 0 :

p+ p → p+ n+ π+

π− + p → n+ π0

3. B(K±) = B(K0) = 0 :

K± → π± + π0

K0 → π+ + π−, π+ + π− + π0

4. B(Λ), B(Σ) = 1 :

π− + p → Λ0 +K0, Σ− +K+

π+ + p → Σ+ +K+, Σ0 + Λ0

5. B(Ξ), B(Ω−) = 1 :

K− + p → Ξ− +K+, Ξ0 +K0, Ω− +K+ +K0

b.) Antibaryons:

6. B(p) = −1 :

p+ p → p+ p+ p+ p

7. B(B) = −1 :

p+ p → n+ n, Λ0 + Λ0, Σ0 + Σ0, Σ± + Σ∓, Ξ+ + Ξ−

c.) Photon:

8. B(γ) = 0 :

p → p+ γ

d.) Leptons: All leptons are produced in pairs, B(e−) = 0 by convention.

9. B(e) = B(µ) = 0 :

γ∗ → e+ + e−, µ+ + µ−

10. B(νe) = B(νµ) = 0 :

n → p+ e− + νe

µ− → e− + νe + νµ

µ+ → e+ + νe + ν−µ

π− → µ− + νµ

π+ → µ+ + νµ

24

References

[1] F. Jegerlehner, Electroweak Theory and LEP Physics,Lectures given in the “Troisieme Cycle de la Physique en Su-isse Romande” at the Ecole Polytechnique Federale Lausanne (seehttp://www-com.physik.hu-berlin.de/ fjeger/books.html)

[2] F. Jegerlehner, The Anomalous Magnetic Moment of the Muon, Springer Tracts in ModernPhysics, Vol. 226, (Springer, Berlin, 2008)

[3] M. Gell-Mann, A. Pais, Phys. Rev. 97 (1955) 1387

[4] S.L. Glashow, Phys. Rev. Lett. 7 (1961) 469

[5] S. Okubo, Phys. Lett. 5 (1963) 165

25

2 Spontaneous symmetry breaking and Nambu–Goldstone bosons

In many cases symmetries are not ideally realized in nature. As an example, isospin invarianceof strong interactions is not exact. It is broken by small mass-splittings among the membersof isospin multiplets. For the nucleon doublet one has (mn − mp)/mp ≃ 1.29/938.27, for thepion triplet (mπ± −mπ0)/mπ0 ≃ 4.59/134.97 and so on. In addition, electromagnetic and weakinteractions violate the isospin symmetry of the strong interactions. Isospin is not conserved, forexample, in decays like π0 → γγ (electromagnetic) or n→ p+ e− + νe (weak).

Besides such explicit breaking one distinguishes spontaneous breaking of symmetries. A sym-metry is said to be spontaneously broken if the ground state exhibits less symmetry than theLagrangian (or Hamiltonian) of the system. In fact it may happen that symmetric equationsof motion have non-symmetric solutions representing states of lower energy than the symmetricones. Though the ground state exhibits non-trivial structure, in the spontaneously broken case,we still call it vacuum.

The phenomenon of spontaneously broken symmetries is well known from condensed matterphysics. Let us consider the Heisenberg model of a ferromagnet, as an example. The modelassumes nearest neighbor spin-spin interactions of spins ~S~r =

(Sx~r , S

y~r , S

z~r

)attached to the sites ~r

of a cubic lattice. The corresponding interaction Hamiltonian

H = −J∑

<~r,~r ′>

~S~r · ~S~r ′

is rotationally symmetric. Since parallel spins are favored energetically, below the critical tem-perature, the system exhibits “spontaneous” magnetization. We may choose it to point in thedirection of the z-axis. The magnetization is the ground state expectation value of the spinvariable ~S

< Sz~r >=< Sz~0 >= M 6= 0 or < ~S~0 >= (0, 0,M) 6= 0

Generally, a local variable which has a non-vanishing ground state expectation value is calledorder parameter. In a state with non-zero magnetization the symmetry of the system is reducedto rotations around the magnetization axis. The original symmetry is “spontaneously broken”.Of course, the magnetization may point in any direction, which means that there are infinitelymany physically equivalent ground states. The different possible ground states are related to eachother by rotations. Once we have chosen a particular ground state to describe the system thesymmetry is spontaneously broken.

2.1 The Goldstone theorem

In abstract terms, we may characterize the situation as follows: Let the Lagrangian of a systembe symmetric with respect to a group G of transformations. If a unique vacuum | 0 > exists,then it must be invariant under G, meaning that | 0 > is a singlet, and the symmetry is exact.In general, however, the ground state may be degenerate such that there exists more than onestate of lowest energy. The set of ground states then must transform as a multiplet of G. For acontinuous symmetry group a continuous “orbit” of vacua is obtained. Each ad hoc choice of oneof the ground states as the “physical vacuum” of the system breaks the symmetry spontaneously.Typically, there exists a local field which transforms non-trivially under G and which has a non-vanishing vacuum expectation value (order parameter):

< 0 | ϕa(x) | 0 >= Fa 6= 0.

26

Since the symmetry is spontaneously broken, there is no symmetry principle which forces a field(or any operator) which has the quantum numbers of the vacuum to have vanishing vacuumexpectation value. Of course we require the vacuum to be symmetric under translations andLorentz transformations. Therefore, the field ϕa(x) must be scalar and Fa must be a constantvector in the internal symmetry space. Let us suppose that ϕa transforms according to thefundamental representation of G = SU(n). By a global transformation

ϕa → ϕ′a = ei∑

iQiωiϕae−i

∑

iQiωi =(ei

∑

i Tiωi

)abϕb

we may arrange things such that only the real part of the nth component has non-vanishingvacuum expectation value:

< 0 | ϕa(x) | 0 >= 0 , a = 1, . . . , n− 1 ; < 0 | ϕn(x) | 0 > 6= 0 .

Since we assume the Lagrangian to have global SU(n) symmetry, we have n2 − 1 conservedHermitian currents jµi (x):

∂µjµ(x) = 0 ; j+µ = jµ

and the generators of the group are represented by the charge operators

Qi =

∫d3x j0(~x, t) ,

dQidt

= 0.

For infinitesimal transformations the transformation law for the field, given above, takes the formof a set of commutation relations

[Qi, ϕa] = (Ti)ab ϕb.

The Ti’s are the generators in the n×n fundamental matrix representation given in Sec. 1. If wetake the vacuum expectation value we obtain

< 0 | [Qi, ϕa] | 0 >= (Ti)ab < 0 | ϕb(x) | 0 >= (Ti)an v.

If the symmetry is not broken spontaneously v = 0 the vacuum must be invariant Qi | 0 >= 0for i = 1, . . . , n2 − 1. Otherwise < 0 | [Qi, ϕa] | 0 > 6= 0 for some i. We denote the subset of Qi’sfor which < 0 | [Qi, ϕa] | 0 > 6= 0 by Qi. Since < 0 | Qiϕa | 0 > − < 0 | ϕaQi | 0 > 6= 0 we musthave Qi | 0 >=| 0′ >i 6= 0. On the other hand, by the symmetry of the Lagrangian, the generatorscommute with the Hamiltonian

[Qi , H] = 0 , i = 1, . . . , n2 − 1.

This implies[Qi,H

]| 0 >= QiH | 0 > −HQi | 0 >= 0

i.e. if | 0 > is an eigenstate of H then Qi | 0 >=| 0′ >i must be an eigenstate of H with thesame eigenvalue if Qi | 0 > 6= 0. The vacuum must be degenerate in this case. Another importantconsequence follows if we consider

< 0 |[Qi,H

]| p > = < 0 | QiH | p > − < 0 | HQi | p >

= p0 < 0 | Qi | p >= p0 i< 0′ | p >= 0.

27

for a complete set of eigenstates | p > of Pµ.

This implies that the new vacuum | 0′ >i= Qi | 0 > is orthogonal to all the states | p >belonging to the Hilbert space with vacuum | 0 >. For each vacuum we get an inequivalentrepresentation of the physical states i.e. the states created from different vacua cannot bemapped by unitary transformations. Each choice of a fixed vacuum | 0 > yields a physicallyequivalent description of the system, however.

Next we consider the conditions

< 0 |[Qi, ϕa

]| 0 >=

∫d3x < 0 |

[j0i (x), ϕa

]| 0 >= ciav 6= 0

which imply that

< 0 | jµi (x)ϕa(y) | 0 > 6= 0.

ϕa(y) contains a creation operator which creates from the vacuum | 0 > a state | p >a. The abovecondition thus is equivalent to

< 0 | jµi (x) | p >a 6= 0.

Using translation invariance and Pµ | p >a= pµ | p >a we obtain

< 0 | jµi (x) | p >a = < 0 | e−iPxjµi (0)eiPx | p >a= < 0 | jµi (0) | p >a eipx

where < 0 | jµi (0) | p >a= fiapµ since it is a function of p only and must be a Lorentz vector.

Here it is important that | p > is a scalar state. Hence

< 0 | jµi (x) | p >a= fiapµeipx with fia 6= 0.

Since jµi (x) is a conserved current we must have

∂µ < 0 | jµi (x) | p >a = < 0 | ∂µjµ(x) | p >a= ifiap

2eipx = 0

and hence p2 = 0. This is a very interesting result saying that the state | p >a must be amass zero state and ϕa(x) must be a massless scalar field. We thus have proven the Goldstonetheorem: Spontaneous breaking of a continuous symmetry implies the existence of zero massbosons, so called Goldstone bosons. How many Goldstone bosons are there? This questionmay be answered easily if we inspect the conditions

< 0 | [Qi, ϕa] | 0 >= (Ti)an v; i = 1, . . . , n2 − 1, a = 1, . . . , n

more closely. Only generators Ti with a non-zero element in the nth column yield a symmetrybreaking condition. Using the representation of the Ti’s given in Sec. 1 we have one of the n− 1diagonal T ′i with a non-zero element in the nth column. We denote the corresponding Qi by Q0

and obtain

< 0 |[Q0, ϕn

]| 0 >= − n− 1√

2n(n− 1)v 6= 0.

28

In addition, there are 2(n− 1) off-diagonal matrices which have either a −i or a 1 at one positionin the last column of rows a = 1 to n − 1. The corresponding Qi’s we denote by Q1

a andQ2a, a = 1, . . . , n− 1. Thus, we have

< 0 |[Q1a, ϕa

]| 0 >= −iv ; < 0 |

[Q2a, ϕa

]| 0 >= v .

The remaining n2 − 1 − 2(n − 1) − 1 = (n − 1)2 − 1 generators have < 0 | [Qa, ϕa] | 0 >= 0 andhence leave the vacuum invariant. These are precisely the generators of the subgroup SU(n− 1)which leaves the vector

Fa =

0

0...

v

invariant:

UF = F ⇒ U =

U 0

0 1

, U ∈ SU(n − 1).

This group is the stability group (or little group) of the vacuum expectation value < 0 | ϕa(x) |0 >= Fa of the scalar multiplet ϕa(x).

The 2(n−1) + 1 broken generators require 2n−1 of the 2n real fields contained in the n complexfields ϕa to be massless i.e. there must be 2n− 1 Goldstone bosons.

If we introduce real fields by ϕa = ϕ1a + iϕ2

a

(ϕ1a = Reϕa, ϕ

2a = Imϕa

)we obtain

< 0 | [Qi, ϕa] | 0 >=< 0 |[Qi, ϕ

1a

]| 0 > +i < 0 |

[Qi, ϕ

2a

]| 0 >

Now, for a real field (and Hermitian generators Q+i = Qi) we have

< 0 |[Qi, ϕ

ka

]| 0 > = < 0 | Qiϕka | 0 > − < 0 | ϕkaQi | 0 >

= < 0 | Qiϕka | 0 > − < 0 | Qiϕka | 0 >∗= i 2 Im < 0 | Qiϕka | 0 >

and hence

< 0 | [Qiϕa] | 0 >= i 2 Im < 0 | Qiϕ1a | 0 > −2 Im < 0 | Qiϕ2

a | 0 >

The non-vanishing expectation values are then

2 Im < 0 | Q0, ϕ2n | 0 > =

n− 1√2n(n− 1)

v

2 Im < 0 | Q1a, ϕ

1a | 0 > = −v

2 Im < 0 | Q2a, ϕ

2a | 0 > = −v

and the corresponding fields must be massless by the argument given before.

29

As a result we have: The complex multiplet ϕa(x) in the fundamental representation of SU(n)exhibits 2n real fields. In the spontaneously broken phase we can choose the internal spaceframe such that exactly one field, ϕ1

n = Reϕn has a non-vanishing vacuum expectation value< 0 | ϕ1

n | 0 >= v > 0. The remaining 2n− 1 must be massless Goldstone boson fields. The fieldϕ′n may have any mass and is not a Goldstone boson even if it would be massless by accident.

A final remark concerning the currents: If a global symmetry is exact the Noether currents

jµi (x) =: jµi (x) :

represented in terms of creation and annihilation operator exhibit terms a+b only (possibly b = a)and no a+b+ or a b terms. Therefore

jµi (x) | 0 >= 0.

In the spontaneously broken phase conditions

< 0 | jµi (x)ϕa(y) | 0 > 6= 0

must hold, such that jµi (x) cannot annihilate the vacuum! How can we understand this? In ourcase ϕn = ϕ′n + v with < 0 | ϕ′n | 0 >= 0 i.e. ϕ′n is the field having “normal” properties ifexpanded in terms of creation and annihilation operators. The contribution of ϕa to the Noethercurrent is

jµi = i : ϕ∗a (Ti)ab↔∂ µ ϕb :

If we shift the field ϕn → ϕn + v we obviously get terms proportional to v∂µϕa which are linearin the fields. This obviously explains why

jµi (x) | 0 > 6= 0

now.

There are 2n− 1 currents which get a linear term

Q0 : jµ0 = i : ϕ∗a(T0

)ab

↔∂ µ ϕb : +2 n−1√

2n(n−1)v∂µϕ

2n

Q1a : jµ1a = i : ϕ∗a

(T 1a

)ab

↔∂ µ ϕb : −2v∂µϕ

1a

Q2a : jµ2a = i : ϕ∗a

(T 2a

)ab

↔∂ µ ϕb : −2v∂µϕ

2a .

and create 2n− 1 different Goldstone bosons from the vacuum.

2.2 Models of spontaneous symmetry breaking

The first field theory models which exhibited spontaneous symmetry breaking have been inventedaround the year 1960 as models for describing the pion-nucleon (π −N) system. Nambu (1960)and Nambu-Jona-Lasinio (1961) proposed the so called Nambu-Jona-Lasinio model. At aboutthe same time Gell-Mann-Levy (1960) proposed the linear σ-model as a description of the samesystem. In both models pions appear as Goldstone bosons. An approximately realized Goldstone

30

mechanism still is the only way to understand why pions are so much lighter than nucleons 5.Later Goldstone (1961) noticed that the appearance of massless states is a general phenomenonrelated to spontaneous symmetry breaking. One distinguishes two different schemes of sponta-neous symmetry breaking. In the linear realization (prototype linear σ-model) a scalar field,which develops a non-vanishing vacuum expectation value, is introduced as a fundamental field inthe Lagrangian. Such models with elementary scalars are often considered to be doubtful as viablephysical models because elementary scalars have not yet been found in nature. The other possi-bility is the dynamical symmetry breaking scheme (prototype Nambu-Jona-Lasinio model)which has no elementary scalars. The order parameter is a composite field like ψψ and theGoldstone bosons are composite fermion – antifermion states similar to the real pions (if theywould be massless). The problem with such models is that our unfortunate inability to treat therelativistic bound state problem makes it hard to make unambiguous predictions.

In the following we consider models with elementary scalars for illustration of spontaneous sym-metry breaking and the Goldstone mechanism.

2.2.1 A model with spontaneous breaking of a discrete symmetry

Consider a real scalar field with self-interaction described by the Lagrangian

L =1

2(∂µϕ)2 − V (ϕ) =

1

2(∂µϕ)2 +

µ2

2ϕ2 − λ

4!ϕ4 .

a) µ2 < 0

•

V (ϕ)

ϕ

V (ϕ)

•

ϕ

b) µ2 > 0

Figure 2.1: Scalar potential a) in the symmetric and b) in the spontaneously broken phase.

5In modern QCD language the Goldstone boson picture of the pions is realized in the chiral limit of vanishinglight quark masses. In this limit the pions are bona fide massless Goldstone particles

31

Stability of the system requires λ > 0. The equation of motion for the field ϕ is

∂µ∂L∂∂µϕ

=∂L∂ϕ

or ϕ = −∂V∂ϕ

= µ2ϕ− λ

3!ϕ3 .

The Lagrangian ha a discrete symmetry Z2 = ±1 : ϕ→ −ϕ.Let us treat ϕ as a classical field for the moment. The equilibrium solution of the system is theone for which ϕ(x) is a constant determined by ∂V

∂ϕ = 0 . If µ2 < 0, we have a unique groundstate solution ϕ(x) = ϕ0 = 0. In the quantized version we then have a unique vacuum | 0 > and< 0 | ϕ(x) | 0 >= 0. m2 = −µ2 > 0 is the physical mass of the field ϕ. Now choose µ2 > 0. Then

∂V

∂ϕ=λ

3!ϕ3 − µ2ϕ = 0

has besides the trivial solution ϕ0 = 0 two non-trivial solutions (see Fig. 2.1)

ϕ± = v± = ±√

6µ2

λ.

Only these two solutions correspond to a minimum of the potential and hence to a ground statesolutions. Obviously now we have two degenerate ground states.

If ϕ(x) is treated as a quantum field we have to choose one of the ground states as the vacuum| 0 >. This choice is fixed if we require, in first approximation the vacuum expectation value tobe equal to one of the classical results

< 0 | ϕ(x) | 0 >= v+ = v > 0

for example.

Notice that a perturbation expansion based on the splitting L = L0 +Lint with L0 = 12 (∂µϕ)2 +

µ2

2 ϕ2 and Lint = − λ

4!ϕ4 does not make sense because this would correspond to an expansion

in terms of negative m2 = −µ2 < 0 solutions. Such Tachyons, neither satisfy the spectrumcondition nor local causality. As we shall see there is a simple way to circumvent this problem. Ifwe would perform a perturbation expansion about the fake free field solution, in every finite orderof perturbation expansion, we would have a tachyon as an artifact of a nonsensical expansion.Only be infinite resummation techniques we would be able to recover the right physical answer.A field which allows for a normal particle interpretation must satisfy

< 0 | ϕ′(x) | 0 >= 0.

Such a field we simply obtain by a shift

ϕ = ϕ′ + v

from the original field ϕ (B. W. Lee 1969). When we rewrite the Lagrangian in terms of ϕ′ weobtain

L(ϕ) = L(ϕ′ + v) = L′(ϕ′)

32

=1

2

(∂µϕ

′)2 +µ2

2

(ϕ′ + v

)2 − λ

4!

(ϕ′ + v

)4

=1

2

(∂µϕ

′)2 +1

2

(µ2 − λv2

2

)ϕ

′2

− λ4!ϕ

′4 − λv

3!ϕ

′3 +

(µ2v − λv3

6

)ϕ′ +

µ2v2

2− λv4

4!.

The mass term of ϕ′ now is given by

m2eff =

λv2

2− µ2 =

λv2

2− λv2

6=λv2

3> 0 !

where we have used v =√

6µ2

λ such that λv2 = 6µ2 .

Now we can set up a perturbation expansion with

L0(ϕ′) =1

2

(∂µϕ

′)2 − m2eff

2ϕ

′2

Lint(ϕ′)

= − λ4!ϕ

′4 − λv

3!ϕ

′3 − c1ϕ′+ c0

and obtain the Feynman rules depicted in Fig. 2.2.

Propagator :

ϕ′ ( +m2eff) ϕ′ i

p2−m2eff+iǫ

+ interaction vertices :

−i λ v

−i λ

−i c1

ϕ′

ϕ′ϕ′

ϕ′

ϕ′ϕ′

ϕ′ϕ′

ϕ′

Figure 2.2: Feynman rules for L (ϕ′)

The vacuum energy must be adjusted to zero: c0 = 0 . The free parameters are λ and m2eff as in

the symmetric case. The vacuum expectation value is determined by

v = +

√3m2

eff

λ

33

and c1 = v(λv2

6 − µ2)

= 0 ! by the condition that < 0 | ϕ′(x) | 0 >= 0.

The symmetry ϕ′ → −ϕ′ is broken now and we have learned that spontaneous symmetry breakingcan reveal a physical mass to a particle.

2.2.2 The Goldstone model

The Goldstone model illustrates spontaneous breakdown of a continuous symmetry. We mayobtain it from the previous model by replacing the real field by a complex field ϕ 6= ϕ∗. TheLagrangian has the form

L = ∂µϕ∗∂µϕ− V (ϕϕ∗) = ∂µϕ

∗∂µϕ+ µ2ϕ∗ϕ− λ

3!(ϕ∗ϕ)2

and is symmetric under global U(1) transformations. We use the equivalent representation interms of a doublet of two real fields

ϕ =

ϕ1

ϕ2

; ϕi = ϕ∗i (i = 1, 2)

which transforms under O(2) ≃ U(1) rotations. The complex (charged) field is then given by

ϕ =1√2

(ϕ1 + i ϕ2) .

With ϕ2 = ϕ21 + ϕ2

2 = 2ϕ∗ϕ the Lagrangian takes the form

L(ϕ) =1

2(∂µϕ)2 +

µ2

2ϕ2 − λ

4!ϕ4 ,

which obviously is symmetric under 2-dimensional rotations

ϕ→ ϕ′ =

ϕ′1

ϕ′2

=

cos θ sin θ

− sin θ cos θ

ϕ1

ϕ2

.

The equation of motion for ϕ is

ϕ = µ2ϕ− λ

3!ϕ2ϕ .

Again for µ2 < 0 we find a unique ground state solution ϕ1 = ϕ2 = 0 and m2 = −µ2 is thecommon mass of ϕ1 and ϕ2 . If µ2 > 0, the solution ϕ0 =

(00

)is unstable. The potential has a

minimum at (see Fig. 2.3)

ϕ20 =

6µ2

λ

and we have a continuous orbit of ground state solutions related by rotations. We now choose, adhoc, one particular ground state solution as the vacuum | 0 > . O(2)-invariance is spontaneously

34

broken now. The particular choice of | 0 > is fixed by specifying the vacuum expectation valueof ϕ :

< 0 | ϕ(x) | 0 >=

0

v

; v > 0 ; v =

√6µ2

λ

The physical effect of this breaking again becomes transparent if we write the Lagrangian in termsof the shifted field

ϕ′(x) = ϕ(x)−

0

v

i.e.

ϕ′1 = ϕ1

ϕ′2 = ϕ2 − v .

We obtain

L(ϕ) = L(ϕ′ + (0

v)) = L′(ϕ′)

=1

2

(∂µϕ

′)2 − 1

2

(λv2

6− µ2

)ϕ

′21 −

1

2

(λv2

2− µ2

)ϕ

′22

− λ4!

(ϕ′)4 − λv

2 · 3!ϕ

′2ϕ′2 − c1ϕ′2 + c0

with the mass terms

m21eff =

λv2

6− µ2 =

λv2

6− λv2

6= 0

m22eff =

λv2

2− µ2 =

λv2

2− λv2

6=λv2

3> 0 !

This is a remarkable result: By spontaneous symmetry breaking the particles have acquireddifferent masses and one of the particles is massless. As we know, the appearance of a Goldstoneboson is a necessary consequence of spontaneous breaking of a continuous symmetry and is notpeculiar to the particular model.

The original Lagrangian which seems to describe the two fields ϕ1 and ϕ2 in a completely sym-metric way, has been shown to describe two neutral scalar particles of unequal mass. This is aclear manifestation of symmetry breaking.

This symmetry breaking is very different, however, from an explicit breaking of the symmetrywhich would result if we would add ad hoc two independent mass terms for the fields ϕ1 and ϕ2.This would yield a model with three independent parameters m1,m2 and λ. In the spontaneouslybroken case one of the fields must be massless and the model has the original number of parameterλ and m2 in spite of the fact that new interaction vertices have been generated by the shift of

the field. v is determined by v =

√3m2

2λ and c1 = v

(λv2

6 − µ2)

= 0.

There is an intuitive way of understanding the existence of a Goldstone boson from the factthat the vacuum is not unique. Since the vacuum is a state of zero energy and momentumdifferent vacuum states can differ only by the presence of a Bose condensate i.e. the presence ofan unspecified number of quanta of zero energy and momentum. Such quanta are possible onlyif there exist massless particles with the quantum numbers of the vacuum. In our example wecan understand intuitively that there will be one such type of Goldstone bosons. The different

35

a) µ2 < 0

•

V (ϕ)

ϕ2

V (ϕ)

•

ϕ2

b) µ2 > 0

ϕ1

ϕ1

Figure 2.3: Potential of the Goldstone model: a) in the symmetric and b) in the spontaneouslybroken phase.

possible values of < 0 | ϕ(x) | 0 >, lie on a circle of radius v. Each point on the circle correspondto a particular choice of the vacuum state. Thus there is one degree of freedom (motion alongthe circle) by which different vacuum states may be connected, and hence one type of masslessbosons.

The curvature of the potential (∂2V

∂ϕi∂ϕk

)∣∣∣∣ϕ=ϕ0

at the classical minimum actually represents the mass matrix of the system. If we diagonalize thematrix we find an eigenvalue zero corresponding to zero curvature of the potential in directiontangential to the ground state orbit.

A final remark may be in order here: we have shown in Subsec. 2.1 that the different degeneratevacua carry physical Hilbert spaces which are totally orthogonal (inequivalent representations).Thus the existence of the valley in the potential which seems to explain the come about of aGoldstone boson as the mode which moves along the bottom of the valley is misleading. Thereare no Goldstone bosons moving from one vacuum to another one. Note that all Goldstonebosons move at the speed of light and carry some momentum. The fact that this momentum maybe arbitrarily small does not change the fact that there cannot be physical transitions betweendifferent degenerate vacua. Similarly, if we consider a Heisenberg ferromagnet which has a netspontaneous magnetization in a given direction (always in the thermodynamic limit, i.e., at infinitevolume ) the spin waves which correspond to the Goldstone excitations of course do not affect(i.e., rotate ) the given ground state of the system.


① Discuss the symmetry breaking of SU(2)flavor (Isospin) and SU(3)flavor (Isospin and

36

Strangeness) in the spin 1/2 baryon octet. Comment the decays Σ0 → Λγ and Σ+ → pγand compare them to the strong decays Σ− → Λπ−(?) and Σ− → nπ−. Use the quarkmodel schema for the discussion.

② In the Goldstone model of spontaneous symmetry breaking two real fields ϕ =(ϕ1ϕ2

)interact

by the Lagrangian L = 12 (∂µϕ)2 + µ2

2 ϕ2 − λ

4! ϕ4. For µ2 > 0 let the ground state be given

by ϕ0 =(0v

), v > 0. Show that the curvature of the potential ∂2V

∂ϕi∂ϕkat the ground state ϕ0

represents the mass matrix of the system.

③ Write down the Feynman rules for the Goldstone model (see Fig. 2.2).

③ Calculate the Noether currents for the Goldstone model, with respect to the unbroken globalsymmetry. Discuss how it is possible that a current couples to the vacuum, i.e. why it canbee that 〈0|jµi (x)|0〉 6= 0.

37

3 Chiral symmetry and quark flavor mixing

3.1 Prolog: QCD as a part of the Standard Model of fundamental interactions

According to present day knowledge, the known “fundamental” interactions of “elementary”particles follow from a local gauge principle (Weyl, Yang-Mills) with the gauge group

GSMlocal = SU(3)c ⊗ SU(2)L ⊗ U(1)Y (3.1)

which is broken through a Higgs mechanism to SU(3)c⊗U(1)em . SU(3)c is the color gauge groupwhich determines the interaction of the color triplets of quarks via an octet of colored gluons.This unbroken color gauge theory is called quantum chromodynamics (QCD) (Fritzsch, Gell-Mann, Leutwyler 1973) and determines strong interaction physics, and in particular, the hadronspectrum and the residual strong interaction between hadrons. Quarks originally emerged asthe building blocks of hadrons in the attempt to classify the hadronic states according to theirflavor (Gell-Mann, Ne’eman, Zweig 1964). The quark model hypothesis required the hadronsto be composite states of quarks with the quarks being permanently confined in the hadrons.The confinement hypothesis declares hadrons to be composite elementary particles. Morespecifically, baryons are three quark states, mesons are quark-antiquark states

Baryons:, p, n,Λ,Σ,∆, . . .

B = (q1q2q3) : s = 1/2 : p : uud s = 3/2 : ∆++ : uuu

n : udd...

...

∆− : ddd

Mesons: π,K, η, ρ, ω, . . .

M = (q1q2) : s = 0 : π+ : ud s = 1 : ρ+ : ud

π0 : 1√2(uu− dd) ρ0 : 1√

2(uu− dd)

π− : du ρ− : du

η : 1√2(uu+ dd) ω : 1√

2(uu+ dd)

Quarks must then be fractionally charged: Qu = 2/3, Qd = −1/3 . A crucial problem showedup for the states which are totally symmetric under permutations of spin and flavor (e.g. ∆++ :u(↑)u(↑)u(↑)). The spin-statistic theorem requires both the quarks (s = 1/2) and the baryons(s = 1/2, 3/2, . . .) to satisfy the Pauli principle. This requires a totally antisymmetric quarkwave function for the quarks in the baryon. This spin-statistics crisis could be solved onlyby assigning a new quantum number, called color, to quarks. Each quark must exist in threecopies, the red(r), green(g), and blue(b) quarks. Since color never has been observed it wasnatural to require SU(3)c color symmetry to be a local symmetry, which implies that colors areindistinguishable. The confinement hypothesis now requires that physical states must be colorsinglets. The singlet condition leads in a very natural way to the baryons

(q1q2q3)color singlet =1√3!εc1c2c3q1c1q2c2q3c3

and mesons

(q1q2)color singlet =1√3δc1c2q1c1 q2c2

38

The wave functions of hadronic states are given now by a product of spatial, color, flavor andspin wave functions. The ground state hadrons have no orbital angular momentum, such thatthe spatial wave function is symmetric under permutations of the constituents. Simultaneouseigenstates of color (singlet), flavor and spin are then easily constructed by considering factors.The calculation of the hadrons as bound states of quarks (i.e. the actual calculation of the spatialwave function) is an unsolved problem. After this digression to hadronic physics we come back

to the other part of of GSMlocal .

The other unbroken subgroup U(1)em of SU(2)L ⊗ U(1)Y defines quantum electrodynamics(QED), describing the interaction of all charged particles with the photon. SU(2)L is the weakisospin gauge group which determines the interaction of the left–handed (V − A) fermion cur-rents, known from weak interaction processes, with a triplet of weak gauge bosons also calledintermediate vector bosons. Finally, the U(1)Y subgroup is needed in order to recover U(1)emwith Q = T3 + Y

2 in the broken phase as we shall see below.

The standard model (SM) is determined essentially by specifying the matter fields and their trans-formation properties under local gauge transformations. A way to “understand” the emergenceof the SM is described in the following.

3.1.1 The matter fields

The “real world” is build from massless spin 1/2 particles the quarks and leptons. Spin 1/2particles in a sense are more fundamental than other particles because they allow to composeparticles of any spin (e.g. 1/2 ⊗ 1/2 = 0 ⊕ 1 etc.). Massless Fermi fields have fixed handednesscalled chirality or helicity. If ψ is a massless Dirac field the left–handed field ψL = 1−γ5

2 ψ and

the right–handed field ψR = 1+γ52 ψ do not mix under Lorentz-transformations, rotations and

space-time translation.

~s⇐−→~p

P←→ ~s⇒−→~p

ψL ψR

The fields ψL and ψR are interchanged under parity transformations. Since ψ is a local fieldsatisfying Einstein causality the left–handed Dirac field of a massless electron denoted by e−Ldescribes a left–handed electron and, simultaneously, a right–handed positron. Similarly, if e−R isthe local field describing a right–handed electron, this field also describes a left–handed positron.Thus e−R ≡ e+L . Therefore, we may consider all massless fields to be left–handed.

According to todays knowledge, matter is made out of colored quarks and leptons which aregrouped into three families6.

As we have argued earlier quarks of different flavors must show up in three replica of red(r),green(g), and blue(b) color. The first family of fermions are

νeL, νeL, e−L , e

+L , uLr, uLg, uLb, u

cLr, u

cLg, u

cLb, dLr, dLg, dLb, d

cLr, d

cLg, d

cLb

where uc denotes the antiparticle of the u-quarks and so on. All stable matter is built fromthese first family quarks and leptons. The two additional families (who has ordered them?) weobtain by replacing (νe, e) with (νµ, µ) and (u, d) with (c, s) and (νe, e) with (ντ , τ) and (u, d)

6The not too long ago observed neutrino oscillations require the neutrinos to have a tiny mass which must bedifferent for the different flavors. This requires the existence of right–handed neutrinos νℓR ≡ νℓL in spite of thefact that they do not couple directly to gauge fields (i.e., they are singlets with respect to the SM gauge group).

39

with (t, b) . They supplement Nature by forms of unstable matter and allow for the phenomenaof flavor mixing and CP -violation. Altogether we know 3× 16 degrees of freedom. If there wouldbe no interactions the (free) Lagrangian of the world would be

Lmatter =∑

a

ψLaiγµ∂µψLa

which exhibits a huge symmetry of global U(48) . In nature a subgroup of this large globalsymmetry group turns out to be a local symmetry

ψLa → U(x)abψLb ; U(x) ∈ Glocal

where Glocal = SU(3)c ⊗ SU(2)L ⊗ U(1)Y . This requires the fields to couple to massless spin 1gauge fields via minimal coupling

∂µψL → DµψL =

(∂µ − i

∑

α

gαTαiVµαi

)ψL

with interaction vertices:

ψ

ψ

Vg

Here, α labels the factors SU(3)c, SU(2)L and U(1)Y to which the Yang-Mills construction ap-plies individually. The emergence of Glocal as a direct product of two simple Lie groups andthe Abelian group U(1)Y is closely related to the phenomenological appearance of strong, weakand electromagnetic interactions as separate phenomena differing in strength and symmetry.The SU(3)c which determines strong interactions distinguishes between triplets of quarks, “an-titriplets” of antiquarks and singlets of leptons and neutrinos. Since the latter do not carry colorthey do not participate in strong interactions i.e. they do not talk to quarks and gluons, the gaugequanta of SU(3)c . The SU(2)L distinguishes between doublets of left–handed particles

(νee−)L, . . .

and singlets of left–handed antiparticles which are usually identified by the right–handed particlese+L ≡ e−R, . . . . This undemocratic treatment of particles and antiparticles is what we know asmaximal parity violation of weak interactions. The Abelian U(1)Y only affects the phases of thefields according to the weak hypercharge assignment Y = 2(Q− T3) .

Notice that a fermion mass term

ψψ = ψLψR + ψRψL

cannot be SU(2)L⊗U(1)Y invariant. It is the parity violating nature of weak interactions whichforbid fermion masses. We summarize the local multiplet structure in the following tables:

“Weak quantum numbers”: Q = T3 + 12Y

40

Doublets Singlets

(νℓ)L (ℓ−)L (u, c, t)L (d, s, b)L (νℓ)R (ℓ−)R (u, c, t)R (d, s, b)R

Q 0 −1 2/3 −1/3 0 −1 2/3 −1/3

T3 1/2 −1/2 1/2 −1/2 0 0 0 0

Y −1 −1 1/3 1/3 0 −2 4/3 −2/3

Table 1: Matter fields and their SU(3)c ⊗ SU(2)L ⊗ U(1)Y quantum numbers.

group representation

SU(3)c 3

qrqgqb

quark color triplets

3∗

qcrqcgqcb

antiquark color triplets

1 leptons, neutrinos

SU(2)L 2 = 2∗(νee−

)

L

,

(u

d

)

L(νµµ−

)

L

,

(cs

)

L

left–handed weak isospindoublets of leptons andquarks (flavor doublets)(

νττ−

)

L

,

(c

b

)

L

1 νeR, e−R, uR, dR

νµR, µ−R, cR, sR

right–handed weak isospinsinglets of leptons andquarks

ντR, τ−R , tR, bR

U(1)Y phase transformations weak hypercharge Y = 2(Q− T3)

Notice: • νeR, νµR and ντR all have zero quantum numbers with re-spect to Glocal (i.e. no couplings to gauge fields).

• the bottom components of the quark doublets are Cabibbo-Kobayashi-Maskawa rotated fields d, s, b (see below).

In the following we denote the weak doublets by Lℓ (ℓ = e, µ, τ) for the leptons and by Lq forthe quarks.

41

3.1.2 The gauge fields

For each factor group of Glocal, local gauge invariance requires the existence of a set of masslessgauge fields in the adjoint representation, which couple minimally to the matter fields. We denotethe gauge fields Vµαi(x) and the gauge couplings as follows:

group gauge fields name coupling

SU(3)c : Gµi ; i = 1, . . . , 8 gluons gs

SU(2)L : Wµa ; a = 1, 2, 3 weakgaugebosons

g

U(1)Y : Bµ g′

The Yang-Mills Lagrangian is given by

LYM = −1

4GµνiG

µνi −

1

4WµνaW

µνa −

1

4BµνB

µν

with field strength tensors

SU(3)c : Gµνi = ∂µGνi − ∂νGµi + gsfiklGµkGνl

SU(2)L : Wµνa = ∂µWνa − ∂νWµa + gεabcWµbWνc

U(1)Y : Bµν = ∂µBν − ∂νBµ .

The non-abelian fields must exhibit self–interactions described by the vertices

gg2

and with coupling strength equal the matter field coupling

Lmatter, int = gψLγµTaψLWµa

where Jµa = ψLγµTaψL are the fermion currents associated with the gauge group.

The non–Abelian gauge field interaction terms for the electroweak gauge group SU(2)L ⊗U(1)Yhave been experimentally confirmed by LEP experiments [16]. The corresponding test for theQCD part will be considered later.

3.2 Chiral transformations, chiral symmetry and the axial-vector anomaly

3.2.1 Chiral fields and the U(1)-axial current

In the zero mass limit a free Dirac particle decouples into two chiral states described by the Weylfields ψL and ψR. This becomes evident if we write the free Dirac Lagrangian in terms of ψL and

42

ψR : ψ = ψL + ψR7

L = ψiγµ∂µψ −mψψ= ψLiγ

µ∂µψL + ψRiγµ∂µψR −m

(ψLψR + ψRψL

)

If m = 0 we observe that L decomposes into two independent Lagrangians for the fields ψL andψR. In this case L is not only invariant under global phase transformations

ψ → eiαψ ; eiα ∈ U(1)V vector group

(infinitesimal: δψ = iαψ, δψ = −iαψ) but also under global chiral transformations

ψ → eiβγ5ψ ; eiβγ5 ∈ U(1)A axial group

(infinitesimal: δψ = iβγ5ψ, δψ = iβψγ5; note the change of sign for δψ!). Since

γ5 ψL = −ψL , γ5 ψR = ψR

the chiral transformation for the fields ψL and ψR reads:

ψL → eiβγ5ψL = e−iβψLψR → eiβγ5ψR = e iβψR

and hence the “chiral” fields transform with opposite chirality (handedness). The conservedNoether currents

jµ(x) = −δψ

δω

∂L∂∂µψ

+∂L∂∂µψ

δψ

δω

for the U(1)V ⊗ U(1)A symmetry group are the vector current(δψδω = iψ

)8

jµ = ψγµψ ; ∂µjµ = 0

and the axial-vector current(δψδω = iγ5ψ

)

jµ5 = ψγµγ5ψ ; ∂µjµ5 = 0 .

Both vectors V µ = ψγµψ and axial vectors Aµ = ψγµγ5ψ are invariant under chiral transforma-tions whereas scalars S = ψψ , pseudoscalars P = iψγ5ψ and tensors T µν = ψσµνψ are not.

7Remember: γ5 is Hermitian and anticommutes with all γ-matrices. Therefore Π± = (1± γ5)/2 are Hermitianprojection operators: Π+ + Π− = 1 , Π2

± = Π± , Π+Π− = Π−Π+ = 0 . Since by definition ψL = Π−ψand ψR = Π+ψ we have ψL = ψ+

Lγ0 = ψ+Π−γ

0 = ψ+γ0Π+ = ψΠ+ and similarly ψR = ψΠ− . ThereforeψLγ

µψR = ψRγµψL = 0 and ψLψL = ψRψR = 0 .

8For massive fields and charged (non-diagonal) currents jµ = ψ1γµψ2 and jµ5 = ψ1γ

µγ5ψ2 the divergences ofthe currents can be easily calculated for free fields: By the Dirac equation γµ∂µψi = −imiψi , ∂µψiγ

µ = imiψi

and hence∂µj

µ =(

∂µψ1

)

γµψ2 + ψ1γµ (∂µψ2) = i(m1 −m2)ψ1ψ2

and∂µj

µ5 =

(

∂µψ1

)

γµγ5ψ2 + ψ1γµγ5 (∂µψ2) = i(m1 +m2)ψ1γ5ψ2 .

Thus jµ is conserved only if m1 = m2 , for jµ5 to be conserved we must require m1 = m2 = 0 .

43

The conserved “charges” which correspond to these conserved currents for free fermions are(a) Q =

∫d3x j0(x0, ~x ) = NR +NL , (b) Q5 =

∫d3x j0(x0, ~x ) = NR −NL ,

where NR is the number of right handed fields and NL the number of left handed fields. Conser-vation of Q corresponds to fermion number conservation, conservation of Q5 means that NR−NL

is preserved.

In reality fermions are interacting, in particular the charged one’s interact electromagnetically.Suppose that ψ describes a massless charged Dirac particle which couples to photons. Themassless QED Lagrangian

L = ψγµ (∂µ − ieAµ)ψ

formally still looks chirally invariant. In fact, however, global chiral symmetry is broken now bythe axial-vector anomaly (Adler, Bell and Jackiw, 1969) [1, 2]

∂µjµ5 =

e2

8π2FµνF

µν 6= 0 (3.2)

where Fµν = ∂µAν − ∂νAµ is the electromagnetic field strength tensor and Fµν = 12ǫµνρσF

ρσ itsdual pseudotensor (parity odd). The pseudoscalar density is a divergence of a gauge dependentpseudovector [9]

FµνFµν = ∂µKµ ; Kµ = 2ǫµρνσA

ρ∂νAσ. (3.3)

This anomaly is a quantum effect which cannot be removed. In particular it cannot be compen-sated by adding a counterterm to the Lagrangian which would restore the symmetry. For QEDthe non-conservation of the jµ5 current poses no problems because photons do not couple to anaxial-vector current. For gauge theories for which gauge fields couple to axial-vector currentsγ5-anomalies are disastrous. If they are present they destroy renormalizability and unitarity. Theproblems are evident if we consider, for example, the Abelian subgroup U(1)Y of hypercharge Yof the electroweak standard model. The latter is not parity conserving and thus does not treatleft-handed and right-handed fields in a democratic way. As we shall see in Sec. 3.1, the leptonichypercharge current has the form

jµY =1

2

(ℓLγ

µℓL + νℓLγµνℓL

)+ ℓRγ

µℓR

where ℓ denotes a lepton field and νℓ its associated neutrino field. This current contributes to theaction a term

A(Y )int = g′

∫d4x jµY (x) Bµ(x) .

Bµ is the U(1)Y gauge field and g′ the corresponding gauge coupling. Under a local gaugetransformation

Bµ(x)→ B′µ(x) = Bµ(x)− ∂µω(x)

the action changes by

δA(Y )int = −g′

∫d4x jµY (x) ∂µω(x)

and after a partial integration

δA(Y )int = −g′

∫d4x

(∂µj

µY (x)

)ω(x)

44

we see that the action cannot be gauge invariant unless the current is conserved. In fact

(∂µj

µY (x)

)6= 0

has an anomaly as we shall see below. In any case, for a gauge theory in order to be renormal-izable we must make sure that axial anomalies are absent. For QED and QCD, which are parityconserving, anomalies are absent because the gauge fields couple to vector currents only. TheSU(2)L⊗U(1)Y electroweak theory is not anomaly save. It can be rendered renormalizable onlyby anomaly cancellation between leptons and quarks [3]. This leads to lepton-quark duality:the electroweak standard model is renormalizable only if fermions appear in lepton-quark families.

3.2.2 The chiral group U(n)V ⊗ U(n)A

In general we have to consider currents associated with non-Abelian symmetries. What we saidabout Abelian models carries over to the non-Abelian ones. Of particular interest is the so calledchiral group

GF = U(NF )V ⊗ U(NF )A ≃ SU(NF )V ⊗ SU(NF )A ⊗ U(1)V ⊗ U(1)A

with NF the number of quark flavors. As we know the pions have odd inner parity which meansthat an effective pion field has the transformation properties of a SU(2)I -triplet

~π(x) = Ψiγ5~τ

2Ψ(x) ; Ψ ≃

p

n

where Ψ is a SU(2)I -doublet of fermions with the quantum numbers of the nucleons, as indicated.This shows that although QCD is parity conserving non-trivial chiral properties are crucial inthe theory of strong interaction. QCD distinguishes between leptons (which do not participate instrong interactions) and colored triplets of quarks and antiquarks it does not distinguish betweendifferent flavors however. Strong interactions therefore exhibit a flavor symmetry. For free quarksknown are the f=u, d, c, s, t and b (6 flavors) quarks which show up in three colors each c=red(r), green (g) and blue(b), we have a Lagrangian

Lq =

6∑

f=1

3∑

c=1

(ψcf i γ

µ∂µψcf −mf ψcfψcf)

To the extend that we can neglect the quark masses this Lagrangian has a global U(18)V ⊗U(18)Asymmetry. If strong interaction is switched on by the minimal substitution

∂µ → Dµ = ∂µδcc′ − igs(λi2

)

cc′Giµ ,

where the gauge fields Giµ (i = 1, . . . , 8) are called gluons, gs is the QCD coupling constant andthe λi are the Gell-Mann matrices (see Sec. 1), the subgroup SU(3)c of color is promoted to alocal symmetry. QCD requires the quark masses to be degenerate in color space. Again, providedwe discard quark masses, there is a symmetry flavor symmetry

[GF , SU(3)c] = 0

where NF = 6. As we increase the number of flavors from NF = 2 to 6, the above symmetry isbroken more and more by increasingly heavy quark masses

45

quark flavor u d s c b t

mass (MeV) ∼5 ∼9 190 1650 4750 172600

In practice at a given energy scale only those flavors are effective in a physical process whichhave masses lower than the given energy. Degrees of freedom with masses heavier than the givenenergy scale do not influence the which means that the flavor symmetry is preserved the betterthe lower the number of flavors is which participate in a given physical process.

Chiral symmetry plays a key role in the physics of the low lying hadrons composed of the lightquarks u, d and s. In the isospin sector, the relevant Noether currents, conserved in the chirallimit, are given by

V aµ =

∑

c

Ψcγµτa

2Ψc ; Aaµ =

∑

c

Ψcγµγ5τa

2Ψc (3.4)

where

Ψc =

uc

dc

is the light quark isospin doublet, c denotes color in the current has to be which is summedover [color singlet currents]. In the following color summation is always understood implicitly.The weak hadronic currents are not diagonal in flavor. Prototype is the neutron β–decay n →p + e− + νe which in the hadronic part is mediated by a d → u quark transition. The currentsthus are of the form

V µ12 = ψ1(x) γµψ2(x) ; Aµ12 = ψ1(x) γµγ5ψ2(x) ,

where ψ1 and ψ2 are field of different masses m1 and m2, respectively. In this cases the Diracequation yields

∂µVµ12(x) = i (m1 −m2) ψψ(x) ; ∂µA

µ12(x) = i (m1 +m2) ψγ5ψ(x) ,

i.e. the vector currents are conserved only if masses are equal, the axial currents only if masses arezero, i.e. in the chiral limit. In particular in isospin SU(2) where mu ∼ 5 MeV and md ∼ 7 MeVthe vector current is practically conserved [exact in the isospin limit mu = md] which goes underthe label “conserved vector current” (CVC). But due to the smallness of the light quark massesalso the axial current is approximately conserved, which goes under the label “partially conservedaxial vector current” (PCAC). If the axial isospin current would be conserved the charged pionscould not decay. In fact the PCAC relation may be formulated in terms of the pion decay constantas follows.

For the chiral SU(2) currents (3.4), in the In Cartesian notation (real pseudoscalar fields) wehave the one pion matrix element of the divergence of the axial current

〈0|∂µAaµ(0)|πb(p)〉 = Fπm2πδab

and correspondingly for the pion field

〈0|ϕa(0)|πb(p)〉 = δab ,

such that

〈0|∂µAaµ(0)|πb(p)〉 = Fπm2π 〈0|ϕa(0)|πb(p)〉 .

46

p1 p2

−(p1 + p2)

k

k + p1 k − p2

igγµTi igγνTj

γλγ5Tk

Figure 3.1: Triangle diagram exhibiting the axial anomaly

The PCAC relation requires this relation between matrixelements (which is nothing but thedefinition of Fπ and that ϕa(x) is an interpolating field for the corresponding pion state |πa(p)〉 )to hold as an operator relation:

∂µAaµ(x) = Fπm2π ϕ

a(x) .

The charged fields we obtain as

A+µ =

(A1µ + iA2

µ

)/√

2 ; ϕ+ =(ϕ1 + iϕ2

)/√

2 .

The hadronic weak charged axial currents thus reads

A+µ =

∑

c

(ψucγµγ5ψdc)/√

2 .

The corresponding axial nucleon current we denote Ja5µ = ΨNγµγ5τa

2 ΨN and the charged axialnucleon current is given by

J+5µ = (ψpγµγ5ψn)/

√2 .

3.2.3 The Adler Bell, Jackiw triangle anomaly

In the following we will discuss the general condition for anomaly freedom of a theory.

In perturbation theory the axial anomaly shows up in closed fermion loops with an odd numberof axial-vector couplings if a non-vanishing γ5-odd trace of γ-matrices like 9

Tr (γµγνγργσγ5) = 4iǫµνρσ

is involved and if the corresponding Feynman integral is not ultraviolet convergent such that itrequires regularization. The simplest diagram exhibiting the axial anomaly is the triangle diagramFig. 3.1 which leads to the amplitude (1st diagram)

T µνλijk (p1, p2) = (−1) i5 Tr (TjTiTk)g2

(2π)4

∫d4k Tr

(1

k/− p/2 + iǫγν

1

k/+ iǫγµ

1

k/+ p/1 + iǫγλγ5

).(3.5)

If we include the Bose symmetric contribution (2nd diagram)

T µνλijk (p1, p2) = T µνλijk (p1, p2) + T νµλjik (p2, p1) (3.6)

9Notice that Tr(∏n

i=1 γµiγ5

)

= 0 for n < 4 and for all n = odd.

47

and impose vector current conservation

p1µTµνλijk (p1, p2) = p2νT

µνλijk (p1, p2) = 0 (3.7)

we obtain the unambiguous regularization independent result

−(p1 + p2)λ Tµνλijk (p1, p2) = i

g2

16π2Dijk 4 ǫµνρσp1ρp2σ 6= 0 (3.8)

with Dijk = Tr (Ti, TjTk) .

This result is independent on the masses of the fermion lines and is not changed by higher ordercorrections. Therefore the result is exact beyond perturbation theory! (Adler and Bardeen,1969) [4].

We may represent the result as an operator identity in configuration space: p1ρ(p2σ) correspondsto a derivative i∂ρ(i∂σ) acting on a gauge field (external leg) Viµ(x) (Vjν(x)) while − (p1 + p2)λcorresponds to the vertex i∂λj

λ5 (x). Because of the permutation symmetry in the two gauge fields

we have to divide by a factor 2. We then obtain:

∂λjλ5k(x) = − g2

32π2Dijk 4 ǫµνρσ (∂ρViµ(x)) (∂σVjν(x)) .

Using the antisymmetry of the ǫ-tensor and renaming summation indices we may rewrite thisresult

∂λjλ5k(x) = − g2

32π2Dijk ǫ

µνρσ (∂ρViµ − ∂µViρ) (∂σVjν − ∂νVjσ)

=g2

16π2Dijk

Gµνi

Gjµν

withGjµν= ∂µVjν − ∂νVjµ and

Gµνi = 1

2ǫµνρσ

Giρσ. The expression for −(p1 + p2)λ T

µνλijk (p1, p2)

is a matrix element of i∂λjλ5 (0) . If terms from other diagrams contributing to other possible

matrix-elements of the axial current are included one finds for the final form of the anomaly

∂λjλ5k(x) =

g2

16π2Dijk G

µνi (x)Gjµν(x) (3.9)

where Giµν(x) is the non-Abelian field strength tensor and Gµνi its dual pseudotensor. As a resultthe condition for the absence of an anomaly reads

Dijk = Tr (Ti, TjTk) = 0 ∀ (ijk) . (3.10)

The matrices Ti are the generators of a gauge group in a representation R under which thefermions transform. What are the general conditions for the absence of anomalies? To answerthis question we need the following basic properties of traces:

i) trace of the transpose AT of a matrix A:

Tr (A) =∑

i

Aii = Tr (AT )

48

ii) trace of the equivalent A′ = SAS−1 of a matrix A

Tr (SAS−1) = Tr (S−1SA) = Tr (A)

since Tr (AB) = Tr (BA). S must be nonsingular.

We have to distinguish two different types of representations, real and complex ones. A represen-tation R is called real if its complex conjugate representation R∗ is equivalent to R : R∗ ≃ Ri.e. R∗ = SRS−1 for some fixed non-singular matrix S. Non-real representation R 6≃ R∗ arecalled complex. The following general statements hold:

A) Real representations are anomaly free: Dijk ≡ 0. SinceR∗ ≃ R we have Dijk(R∗) = Dijk(R).On the other hand in the canonical basis T+

i = Ti, if Ti are the generator of R then−T ∗i = −T Ti are the generators of R∗. Therefore

Dijk(R∗) = −Dijk(R).

ConsequentlyDijk(R) ≡ 0 for real representation. The groups SO(2ℓ+1), ℓ > 1, Sp(2ℓ), G2, F4, E7

and E8 have only real representations and hence are anomaly free. The groups SO(2ℓ), ℓ >1 , except SO(6) ≃ SU(4) are also anomaly free.

B) Groups for which the fundamental representation is real are anomaly free.

One can show that for any representation R

Dijk(R) = Dijk(R0) ·K(R)

where R0 is the fundamental representation. Only the invariant quantity K(R) depends onthe representation, K(R0) = 1.

For the groups SU(2) ≃ SO(3) and E6 one has Dijk(R0) = 0 ; and thus all representationsof these groups are anomaly free.

C) The groups SU(n), n ≥ 3 have complex representations and Dijk(R0) 6= 0. Fermionstransforming under representations of these groups in general lead to anomalies. In orderto avoid anomalies in this case one has to find those representations R for which K(R) = 0i.e. for the SU(n), n ≥ 3, groups one can avoid anomalies only by putting the fermionsinto particular representations. One can easily find these representations. We always canwrite the fermion currents jµi which couple to the gauge fields in terms of left-handed andright-handed fields

jµi = ψLγµTLiψL + ψRγ

µTRiψR

= ψγµ1− γ5

2TLiψ + ψγµ

1 + γ52

TRiψ.

Since ψLγµψR = ψRγµψL = ψLγµγ5ψR = ψRγµγ5ψL = 0 the contributions from the left-handed and right-handed fields in closed fermion loops decouple. The contribution to theanomaly thus is given by the sum of the contributions from left-handed fields and from right-handed fields. If a particular left-handed loop gives an anomaly proportional to Dijk(RL)then the right-handed loop gives an anomaly contribution proportional to −Dijk(RR) be-cause the two contributions differ by a sign at the γµγ5 vertex as γ5

1±γ52 = ±1±γ5

2 . Thusthe condition that no anomalies arise from gauge interactions is that for all i, j, k :

Tr (TLi, TLjTLk)− Tr (TRi, TRjTRk) = 0 . (3.11)

49

For SU(2) the terms are individually zero for any representation. For SU(3) there is noanomaly if the ψL and ψR transform under the same representation. This is the case forQCD where due to TRi = TLi no axial currents couple to the gluons. Since the left-handedantiquarks are in the complex conjugate representation 3∗ of the fundamental representa-tion 3 the corresponding right-handed particle fields also transform under 3: This follows,because under antiparticle conjugation

ψLC→ ψcL = iγ2ψ∗R .

D) The Abelian group U(1) ≃ SO(2) are not anomaly save. The previous argument for SU(n)groups for the U(1) group leads to

D = Tr T 3L0 − Tr T 3

R0 = 0

as a condition for anomaly cancellation. Here T0 denotes the Abelian generator of hypercharge.

For non-simple groups the generators of the factor groups commute with each other. Therefore,the anomaly is given by the sum of the anomalies of the individual subgroups. This means thatanomalies must be absent for each factor group.

For the standard model only the U(1)Y yields a non trivial condition which must be satisfied inorder to have an anomaly free theory! Let us check this condition now.

The matter field are in

doublets

ψ1

ψ2

L

and singlets ψR1, ψR2 .

The hypercharge and charge assignments satisfy:

Yi = 2(Qi − T3i) ; QRi = QLi = Qi ; Q1 −Q2 = 1

such that we have

YL1 = 2(Q1 −1

2) = 2Q1 − 1 , YR1 = 2Q1

YL2 = 2(Q2 +1

2) = 2Q1 − 1 , YR2 = 2Q2 = 2Q1 − 2 .

Thus the anomaly contribution per doublet is given by

D = Y 3L1 + Y 3

L2 − Y 3R1 − Y 3

R2 = 2 (2Q1 − 1)3 − (2Q1)3 − (2Q1 − 2)3 = −12Q1 + 6 .

Consequently, the SU(2)L ⊗ U(1)Y electroweak model with leptons only is not renormalizable(not anomaly free). The anomaly must be canceled by a contribution of opposite size comingfrom the quarks!

For the anomaly factor D we get for a lepton doublet and Nc = 3 colored quark doublets and theassociated singlets

Dleptons +Dquarks = 6− 6Nc (2Qq1 − 1) = 0

50

and since

Qq1 =1

2

(1 +

1

Nc

)= 2/3 for Nc = 3 .

Notice that one could satisfy the anomaly condition also with the nucleon doublet: Nc = 1 , Q1 =1 i.e.

ψ1

ψ2

=

p

n

.

This is an example of ’t Hooft’s anomaly condition: A composite particle must reproduce theaxial anomaly of its fermionic constituents [11]. This may be understood as a consequence of theAdler-Bardeen theorem which states that the anomaly is not renormalized by higher order effectsand hence that the axial anomaly is a known non-perturbation effect.

As a result we find: The electroweak standard model is renormalizable only if fermions are groupedinto lepton-quark families.

Exercises: Section 3.2

① In the SM the leptonic hypercharge current has the form

jµY =1

2

(ℓLγ


)+ ℓRγ

µℓR .

Write it in terms of vector plus axialvector contributions.

② Calculate the trace and the Feynman integral (3.5) and verify (3.8).

3.3 Yukawa interaction of quarks and quark flavor mixing

So far we have considered the Higgs–fermion couplings for each quark–lepton family separately.The fact that there exist three families of fermions which are made up of fields with identicalSU(2)L ⊗ U(1)Y transformation properties allows us to form invariant Yukawa couplings forarbitrary combinations of fields from the different families. Such flavor mixing is known tooccur in the quark sector while for leptons all searches for family mixing have been negative sofar.

For the quarks we have four horizontal vectors in “family space” with identical quantumnumbers with respect to the local gauge group. These are the left–handed and right–handedversions of the up and down family vectors ui = (u, c, t) and di = (d, s, b) and we denote them byuiL, diL, uiR, diR . The general form of the Yukawa term reads

LqYukawa

= −3∑

i,j=1

[GijtqLqiΦtujR +GijbqLqiΦbdjR + h.c.

]

with Gijtq and Gijbq arbitrary complex 3 x 3 matrices. The mass–matrix we obtain by inserting

Φb = v√2

(01

)and Φt = v√

2

(10

). Thus

Lqmass = − v√2

3∑

i,j=1

[GijtquiLujR +Gijbq diLdjR + h.c.

]

51

= −3∑

i,j=1

[mijtquiLujR +mij

bqdiLdjR + h.c.]

with

mij. =

v√2Gij. .

The quark fields considered up to now are in the weak interaction basis in which the matterfield Lagrangian is diagonal in the families. We now have to find the physical fields in the masseigenstate basis in which the mass–matrix is diagonal such that each field has a fixed mass10.To this end we have to perform global unitary transformation on the horizontal vectors. Unitarybecause they have to leave the kinetic terms of the quarks invariant and global because we haveto diagonalize the constant coupling matrices G· q .

We first perform the transformations

ujR → (VuR)jk ukR

djR → (VdR)jk dkR u

d

jL

→ (VL)jk

u

d

kL

of the singlets and the doublet. Since the doublet fields are transformed as a doublet theseunitary transformations do not change the matter field Lagrangian which exhibits terms of theform Lqi · · ·Lqi , uiR · · · uiR and diR · · · diR ! For the mass matrices we obtain

mtq → V +L mtq VuR real diagonal

mbq → V +L mbq VdR

and the transformed matrices are required to be real diagonal each with 3 free parameters (the3 independent masses) left out of the 18 we started with. A unitary 3 x 3 matrix has 9 parametersand with the two matrices VuR and VL we have enough parameters to satisfy the 15 conditionsto make Gtq real diagonal. However, the remaining 3 parameters not used up from VL plus the 9of VdR are not sufficient to diagonalize Gbq. We thus have to transform the (by convention) lowercomponents of the doublets by an independent unitary transformation:

djL → djL = (UCKM)jk dkL

10Notice that the mass-matrices mbq and mtq are not Hermitian in general. We make use of the fact that anysquare matrix can be diagonalized with the help of two unitary transformations. In our case we thus need fourmatrices such that

V +uLmtqVuR = Dtq = diag (mu,mc,mt)

V +dLmbqVdR = Dbq = diag (md,ms,mb)

The matrices may be determined if we multiply the equations with its Hermitian conjugate, for example,

V +uLmtqVuRV

+uRm

+tqVuL = V +

uLmtqm+tqVuL = D2

tq = diag (m2u,m

2c ,m

2t )

and we see that VuL is the matrix which diagonalizes the Hermitian matrix mtqm+tq. This reduces our problem to

a standard eigenvalue problem for Hermitian operators.

52

This however changes the form of Lmatter and generates a family–mixing in the charged current.This leads us to the form of the charged current

JCCµ = (u, c, t) γµ (1− γ5) UCKM

d

s

b

given earlier with the unitary 3× 3 matrix

UCKM =

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

.

Part of the parameters are used up to diagonalize

mbq → U+CKM V +

L mbq VdR real diagonal .

After diagonalization we always assume the fields to be relabeled such that the fields with identicalquantum numbers are ordered with increasing mass:

mu ≤ mc ≤ mt and md ≤ ms ≤ mb .

How many observable parameters are there left ? The phase of a charged field is not observable. Inthe charged current only left–handed fields are present. If we change the phases of the left–handedfields

qL → eiφqqL , φq an arbitrary real number

the CKM-matrix changes according to

V →

e−iφu 0 0

0 e−iφc 0

0 0 e−iφt

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

eiφd 0 0

0 eiφs 0

0 0 eiφb

and thus

Vuidj → exp −i (φui − φdj ) Vuidj .

This generalizes to any number N of families. If all of the 2N fields are transformed with thesame phase Vuidj is not affected and hence there are 2N-1 phases which may be transformed awayby rephasing the left–handed fields. This means that 2N-1 phases of UCKM are not measurable.This is only true if the remaining parts of the Lagrangian are not affected. The neutral currentis not affected by this rephasing because it is diagonal in flavor and handedness. The YukawaLagrangian obviously is not invariant under left–handed rephasing because it necessarily connectsleft–handed and right–handed fields. However, we have the phases of the right–handed fields atour disposal. We may choose these phases to be the same as the one’s of the left–handed fields(for each individual flavor) such that LYukawa remains invariant.

We may count now the number of free parameters which affect the physics. Let us consider Nfamilies. A unitary N × N matrix has N2 parameters. We may compare it with an orthogonalN × N matrix, describing a rotation in N dimensional Euclidean space and exhibiting N(N-1)/2

53

parameters, which may be taken to be the Euler angles. We may parametrize the unitary matrixby the N(N-1)/2 Euler angles, describing the rotations, plus N2-N(N-1)/2 phases. As we haveargued before 2N-1 of these phases are unobservable. Thus we end up with (N-1)(N-2)/2 physicalphases:

number of families number of angles number of phases

N N(N-1)/2 (N-1)(N-2)/2

2 1 0

3 3 1

4 6 3

For N=3 we have 4 relevant parameters. Besides 3 (real) 3-dimensional rotations there remainsone phase undetermined. Thus UCKM may be parametrized in terms of 3 rotation angles and onephase11:

UCKM =

c12c13 s12c13 s13e−iδ13

−s12c23 − c12s23s13eiδ13 c12c23 − s12s23s13eiδ13 s23c13

s12c23 − c12s23s13eiδ13 −c12s23 − s12c23s13eiδ13 c23c13

.

where cij = cos θij and sij = sin θij with i and j being the family labels. Without loss of generalityone may assume all cij and sij to be positive and the phase δ13 to lie in the range 0 ≤ δ13 < 2π.

A non-degenerate phase

δ13 6= 0 , π

leads to complex effective couplings in the charge changing current which violate CP -invariance.It is important to notice that this specific kind of “standard model CP -violation” is only possiblefor more than two families. In fact, the mixing matrix has to satisfy a number of conditions inorder that CP -violation occurs. The basic observation is that a unitary 3 by 3 matrix which hasa zero matrix element somewhere is necessarily real. Therefore mixing must be non-degenerate:

θij 6= 0 ,π

2ij = 12, 23, 13 .

For the same reason, CP -violation can only occur this way if all the states are distinguishable.This means that all states with the same charge must have different masses:

mu < mc < mt and md < ms < mb .

which happens to be so in Nature. Otherwise, suppose the s and the b quarks would have thesame mass, for example, then the Lagrangian would be invariant under U(2) in (s,b)-flavor space.If we perform the U(2) rotation

s′

b′

=

1

X

Vus Vub

|Vub|eiδ′ |Vus|eiδ

s

b

11There are many equivalent parametrizations, here we present the one advocated by the Particle Data Group.

54

with X =√|Vus|2 + |Vub|2 and the phases constraint by unitarity, δ′ = δ+δus−δub+π we obtain

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

d

s

b

=

Vud X 0

Vcd V ′cs V ′cb

Vtd V ′ts V ′tb

d

s′

b′

the u-quark only couples to d and s′ but no longer to the b′ such that the CKM-matrix has a zeroelement and hence would be real.

Due to unitarity, there is no mixing effect in the neutral current. In the weak interaction basis theneutral current is diagonal in helicities, flavors and families and the unitary CKM-transformation

3∑

i=1

diL · · · diL →3∑

i=1

diL · · · diL ≡3∑

i=1

diL · · · diL

leaves its form invariant. This is called the GIM-mechanism explaining the absence of flavor-changing neutral currents (FCNC). In fact, in order to explain the absence of FCNC’s, Glashow,Iliopoulos and Maiani had to propose, in 1970, the existence of a fourth quark, the charm quarkc as a doublet partner of the s quark [25]. At that time only three quarks where known.

The discovery of the charmonium (cc) state J/ψ in 1974 revealed the completeness of the 2ndfamily with the charm quark c. The first 3rd family member showed up in 1975 with the discoveryof the τ . With the observation of the bottomonium (bb) state Υ the existence of the b quarkcould be established. First indications of an unexpectedly heavy top quark came from the firstobservation of B0 ↔ B0 oscillations by the ARGUS Collaboration in 1987 [12]. LEP precisionmeasurements together with SM fits in 1995 had constrained the top mass to

mt = 170± 10+17−19 GeV

assuming a Higgs mass in the range 60 GeV < mH < 1000 GeV. Shortly after in 1995 the topwas discovered at the Tevatron (CDF and D0 at Fermilab) by tt–production [13]. More recenttop mass measurement at the Tevatron [14] determined the rather precise result

mt = 172.6 ± 1.4 GeV , (3.12)

in excellent agreement with the final indirect determination mt = 172.3+10.2−7.6 GeV from the LEP

precision measurements of Z resonance parameters [15].

We summarize the following important consequences:

• i) all masses of quarks, leptons and neutrinos are independent

• ii) the coupling of the Higgs boson to the fermions is universally proportional to each fermionmass, for bosons proportional to the square of each boson mass

• iii) there is quark flavor violation in charge exchange weak interactions, similarly, neutrinosexhibit flavor mixing in the charged currents

• iv) the phase in UCKM is CP-violating and in fact capable of explaining the observed CP-violation in K-decays12 (Cronin and Fitch 1964). At least 3 families are needed to “explain”CP-violation in this way.

12The particle-antiparticle mixing of the neutral kaons K0 ↔ K0 (Gell-Mann und Pais 1955) played a key rolein revealing CP violation as an observable effect. In the B-meson system B0 ↔ B0 mixing plays an analogous role.

55

• v) flavor is conserved in neutral currents (GIM mechanism). This is strikingly supportedby experiment, at least for the light quark flavors.

Note on the actual values of fermion masses:Leptons masses are

me = 0.510998902 ± 0.000000021MeV ; mµ = 105.6583568 ± 0.0000052MeV ; mτ = 1776.99+0.29−0.26MeV .(3.13)

Current quark masses for the light quarks evaluated in the MS scheme at the scale µ = 2 GeVare:

mu ≈ 1.5 − 4.5MeV ; md ≈ 5.0 − 8.5MeV ; ms ≈ 80− 155MeV . (3.14)

The c-quark mass is estimated from charmonium and D masses. The “running” mass in the MSscheme is mc(µ = mc) = 1.290+0.040

−0.045. The range 1.0 − 1.4 GeV for the MS mass corresponds to1.47− 1.83 GeV for the pole mass [converted by two-loop perturbative QCD with αs(µ = mc) =0.39.]. The b-quark mass is estimated from bottomonium and B masses. The “running” massin the MS scheme is mb(µ = mb) = 4.206 ± 0.031. The range 4.0 − 4.5 GeV for the MS masscorresponds to 4.6 − 5.1 GeV for the pole mass [converted by two-loop perturbative QCD withαs(µ = mb) = 0.22.]. The top mass given above is the pole mass.

Empirically the CKM matrix elements may be expanded in λ = sin θCabibbo ≃ 0.22 with thefollowing approximate sizes of the elements

|V | ∼

1 λ λ3

λ 1 λ2

λ3 λ2 1

.

The leptonic CC has some very special properties, which derive from the absence of right-handedneutrinos in them. Among the unsolved neutrino-puzzles we mention: Why are neutrino masses sosmall (see saw mechanism?)? Do neutrinos have unusual magnetic moments? Are there neutrinoswhich are their own antiparticles (Majorana neutrinos)?

The properties of the weak currents have been established in a long history which started withFermi in 1934. Here, we only mention some more recent of the fundamental experimental tests:

• V-A structure of the CC:µ-decay provides the most sensitive clean direct tests for right-handed currents (e.g. SU(2)R⊗SU(2)L ⊗ U(1)B−L extension of the SM). The best limit for the transition amplitude is

AV+A

AV−A< 0.029 (90%CL)

• absence of flavor-changing NC at tree level:

Γ(KL → µ+µ−)/Γ(KL → all) = (9.5+2.4−1.5)× 10−9

Γ(D0 → µ+µ−)/Γ(D0 → all) < 1.1× 10−5

Γ(B0 → e+e−)/Γ(B0 → all) < 3× 10−5

Flavor-changing NC processes are allowed in higher orders (rare processes).

56

For comment about the status of lepton number conservation and direct bounds on neutrinomasses we refer to Sect. 1.6.

Neutrino mixing (ν-oscillations νℓ ↔ νℓ′) turned out to be very hard to establish. Ater decades ofintense research finally neutrino oscillations were confirmed [18–22], which means that neutrinoshave tiny non–degenerate masses and mix [24] similar to the quarks, although the mixing patternlooks very different from quark-mixing13. This also requires the right–handed singlet neutrinos toexist, which, however, are sterile with respect to any gauge interaction. In 2002 R. Davis Jr. andM. Koshiba awarded the Nobel Prize for pioneering contributions to astrophysics, in particularfor the detection of cosmic neutrinos.

At about the same time, when the neutrino puzzle was resolved, the measurements of directCP-violation (ε′) in the K-meson system [CP-LEAR[NA48]/CERN (2002), kTEV/FNL (2008)]and CP-violation in the B-meson system [B factories BaBar/SLAC and Belle/KEK (2001)] wasexperimentally proven to follow excatly the pattern of the Kobayashi-Maskawa 3–family quarkmixing and the resulting CP violation mechanism [23]. The experimental result is sin(2φ1) =

0.78 ± 0.08 where φ1 = arg(−VcdV∗cb

VtdV∗tb

) [17]. M. Kobayashi and T. Maskawa were awarded the

Nobel Prize in 2008, for their prediction (made in 1973) of O(1) (in units of Cabibbo’s λ = sin θc)CP-violaton in the B–meson system.

3.3.1 Flavor mixing pattern

Historically flavor mixing was “observed” first by a comparison of the decays K → µν andπ → µν. Only u, d and s flavors (isospin and strangeness) were known at that time and theapproximate SU(3)flavor symmetry was established. This symmetry is substantially broken bymass splittings within SU(3) multiplets like for the pseudoscalar mesons with mK ≃ 494 MeVand mπ ≃ 140 MeV. Hadronic transition matrix-elements however satisfy SU(3) relations quitewell. Denoting the matrix element between the pseudoscalar meson P and the weak hadroniccurrent hµ(x) by

< 0|hµ(0)|P (p) >= ipµfP

one obtains for the ratio of the decay widths

Γ(K → µν)

Γ(π → µν)=mK

mπ

(1−m2

µ/m2K

1−m2µ/m

2π

)2(fKfπ

)2

≃ 1.3

and thus(fKfπ

)2

≃ 0.075

and not O(1) as suggested by approximate SU(3) symmetry! Cabibbo solved this puzzle bynoting that the strangeness conserving ∆S = 0 part and the strangeness changing ∆S = 1 part

13Present results may be summarized as follows: a) Solar neutrinos: ∆m212 ≈ (7)×10−5 eV2, tan2 Θ12 ≈ 0.4 (large

νe ↔ νµ mixing), sin2 2Θ13 < 0.067. b) Atmospheric neutrinos: ∆m223 ≈ (1.3 − 3.0) × 10−3 eV2, sin2 2Θ23 > 0.9

(large angle νµ ↔ ντ mixing). Main features are:• smallness of ν masses: mν < 1− 2 eV, at least for one mass mν >

√

∆m223 > 0.04 eV,

• hierarchy of ∆m2’s : |∆m212/∆m

223| = 0.01− 0.15,

• no strong hierarchy of masses: |m2/m3| > |∆m12/∆m23| = 0.18+0.22−0.08 ,

• bi–large or maximal mixing between neighboring families (1-2) and (2-3),• small mixing between remote families (1-3),in any case mν ≪ mℓ,mq.

57

of the hadronic current mix in a specific way, described by a rotation:

hµ = hµ∆S=0 cos θc + hµ∆S=1 sin θc

such that the effective couplings for the two processes are

K → µν : GF sin θc

π → µν : GF cos θc

with sin θc ≃ 0.22 and thus

(fKfπ

)2

→ tan2 θc

(fKfπ

)2

≃ 0.0795

(fKfπ

)2

such that the SU(3) relation fK = fπ is satisfied quite well. Of course if one uses the above ratioto fix the Cabibbo angle one has to consider other processes in order to see whether the abovehypothesis makes sense or not. In fact it turned out to be a very successful ansatz, as it turnedout to be universal (Cabibbo universality). All weak processes in the SU(3) sector are describedby one coupling GF up to a rotation: Gµ = GF , Gπ,n = GF cos θc and GK,Λ = GF sin θc whereGµ, Gn and GΛ are the effective couplings in µ–decay, neutron β–decay and Λ–decay, respectively.

As a next step Glashow, Iliopoulos and Maiani [25] introduces the c quark in order to explain theabsence of FCNC’s and thus for the first time considered a 2 family world:

LCC =g

2√

2(u, c) γµ (1− γ5) UCKM

d

s

W µ

with the unitary 2× 2 matrix

U =

Vud Vus

Vcd Vcs

.

For N=2 the quark mixing matrix is automatically real and given by a simple rotation, theCabibbo rotation matrix

cos θc sin θc

− sin θc cos θc

In the 2 family world the hadronic currents are:

CC: J+µ = uγµ (1 − γ5) ( d cos θc + s sin θc)︸︷︷︸

d

Cabibbo

+ cγµ (1− γ5) (−d sin θc + s cos θc)︸︷︷︸s

GIM piece

NC: JZµ = uγµ (vu − auγ5) u+¯dγµ (vd − adγ5) d︸︷︷︸

FCNC

+ cγµ (vu − auγ5) c+ ¯sγµ (vd − adγ5) s GIM piece

= uγµ (vu − auγ5) u+ dγµ (vd − adγ5) d

+ cγµ (vu − auγ5) c+ sγµ (vd − adγ5) s .

58

Without the c quark s would be absent in the CC and if one assumes that in the NC only thefields already present in the CC enter one ends up with a flavor changing NC. Although NC’s hadnot been observed at all (before 1973) such FCNC’s would have had observable consequences.The N=2 mixing scheme sometimes is called Cabibbo universality. Due to the existence of a thirdfamily Cabibbo universality is violated, because the 2 by 2 sub-matrix of the CKM-matrix is notunitary. A comparison of the N=2 and the N=3 mixing schemes in the 2 family world yields:

U N=2 N=3

Vud cos θc c12

Vus sin θc s12

Vcd − sin θc −s12c23Vcs cos θc c12c23

where we used the excellent approximation c13 = 1, as c13 is known to deviate from unity onlyin the fifth decimal place.

The N=2 mixing scheme was extended to N=3 by Kobayashi and Maskawa in 1973 in order toincorporate CP -violation in a natural way.

Empirically the CKM matrix elements have the approximate moduli

|V | ≃

1 λ λ3

λ 1 λ2

λ3 λ2 1

with λ ≃ sin θc ≃ 0.22 given by the sine of the Cabibbo angle. This suggests the Wolfensteinparametrization (by unitarization up to higher order terms)

V =

1− 12λ

2 λ Aλ3 (ρ− iη)

−λ 1− 12λ

2 Aλ2

Aλ3 (1− ρ− iη) −Aλ2 1

+O(λ4)

where A ∼ 1 and ρ2 + η2 < 1. The corresponding quark decay pattern is illustrated in thefollowing diagram:

Note: the u quark is stable, the s and b quarks are metastable. Flavor changing neutral currenttransitions are allowed only as second (or higher) order transitions: e.g. b → s is in fact b →(t∗, c∗, u∗)→ s, where the asterisk indicates “virtual transition”.


① The left (L)- and right (R)-handed fields are given by

ψL =1− γ5

2ψ , ψR =

1 + γ52

ψ .

Show that a mass term is given by

ψψ = ψLψR + ψRψL .

59

X X

X X

t

u

b

s

d

Figure 3.2: The CKM mixing hierarchy (3.15). FCNCs at tree level are forbidden [X].

Discuss helicity mixing for the terms

S = ψψ , P = ψγ5ψ , V µ = ψγµψ , Aµ = ψγµγ5ψ , T µν = ψσµνψ

② Consider two free fields ψ1 and ψ2 with different masses m1 and m2 . Calculate the diver-gences of the vector and axial vector currents

V µ = ψ1γµψ2 and Aµ = ψ1γ

µγ5ψ2

∂µVµ =? ∂µA

µ =?

What is the consequence of this result for the weak currents ? (Hint: Use the Diracequations). Comment on the hadronic charged weak current responsible for β–decay bycomparing the nucleon current with the quark current.

③ Find the form of the quark currents which couple to W and Z and the photon. As a startingpoint use the SU(2)L ⊗ U(1)Y covariant derivative form

Lq = bRiγµ

(∂µ + i

1

3g′Bµ

)bR + tRiγ

µ

(∂µ − i

2

3g′Bµ

)tR

+Lqiγµ

(∂µ − i

1

3

g′

2Bµ − ig

τa2Wµa

)Lq

Check the correctness of the covariant derivatives with the quantum numbers assigned tothe quarks. We have denoted by Lq =

(tb

)L

the left–handed doublet. tR and bR are thecorresponding right–handed singlets. Quark mixing should be ignored.

④ Give numerical values for the Higgs couplings to the various particles. Calculate the decaywidth of the Higgs into a fermion pair and estimate the branching ratios for the differentflavors. Discuss the dependence of the Higgs mass for the range 0 < mH < 2MW . Comparethe Higgs width with the width of known particles.

60

3.4 The Chiral Structure of Low Energy Effective QCD

The flavor symmetries seen in the hadron spectrum (multiplet classification of hadrons) are theapproximate flavor groups SU(Nf ): SU(2) isospin, SU(3) isospin+strangeness etc., just vectorsymmetries and not approximate chiral symmetry SU(Nf )V ⊗ SU(Nf )A as suggested by theQCD Lagrangian. Nambu in 1960 proposed a fundamental solution of this symmetry mismatch:spontaneous breaking of the chiral symmetry

SU(Nf )V ⊗ SU(Nf )A → SU(Nf )V

by a non symmetric vacuum. In the symmetry limit (conserved chiral currents), this necessarilyimplies the existence of a set of massless pseudoscalar Nambu-Goldstone bosons: in SU(2) thepions, in SU(3) the pions, Kaons and the eta meson. Spontaneous symmetry breaking is a non-perturbative phenomenon, e.g. in perturbative QCD one never would get out pions. Fortunately,a firm low energy effective theory of QCD exists and is very well developed: chiral perturbationtheory (CHPT) [Weinberg, Gasser, Leutwyler] an expansion for low momenta p and in the lightcurrent quark masses as chiral symmetry breaking parameters [5, 6]. CHPT is based on thechiral flavor structure SU(3)L ⊗ SU(3)R of the low lying hadron spectrum (u, d, s quark boundstates). The SU(3)V vector currents jµk =

∑ij ψi(Tk)ij γ

µψj as well as the SU(3)A axial currents

jµ5k =∑

ij ψi(Tk)ij γµγ5ψj

14 are partially conserved in the SU(3) sector of the (u, d, s) quarkflavors, and strictly conserved in the chiral limit of vanishing quark masses mu,md,ms → 0,modulo the axial anomaly in the axial singlet current. The partial conservation of the chiralcurrents15 derives from ∂µ(ψ1γ

µψ2) = i(m1 −m2) ψ1ψ2 (CVC in the isospin limit mu = md) and∂µ(ψ1γ

µγ5ψ2) = i(m1 +m2) ψ1γ5ψ2 (PCAC) and the setup of a perturbative scheme is based onthe phenomenologically observed smallness of the current quark masses.

Chiral perturbation theory is the low energy effective form of QCD which essentially exploits thesymmetries of the theory. It has been developed to a high level of sophistication by Gasser andLeutwyler [5]. In the SU(2) flavor sector of u and d quarks, discarding the baryons and 2nd and3rd family quarks, CHPT involves as the basic fields the Goldstone pion fields. We consider thechiral Lagrangian for two flavors in the isospin limit in the following. The chiral expansion is anexpansion in ~

Leff = L2 + ~L4 + ~2L6 + · · ·

and is expanded in powers of derivatives and quark masses. In standard chiral counting one powerof quark mass counts as two powers of derivatives, or momentum p in momentum space. Thepion fields are encoded in the unitary 2× 2 matrix

U = σ + iφ

F 2, σ2 +

φ2

F 2= 1

φ =

π0

√2π+

√2π− −π0

,

where 1 is the unit 2 × 2 matrix and the auxiliary field σ, as indicated, is a nonlinear functionof the pion fields πi fixed by the unitarity of U16. The leading order Lagrangian starts at O(p2)

14Tk (k = 1, . . . , 8) are the generators of the global SU(3) transformations and i, j = u, d, s flavor indices.15Especially in the SU(2) isospin subspace, the small mass splitting |m1 − m2| ≪ m1 + m2 motivates the

terminology: conserved vector current (CVC) and partially conserved axial vector current (PCAC).16In SU(3) there appears an octet of massless pseodoscalar particles (π,K, η). The corresponding U field is the

61

and is given by the nonlinear σ–model Lagrangian in the presence of external fields, which arecommonly represented in terms of matrix fields vµ = viµσ

i, aµ = aiµσi:

L2 =F 2

4〈DµUDµU

† +M2(U + U †

)〉

withDµU = ∂µU − i (vµ + aµ) U + i U (vµ − aµ) (3.17)

the covariant derivative in the presence of the external fields and

M2 = 2Bm (3.18)

where B is proportional to the quark condensate 〈0|uu|0〉. The parameters M and F are theleading order versions of the pion mass and the pion decay constant, respectively:

m2π = M2 [1 +O(m)] , Fπ = F [1 +O(m)] . (3.19)

The external fields viµ and aiµ have to be understood as the external sources for the low energy

representations of the quark currents V iµ and Aiµ, respectively. These low energy effective currents

again are nonlinear in the pion fields and in CHPT again appear expanded in the derivatives ofU and the quark masses. For the axial vector current, for example, one obtains

V iµ =

iF 2

4〈σi(U†DµU + UDµU

†)〉+O(p3) =[

εijkφj∂µφk +O(φ3)

]

+O(p3) ,

Aiµ =

iF 2

4〈σi(U†DµU − UDµU

†)〉+O(p3) =[

−F∂µφi +O(φ3)]

+O(p3) .

which implies the conserved vector current (CVC) and the partially conserved axial vectorcurrent (PCAC) relations. Despite the fact that this Lagrangian is non-renormalizable, onecan use it to calculate matrix elements like in standard perturbation theory. However, unlikein renormalizable theories where only terms already present in the original bare Lagrangian getreshuffled by renormalization, in non-renormalizable theories order by order in the expansion newvertices of increasing dimensions and associated new low energy constants show up and limit thepredictive power of the effective theory. In fact the expansion (3.15) in ~ automatically producesan expansion of the matrix elements in powers of momenta and quark masses. For the matrixelements we are looking for, tree diagrams from L2 generate leading order contributions, whileone–loop diagrams yield terms at next-to-leading order and so on. The occurring divergences inthe one–loop contributions (in d = 4 dimensions) can be absorbed by introducing the effective

unitary 3× 3 matrix

U(Φ) = exp

(

−i√2Φ(x)

F

)

(3.15)

with (Ti the SU(3) generators)

Φ(x) =∑

i

TiΦi =

π0√

2+ η√

6π+ K+

π− −π0√

2+ η√

6K0

K− K0 −2 η√

6

+1√3

η′

η′

η′

(3.16)

where the second term is the diagonal singlet contribution by the η′ meson. The latter is not a Goldstone boson,however it is of leading order in 1/Nc.

62

Lagrangian at O(p4)

L4 =1

4l1〈DµUDµU

†〉2 +1

4l2〈DµUDνU

†〉〈DµUDνU †〉 (3.20)

+1

16l3M

4〈U + U †〉2 +i

2l4M

2〈aµ(DµU −DµU †

)〉

+l5〈FµνR UFL µνU†〉+

i

2l6〈FµνR DµUDνU

† + FµνL DµU†DνU〉+ . . .

with

FµνR,L = ∂µ(vν ± aν)− ∂ν(vµ ± aµ)− i[vµ ± aµ, vµ ± aµ] . (3.21)

We omitted terms which contain external fields only. Since the current considered here are non-singlet currents, 〈vµ〉 = 〈aµ〉 = 0, there is no contribution from the ABJ anomaly at O(p4).

The coupling constants li may be split into a divergent and a finite term

li = γiλ+ lri (µ) (3.22)

where

λ =µd−4

16π2

1

d− 4− 1

2[ln 4π + Γ′(1) + 1]

(3.23)

represents the UV singular factor, and the finite terms are scale-independent by definition

µd

dµli = γi

µd−4

16π2+ µ

d

dµlri (µ) +O(d− 4) = 0 . (3.24)

The finite parts have been determined phenomenologically for the first time by Gasser andLeutwyler [5]. We adopt their notation and use instead of the lri (µ), the finite and scale-independent quantities li,

li =( γi

32π2

)−1lri (µ) − ln

M2

µ2(3.25)

The up to date values for the relevant li and the corresponding γi are listed in Tab. 2.

Table 2: The set of li and corresponding γi which we need for the calculation of the matrixelements in question. The values are taken from [25] for i = 1, 2, and from [5] for all the others.The γi determine the relation between the li and the lri (µ)

i = 1 2 3 4 6

li −1.7 ± 1.0 6.1 ± 0.5 2.9 ± 2.4 4.3 ± 0.9 16.5 ± 1.1

γi 1/3 2/3 −1/2 2 −1/3

For completeness we give the expressions for the pion decay constant and the pion mass up toand including O(p4),

m2π = M2

[1− M2

32π2F 2l3 +O(M4)

], Fπ = F

[1 +

M2

16π2F 2l4 +O(M4)

](3.26)

The mass splitting m2π± −m2

π0 is proportional to (mu −md)2 and thus may be neglected. In the

numerical evaluation, we will use mπ = 139.57 MeV and17 Fπ = 93.1 MeV.

17Fπ is extracted from the decay width Γ(π → µνµ)

63

If one wants to go beyond the next-to-leading order, one has to calculate two–loop diagrams withL2, and one–loop diagrams with one vertex from L4. Again these diagrams will be divergent, butthis is not a problem since at the same order one has contributions from tree diagrams with theL6 Lagrangian (which has been constructed in the case of three lights flavors [26]). By definingappropriately the new coupling constants occurring in this Lagrangian, one is able to remove thedivergences at the next-to-next-to-leading order, and get finite matrix elements.

On remark shuld be made here: in CHPT the low energy constants in first place are free param-eters, which are fixed as input parameters by confronting predictions with measured quantities,i.e., by phenomenological constraints. In fact CHPT represents QCD only for specific values ofthe low energy constants. The latter can be predicted from non-perturbative QCD, as realized indiscretized version by lattice QCD. In future we can expect precise predictions on may of theseconstants, such that CHPT will gain dramatically in its predictive power.

At physical quark masses the value of the condensate is estimated to be 〈mq qq〉 ∼ −(0.098 GeV)4

for q = u, d. The key relation to identify the quark condensates in terms of physical quantitiesis the Gell-Mann, Oakes and Renner (GOR) relation. In the chiral limit the mass operatorsqRuL, or qLuR transform under (3∗, 3) of the chiral group SU(3)L ⊗ SU(3)R. Hence the quarkcondensates would have to vanish identically in case of an exact chirally symmetric world. In factthe symmetry is spontaneously broken and the vacuum of the real world is not chirally symmetric,and the quark condensates do not have to vanish. In order to determine the quark condensates,consider the charged axial currents and the related pseudoscalar density

Aµ = dγµγ5u

P = d iγ5u (3.27)

and the OPE of the product

Aµ(x) P+(y) =∑

i

Ciµ(x− y)Oi(x + y

2) . (3.28)

In QCD we may inspect the short distance expansion and study its consequences. One observationis that taking the VEV only the scalar operators contribute and one obtains the exact relation

〈0|Aµ(x) P+(y)|0〉 =(x− y)µ

2π2 (x− y)4〈0|uu+ dd|0〉 . (3.29)

The spectral representation for the two–point function on the l.h.s. is of the form pµ ρ(p2) andcurrent conservation requires p2 ρ(p2) = 0 such that only the Goldstone modes, the masslesspions, contribute, such that with

〈0|Aµ(0)|π+〉 = i Fπ pµ

〈0|P+(0)|π+〉 = gπ (3.30)

we get

Fπgπ = −〈0|uu+ dd|0〉 . (3.31)

For non-vanishing quark masses the PCAC relation ∂µAµ = (mu +md) P then implies the exactrelation

Fπm2π+ = (mu +md) gπ (3.32)

and the famous GOR relation

F 2πm

2π+ = −(mu +md) 〈0|uu+ dd|0〉 (3.33)

64

follows from the last two relations. Note that the quark condensates must be negative! They are ameasure for the asymmetry of the vacuum in the chiral limit, and thus are true order parameters.If both Fπ and 〈0|uu + dd|0〉 have finite limits as mq → 0 the pion mass square must go to zerolinear with the quark masses

m2π+ = B (mu +md) ; B ≡ − 1

F 2π

〈0|uu+ dd|0〉 ; B > 0 . (3.34)

The deviation from the chiral limit is controlled by CHPT. The quark masses as well as thequark condensates depend on the renormalization scale µ, however, the product 〈0|mq qq|0〉 is RGinvariant as is inferred by the GOR relation.


① Write the PCAC relation in terms of physical parameters (masses and decay constants ofhadrons) and try to understand the Goldberger-Treiman (GT) relation [Phys. Rev. 110(1958) 1178, ibid. 111 (1958) 354]

gA (mp +mn)/2 = Fπ gπpn

where mp and mn are the proton and neutron mass, respectively, and gπpn the effective pionnucleon coupling. The other couplings show up in the matrix elements for neutron β–decay

M(n→ p+ e− + νe) =GF√

2

[up (fV γλ − gAγλγ5)un Lλ

]

and π–decay

M(π− → µ− + νµ) =GF√

2

(−iFπpλ L

λ).

By Lλ = uℓγλ (1− γ5) vνℓ we denoted the leptonic “current”, p is the pion momentum and

GF is the Fermi constant.

② Calculate the decay rate for π0 → γγ using the effective O(p4) coupling

LWZW =α

π

Nc

12Fπ

(π0 +

1√3η8 + 2

√2

3η0

)FµνF

µν ,

which is the Wess-Zumino-Witten [9, 10] Lagrangian. The latter reproduces the ABJanomaly on the level of the hadrons. π0 is the neutral pion field, Fπ the pion decay constant(Fπ = 92.4 MeV). The pseudoscalars η8, η0 are mixing into the physical states η, η′.

References

[1] S. L. Adler, Phys. Rev. 177 (1969) 2426

[2] J. S. Bell, R. Jackiw, Nuovo Cim. 60A (1969) 47

[3] C. Bouchiat, J. Iliopoulos, P. Meyer, Phys. Lett. 38B (1972) 519;D. Gross, R. Jackiw, Phys. Rev. D 6 (1972) 477;C. P. Korthals Altes, M. Perrottet, Phys. Lett. 39B (1972) 546

[4] S. L. Adler, W. A. Bardeen, Phys. Rev. 182 (1969) 1517

65

[5] J. Gasser, H. Leutwyler, Ann. of Phys. (NY) 158 (1984) 142

[6] J. Gasser, H. Leutwyler, Nucl. Phys. B 250 (1985) 517

[7] J. Bijnens, G. Colangelo, J. Gasser, Nucl. Phys. B427 (1994) 427

[8] H.W. Fearing, S. Scherer, Phys. Rev. D53 (1996) 315

[9] J. Wess and B. Zumino, Phys. Lett. B 37 (1971) 95

[10] E. Witten, Nucl. Phys. B 223 (1983) 422

[11] G. ’t Hooft, In: Recent Developments in Gauge Theories, Proceedings of the Summer-Institute, Cargese, France, 1979, ed by G. ’t Hooft et al., NATO Advanced Study InstituteSeries B: Physics Vol. 59 (Plenum Press, New York 1980)

[12] H. Albrecht et al. [ARGUS COLLABORATION Collaboration], Phys. Lett. B 192 (1987)245.

[13] F. Abe, et al. (CDF collaboration), Phys. Rev. Lett. 74, 2626 (1995);S. Abachi, et al. (D collaboration), Phys. Rev. Lett. 74, 2632 (1995).

[14] [Tevatron Electroweak Working Group and CDF Collaboration and D0 Collab.],arXiv:0803.1683 [hep-ex]

[15] LEP Electroweak Working Group (LEP EWWG),http://lepewwg.web.cern.ch/LEPEWWG/plots/summer2006

[ALEPH, DELPHI, L3, OPAL, SLD Collaborations], Precision electroweak measurementson the Z resonance, Phys. Rept. 427 (2006) 257;http://lepewwg.web.cern.ch/LEPEWWG/Welcome.html (March 2008 update); (see also:Tevatron Electroweak Working Group, arXiv:0808.0147 [hep-ex])

[16] LEP Electroweak Working Group (LEP EWWG),http://lepewwg.web.cern.ch/LEPEWWG/lepww/tgc/

[17] K. Abe et al. [Belle Collaboration], Phys. Rev. Lett. 87 (2001) 091802, Phys. Rev. Lett. 89(2002) 071801. B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 87 (2001) 091801,Phys. Rev. Lett. 89 (2002) 201802.

[18] K. Nakamura, in Proceedings of the 25th International Conference on High Energy Physics,Singapore, 2-8 August 1990;D. C. Kennedy, University of Pennsylvania Report No. UPR-0442T, 1990.

[19] K. Eguchi et al., KamLAND Collaboration, Phys. Rev. Lett. 90 (2003) 021802.

[20] Q. R. Ahmad et al., SNO Collaboration, Phys. Rev. Lett. 89, 011301 (2002) 011302.

[21] For a recent review, see for example C. K. Jung, C. McGrew, T. Kajita, and T. Mann, Ann.Rev. Nucl. Part. Sci. 51 (2001) 451.

[22] M. Apollonio et al., Phys. Lett. B 466 (1999) 415; F. Boehm et al., Phys. Rev. D 64 (2001)112001.

[23] N. Cabibbo, Phys. Rev. Lett. 10 (1963) 531.M. Kobayashi, K. Maskawa, Prog. Theor. Phys. 49 (1973) 652.

66

[24] Z. Maki, M. Nakagawa and S. Sakata, Prog. Theor. Phys. 28 (1962) 870.B. Pontecorvo, Sov. Phys. JETP 6 (1957) 429 [Zh. Eksp. Teor. Fiz. 33 (1957) 549], Sov.Phys. JETP 26 (1968) 984 [Zh. Eksp. Teor. Fiz. 53 (1967) 1717].V. N. Gribov and B. Pontecorvo, Phys. Lett. B 28 (1969) 493.S. M. Bilenky and B. Pontecorvo, Phys. Rept. 41 (1978) 225.

[25] S. L. Glashow, J. Iliopoulos, L. Maiani, Phys. Rev. D 2 (1970) 1285

[26] Heavy Flavor Averaging Group (HFAG),http://www.slac.stanford.edu/xorg/hfag/

http://www-cdf.fnal.gov/physics/new/bottom/bottom.html

67

4 Gauge principles and gauge invariance

4.1 Abelian gauge theory: Quantum Electrodynamics

Quarks and leptons, the constituents of matter, interact via gauge fields. Quantum electrody-namics (QED) describes the interaction of charged particles with the photon, which is describedby an Abelian gauge field Aµ(x). As we shall see the form of the interaction may be under-stood as a consequence of a local symmetry: local gauge invariance (Weyl 1929). QED hasbeen tested with extreme accuracy at its quantum effects level (Lamb shift, anomalous magneticmoments) and is the prototype of a very successful quantum field theory.

We assume the reader to be familiar with QED. Here we give a short account of its basic featuresonly. The aim is to remind the reader of some basic problems which one encounters with masslessspin 1 fields and how they are solved. Similar problems will show up in non-Abelian gauge theorieswhich we will considered at a later stage.

We first consider the free photon field Aµ(x) and a free electron field ψα(x). The independentfree Lagrangian densities read:

L0A = Lfree photon = −14 : FµνF

µν :

L0ψ = Lfree electron = : ψα (iγµ∂µ −m)αβ ψβ :

The “: · · · :” prescription means normal ordering, i.e. represent the fields in terms of anni-hilation and creation operators and commute (anticommute) all creation operators to the leftof the annihilation operators. The c-number commutator (anticommutator) terms are omitted(subtracted)18. By this prescription we have subtracted the vacuum density such that now

< 0 | LA,ψ0 (x) | 0 >= 0 .

Notice that the action i∫d4xL(x), for the infinite space-time volume, only may exist after sub-

traction of the vacuum density.

The photon field determines the antisymmetric electromagnetic field strength tensor

Fµν(x) = ∂µAν − ∂νAµ , (4.1)

which is gauge invariant – i.e. an Abelian gauge transformation

Aµ → Aµ − ∂µα(x) , (4.2)

where α(x) is an arbitrary scalar function, leaves Fµν(x) invariant. As we know, if we representFµν(x) as a curl of a vector-potential Aµ, Eq. 4.1, the homogeneous Maxwell equation

∂µFµν = 0 with Fµν =

1

2ǫµνρσFρσ (4.3)

is automatically satisfied. The pseudo-tensor (parity odd) Fµν is the dual of Fµν .

We now consider the free field equations.

18For operators which are bilinear in the (free) fields, the normal ordering prescription is equivalent to thesubtraction of the vacuum expectation value. For higher powers in the fields the relationship between ordinaryproducts and normal products is discussed in my Lausanne lectures.

68

• Field equation for Aµ(x)

The free photon is a solution of the free Maxwell equation, which is the Euler Lagrange equationfor the free photon Lagrangian:

∂µ∂L0A∂∂µAν

=∂L0A∂Aν

⇒ ∂µFµν(x) = 0

or, in terms of the vector-potential,

Aµ(x)− ∂µ (∂νAν(x)) = (gµν − ∂µ∂ν) Aν = 0

If we try to solve this latter equation a fundamental problem shows up. This equation of motiondoes not determine the field Aµ. The differential operator gµν − ∂µ∂ν has no inverse, sinceφµ = ∂µα(x) is a solution with eigenvalue zero.

There is a simple reason for the problem. The field Aµ is not an observable and thereforehas unphysical properties. Aµ is supposed to describe a massless spin 1 particle, which has twophysical degrees of freedom only, the two transversely polarized states. Therefore two componentsof Aµ must be redundant. In particular, Aµ has a scalar component ∂µA

µ = φ which cannot bephysical and must be required to vanish or to decouple from the physical degrees of freedom.

An idea of how to cure the problem we get if we notice that

L0A = −1

4FµνF

µν

is a degenerate quadratic form in Aµ, which means that a change of Aµ does not necessarily changeL0A. In particular we know that a gauge transformation of Aµ leaves L0A unchanged. Obviously,in order to obtain an equation of motion which determines Aµ uniquely we have to break thisdegeneracy. This forces us to add a gauge dependent term to the invariant Lagrangian. Doingso, we fix a particular gauge and loose manifest gauge invariance.

In order to get an idea of what kind of term we may add in order to break the gauge symmetry ofthe Lagrangian without affecting the physics, let us consider the problem on the level of the fieldequation. This is a second order linear partial differential equation. Since it is not sufficient todetermine Aµ uniquely we need some supplementary condition C(A) = 0. The latter should belinear, in order to keep the problem linear, and covariant, both requirements are not mandatory,however. A possibility, actually the only covariant and linear choice, is the use of the Lorentzgauge condition

∂µAµ(x) = 0

as a subsidiary condition. Strictly speaking this condition does not determine uniquely a gauge,because we still can perform a gauge transformation

∂µAµ(x) = 0→ ∂µA

µ(x)−α(x) = 0 if α(x) = 0

which respects the Lorentz condition if we choose a restricted class of gauge functions α(x) whichare solutions of α(x) = 0. In practice we need not bother about this problem further becausewe will see that we get a well defined perturbation expansion if we use the Lorentz condition forgauge fixing.

69

Geometrically the gauge condition picks a hyper-surface in the Aµ field space. Each point on thehyper-surface corresponds to a physically distinct field. Gauge transformations move the fieldorthogonal to the surface. Fields connected by gauge transformations are called gauge copies ofeach other. They are physically equivalent and form so called gauge orbits. The gauge conditionselects the cut point of the gauge orbit with the hyper-surface as one particular representativefield from each gauge orbit.

The gauge condition C(A) = 0 may be imposed by adding a Lagrange-multiplier term λ12C(A)2

to L0A where λ is an arbitrary constant. We obtain the gauge dependent Lagrangian

Lξ0A = L0A + LGF ; LGF = − 1

2ξ(∂µA

µ(x))2 ,

where LGF is the gauge fixing term which lifts the degeneracy of L0A. ξ is called the gaugeparameter and ξ−1 corresponds to the Lagrange multiplier. If we can show that physical pre-dictions, like scattering matrix-elements, for example, are independent of ξ we also have shownthat ∂µA

µ = 0 for what concerns physics. The gauge fixed Lagrangian Lξ0A is now suitable as astarting point for the quantization of the vector-potential while the original invariant Lagrangianwas not.

The modified equation of motion following from the Lagrangian

Lξ0A = −1

4FµνF

µν − 1

2ξ−1 (∂µA

µ(x))2

is

(gµν − (1− ξ−1)∂µ∂ν

)Aν(x) = 0

obtained by adding the extra term

∂µ∂LGF∂∂µAν

= −ξ−1∂ν (∂µAµ(x))

to the previous form of the equation of motion.

Now the free Maxwell equation

∂µFµν = ξ−1∂ν (∂ρA

ρ(x)) 6= 0 !

has no longer its classical form unless ∂µAµ(x) = 0 in some sense. This is in contradistinction to

the Proca field (massive spin 1 field) for which the Proca equation

(( +m2)gµν − ∂µ∂ν

)Aν = 0

automatically implies ∂µAµ(x) ≡ 0.

Notice that the ”vacuum” |0 > also makes troubles. Under a gauge transformation

< 0 | Aµ(x) | 0 > → < 0 | Aµ(x) | 0 > −∂µα(x) < 0 | 0 >=< 0 | Aµ(x) | 0 > −∂µα(x)

which would be a contradiction if Aµ(x) and | 0 > are supposed to have the naive properties, weexcept them to have. It turns out that the “vacuum” of the photon states must be considered

70

as an equivalence class | 0 >Aµ and each gauge representative of a gauge orbit has a differentformal vacuum.

The only clean covariant way to treat the problem is the Gupta-Bleuler formalism. In thisformalism one can show that the physical state space Hphys is characterized by

∂µAµ(+)(x)Hphys = 0

where Aµ(+)(x) denotes the positive frequency part (annihilation term) of the covariant photonfield. The formal Hilbert space, before imposing this Gupta-Bleuler condition, includes unphysicalstates. Only on the physical subspace the physical laws have the “classical” form. For example,the Maxwell equation

∂µFµν = −ejνem (4.4)

is only true in the sense

∂µFµν(+)Hphys = −ejν(+)

em Hphys .

The reason why we need not worry to much about these problems is the fact that we have asimple check of whether or not the physics is gauge invariant: Physical matrix elements mustturn out to be independent of the gauge parameters ξ ! In this case LGF does not affect physicalpredictions and it looks as if ∂µA

µ(x) = 0. Thus gauge invariance is the ”instrument” whichallows to single out physics from technical artifacts.

• Field equation for ψα(x)

The Euler-Lagrange equation for the free electron Lagrangian is the Dirac equation:

∂µ∂L0ψ∂∂µψ

=∂L0ψ∂ψ

⇒ (iγµ∂µ −m)ψ(x) = 0 .

The electromagnetic current of the electron can be constructed from L0ψ as follows:

L0ψ has a global U(1) symmetry: ψ → e−iαψ where α is an arbitrary constant. Then by theNoether theorem there exists a conserved current.

δL0ψ = 0 under

ψ → ψ + δψ ; δψ = −iαψψ → ψ + δψ ; δψ = iαψ

where

δL0ψ = δψ∂L∂ψ

+ δ(∂µψ)∂L

∂(∂µψ)+∂L∂ψ

δψ +∂L

∂(∂µψ)δ(∂µψ) .

For global transformations we have δ∂µψ = ∂µδψ and the equation of motion tells us that

∂L∂ψ

= ∂µ∂L∂µψ

;∂L∂ψ

= ∂µ∂L∂∂µψ

71

and hence

δL0ψ = ∂µ

(δψ

∂L∂∂µψ

+∂L∂∂µψ

δψ

)

= −iα∂µ(ψiγµψ

)= α∂µj

µem = 0

with

jµem = : ψα(γµ)αβψβ : (4.5)

the conserved electromagnetic current.

Notice: For all interactions which do not exhibit derivations of ψ the form of jµem is determinedsolely by L0ψ

• Local gauge invariance and the electromagnetic interaction

We observed that the problems with the photon field Aµ(x) requires local gauge invariance tohold. For the free electron we have another problem which at first seems not to be related to theproblem of quantization of massless spin 1 particles: We know that in quantum mechanics thephase of a wave function is not observable. For a Dirac field we would expect therefore invarianceunder local phase transformations

ψ(x)→ e−ieα(x)ψ(x) . (4.6)

These transformations, which again are related to some redundancy in the description of a particleby a quantum field, correspond to the local gauge transformations discussed before for the photonfield and therefore are also called gauge transformations. However, L0ψ is not locally gaugeinvariant because

∂µψ(x)→ e−ieα(x)∂µψ(x) − ie e−ieα(x)ψ(x)∂µα(x)

and thus

δL0ψ = e ψ(x)iγµ (−i∂µα(x))ψ(x)

= e ψ(x)γµψ∂µα(x) = e jµem∂µα(x) .

A free electron cannot be described in a locally gauge invariant way! The requirement of localgauge invariance implies that “electrons must couple to photons via minimal substitution”. Whichmeans that we have to replace the troublesome derivative

∂µψ → Dµψ

by a covariant derivative Dµψ defined in such a way that it transforms in the same way as ψ:

Dµψ → e−ieα(x)Dµψ (4.7)

under a local gauge transformation of the electron-photon system19

ψ(x) → e−ieα(x)ψ(x)

Aµ(x) → Aµ(x)− ∂µα(x) . (4.8)

19Note that (4.8) must be a simultaneous transformation of the electron and the photon field with identical localgauge function α(x). With separate local functions (4.7) [in conjunction with (4.9)] does not hold and the QEDLagrangian is obviously not manifestly invariant. However, the apparent non–invariance is not real, because oncethe photon couples to the electron the non-invariant terms can always be reabsorbed by a gauge transformation ofthe photon field which cannot (or should not) affect the physics.

72

The requirement (4.7) implies the form

Dµ = ∂µ − ieAµ (4.9)

for the covariant derivative, which thus may be obtained by the so called minimal substitution

∂µ → Dµ = ∂µ − ieAµ (4.10)

to be applied to the free electron Lagrangian. As a consequence

L0ψ → Lψ = L0ψ + ejµemAµ(x) = ψ (iγµDµ −m)ψ

includes automatically a certain type of electron-photon interactions. Thus the principle of localgauge invariance implies the following specific form of the interaction:

Lint = ejµemAµ(x). (4.11)

Obviously Fµν and jµ are gauge invariant objects. Thus one easily checks:

δ (L0ψ + ejµemAµ(x))

= δL0ψ − ejµem∂µα(x)

= ejµem∂µα(x) − ejµem∂µα(x) = 0.

We notice that the form (4.11) of the coupling of photons to the electromagnetic current, whichis prescribed by local gauge invariance, is the crux why it is mandatory to describe the pho-ton, interacting with the charged particles, by the gauge dependent four–potential Aµ (gaugepotential). Note that the two physical photon states with the fixed helicities ±1 transformseparately as an irreducible representation of the Lorentz group. The reducible representationcombining the two photon states into one field would have two independent components. By theabove construction, however, we are forced to describe the photon by a four component field,which necessarily has two superfluous components. This causes a lot of technical complicationspart of which have been addressed above when discussing the free photon field Aµ(x) (for moredetails see my Lausanne lectures).

The result of our discussion may be summarized as follows:

Local U(1) gauge invariance implies electron-photon interaction according to minimal cou-pling. The electromagnetic interaction is described by

LQED = −1

4FµνF

µν − 1

2ξ−1 (∂µA

µ)2 + ψ (iγµDµ −m)ψ

= Lξ0A + L0ψ + ejµem(x)Aµ(x)

Lint = ejµem(x)Aµ(x) . (4.12)

Correspondingly, the field equations for QED read

(iγµ∂µ −m)ψ(x) = −e : Aµ(x)γµψ(x) :(gµν −

(1− ξ−1

)∂µ∂ν

)Aν(x) = −e : ψ(x)γµψ(x) :

(4.13)

The minimally coupled electron-photon system has more symmetry than the free electronsystem, namely, a local one instead of a global one only. Due to the particular form of theinteraction, resulting from the minimal substitution, the global gauge symmetry is promotedto a local gauge symmetry.

73

Let us add here a remark on how the field strength tensor in QED (Abelian Gauge Theory) maybe obtained once we know the covariant derivative: Dµ = ∂µ − ieAµ. Note that both Fµν and[Dµ,Dν ] are covariant antisymmetric tensor linear in the gauge field, in fact they are proportional:

[Dµ,Dν ]Φ(x) = (∂µ − ieAµ) (∂ν − ieAν) Φ(x)

− (∂ν − ieAν) (∂µ − ieAµ) Φ(x)

= ∂µ∂ν Φ− ie (∂µ(AνΦ) +Aµ∂νΦ)− e2AµAν Φ

− µ↔ ν

= −ie (∂µAν − ∂νAµ) Φ(x)

= −ieFµν Φ(x)

holds for an arbitrary field Φ(x); hence

[Dµ,Dν ] = −ieFµν ,

with

Fµν ≡ ∂µAν − ∂νAµ .

This trick will help us to find the non-Abelian field strength tensor in a simple way, below.

Let us come back to the simultaneous local gauge transformation (4.8) under which the classicalQED Lagrangian (i.e., discarding the gauge fixing term in first place) is manifestly invariant: it isimportant to note that we may relax from the manifest invariance requirement. If we transformthe electron field only, for example, the non-invariant term which shows up may be always elim-inated by a gauge transformation of the photon field. The latter is required to leave the physicsunchanged, which has to be proven of course. Thus the formal non–invariance of the Lagrangiannot necessarily implies the non–invariance of the physics. We may thus precise the meaning of(4.8) as follows: to each local gauge transformation of the electron field ψ(x)→ e−ieα(x)ψ(x) thereexists a gauge transformation of the photon field, namely, Aµ(x) → Aµ(x) − ∂µα(x) such thatunder the combined transformation the classical part of the Lagrangian is manifestly invariant.

As we shall see nature frequently makes use of the possibility that particles (the electrons) conspirewith other particles (the photon) in order to enhance the symmetry. In this sense local gaugesymmetries are conspirative symmetries which are only possible by conspiracy of particles ofdifferent kind.

Empirical fact: Nature makes use of the principle of local gauge invariance which is similar tothe equivalence principle known from general relativity. Known fundamental elementary particleinteractions are minimal couplings with respect to an invariance principle:

interaction gauge group quantum numbers

QED U(1)em electric charge

QFD SU(2)L ⊗ U(1)Ybroken→ U(1)em

weak isospin and

weak hypercharge

QCD SU(3)c color

Quantum Flavordynamics QFD we call the combination of the electroweak theory, alsocalled electroweak Standard Model, and QCD, the theory of strong interactions which describes

74

the strong forces between nucleons and other hadrons. In the sections to follow the basic ideasbehind the construction of these theories will be developed.


① Show that the Maxwell equation ∂µFµν = 0 as a field equation for the vector potential

takes the form

Aν(x)− ∂ν (∂µAµ(x)) = 0 .

Show that Aν(x) is not determined by this equation because the operator gµν − ∂µ∂ν hasno inverse. Hint.: ϕµ = ∂µα(x), α(x) an arbitrary scalar function, is a solution of theabove equation with eigenvalue 0.

② Show that for a massive spin 1 field the Proca equation

( +m2

)Aν(x)− ∂ν (∂µA

µ(x)) = 0

implies ∂µAµ(x) ≡ 0 automatically. Comment on the number of degrees of freedom ! Show

that the Proca equation is the Euler-Lagrange equation of the Lagrangian

L = −1

4FµνF

µν +m2

2AµA

µ ; Fµν = ∂µAν − ∂νAµ .

Discuss the invariance properties of L under gauge transformations.

③ Prove that under local gauge transformations

ψ → e−ieα(x)ψ , Aµ(x)→ Aµ(x)− ∂µα(x)

the covariant derivative Dµ = ∂µ − ieAµ has the property: Dµψ transforms identical to ψand ψΓDµψ is gauge invariant provided Dµ commutes with the 4 by 4 matrix Γ.

75

4.2 Non-Abelian gauge theories

Quantum Chromodynamics may be considered as a straight forward non-Abelian generalizationof Quantum Electrodynamics, from a local U(1)em to local SU(3)c symmetry.

4.2.1 Global non-Abelian symmetry

Instead of a single field electron ψ(x) we now consider a multiplet (e.g. a quark triplet) of fields

Φ(x) =

ψ1

ψ2

··

ψn

(x) ; Φ(x)→ Φ′(x) = U Φ(x) ; U ∈ SU(n)

which transforms under SU(n). For a global internal symmetry the Lagrangian, like the free one

L0Φ = Ψ(x) (i γµ∂µ −M) Ψ(x) ,

is invariant under global SU(n) transformations, in particular we may consider infinitesimaltransformations

δΨ = ig∑

i

TiΨ(x)δωi ; δΨ = −ig∑

i

barΨ(x)Tiδωi .

The invariance of L according to Noether implies conserved currents:

δLΨ0 = δΨ∂L∂Ψ

+ δ(∂µΨ

) ∂L∂∂µΨ

+∂L∂Ψ

δΨ +∂L∂∂µΨ

δ (∂µΨ) .

Since the transformation is global (x-independent) δ∂µΨ = ∂µδΨ and by the Euler-Lagrangeequation ∂L

∂Ψ = ∂µ∂L∂∂µΨ

we find

δLΨ0 = ∂µ

δΨ

∂L∂∂µΨ

+∂L∂∂µΨ

δΨ

= −g∑

i

(∂µj

µΨi

)δωi

where

jµΨi = Ψ(x)γµTiΨ(x) ≡ Ψαa(x) (γµ)αβ (Ti)ab Ψβb(x)

are the fermionic SU(n) Noether currents.

Since δLΨ0 = 0 for arbitrary δωi, we indeed must have the currents being conserved

∂µjµΨi (x) = 0 ; i = 1, . . . , n2 − 1 .

76

4.2.2 Local non-Abelian symmetry

While global symmetries give raise to a classification of states according to irreducible representa-tions of the symmetry group and to conserved quantum numbers, local symmetries have a ratherdifferent physical implication. The requirement of local symmetries is a dynamical principlewhich implies that matter fields must be in interaction with massless spin 1 gauge fields in a spe-cific way. The property of local gauge invariance corresponds to the validity of an equivalenceprinciple: the n degrees of freedom of the local field Ψa(x) are locally indistinguishable in thesense that Ψ(x) and U(x)Ψ(x) describe the same physics and

Ψ(x)→ Ψ′(x) = U(x)Ψ(x) , U(x) ∈ SU(n)

is a symmetry of the dynamics. This means that an observer at space-time point x may choose acoordinate frame for the internal degrees of freedom in the multiplet Ψa(x) independently froman observer at a different space-time point x′. At first sight this requirement might look quitenatural and harmless. It has dramatic consequences, however. Essentially, it dictates the form ofthe dynamics once the local transformation laws of the matter fields are known. The equivalenceprinciple for internal symmetries is very similar to the classical equivalence principle of gravity,which implies that gravity emerges as the geometry of space-time.

The freedom to have associated with each space-time point an independent frame for the internalsymmetry space only makes sense if we are able to “synchronize” local frames at different space-time points. In order to be able to actually perform such a synchronization we need carriers ofphysical signals traveling at the universal speed of light, namely, n2 − 1 massless spin 1 bosonsdescribed by a matrix Vµ(x) =

∑i TiVµi(x) which is an element of the Lie-algebra. A change

of local frames between space-time points x and x + dx must be correlated with a local gaugetransformation of Vµ(x). What we need is a statement saying when “a field Ψ(x) does not changebetween x and x+dx”. With other words, we need a definition of parallel displacement. Whenthere are no internal symmetries and Ψ(x) is a real field we would say that the field does notchange between x and x+ dx if

Ψ(x + dx) = Ψ(x) + ∂µΨ(x) dxµ = Ψ(x)

or ∂µΨ(x) = 0. In QED the complex electron field ψ(x) is coupled to the photon field Aµ(x), aU(1) gauge field, and in the coupled electron-photon system the phase ψ(x) = eieα(x) | ψ(x) | hasno physical significance. However, the statement that | ψ(x) | is constant over dx:

∂µ | ψ(x) |= ∂µ

(e−ieα(x)ψ(x)

)= 0

or (∂µ − ie∂µα(x))ψ(x) = 0 is not gauge invariant and must be replaced by the condition thatthe covariant derivative

Dµψ(x) = (∂µ − ieAµ(x))ψ(x) = 0

vanishes. For the non-Abelian SU(n) gauge symmetry this generalizes to

(DµΨ(x))a = (∂µ − igVµ(x))ab Ψb(x) = 0

In Fig. 4.1 we have illustrated the geometrical meaning of the covariant derivative. For infinites-imal dx we compare the fields Ψ(x) at point x and Ψ(x + dx) at point x + dx. Consider thecovariant expansion Ψ(x + dx) = Ψ(x) + DµΨ(x) dxµ . If Ψ(x) satisfies DµΨ(x) = 0 the field is

“parallel”. We denote by Ψ||(dx)(x) the field Ψ(x+dx) which has been shifted parallel from x+dx

to x along the path dx. Then DµΨ(x) dxµ = Ψ||(dx)

(x)−Ψ(x) .

We now discuss in more detail how locally gauge invariant field theories can be constructed.

77

x

x+ dx

Ψ(x)

Ψ||(dx)(x)

Ψ(x+ dx)

DµΨ(x) dxµ

Figure 4.1: Geometrical interpretation of the parallel displacement.

4.2.3 Minimal couplings of the matter fields

Global G (= SU(n)) invariance of LΨ0 follows from the fact that with Ψ(x) also ∂µΨ(x) transformsas a vector:

Ψ(x) → Ψ′(x) = UΨ(x)

∂µΨ(x) → ∂µΨ′(x) = U∂µΨ(x)

when ∂µU = 0. Under local transformations ∂µU 6= 0

∂µΨ(x)→ ∂µΨ′(x) = U(x)∂µΨ(x) + (∂µU(x)) Ψ(x)

= U(x)(∂µ + U−1(x) (∂µU(x))

)Ψ(x)

6= U(x)∂µΨ(x)

does no longer transform vector-like because of the extra term

U−1(x) (∂µU(x)) = ig∑

i

TiViµ ∈ G′.

As indicated, this term is an element of the Lie-algebra G′ of the symmetry group G and afour-vector under Lorentz transformation. Neglecting higher order terms, for infinitesimal trans-formations U = 1 + ig

∑i Tiωi we easily calculate

U−1(x) (∂µU(x)) = ig∑

i

Ti∂µωi(x)

such that Viµ(x) = ∂µωi(x). For finite transformations we may write Viµ(x) =∑

l Λil(ω)∂µωl(x)where the matrix Λil(ω) is given in the Appendix.

When applied to LΨ0 local gauge transformations induce a non-invariant term:

LΨ0 → LΨ0 + ΨiγµU−1 (∂µU) Ψ

= LΨ0 − g∑

i

ΨγµTiΨViµ(x)

= LΨ0 − g∑

i

jµi (x)Viµ(x) .

This term describes the coupling of a set of r = n2 − 1 real vector fields Viµ(x) to the Noethercurrents jµi (x), which are conserved under global transformations.

78

In order to obtain a locally gauge invariant extension of the free system we must introduce a setof r real vector fields Viµ(x) as dynamical variables (physical degrees of freedom) which couple tothe Noether currents:

LΨ0 → LΨ = LΨ0 + g∑

i

jµi (x)Viµ(x)

= LΨ0 + g∑

i

ΨγµTiViµ(x)Ψ

= Ψ (iγµDµ −m) Ψ .

The fields Viµ(x) are called Yang-Mills fields or non-Abelian gauge fields.

Formally LΨ follows from LΨ0 by minimal substitution

∂µ → Dµ = ∂µ − ig Vµ

where

Vµ =∑

i

TiViµ(x) ∈ G′

is an element of the Lie-algebra.

Dµ defines the covariant derivative. LΨ is locally gauge invariant provided DµΨ transformsas a vector

Ψ(x) → Ψ′(x) = U(x)Ψ(x)

DµΨ(x) → (DµΨ)′ = U(x)DµΨ(x)

and hence

D′µU(x) = U(x)Dµ .

This condition fixes the transformation law of the fields Viµ(x):

D′µ = ∂µ − igVµ ′ = U(x)DµU−1(x)

= U(x) (∂µ − igVµ)U−1(x)

= ∂µ − igU(x)VµU−1(x) + U(x)∂µU

−1(x)

= ∂µ − igU(x)

(Vµ −

i

gU−1(∂µU)

)U−1(x) .

Here, we have used ∂µ(UU−1

)= (∂µU)U−1 + U

(∂µU

−1) = 0 or U(∂µU

−1) = − (∂µU)U−1.

Consequently, we find

Vµ → Vµ′ = U(x)

(Vµ −

i

gU−1(∂µU)

)U−1(x) .

Like ∂µΨ(x), Vµ(x) does not transform as a vector since the local transformation law is differentfrom the global one. In fact Vµ has been required to produce a compensating term for thenon-covariant term obtained for ∂µΨ, in order that (∂µ − igVµ)Ψ is a vector.

79

For infinitesimal transformations U = 1 + ig∑

i Tiωi(x) we obtain to linear order in ωi(x):

Vµ′ =

∑

i

TiV′iµ(x) =

∑

i

Ti (Viµ(x) + ∂µωi(x)) + ig∑

k, l

[Tk, Tl]ωkVlµ

=∑

i

Ti (Viµ(x)− gciklVlµωk + ∂µωi)

or

V ′iµ = Viµ − gciklVlµ ωk + ∂µωi .

Since (Tk)il = icikl represent the generators in the adjoint representation we have

δViµ = −gciklVlµ ωk + ∂µωi

= ig(Tk)ilVlµ ωk + ∂µωi

which compares to

δΨa = ig(Tk)abΨbωk

for the matter fields. We notice that Viµ(x) transforms under the adjoint representation up to a

divergence term20. Accordingly the fields Viµ(x) carry SU(n) charge, which is obvious also fromtheir coupling to the charged Noether currents.

As a result, local gauge invariance requires the matter fields to interact with n2 − 1 masslessgauge fields via minimal coupling

LΨ = Ψ (iγµDµ −m) Ψ = LΨ0 + g∑

i

ΨγµTiΨViµ .

The gauge coupling constant g is a free parameter. The interaction vertex is depicted in Fig. 4.2.

Ψ

Ψ

Vg

Figure 4.2: Matter field couplings of a gauge theory.

4.2.4 Non-Abelian gauge field strength tensor, Yang-Mills action

The field strength tensor in QCD (non-Abelian gauge theory) may be constructed from the co-variant derivative: (Dµ)ij = ∂µ δij − ig (Tk)ijGk µ. What we are looking for is a generalization of

20As we know from QED, massless four–component gauge fields necessarily exhibit non–physical degrees offreedom because there are only two physical states the transversal one’s. Hence, the gauge potentials describeamong the physics also redundant stuff. Thats why the transformation laws of the gauge fields under local gaugetransformations are anomalous (by the disturbing divergence term). Attempts to describe gauge interactionsdirectly in terms of the more physical field strength tensor (see below) seem not to be possible. By the constructionpresented before somehow the gauge potentials are quantities which show up in a natural way. As in QED, at theend one has to show that the physical transition matrix elements are gauge invariant and do not depend on theredundancies of the formalism

80

the Abelian field strength tensor, a covariant antisymmetric tensor, this time covariant not onlywith respect to Lorentz transformations, but also with respect to non-Abelian gauge transforma-tions. Since Dµ is a Lorentz vector and satisfies D′µ = U(x)DµU

−1(x) the commutator [Dµ,Dν ]satisfies all the properties required for Gµν .

It is now easy to calculate Gµν . Using

(Dµ)ik = ∂µδik − ig∑

j

(Tj)ikVjµ(x)

we have

(Dµ)ik(Dν)kl = ∂µ∂νδil − ig∑

j

(Tj)il (Vjµ∂ν + Vjν∂µ)

− ig∑

j

(Tj)il∂µVjν − g2∑

j, j′

(Tj)ik(Tj′)klVjµVj′ν

and hence (the symmetric terms drop)

[Dµ,Dν ]il = −ig∑

j

(Tj)il (∂µVjν − ∂νVjµ)

−g2∑

j′, j′′

[T j′, Tj′′

]ilVj′µVj′′ν

= −ig∑

j

(Tj)ilGjµν = −ig (Gµν)il

where, using[Tj′ , Tj′′

]= icj′j′′jTj ,

Giµν = ∂µViν − ∂νViµ + gcijkVjµVkν .

This is indeed the gauge covariant generalization ofGiµν . In absence of matter fields the La-

grangian density

LYM = −1

4

Giµν

Gµνi

−1

2gcikl (∂

µV νi − ∂νV µ

i )VkµVlν

−1

4g2ciklcik′l′V

µk V

νl Vk′µVl′ν

defines the so called pure Yang-Mills theory . It provides the kinetic term and thereforethe dynamics of the gauge fields. However, in contrast to the Abelian fields which are notself interacting, non-Abelian gauge invariance enforces triple and quartic self-interactions of theYang-Mills fields, of course due to the fact that they carry non-Abelian charge.

The corresponding interaction vertices are shown in Fig. 4.3.

gg2

Figure 4.3: Yang-Mills couplings.

81

x x+ dx

x+ dy x + dx+ dy

Ψ||(dx,dy)(x)

Ψ||(dy,dx)(x)

Ψ(x+ dx+ dy)

[Dµ,Dν ]Ψ(x) dxµ dyν

Figure 4.4: Geometrical interpretation of the field strength tensor.

If we include the matter fields we have the complete locally gauge invariant Lagrangian density

Linv = −1

4

∑

i

GiµνGµνi + Ψ (iγµDµ −m) Ψ

with one coupling constant g as a free parameter. The strengths of the three different interactionvertices are fixed by the same gauge coupling constant.

Let me add a remark about the geometrical interpretation of the field strength tensor which derivesfrom the one of the covariant derivative. To this end we consider an infinitesimal parallelogramof points x, x+ dx, x+ dy and x+ dx+ dy and a field Ψ(xi) at the different points. In order toshift Ψ(x+dx+dy) parallel to the point x we have two possible paths to follow along the sides ofthe parallelogram. We denote these paths by (dx, dy) and (dy, dx). The parallel displaced fields

are Ψ||(dx,dy)(x) and Ψ

||(dy,dx)(x) . We now calculate the difference of these two fields. To this end

we perform a covariant expansion along the two paths:

Ψ(x+ dx+ dy) = Ψ(x+ dy) +DµΨ(x+ dy) dxµ

= Ψ(x) +DνΨ(x) dyν +DµΨ(x) dxµ +DµDνΨ(x) dxµ dyν

Ψ(x+ dx+ dy) = Ψ(x+ dx) +DνΨ(x+ dx) dyν

= Ψ(x) +DµΨ(x) dxµ +DνΨ(x) dyν +DνDµΨ(x) dxµ dyν

For the difference we obtain

Ψ||(dy,dx)(x)−Ψ

||(dx,dy)(x) = [Dµ,Dν ]Ψ(x) dxµ dyν = −igGµνΨ(x) dxµ dyν

exhibiting the field strength tensor as a curvature tensor . If the field strength is non-vanishingthe parallel-displacements of a vector along different paths yield a different result. For infinitesi-mal shifts the difference vector is proportional to the original vector to the field strength and tothe area of the parallelogram. The curvature is illustrated in Fig. 4.4.

4.2.5 Equations of motion and currents

Given the invariant Lagrangian

Linv = −1

4

∑

i

GiµνGµνi + Ψ

(iγµ

(∂µ − ig

∑

i

TiViµ

)−m

)Ψ

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the Euler-Lagrange equations

∂µ∂L∂∂µΨ

=∂L∂Ψ

and ∂µ∂L

∂∂µViν=

∂L∂Viν

read

(iγµ∂µ −m) Ψ = −gγµTiViµΨ

and

∂µGµνi = −g

(jνΨi + jνVi

)= −gjνi

with

jνΨi = ΨγνTiΨ Noether current of the matter field

jνVi = ciklGνρk Vlρ current of the gauge field

The total current jνi = jνΨi + jνVi is obviously conserved

∂νjνi = −1

g∂ν∂µG

µνi = 0

because ∂ν∂µ is symmetric whereas Gµνi is antisymmetric in (µν).

The equations of motion may be written in a manifestly gauge invariant form

(iγµDµ −m) Ψ(x) = 0

DµGµνi (x) = −gJνi (x)

where

DµGµνi = ∂µG

µνi − gciklVkµG

µνl .

The gauge invariant form is simply obtained by adding the appropriate terms on both sides ofthe equations of motion given before. Notice that DµG

µνi and hence Jνi , given by

Jνi = jνi − ciklVkµGµνl = jνΨi = ΨγνTiΨ,

are vectors under local gauge transformations.

The covariant current coincides with the matter field current which is not conserved: ∂νJνi 6= 0.

On the other hand the conserved current

jνi = ΨγνTiΨ + ciklVkµGµνl = −1

g∂µG

µνi

is obviously not covariant because it is the ordinary derivative of a vector. We then arrive at theconclusion:

In a locally gauge invariant theory a covariant conserved current with respect to thegauge symmetry does not exist.

83

This tells us that local symmetries are not symmetries in the usual global sense, like Poincareinvariance, Isospin invariance etc. . Local symmetries are dynamical symmetries and a conse-quence of the validity of an equivalence principle. Global symmetries describe algebraic propertiesof a system, only.

Summary (of subsection 4.2)

1. If we require n matter fields (ψ1, . . . ψn) to be locally indistinguishable, such that

Ψ(x) =

ψ1

...

ψn

→ Ψ′(x) = U(x)Ψ(x) ; U(x) = exp ig

∑

i

Tiωi(x) ∈ SU(n)

is a local symmetry of the system, the matter fields must couple minimally to a set ofr = n2 − 1 massless spin 1 gauge fields Viµ(x):

LΨinv = Ψ(x) (iγµDµ −m) Ψ(x)

with covariant derivative

Dµ = ∂µ − ig∑

i

TiViµ(x) .

All matter fields ψ1 . . . ψn must have identical mass and spin.

2. The locally gauge invariant Lagrangian of the gauge fields must be of the form

LYM = −1

4

∑

i

GiµνGµνi

with covariant field strength tensor

Giµν = ∂µViν(x)− ∂νViµ(x) + gciklVkµ(x)Vlν(x)

The non-Abelian gauge fields must be self-interacting in the specific way as prescribed byLYM . A mass term

M2

2

∑

i

Viµ(x)V µi (x)

for the gauge bosons is not admitted.

3. Under infinitesimal gauge transformations the fields transform as

δΨa = ig (Ti)abΨbδωi , δΨa = −ig Ψb(Ti)baδωi

with (Ti)ab the generators of SU(n) in the fundamental representation.

δVkµ = ig(Ti

)klVlµδωi + ∂µδωi , δGkµν = ig

(Ti

)klGlµνδωi

with(Ti

)kl

= −icikl the generators of SU(n) in the adjoint representation.

84

A final remark concerns the generalization of the Yang-Mills construction to other gauge groupsG. The necessary and sufficient conditions for the Yang-Mills construction to be possible are thatthe structure constants cikl

i) satisfy the Jacobi-identity

ii) are fully antisymmetric.

Whereas i) is automatic for finite matrices (finite dimensional representations) but may not holdfor infinite matrices, the condition ii) is more restrictive. It holds for compact semi-simple Liegroups. Compact mean that the parameter space has a finite volume

∫ ∏i dωi = Vω <∞. Semi-

simple means G is a product of simple Lie-groups e.g. G = SU(3) ⊗ SU(2) ⊗ U(1). SimpleLie-groups are those which cannot be decomposed into invariant subgroups. For each simplesubgroup of a non-simple group G there would be a set of non-Abelian or Abelian (U(1)) gaugefields and an independent coupling constant.

Notice that the conditions i) and ii) are really non-trivial. As an example the Poincare group isnon-compact and has no finite dimensional unitary representation.

Outlook: For a long time, after Yang and Mills had proposed to extend local gauge invariancefrom Abelian to non–Abelian symmetry groups, non–Abelian gauge theories were consideredto be unphysical, because they required the existence of multiplets of massless spin 1 bosons,which were known not to exist in Nature. At that time it was not known that there are twoways out of the dilemma. One is the Higgs mechanism where the gauge bosons acquire a massby “spontaneous symmetry breaking of the local gauge symmetry”. Today we know that theelectroweak gauge group SU(2)L ⊗ U(1)Y , of weak isospin and weak hypercharge Y , is brokendown to the Abelian electromagnetic gauge group U(1)em. Three out of the 3+1 gauge bosonsacquire a mass and the remaining massless state is the photon. The weak gauge bosons W±

and Z in fact turned out to be very heavy (about 80 GeV and 91 GeV, respectively) and werediscovered in 1983 at the CERN pp collider. The LEP e+e− storage ring at CERN, in operationsince 1989, is a Z factory and produces millions of Z’s.

The other “solution” is confinement. Unbroken non-Abelian gauge theories are asymptoticallyfree (Politzer, Gross and Wilzcek 1973), which means that they have small effective coupling athigh energies (short distances) but strong effective coupling at low energies (large distances). Weknow that the strong interactions of hadrons are described by an unbroken color SU(3)c localgauge theory, called quantum chromodynamics (QCD) (Gell-Mann, Fritzsch and Leutwyler 1973).The matter fields are the colored quarks triplets which interact through the octet of massless gaugefields, called gluons. Quarks and gluons are permanently confined inside of the hadrons. This isanother mechanism which hides massless gauge bosons from the physical spectrum.

Obviously the earlier conclusion that Yang-Mills theories are not of relevance for physics waspremature.


① If Vµ = igU−1(x) (∂µU(x)) the field is called a “ pure gauge” field. Show that in this case

Gµν ≡ 0 .

② Show that a mass term M2

2

∑i ViµV

µi cannot be locally gauge invariant.

85

③ Show that LYM can be written in the form of a trace

LYM = −1

2Tr (GµνG

µν)

④ Prove the validity of the “homogeneous Maxwell equation” or “Bianchi-identity”:

DρGµν + terms cyclic in (ρµν) ≡ 0

86

5 Quantization of gauge theories

Quantum chromodynamics, the modern theory of the strong interactions, is a non–Abelian gaugetheory with gauge group SU(3)c consisting of unitary 3 × 3 matrices of determinant unity. Thecorresponding internal degrees of freedom are called color. The generators are given by the basisof hermitian traceless 3 × 3 matrices Ti, i = 1, · · · 8. Quarks transform under the fundamen-tal 3–dimensional representation 3 (quark triplets) antiquarks under the complex conjugate 3∗

(antiquark anti–triplets). The requirement of local gauge invariance with respect to SU(3)c trans-formations implies that quark fields ψi(x) must couple to an octet of gauge fields, the gluon fieldsGµj , j = 1, · · · , 8, and together with the requirement of renormalizability this fixes the form ofthe interactions of the quarks completely: in the free quark Dirac–Lagrangian we have to replacethe derivative by the covariant derivative

∂µψ(x)→ Dµψ(x) , (Dµ)ik = ∂µδik − i gs∑

j

(Tj)ikGµj(x)

where gs is the SU(3)c gauge coupling constant. The dynamics of the gluon fields is controlledby the non–Abelian field strength tensor

Gµνi = ∂µGνi − ∂νGµi + gscijkGµjGνk

where cijk are the SU(3) structure constants obtained from the commutator of the generators[Ti, Tj ] = i cijk Tk. The locally gauge invariant Lagrangian density is then given by

Linv = −1

4

∑

i

Giµν Gµνi + ψ (iγµDµ −m) ψ .

We split Linv into a free part L0 and an interaction part Lint which is taken into account asa formal power series expansion in the gauge coupling gs. The perturbation expansion is anexpansion in terms of the free fields described by L0. The basic problem of quantizing mass-less spin 1 fields is familiar from QED. Since LYM is gauge invariant, the gauge potentials Giµcannot be uniquely determined from the gauge invariant field equations. Again one has to breakthe gauge invariance, now, for a SU(n) gauge group, by a sum of r = n2−1 gauge fixing conditions

Ci(G) = 0 , i = 1, · · · , r . (5.1)

It is known from QED that the only relativistically invariant condition linear in the gauge potentialwhich we can write is the Lorentz condition. Correspondingly we require

Ci(G) = −∂µ Gµi (x) = 0 , i = 1, · · · , r .

It should be stressed that a covariant formulation is mandatory for calculations beyond the treelevel. We are thus lead to break the gauge invariance of the Lagrangian by adding the gaugefixing term

LGF = − 1

2ξ

∑

i

(∂µ Gµi (x))

2

with ξ a free gauge parameter. Together with the term LG0 from Linv we obtain for the bilineargauge field part

LG,ξ0,i = −1

4(∂µ Gi ν − ∂ν Gi µ)2 − 1

2ξ(∂µ G

µi (x))

2

87

which now uniquely determines a free gauge field propagator. Unlike in QED, however, LGF

breaks local gauge invariance explicitly and one has to restore gauge invariance by a compen-sating Faddeev-Popov term (Faddeev and Popov 1967). The Faddeev-Popov trick consistsin adding further charged ghost fields ηi(x) and ηi(x), the so called Faddeev-Popov ghosts,which conspire with the other ghosts in such a way that physical matrix elements remain gaugeinvariant. Unitarity and renormalizability are then restored. The FP–ghosts must be masslessspin 0 fermions. For the unphysical ghosts this wrong spin–statistics assignment is no obstacle.The Faddeev-Popov term must be of the form

LFP = ηi(x)Mikηk(x) (5.2)

where

Mik =∂Ci(G)

∂Gjµ(x)(Dµ)jk = −∂µ (∂µδik − gcikjGjµ(x)) (5.3)

= −δik + gcikjGjµ(x) ∂µ + gcikj (∂µGjµ(x)) . (5.4)

By partial integration of SFP =∫

d4x LFP(x) we may write

LFP = ∂µηi∂µηi − gcikj (∂µηi)Cjµηk

which describes massless scalar fermions in interaction with the gauge fields.

5.1 The QCD Feynman rules

The complete Lagrangian for a quantized Yang-Mills theory is

Leff = Linv + LGF + LFP .

The free (bilinear) part

L0 = L0(G) + L0(ψ) + L0(η) (5.5)

with

L0(G) =1

2Giµ

[(gµν −

(1− 1

ξ

)∂µ∂ν

)δik

]Gkν (5.6)

L0(ψ) = ψαa

[((iγµ)αβ ∂µ −mδαβ

)δab

]ψβb (5.7)

L0(η) = ηi [(−) δik] ηk (5.8)

determines the free propagators, the differential operators in the square brackets being the inversesof the propagators. By Fourier transformation the free propagators are obtained in algebraic form(i.e. the differential operators are represented by c–numbers) in momentum space. Inverting thesec–number matrices we obtain the results depicted in Fig. 5.1.

The interaction part of the Lagrangian is given by

Lint = gsψγµTiψGiµ −

1

2gscikl (∂

µGνi − ∂νGµi ) GkµGlν

− 1

4g2sciklcik′l′G

µkG

νl Gk′µGl′ν − gscikj (∂µηi) Gjµηk

with a single coupling constant gs for the four different types of vertices.

88

While the formal argumentation which leads to the construction of local gauge theories looksnot too different for Abelian and non–Abelian gauge groups, the physical consequences are verydifferent and could not be more dramatic: in contrast to Abelian theories where the gauge field isneutral and exhibits no self–interaction, non–Abelian gauge fields necessarily carry non–Abeliancharge and must be self–interacting. These Yang-Mills self–interactions are responsible for theanti–screening of the non–Abelian charge, known as asymptotic freedom (AF) (see end of section).It implies that the strong interaction force gets weaker the higher the energy, or equivalently,the shorter the distance. While it appears most natural to us that particles interact the lessthe farther apart they are, non–Abelian forces share the opposite property, the forces get thestronger the farer away we try to separate the quarks. In QCD this leads to the confinement ofthe constituents within hadrons. The latter being quark bound states which can never be brokenup into free constituents. This makes QCD an intrinsically non–perturbative theory, the fields inthe Lagrangian, quarks and gluons, never appear in scattering states, which define the physicalstate space and the S–matrix. QED is very different, it has an perturbative S-matrix, its properdefinition being complicated by the existence of the long range Coulomb forces. Nevertheless, thefields in the QED Lagrangian as interpolating fields are closely related to the physical states, thephotons and leptons. This extends to the electroweak SM, where the weak non–Abelian gaugebosons, the W± and the Z particles, become massive as a consequence of the breakdown of theSU(2)L gauge symmetry by the Higgs mechanism. Also the weak gauge bosons cannot be seenas scattering states in a detector, but this time because of their very short lifetime. Due to itsnon–perturbative nature, precise predictions in strong interaction physics are often difficult, ifnot impossible. Fortunately, besides perturbative QCD which applies to hard subprocesses, non–perturbative methods have been developed to a high level of sophistication, like chiral perturbationtheory (CHPT) and QCD on a Euclidean space–time lattice (lattice QCD). Chiral perturbationtheory is based on the low energy structure of QCD: in the limit of vanishing quark masses QCDhas a global SU(Nf )V ⊗SU(Nf )A⊗U(1)V symmetry (chiral symmetry). Thereby the SU(Nf )Asubgroup turns out broken spontaneously, which, in the isospin limit Nf = 2, mu = md = 0,implies the existence of a triplet of massless pions (Goldstone bosons). U(1)V is responsiblefor baryon number conservation, whereas in contrast U(1)A is broken by the Adler-Bell-Jackiwanomaly.

The derivation of the proper quantization of QCD has been very formal and needs better justifi-cation. In fact the products of field operators, which are used in particular in the Lagrangian arenot well defined on the quantum level. This becomes immediately clear when we try calculationsof leading quantum corrections. What one has to do first is a proper definition of the theory,means the theory has to be regularized, such that singularities are absent at least order by orderin the perturbative expansion. Our aim is to discuss dimensional regularization before we aregoing to calculate one–loop counterterms and leading renormalization group coefficients, whichplay a key role in any phenomenological application of QCD.


① Spin 1 propagators are symmetric second rank Lorentz tensors proportional to gµν andkµkν , where k is the four momentum of the spin 1 boson. Show that

T µν = gµν − kµkν

k2; Lµν =

kµkν

k2

are projection tensors to the transversal and the longitudinal field components. Verify

89

the projector property of T and L. Use the decomposition

Kµν = AT µν +B Lµν

for the bilinear part of the L0G = GµKµνGν of the free part of the Lagrangian (what are

A and B in case of the gluon), calculate the propagator (inverse of the kernel K) using theprojection technique. Formulate the condition under which the inverse exists.

② Given the Yang-Yills Lagrangian in coordinate space derive the triple gluon coupling

−igscijk gµν (p2 − p1)ρ + gµρ (p1 − p3)ν + gνρ (p3 − p2)µ

and the quartic gluon coupling

−g2s cnijcnkl (gµρgνσ − gµσgνρ) + cnikcnjl (gµνgρσ − gµσgνρ) + cnilcnjk (gµνgρσ − gµρgνσ)

in momentum space.

③ According to Faddeev and Popov the fatal violation of gauge invariance by the gauge fixingterm can be avoided by taking into account the functional determinant obtained in thefunctional integral under a gauge transformation of the fields (integration variables). If wedefine the functional integral as follows, with a Faddeev-Popov determinant,

∫DGµaDet

(δCaδωb

)ei∫

(

Linv− 12ξC2

a

)

d4x

one easily checks that now the functional integral is independent on the specific choice ofthe gauge function Ca. Check this!

By introducing anticommuting scalars, the FP ghost fields ηa and ηa, we may represent theFP-determinant as a Berezin integral over Grassmann variables (algebra of anticommutingc-numbers)

Det

(δCaδωb

)=

∫DηDηei

∫

LFPd4x

with

LFP = ηaMabηb ; Mab.=δCaδωb

.

Check that this agrees with the expression adopted above.

④ Becchi-Rouet-Stora (BRS) symmetry: given the “quasi invariant” effective Lagrangian

Leff = Linv + LGF + LFP ,

the local gauge invariance of the functional integral (path integral quantization)∫DGDηDη ei

∫

Leffd4x (5.9)

yields relations between Green functions, the Slavnov-Taylor (ST) identities. They gener-alize the Ward-Takahashi (WT) identities, relations between Green functions which derivefrom global symmetries. In contrast to canonical quantization, which relies on the nongauge-invariant splitting into a free and an interacting part of the Lagrangian, the pathintegral formulation work with the complete effective Lagrangian.

The ST-identities provide the tool needed for proofs of

90

i) gauge invariance

ii) unitarity

iii) renormalizability

of the S-matrix. ST-identities may be obtained from the BRS-symmetry of Leff .

Convince yourself that the BRS procedure (described in the following) to control gauge in-variance indeed works. The idea behind BRS-symmetry is to dispose of the as yet undefinedtransformation properties of the FP-ghost fields η and η such that

δBRSLeff = 0 . (5.10)

In order to achieve this it is natural to demand the following relations to hold:

i) δLinv = 0

ii) δLGF = −1ξCaMabωb

iii) δLFP = δηMabηb + ηaδ(Mabηb)

= −δLGF

iv) DGDηDη invariant .

A solution for this set of conditions may be obtained as follows:1) Introduce anticommuting global c−number variables δλ, δλ anticommuting with η and η,and identify ωb = ηbδλ. Thus

• δGa = Dabηbδλ

where Ga can be a gauge field, a scalar or a Fermi field with Dabηb given by

δGaµδωb

ηb.= Dµabηb

in case of the gauge field.

2) Assume η to transform according to the regular representation, thus

• δηa = −12gfabcηbηcδλ

where a permutation symmetry factor 1/2 (antisymmetry of fabc and anticommutativity ofthe η’s) has been taken into account. 1) and 2) imply δ(Mabηb) = 0 . We thus take thefreedom to choose: 3) The field η transforms as

• δηa = −1ξCaδλ

such that conditions i) to iii) are satisfied. One can show iv) to be true for the above set oftransformations which define the BRS-transformation.ST-identities:The BRS invariance of Leff allows a simple derivation of the ST-identities. Performing achange of integration variables in the functional integral does not change the value of theintegral. See my TASI Lectures Renormalizing the Standard Model for more details.

91

a). Quark propagator

: ∆ψF (p)αβ, ab =

(1

/p−m+iε

)αβδab

b). Massless gluon propagator

: ∆GF (p, ξ)µνik = −

(gµν − (1− ξ)pµpν

p2

)1

p2+iεδik

c). Massless FP–ghost propagator

: ∆ηF (p)ik = 1

p2+iεδik

d). Quark–gluon coupling

:= gs (γµ)αβ (Ti)ab

e). Triple gluon coupling

:= −igscijk gµν (p2 − p1)ρ + gµρ (p1 − p3)ν + gνρ (p3 − p2)µ

f). Quartic gluon coupling

:= −g2s

cnijcnkl (gµρgνσ − gµσgνρ)+cnikcnjl (gµνgρσ − gµσgνρ)+cnilcnjk (gµνgρσ − gµρgνσ)

g). FP–ghost gluon coupling

:= −igscijk (p3)µ

p

α, a β, b

p

µ, i ν, k

p

i k

µ, i, p1

α, a, p3

β, b, p2

µ, i, p1ρ, k, p3

ν, j, p2

µ, i ν, j

ρ, kσ, l

µ, i, p1k, p3

j, p2

Figure 5.1: Feynman rules for QCD. Momenta at vertices are chosen ingoing

92

6 Dimensional regularization

In local QFT field products (monomials in the fields) which show up at the classical Lagrangianlevel (usually the starting point for the quantization) are ill defined due to short distance singu-larities showing up when going to coinciding space–time points: limx→y ϕ(x)ϕ(y).

When calculating perturbative contributions, to a Green function say, associated with a particularFeynman diagram Γ one has to consider an expression like

ΠV ∈Γ PV (∂) Πℓ∈Γ Pℓ(∂) ∆F(xiℓ − xfℓ) (6.1)

to be integrated over all interaction vertices V ∈ Γ. The lines labeled by ℓ ∈ Γ representthe Feynman propagators Pℓ(∂) ∆F(xiℓ − xfℓ). The polynomial Pℓ depends of the spin of theparticle and the polynomial PV represents the possible derivative couplings at the vertex V (inmomentum space Pℓ(∂) and PV (∂) are polynomials of the appropriate momenta). All singularitiesin coordinate space are at coinciding points (representing the vertices), which means that theproblem concerns a local distribution

ΠℓPω(∂) δ(xiℓ − xfℓ) , (6.2)

where Pω(∂) is a polynomial in the derivatives with respect to the arguments of the delta functions.For a renormalizable QFT the degree of the polynomial is not higher than 2 (for a proof seeIntroduction to the theory of quantized fields by N.N. Bogolyubov, D.V. Shirkov. 1980.) andsubtracting (6.2) from (6.1) with appropriate values for the coefficients of the polynomial, removesall singularities and determines the renormalized QFT order by order in perturbation theory. Hismeans that the bare theory is well defined almost everywhere, except from a low dimensionalsubspace of coinciding vertex points. Renormalization we call the procedure to make the theorywell defined everywhere, i.e. to perform a continuous extension to the subspace.

In momentum space the short distance singularities show up as ultraviolet (UV) singularities,which manifest themselves as a polynomial in the momenta (at most quadratic in a renormalizabletheory), however with singular coefficients. In order to get the perturbative expansion welldefined, the theory has to be regularized such that one has a well defined staring point. Fornon-Abelian gauge theories one needs a regularization which preserves in particular the gaugesymmetry as much as possible. Dimensional regularization makes use of the fact that a QFTbecomes less singular as the dimension of space–time gets lower (than 4), without changing basicsymmetry properties (identical Feynman rules) of the theory. In the following we describe inshort the basic mechanism how it works.

Dimensional regularization of theories with spin is defined in three steps.

1. Start with Feynman rules formally derived in d = 4.

2. Generalize to d = 2n > 4. This intermediate step is necessary in order to treat the vector andspinor indices appropriately. Of course it means that the UV behavior of Feynman integrals atfirst gets worse.

1) For fermions we need the d = 2n–dimensional Dirac algebra:

γµ, γν = 2gµν1 ; γµ, γ5 = 0

where γ5 must satisfy γ25 = 1 and γ+5 = γ5 such that 12 (1± γ5) are the chiral projection matrices.

93

The metric has dimension d

gµνgµν = gµµ = d ; gµν =

1 0 · · ·0 −1...

. . .

−1

. (6.3)

By 1 we denote the unit matrix in spinor space. In order to have the usual relation for the adjointspinors we furthermore require

γµ+ = γ0γµγ0 .

Simple consequences of this d–dimensional algebra are:

γαγα = d 1

γαγµγα = (2− d) γµ

γαγµγνγα = 4gµν 1 + (d− 4) γµγν

γαγµγνγργα = −2γργνγµ + (4− d)γµγνγρ etc.

(6.4)

Traces of strings of γ–matrices are very similar to the ones in 4–dimensions. In d = 2n dimensionsone can easily write down 2d/2–dimensional representations of the Dirac algebra. Then

Tr(1) = f(d) = 2d/2

Tr∏2n−1i=1 γµi(γ5) = 0

Tr γµγν = f(d) gµν

Tr γµγνγργσ = f(d) (gµνgρσ − gµρgνσ + gµσgνρ) etc.

(6.5)

One can show that for renormalized quantities the only relevant property of f(d) is f(d) → 4for d → 4. Very often the convention f(d) = 4 (for any d) is adopted. Bare quantities and therelated minimally subtracted MS or modified minimally subtracted MS quantities (see below forthe precise definition) depend upon this convention (by terms proportional to ln 2).

In anomaly free theories we can assume γ5 to be fully anticommuting! But then

Tr γµγνγργσγ5 = 0 for all d 6= 4! (6.6)

The 4–dimensional object

4iεµνρσ = Tr γµγνγργσγ5 for d = 4 (6.7)

cannot be obtained by dimensional continuation if we use an anticommuting γ5 (see Jegerlehner2001 and references therein).

Since fermions do not have self interactions they only appear as closed fermion loops, whichyield a trace of γ–matrices, or as a fermion string connecting an external ψ · · · ψ pair of fermionfields. In a transition amplitude |T |2 = Tr (· · ·) we again get a trace. Consequently, in principle,we have eliminated all γ’s! Commonly one writes a covariant tensor decomposition into invariantamplitudes, like, for example,

= iΓµ = −ie

γµA1 + iσµν qν

2mA2 + γµγ5A3 + · · ·

f

f

γ

94

where µ is an external index, qµ the photon momentum and Ai(q2) are scalar form factors.

2) External momenta (and external indices) must be taken d = 4 dimensional, because thenumber of independent “form factors” in covariant decompositions depends on the dimension,with a fewer number of independent functions in lower dimensions. Since four functions cannotbe analytic continuation of three etc. we have to keep the external structure of the theory ind = 4. The reason for possible problems here is the non–trivial spin structure of the theory ofinterest. The following rules apply:

External momenta : pµ = (p0, p1, p2, p3, 0, · · · , 0) 4− dimensional

Loop momenta : kµ = (k0, · · · kd−1) d − dimensional

k2 = (k0)2 − (k1)2 − · · · − (kd−1)2

pk = p0k0 − ~p · ~k 4− dimensional etc.

3. Interpolation in d to complex values and extrapolation to d < 4.

Loop integrals now read

µ4−d∫

ddk

(2π)d· · ·

with µ an arbitrary scale parameter. The crucial properties valid in DR independent of d are:(F.P. = finite part)

a)∫

ddkkµf(k2) = 0

b)∫

ddkf(k + p) =∫

ddkf(k)

which is not true with UV cut− off ′s

c) If f(k) = f(| k |) :∫

ddkf(k) = 2πd/2

Γ(d2)

∫∞0 drrd−1f(r)

d) For divergent integrals, by analytic subtraction :

F.P.∫∞0 drrd−1+α ≡ 0 for arbitrary α

so called minimal subtraction (MS). Consequently

F.P.∫

ddkf(k) = F.P.∫

ddkf(k + p) = F.P.∫

dd(λk)f(λk) .

This implies thatdimensionally regularized integrals behave like convergent integralsand formal manipulations are justified. Starting with d sufficiently small, by partial integration,one can always find a representation for the integral which converges for d = 4− ǫ , ǫ > 0 small.

In order to elaborate in more detail how DR works in practice, let us consider a genericone–loop Feynman integral

Iµ1···µmΓ (p1, · · · , pn) =

∫ddk

∏mj=1 k

µj

∏ni=1((k + pi)2 −m2

i + iε)(6.8)

which has superficial degree of divergence

d(Γ) = d+m− 2n ≤ d− 2 (6.9)

95

where the bound holds for two– or more–point functions in renormalizable theories and for d ≤ 4.Since the physical tensor and spin structure has to be kept in d = 4, by contraction with externalmomenta or with the metric tensor gµiµj it is always possible to write the above integral as a sumof integrals of the form

Iµ1···µm′Γ (p1, · · · , pn′) =

∫ddk

∏m′

j=1 kµj

∏n′i=1((k + pi)2 −m2

i + iε)(6.10)

where now µj and pi are d = 4–dimensional objects and

ddk = d4k dd−4k = d4k ωd−5 dω dΩd−4 . (6.11)

In the d− 4–dimensional complement the integrand depends on ω only! The angular integrationover dΩd−4 yields

∫dΩd−4 = Sd−4 =

2πǫ/2

Γ(ǫ/2); ǫ = d− 4 , (6.12)

which is the surface of the d − 4–dimensional sphere. Using this result we get (discarding thefour–dimensional tensor indices)

IΓ(pi) =

∫d4kJΓ(d, p, k) (6.13)

where

JΓ(d, p, k) = Sd−4

∫ ∞

0dωωd−5f(p, k, ω) . (6.14)

Now this integral can be analytically continued to complex values of d. For the ω–integration wehave

dω(Γ) = d− 4− 2n (6.15)

i.e. the ω–integral converges if

d < 4 + 2n . (6.16)

In order to avoid infrared singularities in the ω–integration one has to analytically continue byappropriate partial integration. After p–fold partial integration we have

IΓ(pi) =2π

d−42

Γ(d−42 + p)

∫d4k

∫ ∞

0dωωd−5+2p

(− ∂

∂ω2

)pf(p, k, ω) (6.17)

where the integral is convergent in 4− 2p < Re d < 2n−m = 4− d(4)(Γ) ≥ 2 .For a renormalizable theory at most 2 partial integrations are necessary to define the theory.

6.1 Tools for the Evaluation of Feynman Integrals

6.1.1 ǫ = 4− d Expansion, ǫ→ +0

For the expansion of integrals near d = 4 we need some asymptotic expansions of Γ–functions:

Γ(1 + x) = exp

[−γ x+

∞∑

n=2

(−1)n

nζ(n)xn

]|x| ≤ 1 (6.18)

ψ(1 + x) =d

dxln Γ(1 + x) =

Γ′(1 + x)

Γ(1 + x)

|x|<1= −γ +

∞∑

n=2

(−1)nζ(n)xn−1 (6.19)

96

where ζ(n) denotes Riemann’s Zeta function. The defining functional relation is

Γ(x) =Γ(x+ 1)

x, (6.20)

which for n = 0, 1, 2, · · · yields Γ(n+ 1) = n! with Γ(1) = Γ(2) = 1. Furthermore we have

Γ(x) Γ(1− x) =π

sinπx(6.21)

Γ(1

2+ x) Γ(

1

2− x) =

π

cos πx. (6.22)

Important special constants are

Γ(1

2) =

√π (6.23)

Γ′(1) = −γ ; γ = 0.577215 · · · Euler′s constant (6.24)

Γ′′(1) = γ2 + ζ(2) ; ζ(2) =π2

6= 1.64493 · · · (6.25)

As a typical result of an ǫ–expansion, which we should keep in mind for later purposes, we have

Γ(1 +ǫ

2) = 1− ǫ

2γ + (

ǫ

2)2

1

2(γ2 + ζ(2)) + · · · (6.26)

6.1.2 Bogolubov-Schwinger Parametrization

Suppose we choose for each propagator an independent momentum and take into account mo-mentum conservation at the vertices by δ–functions. Then, for d = n integer, we use

i)

i

p2 −m2 + iε=

∫ ∞

0dα e−iα(m

2−p2+iε)

ii)

δ(d)(k) =1

(2π)d

∫ +∞

−∞ddx eikx

and find that all momentum integrations are of Gaussian type. The Gaussian integrals yield

∫ +∞

−∞ddkP (k)ei(ak

2+2b(k·p)) = P

(−i

2b

∂

∂p

)( πia

)d/2e−i b

2/a p2

for any polynomial P. The resulting form of the Feynman integral is the so called Bogolubov-Schwinger representation.

6.1.3 Feynman Parametric Representation

Transforming pairs of α–variables in the above Bogolubov-Schwinger parametrization accordingto (l is denoting the pair (i, k))

(αi, αk)→ (ξl, αl) : (αi, αk) = (ξlαl, (1− ξl)αl)

97

⊗

⊗

⊗

⊗

(6.28)

Im q0

Re q0

C

R

Figure 6.1: Wick rotation in the complex q0–plane. The poles of the Feynman propagator areindicated by ⊗’s. C is an integration contour, R is the radius of the arcs.

∫ ∞

0

∫ ∞

0dαidαk · · · =

∫ ∞

0dαl αl

∫ 1

0dξl · · · ,

the integrals are successively transformed into∫ 10 dξ · · · integrals and at the end there remains

one α–integration only which can be performed using

∫ ∞

0dα αa e−αx = Γ(a+ 1)x−(a+1) .

The result is the Feynman parametric representation. If L is the number of lines of a diagram,the Feynman integral is L− 1–dimensional.

6.1.4 Euclidean Region, Wick–Rotations

The basic property which allows us to perform a Wick rotation is analyticity which derives fromthe causality of a relativistic QFT. In momentum space the Feynman propagator

1

q2 −m2 + iε=

1

q0 −√~q 2 +m2 − iε

1

q0 +√~q 2 +m2 − iε

=1

2ωp

1

q0 − ωp + iε− 1

q0 + ωp − iε

is an analytic function in q0 with poles at q0 = ±( ωp − iε)21 where ωp =√~q 2 +m2. This

allows us to rotate by π2 the integration path in q0, going from −∞ to +∞, without crossing any

singularity. In doing so, we rotate from Minkowski space to Euclidean space

q0 → −iqd ⇒ q = (q0, q1, . . . , qd−2, qd−1)→ q = (q1, q2, . . . , qd−1, qd) (6.27)

and thus q2 → −q2. This rotation to the Euclidean region is called Wick rotation.

More precisely: analyticity of a function f(q0, ~q ) in q0 implies that the contour integral

∮

C(R)dq0 f(q0, ~q ) = 0

21Note that because of the positivity of ~q 2 +m2 for any non–vacuum state, we have ωp − iε =√

~q 2 +m2 − iε inthe limit limε→0, which is always understood. The symbolic parameter ε of the iε prescription, may be scaled byany fixed positive number.

98

for the closed path C(R) in Fig. 6.1 vanishes. If the function f(q0, ~q ) falls off sufficiently fast atinfinity, then the contribution from the two “arcs” goes to zero when the radius of the contourR→∞. In this case we obtain

∞∫

−∞

dq0 f(q0, ~q ) +

−i∞∫

+i∞

dq0 f(q0, ~q ) = 0

or

∞∫

−∞

dq0 f(q0, ~q ) =

+i∞∫

−i∞

dq0 f(q0, ~q ) = −i

+∞∫

−∞

dqd f(−iqd, ~q ) ,

which is the Wick rotation. At least in perturbation theory, one can prove that the conditionsrequired to allow us to perform a Wick rotation are fulfilled.

We notice that the Euclidean Feynman propagator obtained by the Wick rotation

1

q2 −m2 + iε→ − 1

q2 +m2(6.29)

has no singularities (poles) and an iε–prescription is not needed any longer.

In configuration space a Wick rotation implies going to imaginary time x0 → ix0 = xd suchthat qx→ −qx and hence

x0 → −ixd ⇒ x2 → −x2 , x → −∆x , i

∫ddx · · · →

∫ddx · · · . (6.30)

While in Minkowski space x2 = 0 defines the light–cone x0 = ±|~x|, in the Euclidean region x2 = 0implies x = 0. Note that possible singularities on the light–cone like 1/x2, δ(x2) etc. turn intosingularities at the point x = 0. This simplification of the singularity structure is the merit ofthe positive definite metric in Euclidean space.

In momentum space the Euclidean propagators are positive (discarding the overall sign) and anyFeynman amplitude in Minkowski space may be obtained via

IM (p) = (−i)Nint(−i)V−1 IE(p)∣∣p4=ip0 ; m2→m2−iε (6.31)

from its Euclidean version. Here, Nint denotes the number of internal lines (propagators) and Vthe number of vertices if we use the substitutions (convention dependent)

1

p2 −m2 + iε→ 1

p2 +m2; igi → i (igi) = −gi ;

∫ddk →

∫ddk

to define the Euclidean Feynman amplitudes. By gi we denote the gauge couplings.

For the dimensionally regularized amplitudes, where potentially divergent integrals are definedvia analytic continuation from regions in the complex d–plane where integrals are manifestlyconvergent, the terms from the arc segments can always be dropped. Also note that dimensionalregularization and the power counting rules (superficial degree of divergence etc.) hold irrespec-tive of whether we work in d–dimensional Minkowski space–time or in d–dimensional Euclideanspace. The metric is obviously not important for the UV–behavior of the integrals.

99

6.2 Scalar One–Loop Integrals

Here we apply our tools to the simplest scalar one–loop integrals (p.i.= partial integration).

= µ4−d

(2π)d

∫

ddk 1k2+m2 = µ4−d(4π)−d/2

∫

∞

0 dαα−d/2e−αm2

convergent for d < 2 ∗ ∗ ∗ 1

p.i.= −2m2

d−2µ4−d(4π)−d/2

∫

∞

0 dα1−d/2e−αm2

convergent for d < 4

= −2m2(4π)−d/2 Γ(2−d/2)d−2

(

m2

µ2

)d/2−2

= −2m2(4π)−2 2ǫ Γ(1 + ǫ

2 ) 12−ǫe

ǫ2(ln 4π−ln m2

µ2 )

ǫ→+0≃ m2(4π)−2

2ǫ − γ + 1 + ln 4π − ln m2

µ2

+O(ǫ)

= µ4−d

(2π)d

∫

ddk 1k2+m2

1

1(k+p)2+m2

2

= µ4−d(4π)−d/2∫

∞

0 dα1dα2(α1 + α2)−d/2e

−(α1m21+α2m2

2+α1α2

α1+α2p2)

α1 = xλ ; α2 = (1− x)λ= µ4−d(4π)−d/2Γ(2− d

2)∫ 10 dx(xm2

1 + (1− x)m22 + x(1− x)p2))d/2−2

convergent for d < 4

= (4π)−2 2ǫ Γ(1 + ǫ

2)eǫ2

ln 4π ∫ 10 dxe

−ǫ2

lnxm2

1+(1−x)m2

2+x(1−x)p2

µ2

ǫ→+0≃ (4π)−2

2ǫ − γ + ln 4π −

∫ 10 dx ln

xm21+(1−x)m2

2+x(1−x)p2

µ2

+O(ǫ)

= µ4−d

(2π)d

∫

ddk 1k2+m2

1

1(k+p

1)2+m2

2

1(k+p

1+p

2)2+m2

3

convergent for d = 4ǫ→+0≃ (4π)−2

∫

∞

0 dα1dα2dα31

(α1+α2+α3)2 e−(α1m21+α2m2

2+α3m23)

×e−

α1α2p21+α2α3p2

2+α3α1p2

3α1+α2+α3

α1 = xyλ ; α2 = x(1− y)λ ; α3 = (1− x)λ ; α1 + α2 + α3 = λ

= (4π)−2∫ 10 dydxx 1

N

m

p

m1

m2p

p3

p1

p2

m1

m3

m2

N = x2y (1− y)p21+ x (1− x)(1− y)p2

2+ x (1− x) yp2

3+ xym2

1 + x (1− y) m22 + (1− x) m2

3

Figure 6.2: Standard scalar one–loop integrals∗

∗A direct integration of the one–loop tadpole yields

m2(4π)−d/2Γ(1− d/2)

(

m2

µ2

)d/2−2

which by virtue of Γ(1 − d/2) = −2Γ(2 − d/2)/(d − 2) is the same analytic function as the one obtained via thepartial integration method.

100

Standard Scalar One–loop Integrals (m2=m2 − iε).

m

p= µǫ0

∫ddk

(2π)d1

k2 −m2

.= − i

16π2A0(m) , (6.32)

defines the standard tadpole type integral , where

A0(m) = −m2(Reg + 1− lnm2) (6.33)

with

Reg =2

ǫ− γ + ln 4π + lnµ20 ≡ lnµ2 . (6.34)

The last identification defines the MS scheme of (modified) minimal subtraction.

m

mp

1

2

= µǫ0

∫ddk

(2π)d1

(k2 −m21)((k + p)2 −m2

2)) .

=i

16π2B0(m1,m2; p2) , (6.35)

defines the standard propagator type integral , where

B0(m1,m2; s) = Reg−∫ 1

0dz ln(−sz(1− z) +m2

1(1− z) +m22z − iε) . (6.36)

m

m

mp

p

p

1

3

2

3

1

2

= µǫ0

∫ddk

(2π)d1

(k2 −m21)((k + p1)2 −m2

2

) ((k + p1 + p2)2 −m2

3

)

= − i

16π2C0(m1,m2,m3; p

21, p

22, p

23) , (6.37)

defines the standard form factor type integral , where

C0(m1,m2,m3; s1, s2, s3) =

∫ 1

0dx

∫ x

0dy

1

ax2 + by2 + cxy + dx+ ey + f(6.38)

with

a = s2, d = m22 −m2

3 − s2,b = s1, e = m2

1 −m22 + s2 − s3,

c = s3 − s1 − s2, f = m23 − iε .

Remark: the regulator term Reg in (6.34) denotes the UV regulated pole term 2ǫ supplemented

with O(1) terms which always accompany the pole term and result from the ǫ–expansion of thed–dimensional integrals. While in the MS scheme just the poles 2

ǫ are subtracted, in the modifiedMS scheme MS also the finite terms included in (6.34) are subtracted. The dependence on theUV cut–off 2

ǫ in the MS scheme defined by Reg ≡ lnµ2 is reflected in a dependence on the MSrenormalization scale µ.

The UV –singularities (poles in ǫ at d=4) give rise to finite extra contributions when they aremultiplied with d (or functions of d) which arise from contractions like gµµ = d , γµγµ = d etc.For d→ 4 we obtain:

dA0(m) = 4A0(m) + 2m2 , dB0 = 4B0 − 2 . (6.39)

The explicit evaluation of the scalar integrals (up to the scalar four–point function) is discussedin ’t Hooft-Veltman 1979 (see also Davydychev, Kalmykov 2001, Fleischer, Jegerlehner, Tarasov2003).

101

Figure 6.3: R(z) as a function of z in the integration range I : 0 ≤ z ≤ 1. a) for s < (m1 −m2)2

R(z) is real and has no real zeros in I; the integral is real for real s < (m1 − m2)2 . b) for

s > (m1 + m2)2 R(z) has two distinct real zeros within I; the integral is complex for reals > (m1 +m2)

2.

6.3 Selected Properties of Scalar Integrals

The general scalar two–point function is given by (see Exercises to this Section) by

B0(m1,m2; s) = Reg + 2− 1

2lnm2

1 −1

2lnm2

2 −1

2

m21 −m2

2

slnm2

1

m22

+

√X2s ln

m21+m

22−s+

√X

m21+m

22−s−

√X

; s < (m1 −m2)2

−√−Xs arctan

√−X

m21+m

22−s

; (m1 −m2)2 < s < (m1 +m2)2

√X2s

(ln

s−m21−m2

2−√X

s−m21−m2

2+√X

+ 2πi)

; (m1 +m2)2 < s

(6.40)

where

X = s2 − 2s (m21 +m2

2) + (m21 −m2

2)2

= s2 (1− y+) (1− y−) ; y± =(m1 ±m2)

2

s.

In order to understand the main properties let us look in Fig. 6.3 at the argument R(z) in thedefining integral

B0(m1.m2; s) = Reg−1∫

0

dz lnR(z) .

We note thatImB0(m1,m2; s) = 0 for s < (m1 +m2)

2 .

If the energy exceeds the production threshold√s = m1 +m2 the particles of masses m1 and m2

may be produced as real states, i.e. the intermediate states in the loop diagramm

mp

1

2

⇒m

mp

1

2

may exist as real physical states. B0 then must exhibit an imaginary part (compare this with theunitarity relation discussed in Sect. ?? below)

ImB0(m1,m2; s) = 2π i

√X

2sfor s > (m1 +m2)

2 . (6.41)

102

We note that B0(m1,m2; s) exhibits a cut in the complex s–plane for s > (m1 + m2)2. Thephysical sheet is Im s > 0 i.e. s+ iε

s physical edge

Im s

Re s

6.4 Tensor Integrals

In dimensional regularization also the calculation of tensor integrals is rather straight forward.Sign conventions are chosen in accordance with the Passarino-Veltman (1979) convention. Invari-ant amplitudes are defined by performing covariant decompositions of the tensor integrals, whichthen are contracted with external vectors or with the metric tensor. A factor i/16π2 is taken outfor simplicity of notation, i.e.

∫

k· · · = 16π2

i

∫ddk

(2π)d· · · .

1) One point integrals:By eventually performing a shift k → k + p of the integration variable we easily find the followingresults:

∫k

1(k+p)2−m2 = −A0(m)

∫k

kµ

(k+p)2−m2 = pµA0(m)

∫k

kµkν

(k+p)2−m2 = −pµpνA21 + gµνA22

(6.42)

A21 = A0(m)

A22 = −m2

dA0(m)

ǫ→0≃ −m2

4A0(m) +

m4

8

2) Two point integrals: the defining equations here are

∫k

1(1)(2) = B0(m1,m2; p

2)

∫k

kµ

(1)(2) = pµB1(m1,m2; p2)

∫kkµkν

(1)(2) = pµpνB21 − gµνB22 ,

(6.43)

where we denoted scalar propagators by (1) ≡ k2 −m21 and (2) ≡ (k + p)2 −m2

2. The simplestnon–trivial example is B1. Multiplying the defining equation with 2pµ we have

2p2B1 =

∫

k

2pk

k2 −m21 + iε

1

(p+ k)2 −m22 + iε

(6.44)

and we may write the numerator as a difference of the two denominators plus a remainder whichdoes not depend on the integration variable:

2pk = (p + k)2 − k2 − p2 = [(p+ k)2 −m22]− [k2 −m2

1]− (p2 +m21 −m2

2)

103

After canceling the square brackets against the appropriate denominator we obtain

B1(m1,m2; p2) =1

2p2A0(m2)−A0(m1)− (p2 +m2

1 −m22) B0(m1,m2; p2)

(6.45)

A further useful relation is

B1(m,m; p2) = −1

2B0(m,m; p2) .

In a similar way, by contracting the defining relation with pν and gµν we find for arbitrarydimension d

B21 = 1(d−1) p2

(1− d/2)A0(m2)− d/2(p2 +m2

1 −m22)B1 −m2

1B0

B22 = 12(d−1)

A0(m2)− (p2 +m2

1 −m22)B1 − 2m2

1B0

.

Expansion in d = 4− ǫ, ǫ→ 0 yields

B21 = −13p2

A0(m2) + 2(p2 +m2

1 −m22)B1 +m2

1B0 + 1/2(m21 +m2

2 − p2/3)

B22 = 16

A0(m2)− (p2 +m2

1 −m22)B1 − 2m2

1B0 − (m21 +m2

2 − p2/3) (6.46)

where the arguments of the B–functions are obvious.

3) Three point integrals: for the simplest cases we define the following invariant amplitudes

∫k

1(1)(2)(3) = −C0(m1,m2,m3; p21, p

22, p

23)

∫k

kµ

(1)(2)(3) = −pµ1C11 − pµ2C12

∫k

kµkν

(1)(2)(3) = −pµ1pν1C21 − pµ2pν2C22 − (pµ1pν2 + pµ2p

ν1)C23 + gµνC24

(6.47)

where p3 = −(p1 + p2), (1) ≡ k2 −m21, (2) ≡ (k + p1)

2 −m22 and (3) ≡ (k + p1 + p2)2 −m2

3.

The C1i’s can be found using all possible independent contractions with p1µ,ν , p2µ,ν and gµν . Thisleads to the equations

p21 p1p2

p1p2 p22

︸︷︷︸X

C11

C21

=

R1

R2

with

R1 = 12(B0(m2,m3; p

22)−B0(m1,m3; p23)

−(p21 +m2

1 −m22)C0

)

R2 = 12

(B0(m1,m3; p23)−B0(m1,m2; p

21)

+ (p21 − p23 −m22 +m2

3)C0

).

The inverse of the kinematic matrix of the equation to be solved is

X−1 =1

DetX

p22 −p1p2−p1p2 p21

, DetX

.= p21p

22 − (p1p2)

2

104

and the solution reads

C11 =1

DetX

p22R1 − (p1p2)R2

C12 =1

DetX

−(p1p2)R1 + p21R2

. (6.48)

The same procedure applies to the more elaborate case of the C2i’s where the solution may bewritten in the form

C24 = −m21

2C0 +

1

4B0(2, 3) − 1

4(f1C11 + f2C12) +

1

4(6.49)

C21

C23

= X−1

R3

R5

;

C23

C22

= X−1

R4

R6

(6.50)

with

R3 = C24 − 12 (f1C11 +B1(1, 3) +B0(2, 3))

R5 = −12 (f2C11 +B1(1, 2) −B1(1, 3))

R4 = −12 (f1C12 +B1(1, 3) −B1(2, 3))

R6 = C24 − 12 (f2C12 −B1(1, 3))

andf1 = p21 +m2

1 −m22 ; f2 = p23 − p21 +m2

2 −m23 .

The notation used for the B–functions is as follows: B0(1, 2) denotes the two point functionobtained by dropping propagator 1

(3) from the form factor i.e.∫k

1(1)(2) and correspondingly for

the other cases.

6.5 Special Integrals for massless propagators

We are again adopting the notation

∫

k· · · = 16π2

i

∫ddk

(2π)d· · ·

The base integral we consider are the two–point functions with arbitrary powers of the denomi-nators:

I(r, s;m1,m2; p2) =

∫

k

1

(1)r(2)s=

1

(r − 1)!

1

(s− 1)!∂r−11 ∂s−12

∫

k

1

(1)(2)

with (1) = k2−m21 + i0, (2) = (k+ p)2−m2

2 + i0, ∂1 ≡ ∂/∂m21 and ∂2 ≡ ∂/∂m2

2. In d dimensions

∫

k

1

(1)(2)= Bd

0(m1,m2; p2)

= (4π)2−d/2∫ ∞

0dα1dα2 (α1 + α2)−d/2 e

−(

α1m21+α2m2

2−α2α2α1+α2

p2)

.

105

Utilizing this representation we may write (note the symmetry I(s, r) = I(r, s) when m1 = m2)

I(r, s) =(−1)r+s

(r − 1)!(s − 1)!(4π)2−d/2

∫ ∞

0dα1dα2 α

r−11 αs−12 · · ·

=(−1)r+s

Γ(r) Γ(s)(4π)2−d/2

∫ ∞

0dλλr+s−d/2−1

×∫ 1

0dxxr−1 (1− x)s−1 e−λ(xm

21+(1−x)m2

2−x(1−x) p2)

= (4π)2−d/2(−1)r+s

Γ(r) Γ(s)Γ(r + s− d/2)

×∫ 1

0dxxr−1 (1− x)s−1

(xm2

1 + (1− x)m22 − x(1− x) (p2 + i0)

)d/2−r−s.

Concerning the iε–prescription we note that m2i → m2

i − iε yields xm21+(1−x)m2

2−x(1−x) p2 →xm2

1+(1−x)m22−x(1−x) p2− iε, which is fully equivalent to xm2

1+(1−x)m22−x(1−x) (p2+iε)

as both x ≥ 0 and (1− x) ≥ 0 in the integration range.

We are interested in special zero mass integrals which frequently show up in QCD calculations:1. m2

1 = m2 ; m22 = 0 ; p2 = 0: Using

∫ 1

0dx · · · = (m2)d/2−r−s

∫ 1

0dxx

d2−s−1 (1− x)s−1

= (m2)−ε/2 (m2)2−r−sB(d

2− s, s)

where B(α, β) = Γ(α)Γ(β)Γ(α+β) is the well known Beta–function. We thus find

I(r, s;m2, 0; 0) = (4π)ε/2 (m2)−ε/2 (−m2)2−r−sΓ(r + s− d/2)Γ(d/2 − s)

Γ(r)Γ(d/2).

2. m21 = m2

2 = 0: Using

∫ 1

0dx · · · = (−(p2 + i0))d/2−r−s

∫ 1

0dxx

d2−s−1 (1− x)d/2−r−1

= (−(p2 + i0))−ε/2 (−1)r+s (p2)2−r−sB(d

2− s, d

2− r) ,

in this case we obtain

I(r, s; 0, 0; p2) = (4π)ε/2 (−(p2 + i0))−ε/2 (p2)2−r−sΓ(r + s− d/2)

Γ(d− r − s)Γ(d/2 − s)

Γ(s)

Γ(d/2− r)Γ(r)

.

Here we present an application of case 1. above. Note that while∫

k

kµ1 · · · kµ2n+1

(k2 −m2)r= 0 ; n = 0, 1, 2, · · ·

we have∫

k

kµ1 · · · kµ2n(k2 −m2)r

= gµ1µ2 · · · gµ2n−1µ2n A2n,r(m2) .

From the last relation we obtain

dnA2n,r =

∫

k

(k2)n

(k2 −m2)r= (4π)ε/2 (m2)−ε/2 (−m2)2−r+n

Γ(r − n− d/2)Γ(d/2 + n)

Γ(r)Γ(d/2),

106

and, as a limit m→ 0, we find that

A2n,r(0) = 0 for n ≥ r .

For n < r, evaluating

∫

k

1

(k2 −m2)r=

1

(r − 1)!∂r−11

∫

k

1

k2 −m2

= (4π)2−d/21

(r − 1)!∂r−11

∫ ∞

0dαα−d/2 e−αm

2= (4π)ε/2

(−1)r−1

Γ(r)

∫ ∞

0dααr−d/2−1 e−αm

2

= (4π)ε/2(−1)r−1

Γ(r)Γ(r − d/2) (m2)−(r−d/2) = −(4π)ε/2 (m2)−ε/2 (−m2)2−r

Γ(r − d/2)

Γ(r),

the limit m→ 0 exists, and is vanishing for all d with Re d > 2r:

limm→0

∫

k

1

(k2 −m2)r≡ 0 Re d > 2r .

Then, by analytic continuation,

∫

k

1

(k2)r= 0 ∀d .

Together with the previous result this proves that in DR

∫

k

kµ1 · · · kµn(k2)r

= 0 ∀n, r .

Tensor integrals:Let us denote by (pµ1 , · · · , pµm) the rank m traceless symmetric tensor, i.e. any contraction withgµiµk for i, k ∈ (1, · · · ,m) vanishes:

(pµ1) = pµ1 ,

(pµ1pµ2) = pµ1pµ2 − 1

dgµ1µ2 p2 ,

etc.

We define tensor integrals Im(r, s) by

I(µ1,···,µm)(p) =

∫

k

(kµ1 · · · kµm)

k2r(k + p)2s= (pµ1 · · · pµm) Im(r, s) , (6.51)

which will show up in QCD one-loop calculations below. We may write them as derivatives withrespect to the external momentum [∂µi ≡ ∂/∂pµi ] as follows:

(r −m− 1)!

(r − 1)!

1

2m(∂µ1 · · · ∂µm)

∫

k

1

(k)2r−2m(k + p)2s

=(r −m− 1)!

(r − 1)!

1

2m(∂µ1 · · · ∂µm)

∫

k

1

(k − p)2r−2m(k)2s

=

∫

k

((k − p)µ1 · · · (k − p)µm)

(k − p)2r(k)2s=

∫

k

(kµ1 · · · kµm)

(k)2r(k + p)2s.

107

Using the representation derived above for the integrals I(r, s), we therefore may write

I(µ1,···,µm)(p) =(r −m− 1)!

(r − 1)!

1

2m(∂µ1 · · · ∂µm) I(r −m, s)

=(−1)r+s

Γ(r) Γ(s)

1

2m(4π)2−d/2

∫ ∞

0dλλr+s−m−d/2−1

×∫ 1

0dxxr−m−1 (1− x)s−1 (∂µ1 · · · ∂µm) e−λ(···−x(1−x) p

2)

=(−1)r+s

Γ(r −m)Γ(s)

Γ(r −m)

Γ(r)(4π)2−d/2

×∫ ∞

0dλλr+s−d/2−1 (pµ1 · · · pµm)

∫ 1

0dxxr−1 (1− x)s+m−1 e−λ(···−x(1−x) p

2) .

Where we have used (∂µ1 · · · ∂µm) e−λ (···−x(1−x) p2) = 2m λm (pµ1 · · · pµm)xm (1−x)m e−λ(···− x(1−x) p2).

After performing the λ–integration, we may read off the representation of the tensor integral

Im(r, s) =(−1)r+s

Γ(r −m)Γ(s)

Γ(r −m)

Γ(r)(4π)2−d/2 Γ(r + s− d/2)

×∫ 1

0dxxr−1 (1− x)s+m−1 (xm2

1 + (1− x)m22 − x (1− x) p2)d/2−r−1 ,

and for m21 = m2

2 = 0 using∫ 10 dxxd/2−s−1 (1 − x)d/2+m−r−1 = B(d/2 − s, d/2 + m − r) the

x–integral simplifies to

(−(p2 + i0))d/2−r−s × B(d/2− s, d/2 +m− r) .

The massless tensor integrals then are given by

Im(r, s) = (4π)ε/2 (−(p2 + i0))−ε/2 (p2)2−r−sΓ(r + s− d

2)

Γ(d+m− r − s)Γ(d2 − s)

Γ(s)

Γ(d2 +m− r)Γ(r)

.

For ε→ 0, the singular plus finite parts in any case can be represented in terms of massless B0’sand constants. We collect them in the following

Table for massless integrals: (use with care [see below])

Iµν...(p, r, s).=

∫

k

kµkν · · ·(k2 + iε)r((k + p)2 + iε)s

a): I(p, r, s) = i16π2 (p2)−(r+s−2) I0(r, s)

b): Iµ(p, r, s) = i16π2 (p2)−(r+s−2) pµ I1(r, s)

c): Iµν(p, r, s) = i16π2 (p2)−(r+s−2)

(pµpν I21(r, s) + gµν p2 I22(r, s)

)

(r, s) I0(r, s) I1(r, s) I21(r, s) I22(r, s)

(1, 1) B[UV ]0 −1

2B[UV ]0

13B

[UV ]0 + 1

18 − 112B

[UV ]0 − 1

18

(2, 1) −B[IR]0 + 2 −1 1

214B

[UV ]0

(1, 2) −B[IR]0 + 2 B

[IR]0 − 1 −B0 + 1

214B0

(3, 1) −1 12B

[IR]0 − 1

212 −1

4B[IR]0 + 1

4

(2, 2) −2B[IR]0 + 2 B

[IR]0 − 1 −B[IR]

012

(1, 3) −1 −12B

[IR]0 + 3

2 B[IR]0 − 3

2 −14B

[IR]0 + 1

4

108

B0 = B0(0, 0; p2) = Reg + 2− ln(−(p2 + i0)

)

= Reg + 2− ln p2 + iπ (p2 > 0)

Reg =2

ε− γ + ln 4π + lnµ20 ; ε = 4− d .

Using the massless integrals above some care is in necessary. In fact the ε–poles are not necessarilyUV poles, in most cases they turn out to be IR poles. For the integrals (6.51) considered above,the correct interpretation of the ε–poles follows from the respective IR and UV convergenceregions of the integrals in d. I(µ1,···,µm)(p) is UV convergent for d + m − 2r − 2s < 0 and IRconvergent for d+m− 2r > 0 and d− 2s > 0. Thus the convergence region is

2s, 2r −m < d < 2r + 2s −m .

If dmax = 2r+2s−m < 4 Im(r, s) has an UV pole:(1ε

)UV

; if dmin = max(2s, 2r−m) > 4 Im(r, s)

has an IR pole:(1ε

)IR

. In boundary cases we have dmin = dmax = 4 the integral has both typesof poles, and it must be defined via an extra regularization as discussed before.

• For 2s ≥ 4 or 2r −m ≥ 4 the ε–pole is IR–pole provided 2r + 2s−m > 4.

• For 2s, 2r −m < 0 the ε–pole is UV–pole.

• Boundary cases are: 2r = m, s = 2 and s = 0, 2r = m+ 4.

A typical boundary case is

∫

k

1

k4=

16π2

i

2πd/2

(2π)d Γ(d/2)× −i

∫ ωmax

ωmin

dω ωd−5

= −2(4π)−ε/2

Γ(d/2)

ωd−4

d− 4

∣∣∣∣max

min

= −2 (1 + ε/2 (γ − ln 4π + 1))

(−1

ε+ lnω

)∣∣∣∣max

min

=

2

ε− γ + ln 4π − 1− lnω2

∣∣∣∣max

min

= (Reg)UV − (Reg)IR = ” lnω2min

ω2max

” .

The regulator terms are defined by

(Reg)UV =2

ε− γ + ln 4π − lnµ2UV

(Reg)IR =2

ε− γ + ln 4π − lnµ2IR

where µ2UV ≫ µ2IR. Thus strictly speaking we have

∫

k

1

k4= (Reg)UV − (Reg)IR = ” B

[UV ]0 −B[IR]

0 ” ,

while usually in DR we assign zero [which corresponds to the arbitrary choice µ2UV = µ2IR] to suchscale-less integrals.

109

An example:

∫

k

2kp

k4(k + p)2=

∫

k

1

k4−∫

k

1

k2(k + p)2− k2

∫

k

1

k4(k + p)2

−2 = ” 0 ”−B[UV ]0 −

(−B[IR]

0 + 2)

−2 = ” B[UV ]0 −B[IR]

0 ”−B[UV ]0 −

(−B[IR]

0 + 2)

Thus the “trick” to distinguish B[UV ]0 and B

[IR]0 allows for a separate check of UV and IR singular

terms. This, in principle, always should be done.


① Prove the relations (6.4) and (6.5) for the d dimensional Dirac algebra.

② Show that the γ5–odd traces are not cyclic in the d dimensional Dirac algebra. Note thatfor a mathematically well defined set of matrices a trace trivially is cyclic. In particularprove the claim (6.6). Hint: calculate Tr γαγ

µγνγργσγαγ5 and join the connected γ’s suchthat one may use γαγ

α = d. Note that there are two ways to join these γ’s.

③ Prove the relations (6.39) and derive the corresponding relations for the tensor integralsB1, B21, B22 as well as for the three point amplitudes C1i and C2i. Verify (6.46).

④ Calculate B0(m1,m2; p2) performing the 1-dimensional integral (6.36) in terms of elemen-

tary functions.

110

7 One–Loop Renormalization

7.1 The quark self–energy

Next we study the full propagator of a Dirac fermion f

iS′f (x− y) = 〈0|Tψf (x)ψf (y)

|0〉

in momentum space, where the free Dirac propagator is given by iSf (p) = i/(p/ − mf ). Thepropagator has the structure of a repeated insertion of the 1PI self–energy −iΣf (p) [diagonal incolor space, 4 x 4 matrix in spinor space]

f f = + +

f+···

i S′f (p) ≡ i

p/−mf+

i

p/−mf(−iΣf )

i

p/−mf

+i

p/−mf(−iΣf )

i

p/−mf(−iΣf )

i

p/−mf+ · · ·

=i

p/−mf

1 + (ΣfSf ) + (ΣfSf )2 + · · ·

=i

p/−mf

1

1− ΣfSf

=

i

p/−mf − Σf.

The last relation follows by taking the inverse:

−iS′−1f = −i (1− ΣfSf ) S−1f = −i

(S−1f − Σf

).

This demonstrates that the Dyson series is a geometric progression of matrix insertions whichcan be summed in closed form and the inverse full fermion propagator reads

−iS′−1f = = + + · · ·

= −i p/−mf − Σf (p) .

The self–energy is given by an expansion in a series of 1PI diagrams

−iΣf (p) ≡ ’= + · · · , (7.1)

the prime (′) on the “blob” indicates the non-trivial part. The covariant decomposition of Σf (p)for a massive fermion takes the form

Σ(p) = mf Σ1(p2) + (p/−mf ) Σ2(p2) = p/

(A(p2,mf , · · ·)

)+mf

(B(p2,mf , · · ·)

),

where A and B, as well as Σ1 = A + B and Σ2 = A, are Lorentz scalar functions which dependon p2 and on all parameters (indicated by the dots) of a given theory. In vector–like theories,like QED and QCD, no parity violating γ5 terms are present, and the pole of the propagator,or, equivalently, the zero of the inverse propagator, is given by a multiple of the unit matrix inspinor space:

p/ = mf ,where m2f = sP

defines the “pole mass” of the fermion in the p2–plane

p/−mf − Σf (p)|p/=mf= 0 .

111

We are ready now to calculate the quark self–energy in the one–loop approximation. We dropthe index f for the mass in the following. We have to calculate22

Σ(p) = il+p

l

p

= i i4g2µε (TiTk)ab

∫ddℓ

(2π)dγρ

p/+ ℓ/+m

(p + ℓ)2 −m2 + iεγσ(−gρσ + (1− ξ) ℓρℓσ

ℓ2

)1

ℓ2 + iεδik ,

where δik (TiTk)ab = (TiTi)ab = δab C2(R) is a SU(3) Casimir operator, R denotes the represen-tation of the Ti’s which here is the fundamental one, for which we easily obtain C2(R) = 4

3 . Inthe following we separate out a common factor N :

N = −i g2µε δab C2(R)i

16π2,

where the last factor is a normalization of the integrals used below (see our definition of thestandard integrals (6.33), (6.35)). The gluon propagator has two terms: the1st term: has an integrand

1

ℓ21

(ℓ+ p)2 −m2(γα(ℓ/+ p/)γα +md︸︷︷︸

(2−d) (ℓ/+p/)+md

) ,

such that, in terms of the standard integrals introduced in the preceding section, the integrationyields the term

T1 = ((2− d) p/+md) B0(0,m2; p2) + (2− d) p/B1(0,m2; p2)

with

B1(m1,m2; p2) =1

2p2−A0(m1) +A0(m2)− (p2 +m2

1 −m22)B0(m1,m2; p2)

.

Expanding in ε = 4− d we have to take into account (see (6.39))

dB0 = 4B0 − 2 ; dB1 = 4B1 + 1 .

The first term then is given by

T1 = m (4B0 − 2) + p/ (1− 2 (B0 +B1))

= m (4B0 − 2) + p/

(1− A0(m)

p2− p2 +m2

p2B0

)

with

B0 = Reg + 2− lnm2 +m2 − p2p2

ln

(1− p2 + i 0

m2

).

22In case infrared singularities show up we use a tiny (smaller than any other mass scale) gluon mass as an IRregulator and then work with a gluon propagator of the form (’t Hooft gauge for massive spin 1 bosons)

Dρσ(k) = −(

gρσ − (1− ξ)kρkσ

k2 − ξm2g

)

1

k2 −m2g + iε

.

112

2nd term: has the integrand

−(1− ξ) 1

ℓ41

(ℓ+ p)2 −m2(ℓ/ (ℓ/+ p/+m) ℓ/︸︷︷︸

X

) .

Looking at

X = ℓ/ (ℓ/+ p/+m) ℓ/ = ℓ2ℓ/+ ℓ2m− ℓ2p/+ (2pℓ) ℓ/

and writing (2pℓ) as a difference of the denominators (1) = ℓ2 and (2) = (ℓ + p)2 −m2 plus aremainder (2pℓ) = (2)− (1) − (p2 −m2) we have

X = ℓ2ℓ/− ℓ/ℓ2 − (p/−m) (1) + ℓ/ (2)− ℓ/ (p2 −m2) .

Herewith the integrand of the 2nd term reads

−(1− ξ)−(p/−m)

1

(1)(2)+

ℓ/

ℓ4− (p2 −m2)

ℓ/

(1)2(2)

,

and by integration we obtain as a 2nd term

T2 = (1− ξ)

(p/−m)B0 + (p2 −m2) I,

where (see Sect. 6.5)

pµI.=

∫

ℓ

ℓµ

ℓ4 ((ℓ + p)2 −m2)

with

I = − 1

p2

(1 +

m2

p2ln

(1− p2 + i0

m2

)).

Note that the second term in the integrand is odd in the integration momentum and hence doesnot contribute to the integral. So far we have calculated the bare regularized one–loop quarkself-energy

Σ(p) = mΣ1 + (p/−m) Σ2 ,

where

Σ1 = 4B0 − 1− A0(m)p2− p2+m2

p2B0 + (1− ξ) (p2 −m) I ,

Σ2 = 1− A0(m)p2− p2+m2

p2B0 + (1− ξ) (B0 + (p2 −m) I) ,

(7.2)

with a common factor

g2

16π2δab C2(R) .

In the bare result the mass m and the coupling g actually have to be identified with the bareparameters m0 and g0. By m and g we will denote the renormalized parameters in the nextparagraph. The next step is renormalization.

Renormalization

After regularization and ε–expansion in d = 4 − ε we are left with regularized bare amplitudeswhich, at the one–loop level, exhibit simple poles 1

ε in ε as ε → 0, i.e., if we want to go to the

113

physical theory in d = 4 dimensions. Note that a QFT is defined in terms of the bare Lagrangian,which is the true Lagrangian of the theory. Renormalization is a reparametrization of the baretheory, in terms of renormalized parameters and fields. In fact a multiplicative renormalizationof parameters and fields, in a renormalizable theory like QCD, is able to absorb all singularitiesand to obtain a renormalized parametrization which has a finite limit ε → 0. In the limit theregularization is removed and we and the true d = 4 physical QFT is “recovered”.

The renormalized full quark propagator is given by

S′−1F ren = ZF S

′−1F0 = ZF (p/−m0 − Σ(p))

= (p/−m) + (ZF − 1) (p/−m)− ZF (δm + Σ)

=-1

+ ⊗ +’

= (p/−m) +(δZ

(n)F (p/−m)− δm(n)

)− Σ

′(n)

where δZF = ZF − 1 and δm = m0−m, m = mren the renormalized mass. The n-th order coun-terterms are fixed by the following renormalization conditions at that order after the calculationof Σ

′(n) which includes all loop diagrams at order n including loop diagrams with counterterminsertions from counterterms up to order n− 1.

1) Mass renormalization. By m we denote the “physical mass” in the following. Becauseof confinement and the corresponding non-existence of free quarks, a quark mass is not it-self an observable quantity. However, it is a symmetry breaking parameter, breaking chiralSU(Nf )V ⊗ SU(Nf )A in a particular way. An example how quark masses can be given a precisephysical meaning we have discussed at the end of Sect. 3.4 . Different possible mass definitionscorresponding to different renormalization schemes. Since one can calculate one version of massfrom the other, one usually starts with a convenient convention. At least for the light quarksone usually adopts so called modified minimal subtraction scheme (renormalized running mass inMS scheme), which we will introduce below. First we treat the quark mass as pole mass, like alepton mass, defined by the location of the pole of the propagator. Therefore, we look at

limp/→m

δm+ Σ(p) = δm+mΣ1 + (p/−m) Σ2

→ δm+mΣ1 = 0

which yield the mass counterterm

δm

m= −Σ1(p

2 = m2) = 3A0(m)

m2− 1 .

We have used

B0(0,m;m2) = 1− A0(m)

m2,

and we may write

δm

m= −3 Reg + 3 lnm2 − 4 .

Note that p/→ m implies p2 → m2. We furthermore note that,

• first, the mass counterterm is gauge invariant (it is the on–shell value of an amplitude andhence a ”quasi S–matrix element”, albeit only the perturbative one, which is far from anytrue QCD S–matrix element describing real processes between hadrons),

114

• second, the term T2 does not contribute at all.

Later we will represent the mass–shift, which looks like an additive renormalization, as a multi-plicative renormalization factor

m0 = Zmmren = m+ δm = m

(1 +

δm

m

),

i.e.,

Zm = 1 +δm

m; δZm =

δm

m.

2)Wave–function renormalization. Again we try the academic exercise to make to “on–shell”residue of the propagator pole to be unity:

δZF (p/−m)− (δm + Σ(p)) → 0 as p/→ m .

This fixes the wave–function renormalization constant ZF . To determine it we have to expandthe self–energy about the mass–shell:

δm + Σ(p) = δm+mΣ1 + (p/−m) Σ2

= δm+mΣ1(p2 = m2)

+ m (p/−m) (p/+m)∂Σ1

∂p2

∣∣∣∣p2=m2

+ (p/−m) Σ2|p2=m2 +O(p2 −m2)

=

(2m2 ∂Σ1

∂p2+ Σ2

)∣∣∣∣p2=m2

(p/−m) +O(p2 −m2) .

We then read off that the choice

δZF =

(Σ2 + 2m2 ∂Σ1

∂p2

)∣∣∣∣p2=m2

ξ=1= −Reg + lnm2 − 4 + 4 ln

m

mg

formally makes the residue of the renormalized propagator equal to 1. The last explicite expressionis given for the Feynman gauge ξ = 1, for simplicity. Two points are crucial here,

• first, the wave–function renormalization factor is gauge dependent (as the field it renormal-izes itself),

• second, for a charged (here color as the charge) field (particle) the wave-function renormal-ization is infrared (IR) singular, i.e. the residue of the pole does not exist.

We therefore have adopted an infinitesimally small (i.e. we take the limit mg → 0 whenever itexists) gluon mass as an IR regulator. This is a well known problem of QED, and the fake IRsingularities must cancel in appropriately defined physical observables (Bloch-Nordsieck prescrip-tion).

Concerning the gauge dependence of the field renormalization, it is important to note that the on–shell limit of the bare Green functions (defining the S–matrix elements) is gauge invariant only

115

after the gauge–dependent wave–function renormalization has been performed. The on–shelllimit of the bare quantities itself is not gauge–invariant, in general.

In particular in QCD, the IR problem is an artificial problem, since there exist no asymptoticallyfree quarks in nature anyway. We therefore will adopt (modified) minimal subtraction in thefollowing: just make the residue of the pole ultraviolet (UV) finite via the choice

δZF = ”δZF ”|UV singular part .

We thus just collect the regulator terms Reg:

(Σ1)UV =

(4 +

m2

p2− 1− m2

p2

)Reg = 3 Reg ,

(Σ2)UV =

(m2

p2− 1− m2

p2+ (1− ξ)

)Reg = −ξReg ,

and hence

(δZF )MS = −ξReg .

As a result of the quark propagator renormalization we obtained

δZm = g2

16π2 C2(R)(−3 Reg + 3 lnm2 − 4

),

(δZF )MS = g2

16π2 C2(R) (−ξReg) ,(7.3)

with (γ here is Euler’s constant)

Reg =2

ε− γ + ln 4π + lnµ2 .

In the MS renormalization scheme the renormalized quantities are obtained formally by thesubstitution

Reg→ lnµ2 .

This has the effect that all ln p2 or lnm2 terms only show up in the form ln p2

µ2and ln m2

µ2, i.e., µ

is the MS renormalization scale (reference–scale).

7.2 The gluon self–energy

Also the full gluon propagator is given by a Dyson series of self–energy insertions

g g= + +

g+ · · ·

i D′µν(k) ≡ iDµν(k) + iDµρ(k) (−i Πρσ) iDσν(k)

+iDµρ(k) (−i Πρσ) iDσρ′(k)(−i Πρ′σ′

)iDσ′ν(k) + · · ·

= iDµρ(k)

1 + (ΠD) νρ + (ΠD)2 ν

ρ + · · ·

= iDµρ(k)

(1

1−ΠD

) ν

ρ

.

Again we get a geometrical progression which allows for a closed summation, and again this turnsout to be very important. The resummed result shows that the full propagator has a simple

116

pole in k2 only, like the free propagator, and no multi–poles as it might look like before theresummation has been performed.

If we multiply the last relation from the right with its inverse we obtain D′ (1 − ΠD)D−1 = 1which means that

D′−1 = D−1 −Π .

The one–particle irreducible (1PI) self–energy function

−iΠµν(k) ≡ ’(7.4)

is also called the vacuum polarization tensor . Its covariant decomposition reads

Πµν(k) = gµν Π1(k2) + kµkν Π2(k2) .

We have suppressed the color indices as the propagator is diagonal in the color as in the free casewhere with i, k color indices

D−1µνik (k) = −(gµν k2 − (1− ξ−1) kµkν) δik .

The Slavnov-Taylor (ST) identity, related to ”∂µGµia(x) = 0 ”, in momentum space reads

kν D′−1ik µν(k) = −kµ

(ξ−1k2 + Π1(k

2) + k2 Π2(k2))δik

= −kµ ξ−1k2 δik ,

which means that only the trivial lowest order part, stemming from the gauge fixing, preventsits vanishing. This requires the transversality of the non-trivial vacuum polarization tensor Π1 =−k2 Π2 and hence

Πµνik (k) =

(kµkν − k2 gµν

)Π(k2) δik

where we denoted by Π2(k2) = Π(k2). It means that no mass is generated by higher ordercorrections and hence mass renormalization is absent. We thus have the following form for theinverse full gluon propagator

D′−1µνik = −δik

(gµν k2 − kµkν

) (1−Π(k2)

)+ 1

ξ kµkν

(7.5)

We now proceed to calculate the gluon self–energy at one-loop:

−i Πµνik (k) =

l

l+k

k (1)+

(2)

+

η

_

η

(3)+

q_

q

(4)

Note that tadpole diagrams do not appear as TrTi = 0, fikl δkl = 0 and 〈Gµi〉 = 0. We nowcalculate the first diagram. Note that the first 3 diagrams have no correspondence in QED,where only the last type (fermion loops) is present. The most important preparation for a correctcalculation is a proper detailed labeling of the lines and vertices:

117

i ×

l

l+k

k

(i, µ)

(l, σ)

(j, ρ)

(k, ν)

(l, σ′)

(j, ρ′)

= i i41

2(i g)2

∑

jl

fijl fkjl ×∫

ddℓ

(2π)d1

ℓ2 + iε

1

(ℓ + k)2 + iε×

×(gρρ′ − (1− ξ) ℓρℓρ′

ℓ2

) (gσσ′ − (1− ξ) (ℓ+ k)σ(ℓ+ k)σ′

(ℓ + k)2

)

×gµρ (ℓ− k)σ + gµσ(ℓ + 2k)ρ − gρσ(2ℓ+ k)µ

×−gνρ′ (ℓ− k)σ

′ − gνσ′(ℓ + 2k)ρ′+ gρ

′σ′(2ℓ+ k)ν.

Again we single out the common factor:

N(1) ik = −i g2

2 C2(G) δiki

16π2 ,where the last factor comes from the definition of the integrals (see (6.35)). The group factor is∑

jl fijl fkjl = Tr (titk) = δik C2(G) and one finds C2(G) = n for SU(n). ti are the generators inthe adjoint representation.

1st term: with the gρρ′gσσ′ · · · term of the product of the gluon propagators we have in thenumerator:

(gµρ (ℓ− k)σ + gµσ(ℓ + 2k)ρ − gρσ(2ℓ+ k)µ) (gρσ(2ℓ + k)ν − gνσ(ℓ + 2k)ρ − gνρ(ℓ− k)σ)

= (6− 4d) ℓµℓν + (3− 2d) (ℓµkν + kµℓν) + (6− d) kµkν − gµν (4 k2 + ℓ2 + (ℓ+ k)2) .

Denoting propagators by (1) = ℓ2+iε and (2) = (ℓ+k)2+iε, the integrals we need are the following:

∫

ℓ

ℓµℓν

(1)(2)= kµkν B21 − gµν B22 ;

∫

ℓ

ℓµ

(1)(2)= kµB1 .

As massless integrals these are particularly simple

B22(0, 0; k2) = k2(

1

12B0 +

1

18

),

B21(0, 0; k2) =1

3B0 +

1

18,

B1(0, 0; k2) = −1

2B0 ,

B0 = B0(0, 0; k2) =

∫

ℓ

1

(1)(2).

Integrating the 1st term with the numerator given above we arrive at:

T1 = kµkν

(6− 4d)B21 − 2 (2d − 3)B1 + (6− d)B0

+gµν

(4d− 6)B22 − 4k2B0

,

118

where∫

ℓ

1

ℓ2= 0 ;

∫

ℓ

1

(ℓ + k)2= 0

has been used. Using the above relations for the tensor integrals we have

T1 = kµkν

113 B0 + 1

9

− k2 gµν

196 B0 + 1

9

. (7.6)

2nd term:23 the mixed term (ξ − 1)ℓρℓρ′ℓ2

gσσ′ × 2 of the gluon propagator product yields in thenumerator:

2 (1 − ξ)gµν (ℓ2 + 2 ℓk)2 + kµkν ℓ2 − ℓµℓν (ℓ2 + 2 ℓk − k2)− (ℓµkν + kµℓν) (ℓ2 + 3 ℓk)

1

ℓ2× · · ·

where we used

ℓρℓρ′ gσσ′

· · ·· · ·

= −gµσ (ℓ2 + 2 ℓk) − ℓµℓσ − ℓµkσ − kµℓσ

gνσ (ℓ2 + 2 ℓk)− ℓνℓσ − ℓνkσ − kνℓσ

.

Note that ℓ2 + 2 ℓk = (ℓ+ k)2 − k2 and, in accord with dimensional regularization properties, allno-scale integrals

∫

ℓ

ℓµ1 · · ·(ℓ2)r

≡ 0 .

We then are left with the following non-trivial integrations.a) First consider the term

gµν (ℓ2 + 2 ℓk)1

ℓ4 (ℓ + k)2= gµν

(ℓ2 + 2 ℓk

ℓ4− k2 1

ℓ4+ k4

1

ℓ4 (ℓ+ k)2

).

While the first two terms in DR vanish upon integration, the third yields the integral

I0(2, 1) = k2∫

ℓ

1

ℓ4 (ℓ + k)2= −B0 + 2

(see Table of Sect. 6.5). The complete term a) then yields

T(a)2 = gµν (−B0 + 2) . (7.7)

b) The next term is

kµkνℓ2

ℓ4 (ℓ+ k)2= kµkν

1

ℓ2 (ℓ+ k)2,

which yields the integral

T(b)2 = kµkν B0 . (7.8)

23The symmetric term (ξ − 1) gρρ′(ℓ+k)σ(ℓ+k)σ′

(ℓ+k)2yields the same result: check ℓ + k = ℓ′, k = −k′ such that

ℓ− k = ℓ′ + 2k′, 2ℓ+ k = 2ℓ′ + k′ and ℓ+ 2k = ℓ′ − k′ (ℓ = ℓ′ + k′) as well as ρ↔ σ and ρ′ ↔ σ′ .

119

c) The next term we have reads

−ℓµℓν (ℓ2 + 2 ℓk − k2)1

ℓ4 (ℓ+ k)2= −ℓ

µℓν

ℓ4+ 2k2

ℓµℓν

ℓ4 (ℓ+ k)2,

with a vanishing integral for the first term. The second integral is

k2∫

ℓ

ℓµℓν

ℓ4 (ℓ+ k)2= kµkν I21(2, 1) + gµν k2 I22(2, 1) ,

= kµkν1

2+ gµν k2

1

4B0

and hence

T(c)2 = kµkν + gµν k2

1

2B0 . (7.9)

d) The last term is

−(ℓµkν + kµℓν) (ℓ2 + 3 ℓk)1

ℓ4 (ℓ + k)2µ↔ν= −2kµ

ℓν

ℓ2 (ℓ+ k)2− 3kµ

ℓν 2 ℓk

ℓ4 (ℓ+ k)2

= −2kµℓν

ℓ2 (ℓ + k)2− 3kµ

ℓν

ℓ4+ 3kµ

ℓν

ℓ2 (ℓ+ k)2+ 3kµ k2

ℓν

ℓ4 (ℓ+ k)2

where we have used the symmetry in µ↔ ν. The first and third term combine while the secondterm vanishes upon integration. One thus obtains

kµkν I1(1, 1) + 3kµkν I1(2, 1) ,

and hence

T(d)2 = kµkν

(−1

2B0 − 3

). (7.10)

Collecting all terms we have

T2 = 2 (1 − ξ)(kµkν − gµν k2

) (12B0 − 2

), (7.11)

which is purely transversal.

3rd term: is due to the product (1− ξ)2 1ℓ2 (ℓ+k)2

ℓρℓρ′ (ℓ+ k)σ (ℓ+ k)σ′ of the second terms of the

gluon propagators which yields in the numerator:

ℓρℓρ′ (ℓ+ k)σ (ℓ+ k)σ′

· · ·· · ·

= −kµ (ℓk)− ℓµ k2

kν (ℓk)− ℓν k2

= −kµkν (ℓk)2 − ℓµℓν (k2)2 + (kµℓν + ℓµkν) k2 (ℓk)

=(−kµkν kρkσ − gµρgνσ (k2)2 + (kµgνσkρ + gµρkνkσ) k2

)× ℓρℓσ

and using∫

ℓ

ℓρℓσ(ℓ2)2 ((ℓ + k)2)2

=(kρkσ I21(2, 2) + gρσ k

2 I22(2, 2))× 1

(k2)2,

we note that

kρkσ (· · ·) = −kµkν (k2)2 − kµkν (k2)2 + 2 kµkν (k2)2 = 0 ,

gρσ k2 (· · ·) = −kµkν (k2)2 − gµν (k2)3 + 2 kµkν (k2)2 ,

120

such that the 3rd term reads

(1− ξ)2(kµkν − gµν k2

)I22(2, 2) ,

or

T3 = 2 (1− ξ)2(kµkν − gµν k2

) (12

). (7.12)

Also this term is purely transversal.The result for diagram (1) thus reads

Πµνik (1) = δik

116π2

g2

2 C2(G)×kµkν

(113 B0 + 1

9 + (1− ξ) (B0 − 4) + (1− ξ)2 12

)

−gµν k2(196 B0 + 1

9 + (1− ξ) (B0 − 4) + (1− ξ)2 12

).

Diagram (2) is vanishing because the self-closing line is massless, according to the features ofdimensional regularization, which assigns

∫

ℓ

(gρσ − (1− ξ) ℓρℓσ

ℓ2

)1

ℓ2≡ 0 .

The third diagram (3) is again non-trivial and was actually the missing piece before the Faddeev-Popov Lagrangian got established. We have

i ×η

_

η(i, µ) j

l

(k, ν)j

l

= i i4 (−1) (−i g)2∑

jl

fijl fklj ×∫

ddℓ

(2π)dℓµ (ℓ + k)ν

1

ℓ2 + iε

1

(ℓ+ k)2 + iε,

where the factor (−1) is due to Fermi statistics of the FP ghosts. The common factor here is:N(3) ik = −i g2 C2(G) δik

i16π2 .

The integrals we need are∫

ℓ

ℓµℓν

ℓ2 (ℓ+ k)2= kµkν I21(1, 1) + gµν k2 I22(1, 1)

= kµkν(

1

3B0 +

1

18

)+ gµν k2

(− 1

12B0 −

1

18

),

∫

ℓ

ℓµ

ℓ2 (ℓ+ k)2= kµ I1(1, 1) = kµ

(−1

2B0

),

and we easily find

Πµνik (3) = δik

116π2 g

2 C2(G)

kµkν

(−1

6B0 + 118

)− gµν k2

(112B0 + 1

18

).

The sum of the two non-vanishing “pure gauge” diagrams yields

Πµνik [(1) + (2) + (3)] = δik

g2

32π2 C2(G)×kµkν

(103 B0 + 2

9 + (1− ξ) (B0 − 4) + (1− ξ)2 12

)

−gµν k2(103 B0 + 2

9 + (1− ξ) (B0 − 4) + (1− ξ)2 12

)(7.13)

121

and, as it should be, turns out to be transversal! Note that the individual diagrams do not havethis property. The transversality means that the gluon self–energy insertion actually satisfies theST–identity. The latter must hold for the pure gauge part separately, without the quarks.

We are left with the quark loop

i ×q_

q(i, µ)

ℓ

(k, ν)

= i i4 (−1) g2 Tr (TiTk)×∫

ddℓ

(2π)dTr (γµ (ℓ/+m) γν (ℓ/+ k/+m))

[ℓ2 −m2 + iε] [(ℓ + k)2 −m2 + iε],

where the factor (−1) is due to Fermi statistics of the quarks. The spinor space trace yields

Tr (· · ·) = Tr (γµℓ/γν (ℓ/+ k/)) +m2Tr γµγν

= Tr(1) ·ℓµ (ℓ+ k)ν − gµν (ℓ2 + ℓk) + (ℓ + k)µ ℓν +m2gµν

.

To proceed we may take advantage of the transversality kνΠµν = 0 of the fermion loop contribu-tion, which is easy to check. We may write

kν

(γµ

1

ℓ/−m γν1

ℓ/+ k/−m

)

= γµ1

ℓ/−m k/1

ℓ/+ k/ −m = γµ1

ℓ/−m [(ℓ/+ k/−m)− (ℓ/−m)]1

ℓ/+ k/ −m

= γµ(

1

ℓ/−m −1

ℓ/+ k/ −m

)∫

ℓ= 0 .

In the last step we used∫

ℓ

1

ℓ/−m =

∫

ℓ

1

ℓ/+ k/ −m ,

which follows by a shift of the integration variable. The latter is possible in DR. This is crucialhere.

As a result we have shown that the quark loop contribution is transversal

Πµνik (4) =

(kµkν − gµν k2

)Π(4) ik(k2) = N4 ik


)Π(4)(k

2) ,

where N4 ik is the common factorN4 ik = −i g2 T (R) f(d) δik

i16π2 ,

and we denoted Tr1 = f(d) (see (6.5) and comments there). The group factor follows from

TrTiTk =∑

a,b

(Ti)ab (Tk)ba = δik T (R) ,

with Ti the generators of SU(n) in the fundamental representation, on finds T (R) = 12 for SU(n)

per flavor, i.e.∑

f T (R) =Nf

2 .

Utilizing the transversality, we may contract the vacuum polarization tensor with gµν to obtaina scalar amplitude:

gµν Πµνik (4) = N4 ik k

2 (1− d) Π(4)(k2)

= N4 ik ×∫

ℓ

1

(1)(2)

(2 (ℓ2 + ℓk)− d (ℓ2 + ℓk) +m2 d

)

122

with (1) = ℓ2−m2+iε and (2) = (ℓ+k)2−m2+iε . The numerator in the integral may be rewrittenin terms of inverse scalar propagators which will cancel against the propagator denominators:

(2− d) (ℓ2 + ℓk) +m2 d

= (2− d)

[(ℓ2 −m2)︸︷︷︸

(1)

+m2 +1

2

((ℓ+ k)2 −m2

︸︷︷︸(2)

− (ℓ2 −m2)︸︷︷︸(1)

−k2)]

+m2 d

Using this we obtain the integral

X = (2− d)

[∫

ℓ

1

(2)+m2

∫

ℓ

1

(1)(2)+

1

2

∫

ℓ

1

(1)− 1

2

∫

ℓ

1

(2)− k2

2

∫

ℓ

1

(1)(2)

]+ dm2

∫

ℓ

1

(1)(2)

= (2− d)

[−A0 +m2B0 −

1

2A0 +

1

2A0 −

k2

2B0

]+ dm2B0

and using dB0 = 4B0 − 2, dA0 = 4A0 + 2m2 we furthermore may simplify

X = 2A0 + 2m2 − 2m2B0 + 2m2 + k2B0 − k2 + 4m2B0 − 2m2

= 2m2 + 2A0 + 2m2B0 + k2B0 − k2= k2 (1− d) Π(4)(k

2) = −3k2 Π(4) + k2 ε Π(4) .

The last term may be worked out

limε→0

k2 ε Π(4) = k2 ε[−2m2 + 2m2 + k2

] 2

ε

−1

3k2= −2

3k2 ,

and we end up with

Π(4) =−1

3k2

2A0 + 2m2B0 + k2B0 + 2m2 − k2 +

2

3k2

such that

Π(4) ik(k2) = −δik g2

16π2 T (R) f(d) 13k2

2A0 + 2m2B0 + k2B0 + 2m2 − 1

3k2

(7.14)

with A0 = A0(m), B0 = B0(m,m; k2). Using the special value

B0(m,m; 0) = −1− A0(m)

m2

we may check that the gluon indeed remains massless:

limk2→0

k2 Π(4) ik(k2) = · · ·

2A0 + 2m2 + 2m2B0(m,m; 0)

= 0 .

Like for the photon in QED there is no mass-renormalization for the gluon in QCD. We now havethe complete one–loop self-energy for the gluon, which is given by

Πµνik (k) =


)δik Π(k2) ,

with scalar amplitude

Π(k2) = g2

32π2

C2(G)

(103 B0(0, 0; k2) + 2

9 + (1− ξ)(B0(0, 0; k2)− 4

)+ (1− ξ)2 1

2

)

−4T (R) f(d)3k2

∑q

(A0(mq) +

(m2q + k2

2

)B0(mq,mq; k

2) +m2q − k2

6

).

(7.15)

123

The standard one–loop integrals used here are given by

A0(mq) = −m2q

(Reg + 1− lnm2

q

)

B0(0, 0; k2) = Reg + 2− ln(−(k2 + i0)

)

B0(mq,mq; k2) = Reg + 2− lnm2

q + 2

(4m2

q

k2− 1

)G

(4m2

q

k2

)

where

G(y) =1

2√

1− y ln

√1− y + 1√1− y − 1

. (7.16)

For small momenta the leading terms of the asymptotic expansion reads

B0(mq,mq; k2) = Reg− lnm2

q +1

6

k2

m2q

+ · · · ; k2 → 0 .

Renormalization of the gluon propagator

The structure of the gluon propagator and its renormalization may be summarized by the followingrelations:

D′−1µνik ren = Z3D

′−1µνik 0 =


)δik Z3

(1−Π(k2)

)− δik

1

ξrenkµkν

= D−1µνik + δik(kµkν − gµν k2

)((Z3 − 1)− Z3 Π)

= D−1µνik +D−1µνik trans δZ(n)3 −Π

′µν (n)ik trans

=-1

+ ⊗ +’

δZ(n)3

We have chosen ξren = Z−13 ξ0. This implies that the gauge fixing Lagrangian LGF remainsunrenormalized, as the field renormalization is precisely compensated by the gauge parameterrenormalization. Again the non-trivial irreducible piece Π′ includes diagrams with counterterm

insertions of counterterms up to order (n−1). The counter term δZ(n)3 at order (n) is determined

such that the full (inverse) propagator is finite at this order. Once again we note that there is nomass renormalization since

limk2→0

k2 Π(k2) = 0 .

The wave function renormalization

δZ3 = Z3 − 1 = limk2→0

Π(k2) ,

for the colored gluons, like for the colored quarks, is in trouble, because the on–shell limit doesnot exist! Since the gluons are color charged (and hence self–interacting) massless objects, thereis an infrared problem24. Again minimal subtraction is possible and commonly accepted. Therequirement is not that the residue of the pole should by normalized to unity, but just to be finite:thus

δZ3 = Π(k2)∣∣singular part

at k2 = 0 ,

24This is in contrast to the charge neutral photon in QED, in which case the on–shell limit exists.

124

which yields

(δZ3)MS = g2

32π2

C2(G)

(103 + (1− ξ)

)− 4T (R) f(d)

3k2∑

q

(−m2

q +m2q + k2

2

)Reg ,

such that, adopting the scheme f(d) = 4 (convention!),

(δZ3)MS = g2

32π2

C2(G)

(133 − ξ

)− 8

3 T (R)Nf

Reg (7.17)

is our final result for the renormalization constant(s) related to the gluon propagator.

7.3 The Faddeev-Popov ghost self–energy

Since the FP ghosts have to obey Fermi statistics, the structure resembles the one of a quarkpropagator, however, it is much simpler as it is a scalar ghost. The Dyson series of self–energyinsertions here is given by

η η= + +

η+ · · ·

i D′(k) ≡ i D(k) + i D(k)

(−i Π i D

)(k) + i D(k)

(−i Π i D

)2(k) + · · ·

= i D(k)

(1

1− Π D

).

The self–energy itself is obtained by an expansion in a series of 1PI diagrams

−i Πik(k) ≡ ’= + · · · . (7.18)

The inverse full propagator is

D′−1ik = D−1ik − Πik with D−1ik = k2 δik .

In this case there is one diagram to be calculated at one loop:

i ×l

l+k

k

i

(l, ρ)

j

k

(l, ρ′)

j

= i i41

2(−i g)2

∑

lj

flij fljk × (−1) ×

∫ddℓ

(2π)dℓρ (−k)ρ

′

ℓ2 + iε

(gρρ′ − (1− ξ) (ℓ+ k)ρ(ℓ+ k)ρ′

(ℓ + k)2

)1

(ℓ + k)2 + iε

The common factor here reads:Nik = i g2 C2(G) δik

i16π2 .

We again consider the two gluon propagator terms in turn.1st term: gρρ′

∫

ℓ

ℓk

ℓ2 (ℓ + k)2=

1

2

∫

ℓ

(ℓ+ k)2 − ℓ2 − k2ℓ2 (ℓ + k)2

=1

2

∫

ℓ

1

ℓ2− 1

2

∫

ℓ

1

(ℓ + k)2− k2

2

∫

ℓ

1

ℓ2 (ℓ+ k)2

= −k2

2B0 .

125

The first two integrals are zero in DR (no scale).2nd term: (1− ξ) (ℓ + k)ρ (ℓ + k)ρ′/(ℓ + k)2

The integral to be considered is

∫

ℓ

(ℓ2 + ℓk) (ℓk + k2)

ℓ2 (ℓ+ k)4.

The term ∫

ℓ

ℓ2 (ℓ+ k)k

ℓ2 (ℓ + k)4=

∫

ℓ

ℓk

ℓ4= 0

obviously vanishes after a shift of the integration momentum. Next we have

∫

ℓ

ℓk (k2 + ℓk)

ℓ2 (ℓ + k)4= k2

∫

ℓ

ℓk

ℓ2 (ℓ + k)4+

1

2

∫

ℓ

ℓk

ℓ2 (ℓ + k)4((ℓ + k)2 − ℓ2 − k2)

=k2

2

∫

ℓ

ℓk

ℓ2 (ℓ+ k)4+

1

2

∫

ℓ

ℓk

ℓ2 (ℓ+ k)2− 1

2

∫

ℓ

ℓk

(ℓ + k)2

=k2

2I1(1, 2) +

k2

2I1(1, 1) + 0 =

k2

2

(B0 − 1− 1

2B0

),

and thus

Πik =g2

32π2δik C2(G) k2

(B0 + (1− ξ)

(1

2B0 − 1

)).

Like the gluon, the ghost remains massless

limk2→0

Πik = 0 .

Here

B0 = B0(0, 0; k2) = Reg + 2− ln(−(k2 + i0)

).

Renormalization follows according to:

D′−1ik ren = Z3 D

′−1ik 0 = Z3

(δik k

2 − Πik

)

= δik k2(

1 + Z3 − 1− Z3b)

= δik k2(

1 + δZ(n)3 − b′(n)

)

=-1

+ ⊗ +’

δZ(n)3

with b : Πik.= δik k

2 b(k2) .

While there is no mass renormalization, we need the wave function renormalization

Z3 − 1 = b(k2)∣∣∣singular part

.

Explicitely, we have (δZ3

)MS

= g2

32π2 C2(G) 3−ξ2 Reg (7.19)

as a result for the counterterm needed for the renormalization of the FP ghost propagator.

126

7.4 The quark-gluon vertex

At the tree level (Born approximation) we have the quark–gluon vertex

:= i g γµ (Ti)bc

i, µ

k

b, q

c, p

and there are two diagrams contributing to the one–loop correction

+

(1) (2)

We first calculate diagram (1):

−i ×k

i, µρ

σ

ℓ, j

b, q

c, p

b′

c′

= −i i6 g3∑

j

(Tj)bb′ (Ti)b′c′ (Tj)c′c × (−1)×

∫ddℓ

(2π)d1

ℓ2 + iε

1

ℓ21 −m2 + iε

1

ℓ22 −m2 + iε(γρ(ℓ/2 +m)γµ(ℓ/1 +m)γσ)

(gρσ − (1− ξ) ℓρℓσ

ℓ2

).

Common factor is:N1 = −i g3

(− 1

2n

)(Ti)bc

i16π2 .

To be evaluated

TjTiTj = TjTjTi + i fijlTjTl = C2(R)Ti +1

2i fijl (TjTl + TlTj)

= C2(R)Ti −1

2fijlfjlk Tk =

(C2(R)− 1

2C2(G)

)Ti = − 1

2nTi ,

where we have used the SU(n) relations

∑

j

(TjTj)bc′ = C2(R) δbc′ ; C2(R) =n2 − 1

2n,

∑

jl

fijlfkjl = C2(G) δik ; C2(G) = n .

We consider the two terms from the gluon propagator separately.1st term: gρσThe numerator reds:

X1 = γα(ℓ/2 +m)γµ(ℓ/1 +m)γα = (6− d) ℓ/2γµℓ/1

+ 4 γµ ℓ1ℓ2 − 4 (ℓ/2ℓµ1 + ℓµ2ℓ/1) +m ((d− 4)(ℓ/2γ

µ + γµℓ/1) + 4 (ℓµ1 + ℓµ2 )) +m2 (2− d) γµ .

In order to keep the number of amplitudes small we only consider the one’s which directly con-tribute to the on–shell matrix elements, i.e. we consider the expression above sandwiched

u(q) · · · u(p)

127

between the external spinors (=one–particle wave functions) and use the Dirac equation:

q/u(q) = mu(q) ; p/u(p) = mu(p) .

Doing so we obtain

X1 = (d− 2) γµℓ2 − 2γµk2 + (4− 2d) ℓ/ℓµ + 4γµℓQ− 4ℓ/Qµ + 4mℓµ + 4m2γµ

= (d− 6)γµℓ2 + 2γµ (ℓ21 −m2) + 2γµ(ℓ22 −m2) + 4γµpq + 2(2− d)ℓµℓ/− 4ℓ/Qµ + 4mℓµ ,

where we used: Q.= p+ q, p2 = q2 = m2, 2ℓQ = (ℓ21 −m2) + (ℓ22 −m2)− 2ℓ2, 2pq = 2m2 − k2.

The integral related to the 1st term then yields [(1) = ℓ21 −m2, (2) = ℓ22 −m2, (3) = ℓ2]

T1 =

∫

ℓ

γµ[(d− 6)

1

(1)(2)+ 2

1

(1)(3)+ 2

1

(2)(3)+ 4 pq

1

(1)(2)(3)

]

−4 (Qµγα −mgµα)ℓα

(1)(2)(3)+ 2 (2− d) gµαγβ

ℓαℓβ(1)(2)(3)

= γµ[(d− 6)B0(m,m; k2) + 2B0(0,m; p2) + 2B0(0,m; q2)− 4 pq C0(0,m,m; p2, k2, q2)

]

+4 (Qµγα −mgµα) (pαC11 + kα C12)

+2 (d− 2) gµαγβ (pαpβ C21 + kαkβ C22 + (pαkβ + kαpβ)C23 − gαβ C24) ,

where we adopted Veltman’s convention (sign conventions corresponds to Euclidean metric) forthe tensor integrals (6.47)

∫

ℓ

ℓα(1)(2)(3)

= −pαC11 − kα C12 ,

∫

ℓ

ℓαℓβ(1)(2)(3)

= −pαpβ C21 − kαkβ C22 − (pαkβ + kαpβ)C23 + gαβ C24 .

Using once more the on–shell conditions (applying the Dirac equation), we have

4 (Qµγα −mgµα) pα = 4 (Qµ − pµ)m = 2m (Q + k)µ ,

4 (Qµγα −mgµα) kα = −4mkµ ,

gµαγβ pαpβ = mpµ = m(Q− k)µ

2,

gµαγβ kαkβ = 0 ,

gµαγβ (pαkβ + kαpβ) = mkµ ,

gµαγβ gαβ = γµ ,

and we obtain

T1 = γµ

(d− 6)B0(m,m; k2) + 2B0(0,m; p2) + 2B0(0,m; q2)− 4 pq C0 − 2 (d− 2)C24

+mQµ

2C11 + (d− 2)C12

+mkµ

2C11 − 4C12 − (d− 2)C12 + C23

.

The tensor integrals may be written in terms of C0, B0’s and A0’s according to (6.48)–(6.50).Here we only note that all invariant Ci functions are UV finite except for C24, for which

(C24)UV =1

4Reg .

128

Together with the Reg terms from the B0’s (B0 = Reg + · · ·) we find

(T1)UV = γµ Reg .

On–shell q2 = p2 = m2 and In the relativistic limit m→ 0 one obtains C24 ≃ 14 (B0(0, 0; k2) + 1)

such that

T1 ≃ γµ

4B0(0,m;m2)− 3B0(0, 0; k2)− 2 + 2 k2 C0(0,m,m; q2, k2, p2)

.

However, the on–shell limit does not exist! There is an infrared problem C0(0,m,m; q2, k2, p2)has a logarithmic IR singularity for q2 → m2 and p2 → m2 .2nd term: −(1− ξ) ℓ/ (ℓ/2 +m) γµ (ℓ/1 +m) ℓ/ 1

ℓ2

Considering first the numerator: if we utilize the on–shell conditions

u(q) (q/−m) · · · = 0 and · · · (p/−m)u(p) = 0 ,

respectively, we may write

ℓ/ (ℓ/2 +m) γµ (ℓ/1 +m) ℓ/

= (ℓ/2 −m)(ℓ/2 +m)γµ(ℓ/1 +m)(ℓ/1 −m) = (ℓ22 −m2) γµ (ℓ21 −m2)

and the quark propagators drop out. The remaining integral is IR divergent

−(1− ξ)∫

ℓ

1

ℓ4γµ .

In DR such integrals are zero, however, it mixes up UV and IR singularities, which cancel againsteach other. It should be noted that off-shell there is no IR problem; one thus has to study fromcase to case carefully how limits are taken. Alternatively, one may use an infrared regularization,e.g. in infinitesimally small gluon mass: the inverse propagator in the Lagrangian in momentumspace has the form

−(gµν (k2 −m2

g)− (1− 1

ξ) kµkν

)

= −(gµν − (1− 1

ξ)

kµkνk2 −m2

g

)(k2 −m2

g)

such that the propagator takes the form

−(gµν − (1− ξ) kµkν

k2 − ξ m2g

)1

k2 −m2g + iε

.

As a result∫

ℓ

1

ℓ4→∫

ℓ

1

ℓ2 −m2g

1

ℓ2 − ξ m2g

= B0(mg,√ξmg; 0)

and we find

T2 = (1− ξ)B0(· · ·) = (1− ξ) (Reg + 1− lnm2g) + ξ ln ξ ,

which is UV and IR singular! This phenomenon is known from QED. Because of the masslessphoton the interaction has infinite range (Coulomb interaction) and one cannot take the charged

129

leptons on–shell, because they never become truly free25. In perturbative QCD the interactionwith the massless gluons prevents the on–shell limit to exist for all color charged “perturbativestates”, i.e., for quarks as well as for the gluons. Due to confinement of QCD, which onlyallows colorless hadrons as physical states, the non-existence of the on–shell limit in fact is anacademic problem as free quarks and gluons are not physical. For off-shell quarks there are noIR singularities, however the UV singularity is always there. With

(T2)UV = γµ (−(1− ξ) Reg)

we find the complete UV term for diagram (1):

(T1 + T2)UV (1) = γµ ξReg .

We may now summarize the result for the vertex form factor from diagram (1):

g γµ (Ti)bc (1 + δg) for p2 = q2 = m2 (mg finite infinitesimal)

with

δg(1) =g2

16π2

(C2(R)− 1

2C2(G)

)

×

4B0(0,m;m2)− 3B0(m,m; k2)− 2 + 2 (k2 − 2m2)C0(mg,m,m;m2, k2,m2)

−(1− ξ)B0(mg,√ξmg; 0)

Explicitely we have:

B0(0,m;m2) = Reg + 2− lnm2

B0(m,m; k2) = Reg + 2− lnm2 + 2

(4m2

k2− 1

)G

(4m2

k2

)

B0(0,m; k2) ≃ Reg + 2− ln(−(k2 + i0)

); m→ 0

B0(mg,√ξmg; 0) = Reg + 1− lnm2

g +ξ

1− ξ ln ξ

C0(mg,m,m;m2, k2,m2) =2

k2√

1− y

lnx ln

m

mg+ Sp(−x) +

π2

12− 1

4ln2 x+ lnx ln(1 + x)

where y =4m2

k2; x =

√1− y − 1√1− y + 1

; Sp(x) = −∫ 1

0

dt

tln(1− xt) .

The new function Sp(x) is called dilogarithm or Spence function. The UV singular term reads

δg(1)UV =

g2

32π2

(C2(R)− 1

2C2(G)

)2ξReg .

25As we know from QED, it is a matter of looking for well defined realistic observables to avoid such singularities.One solution is the Bloch-Nordsieck prescription, which requires to include soft real photons in addition to thevirtual ones, since they are indistinguishable experimentally.

130

We now calculate diagram (2): (we denote ℓ2 = ℓ+ q and ℓ1 = ℓ+ p)

−i ×k

i, µρ

σ

−ℓ, a

b, q

c, p

j, ρ′

l, σ′

ℓ2

ℓ1

= −i i6 (ig) g2∑

jl

fijl (Tj)ba (Tl)ac ×∫

ddℓ

(2π)dγρ

(−ℓ/+m)

ℓ2 −m2γσ × 1

ℓ21

1

ℓ22

×gρ

′σ′ (ℓ1 + ℓ2)µ − gµρ′(ℓ2 + k)σ′ − gµσ′(ℓ1 − k)ρ

′

×(gρρ′ − (1− ξ) ℓ2ρℓ2ρ′

ℓ22

)(gσσ′ − (1− ξ) ℓ1σℓ1σ′

ℓ21

).

Common factor here is:N2 = −i g

3

2 C2(G) (Ti)bci

16π2 ,and we already used

fijl TjTl =1

2fijl Tj, Tl+

i

2fijlfjlk Tk =

i

2C2(G) δik Tk .

We again sort contributions according to the terms of the gluon propagators. This time we havea product of two propagators.1st term: gρρ′gσσ′

for the numerator yields

X1 = γρ (−ℓ/+m) γσ ×gρσ (ℓ1 + ℓ2)

µ − gµ ρ(ℓ2 + k)σ − gµ σ(ℓ1 − k)ρ

= 2 (d − 2) ℓ/ℓµ + (d− 2) ℓ/Qµ + 2mdℓµ +mdQµ + γµℓ/ (ℓ/+ 2q/− p/) + (ℓ/+ 2p/− q/) ℓ/γµ

−m (2ℓµ + 4Qµ) + 6m2γµ

= 2 (d − 2) ℓ/ℓµ + (d− 6) ℓ/Qµ + 2mdℓµ +m (d− 4)Qµ + 2γµℓ2 + 4 ℓQγµ + 6m2γµ .

Here we again have used the on–shell condition and replaced p/ = m and q/ = m when sandwichedbetween the external quark spinors and collected different terms. If we use 2ℓQ = ℓ21+ℓ22−2ℓ2−2m2

and p2 = m2 and q2 = m2 we obtain

X1 = −2γµ (ℓ2 −m2) + 2γµℓ21 + 2γµℓ22 + (d− 6) ℓ/Qµ + 2 (d− 2) ℓ/ℓµ + 2mdℓµ +m (d− 4)Qµ .

Denoting (1) = ℓ21, (2) = ℓ22 and (3) = (ℓ2 −m2) we may write

T1 =

∫

ℓ

γµ[−2

1

(1)(2)+ 2

1

(1)(3)+ 2

1

(2)(3)

]

+m (d− 4)Qµ1

(1)(2)(3)+ ((d− 6)Qµγα + 2mdgµα)

ℓα(1)(2)(3)

+ 2 (d− 2) gµαγβℓαℓβ

(1)(2)(3)

= γµ[−2B0(0, 0; k2) + 2B0(m, 0; p2) + 2B0(m, 0; q2)

]

−m (d− 4)Qµ C0(m, 0, 0; p2, k2, q2)

− ((d− 6)Qµγα + 2mdgµα) (pαC11 + kα C12)

−2 (d − 2) gµαγβ (pαpβ C21 + kαkβ C22 + (pαkβ + kαpβ)C23 − gαβ C24) .

131

Setting p = (Q − k)/2, q = (Q + k)/2 and applying the on–shell condition (Dirac equation) wefinally have

T1 = γµ[2 (d− 2)C24 − 2B0(0, 0; k2) + 2B0(m, 0; p2) + 2B0(m, 0; q2)

]

+mQµ [(4− d)C0 + (6− 2d)C11 − (d− 2)C21]

+mkµ [dC11 − 2 dC12 + (d− 2)C21 − 2 (d− 2)C23] ,

where the last two terms are finite such that

(T1)UV (2) = γµ 3 Reg .

2nd term: −(1− ξ) gρρ′ ℓ1σℓ1σ′ℓ21

(+ sym ρ, ℓ1 ↔ σ, ℓ2) from the product of gluon propagators.

Here we obtain a numerator

X2 = −(1− ξ) γρ (−ℓ/+m) ℓ/11

ℓ21(+ sym ℓ1 ↔ ℓ2)

×ℓρ1 (ℓ1 + ℓ2)

µ − gµρ (ℓ22 − k2︸︷︷︸ℓ1(ℓ2+k)

)− ℓµ1 (ℓ1 − k)ρ

= −(1− ξ) 1

ℓ21×ℓ/1 (−ℓ/+m) ℓ/1ℓ

µ2 − γµ (−ℓ/+m) ℓ/1 (ℓ22 − k2) + k/ (−ℓ/+m) ℓ/1ℓ

µ1

,

and using the on–shell relations ℓ/1u(p) = (ℓ/+m)u(p), u(q)k/u(p) = 0 etc. we arrive at

X2 = (1− ξ) 1

ℓ21(ℓ2 −m2) ×

ℓ/1︸︷︷︸

(ℓ/2−k/)=ℓ/2

ℓµ2 − γµ (ℓ22 − k2) + k/︸︷︷︸=0

ℓµ1

.

We thus have to evaluate

T2 = (1− ξ)∫

ℓ

ℓ/2ℓ

µ2

ℓ41ℓ22

− γµ 1

ℓ41+ γµ k2

1

ℓ41ℓ22

.

First we note that here we have no IR problem if we consider the two IR singular terms together,indeed (ℓ22 − k2)/ℓ41/ℓ22 − ℓ1(ℓ2 + k)/ℓ41/ℓ

22 is O.K. The integrals are

∫

ℓ1

1

ℓ41= 0

k2∫

ℓ1

1

ℓ41ℓ22

∣∣∣∣ℓ2=ℓ1+k

= I0(2, 1) = −B0 + 2

∫

ℓ2

ℓ/2ℓµ2

ℓ41ℓ22

∣∣∣∣ℓ1=ℓ2−k

= k/kµ

k2I21(1, 2) + γµ I22(1, 2)︸︷︷︸

14B0

.

The contribution from the first term of the last integral vanishes on–shell (k/ = 0). Again thereis no IR problem in the γµ amplitude. Also note ℓ/2ℓ

µ2 → ℓ/1ℓ

µ1 + terms proportional to k/, which

vanish on–shell. We thus arrive at

T2 = 2 (1− ξ) γµ

1

4B0(0, 0; k2)−B0(0, 0; k2) + 2

with a factor 2 for including the symmetric term (ℓ1 ↔ ℓ2) and thus

(T2)UV (2) = γµ(−(1− ξ) 3

2Reg

).

132

3rd term: (1− ξ)2 ℓ/2 (−ℓ/+m) ℓ/11ℓ21

1ℓ22, from the product of the two gluon propagators, yields in

the numerator

X3 = (1− ξ)2 ℓ/2 (−ℓ/+m) ℓ/11

ℓ21

1

ℓ22

×

(ℓ1ℓ2)︸︷︷︸12(ℓ21+ℓ

22−k2)

(ℓ1 + ℓ2)µ − ℓµ2 (ℓ2(ℓ2 + k))︸︷︷︸ℓ22−k2

−ℓµ1 ((ℓ1 − k)ℓ2)︸︷︷︸ℓ21−k2

= −(1− ξ)2 1

2

ℓ2 −m2

ℓ21ℓ22

ℓ/2[= ℓ/1 on − shell] ·k2 (ℓ1 + ℓ2)µ + ℓ21k

µ − ℓ22kµ,

which yields

T3 = −(1− ξ)22

∫

ℓ

k2ℓ/1ℓ

µ1

ℓ41ℓ42

+ k2ℓ/2ℓ

µ2

ℓ41ℓ42

+ kµℓ/1ℓ21ℓ

42

− kµ ℓ/2ℓ41ℓ

22

.

Here we have no IR problem in the γµ amplitude. The problematic terms may be written as

ℓ/1ℓµ2

ℓ41ℓ42

andℓ/2ℓ

µ1

ℓ41ℓ42

.

Between spinors only γµ terms survive as u(q)k/u(p) = 0. Integration variables are ℓ1 withℓ2 = ℓ2 + k or ℓ2 with ℓ1 = ℓ2 − k. Only terms gµαγα = γµ of the integrals:

k2∫

ℓ1

ℓ/1ℓµ1

ℓ41ℓ42

= k/kµ

k2I21(2, 2) + γµ I22(2, 2)︸︷︷︸

12

contribute and with twice the same term we arrive at

T3 = γµ(−(1− ξ)2

2

)finite!

and hence

(T3)UV (2) = 0 .

Herewith we have the result for the vector form–factor from diagram (2) for q2 = p2 = m2

δg(2) =g2

16π2

(C2(G)

2

)×

4C24(m, 0, 0;m2, k2,m2)− 1− 2B0(0, 0; k2) + 4B0(m, 0;m2)

−2 (1− ξ)(

3

4B0(0, 0; k2)− 2

)− (1− ξ)2 1

2

=g2

32π2C2(G)

2(6− 3 (1 − ξ)) Reg + finite

where 4C24 → B0(0, 0; k2) + 1 for m→ 0. The ε–pole term reads

δg(2)UV =

g2

32π2C2(G)

2(6− 3 (1− ξ)) Reg .

So far we only calculated the dimensionally regularized bare amplitudes. In order to obtain thephysical amplitudes for d = 4 one has to perform the renormalization and take the limit ε → 0.As we know, this limit only exists after renormalization.

133

Renormalization of the quark–gluon vertex

Again we first do what one would do in QED, namely, we try the on–shell renormalization. Therenormalized vertex on the mass shell is renormalized to the lowest order (tree level) vertex

= i gren γµ (Ti)bc =

√Z3Z

F3

( )

OS

=√Z3Z

F3

(+ + + · · ·

)

OS

= i√Z3Z

F3 g0 γ

µ (Ti)bc + · · · = i√Z3Z

F3 Zg gren γ

µ (Ti)bc + · · ·= i gren Γµi bc ren

where OS means on–shell: q2 = p2 → m2 and k2 → 0, and we have used the parameter renor-malization relations

g0 = gren + δg = gren

(1 +

δg

g

)= Zg gren .

We again denote δZi = Zi − 1. The expression above is then expanded in gren, which we usuallysimply write g, while in the bare calculation before the renormalization g meant actually the bareg0. The expansion yields

Γµi bc ren = (Ti)bc

(γµA1 +QµA2 + kµA3

)

and we may write the only amplitude A1 which picks external renormalization factors (the latteronly enter proportional to the Born term)

A1 = 1 +1

2δZ3 + δZF

3 +δg

g+ + + · · ·

In the OS renormalization scheme, after factorization of the renormalized vertex (defining therenormalized coupling constant), the renormalization condition for A1 reads

(A1)OS ≡ 1 .

This fixes the charge renormalization counterterm to be given by

δg

g= δZg = −

(1

2δZ3 + δZF

3 + + + · · ·)

OS

.

In QCD one mainly applies the (modified) minimal subtraction scheme (MS –scheme) definedessentially by

(δZg)MS = (δZg)UV singular part

and collecting the different terms we have calculated above we obtain

(δZg)MS = − g2

32π2

C2(G)

2

(10

3+ (1− ξ)

)− 4

3T (R)Nf +C2(R) (−2ξ) +C2(R) (2ξ)

−C2(G)

2(2ξ) +

C2(G)

2(6− 3(1 − ξ))

Reg

134

and hence

(δZg)MS = 1− g2

32π2

113 C2(G)− 4

3 T (R)Nf

Reg (7.20)

a gauge invariant result! It was first published by Politzer, Gross and Wilczek in 1973. It maybe not completely clear at this point why this result was honored with the Nobel prize in 2004!

Using the Slavnov-Taylor identities for the Z–factors we have a complete set of one-loop renormal-ization constants for QCD. They are the basic input or the renormalization group to be discussednext.

Summary of minimal subtraction renormalization constants:

Zq = ZF = 1 +g2

16π2(−ξ C2(R)) Reg

ZG = Z3 = 1 +g2

32π2

(C2(G)

(13

3− ξ)− 8

3T (R)Nf

)Reg

Zη = Z3 = 1 +g2

32π2

(1

2C2(R) (3− ξ)

)Reg

Zg = 1 +g2

16π2

(11

3C2(R)− 4

3T (R)Nf

)Reg (7.21)

Zm = 1 +g2

16π2(−3C2(R)) Reg

Remark on MS renormalization:In case we have omitted the factors µε/2 needed to define the dimensionless bare coupling g0 =g0 µ

−ε/2 in d 6= 4, we have to identify g0 with g0 and a bare amplitude at one loop [(1)] has theform

A0 = 1 + g20

(a(1)0s Reg + a

(1)0r

)

= 1 + g20 µε(a(1)0s Reg + a

(1)0r

),

with a(1)0s the coefficient of the singular term (ε–poles) and b

(1)0r a regular one. The effect of the

scale factors µ4−d (which usually is included in the definitions of the standard integrals (seeFig. 6.2)) is a substitution:

µε Reg = eε2lnµ2 Reg ∼ Reg +

ε

2lnµ2 Reg

∼ Reg + lnµ2 as ε→ 0

where we used

Reg =2

ε− γ + ln 4π .

Since g0 = gren + δg (we simply write g for gren in the following ) with δg/g = O(g2) the bareamplitude may be written in terms of the renormalized coupling as

A0 = 1 + g2(a(1)0s Reg + a

(1)0s lnµ2 + b

(1)0r

).

The renormalized amplitude for “minimal subtraction” then reads

A(MS)ren = 1 + g2

(a(1)0s lnµ2 + b

(1)0r

),

135

which means that renormalized quantities in the MS scheme follow from the bare ones by thesubstitution

Reg→ lnµ2 [MS] or2

ε→ lnµ2 [MS] .

Appendix on group factors for SU(3)

The basic relations are

[Ti, Tk] = i fikl Tl ; Ti, Tk = dikl Tl +1

3δik

Further useful relations are:

Tr TiTkTl =i

4fikl +

1

4dikl ,

∑

ikl

d2ikl =40

3;∑

ikl

f2ikl = 24

and

∑

dei

εade (Ti)bd (Ti)ec = −2

3εabc .

Casimir invariants are the following:1. Casimir of the regular representation of G = SU(n):

C2(G) δik =∑

lm

film fklm ; C2(G) = n

2. Let r be the dimension of the group G (number of generators) and d(r) the dimension of therepresentation R. With

T (R) δik = Tr (Ti Tk) =∑

ab

(Ti)ab (Tk)ba ,

the Casimir of this representation then is

C2(R) =r

d(r)T (R) ; C2(R) δab =

(∑

i

Ti Ti

)

ab

For R = fundamental representation in particular

T (R) =1

2; C2(R) =

n2 − 1

2n.

For QCD we have

G = SU(3)c ; C2(G) = 3 ; T (R) =1

2Nf ; C2(R) =

4

3.


136

① Calculate the group invariants C2(G) and C2(R) using the known properties of the SU(n)groups.

② Derive from our one-loop QCD results the corresponding ones for QED. Discuss main dif-ferences.

③ Calculate the integral C0(mg,m,m;m2, k2,m2) in the regime m2g ≪ m2, |p2| using the

representation (6.38). Discuss typical properties of the result.

④ Elaborate the following statement [from ’t Hooft and Veltman’s Diagrammar].Statement: Order-by-order renormalization is not equivalent to “throwing away ε–poles andtheir residues” of the unrenormalized S–matrix.

Illustrate this for the following example of a double lepton–loop contribution to the photonself–energy:

Up to a factor p2 gµν − pµpν, the irreducible self–energy bubble may be written in the form

:= f(p2) =1

d− 4f1(p

2) + f2(p2) + (d− 4) f3(p3)

What is the correct two–loop contribution from the factorizable double–loop above?

137

8 The renormalization group, QCD and asymptotic freedom

The renormalization group is the best example which proves that the renormalization problem inQFT is not just a technical problem (some people refuse QFT or at least its perturbative definitionand treatment as an ill defined or mathematically ambiguous field) but has a deep physics reason.The renormalization group on the one hand is intimately related to renormalization as o tool toremove singularities and on the other hand controls the scale dependence of physics. Before wegive a systematic formal derivation of the renormalization group (RG), we present a simplifiedoutline of the basic ideas and features. After our discussion of the basics of renormalizationand radiative corrections by loop effects (quantum fluctuations), an obvious and simple questionwhich we may naturally ask is: how does the theory depend on the scale parameter µ? and whatrole does it play in physics.

The Renormalization Group is a key instrument to control:

• the dependence on renormalization scale (parameter µ in the MS scheme),

• the resummation of the perturbation expansion (summing leading-, subleading-, · · · loga-rithms); RG–improved perturbation expansion,

• the physical interpretation of the renormalized parameters (important in case when noS–matrix exists [in perturbation theory]),

• the renormalization scheme dependence,

• the asymptotic behavior (for large momenta [asymptotically free theories like QCD] or smallmomenta [infrared free theories like QED]).

• The RG is to be considered as the Dilatation Ward identity; Dilatation symmetry impliesconformal symmetry but these possible symmetries [strictly renormalizable massless QFTat classical level] are always broken by renormalization effects [renormalization anomalysimilar to triangle anomaly].

• Further notions: renormalization fixed points, anomalous dimensions, asymptotic freedom,Bjorken scaling etc.

8.1 Renormalization Group: the general solution

The objects of our interest are the building blocks of a QFT, the bare dimensionally regularizedd = 4− ε (ε > 0) dimensional 1PI vertex–functions (amputated Green functions):

Γ(nG,2nF )0 (p; g0,m0, ξ0)ε

with

g0 = g0µε/2

the bare coupling and

dimΓ = d− nGd− 2

2− 2nf

d− 1

2

138

the canonical dimension of the vertex–function. The bare vertex functions under a change ofscale are homogeneous functions when we rescale all parameters according to their dimensionsincluding the “reference scale parameter” µ:

Γ(nG,2nF )0 (κ p; g0 (κµ)ε/2, κm0, ξ0)ε = κdimΓ Γ

(nG,2nF )0 (p; g0 (µ)ε/2,m0, ξ0)ε

(engineers scaling). The renormalized vertex–functions needed for real physics in d = 4 dimensionswe obtain by renormalization of the fields and parameters

G0 =√Z3Gren ; ψ0 =

√ZF3 ψren ; η0 =

√Z3 ηren,

g0 = Zg gr ; m0 = Zmmr ; ξ0 = Z3ξr .

This translates into the relation

Γ(nG,2nF )0 (p; g0 (µ)ε/2,m0, ξ0)ε = (Z3)

−nG/2ε

(ZF3

)−nF

εΓ(nG,2nF )ren (p; gr,mr, ξr, µ)ε ,

for the vertex–functions. The renormalization constants Zi must be chosen such that

limε→0

Γ(nG,2nF )ren (p; gr,mr, ξr, µ)ε exists .

As a consequence of the renormalization Γren depends in a very non–trivial manner on the scaleparameter µ, also in the limit ε→ 0. While a complete rescaling of all dimensionful parameters

Γ(nG,2nF )ren (κ p; gr, κmr, ξr, κ µ)ε = κdimΓ Γ(nG,2nF )

ren (p; gr ,mr, ξr, µ)ε ,

shows a perfect homogeneous scaling law, scaling of the physical parameters only, together withthe momenta, spoils the previous relation

Γ(nG,2nF )ren (κ p; gr, κmr, ξr, µ)ε 6= κdimΓ Γ(nG,2nF )

ren (p; gr ,mr, ξr, µ)ε .

It means that the homogeneity is broken and this actually represents another renormalizationanomaly: quantum fluctuations spoil naive, so called canonical, scaling laws expected at theclassical level. The actual dependence of Γren on µ, we obtain from the renormalization equationabove by applying the chain rule of differentiation (the µ–dependence is not only continuous butalso differentiable), and the fact that for fixed g0 Γ is completely independent of µ:

µ∂Γ0

∂µ

∣∣∣∣g0

≡ 0 .

This translates into a highly non-trivial relation, the RG, for the dependence of the renormalizedvertex–functions on the renormalized parameters26:

µ∂

∂µ+ β

∂

∂gr+ ω

∂

∂ξr+ γmmr

∂

∂mr− nG γG − 2nF γF

Γ(nG,2nF )ren (p; gr,mr, ξr, µ)ε = 0 ,

26Explicitly,

µ∂

∂µΓ(nG,2nF )0 (p; g0, m0, ξ0)

∣

∣

∣

∣

g0

=

(

µ∂

∂µ(Z3)

−nG/2

)∣

∣

∣

∣

g0

(

ZF3

)−nF

Γ(nG,2nF )ren

+ (Z3)−nG/2

(

µ∂

∂µ

(

ZF3

)−nF

)∣

∣

∣

∣

g0

Γ(nG,2nF )ren + (Z3)

−nG/2(

ZF3

)−nF

µ∂

∂µΓ(nG,2nF )ren

∣

∣

∣

∣

g0

.

Using

µ∂

∂µ(Z3)

−nG/2

∣

∣

∣

∣

g0

= −nG

2(Z3)

−nG/2 Z−13 µ

∂

∂µZ3

∣

∣

∣

∣

g0

= −nG

2(Z3)

−nG/2 µ∂

∂µlnZ3

∣

∣

∣

∣

g0

= −nG γG (Z3)−nG/2 ,

139

the famous RG–equation, which characterizes the renormalization properties of any QFT. Weremind the reader that there is a Slavnov-Taylor identity which requires

ξ0 = Z3ξr ,

with Z3 the gauge field renormalization constant. This implies

ω = −2ξr γG

such that in the Landau–gaugeξr = 0

all RG–coefficient functions as dimensionless functions only depend on the renormalized couplinggr:

β = β(gr) ; γi = γi(gr) ; ω ≡ 0 Landau gauge .

Because the physical parameter renormalization factors Zg and Zm can be defined in a gaugeindependent way, also β and γm may always be defined in a gauge invariant manner, i.e. to be ξindependent. In order to determine the renormalization group coefficients, we have to make useof the fact that the bare quantities depend of µ only via

g0 = g0 µε/2 ,

such that for any bare function F (g0, µ)27

µ∂

∂µF(g0 = g0 µ

ε/2)∣∣∣∣g0

=

(µ∂

∂µ− ε

2g0

∂

∂g0

)F (g0, µ)

.= Dµ,ε F (g0, µ) .

Furthermore, we note that

F−1 Dµ,ε F = Dµ,ε lnF ; Dµ,ε µ = µ .

From the chain rule we identify the coefficient functions as

β = Dµ,ε gr = gr

(−ε

2+ε

2g0

∂

∂g0lnZg

)

γG = Dµ,ε ln√Z3 = −ε

4g0

∂

∂g0lnZ3(g0, ξ0)

and the corresponding relation for ZF3 together with

µ∂

∂µΓren(p; gr,mr, ξr, µ)

∣

∣

∣

∣

g0

= µ∂

∂µΓren(p; gr,mr, ξr, µ) + µ

∂gr∂µ

∣

∣

∣

∣

g0

∂

∂grΓren(p; gr,mr, ξr, µ)

+ µ∂ξr∂µ

∣

∣

∣

∣

g0

∂

∂ξrΓren(p; gr, mr, ξr, µ) + µ

∂mr

∂µ

∣

∣

∣

∣

g0

∂

∂mrΓren(p; gr,mr, ξr, µ)

where the derivatives of the vertex–functions are taken at fixed renormalized parameters, the ones of the parameters

and counterterms at fixed g0. With µ∂gr∂µ

∣

∣

∣

g0

= β etc. the result follows.

27Using

µ∂

∂µF (g0, µ)

∣

∣

∣

∣

g0

= µ∂

∂µF (g0, µ)

∣

∣

∣

∣

g0

+ µ∂g0∂µ

∣

∣

∣

∣

g0

∂

∂g0F (g0, µ)

∣

∣

∣

∣

µ

with

µ∂(g0 = g0µ

−ε/2)

∂µ

∣

∣

∣

∣

g0

= − ε2g0µ

−ε/2 = − ε2g0 .

140

γF = Dµ,ε ln√ZF3 = −ε

4g0

∂

∂g0lnZF

3 (g0, ξ0)

γmmr = Dµ,εmr =ε

2mr g0

∂

∂g0lnZm(g0, ξ0)

ω = Dµ,ε ξr = −ε2g0

∂

∂g0ξr = −2 ξr γG .

They all have finite limits for ε → 0. The right hand sides may be recalculated in terms ofrenormalized parameters, where in MS type renormalization schemes we have

Zi = 1 +

∞∑

n=1

zi,n(g, ξ)

εn,

which together with the chain rule implies that the coefficient functions are determined by thezi,1(g, ξ) alone:

β(g) = g2 g

∂∂g zg,1(g)

γm(g) = 12 g

∂∂g zm,1(g)

γG(g) = −14 g

∂∂g z3,1(g, ξ)

γF (g) = −14 g

∂∂g z

F3,1(g, ξ) .

(8.1)

The coefficients of the ε–poles in higher orders are determined by recurrence relations like

g

2g∂

∂gzg,n+1(g) = β(g)

zg,n + g

∂

∂gzg,n

1

2g∂

∂gzm,n+1(g) =

(β(g)

∂

∂g+ γm(g)

)zm,n

1

2g∂

∂gz3,n+1(g, ξ) =

(β(g)

∂

∂g− 2γG ξ

∂

∂ξ− 2γG(g)

)z3,n

1

2g∂

∂gzF3,n+1(g, ξ) =

(β(g)

∂

∂g− 2γG ξ

∂

∂ξ− 2γF (g)

)zF3,n .

From the results for the one–loop renormalization counterterms (see previous section) we mayread off the UV singular parts of the counter terms:

Zg = 1− g2

16π2

(11 − 2

3 Nf

)1ε , Zm = 1− g2

16π2 8 1ε ,

Z3 = 1 + g2

16π2

(13 − 3ξ − 4

3 Nf

)1ε , ZF

3 = 1− g2

16π2 ξ83

1ε ,

from which the leading terms of the RG coefficient functions may be easily read off.

The RG equation is a partial differential equation which is homogeneous and therefore can besolved easily along so called characteristic curves. Let s parameterize such a curve, such that allquantities become functions of the single parameter s: g = g(s), m = m(s), µ = µ(s) and

dΓ

ds(p; g(s),m(s), µ(s)) =

dµ

ds

∂

∂µ+

dg

ds

∂

∂g+

dm

ds

∂

∂m

Γ = nγ Γ ,

with

dµ

ds= µ ,

dg

ds= β(g) ,

dm

ds= mγm(g) ,

which is a set of ordinary differential equations the solution of which is solving the RG equation.For simplicity of notation and interpretation we have assumed the Landau gauge ξ = 0 and weabbreviated nGγG + 2nF γF = nγ. The successive integration then yields

141

1) dµ

ds= µ lnµ = s+ constant µ = µ0 es = µ0 κ ,

where κ = es is a scale dilatation parameter

2)dg

ds= β(g)

dg

β(g)= ds =

dµ

µ

ln(µ/µ0) = lnκ =

g(κ)∫

g

dg′

β(g′),

which is the implicit definition of the running coupling g(κ) with g = g(1) the coupling atreference scale µ0 and g(κ) = g(µ/µ0) the coupling at scale µ.

3)dm

ds= mγm

dm

m= γm(g) ds = γm(g)

dg

β(g)

lnm+ constant =

g(κ)∫

g

γ(g′) dg′

β(g′),

or

m(κ) = m exp

g(κ)∫

g

γ(g′) dg′

β(g′),

4)dΓ

ds= nγ(g) ds = nγ(g)

dµ

µ= nγ(g)

dg

β(g)

Γ(κ) = Γ exp

n

g(κ)∫

g

γ(g′) dg′

β(g′)

= Γ z3(g, κ)nG zF3 (g, κ)2nF ,

with Γ = Γ(1), and

z3(g, κ) = exp

g(κ)∫

g

γG(g′) dg′

β(g′), zF3 (g, κ) = exp

g(κ)∫

g

γF (g′) dg′

β(g′).

Altogether, we may write this as an equation which describes the response of the theory withrespect to a change of the scale parameter µ:

Γ (p; g,m, µ/κ) = z3(g, κ)−nG zF3 (g, κ)−2nF Γ (p; g(κ),m(κ), µ) .

Thus a change of the scale parameter µ is equivalent to a finite renormalization ofthe parameters and fields and together with the homogeneity relation we have for the vertexfunctions with scaled momenta

Γ (κp; g,m, µ) = κdimΓ Γ

(p; g(κ),

m(κ)

κ,µ

κ

)

= κdimΓ z3(g, κ)−nG zF3 (g, κ)−2nF Γ

(p; g(κ),

m(κ)

κ, µ

),

which is the basic relation for a discussion of the asymptotic behavior.

142

β(g)

g

g∗

−

→ ←

QCD

a) β(g)

gg∗

+

→ ←

QEDb)

Figure 8.1: RG fixed points are zeros of the β–function: a) UV fixed points, b) IR fixed points

8.2 Asymptotic Behavior

Two regimes are of interest, the high energy (ultraviolet) behavior and the low energy (infrared)behavior.

8.2.1 Short distance or UV behavior

The ultraviolet behavior, which determines the short distance properties, is obtained by choosingκ|p| ≫ m,µ thus

lnκ =

g(κ)∫

g

dg′

β(g′)→ +∞ ; κ→∞ .

However, the integral can only become divergent for finite g(κ) if β(g) has a zero at limκ→∞ g(κ) =g∗: more precisely, in the limit κ → ∞ the effective coupling has to move to a fixed pointg(κ) → g∗− if finite, and the fixed point coupling is characterized by β(g∗−) = 0, β′(g∗−) < 0.Thus g∗− is an ultraviolet fixed point coupling. Note that by dilatation of the momenta atfixed m and µ, the effective coupling is automatically driven into a fixed point, a zero of theβ–function with negative slope, if it exists. If g∗− = 0 we have asymptotic freedom. This ishow QCD behaves, which has a β–function

βQCD(gs) = −gs(β0

(g2

16π2

)+ β1

(g2

16π2

)2

+ · · ·)

with β0 > 0 (see Fig. 8.1 a). QCD will be considered in more detail later on.

A possible fixed point is accessible in perturbation theory provided g∗ is sufficiently small, suchthat perturbation theory is sufficiency “convergent” as an asymptotic series. One may thenexpand about g∗:

β(g) = (g − g∗−) β′(g∗−) + · · ·γ(g) = γ∗ + (g − g∗−) γ′(g∗−) + · · ·

and provided β′(g∗−) 6= 0 we have

a(g, κ) = exp

g(κ)∫

g

γ(g′)β(g′)

dg′ = exp

g(κ)∫

g

γ(g∗−)

β(g′)dg′ · r(g, κ)

= κγ∗r(g, κ)

143

where the non–singular remainder

r(g, κ) = exp

g(κ)∫

g

(γ(g′)− γ∗)β(g′)

dg′

in the limit of large κ yields a finite scale independent wave function renormalization

limκ→∞

r(g, κ) = r(g,∞) .

We thus find the asymptotic from

Γ(κp; g,m, µ)→∼ κd

(κdG rG(g,∞)

)−nG(κdF rF (g,∞)

)−2nF

Γ(p; g∗−, 0, µ)

which exhibits asymptotic scaling. In the first place it is given by the vertex functions of amassless theory. As expected, at high energies masses may be neglected, however on the expensethat another mass scale remains in the game, the scale parameter µ. The first factor κd is trivialand is due to the d–momentum conservation which was factored out. Then each field exhibitsa homogeneous (power–like) behavior in the dilatation factor κ, the exponent of which exhibitsan anomalous dimension as a consequence of the dynamics of the theory:

dG =d− 2

2+ γ∗G , dF =

d− 1

2+ γ∗F .

The first term is the naive or engineers dimension the second part is the anomalous part whichis a quantum effect, a relict of the breaking of scale invariance, when g 6= g∗. While naively wewould expect that in d = 4 dimensions the massless theory has scaling: for example a scalar two–point function, the only dimensionful physical quantity being the momentum, one would expectG(p; g) ∼ 1/p2 as G has dimension 2. However, if there would be a non–trivial UV fixed pointone would have G(p, g, µ) ∼ (µ2)γ

∗/(p2)1+γ

∗( γ∗ > 0) which shows the role and unavoidability

of the scale parameter µ, which has to eat up the extra dimension γ∗ induced by the dynamicsof the theory. Otherwise only truly free theories could have scaling, called canonical scaling inthis case. The discovery of asymptotic freedom of QCD is the prime example of a dynamicaltheory, notabene of the theory of strong interactions, exhibiting asymptotic canonical scaling(Bjorken scaling) of liberated quarks (quark parton model). The latter was discovered before inthe pioneering investigations concerning Deep Inelastic Scattering (DIS) of electrons on protonsand bound neutrons by Friedman, Kendall and Taylor (Nobel prize 1990). These experimentshave been of essential importance for the development of the quark model and to the discoveryof QCD as the theory of the strong interactions.

8.2.2 Long distance or IR behavior

The infrared behavior corresponds to the long distance properties of a system. Here the regimeof interest is κ|p| ≪ m,µ and the discussion proceeds essentially as before: now

lnκ =

g(κ)∫

g

dg′

β(g′)→ −∞ ; κ→ 0 ,

is required. Now as κ→ 0 the effective g(κ) → g∗+ where g∗+ is a zero of the β–function withpositive slope, see Fig. 8.1 b), β(g∗+) = 0 and β′(g∗+) > 0. This is the typical situation in the

144

construction of low energy effective theories, particularly in the discussion of critical phenomenaof statistical systems (keywords: critical behavior, critical exponents, scaling laws, universality).If g∗+ = 0 the effective theory is infrared free (the opposite of asymptotic freedom), also calledGaussian (Gaussian fixed point). Here the well known examples are QED

βQED(e) =e3

12π2

∑

f

NcfQ2f + · · ·

or the self–interacting scalar field φ4–theory

β(λ) = −ελ+3λ2

16π2+ · · ·

in d = 4 dimensions. For QED the running coupling to leading order thus follows from

lnκ =

e(κ)∫

e

1

β(e′)de′ =

12π2∑f NcfQ

2f

e(κ)∫

e

1

(e′)3de′ =

24π2∑f NcfQ

2f

(1

e2− 1

e(κ)2

)

where the sum extends over all light flavors f : mf < µ28. The running fine structure constantthus at leading order is given by

α(µ) =α

1− 2α3π

∑f NcfQ

2f lnµ/µ0

where µ0 is the scale where the lightest particle starts to contribute, which is the electronµ0 = me. We then may identify α(µ0) = α the classical low energy value of the fine structureconstant, with the proviso that only logarithmic accuracy is taken into account (see below). Therunning α is equivalent to the Dyson summation of the transversal part of the photon self–energyto the extent that only the logs are kept. The RG running takes into account the leading radiativecorrections in case the logs are dominating over constant terms, i.e., provided large scale changesare involved.

As α(µ) is increasing with µ in the resummed perturbation theory running coupling exhibits apole, the so called Landau pole at which the coupling becomes infinite: lim

µ<→µL

α(µ) =∞ The

“fixed point” very likely is an artifact of perturbation theory, which of course cease to be validwhen the one–loop correction approaches 1. What this tells us is that we actually do not knowwhat the high energy asymptotic behavior of QED is.

We must stress here that the discovery of the RG (Stueckelberg, Petermann 1953, Gell-Mann,Low 1954, Bogoliubov, Shirkov 1957, Ken Wilson 1971) had and still has far reaching conse-quences for understanding the scaling structure of QFTs. Equally important the RG provides anindispensable tool for the resummation of leading quantum effects, without running into conflictwith basic QFT properties like unitarity and locality. In contrast, Dyson resummation of propa-gator effects, for example, leads to such conflicts, as it treats propagators and vertices on differentfooting. Anomalous dimensions (anomalous power laws) would never have been seen in an orderby order perturbative treatment. In condensed matter physics it took 70 years to understandlong distance scaling because of lack in understanding the emergence of anomalous dimensions.Wilson’s version of the RG finally provided the solution. Due to poor convergence (numerically

28This latter restriction takes into account the decoupling of heavy flavors, valid in QED and QCD. Since in theMS scheme, i.e., renormalization by the substitution Reg → lnµ2, which we are considering here, decoupling isnot automatic, one has to impose it by hand. At a given scale one is thus considering an effective theory, whichincludes only those particles with masses below the scale µ.

145

relatively large coupling) perturbative QCD would be almost useless, if renormalization groupimprovement (running coupling, running masses, anomalous dimensions) would not be available.In Grand Unified Theories the RG is a key tool for understanding possible coupling unification.

Last but not least, also in QED, for example in high precision calculations of g−2, where leading5–loop contributions are required to match the experimental precision, the RG is an indispensabletool which allows us to calculate in a simple way leading higher order effects, just by replacingthe fixed order-by-order coupling by the running one. In QED vertex corrections cancel againstelectron wave function renormalization contributions, a famous QED Ward-Takahashi identity,and only photon vacuum polarization (VP) effects contribute to the renormalization of the charge.Therefore, in QED the running α is equivalent to the Dyson summation of the transversal partof the photon self–energy to the extent that only the logs are kept. The RG running takes intoaccount the leading radiative corrections in case the logs are dominating over constant terms, i.e.,provided large scale changes are involved.

An example: in the calculation of the contributions from electron loops in photon propagatorsto the muon anomaly aµ, such large scale changes from me to mµ are involved and indeed onemay calculate such two–loop contributions starting from the lowest order result

a(2)µ =α

2πvia the substitution α→ α(mµ)

where

α(mµ) =α

1− 23απ ln

mµ

me

= α

(1 +

2

3

α

πlnmµ

me+ · · ·

)

such that we find

a(4) LLµ (vap, e) =1

3lnmµ

me

(απ

)2

which indeed agrees with the leading log result obtained by the direct calculation. In the cal-culation of aµ only the electron VP insertions are governed by the RG and the correspondingone–flavor QED β–function has been calculated to three loops

β(α) =2

3

(απ

)+

1

2

(απ

)2− 121

144

(απ

)3+ · · ·

which thus allows to calculate leading αn (lnmµ/me)n, next–to–leading αn (lnmµ/me)

n−1 andnext–to–next–to–leading αn (lnmµ/me)

n−2 log corrections.

In d = 4 QFTs we are usually limited to perturbative methods. This also applies to RG appli-cations. Since the g = 0 is always a RG fixed point the RG allows us to study asymptoticallyfree theories at large enough energies (QCD) and infrared free theories (QED) at long distances.In the electroweak SM the initial couplings are weak enough such that calculations up to ratherhigh energies usually are possible without problems.

Summary: the RG of QCD in Short

The renormalization group, introduced above, for QCD plays a particularly important role for aquantitative understanding of AF as well as a tool for improving the convergence of the pertur-bative expansion. For QCD the RG is given by

µd

dµgs(µ) = β (gs(µ))

µd

dµmi(µ) = −γ (gs(µ)) mi(µ)

146

withβ(g) = −β0

g3

16π2− β1

g5

(16π2)2+O(g7)

γ(g) = γ0g2

4π2+ γ1

g4

(4π2)2+O(g6)

where, in the MS scheme,

β0 = 11− 23Nf ; γ0 = 2

β1 = 102− 383 Nf ; γ1 = 101

12 − 518Nf

and Nf is the number of quark flavors. The RG for QCD is known to 4 loops. It allows to defineeffective parameters in QCD, which incorporate the summation of leading logarithmic (1–loop),next–to–leading logarithmic (2–loop), · · · corrections (RG improved perturbation theory). Thesolution of (8.5) for the running coupling constant αs(µ) = g2s(µ)/(4π) yields

4π

β0αs(µ)− β1β20

ln

(

4π

β0αs(µ)+β1β20

)

=

lnµ2/µ20 +

4π

β0αs(µ0)− β1β20

ln

(

4π

β0αs(µ0)+β1β20

)

≡ lnµ2/Λ2

with reference scale (integration constant)

ΛQCD = Λ(Nf )

MS= µ exp

− 4π

2β0αs(µ)

(

1 +αs(µ)

4π

β1β0

lnβ0αs(µ)

4π + β1

β0αs(µ)

)

which can be shown easily to be independent of the reference scale µ. It is RG invariant

µd

dµΛQCD = 0 , (8.2)

and thus QCD has its own intrinsic scale ΛQCD which is related directly to the coupling strength(dimensional transmutation). This is most obvious at the one–loop level where we have the simplerelation

αs(µ) =1

β04π ln µ2

Λ2

.

Thus ΛQCD incorporates the reference coupling αs(µ0) measured at scale µ0 in a scale invariantmanner, i.e., each experiment measures the same ΛQCD irrespective of the reference energy µ0 atwhich the measurement of αs(µ0) is performed.

The solution of (8.5) for the effective masses mi(µ) reads

mi(µ) = mi(µ0)r(µ)

r(µ0)≡ mir(µ)

with

r(µ) = exp−2

γ0β0

ln4π

β0αs(µ)+

(

γ0β0

− 4γ1β1

)

ln(1 +β1β0

αs(µ)

4π)

.

Note that also the mi are RG invariant masses (integration constants) and for the masses play arole similar to ΛQCD for the coupling. The solution of the RG equation may be expanded in the

large log L ≡ ln µ2

Λ2 , which of course only makes sense if L is large (µ≫ Λ),

αs(µ) =4π

β0 L

1− β1

β20

ln(L+ β1β20)

L+ · · ·

mi(µ) = mi

(L

2

)− γ0β0

(1− 2β1γ0

β30

lnL+ 1

L+

8γ1β20L

+ · · ·)

.

147

Figure 8.2: The QCD running coupling “constant” as a function of the energy scale.

If L is not large one should solve (8.2) or its higher order version numerically by iteration forαs(µ). For the experimental prove of the running of the strong coupling constant see Fig. 8.2.We note that in agreement with RG improved perturbative QCD the experimentally determinedvalues of the strong coupling constant at different energy scales show huge effects as αs variesfrom αs(Mτ ) = 0.322 ± 0.03 at the τ mass near 2 GeV to αs(MZ) = 0.1183 ± 0.0027 at the Zmass MZ = 91.1876 ± 0.0021 GeV.

The non-perturbative calculations in lattice QCD are able to demonstrate a surprisingly goodagreement with perturbative results. The corresponding result from the Alpha Collaboration isshown in Fig. 8.5 below.

Exercise: Consider the 2–loop QCD β–function as a function of αs and discuss its dependence onthe number of quark flavors Nf .

8.3 Universality of the asymptotic behavior

What happens to the RG if we use a different renormalization procedure, which leads to areparametrization of the Green’s functions. What happens is that the Zi renormalization con-stants change together with the redefinition of the parameters:

Zi → Zi = Fi Zi ; Fi a finite function . (8.3)

Of course, the reparametrization functions Fi are finite since the UV singularities of the baretheories are identical. For simplicity we assume here a mass independent reparametrization ofthe type

(g,m, µ) → (g, m, µ) ; Zi(g0, ξ0) = Fi(g0, ξ0)Zi(g0, ξ0) ,

148

which, by applying the chain rule of differentiation yields,

β(g, ξ) = β(g)∂g

∂g+ ω(g, ξ)

∂g

∂ξ

γ(g, ξ) = γ(g)−(β(g)

∂ ln Fm∂g

+ ω(g, ξ)∂ ln Fm∂ξ

)

ω(g, ξ) = ω(g, ξ)F−13 − ξ(β(g)

∂ ln F3

∂g+ ω(g, ξ)

∂ ln F3

∂ξ

)

γi(g, ξ) = γ(g)i −1

2

(β(g)

∂ ln F i3∂g

+ ω(g, ξ)∂ ln F i3∂ξ

), i = G,F .

Note that in case we chose gauge independent physical parameter renormalizations the last termsof the first two equations are absent, because then ∂g/∂ξ = 0 and ∂Fm/∂ξ = 0. Remember thatin contrast, the field renormalization factors in general are gauge dependent. If we again chosefor simplicity the Landau–gauge, we have

ξ = 0⇒ ω = 0 such that ξ = 0⇒ ω = 0 . (8.4)

Beyond the special assumptions made, it turns out that RG fixed points and anomalous dimen-sions are universal in the following sense: If a fixed point g = g∗ exists with β(g∗) = 0 then inthe new scheme there exists fixed point coupling g∗ = g(g∗) with β(g∗) = 0 and the above chainrule relations imply

β′(g∗) = β′(g∗) , γi(g∗) = γi(g

∗) ; i = m,G,F .

This proves the universality of asymptotic behavior. Universality plays a very important role inthe theory of critical phenomena describing the long range behavior of of statistical mechanicssystems near the critical pint of a second or higher order phase transition. In statistical physicsuniversality hold under much more general conditions (Wilson’s renormalization semi group).

8.4 RG-invariance of physical observables

Let R(s) be an observable (a measurable quantity) where√s stands for a set of scales like

the invariant energy, momentum transfers etc. Assume we have calculated R(s) in a specificrenormalization scheme (i.e. for a given parametrization) where

R(s) = R(s; g,m, ξ, µ) .

As an observable R(s) must be independent of µ and of ξ: which means

µd

dµR =

µ∂

∂µ+ β

∂

∂g+ ω

∂

∂ξ+ γmm

∂

∂m

R(s; g,m, ξ, µ) = 0

d

dξR =

∂

∂ξ+ ρ

∂

∂g+ τmm

∂

∂m

R(s; g,m, ξ, µ) = 0 .

This property is called GR–invariance and is special because there is no amplitude renormalizationanalogous to the wave function renormalization factors. The parameters are, as far as theyexplicitly show up (e.g. in a reasonable renormalization scheme no explicit gauge parameterdependence would show up), all dependent of each other, with

β = µ∂

∂µg ; ω = µ

∂

∂µξ ; γ m = µ

∂

∂µm ;

ρ =∂

∂ξg ; τ m =

∂

∂ξm .

It means that the two coupled equations (8.5) eliminate the two redundant parameters µ and ξ.As it should be, the theory only has g and m as relevant physical parameters.

149

8.5 The “running” parameters of QCD

The relevant QCD parameters are the strong interaction coupling constant gs(µ) or αs(µ).= g2s(µ)

4π ,respectively, and the quark masses mi(µ) (i = u, d, s, c, b, t). For QCD the RG is given by

g(κ) : lnκ =g(κ)∫g(1)

1β(g′) dg′ ⇔ µ d

dµgs(µ) = β (gs(µ))

mi(κ) : m(κ)m(1) = exp

g(κ)∫g(1)

γm(g′)β(g′) dg′ ⇔ µ d

dµmi(µ) = −γm (gs(µ)) mi(µ)

with

β(g) = −β0g3

16π2− β1

g5

(16π2)2+O(g7)

γm(g) = γ0g2

4π2+ γ1

g4

(4π2)2+O(g6)

where, in the MS scheme,

β0 = 11− 23Nf ; γ0 = 2

β1 = 102− 383 Nf ; γ1 = 101

12 − 518Nf

and Nf is the number of quark flavors.

Minimal subtraction type renormalization schemes are mass m and gauge parameter ξ indepen-dent. The following theorem holds.Theorem: For any m and ξ independent renormalization scheme the leading RG coefficients b0,b1 and c0 in the perturbative expansions

β(g) = b0 g3 + b1 g

5 + · · · ; γm(g) = c0 g2 + · · ·

are renormalization scheme independent and thus universal.

Proof: the perturbative expansion is an expansion in α = g2

4π , which means that corrections areO(g2) for succeeding terms in the expansion. The couplings of different schemes are thus relatedby

g = g + a1 g3 + · · · ⇒ g = g − a1 g3 + · · · .

The change of scheme (g,m)→ (g, m) yields

β(g) = β(g)∂g

∂g; γm(g) = γm(g) − β(g)

∂ lnFm∂g

,

where Fm = 1 + f1 g2 + · · ·. The explicit calculation then yields

b0 = b0 , b1 = b1 and c0 = c0

as inferred by the theorem. This theorem in particular applies for going from the minimal tothe modified minimal subtraction scheme MS → MS ! It thus covers only a very limited set ofrenormalization schemes, which, however, includes the formally simple (but not very physical)MS scheme which is utilized in most QCD applications.

150

8.6 αs in perturbation theory

Here we work out the effective strong interaction coupling constant in pQCD, in successive ap-proximations obtained by taking perturbative approximations for the coefficient functions β, γetc. in solving the RG equations.

a) Leading order O(αs)

With the lowest order beta function we find

lnκ =

g(κ)∫

g(1)

dg′(− β0

(4π)2g′3) =

12β0(4π)2 g

′2

∣∣∣∣∣

g(κ)

g(1)

,

i.e.,

4π

(g(κ))2− 4π

(g(1))2=β04π

lnκ2 .

This we may write in the form

α−1s (µ) = α−1s (µ0) + β04π lnµ2/µ20 (8.5)

or

αs(µ) =αs(µ0)

1 + αs(µ0)β04π lnµ2/µ20

= αs(µ0) ·∞∑

k=0

(−αs(µ0)

β04π

lnµ2

µ20

)k. (8.6)

The last reexpanded result of the RG–solution demonstrates that the latter corresponds to aresummation of the leading log’s in the perturbative expansion.

Exercise: Study the convergence properties of the RG-resummation.

Example: we may determine αs(MZ) at the Z boson mass scale MZ = 91.1876 GeV, using the

observable R(s) =∑f≤√s

Q2fNcf ·

1 + c1

αs(s)π

(see below), where for s≫ 4m2

f we have c1 = 1,and

given R(MZ) ≃ 3.811±0.003 as an (pseudo-) experimental value. We may then calculate αs(MB)at the B meson mass scale MB ≃ 5.3 GeV using the RG. Note that in this energy region Nf = 5

is the number of active flavors. To lowest order R(0)Nf

=∑Q2fNcf is given by R

(0)3 = 2, R

(0)4 = 3 1

3

and R(0)5 = 3 2

3 . Using R(1)(MZ) = R(MZ), we obtain α(1)s (MZ) = 0.124 ± 0.002 and using

Eq. (8.5) we find α(1)s (MB) = 0.218±0.006. Not only the coupling gets stronger as we go to lower

energies, also the uncertainty gets magnified correspondingly.

The QCD–scaleWe note that in the 1–loop approximation α−1s (µ) plotted against lnµ2/µ20 is a straight line, asillustrated in Fig. 8.3:

α−1s (µ) = α−1s (µ0) +β04π

lnµ2/µ20 .

A very interesting point to be stressed here is the fact that the effective coupling and the scale

151

Figure 8.3: α−1s (µ) vs. lnµ2/µ20 in the one–loop approximation a straight line.

determine each other (up to neglected higher order effects). It is therefore natural to define anintrinsic QCD scale Λ = ΛQCD by

α−1s (µ0).=β04π

lnµ20/Λ2 ,

where Λ < µ0 . Consequently, we have

α−1s (µ) =β04π

lnµ2/Λ2

for arbitrary values of µ and we generally obtain

α−1s (µ) = 1β04π

lnµ2/Λ2. (8.7)

The relation between the QCD coupling αs and the QCD scale Λ is called dimensional transmutation.If µ0 =

√s0 is the invariant energy at which the coupling αs(µ0) is measured, then (in the con-

sidered approximation) the QCD scale is given by

ΛQCD =√s0 exp

(− 4π

2β0αs(√s0)

)

as an intrinsic physical scale of QCD. Dimensional transmutation is a quantum phenomenonrelated to the breaking of scale (and hence of conformal) invariance:

coupling scale

αs(µ) ⇔ ΛQCD

(dimensionless) (dimension of a mass)

152

Figure 8.4: αs(E) as a function of the energy scale E. As E approaches the QCD scale Λ ≃123 MeV, the effective coupling diverges (so called Landau pole), signaling the breakdown ofperturbation theory as we go to too low energies.

For the example given above we find for Nf = 5 flavors at one–loop

ΛQCD ≃ (0.123 ± 0.013) GeV .

Exercise: Elaborate on the question where is the best scale to determine the reference stronginteraction coupling constant. Compare the situation with the fine structure constant in QED.

• The crucial point is the following: ΛQCD is independent of µ at which it was determined,in otherwords, different physicists determine the same ΛQCD also if they measure the reference couplingat different energies!

Verification:

µd

dµΛQCD = µ exp

(− 4π

2β0αs(µ)

)

+ µ exp

(− 4π

2β0αs(µ)2

)× 4π

2β0αs(µ)µ

d

dµαs(µ)

= 0 (8.8)

since

µd

dµαs(µ) =

2g

4πµ

dg

dµ=

2g

4πβ(g) = −2β0

4παs(µ)2 (8.9)

in the given approximation.

Note: all “derived” QCD quantities, like αs(µ) or ΛQCD , extracted by confronting a pQCDcalculation (theory) with a measured quantity (experiment) depend on the approximation used.This is of course generally true when confronting theory and experiment. However, because of

153

the relatively poor convergence of pQCD (much larger coupling strength in QCD vs. QED) instrong interaction physics numerical values for derived quantities usually depend substantially onthe renormalization scheme and the order of perturbation theory used.

b) Next to leading order O(α2s)

The solution of the RG equation with the next to leading order beta function Eq. (8.5) we mayeasily find (see c) below)

4π

β0 αs(µ)− β1β20

ln

(4π

β0 αs(µ)+β1β20

)

= lnµ2/µ20 +4π

β0 αs(µ0)− β1β20

ln

(4π

β0 αs(µ0)+β1β20

)

= lnµ2/Λ2

with

Λ = ΛQCD = Λ(Nf )

MS= µ exp

(−1

2

(4π

β0 αs(µ)− β1

β20

ln(

4πβ0 αs(µ)

+ β1β20

))). (8.10)

Again, Λ is RG–invariant in the given approximation and Λ ≤ µ as (β0 > 0 , αs(µ) > 0).

Remember: ΛQCD depends on

1) the renormalization scheme (MS , MOM, · · ·)

2) the number of flavors Nf = 3, 4, 5, 6

3) further approximations made when solving the RG–equation (see below).

An approximation made frequently is an expansion in powers of the large logarithm

L.= lnµ2/Λ2

when µ ≫ Λ. Note that this is a further approximation, beyond truncating the perturbativeseries. One easily works out

αs(µ) =4π

β0L

1 +

β1β20

ln(L + β1

β20

)

L+ · · ·

−1

.

For the effective mass at next to leading order the solution of the RG–equation takes the form

mi(µ) = mi(µ0)r(µ)

r(µ0)

with

r(µ) = exp−2(γ0β0

ln 4πβ0 αs(µ)

+(γ0β0− 4γ1

β1

)ln(1 + β1

β0

αs(µ)4π

)(8.11)

and an expansion in L−1 yields

mi(µ) = mi

(L

2

)− 2γ0β0

(1− 2β1γ0

β30

L+ 1

L+

8γ1β20L

+ · · ·)

.

Like ΛQCD the mI are RG–invariant (serving as integration constants). As a corollary we maycheck that

m(Nf )i (µ)

m(Nf )k (µ)

=mi

mk.

154

c) Higher orders We first rewrite the RG–solution for g in terms of αs:

lnκ =

g(κ)∫

g(1)

dg′

β(g′)⇒ lnκ2 =

αs(µ)∫

αs(µ0)

dα

β(α)(8.12)

where

β(α) =gβ(g)

4π= −b0 α2 − b1 α3 − · · ·

with

b0 =β04π

, b1 =β1

(4π)2

We may write

1

β= − 1

b0 α2

1

1 + b1b0α

+

(1

β(α)+

1

b0 α2 + b1 α3

),

where the first two (leading plus next–to–leading) terms may be written as

1

β(α)(2)=

1

−b0 α2 − b1 α3=

1

−b0 α2

1

1 + b1/b0 α

= − 1

b0 α2+b1b20

(1

α− 1

α+ b0b1

).

Integration then yields

lnκ2 =

(1

b0 α− b1b20

ln

(1

b0 α+b1b20

)− b1b20

lnb20b1

)∣∣∣∣αs(µ)

αs(µ0)

+F (αs(µ))− F (αs(µ0))

with

F (αs) =

αs∫

0

dα

(1

β(α) + 1b0 α2+b1 α3

).

Note that the last integrand is

β(α) + b0 α2 + b1 α

3

(b0 α2 + b1 α3)β(α)= O(1) for small α

i.e, it is regular at α = 0. Thus the third term of Eq. (8.13) contributes

αs(µ)∫

αs(µ0)

dα (· · ·) =

0∫

αs(µ0)

dα −0∫

αs(µ0)

dα

(· · ·) = F (αs(µ))− F (αs(µ0)) .

In contrast to the singular leading term 1/αs and next–to–leading term lnαs, the function F (αs)is regular and has a convergent power series expansion with F (αs) = O(αs) as αs → 0.

155

The result thus may be written in as

f(αs(µ)).=

4π


ln

(4π

β0 αs(µ)+β1β20

)+ F (αs(µ))

= lnµ2/µ20 + f(αs(µ0)).= lnµ2/Λ2 . (8.13)

Note that f(αs(µ0)) = lnµ20/Λ2, and that the extra higher order term (contributions to β beyond

2–loop) just amount to a redefinition of ΛQCD:

4π


ln

(4π

β0 αs(µ)+β1β20

)

︸︷︷︸lnµ2/Λ2

(2)(µ)

+ F (αs(µ))︸︷︷︸ln Λ2

(2)(µ)/Λ2

The consequence of the analysis is the following:

1) Λexact is invariant under the exact RG, i.e., strictly renormalization independent

2) since F (αs(µ)) depends on the renormalization scale µ the finite order pQCD Λ’s like Λ(2)(µ)must depend on µ.

Let us stress once more: the “measured” reference coupling αs(µ0) is obtained by comparing anobservable, an experimentally determined number, with a scheme dependent finite order pQCDprediction (usually RG improved), and thus itself depends on the precise theory input. For givenαs(µ0), furthermore ΛQCD and αs(µ) depend in a essential manner on the approximation used.

Note: for self-consistency one should strictly use the same scheme and the same order of RG–improved pQCD for both extracting αs(µ0) from the data and for the prediction of αs(µ).

The following example may illustrate the sensitivity to higher order effects: we consider QCDwith Nf = 3 flavors a scale µ0 = 2 GeV in the MS scheme and going from two to three loops.We then have

β0 = 11− 2

3Nf = 9

β1 = 102 − 38

3Nf = 64

β2 =2857

2− 5033

18Nf +

325

54N2f =

3863

6≃ 643.833 . . .

and thus

β(α) = −b0 α2 − b1 α3 − b2 α4 − · · ·

with

b0 =β04π≃ 0.716 ; b1 =

β1(4π)2

≃ 0.405 ; b2 =β2

(4π)3≃ 0.324 .

The regular higher order (here 3–loop) correction term stems from integrating

1

β(α)+

1

b0 α2 + b1 α3=β(α) + b0 α

2 + b1 α3

β(α) (b0 α2 + b1 α3)≃ b2b20

and thus

F (αs(µ0)) ∼αs(µ0)∫

0

dαb2b20

=b2b20αs(µ0) = ln

Λ2(2)

Λ2(3)

≃ 0.203 .

156

The reference scale chosen in our example µ0 = 2 GeV is close to the τ lepton mass Mτ =(1.77684 ± 0.00017) GeV at the boarder of applicability of pQCD. Here we may use αs(Mτ )extracted from hadronic τ decays: αs(µ0) ≃ αs(Mτ ) ≃ 0.322 ± 0.030. We then obtain

Λ(2)

Λ(3)= exp

(b22b20

αs(µ0)

)≃ 1.11 ,

which is a 11 % correction. More precisely, using the exact value F (αs(µ0)) ∼ 0.170± 0.013, thecorrection is 9 %. As a typical result we find for Nf = 3

Λ(3)

MS= (355 ± 55) MeV , while Λ

(2)

MS= (387 ± 66) MeV

Present day pQCD analyses generally use the known 4–loop RG results (van Ritbergen, Ver-maseren, Larin 1997, Chetyrkin 2004, Czakon 2004). Numerical routines including these resultsare available ( e.g. RHAD by Harlander, Steinhauser 2002)

d) the non-perturbative αs(µ) from lattice QCD:Discretized QCD on a d = 4 Euclidean “space–time” lattice allows to calculate αs in a non–perturbative manner by numerical simulation. The QCD path integral is directly evaluated bymeans of Metropolis Monte Carlo integration (with appropriate importance sampling of the gaugeconfigurations). Results from the Alpha Collaboration translated into the MS scheme are sownin Fig. 8.5.

Exercise: Work out the 2–loop result for the mass renormalization factor Eq. (8.11) using thetechniques outlined above.

8.7 How does the RG work for asymptotically free theories?

Because of its importance for principal as well as practical purposes it may be helpful to illustrateonce more in a slightly different way what is the essence of RG improved perturbation theory.Let us consider as a simple example the two–point function G(2)(p; g, 0, µ) of the massless theory.Perturbation theory at k-th order yields a leading contribution

Ck

(αsπ

lnp2

µ2

)k; αs

.=g2

4π

and hence

G(2)(p; g, 0, µ) =

∞∑

k=0

Ck

(αsπ

lnp2

µ2

)k+ · · ·

Now let p = κp0 where p0 = O(µ). For large κ≫ 1 we obtain

αs lnp2

µ2≃ αs lnκ2 + αs ln

p20µ2

with a negligible second term, and αs lnκ2 acts as an effective coupling:

G(2)(κp0; g, 0, µ) =

∞∑

k=0

Ck

(αsπ

lnκ2)k

+ · · · =∞∑

k=0

Ck

(αs(κ)

π

)k+ · · ·

Obviously the naive perturbation series only makes sense if

αs lnκ2 < 1 (better ≪ 1) ,

157

strong coupling

(confinement)

⇑ =⇒ asymptotic freedom

Figure 8.5: The strong interaction coupling αs(µ) as a function of the renormalization scaleobtained non–perturbatively from lattice QCD vs. the 3–loop pQCD prediction. The non-

perturbative lattice result for ΛQCD for Nf = 2 is Λ(2)

MS= 245(16)(16) MeV. From Ref. [1]

(ALPHA Collaboration).

i.e., for large κ the naive perturbation expansion breaks down, in spite of asymptotic freedomwhich tells us that the effective coupling should become small. The latter fact can only beobtained by applying the RG. As a solution of the latter we have

G(2)(κp0; g, 0, µ) = κ2d0G a(κ, g)2 G(2)(p0; g(κ), 0, µ)

with

αs(κ) =αs

1 + αsβ04π lnκ2

≪ 1

which means that up to calculable singular factors κ2d0G a(κ, g)2 we obtain

G(2)(p0; g(κ), 0, µ) =∞∑

k=0

Ck

(αs(κ)

πlnp20µ2

)k+ · · · ,

which actually converges the better the larger κ is! The important point here is that ln p20/µ2 =

O(1) is kept fixed. Note that the order by order (i.e. not RG resummed) pQCD result does notsuggest the correct structure of the theory in the asymptotic region.

158

Exercises I: Section 8

① Consider the 2–loop QCD β–function as a function of αs and discuss its dependence on thenumber of quark flavors Nf .

② Study the convergence properties of the RG-resummation Eq. (8.6) (see also Subsect. 8.7).

③ Elaborate on the question ”Where is the best scale to determine the reference strong in-teraction coupling constant?”. Compare the situation with the fine structure constant inQED.

④ Work out the 2–loop result for the mass renormalization factor Eq. (8.11) using the tech-niques outlined in the text.

8.8 Decoupling of heavy states and the decoupling theorem

Mass independent renormalization schemes are not very physical, because they do not take intoaccount the physical mass dependence of full QCD with Nf = 6. In a more physical renormaliza-tion scheme like on-shell renormalization in QED one observes automatic decoupling of heavierstates and fields. For example one can do very precise Lamb–shift calculations in atomic spectrawithout any knowledge about the existence of the muon or even of the τ–lepton. Similarly, lowenergy hadron physics like the existence of the pions and their properties certainly do not dependon the existence of the charm, bottom or top quark. Even the much lighter strange quark whichis much closer to the up and down quarks, but still clearly heavier, has no substantial effect onthe pion system. Now, applying massless renormalization, means all quarks are taken massless infirst place, which of course would change real physics at the low scales. Unfortunately, in QCDwe cannot apply in a physical sound way on–shell renormalization, because in pQCD a physicalS–matrix does not exist, the physical states are hadrons not quarks and gluons. So what to do?Formally, one can try to implement to some extent an on-shell renormalization procedure, how-ever, as in QED already, infrared singularities spoil also this possibility because of the masslessgluons. An infinitesimally small gluon mass allows to get IR regulated answers for intermediatequantities which are not infrared save by themselves. One advantage of such a scheme would bethat one would actually obtain automatic decoupling, because each state or field is taken intoaccount only at its intrinsic scale.

For a long time is has been taken for granted that the physics at low energies is not influencedin any sense by the possible existence of particles at much higher energy scales. In a quantumfield theory though this is not trivial because heavy particles in general contribute to quantumcorrections as they appear in loops as virtual particles. Let us consider QCD, specifically. In theQCD Lagrangian only quarks have mass. However, the scale of quark masses covers a very largerange from a few MeV of the up and down quarks to almost 200 GeV for the top quark. Thestrong coupling constant is a parameter entering the QCD Lagrangian independent of the quarkmasses. In contrast to electroweak theory where couplings and masses, generated by the Higgsmechanism, are strongly correlated. Unlike in electroweak theory, in QCD we can choose theparticle masses arbitrary heavy at a fixed value of the coupling constant. As seen from the worldof the light particles a heavy quark can contribute to the light quark and gluon amplitudes at the2–loop level only, by diagrams of the type shown in Fig. 8.6. In QCD like theories29, at the level ofthe bare theory, masses of a heavy quark for example only enter in the denominator of the quarkpropagator. The latter, in principle, may be expanded as 1/(q/ −mt) = (q/ + mt)/(q

2 − m2) ∼−1/mt (1+q//mt)(1+q2/m2

t+· · ·). However, such an expansion makes little sense in a loop integral

29the other well known example is QED, where the leptons play the role of the quarks

159

Figure 8.6: Heavy virtual particles (fat lines) in light field (thin lines) amplitudes. In the “lightsector’ the heavy fields only appear “integrated out”.

where q, or a linear combination of q with other momenta, is a loop momentum to be integratedover. Indeed, the expansion produces fake UV singularities. This shows that a slightly moresophisticated consideration is necessary. Such an analysis was performed by Symanzik 1973 [13]and Appelquist and Carazzone 1975 [14]. The result is known as the Appelquist–Carazzonetheorem, which we present in the following.

The question is how heavy particles of mass Mi ≫ µ manifest themselves in the energy range√s ≤ µ. An example are the RG–coefficients in the MS scheme:

β0 = 11− 23 Nf ; γ0 = 2

β1 = 102− 383 Nf ; γ1 = 101

12 − 518 Nf

What is Nf to be taken here? In a mass independent renormalization scheme each flavor yieldsa mass independent contribution, thus for full QCD Nf = 6. One could ask the question: is itpossible to determine Nf from the experimentally determined evolution of αs(µ) as a function ofµ? In massless renormalization the theory at low energies seems to look like full QCD far beyondthe top mass scale, beyond 0.5 TeV say. Is this reasonable? The decoupling theorem helps answersuch questions.

The Decoupling Theorem: if all external momenta of a process or a corresponding amplitude aresmall relative to the mass M of a heavy state, then the Green functions of the full theory whichhowever exhibit light fields only as external lines differ from the theory which has no heavy fieldsat all only by finite renormalizations of couplings, masses and fields of the light theory, up toterms which are suppressed by inverse powers of the heavy mass. Thus further corrections are ofthe form (µ/M)x with x ≥ 1.

It means that only the renormalization subtraction constants are dependent on M (logarithms)and this M–dependence gets renormalized away by physical subtraction conditions. In otherwords, effects from degrees of freedom which are not “excited” at a given scale µ can be dropped.Essentially, we may always take Nf to be the number of flavors with masses mi < µ.

In view of the decoupling theorem what can we say about the MS renormalization scheme appliedto full QCD? In addition to typical large logarithms ln s/µ2 (µ the renormalization scale ,

√s a

typical physical process energy) also even larger logarithms lnM2/µ2 or lnM2/s are showing upwhich are fake (an artifact of massless renormalization) and potentially affect the convergence ofthe perturbation expansion. Such a scheme thus is not recommendable to do physics at scalesµ,√s ≪ M . It is thus certainly more adequate to drop the heavy field from the consideration

from the very beginning and consider an effective theory exhibiting the light fields only. Thedrawback will be that different effective theories are needed, the parametrizations of which haveto be made consistent by hand at the “matching points” (see below).

160

8.9 MOM scheme, automatic decoupling

Momentum (MOM) subtraction schemes are a generalized version of the on–shell scheme con-ventionally used in QED (more generally in electroweak theory), where quasi physical S-matrixelements are subtracted to determine the physical meaning of the parameters. This was the routewe followed in Sect. 7, when discussing one–loop renormalization besides the MS renormalizationalso considered there. MOM schemes are mass–dependent, like the on–shell scheme but avoidIR singularities by not subtracting directly on the mass–shell (which leads necessarily to IR sin-gularities in the wave–function renormalization factors) but at some Euclidean reference scale.More precisely the conditions applied read:

’ Πren(k2)∣∣k2=−µ2 = 0

’ Σ1 ren(p2)∣∣p2=−µ2 = Σ2 ren(p2)

∣∣p2=−µ2 = 0

’ bren(q2)∣∣q2=−µ2 = 0

’Γ

′ρσµkli ren(k, p, q)

∣∣∣sp:−µ2

= 0 MOMI

’Γ

′µi,bc ren(k, p, q)

∣∣∣k2=p2=q2=−µ2

= 0 MOMII

’Γ

′µi,kl ren(k, p, q)

∣∣∣k2=p2=q2=−µ2

= 0 MOMIII .

The prime (’) indicates that the non–trivial higher order part only is to be taken. By “sp” wedenote the symmetry point, for a symmetric 3–point function (like ggg) sp: pipj = 1

2 (3 δij−1)µ2,for a symmetric 4–point function (e.g gggg) sp: pipj = 1

3 (4 δij − 1)µ2. The expressions areobviously symmetric and take into account 4–momentum conservation of the three and fourmomenta,respectively.

In a MOM scheme we obtain mass dependent renormalization constants

ZiMOM = Zi(g, ξ,mi

µ)

and the RG takes the form

µ ∂∂µg = β(g, ξ, mi

µ ) ; β = −β0 g3

16π2 + · · ·µ ∂∂µmi = −γ(g, ξ, mi

µ )mi ; γ = γ0g2

4π2 + · · ·

of a coupled system, which can be solved by numerical integration. We should stress that MOMschemes using off–shell subtraction (to avoid IR problems) per se are leading to gauge dependenteffective parameters and gauge dependent RG coefficients. Once more we encounter here a situ-ation where unphysical redundancy of the theoretical formalism cannot be avoided. There is noreal problem though, as only true observable quantities must be free of all these diseases.

It is not too difficult to investigate explicitly the mass effects in the renormalization constants.Physical mass effects in the coupling constant renormalization defined via the gqq–vertex weactually have calculated in Sect. 7.4. In QCD like in any non-Abelian gauge theory the gaugecoupling constant gs may be defined in different ways. At the tree level we may choose the ggg

161

(MOMI), the gqq (MOMII) or even the unphysical gηη (MOMIII) vertex. Let us compare thefirst two here. At one loop in the Landau gauge ξ = 0 one obtains

β0 = 11− 2

3

∑

f

K

(m2f

µ2

); γi0 = 2 ·G

(m2i

µ2

)

where in the MOMI we have

K = Kggg(x2) = 1− 18x2 +

36x4√1 + 4x2

ln

√1 + 4x2 + 1√1 + 4x2 − 1

−2

1/3∫

0

dy ρ(y)

[x2

y + x2− 3x4

(y + x2)2

]

with

ρ(y) =

2√

3 arctan(√

3 1−√1−4 y1+3√1−4 y

); 0 ≤ y ≤ 1/4

2π√3

; 1/4 ≤ y ≤ 1/3 .

For MOMII we have the simpler expression

K = Kgqq(x2) = 1− 6x2 +

12x4√1 + 4x2

ln

√1 + 4x2 + 1√1 + 4x2 − 1

.

The mass renormalization we may get from the quark self–energy function

G(x2) = 1− x2 ln

(1 +

1

x2

).

We easily evaluate

K(x2) , G(x2) → 1 for m2i ≪ µ2 : β0 → β0MS(Nf )

K(x2) , G(x2) → 0 for m2i ≫ µ2 : β0 → β0MS(Nf − 1) .

For the mass renormalization in the heavy limit we obtain

γi0mi∂

∂mi→ 1

2

µ2

m2i

mi∂

∂mi→ 0 .

The behavior of the physical threshold functions which switch on or off the contribution fromthe heavy field is depicted in Fig. 8.7. The MS scheme in contrast requires decoupling by hand,which means

K(x2) , G(x2) → Θ(µ2 −m2i ) .

with the Heaviside Θ–function. At a given scale√s = µ, e.g. an energy E =

√s involved in a

process, as an approximation one may consider the theory L(Nf )QCD, with Nf the number of “excited”

flavors: 2µi<∼µ (i = 1, 2, · · · , Nf ) which yields a particular parametrization (convention!) validin the range 2mNf

<∼µ<∼2mNf+1. This is a parametrization which optimizes the convergence ofpQCD in that particular range. Crossing a new threshold at µNf+1 ≃ 2mNf+1 we go to a neweffective theory

L(Nf )QCD → L

(Nf+1)QCD (8.14)

162

Figure 8.7: Threshold behavior of MOM RG coefficient functions.

with matching conditions [15] (to be discussed below) at the new threshold. The thresholdstypically are (not very well defined)

qq − threshold typical mass Nf → Nf + 1

(s0)ud ≃ 4m2π (≃ 0.1 GeV2) 2

(s0)ss ≃ 4m2K (≃ 1.0 GeV2) 2→ 3

(s0)cc ≃ M2J/ψ (≃ 10. GeV2) 3→ 4

(s0)bb ≃ M2Υ (≃ 100 GeV2) 4→ 5

(s0)tt ≃ 4m2t ≃ (350 GeV)2 5→ 6 .

Often in the past as a condition the transition (8.14) has been supplemented by a requirementof continuity of the parameters when crossing the threshold: e.g. continuity of αs(µ) at µ0 =√sNf+1, which yields Λ

(Nf+1)

MSwhen Λ

(Nf )

MSis known: (this prescription is ad hoc and know

accepted to be obsolete!)

lnµ0

Λ(Nf )

MS

=4π

2β0αs(µ0)

(1− αs(µ0)

4π

β1β0

ln

(4π

β0αs(µ0)+β1β20

))

lnµ0

Λ(Nf+1)

MS

=4π

2β+0 αs(µ0)

(1− αs(µ0)

4π

β+1β+0

ln

(4π

β+0 αs(µ0)+

β+1β+20

))

where(β+0 , β

+1 ) = (β0, β1)|Nf→Nf+1 .

Results are displayed in Fig. 8.8.

163

Figure 8.8: Matching by continuity (obsolete!): input αs(MZ) = 0.1189; Λ(5)

MS= 223 MeV, Λ

(4)

MS=

342 MeV, Λ(3)

MS= 435 MeV. Furthermore: αs(2mb) = 0.1817 and αs(2mc) = 0.2569. Thresholds:

3→ 4 at 3.2 GeV and 4→ 5 at 9.6 GeV.

8.10 Nf - flavor effective QCD and matching conditions

Given the problems with MOM scheme renormalization, the alternative way, which is commonlyused by QCD practitioners, is the MS scheme together with decoupling by hand. As QCD byitself is a vector–like theory (no axial currents participate in QCD dynamics, although weak axialcurrents of course play a role by interference between weak and strong interaction processes)QCD is well defined for whatever the number of flavors we take. As the physical thresholds µthres(again in QCD the notion of a threshold is not very well defined) we may either consider thequark masses mc ≃ 1.3 GeV, mb ≃ 4.2 GeV and mt ≃ 171 GeV or the thresholds for charm,bottom or top pair–production in e+e−–annihilation, which lie at about twice the correspondingquark mass, or more physically vector meson masses MJ/ψ ≃ 3.1 GeV and MΥ ≃ 9.5 GeV, whichare the lowest flavor neutral cc and bb states, respectively. The top is so unstable that it decaysweakly before it is able to form a hadronic bound state and one uses naturally 2mt as a pairproduction threshold.

What one then does is the following: at the energy region between two successive thresholdsMNf

and MNf+1 one takes Nf–flavor QCD as an effective theory, i.e., the first Nf quarks arethe light ones (theory of the light fields) while the other heavier fields are completely ignored.So, for scales E =

√s with MNf

< E ≤ MNf+1 one solves the RG-equations for the Nf flavors,for MNf+1 < E ≤ MNf+2 with Nf + 1 and so forth. This is what we have done before when wediscussed the running of QCD parameters. The crucial question is what is the proper procedureto continue the effective theory when we pass a threshold and go form Nf to Nf + 1 flavors. Inother words, we are looking for the appropriate matching conditions30.

30One should keep in mind that in many of the earlier papers the matching has not been implemented in aconsistent way. For example, as mentioned earlier, just requiring continuity of the parameters at µthres is not aproper matching prescription.

164

What it means is that we are comparing two different theories which exhibit their own parametersand renormalization constants. The condition is that at the threshold the two theories have toagree. In pQCD one set of parameters is a perturbative series in the other set of parameters:(see [17])

aNf+1(µthres) = aNf(µthres)

[1 +

∞∑

k=1

Ck(x) akNf(µthres)

]

mq,Nf+1(µthres) = mq,Nf(µthres)

[1 +

∞∑

k=1

Hk(x) akNf(µthres)

]

where the coefficients depend on x = lnµ2thres/m2 where m is an RG-invariant mass of the heavy

quark which is switched on at the threshold.

All logarithmic terms are actually fixed by the fact that aNfand mq,NF

satisfy the RG equationsof the effective theory with Nf flavors:

da

d lnµ2= β(a) = −a2

(β0 + β1 a+ β2 a

2 + β3 a3)

+O(a6)

d lnmq

d lnµ2= γm(a) = −a

(γ0 + γ1 a+ γ2 a

2 + γ3 a3)

+O(a5) ,

where µ is to be taken in units of a RG invariant reference scale, like ΛQCD or a RG invariantmass m. The coefficients of the QCD beta function have been calculated recently in the MSscheme up to four loops31 (see [2]– [12] )

β0 =1

4

[11− 2

3Nf

], β1 =

1

16

[102 − 38

3Nf

],

β2 =1

64

[2857

2− 5033

18Nf +

325

54N2f

],

β3 =1

256

[(149753

6+ 3564ζ3

)−(

1078361

162+

6508

27ζ3

)Nf

+

(50065

162+

6472

81ζ3

)N2f +

1093

729N3f

], (8.15)

and also the coefficients of the quark-mass anomalous dimension have been calculated at the sameorder,

γ0 = 1 , γ1 =1

16

[202

3− 20

9Nf

],

γ2 =1

64

[1249 +

(−2216

27− 160

3ζ3

)Nf −

140

81N2f

],

γ3 =1

256

[4603055

162+

135680

27ζ3 − 8800ζ5

+

(−91723

27− 34192

9ζ3 + 880ζ4 +

18400

9ζ5

)Nf

+

(5242

243+

800

9ζ3 −

160

3ζ4

)N2f +

(−332

243+

64

27ζ3

)N3f

]. (8.16)

31In this section we adopt a slightly different normalization for the RG coefficients, the one used in the relevantpapers on the subject, which are quoted in the references to this chapter. All factors 1/π are absorbed intoa = αs/π, all remaining factors into the coefficients.

165

Here ζn is the Riemann zeta-function ( ζ2 = π2/6, ζ3 = 1.202056903 . . ., ζ4 = π4/90 and ζ5 =1.036927755 . . .) and Nf is the number of quark flavors with mass lower than the renormalizationscale µ.

Taking the derivatives of the matching equations Eqs. (8.15), denoting derivatives by dots ( ˙ )and Nf + 1 quantities by hats (ˆ), we obtain

˙a = a [1 +∑

k

Ck ak] + a [

∑

k

Ck ak] + a [

∑

k

k Ck a ak−1]

˙m = m [1 +∑

k

Hk ak] +m [

∑

k

Hk ak] +m [

∑

k

kHk a ak−1] (8.17)

A comparison of the coefficients of the powers ak then yields

a2 : C1 = β0 − β0a3 : C2 = β1 − β1 + 2 (β0 − β0)C1

a4 : C3 = β2 − β2 + (2β1 − 3β1)C1 − β0 C21 + (3β0 − 2β0)C2

...

ma : H1 = γ0 − γ0ma2 : H2 = γ1 − γ1 − γ0C1 + (β0 + γ0 − γ0) H1

ma3 : H3 = γ2 − γ2 − 2 γ1 C1 − γ0 C2 + (β1 + γ1 − γ1 − γ0 C1) H1 + (2β0 + γ0 − γ0) H2

...

or, inserting the coefficient functions,

C1 =1

6

C2 =19

24+

1

3C1

C3 =7387

1728− 325

1728Nf −

(4− 19

24Nf

)C1 −

(31

12− 1

6Nf

)C21 +

(37

12− 1

6Nf

)C2

...

H1 = 0

H2 =5

36− C1 +

(11

4− 1

6Nf

)H1

H3 =1697

1296+

5 ζ36

+35

648Nf −

(293

36− 5

18Nf

)C1 − C2

+

(469

72− 19

24Nf

)H1 − C1H1 +

(11

2− 1

3Nf

)H2

... (8.18)

The system can be solved and has a unique solution up to the integration constants. The latterhave to be determined by a direct calculation of the self-energies and vertex functions whichdefine Zg and Zm. In fact for the leading terms C1 and H1 the integration constants vanish,i.e. c1,0 = 0 and h1,0 = 0. This we will use in the following. Integration of the above system ofequations for Ck yields, (x = lnµ2/Λ2)

C1 =1

6x

166

C2 = c2,0 +19

24x+

1

36x2

C3 = c3,0 + c2,0

(37

12− 1

6Nf

)x+

7387

1728x− 325

1728Nf x+

511

576x2 +

1

216x3

...

H1 = 0

H2 = h2,0 +5

36x− 1

12x2

H3 = h3,0 + h2,0

(11

2− 1

3Nf

)x− c2,0 x

+1697

1296x +

5 ζ36x +

35

648Nf x−

299

432x2 − 35

216x3 +

1

108Nf x

3

...

(8.19)

As a result we learn that most of the matching terms are just determined by the RG. Not soeasy is the evaluation of the integration constants. Their determination requires to calculatethe renormalization constants gs0 = Zg gs, mq0 = Zmmq, ψq0 =

√ZF ψq, G

µa0 =

√Z3G

µa and

η0 =√Z3 ηa once in the full and once on the effective theory, corresponding quantities we denote

by a prime, such that the effective parameters read

α′s(µ) =

(ZgZ ′g

ζ0g

)2

αs(µ) = ζ2g αs(µ)

m′q(µ) =ZmZ ′m

ζ0mmq(µ) = ζmmq(µ)

in terms of the full QCD parameters. The relative renormalization factors ζi are finite andmediate the corresponding change of the renormalization scheme. What we need to know is whathappens when we take the heavy field masses in full QCD to go to infinity and compare theresult with what one obtains for the same Green functions in the Nf - effective theory. Thisprovides a matching up to terms of order O(1/M). As usual, for simplicity, we adopt dimensionalregularization together with the MS scheme and the convention Tr1 = f(d) = 4 for the trace ofthe unit matrix in Dirac spinor space. This implies c1,0 = d1,0 = 0, which we have used alreadyabove. The other non-trivial constant depend on the choice of the RG invariant reference scale.

If the RG-invariant MS mass is used as a reference scale, that is m = m(m) the coefficients oneobtains are [10,16,18]

c2,0 = −11

72, c3,0 =

82043

27648ζ3 −

575263

124416+

2633

31104nf , d2,0 = − 89

432. (8.20)

If the pole mass is used as a reference scale, that is m = M the coefficients one obtains are [16,18]

c2,0 =7

24

c3,0 =80507

27648ζ3 +

1

9ζ2 (2 log(2) + 7) +

68849

124416− nf

9

(ζ2 +

2479

3456

). (8.21)

while d2,0 does not change, and d3,0, when known, has to be shifted by +10/27.

In [18] the relations Eq. 8.20 are directly calculated and given in compact numerical form. Thenumber of light flavors is denoted by nl (corresponding to our Nf above). If the RG-invariant

167

Figure 8.9: Matching via observables in different orders of the perturbation expansion. In-put: αs(MZ) = 0.1189, mc = 1.3 GeV, mb = 4.2 GeV. The dashed vertical lines markthe threshold positions. The discontinuities are quite substantial and clearly visible. Now:

Λ(5)

MS= 219 MeV, Λ

(4)

MS= 330 MeV, Λ

(3)

MS= 411 MeV. Furthermore: αs(2mb) = 0.1891 and

αs(2mc) = 0.2887, while the OS masses read Mc = 1.15 GeV and Mb = 4.48 GeV.

MS mass µh = m(m) is used as a reference scale one obtains

(ζMSg )2

(µ=µh)= 1 + 0.152778 a2s(µh) + a3s(µh) (0.972057 − 0.0846515 nl)

+ a4s(µh)(5.17035 − 1.00993 nl − 0.0219784 nl

2), (8.22)

1

(ζMSg )2

(µ=µh)= 1 − 0.152778 a′ 2s (µh) + a′ 3s (µh) (−0.972057 + 0.0846515 nl)

+ a′ 4s (µh)(−5.10032 + 1.00993 nl + 0.0219784 nl

2). (8.23)

If the perturbative pole mass m = M is used as a reference scale, the result is given by

(ζOSg )2

(µ=M)= 1 − 0.291667 a2s(M) + a3s(M) (−5.32389 + 0.262471 nl)

+ a4s(M)(−85.875 + 9.69229 nl − 0.239542 nl

2), (8.24)

1

(ζOSg )2

(µ=M)= 1 + 0.291667 a′ 2s (M) + a′ 3s (M) (5.32389 − 0.262471 nl)

+ a′ 4s (M)(86.1302 − 9.69229 nl + 0.239542 nl

2). (8.25)

After proper matching of Nf - to Nf + 1-flavor effective QCD, the result for the running MScoupling is displayed in Fig. 8.9, which compares with Fig. 8.8. The discontinuities across thematching points at a given order of the perturbation expansion are a result of the change of theeffective Lagrangian (discontinuous change of Nf ). In fact the discontinuities are small relative to

168

Figure 8.10: The discontinuous behavior of αMSs across the charm threshold chosen at 3.2 GeV.

169

the running effects as is shown here for passing the charm threshold in Fig. 8.10. In this approachthe matching observable is a given number at each selected matching point, such that the “re-

predicted” observable is continuous, while αNfs (E) is a scale dependent effective parameter of the

Nf -flavor effective theory, which, depending on the renormalization scheme, in general is not anobservable itself.

8.11 Renormalization Scheme Dependence

We already have encountered a variety of renormalization schemes (RS), the choice each wasmotivated by formal or physical arguments. Different RS’s represent different parametrization ofa given theory and at the end of the same physics. In finite order perturbation theory, a changeof a RS formally correspond to a reshuffling of higher order terms in the prediction of variousquantities. Predictions of physical quantities of course should not depend on the specific choice ofthe input parameters and they in fact do not if we include all orders of the perturbation expan-sion. Actually, the reparametrization invariance is inferred by renormalization group invariance.However, practical perturbative calculations are approximations obtained by truncation of theperturbation series. The accuracy of the finite order approximations depends on the choice ofthe input parameters i.e. finite order results are scheme dependent. Let us illustrate this pointby an example: Suppose we compute a matrix element M in the αs-scheme (I) to one-loop orderyielding a result

M (1) = αns C [1 + b αs].

Now, suppose we calculate the same quantity in the α′s-scheme (II) which amounts to a replace-ment of αs by α′s = αs (1 + ∆αs) i.e. to one-loop order α′s = αs[1 + aαs] and

M ′(1) = α′ns C [1 + b′ α′s].

Inserting α′s we get

M ′(1) = M (1) + δM

with b′ = b− na and

δM = αns C

[(n(n− 1)

2a2 + (n+ 1)ab′)α2

s + · · · + an+1b′αn+2s

].

Thus the result differs by δM . If we do not actually calculate the higher orders

δM = M ′(1) −M (1)

must be considered as an uncertainty due to unknown higher order effects. In QCD in particular,RG resummation is mandatory and reduces the SD substantially. Nevertheless, we are alwaysdealing with approximations which also in RG improved finite order pQCD are never RG invari-ant. The residual µ–dependence of a prediction is an important measure for the quality of anapproximation. The study of the scheme dependence of resummation improved results is a wayto estimate missing higher order contributions (educated guess). Of course only an actual n-loopcalculation can tell us what the full n-loop answer is.

Let us summarize the content of this subsection by the following conclusions:

• If a physical quantity is calculated with different input parameters the answer is the sameif we calculate it to arbitrary high orders. Thus RS independence only applies for the exactresult.

170

However:

• Calculating a quantity to a given order the omitted higher order terms differ for differentparametrizations. This leads to a scheme dependence of the result (approximate) due todifferent truncation errors. Thus RS dependence is is the result of omitted higher orderterms which depend on the RS.

• Differences can also be due to different resummation prescriptions (see below).

• Note that the “convergence” of the asymptotic perturbation expansion may be substantiallyaffected by the choice of the RS (remember that the perturbation series as an asymptoticseries diverges beyond some unknown order).

Here we look at a few very simple lowest order examples where effects are particularly large (loworder=large truncation errors):

1) Various definitions of αs(µ):

MS

MS

β0, β1 identical, β2 different

MOMI

MOMII mass dependent, but also gauge dependent

MOMIII

... ???

2) Identification of µ with a physical scale√s:

If µMS =√s what is µMS, µMOM, · · · ???

For example: we have

lnµ2MS = lnµ2MS

+ γ − ln 4π

thus

µMS/µMS = expln 4π − γ

2≃ 2.66

which tells us that we should be aware of the fact that the identification of the renormalizationscale with a physical scale is fairly arbitrary (largely a commonly agreed convention!).Values for µMOM/µMS at one-loop (Celmaster, Gonsalves 1979):

Gauge Nf MOMI MOMII MOMIII

Landau 3 2.46 2.10 2.33

(ξ = 0) 4 2.16 2.09 2.33

5 1.85 2.07 2.33

Feynman 3 2.07 1.83 2.69

(ξ = 1) 4 1.79 1.80 2.73

5 1.51 1.76 2.76

171

• For different processes or observables, in general, different parametrizations are optimalwith respect to “convergence” properties.

• However, is makes not sense to consider isolated processes described in a particular scheme.To see whether the theory works requires a convention for a universal parametrization, forexample the MS one, for its simplicity and relatively good convergence properties, togetherwith the decoupling by hand prescription.

In any case: specification of the RS is indispensable, as otherwise numbers make little sense.What we always need to know:

• renormalization scheme

• identification of the scale with the scale of a process

• order of perturbation theory used

• number of flavors used (with MS )

More interesting and realistic examples of renormalization scheme dependence will be given inthe later course of these lectures.

As a last remark, let me mention that the well established “running” of the strong interactionconstant, as established by experimental results collected in Fig. 8.11, made tremendous progressthanks to the LEP experiments at CERN, especially by high precision hadron physics at theZ peak and in hadronic decays of τ leptons produced with high statistics at LEP. The energyrange spans scales form Mτ ∼ 1.8 GeV to the Z mass MZ ∼ 91.19 GeV and at LEP-II up to200 GeV. Pre LEP status of αs measurements is shown in Fig. 8.12. Besides the e+e− physics atLEP, deep inelastic eN–scattering experiments (H1 and ZEUS) with HERA at DESY (see below)contributed significantly to the progress in determining αs.

Epilogue on the RG and running couplings:

As we have learned the RG and the running of couplings and other parameters is intimatelyrelated to the UV singularities of the bare QFT. Very often, the occurrence of UV and theneed for regularization and renormalization is considered as something obsolete, a lack of havinghad the right tricks to avoid such mathematical diseases and in fact there have been a number ofproposals which technically avoid the need to deal with such singularities like the BPHZ approachor methods based on the use of Cutkosky cutting rules. From the point of view of physics we havelearned that the UV singularities are not an artifact of missing the appropriate mathematics, theymust be there for deep physics reasons, namely, that quantum effects break scaling properties.For any massless renormalizable theory dilatation invariance and hence conformal invariance isbroken by the dynamics, and there is a one–to–one correspondence of between the would–besingularities and the renormalization group coefficients. In fact it also “proves” that nature has abuilt in regularization, presumably a sort of Planck cut-off. What we observe as a renormalizableQFT is a low energy effective theory, the low energy tail of a structure which at the Planck scalehas a built in fundamental cut-off. It is well known that non–Abelian local gauge field theoriesare a natural consequence of such a scenario (see [20] and references therein).

172

Figure 8.11: Present experimental status of the QCD running coupling. Shown are RG solutionswith αs(Mτ ) extracted from τ–decays and the one with αs(MZ) measured at LEP as start values.The corresponding central values agree at the 1 σ level.

Figure 8.12: Status of αs measurements 1989, before LEP went into operation. The curves

show the best estimate for the strong coupling α(5)s (MZ) = 0.11 ± 0.01 (corresponding to Λ

(5)

MS=

140 ± 60 MeV).[Taken from [19]].

173

Exercises II: Section 8

⑤ Solve the matching relations Eq. (8.15) by power expansion in a on both sides of the equationfor the first three non-trivial terms. While the first two terms can be worked out by hand,the third conveniently may be be calculated using computer algebra (maple, mathematicaor maxima).

⑥ Verify the numerical formulae (8.22) and (8.24) by evaluating the corresponding analyticalexpressions.

⑦ Given αs(MZ) = 0.1215± 0.0012 calculate αs(Mτ ) using the 4–loop RG equations togetherwith 3–loop matching. This is the consistent combination of the two aspects: explain thisstatement (i.e. why don’t we need 4–loop matching relations as well?).

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175

8.12 General O(α2) framework for predicting R(s)

Quarks are charged particles and couple to the photon via the quark contribution to the electro-magnetic current. The latter is given by

Jγµ =∑

i

(2

3uiγµui −

1

3diγµdi

)(8.1)

where i = 1, 2, 3 labels the three quark doublets: ui

di

=

u

d

,

c

s

,

t

b

.

The Lagrangian density for strong and electromagnetic interactions is given by

Lstrong&electromagnetic = LQCD + eJγµAµ − 1

4FµνF

µν .

Since α = e2

4π = (137.036)−1 is small, it will be sufficient to consider the one–photon exchangeapproximation in many cases. In the following we denote by X a hadronic state (mesons, baryons)of appropriate quantum numbers. We then are interested in the S–matrix element

〈X|S|e+e−〉 = 〈X|T

ei∫

d4x (LQCD,int+LQED,int)|e+e−〉

of the process

e+e− → X . (8.2)

It should be stressed that we consider here pure hadronic states X only, which means a state whichis obtained (e.g. in a lattice QCD calculation) in QCD alone. The much weaker electromagneticand weak interactions are switched off, for what concerns X. Also experimentally we assume hata hadronic state can be unambiguously identified32. Additional photons radiated by hadrons arecorrections which count as O(α3) effect, which is discarded in this leading O(α2) approximation.Since α is small a systematic expansion in α is always assumed in the following. As perturbativeQCD fails at low energies, we treat strong interaction effects non-perturbatively in a first step.Later we will argue where pQCD can be applied.

Since we are interested in the process (8.2), where at this order no external photon occurs, theprocess must be second order in QED. We thus have in lowest non–trivial order O(α2)

〈X|S|e+e−〉 = − 1

2!

∫d4x1d4x2 〈X|T

LQED,int(x1),LQED,int(x2) ei

∫

d4xLQCD,int

|e+e−〉

where for our purpose (we do not consider muons and taus here)

LQED,int =(−eψeγµψe + eJγµ

)Aµ = eJµA

µ .

In addition the photons must appear contracted such that

〈X|S|e+e−〉 = −e2

2!

∫d4x1d

4x2 iDµν〈X|TJµ(x1), Jν(x2) ei

∫

d4xLQCD,int

|e+e−〉 , (8.3)

where Dµν denotes the photon propagator (see Fig. 8.1).

32Especially at low energies this is not completely trivial for states which have masses in a similar range like π+π−–and µ+µ−–pairs produced at the same time in e+e−–annihilation. At high energies hadronic states typically havehigh multiplicities (hadronic jets) and can be neatly distinguished from lepton–pair production. Higher multiplicitylepton states are of higher order in the electroweak couplings and hence are suppressed due to the small coupling.

176

γ

e−

e+

hadrons

Figure 8.1: Hadron production in electron–positron annihilation.

The product of the two currents yields four pieces

Jµ(x1)Jν(x2) =Jµlep(x1)Jνlep(x2) + Jµlep(x1)Jνhad(x2) + Jµhad(x1)Jνlep(x2) + Jµhad(x1)Jνhad(x2)

.

The electron and positron states are created by external LSZ fields ψe,ext and ψe,ext which contractwith the fields of the currents, where the LSZ states are represented by

i∫

d4y ψ(y)(iγµ←∂ µ +me)u(p−, s−) e−ip−y incoming electron

i∫

d4y e−ip+y v(p+, s+) (iγµ→∂ µ −me)ψ(y) incoming positron .

The states |e+e−〉 and |X〉 are to be considered as products in the Hilbert space

Hlep, had = Hlep ⊗Hhad ,

i.e.,

|e+e−〉 = |e+e−〉lep ⊗ |0〉had|X〉 = |0〉lep ⊗ |X〉had .

In detail:

〈X|Jµ(x1)Jν(x2)|e+e−〉= 〈X|0〉had〈0|Jµlep(x1)Jνlep(x2)|e+e−〉lep + 〈X|Jµhad(x1)|0〉had〈0|Jνlep(x2)|e+e−〉lep+ 〈X|Jνhad(x2)|0〉had〈0|Jµlep(x1)|e+e−〉lep + 〈X|Jµhad(x1)J

νhad(x2)|0〉had〈0|e+e−〉lep

= 2 〈X|Jµhad(x1)0〉had〈0|Jνlep(x2)|e+e−〉lep ,

because, 〈X|0〉had = 0 as well as 〈0|e+e−〉lep = 0 and by the symmetry x1, µ↔ x2, ν. Furthermore,we use the Fourier representation of the photon propagator

Dµν(x1 − x2) =1

(2π)4

∫d4q e−i q(x1−x2)Dµν(q)

and translational invariance

Jµ(x) = U(1, x)Jµ(0)U−1(1, x) = eiPx Jµ(0) e−iPx

with P the total four momentum operator. Assuming that X is an eigenstate of momentum pXwe may use

e±iPx|0〉 = |0〉e−iPx|X〉 = e−i pXx|X〉

e−iPx|e+e−〉 = e−i pe+e−x|e+e−〉 ,

177

such that altogether we have

〈X|S|e+e−〉 = −e2 i

(2π)4

∫d4q Dµν(q) · 〈X|Jµhad(0)|0〉〈0|Jνlep(0)|e+e−〉

·∫

d4x1 d4x2 e−iqx1 eiqx2 eipXx1 e−ipe+e−x2

= −i e2 (2π)4 δ(4)(pX − p+ − p−) Dµν(q)× 〈X|Jµhad(0)|0〉〈0|Jνlep(0)|e+e−〉 .

We have used here that the xi integrals over the exponentials yield a product of two delta functions

(2π)4 δ(4)(pX − q) × (2π)4 δ(4)(q − pe+e−)

which allows to perform the q integration trivially, leaving the overall four–momentum conserva-tion and q is fixed to q = pX = pe+e− . Current conservation of the leptonic current qνJ

νlep = 0 and

the hadronic one qµJµhad = 0 implies that only the gµν–part of the photon propagators contributes

to the matrix element:

Dµν(q)→ −gµν1

q2 + iε.

For the relevant T–matrix element, defined by S = 1 + i (2π)4 δ(4)(∑pi) we find

Te+e−→X =e2

q2〈X|Jµhad(0)|0〉〈0|Jµ lep(0)|e+e−〉

where the leptonic matrix element is given at leading order by

〈0|Jµ lep(0)|e+e−〉 = −v(p+, s+) γµ u(p−, s−)

and therefore

T (e+e− → γ∗ → X) = −e2

q2v(p+, s+) γµ u(p−, s−) · 〈X|Jµhad(0)|0〉 . (8.4)

The total cross–section is

σh(s) = Σ

∫

Γdσ(e+e− → X, s = (p+ + p−)2)

with the differential one given by ( denoting dµ(p) = d3p(2π)3 2ω(p)

, ω(p) =√m2 + ~p 2 )

dσ =(2π)4 δ(4)(p+ + p− − pX)

2√λ(s,m2

e,m2e)

|T |2 dµ(p′1) · · ·

For unpolarized e+e−-beams, the square matrix element is given by the average over the initialspins of e− and e+ and can easily be calculated by standard techniques:

|T |2 =1

4

∑

s±

|T |2 =e4

q4ℓµν h

µν (8.5)

where

ℓµν =1

4

∑

s±

v(p+, s+) γµ u(p−, s−) (v(p+, s+) γν u(p−, s−))∗

=1

4Tr (p/+ −me) γµ (p/− +me) γν

=1

2

qµqν − q2 gµν − (p+ − p−)µ(p+ − p−)ν

178

=∑

n

2 Im

A, p1

B, p2

A, p1

B, p2

=∑

n

2 ImA, p A, p

Figure 8.2: Optical theorem for scattering and propagation.

and

hµν = 〈0|Jµ(0)|X〉〈X|Jν (0)|0〉 .

Writing√λ(s,m2

e,m2e) = s

√1− 4m2

e/s, we obtain

σh(s) =α2

2s316π2√

1− 4m2e/s

ℓµν Σ

∫

X(2π)4 δ(4)(p+ + p− − pX)hµν

where summation/integration is over the hadronic final states. The leptonic tensor ℓµν describesthe initial state and is fixed. For what concerns the hadronic part we try to be general beforewe will resort to pQCD which only applies at sufficiently high energy. At this point the opticaltheorem helps to proceed at the non-perturbative level.

Exploiting unitarity:

The optical theorem derives directly from the unitarity of the S–matrix. For T–matrix elementsthe unitarity condition reads

iT ∗if − Tfi

= Σ

∫

n(2π)4 δ(4)(Pn − Pi) T ∗nfTni , (8.6)

and the optical theorem, is obtained from this relation in the limit of elastic forward scattering|f〉 → |i〉 where

2 Im Tii = Σ

∫

n(2π)4 δ(4)(Pn − Pi) |Tni|2 . (8.7)

Graphically, this relation may be represented as shown in Fig. 8.2. For the photon propagatorit tells us that the imaginary part of the photon propagator is proportional to the total crosssection σtot(e

+e− → γ∗ → anything) (“anything” means any possible state):

ImΠ′γ(s) =s

4πασtot(e

+e− → anything) . (8.8)

In first place, applying the optical theorem to e+e−–annihilation we have

2 Im T (e+e− → e+e−)forward = (2π)4∑

QN ′s

∫Πni=1 dµ(p′i) δ

(4)(P ′n − Pi) · |T (p′1, · · · , p′n|p+, p−)|2

179

e+

e−

γe+

e−

γhadrons

X a) b)

Figure 8.3: The optical theorem in e+e−–annihilation: a) including the QED part, b) the QCDpart isolated.

which in the given one–photon exchange approximation has the structure shown in Fig. 8.3.The sum extends over spin and other possible quantum numbers (QN). The isolated QCD part,representing (2π)4 δ(4)(p+ + p−− pX)hµν , thus corresponds to the hadronic excitations (hadronicvacuum polarization) in the photon propagator. The hadronic part corresponds to the photonpropagator after amputation of the external photons Aµ(x) → Jµ(x), where only the hadronicpart Jµhad(x) is of interest here. The hadronic “blob” b) of Fig. 8.3 thus is given by the hadroniccurrent correlator

Πµν(q) = i

∫d4x ei qx〈0|Jµhad(x)Jνhad(0)|0〉

= −(gµν q2 − qµqν

)Π′γ(q2) . (8.9)

Note that Π′γ(q2) times e2 is the usual photon self-energy. We have used transversality which is aconsequence of the current conservation. The hadronic photon vacuum polarization thus is givenby a single scalar amplitude Π′γ(q2) = e2 Π′γ(q2).

The physical thresholds of hadron production in e+e− → X are

X = π+π−, ρ, π0π+π−, ω,K+K−,K0K0, φ, · · ·with lowest threshold at s = 4m2

π±33. The vector mesons ρ, ω and φ correspond to π+π−,

π0π+π− and KK resonance peaks, respectively.

We now may continue our cross–section calculation using

(2π)4 δ(4)(p+ + p− − pX)hµν = 2 Im Πµν(q) = 2(qµqν − q2 gµν

)Im Π′γ(q2) ,

such that

σh(s) =α2

s38π2√

1− 4m2e/s

qµqν − q2 gµν − (p+ − p−)µ(p+ − p−)ν

(qµqν − q2 gµν

)Im Π′γ(q2) .

Contracting the tensors and working out the scalar products: with q = p+ + p−, p2+ = m2e,

p2− = m2e, q · (p+ − p−) = p2+ − p2− = 0, q2 = (p+ + p−)2 = 2m2

e + 2p+p− or 2p+p− = q2 − 2m2e

and (p+ − p−)2 = 2m2e − 2p+p− = 4m2

e − q2 we get

ℓµν · 2(qµqν − q2 gµν

)= 2 q2

(1 +

2m2e

q2

)

and hence

σh(s) =α2

s

16π2√1− 4m2

es

(1 +

2m2e

s

)Im Π′γ(s)

≃ 16π2α2

sIm Π′γ(s) since s ≥ 4m2

π ≫ m2e .

33If we would include higher order electromagnetic interaction, like final state radiation (FSR) from hadrons, thelowest possible state exhibiting a hadron would be π0γ with threshold at s = m2

π0 . In our counting this is a O(α3)effect not included in our leading order consideration.

180

We may compare this result with the Born level muon–pair production cross–section, in the highenergy approximation given by

σ(e+e− → µ+µ−) ≃ 4πα2

3 s.

Exercise: Calculate the exact Born cross section for e+e− → µ+µ−, by replacing the hadronictensor hµν by the leptonic one ℓµν of the muon. What is the difference to e+e− → e+e−?

It is common use to represent the hadronic production cross–section σh(s) in units of the leptonicpoint cross section. This leads to the definition of the so called R–ratio function

R(s).=

σtot(e+e− → hadrons)

σ0(e+e− → µ+µ−)

=σh(s)(4πα2

3s

) =16π2α2

4πα2· 3 s 1

sIm Π′γ(s)

= 12πIm Π′γ(s) = 12π2 ρ1(s) (8.10)

where ρ1(s) = 1π Im Π′γ(s) has the meaning of a spectral density (see below).

As we will see below the function R(s) has the asymptotic property

R(s)→ constant ; s→∞

which means that the imaginary part of the vacuum polarization function Π′γ(s) tends to a non–vanishing constant at time–like infinity.

Exploiting analyticity:

Causality34 together with unitarity imply analyticity Π′γ(s) in the complex s–plane except for thecut along the positive real axis starting at s ≥ 4m2, where m is the lightest state which can bepair–produced. Cauchy’s integral theorem tells us that the contour integral, for the contour Cshown in Fig. 8.4, satisfies

Fi(s) =1

2πi

∮

C

ds′F (s′)s′ − s .

We first note that by the causal iε–prescription of the time–ordered Green functions the imaginaryparts of the analytic amplitudes change sign when s→ s∗. Therefore F (s∗) = F ∗(s), which is theSchwarz reflection principle. Since F ∗(s) = F (s∗) the contribution along the cut may be writtenas

limε→0

(F (s+ iε)− F (s− iε)) = 2 i Im F (s) ; s real , s > 0 (8.11)

and hence for R→∞

F (s) = limε→0

F (s+ iε) =1

πlimε→0

∞∫

4m2

ds′Im F (s′)s′ − s− iε

+ C∞ . (8.12)

34Causality is implemented in a QFT by the iε prescription required by the time–ordered Green functions, whichrelate to the S–matrix elements via the LSZ formulae.

181

Im s

Re s

CR

|s0

Figure 8.4: Analyticity domain and Cauchy contour C for the lepton and quark form–factors(vacuum polarization). C is a circle of radius R with a cut along the positive real axis fors > s0 = 4m2 where m is the mass of the lightest particles which can be pair–produced.

In all cases where F (s) falls off sufficiently rapidly as |s| → ∞ the boundary term C∞ vanishesand the integral converges. This may be checked order by order in perturbation theory. In thiscase the “un–subtracted” dispersion relation (DR)

F (s) =1

πlimε→0

∞∫

4m2

ds′Im F (s′)s′ − s− iε

uniquely determines the function by its imaginary part. A technique based on DRs is frequentlyused for the calculation of Feynman integrals, because the calculation of the imaginary part issimpler in general. The real part which actually is the object to be calculated is given by theprincipal value (P) integral

Re F (s) =1

πP∞∫

4m2

ds′Im F (s′)s′ − s ,

which is also known under the name Hilbert transform.

In summary: analyticity implies the validity of a dispersion relation (DR). Using the unitarityproperties, this DR determines the real part of the vacuum polarization amplitude in terms ofits imaginary part. In fact the electromagnetic current correlator exhibits a logarithmic UVsingularity and thus requires one subtraction such in place of the unsubtracted DR

Π′γ(q2) =1

π

∞∫

0

dsIm Π′γ(s)

s− q2 − iε, (8.13)

we find

Π′γ(q2)− Π′γ(0) =q2

π

∞∫

0

dsIm Π′γ(s)

s (s− q2 − iε). (8.14)

In a renormalizable QFT UV divergencies in physical amplitudes are at most quadratic. Thereforeat most two subtractions (mass– and wavefunction–renormalization, or vertex renormalization)are needed to get any DR finite. In low energy effective theories the number of subtractionsneeded corresponds to the number of low energy constants at the given order of the expansion(powers in p/Λ; p process specific momentum, Λ, typically ΛQCD or Mp the proton mass).

182

Exercise: Verify that the subtraction changes the DR of the form (8.13), assumed to converge orsupplemented with a cut–off, to (8.14), which we call a subtracted DR.

Physical interpretation (and an application):

As we know the full photon propagator is given by the geometrical progression of self–energyinsertions −iΠγ(q2). The corresponding Dyson summation implies that the free propagator isreplaced by the dressed one

iDµνγ (q) =

−igµν

q2 + iε→ iD

′µνγ (q) =

−igµν

q2 + Πγ(q2) + iε

modulo unphysical gauge dependent terms. By U(1)em gauge invariance the photon remainsmassless and hence we have Πγ(q2) = Πγ(0) + q2 Π′γ(q2) with Πγ(0) ≡ 0. As a result we obtain

iD′µνγ (q) =

−igµν

q2 (1 + Π′γ(q2))+ gauge terms

where the “gauge terms” will not contribute to gauge invariant physical quantities, and need notbe considered further.

Including a factor e2 and considering the renormalized propagator (wave function renormalizationfactor Zγ) we have

i e2 D′µνγ (q) =

−igµν e2 Zγ

q2(1 + Π′γ(q2)

) + gauge terms

which in effect means that the charge has to be replaced by a running charge

e2 → e2(q2) =e2Zγ

1 + Π′γ(q2).

The wave function renormalization factor Zγ is fixed by the condition that at q2 → 0 one ob-tains the classical charge (charge renormalization in the classical Thomson limit). Thus therenormalized charge is

e2 → e2(q2) =e2

1 + (Π′γ(q2)−Π′γ(0))(8.15)

where the lowest order diagram in perturbation theory which contributes to Π′γ(q2) is

γ γf

f

and describes the virtual creation and re–absorption of fermion pairs γ∗ → e+e−, µ+µ−, τ+τ−, uu, dd,· · ·→ γ∗ .

In terms of the fine structure constant α = e2

4π Eq. (8.15) reads

α(q2) =α

1−∆α; ∆α = −Re

(Π′γ(q2)−Π′γ(0)

).

The various contributions to the shift in the fine structure constant come from the leptons (lep= e, µ and τ) the 5 light quarks (u, d, s, c, and b and the corresponding hadrons = had) andfrom the top quark:

∆α = ∆αlep + ∆(5)αhad + ∆αtop + · · ·

183

Figure 8.5: Shift of the effective fine structure constant ∆α as a function of the energy scale inthe space–like region q2 < 0 (E = −

√−q2). In the sum “lepton+quarks” the band indicates the

hadronic uncertainty.

Also W–pairs contribute at q2 > M2W . While the other contributions can be calculated order

by order in perturbation theory the hadronic contribution ∆(5)αhad exhibits low energy stronginteraction effects and hence cannot be calculated by perturbative means. Here the dispersionrelations play a key role as discussed already.

The leptonic contributions are calculable in perturbation theory. Using our result Eq. (7.15) forthe gluon self–energy, we easily obtain the photon self–energy by the replacements: g → eQℓ,C2(G) = 0 (photon is Abelian) and T (R) = 1 (QED is Abelian)35. As a result, the free leptonloops yield

∆αlep(q2) =

=∑

ℓ=e,µ,τ

α3π

[−5

3 − yℓ + (1 + yℓ2 )√

1− yℓ ln(|√1−yℓ+1√1−yℓ−1 |

)]

=∑

ℓ=e,µ,τ

α3π

[−8

3 + β2ℓ + 12βℓ(3− β2ℓ ) ln

(|1+βℓ1−βℓ |

)]

=∑

ℓ=e,µ,τ

α3π

[ln(|q2|/m2

ℓ

)− 5

3 +O(m2ℓ/q

2)]

for |q2| ≫ m2ℓ

≃ 0.03142 for q2 = M2Z

where yℓ = 4m2ℓ/q

2 and βℓ =√

1− yℓ are the lepton velocities. This leading contribution isaffected by small electromagnetic corrections only in the next to leading order. The leptoniccontribution is actually known to three loops [4, 5] at which it takes the value

∆αleptons(M2Z) ≃ 314.98 × 10−4.

As already mentioned, in contrast, the corresponding free quark loop contribution gets substan-tially modified by low energy strong interaction effects, which cannot be calculated reliably byperturbative QCD. The hadronic contribution to the shift in the fine structure constant one has

35The final result for the renormalized photon vacuum polarization for a lepton of mass m then reads

Π′γ ren(q

2) =α

3π

5

3+ y − 2 (1 +

y

2) (1− y)G(y)

with y = 4m2/q2 and G(y) = 12√

1−yln

√1−y+1√1−y−1

.

184

to evaluate by means of the DR

∆αhad(q2) = −Re Π′γ(q2) + Π′γ(0) = − e2 q2

12π2P∞∫

4m2π

dsR(s)

s (s− q2 − iε),

using the experimental e+e− → hadorns data in the non-perturbative regions. Within experimen-tal errors the hadronic shift is uniquely determined by σh(s), which has been measured directlyup to about 40 GeV. At higher energies large γ − Z mixing effects come into play. In regionswhere pQCD applies, in particular for the experimentally not explored high energy tail, R(s) canbe safely calculated in pQCD.

Vacuum polarization effects are large when large scale changes are involved (large logarithms) andbecause of the large number of light fermionic degrees of freedom as we infer from the asymptoticform in perturbation theory

∆αpert(q2) ≃ α

3π

∑

f

Q2fNcf

(ln|q2|m2f

− 5

3

); |q2| ≫ m2

f .

Fig. 8.5 illustrates the running of the effective charges at lower energies in the space–like region.Typical values are ∆α(5GeV) ∼ 3% and ∆α(MZ) ∼ 6%, where about ∼ 50% of the contributioncomes from leptons and about ∼ 50% from hadrons. Note the sharp increase of the screeningcorrection in Fig. 8.5 which implies large corrections at relatively low energies.

Summary: In the one photon exchange approximation, i.e. at O(α2), the total hadronic pro-duction cross–section in electron–positron annihilation

σh(e+e− → γ∗ → hadrons) :

e+

e−

γX hadrons

determines the imaginary part of the photon vacuum polarization amplitude

: Πµν(q) =(

qµq

ν − q2g

µν)

Π′

γ had(q2)

γ γhad

q

σh =16π2α2

sIm Π′γ had .

Usually one represents the e+e− → hadrons data in terms of

R(s) =σh(s)

σ0(e+e− → µ+µ−)= 12π Im Π′γ had .

The hadronic shift of the effective fine structure constant at scale M is given by

∆αhad(M2) = −αM2

3πRe

∞∫

4m2π

dsR(s)e+e−→γ∗→had

s (s−M2 − iε).

185

8.12.1 Current correlators and the Kallen-Lehmann spectral representation

We have seen that dispersion relations play a key role in the attempt to understand relationshipsbetween physical quantities in a non–perturbative framework. The Kallen-Lehmann spectralrepresentation sheds more light on how general QFT properties can be exploited in this context.We therefore present a short digression on the subject here.

We consider a current correlator36

〈0|Jµ(x)Jν(0)|0〉 =∑

α

∫d4p 〈0|Jµ(x)|p, α〉〈p, α|Jν(0)|0〉

and have applied the completeness relation

∑

α

∫d4p|p, α〉〈p, α| = 1 ,

where p is the total four–momentum and α denotes an appropriate set of quantum numberswhich characterize the state |p, α〉. In addition, as usual, we use translational invariance Jµ(x) =eiPx Jµ(0) e−iPx together with the fact that |0〉 and |p, α〉 are eigenstates of the total energy-momentum operator P in order to simplify the two–point function.

Next we define the spectral function tensor

ρµν.= (2π)3

∑

α

〈0|Jµ(0)|p, α〉〈p, α|Jν(0)|0〉

which has the following properties:1) Relativistic covariance implies

ρµν = ρ1(p2) gµν + ρ2(p

2) pµpν .

2) The spectral condition (physical state must have p0 ≥ 0 and p2 ≥ 0 and, unless the state isthe vacuum, p = (p0, ~p) 6= (0,~0)) requires

ρµν = Θ(p0) Θ(p2) · · · = Θ(p0)

∫ ∞

0ds δ(s− p2) · · ·

which implies

ρµν = Θ(p0)

∫ ∞

0ds δ(s− p2) ρ1(s) gµν + ρ2(s) p

µpν

=

∫ ∞

0dsΘ(p0) δ(p2 − s)

(pµpν − p2 gµν

)ρ1(s) + pµpν ρ0(s)

,

with

ρ1 = − ρ1(s)s

; ρ0 = ρ2(s) +ρ1(s)

s.

The scalar functions ρ1(s) and ρ0(s) denote the transversal and the longitudinal spectral functions,respectively.3) Current conservation (in case it applies)

pµ ρµν = 0 ⊲⊳ ρ0(s) ≡ 0 .

36Specifically, the object we consider here is called the positive frequency part of the current correlator function.

186

We furthermore can prove the positivity of the spectral function ρ1(s):

ρµν = (2π)3∑

α

〈0|Jµ(0)|p, α〉〈p, α|Jν(0)|0〉

= Θ(p0)

∫ ∞

0ds δ(p2 − s)

(pµpν − p2 gµν

)ρ1(s)

ρ00 = (2π)3∑

α

〈0|J0(0)|p, α〉〈p, α|J0(0)|0〉

= (2π)3∑

α

∣∣〈0|J0(0)|p, α〉∣∣2 ≥ 0

= Θ(p0)

∫ ∞

0ds δ(p2 − s) ~p 2 ρ1(s) = Θ(p0) Θ(p2) ~p 2 ρ1(p

2)

This implies ρ1(p2) ≥ 0 and real.

Exercise: Discuss the positivity properties of the functional P (s) defined as the derivative of

P (s) = P∫ ∞

s0

ds′R(s′)

s′(s′ − s) ,

where R(s) is a positive ordinary function. As a distribution (or linear functional) P (s) has aderivative. Formally,

P (s) =dP (s)

ds= P∫

ds′R(s′)

(s′ − s)2 .

With these general properties we may write

〈0|Jµ(x)Jν(0)|0〉 =

∫d4p e−i px

1

(2π)3ρµν(p)

=

∫ ∞

0ds −gµν ρ1(s) + ∂µ∂ν (ρ1(s) + ρ0(s)) ×

1

(2π)3

∫d4p e−i px Θ(p0) δ(p2 − s)

︸︷︷︸+i∆+(x;s)

= −i

∫ds

(∂µ∂ν − gµν) ∆+(x; s) ρ1(s) + ∂µ∂ν ∆+(x; s) ρ0(s),

the Kallen-Lehmann representation for the positive frequency correlator in coordinate space.Note that i ∆+(x;m2) is the positive frequency part of a scalar free field propagator of massm. The current commutator function 〈0| [Jµ(x)Jν(0)] |0〉 and the time–ordered Green function〈0|T Jµ(x)Jν(0) |0〉 are then given by the same representation (same spectral functions) with thereplacements ∆+(x; s)→ ∆(x; s), the free scalar commutator function, and ∆+(x; s)→ ∆F (x; s),the free scalar Feynman propagator, respectively.

The application to the hadronic vacuum polarization is the following: the Kallen-Lehmann rep-resentation is given by

Πµν(q) = i

∫d4x ei qx 〈0|T Jµ(x)Jν(0) |0〉

= −∫ ∞

0ds(qµqν − q2 gµν

)ρ1(s)

1

q2 − s+ iε.

Using

1

q2 − s+ iε= P 1

q2 − s − iπ δ(q2 − s) ,

187

where P denotes the principle value prescription, we obtain

2 Im Πµν(q) = 2π

∫ ∞

0ds(qµqν − q2 gµν

)ρ1(s) δ(q

2 − s) .

This means

1

πIm Πµν(q) =

(qµqν − q2 gµν

)ρ1(q

2)

or

Im Π′γ(q2) = π ρ1(q2) .

A last remark here: the positivity of ρ1(s) discussed above implies that Π′γ(s) is a real analytic

function with the property: for real q2 < 0 we get Im Π′γ(s) = 0, for real q2 < 4m2 the amplitude

Π′γ(q2) is real Π′γ(q2) = Π′∗γ (q2) and more generally: Π

′∗(q2) = Π′γ(q2∗). This is an alternativederivation of the properties usually inferred using T–matrix element unitarity relations includingthe optical theorem.

8.13 High energy hadron-production in e+e−–annihilation

At sufficiently high energy E =√s we expect to be able to perform a perturbative expansion in the

effective strong coupling constant αs(E), as, due to asymptotic freedom, the latter decreases withincreasing energy. The photon at short distances directly couples to the quarks q = u, d, s, c, b, tand by applying perturbation theory we actually calculate the production of colored qq–pairs,which we know do not exist as free asymptotic states. Since only hadrons like pions, Kaons, vectormesons etc are produced in reality, what we calculate in pQCD is therefore not precisely whatwe would like to know. For appropriate observables, like the total inclusive cross–section, it isexpected or at least hopped that the hadronization of the quarks into hadrons does not affect thequark production cross section too much. Since what primary is produced in any case is a virtualquark–antiquark pair (up to higher orders in α) which has to hadronize in any case, we expect thequark production rate should rather well account for what we get when we sum over any type ofhadron state observed. However, we know that the existence of sharp and pretty huge resonanceexcitations (see later) tells us that the hoped for “quark–hadron duality” cannot be local (i.e.hold bin by bin in the energy), because the quark production cross–section is a smooth non-resonant function up to the known threshold “steps”. In any case quark hadron duality37 [7, 8]is the bridge between pQCD based theory and the QCD state space explored in experiments.The precise formulation states that for sufficiently large s the average non–perturbative hadroncross–section equals the perturbative quark cross–section:

σ(e+e− → hadrons)(s) ≃∑

qσ(e+e− → qq)(s) , (8.16)

where the averaging extends from threshold up to the given s value which must lie far enoughabove a threshold (global duality). Approximately, such duality relations then would hold forenergy intervals which start just below the last threshold passed up to s. Qualitatively, such abehavior is clearly visible in the data. However, for precise reliable predictions it has not yetbeen possible to quantify the accuracy of the duality conjecture. A quantitative check wouldrequire much more precise cross–section measurements than the ones available today. As we will

37Quark–hadron duality was first observed phenomenologically for the structure function in deep inelasticelectron–proton scattering [6].

188

see quark–hadron duality is obviously there, however, at present, it is not known to what extendit can safely be used as a tool for precision physics.

In the following e will try to quantify these ideas. We first are going to discuss the calculations ofR(s) in perturbative QCD. This provides one of the most important testing grounds for pQCD.Practically, what we have to do is to calculate the hadronic current correlators in perturbationtheory as a power series in αs/π. According to the general analysis presented above, the objectof interest is

R(s) = 12π Im Π′γ had(s) ; Πγ had(q) :’

, (8.17)

where the “blob” is the hadronic one, in pQCD represented by valence quark loops dressed inall possible ways by gluons and possible sea quark loops. The leading plus next to leading QCDperturbation expansion diagrammatically is given by

’= + ⊗

+ + +

+ ⊗ + ⊗ +⊗

+⊗

+ · · ·(8.18)

Lines show external photons, propagating quarks/antiquarks andpropagating gluons. See Fig. (5.1) for the Feynman rules of QCD. The vertices ⊗ are markingrenormalization counter term insertions. They correspond to subtraction terms which render thedivergent integrals finite.

In QED (the above diagrams with gluons replaced by photons) the phenomenon of vacuumpolarization was discussed first by Dirac [9] and finalized at the one–loop level by Schwinger [10]and Feynman [11]. Soon later Jost and Luttinger [12] presented the first two–loop calculation.

In 0th order in the strong coupling αs we have

2 Im =2

(8.19)

which is proportional to the the free quark–antiquark production cross–section [13]. This approx-imation corresponds to the so called Quark Parton Model (QPM), describing free quarks withthe strong interaction turned off. The QPM is expected to provide a good approximation in thehigh energy limit of QCD. The leading order result can be obtained in two ways from earlier cal-culations. In Sect. 7.2 we calculated the gluon self-energy. From the latter the photon self-energymay be obtained by taking into account only the last of the four diagrams, the one with thequark loop, now with T (R) replaced by unity: T (R)→ 1 and g → eQq with Qq the charge of thequark in units of the positron charge e. In Eq. (7.15) we replace g → eQq, C2(G) = 0 (photon isAbelian) and T (R) = 1 (QED is Abelian). Furthermore we need the imaginary part only, whichis finite such that f(d) = 4, in any case. The self-energy amplitude Π(k2) of (7.15) correspondsto e2 Π′γ had(q2) considered here, since Π′γ had is defined as a current correlator without coupling

factor e from the QED–vertex e JµAµ. Also note that by definition the common factor e2 cancelsin the cross section ratio R(s) according to the definition (8.10). In addition we have to sum overthe color of the quarks which yields a factor Nc = 3 and the quark flavors denoted by f in the

189

following. We then obtain

Im Π′γ had(q2) = Nc

∑

f

Q2f

12π2(1 + 2m2

f/q2)

ImB0(mf ,mf ; q2)

and the imaginary part of B0 is given by Eq. (6.41): Im B0 = π√

1− 4m2f/q

2. The mass

dependent threshold factor is a function of the center of mass quark velocity

vf =

(1−

4m2f

s

)1/2

and, using (8.17), we may write

R(s) = Nc

∑

f

Q2f

12π212π2

(1 + 2m2

f/s) √

1− 4m2f/s [s ≥ 4m2

f ]

= Nc

∑

f

Q2f

vf2

(3− v2f

)Θ(s− 4m2

f ) .

The other possibility to get this result is to calculate the Born cross section for the productionof a quark-antiquark pair σ(e+e− → qq), directly, by taking free quarks X = qq in the matrixelement (8.4). We then have

〈X|Jµhad(0)|0〉 = Qq uq(pq, sq) γµ vq(pq, sq) (8.20)

and hµν in (8.5) is given by an expression like ℓµν , with q unchanged (photon momentum) andp+ → pq and p− → pq as outgoing momenta. Up to different charge and masses and a colorsummation, the cross section σ(e+e− → qq) agrees formally with the lepto–production cross

section σ(e+e− → µ+µ−) = 4πα2

3svµ2 (3−v2µ) with vµ the muon velocity (corresponding to Rµ(s) =

vµ2 (3− v2µ) with Rµ(s)→ 1 as s≫ m2

µ).

A first assessment of the lowest order result, which is O(α0s), i.e. strong interaction switched

off! is in order here. The result in any case should make sense at high enough energy. What ishigh enough, actually was the big question at the time when QCD became the candidate stronginteraction theory. Besides αs ≪ 1 we also require 4m2

f/s≪ 1 to be away from resonances whichusually show up in the threshold regions. We then are dealing with the quark–parton modelwhich, with Nc = 3, Qu,c,t = 2

3 and Qd,s,b = −13 , predicts the following constant R–values:

Nf 3 4 5 6

quarks uds udsc udscb udscbt

R 2 3 12 3 2

3 5

range 1.8 - 3.73 GeV 4.8 - 10.52 GeV 11.20 - (2mt-10.0) GeV (2mt+10.0) GeV - ∞

The ranges refer to regions where the cross sections vary slowly, except for the lowest (usually3) very narrow resonances, which appear the lie below the corresponding flavor thresholds. Thegaps are regions of strongly varying data, where applying pQCD makes little sense. In the“perturbative regions” the lowest order pattern (the step wise increase) may clearly be identifiedin the data as seen in Fig. 8.6. Most remarkably, the color factor 3 is unambiguously confirmedby the data (it resolved the unitarity crisis of 1974! at the end). Quark-hadron duality seems

190

Figure 8.6: Test of pQCD prediction of R with some more recent data in the non-resonant regions.The spikes show the sharp J/ψ (cc) and Υ (bb) resonances.

well at work. QCD corrections are fairly small, which means that the effective strong couplingconstant is relatively small. We consider these corrections next.

Including the presently known higher order terms the perturbative result for R(s) is given by [14–16]

R(s)pert = Nc

∑

f

Q2f

vf2

(3− v2f

)Θ(s− 4m2

f )

×

1 + ac1(vf ) + a2c2 + a3c3 + a4c4 + · · ·

(8.21)

where a = αs(s)/π and, assuming 4m2f ≪ s, i.e. in the massless approximation

c1 = 1

c2 = C2(R)

− 3

32C2(R)− 3

4β0ζ(3)− 33

48Nf +

123

32Nc

=365

24− 11

12Nf − β0ζ(3) ≃ 1.9857 − 0.1153Nf

c3 = −6.6368 − 1.2002Nf − 0.0052N2f − 1.2395 (

∑

f

Qf )2/(3∑

f

Q2f )

c4 = 135.8 − 34.4Nf + 1.88N2f − 0.010N3

f − π2β20(

1.9857 − 0.1153Nf +5β16β0

),

with β0 = (11 − 2/3Nf )/4, β1 = (102 − 38/3Nf )/16. All results are in the MS scheme. Nf =∑f :4m2

f≤s1 is the number of active flavors. There is a mass dependent threshold factor in front

of the curly brackets and the exact mass dependence of the first correction term

c1(v) =2π2

3v− (3 + v)

(π2

6− 1

4

)

is singular (Coulomb singularity due to soft gluon final state interaction) at threshold. Thesingular terms of the n–gluon ladder diagrams

· · ·

191

Figure 8.7: Compilation of a fairly complete collection of e+e− → hadrons data [from Ref. [18]]in comparison with up-to-date pQCD predictions for R(s). Only statistical errors of data pointsare shown. The uncertainty band of the theory is due to the uncertainties in QCD parameters(mainly αs).

exponentiate and thereby remove the singularity [17]:

1 + x→ 2x

1− e−2x; x =

2παs3v

(1 + c1(v)

αsπ

+ · · ·)→

(1 + c1(v)

αsπ− 2παs

3v

)4παs

3v

1

1− exp−4παs

3v

. (8.22)

Since pQCD in any case should not be applied near threshold, the above remark is somewhatacademic. In contrast to QED, where Coulomb interaction problems are there and have to becured appropriately.

Mandatory is applying renormalization group improvement of pQCD calculations: the couplingαs and the masses mq have to be understood as running parameters

R

(m2

0f

s0, αs(s0)

)= R

(m2f (µ2)

s, αs(µ

2)

); µ =

√s . (8.23)

where√s0 is a reference energy. RG resummation dramatically improves the convergence of per-

turbative approximations. Mass effects are important once one approaches a threshold from the

192

Im s

Re s←− asymptotic freedom (pQCD)|s0

Figure 8.8: Analyticity domain for the photon vacuum polarization function. In the complexs–plane there is a cut along the positive real axis for s > s0 = 4m2 where m is the mass of thelightest particles which can be pair–produced.

perturbatively save region sufficiently far above the thresholds where mass effects may be safelyneglected. They have been calculated up to three loops by Chetyrkin, Kuhn and collaborators [19]and have been implemented in the FORTRAN routine RHAD by Harlander and Steinhauser [20].

In Fig. 8.7 we have collected the e+e−–annihilation data into hadrons and compare the datawith state of the art pQCD calculations of R(s) (for the experimental data [?, 21, 24–29]). Theagreement between theory and experiment is indeed remarkable in view of the conceptual prob-lems with the hadronization. It should be mentioned that statistical errors only are shown inthe plot. Typically, pQCD fails at low energy and in quark threshold regions, where hadronicresonances show up. The latter usually are parametrized by Breit-Wigner resonance shapes (seebelow).

Where would we expect that we can trust the perturbative result? Perturbative QCD is supposedto work best in the deep Euclidean region38 away from the physical region characterized by thecut in the analyticity plane Fig. 8.8. Fortunately, the physical region to a large extent is accessibleto pQCD as well as we have seen, provided the energy scale is sufficiently large and one looks forthe appropriate observable, which in our case is the inclusive total hadronic cross–section.

As we have shown earlier the total cross–section represented by the imaginary part of the vacuumpolarization function Π(q2) corresponds to the jump across the cut. On the cut we have thethresholds of the physical states, with lowest lying channels: π+π−, π0π+π−, · · · and resonancesρ, ω, φ, J/ψ · · ·, Υ · · ·, · · ·. QCD is confining the quarks (a final proof of confinement is yetmissing) in hadrons. In any case the quarks hadronize (see Fig. 8.9), a highly non–perturbativephenomenon which is poorly understood in detail.

γ

π+

π−

π+

π−

π0

u

u

d

d

Figure 8.9: Hadron production in low energy e+e−–annihilation: the primarily created quarksmust hadronize. The shaded zone indicates strong interactions via gluons which confine thequarks inside hadrons.

While at higher energies pQCD works sufficiently far away from thresholds and resonances, i.e. in

38This is directly accessible in Deep Inelastic electron–proton Scattering (DIS), which we will learn about later.

193

Ve+

e−

q

q

γ e+ e−

q

q

θ

Figure 8.10: Fermion pair production in e+e−–annihilation. The lowest order Feynman diagram(left) and the same process in the c.m. frame (right). The arrows represent the spatial momentumvectors and θ is the production angle of the quark relative to the electron in the c.m. frame.

regions where R(s) is a slowly varying monotonic function, neither the physical thresholds nor theresonances are obtained with perturbation theory! Not to talk about the pions as quasi-Goldstonemodes of the spontaneous breakdown of chiral symmetry. We note that the perturbative quark–pair thresholds in (8.21) do not nearly approximate the physical thresholds for the low energyregion below about 2 GeV, as we will see. Less problematic is the space–like (Euclidean) region−q2 → ∞, since it is away from thresholds and resonances. The best monitor for a comparisonbetween theory and experiment has been proposed by Adler [30] long time ago: the so calledAdler–function, up to a normalization factor, the derivative of the vacuum polarization functionin the space–like region (see Sect. 8.14.1 below). In any case only the data presently can tell us towhat extent we understand strong interaction physics and in many cases a semi–phenomenologicalapproach is very successful.

Let us elaborate a little bit more about “parton physics” in e+e−–annihilation. At higher energieshighly energetic partons, quarks and/or gluons, are produced and we learned how pQCD is ableto give the correct prediction of σtot(e

+e− → γ∗ → hadrons) in non–resonant regions, in the senseof quark–hadron duality (8.16). However, the consequences of the validity of pQCD are morefar–reaching. According to perturbation theory the production of hadrons in e+e−–annihilationproceeds via the primary creation of a quark–antiquark pair which must hadronize as shownin Fig. 8.9, in the simplest cases. The elementary hard process tells us that in a high energycollision of positrons and electrons (in the center of mass frame) q and q are produced with highmomentum in opposite directions (back–to–back), as illustrated in Fig. 8.10. The differentialcross–section, up to a color factor the same as for e+e− → µ+µ−, reads

dσ

dΩ(e+e− → qq) =

3

4

α2s

s

∑Q2f

(1 + cos2 θ

)(8.24)

typical for an angular distribution of a spin 1/2 particle. Indeed, the quark and the antiquarkseemingly hadronize individually in that they form jets [31]. Jets are bunches of hadrons whichconcentrate in a relatively narrow angular cone. This in spite of the fact that the quarks haveunphysical charge and color. True physical states only can have integer charge and must be colorsinglets. Apparently, while charge and color have enough time to recombine into color singlets ofinteger charge, the momentum apparently has not sufficient time to distribute isotropically in thehadronization process. The extra quarks needed to form physical states are one or more virtualpairs pulled out of the vacuum and carried along by the primary quarks (see Fig. 8.9). As a rulepQCD is applicable to the extent that “hard partons”, quarks or gluons, may be interpreted asjets. Fig. 8.11 illustrates such qq (two–jet event) and qqg (three–jet event) jets. Three jet eventsproduced with the electron positron storage ring PETRA at DESY in 1979 revealed the existenceof the gluon. The higher the energy the narrower the jets, quite opposite to expectations atpre QCD times when most people believed events with increasing energy will be more and moreisotropic multi–hadron states.

It is one of the spectacular and at the beginning completely unexpected predictions of QCD,

194

Figure 8.11: Two and three jet event first seen by TASSO at DESY in 1979.

that the energy get collimated more and more into narrower and narrower jets as the enegyincreases. However, as the energy increases also the number of primary partons increases suchthat the energy distributes among a larger number of jest, which then brings back a more isotropicdistribution.

8.14 Non–Perturbative Effects, Operator Product Expansion

Here we have to address the question: what are the contributions to R(s) from the quark con-densates which are the order parameters of the spontaneous breakdown of chiral symmetry. Suchnon–perturbative (NP) effects are parametrized as prescribed by the Operator Product Expansion(OPE)39 of the electromagnetic current correlator [34]

Π′NPγ (Q2) =

4πα

3

∑

q=u,d,s

Q2qNcq ·

[1

12

(1− 11

18a

) 〈αsπ GG〉Q4

+ 2

(1 +

a

3+

(11

2− 3

4lqµ

)a2) 〈mq qq〉

Q4(8.28)

+

(4

27a+

(4

3ζ3 −

257

486− 1

3lqµ

)a2) ∑

q′=u,d,s

〈mq′ q′q′〉Q4

]+ · · ·

where a ≡ αs(µ2)/π and lqµ ≡ ln(Q2/µ2). 〈αs

π GG〉 and 〈mq qq〉 are the scale–invariantly de-fined condensates. Sum rule estimates of the condensates yield typically (large uncertainties)〈αsπ GG〉 ∼ (0.389 GeV)4, 〈mq qq〉 ∼ −(0.098 GeV)4 for q = u, d , and 〈mq qq〉 ∼ −(0.218 GeV)4

39The operator product expansion (Wilson short distance expansion) [32] is a formal expansion of the product oftwo local field operators A(x)B(y) in powers of the distance (x− y) → 0 in terms of singular coefficient functionsand regular composite operators:

A(x)B(y) ≃∑

i

Ci(x− y)Oi(x+ y

2) (8.25)

where the operators Oi(x+y2

) represent a complete system of local operators of increasing dimensions. The coeffi-cients may be calculated formally by normal perturbation theory by looking at the Green functions

〈0|TA(x)B(y)X|0〉 =N∑

i=0

Ci(x− y) 〈0|TOi(x+ y

2)X|0〉 +RN (x, y) (8.26)

constructed such that

RN → 0 as (x− y)aN ; (x− y)2 < 0 , aN < aN+1 ∀ N (8.27)

(asymptotic expansion). By X we denoted any product of fields suitable to define a physical state |X〉 via the LSZreduction formula. A detailed explanation will be given in Sect. ?? later (see also [33])

195

for q = s . Note that the above expansion is just a parametrization of the high energy tail ofNP effects associated with the existence of non–vanishing condensates. The dilemma with theOPE in our context is that it works for large enough Q2 only and in this form fails do describeNP physics at lower Q2. Once it starts to be numerically relevant pQCD starts to fail because ofthe growth of the strong coupling constant. In R(s) NP effects as parametrized by (8.28) havebeen shown to be small in [35–37]. Note that the quark condensate, the vacuum expectationvalue (VEV) 〈Oq〉 of the dimension 3 operator Oq .

= qq, is a well defined non–vanishing orderparameter in the chiral limit of QCD. In pQCD it is vanishing to all orders. In contrast theVEV of the dimension 4 operator OG .

= αsπ GG is non–vanishing in pQCD but ill–defined at first

as it diverges like Λ4 in the UV cut–off. OG contributes to the trace of the energy momentumtensor40 [38–40]

Θµµ =

β(gs)

2gsGG + (1 + γ(gs))

mu uu+md dd+ · · ·

(8.31)

where β(gs) and γ(gs) are the RG coefficients (8.5) and in the chiral limit

εvac = −β032

+O(αs)

〈OG〉 (8.32)

represents the vacuum energy density which is not a bona fide observable in a continuum QFT.In the Shifman-Vainshtein-Zakharov (SVZ) approach [34] it is treated to represent the soft partwith respect to the renormalization scale µ, while the corresponding OPE coefficient comprisesthe hard physics from scales above µ.

Note that in the chiral limit mq → 0 the trace (8.31) does not vanish as expected on the classicallevel. Thus scale invariance (more generally conformal invariance) is broken in any QFT unlessthe β–function has a zero. This is another renormalization anomaly, which is a quantum effectnot existing in a classical field theory. The renormalization group is another form of encodingthe broken dilatation Ward identity. It’s role for the description of the asymptotic behavior ofthe theory under dilatations (scale transformations) has been discussed in Sect. 8, where it wasshown that under dilatations the effective coupling is driven into a zero of the β–function. For anasymptotically free theory like QCD we reach the scaling limit in the high energy limit. At finiteenergies we always have scaling violations, as they are well known from deep inelastic electronnucleon scattering. In e+e−–annihilation the scaling violation are responsible for the energydependence (via the running coupling) of R(s) in regions where mass effects are negligible.

40In a QFT a symmetric energy momentum tensor Θµν(x) should exist such that the generators of the Poincaregroup are represented by (see cf. [1])

Pµ =

∫

d3xΘ0µ(x) ,Mµν =

∫

d3x (xµ Θ0ν − xν Θ0µ) (x) . (8.29)

This corresponds to Noether’s theorem (see (1.5.2)) for the Poincare group. In a strictly renormalizable masslessQFT which exhibits only dimensionless couplings classically one would expect the theory to be conformally in-variant. The energy momentum tensor then would also implement infinitesimal dilatations and special conformaltransformations. That is, the currents

Dµ(x) = xρ Θµρ ; Kµν = 2 xρ xν Θµρ − x2Θµν (8.30)

ought to be conserved, which requires the trace of the energy momentum tensor to vanish Θµµ = 0 .

196

Figure 8.12: Perturbative coefficient functions Hi for the Adler function in full QCD at 2 [left]and 3 [right] loops. The curvature is entirely due to the non-zero masses. In the massless limitthe Hi’s are normalized to unity. From [37].

8.14.1 Testing non–perturbative hadronic effects via the Adler function

The non-perturbative Adler function related to the photon vacuum polarization can be calculatedin terms of experimental e+e− annihilation data by the dispersion integral

D(Q2) = Q2

(∫ E2cut

4m2π

Rdata(s)

(s+Q2)2ds

+

∫ ∞

E2cut

RpQCD(s)

(s +Q2)2ds

). (8.33)

Here Q2 = −q2 is the squared Euclidean momentum transfer and s the center of mass energysquared for hadron production in e+e−–annihilation. Formally the Adler function is defined asthe derivative of the shift in the fine structure constant

D(Q2)

Q2= (12π2)

dΠ′γ (q2)

dq2= −3π

α

d

dq2∆αhad(q2) ,

evaluated in the Euclidean at Q2 = −q2. Π′γ (q2) is the photon vacuum polarization amplitudedefined by Eq. (8.9). The perturbative result is given in [37]. Crucial for this prediction areknown full massive QCD results [41–43]. Note that the main Q2 dependence of D(Q2) is due tothe quark masses mc and mb. Without mass effects, up to small effects from the running of αs,D(Q2) = 3

∑f Q

2f (1 +O(αs)) is a constant depending on the number of active flavors.

We write the contributions as a loop expansion

D(Q2) = D(0)(Q2) +D(1)(Q2) +D(2)(Q2) + · · ·+DNP(Q2)

At one loop, taking the derivative of (8.16) with appropriate coefficient, we obtain

D(0)(Q2) =∑

f

Q2fNcfH

(0)

197

with H ≡ (12π2)˙Π′V ,

˙Π′V ≡ −s dΠ′V /ds , in terms of the vector current amplitude Π′V .

Explicitly,

H(0) = 1 +3y

2− 3y2

4

1√1− y ln ξ y = 4m2

f/s , ξ =

√1− y − 1√1− y + 1

where ξ is taking values 0 ≤ ξ ≤ 1 for s ≤ 0 .

Asymptotically we have the expansion

H(0) →

15Q2

m2f− 3

70

(Q2

m2f

)2

+ 1105

(Q2

m2f

)3

+ · · · Q2 ≪ m2f

1− 6m2

f

Q2 − 12

(m2

f

Q2

)2

lnm2

f

Q2 + 24

(m2

f

Q2

)3(ln

m2f

Q2 + 1

)+ · · · Q2 ≫ m2

f

(8.34)

and this behavior determines the quark parton model (QPM) (leading order QCD) property ofthe Adler function: heavy quarks (m2

f ≫ Q2) decouple like Q2/m2f while light modes (m2

f ≪ Q2)

contribute Q2fNcf to D(0).

At two loops we have the known analytic result (Broadhurst 85 and others)

D(1)(Q2) =αs(Q

2)

π

∑

f

Q2fNcf H

(1)

where H(1) = (12π2)˙Π

′(2)

V (−Q2,m2f ). At three loops the result is known as low and large mo-

mentum expansion (Chetyrkin, Harlander, Kuhn, Steinhauser 96/97)

D(2)(Q2) =

(αs(Q

2)

π

)2∑

f

Q2fNcf H

(2)

where H(2) = (12π2)˙Π

′(3)

V (−Q2,m2f ). Both series expansions diverge at the boundary of the circle

of convergence Q2 = 4m2, which means that we have a problem in the region where mass effectsare of the order of unity in the Euclidean region. Here we apply a conformal mapping (Schwinger48)

y−1 =−Q2

4m2→ ω =

1−√

1− 1/y

1 +√

1− 1/y

from the complex negative q2 half–plane to the interior of the unit circle |ω| < 1 together with Paderesummation (Fleischer and Tarasov 94). The Pade approximant provides a good estimation tomuch higher values of 1/y up to about 1/y ∼ 4. This is displayed in Fig. 8.12. Pade improvementallows us to obtain reliable results also in the relevant Euclidean “threshold region”, around y = 1.

We also include the 4–loop [44,45] and 5–loop [46] contributions in the high energy limit (masslessapproximation)

D(Q2) ≃ 3∑

f

Q2f

(1 + a+ d2a

2 + d3a3 + d4a

4 + · · ·)

(8.35)

with a = αs(Q2)/π,

d2 = 1.9857 − 0.1153Nf ,

d3 = 18.2428 − 4.2159Nf + 0.0862N2f − 1.2395 (

∑Qf )2/(3

∑Q2f ) ,

d4 = −0.010N3f + 1.88N2

f − 34.4Nf + 135.8 .

198

Figure 8.13: The “experimental” non-perturbative Adler–function versus theory (pQCD + NP).The error includes statistical + systematic here (in contrast to most R-plots showing statisticalerrors only!). “[5-loop]” indicates that 4- and 5-loop contribution in the massless limit are takeninto account. For more details see Ref. [37].

The corresponding formula for R(s) only differs at the 4–loop and 5–loop level due to the effectfrom the analytic continuation from the Euclidean to the Minkowski region which yields rR3 =

d3 − π2β20 d13 with β0 = (11 − 2/3Nf )/4, d1 = 1 and rR4 = d4 − π2β20(d2 + 5β1

6β0d1

)with β1 =

(102−38/3Nf )/16. Numerically the 4–loop term proportional to d3 amounts to −0.0036% at 100GeV and increases to about 0.32% at 2.5 GeV. The higher order massless results only improvethe perturbative high energy tail (see Fig. 8.13). Towards low Q2 we also approach the Landaupole of αs(Q

2), present typically in MS type schemes, and pQCD ceases to “converge”.

The Adler–function is a good monitor to compare the pQCD as well as the NP results withexperimental data. Fig. 8.13 shows that pQCD in the Euclidean region works very well for√Q2>∼2.5 GeV [37]. The NP effects just start to be numerically significant where pQCD starts

to fail. Thus, no significant NP effects can be established from this plot.

Note that what we used here is not the MS scheme but full massive QCD in the backgroundMOM scheme [43]. The proper curvature is the key point in this analysis and the curvatureis primarily due to the mass dependence. In the Euclidean region the latter is smooth ansmomotonic unlike in the time–like region where we have step like behavior at thersholds. Thegross feature shape is obtained in the QPM, but numerically it fails badly. The first correction ofO(αs) improves the situation dramatically, however, starts to fail at rather high energies (belowabout 20 GeV) already, and approaching the non-perturbative regime below about 2 GeV starts todiverge (nearby Landau pole). At 3–loops for the first time we get a surprisingly good perturbativedescription of the data, down to not far above 2 GeV, where again the breakdown of pQCD ishitting us. In view of the fairly large differences between the different perturbative orders, amassive 4–loop calculation is expected to provide further substantial improvement. Needless tosay, the good agreement betwen theory and experiment for the Adler function as a test observablealso strenghtens our confidence in pQCD results for other onservables.

199

Figure 8.14: The modulus square of pion form factor |Fπ|2 (which is proportional to cross–sectionσh(s)) as a function of E =

√s. The low energy domain is dominated by the channel e+e− →

π+π−, which developes the ρ–resonance. The ρ−ω mixing, due to isospin breaking by mu 6= md,is distorting the ideal Breit-Wigner resonance shape of the ρ. The ratio |Fπ(E)|2/|Fπ(E)|2fitshows the fairly good compatibility of the newer measurements relative to a CMD-2 fit. Dashedhorizontal lines mark ± 10%.

8.15 Some considerations on hadron production at low energy

Certainly, quark-antiquark pair production is far from describing experimental cross–section dataat low energies, where “infrared slavery” (confinement) is the dominating feature. In Fig. 8.14 weshow a compilation of the low energy data in terms of the pion form factor. The latter is definedby

σππ(s) =πα2

3s(βπ)3|Fπ(s)|2 . (8.36)

with βπ = (1 − 4m2π/s)

1/2 the pion veclocity in the CM frame. Expressed in terms of R(s), theform factor is given by

|Fπ(s)|2 = 4Rππ(s) (βπ)−3 . (8.37)

Low energy hadro-production is dominated by the lightest hadrons, the isospin SU(2) triplet(π+, π0, π−) of pions, pseudoscalar spin 0 mesons of masses: mπ± = 139.75018(35) MeV, mπ0 =134.9766(6) MeV. As elaborated earlier, spontaneous breakdown of the nearby chiral symmetry

200

of QCD, an intrinsically non–perturbative phenomenon, shapes the hadron physics in the lowenergy tail. Pions as quark–antiquark color singlet bound states exhibit electromagnetic and weakinteractions via the quarks. This is particularly pronounced in the case of the neutral π0 whichdecays electromagnetically via π0 → γγ and has a much shorter life time τπ0 = 8.4(6) · 10−17 secthan the charged partners which can decay by weak interaction only, according to π+ → µ+νµ,and hence live longer by almost 10 orders of magnitude τπ± = 2.6033(5) · 10−8 sec.

As perturbation theory fails, chiral perturbation theory is limited to the very low energy tail onlyand lattice QCD so far is not able to control time-like processes, one has to resort to modelswhich are inspired by known properties of QCD as well as known phenomenological facts. Asan example, electromagnetic interactions of charged pions treated as point–particles would bedescribed by scalar QED, as a first step in the sense of a low energy expansion. In fact, in thethreshold region e+e− → π+π− is the dominating process. However, at slightly higher energies,the ρ meson represents a prominent π+π−–resonance peak, showing a resonance enhancement byabout a factor 50. Here, it looks natural to apply a vector–meson dominance (VMD) like model(see below). Note that in photon–hadron interactions the photon mixes with hadronic vector–mesons like the ρ0. But also nearby resonances like ρ0 and ω are mixing and thereby destortingundisturbed resonance patterns (see Fig. 8.14). In the following we are going to discuss some ofthe relevant models in some detail.

8.15.1 Pions and Scalar QED

Pions seen in a particle detector in may respects behave like point-particles , Wigner states in thesense of relativistic quantum mechanics. Such states and their associated LSZ interpolating fieldswere intruduced in Sect. 1.5. The effective Lagrangian for the electromagnetic interaction of acharged point–like pion described by a complex scalar field ϕ follows from the free Lagrangian

L(0)π = (∂µϕ)(∂µϕ)∗ −m2πϕϕ

∗ (8.38)

via minimal substitution ∂µϕ → Dµϕ = (∂µ + ieAµ(x)) ϕ, which replaces the ordinary by thecovariant derivative, and which implies the scalar QED (sQED) Lagrangian

LsQEDπ = L(0)π − ie(ϕ∗∂µϕ− ϕ∂µϕ∗)Aµ + e2gµνϕϕ

∗AµAν .

Thus gauge invariance implies that the pions must couple via two different vertices to the elec-tromagnetic field, and the corresponding Feynman rules are given in Fig. 8.15.

In sQED the contribution of a pion loop to the photon VP is given by

−i Πµν (π)γ (q) = + . (8.39)

The bare result for the transversal part, defined as in (8.9), reads

Π(π)γ (q2) =

e2

48π2

B0(m,m; q2)

(q2 − 4m2

)− 4 A0(m)− 4m2 +

2

3q2

(8.40)

with Πγ(0) = 0. We again calculate the renormalized transversal self–energy Π′γ(q2) = Πγ(q2)/q2

which is given by Π′γren(q2) = Π

′γ(q2)−Π

′γ(0). The subtraction term

Π′(π)γ (0) =

−e248π2

A0(m)

m2+ 1

201

(1) Pion propagator

: i∆π(p) = i(p2 −m2

π + iε)−1

(2) Pion–photon vertices

: = − i e (p+ p′)µ , : = 2 i e2 gµν

p

Aµϕ+

ϕ−

p′

p

Aµ

Aν

ϕ+

ϕ−

Figure 8.15: Feynman rules for sQED. p is incoming, p′ outgoing.

is the π± contribution to the photon wavefunction renormalization and the renormalized transver-sal photon self–energy reads

Π′ (π)γren(q2) =

α

6π

1

3+ (1 − y)− (1− y)2 G(y)

where y = 4m2/q2 and G(y) given by (7.16). For q2 > 4m2 there is an imaginary or absorptivepart given by substituting

G(y)→ ImG(y) = − π

2√

1− ysuch that

Im Π′ (π)γ (q2) =

α

12(1− y)3/2

and for large q2 is 1/4 of the corresponding value for a lepton. According to the optical theoremthe absorptive part may be written in terms of the e+e− → γ∗ → π+π− production cross–sectionσπ+π−(s) as

Im Π′ hadγ (s) =

s

4πασhad(s)

which hence we can read off to be

σπ+π−(s) =πα2

3sβ3π

again with βπ =√

(1− 4m2π/s) the pion velocity. This means that sQED predicts

Fπ(s) = 1 , (8.41)

in view of Fig. 8.14 a complete failure up to 50 : 1. Therefore sQED in general only can makesense if we replace e → e Fπ(q2), e2 → e2 |Fπ(q2)|2 and take the form factor from experimentor from some more realistic model like the VMD model and improvements thereof. It meansthat the bound state nature of the charged pion is taken care off by introducing a non-trivial

202

q2–dependent pion form factor. Note that in the low energy limit (classical limit) electromagneticcurrent conservation implies

Fπ(0) = 1 , (8.42)

a mandatory constraint.

Often one has to resort to sQED in particular in connection with the soft photon radiationproblem of charged particles, where sQED provides a good description of the problem. However,in calculating the pion contribution the photon vacuum polarization, we would include hardphotons when we apply it in the region of interest above the π+π− production threshold to about1 GeV or even higher. In conclusion: sQED in the classical limit povides the exact describtion ofphoton radiation, in particular for what concerns the IR problem (Bloch-Nordsieck prescription).The hard end of the radiation spectrum is also known exactly, as given by the quark–photoninteraction (QED of quarks). An experimental “proof” that the latter is correct is the reasonablygood QPM prediction of R(0)(s), discussed before. The intermediate range is presently not wellunder control, however.

8.15.2 The Vector Meson Dominance Model

As strict sQED fails badly as soon as the ρ meson, with mass about 776 MeV, is approacched,another phenomenologically motivated mechanism comes into play. Actually Fig. 8.14 suggeststhat the photon itself has a hadronic component described in leading approximation by the ρ0 viaa direct coupling, i.e., by γ− ρ0–mixing. Indeed in photon–hadron interactions the photon mixeswith hadronic vector–mesons like the ρ0. The naive VMD model attempts to take into accountthis hadronic dressing by replacing the photon propagator as

i gµν

q2+ · · · → i gµν

q2+ · · · −

i (gµν − qµqν

q2)

q2 −m2ρ

=i gµν

q2m2ρ

m2ρ − q2

+ · · · ,

where the ellipses stand for the gauge terms. Of course real photons q2 → 0 in any case remainundressed and the dressing would go away for m2

ρ → ∞. The main effect is that it provides adamping at high energies with the ρ mass as an effective cut–off (physical version of a Pauli-Villars cut–off). However, the naive VMD model does not respect chiral symmetry properties.More precisely, the hypothesis of vector–meson dominance [47] relates the matrix element of thehadronic part of the electromagnetic current jhadµ (x) to the matrix element of the source density

J (ρ)(x) of the neutral vector meson ρ0 by

〈B|jhadµ (0)|A〉 = −M2ρ

2γρ

1

q2 −M2ρ

〈B|J (ρ)µ (0)|A〉

where q = pB − pA, pA and pB the four momenta of the hadronic states A and B, respectively,Mρ is the mass of the ρ meson. So far our VMD ansatz only accounts for the isovector part, butthe isoscalar contributions mediated by the ω and the φ mesons may be included in exactly thesame manner, as shown in Fig. 8.16. The key idea is to treat the vector meson resonances like theρ as elementary fields in a first approximation. Free massive spin 1 vector bosons are describedby a Proca field Vµ(x) satisfying the Proca equation ( + M2

V ) Vµ(x) − ∂µ (∂νVν) = 0, which is

designed such that it satisfies the Klein-Gordon equation and at the same time eliminates theunwanted spin 0 component: ∂νV

ν = 0. In the interacting case this equation is replaced by acurrent–field identity (CFI) [47]

( +M2V ) Vµ(x)− ∂µ (∂νV

ν) = gV J(V )µ (x)

203

=∑

V =ρ0, ω, φ,···

B

A

B

AV

γγ

=M2

V

2γV

; =−1

q2 −M2V

.

Figure 8.16: The vector meson dominance model. A and B denote hadronic states.

where the r.h.s. is the source mediating the interaction of the vector meson and gV the coupling

strength. The current should be conserved ∂µJ(V )µ (x) = 0. The CFI then implies

〈B|Vµ(0)|A〉 = − gVq2 −M2

V

〈B|J (V )µ (0)|A〉 (8.43)

where terms proportional to qµ have dropped due to current conservation. The VMD assumesthat the hadronic electromagnetic current is saturated by vector meson resonances41

jhadµ (x) =∑

V=ρ0, ω, φ,···

M2V

2γVVµ(x)

such that, e.g.

〈ρ(p)|jhadµ (0)|0〉 = ε(p, λ)µM2ρ

2γρ, p2 = M2

ρ . (8.45)

The mass–dependent factor M2V must be there for dimensional reasons, γV is a coupling constant

introduced in this form by convention. The VMD relation (8.43) thus derives from the CFI andansatz (8.45). The VMD model is known to describe the gross features of the electromagneticproperties of hadrons quite well, most prominent example are the nucleon form factors.

A way to incorporate vector–mesons ρ, ω, φ, . . . in accordance with the basic symmetries of QCDis the Resonance Lagrangian Approach (RLA) [52,53], an extended version of CHPT [54,55] (seeSect. 3.4), which also implements VMD in a consistent manner. In the flavor SU(3) sector, similarto the pseudoscalar field Φ(x) (3.16), the SU(3) gauge bosons conveniently may be written as a3× 3 matrix field

Vµ(x) =∑

i

TiVµi =

ρ0√2

+ ω8√6

ρ+ K∗+

ρ− −ρ0√2

+ ω8√6

K∗0

K∗− K∗0 −2 ω8√

6

µ

(8.46)

in order to keep track of the appropriate SU(3) weight factors.

41In large Nc QCD [48–50] all hadrons become infinitely narrow, since all widths are suppressed by powers of1/Nc, and the VMD model becomes exact with an infinite number of narrow vector meson states. The large-Nc

expansion attempts to approach QCD (Nc = 3) by an expansion in 1/Nc. In leading approximation in the SU(∞)theory R(s) would have the form [51]

R(s) =9π

α2

∞∑

i=0

Γeei Mi δ(s−M2

i ) . (8.44)

204

8.15.3 Mass and Width of Vector Mesons

All hadronic vector mesons are unstable, usually decaying predominantly into hadrons, like ρ0 →π+π− or ω → π+π−π0, and therefore the zero width approximation breaks down near resonance.For a more realistic describtion of the vector mesons their finite width has to be taken intoaccount. We may understand the finite width as a self-energy effect, via the Dyson summationof the self-energy insertions. The procedure is very similar to the one discussed for the gluonpropagator Sect. 7.2, except that we are dealing with a massive spin 1 boson here. As usualwe decompose the spin 1 vector boson self-energy Πµν(q2,m2, · · ·) into a transverse Π(q2) and alongitudinal L(q2) part

Πµν(q2,m2, · · ·) =

(gµν −

qµqνq2

)Π(q2) +

qµqνq2

L(q2), (8.47)

where for the definition of the mass and width only the tranverse part is relevant. We then onlyconsider the transversal part, and suppress the transversal projector, which is an overall factor.The full or dressed propagator is then given by the geometrical progression

= + + + · · ·

=−i

q2 −m2+

−i

q2 −m2(−i Π)

−i

q2 −m2+

−i

q2 −m2(−i Π)

−i

q2 −m2(−i Π)

−i

q2 −m2+ · · ·

=−i

q2 −m2

1 +

( −Π

q2 −m2

)+

( −Π

q2 −m2

)2

+ · · ·

=−i

q2 −m2

1

1 + Πq2−m2

=

−i

q2 −m2 + Π(q2)≡ −i D(q2) . (8.48)

The dressed propagator then reads

Dµν(q) =−i

q2 −m2 + Π(q2)

(gµν −

qµqνq2

)+qµqνq2· · · . (8.49)

The position of the pole M2 of the propagator of a massive spin 1 boson in a quantum field theoryis a solution for q2 at which the inverse of the connected full propagator equals zero, i.e.,

M2 −m2 + Π(M2,m2, · · ·) = 0, (8.50)

where Π(q2, · · ·) is the transversal part of the one-particle irreducible self-energy. The latterdepends on all parameters of the theory but, in order to the keep notation simple, we haveindicated explicitly only the dependence on the external momentum q and in some cases also m,where m is the mass of the particle under consideration. This can be either the bare mass m0 orthe renormalized mass defined in some particular renormalization scheme.

Generally, the pole M2 is located in the complex plane of q2 and has a real and an imaginarypart. We write

M2 ≡M2 − iMΓ. (8.51)

The real part of (8.51) defines M which we call the pole mass, while the imaginary part is relatedto the width Γ of the particle. This is the natural generalization of the physical mass of a stableparticle, which is defined by the mass of its asymptotic scattering state. M always denotes thepole mass (≡ on-shell mass).

205

We note the following properties of Πρ:1) Πρ(q

2) is complex

Πρ = Re Πρ + i Im Πρ

when q2 > 4m2π, taking into account only the strong interaction under which ρ decays into

pion–pairs. The relevant physical parameters are Mρ = (775.8 ± 0.5) MeV, mπ± = (139.57018 ±0.00035) MeV and Γρ = (145.14±1.50) MeV. There is also a tiny direct electromagnetic ρ→ e+e−

channel which has a branching fraction Br(ρ→ e+e−) = (4.67 ± 0.09) · 10−5, which is not takeninto account here.

As disussed earlier, the imaginary part corresponds to the cut diagram

mπ

mπ

q

with physical intermediate state. Consequently, the ρ is an unstable particle and on the mass-shellq2 = M2

ρ of the ρ we have

Im Πρ(q2 = M2

ρ ) = Mρ Γρ 6= 0 (8.52)

defining the finite width Γρ of the ρ-meson. The real part Re Πρ is UV-divergent and requiresrenormalization: The bare propagator

Dρ b =1

q2 −M2ρ b

with Mρ b the bare ρ mass in modified by self–energy effect to

Dρ b =1

q2 −M2ρ b + Πρ(q2)

and required mass and wave–function renormalization:

M2ρb = M2

ρ + δM2ρ

where δM2ρ is the mass counter-term fixed by the condition:

Re[q2 −M2

ρ − δM2ρ + Πρ(q

2)] ∣∣∣q2=M2

ρ= 0

mδM2

ρ = Re Πρ(M2ρ )

(8.53)

this removes the quadratically divergent term from the ρ self-energy, and defines the physicalmass by

M2ρ = Re M2

ρ . (8.54)

The remaining logarithic UV divergence is removed by the condition that

• the real part of the residue of the propagator pole must be normalized to one.

206

If we expand the self-energy at the pole

Πρ(q2) ≃ Πρ(M

2ρ ) + (q2 − M2

ρ )dΠρ

dq2(M2

ρ ) + · · · ; q2 → M2ρ

we obtain, using δM2ρ = Re Π(M2

ρ ), MρΓρ = Im Π(M2ρ ),

Dρ =1

q2 −M2ρ +

(Πρ(q2)− Re Πρ(M2

ρ ))

=1

q2 − M2ρ

1

1 +dΠρ

dq2(M2

ρ )+O(q2 − M2

ρ )

and the residue of the pole can be read off. If we now perform the field renormalization andconsider the propagator of the renormalized field Dρ ren = Z−1ρ Dρ bare i. e.

1

q2 −M2ρ + Πρ ren(q2)

=1

Zρ· 1

q2 −M2ρ +

(Πρ(q2)−Re Πρ(M2

ρ )) (8.55)

which is required to have residue one and thus

Zρ = Re

[1 +

dΠρ

dq2(M2

ρ )

]−1. (8.56)

All together we have

Dρ ren =1

q2 −M2ρ + Πρ ren(q2)

=1

q2 − M2ρ

+O(q2 − M2ρ )

which is he result we were looking for. Note that off-pole corrections may be substantial, inpariclular, because Γρ/Mρ ∼ 0.2 is not small. Also notice that we have not used here that theself–energy is a perturbative correction, as usual in perturbative problems. Neither pQCD norCHPT do apply here.

In summary: the unstable spin 1 vector meson is described by a propagator exhibiting a pole inthe complex q2-plane. The pole

M2ρ ≡

(q2)pole

= M2ρ − i Mρ Γρ (8.57)

is characterized by the correspondence

physical mass ⇐⇒ real part of location of propagator pole

width ⇐⇒ imaginary part of the location of the pole .

8.15.4 ρ0 − γ–Mixing

The simple relation between the full propagator and the irreducible self-energy only holds if thereis no mixing, like for the charged ρ±. In the neutral sector, because of γ − ρ0 mixing, we cannot

207

consider the ρ0 and γ propagators separately. They form a 2×2 matrix propagator, so that (8.50)is modified into42

sP −m2ρ0 −Πρ0ρ0(sP )−

Π2γρ0(sP )

sP −Πγγ(sP )= 0 , (8.60)

with sP = M2ρ0 . The above considereatio remains true if we denote the self-energies by ΠV

(V = ρ0, ρ±) withΠρ±(p2, · · ·) = Πρ+ρ−(p2, · · ·)

and

Πρ0(p2, · · ·) = Πρ0ρ0(p2, · · ·) +Π2γρ0(p2, · · ·)

p2 −Πγγ(p2, · · ·) .

Thus, formally, the form (8.50) applies for both the ρ± and the ρ0.

8.15.5 Vector–Meson Production Cross–Sections

The cross–section for spin 1 boson production in e+e−–annihilation is described in full deail inSect. 14 of [1]. We have to replace (8.20) by (8.45) and use the completeness relation

∑

λ

ε(p, λ)µ ε∗(p, λ)ν =

(−gµν −

pµpνM2ρ

)

for the massive spin 1 boson polarization vectors ε(p, λ), to calculate the hadronic tensor

hµν =1

3

∑

λ

ε(p, λ)µM2ρ

2γρε∗(p, λ)ν

M2ρ

2γρ=

(−gµν −

pµpνM2ρ

)M4ρ

2γ2ρ,

needed the caculate the spin average |T |2 in (8.5).

The result for the e+e− → ρ0 cross–section and some useful approximations are the following:

42The simplest way to treat this problem is to start from the inverse propagator given by the irreducible self-energies (sum of 1pi diagrams). Again we restrict ourselves to a discussion of the transverse part and we take outa trivial factor −i gµν in order to keep notation as simple as possible. With this convention we have for the inverseγ − ρ propagator the symmetric matrix

D−1 =

k2 +Πγγ(k2) Πγρ(k

2)

Πγρ(k2) k2 −M2

ρ +Πρρ(k2)

(8.58)

Using 2× 2 matrix inversion

M =

a b

b c

⇒M−1 =1

ac− b2

c −b−b a

we find for the propagators

Dγγ =1

k2 +Πγγ(k2)− Π2γρ(k

2)

k2−M2ρ+Πρρ(k2)

≃ 1

k2 +Πγγ(k2)

Dγρ =−Πγρ(k

2)

(k2 +Πγγ(k2))(k2 −M2ρ +Πρρ(k2))− Π2

γρ(k2)≃ −Πγρ(k

2)

k2 (k2 −M2ρ )

Dρρ =1

k2 −M2ρ +Πρρ(k2)− Π2

γρ(k2)

k2+Πγγ(k2)

≃ 1

k2 −M2ρ +Πρρ(k2)

. (8.59)

These expressions sum correctly all the reducible bubbles. The approximations indicated are the one-loop results.The extra terms are higher order contributions.

208

• Breit-Wigner resonance: field theory version

The field theoretic form of a Breit-Wigner resonance obtained by the Dyson summation of amassive spin 1 transversal part of the propagator in the approximation that the imaginary partof the self–energy yields the width by Im ΠV (M2

V ) = MV ΓV near resonance.

σBW (s) =12π

M2R

Γe+e−

Γ

sΓ2

(s −M2R)2 +M2

RΓ2.

• Breit-Wigner resonance

The resonance cross–section from a classical non–relativistic Breit-Wigner resonance is given by

σBW (s) =3π

s

ΓΓe+e−

(√s−MR)2 + Γ2

4

.

• Narrow width resonance

The narrow width approximaion for a zero width resonance reads

σNW(s) =12π2

MRΓe+e−δ(s −M2

R) .

Exercise: Derive the non-relativistic and the narrow width approximation from the field theoryversion of the resonance cross–section.

For broad resonances the different parametrizations of the resonance in general yield very differentresults. Therefore, it is important to know how a resonance was parametrized to get the resonanceparameters like MR and Γ. For narrow resonances, which we will have to deal with later, resultsare not affected in a relevant way by using different parametrizations. Note that for the broadnon–relativistic ρ meson the classical BW parametrization works well, not however a narrowwidth approximation.

Due to isospin breaking of the strong interactions (md −mu mass difference as well as electro-magnetic effects Qu = 2/3 6= Qd = −1/3) the ρ and ω mix and more sophisticated parametriza-tions must be applied, like the Gounaris-Sakurai parametrization [56] based on the vector mesondominance (VMD) model. More appropriate is a parametrization which relies on first principleconcepts only, the description by unitarity, analyticity and constrained by chiral perturbationtheory. In the following we will quantify these issues in more detail.

8.15.6 ρ0 − ω–Mixing

The electromagnetic form factor of the pion Fπ(s) usually is defined in an idealized world ofstrong interactions with two quark flavors (u and d) only, and electroweak interactions switchedoff. In γ∗ → π+π− the pion form factor Fπ(s) is defined as a matrix element of the externalelectromagnetic current and has an isovector part I = 1 as well as an isoscalar part I = 0. Thelatter is due to isospin breaking by the mass difference of the u and d quarks: mu − ms 6= 0,which leads to ρ− ω mixing:

|ρ〉 = |ρ0〉 − ε|ω0〉 , |ω〉 = |ω0〉+ ε|ρ0〉 , (8.61)

209

Figure 8.17: CMD-2 data for |Fπ|2 in ρ−ω region together with Gounaris-Sakurai fit. Left beforesubtraction right after subtraction of the ω.

where |ω0〉 and |ρ0〉 are the pure isoscalar and isovector states, respectively, and ε is the ρ − ωmixing parameter. Then, in the energy region close to the ρ(776)– and ω(783)–meson masses,the form factor can be written as

Fπ(s) ≃[

Fρs−M2

ρ

+ εFω

s−M2ω

] [Fρ−M2

ρ

+ εFω−M2

ω

]−1

≈ −M2ρ

s−M2ρ

[1 + ε

Fω(M2ω −M2

ρ )s

FρM2ω(s−M2

ω)

], (8.62)

where we only keep the terms linear in ε. The quantities Mω and Mρ are complex and containthe corresponding widths.

The mixing is responsible for the typical distortion of the ρ–resonance (see Figs. 8.17 and 8.14),which originally would be a pure isospin I = 1 Breit-Wigner type resonance. The pion form factor(8.62) is the basic ansatz for the Gounaris-Sakurai formula [56] which is often used to representexperimental data by a phenomenological fit (see e.g. [24]).

8.15.7 Gounaris-Sakurai Parametrization of Fπ(s)

Based on VMD ideas and ρ−ω–mixing, Gounaris and Sakurai [56] proposed as a parameterizationof Fπ (see [25, 56] for details) (FSR not included; BW=Breit-Wigner; BWGS=Gounaris-Sakuraimodified Breit-Wigner):

Fπ(s) =BWGS

ρ(770)(s) ·(1+δ BWω(783)(s)

1+δ

)+ β BWGS

ρ(1450)(s) + γ BWGSρ(1700)(s)

1 + β + γ, (8.63)

where we included also the higher resonances ρ′ and ρ′′. This formula often is used to represent

experimental pion–pair production data. We may also use this formula to extract the isovectorpart of the square of the pion form factor |Fπ|2 I=1

(e+e−)(s) by setting the mixing parameter δ = 0.The result is displayed in Fig. 8.17, and illustrates the effect of ρ− ω mixing at the ρ–resonance.For the ρ(770), ρ(1450) and ρ(1700) the GS parametrization is used [56]:

BWGSρ(Mρ)

=M2ρ (1 + d · Γρ/Mρ)

M2ρ − s+ f(s)− iMρΓρ(s)

, where

210

where Mρ,Γρ denote mass and width of the ρ–meson and

f(s) = ΓρM2ρ

p3π(M2ρ)

[p2π(s)

(h(s)− h(M2

ρ))

+ (M2ρ − s) p2π(M2

ρ)dh

ds

∣∣∣∣s=M2

ρ

]. (8.64)

Here pπ(s) = 12

√sβπ(s) denotes the pion momentum with βπ =

√1− 4m2

π/s the pion velocity inthe c.m. frame,

h(s) =2

π

pπ(s)√s

ln

√s+ 2pπ(s)

2mπ, (8.65)

such that

dh

ds

∣∣∣∣s=M2

ρ

= h(M2ρ )[(8p2π(M2

ρ))−1 − (2M2

ρ)−1]+ (2πM2

ρ)−1 . (8.66)

The constant d is chosen to satisfy BWGSρ(Mρ)

(0) = 1:

d =3

π

m2π

p2π(M2ρ)

lnMρ + 2pπ(M2

ρ)

2mπ+

Mρ

2πpπ(M2ρ)− m2

πMρ

πp3π(M2ρ)

. (8.67)

For the energy dependence of the ρ(770) width, the P-wave phase space is taken:

Γρ(s) = Γρ

[pπ(s)

pπ(M2ρ)

]3 [M2ρ

s

]1/2. (8.68)

For the ω(782)-meson contribution the simple Breit-Wigner parametrization with a constantwidth is used. We further note that the leptonic width is given by

Γρ→e+e− =2α2p3π

(M2ρ

)

9MρΓρ

(1 + d · Γρ/Mρ)2

(1 + β)2. (8.69)

and the branching ratio of the ω → π+π− decay by

Br(ω → π+π−) =2α2p3π(M2

ω)

9MωΓω→e+e−Γω

∣∣∣BWGSρ(770)(M

2ω)∣∣∣2 |δ|2

(1 + β)2. (8.70)

8.15.8 The Theory of the Pion Form Factor in the Threshold Region

We first concentrate on the threshold behavior of Fπ(s). Experimental data are poor below about400 MeV because the cross section is suppressed near the threshold by phase space. Here the lowenergy structure of QCD as represented by chiral perturbation theory (CHPT), which we haveoutlined in Sect. 3.4, provides at appropriate “from first principles” framework. Applications tothe the pion form factor have been discussed in [55,57,58].

We are interested in low energy pion production matrix elements of the form

〈πi1(p1) · · · πin(pn) out|Iiµ(0)|0〉 (8.71)

where the current Iiµ = V iµ = q T i γµq when n is even and Iiµ = Aiµ = q T i γµγ5q when n is odd.

V iµ and Aiµ are the vector and axial vector currents, respectively, belonging to the chiral flavor

group SU(2)V ⊗ SU(2)A, which is exact only in the chiral limit when quark masses are taken

to be zero. The SU(2) generators are given by T i = σi

2 in terms of the Pauli matrices σi. In

211

the chiral limit the chiral group is spontaneously broken down to SU(2)V and according to theGoldstone theorem three massless Goldstone bosons must exist: the pions: π±, π0. The brokenphase is characterized by the light quark condensates 〈qq〉 6= 0 q = u, d as order parameters. Ifquark masses are switched on the pions acquire a mass, which however remains small because theu and d quarks turn out to be surprisingly light in the real world (see (3.14) ). At physical quarkmasses the value of the condensate is estimated to be 〈mq qq〉 ∼ −(0.098 GeV)4 for q = u, d.

The isospin symmetry limit corresponds to mu = md = m = 12 (mu+md) which in most practical

cases is a rather good approximation as mu−md is small. Since the quark charges are Qu = 2/3and Qd = −1/3, isospin is broken by electromagnetism at O(α). More precisely, in the SM thecharge is given by the Gell-Mann–Nishijima relation Q = T3 + Y

2 , and thus the electromagneticcurrent is of the form

jµ hadem = ψγµ

Y + σ32

ψ

with ψ =(ud

)the 1st family quark isospin doublet. The form shows that it is a mixture of an

isoscalar part proportional to the hypercharge Y and an isovector part proportional to the 3rdcomponent of an isovector. Similarly, the difference in the weak isospin charges T3u = 1/2 andT3d = −1/2 leads to isospin breaking by the weak interactions. Typical isospin violation effectsare the mπ± −mπ0 electromagnetic mass difference or ρ− ω–mixing, discussed below.

a) The vector form factor in CHPT:The hadronic matrix element of interest in pion pair production is

〈πi(p1)πj(p2) out|V kµ (0)|0〉 = iεijk (p1 − p2)µ FV (s) (8.72)

with s = (p1 + p2)2. The vector form factor FV (s) has been calculated in CHPT in [55] (one–

loop), [57] (two–loop numerical) and [58] (two–loop analytical). The last reference gives a compactanalytical result

FV ( s ) = 1 +1

6〈r2〉πV s + cπV s

2 + fUV

(s

m2π

)(8.73)

fUV (x ) =m2π

16π2f2π

x

9

(1 + 24π2σ2J(x)

)− x2

60

(8.74)

+

(m2π

16π2f2π

)2 [

l2 − l1 +l62

+6l4x

]x2

27

(1 + 24π2σ2J(x)

)− x2

30l4

+3191

6480x2 +

223

216x − 16

9− π2 x

540( 37x + 15 )

+4π2

27( 7x2 − 151x + 99 ) J (x) +

2π2

9x(x3 − 30x2 + 78x− 128 )K1(x)

+ 8π2(x2 − 13

3x− 2

)K4(x)

.

Where we have used the following functions:

J(x ) =1

16π2(F (x ) + 2 ) ,

K1(x ) =1

16π2F 2(x)

σ2,

K4(x ) =1

16π2F (x)

xσ2+

1

32π21

xσ2

[F 2(x)

σ2+ π2

](8.75)

+1

48π21

xσ2

1

xσ4[F 3(x) + π2σ2F (x)

]− π2

+

1

192− 1

32π2,

212

with

F (x ) = σ lnσ − 1

σ + 1, σ =

√1− 4/x .

The functions Ki(x ) and J(x ) are analytic everywhere apart from a branch cut from 4 to ∞,and go to zero as x→ 0.

All the constants which occur in fUV ( s/m2π ) are known. The subtraction constants 〈r2〉πV and cπV

are calculable in CHPT and can be expressed in terms of the low energy constants lri (µ), chirallogs, and the new low energy constants which appear in L6. With this representation of thesubtraction constants given by CHPT one automatically satisfies the relevant Ward identities, upto the order at which one is working. We do not give this explicit representation here because upto now there is no information on the numerical value of the new L6 low energy constants. In thefuture, with more accurate data on various low energy processes, and more two loop calculationavailable, one could try to pin down at least some of them, but this will require a considerableamount of work and it is beyond the scope of this introduction. A fit to the space–like NA7data [59] with the expression (8.73) leaving 〈r2〉πV and cπV as free parameters, and including thetheoretical error, leads to

〈r2〉πV = 0.431 ± 0.020 ± 0.016 fm2

cπV = 3.2 ± 0.5 ± 0.9 GeV−4 (8.76)

where the first and second errors indicate the statistical and theoretical uncertainties, respectively.The central value of cπV is rather close to the value obtained by resonance saturation, cπV =4.1 GeV−4 [57].

The crucial point here is that the threshold behavior is severely constrained by the chiral structureof QCD via the rather precise data for the pion form factor in the space–like region. The space–like fit provides a good description of the data in the time-like region. Pure chiral perturbationtheory is able to make predictions only for the low energy tail of the form factor. However,theory in this case can do much more, by exploiting systematically analyticity, unitarity andthe properties of the chiral limit. This relates space–like data, ππ–scattering phase shifts andtime–like data in a manner leading to severe theoretical constraints! This we will consider next.

b ) The pion form factor a la Omnes–Muskhelishvili:Some basic concepts we need for a “from first principles” study of the pion form factor we alreadyencountered in the first part of this Section. In particular, the role of analyticity and unitaritywe have discussed in some detail above. First attempts and ideas to model Fπ(s) followed inSects. 8.15.1 and 8.15.2. The electromagnetic form factor of the pion Fπ(s) in first place isdefined in a world with strong interactions and two quark flavors (u and d) only, electroweakinteractions are switched off. Fπ(s) has an isovector part I = 1 as well as an isoscalar part I = 0.The latter is due to isospin breaking by the mass difference of the u and d quarks: mu−ms 6= 0,which leads to ρ − ω mixing outlined before in Sect. 8.15.6. The mixing is responsible for thetypical distortion of the ρ–resonance (see Fig. 8.14), which originally would be a pure isospinI = 1 Breit-Wigner type resonance. The isospin symmetry limit is also the starting point of ourtheoretical consideration.

The corresponding electromagnetic vector current form factor Fπ(s) has the following proper-ties [60,61]:i) Fπ(s) is an analytic function of s in the whole complex s–plane, except for a cut on the positivereal axis for 4m2

π ≤ s <∞. If we approach the cut from above s→ s+ iε, ε > 0, ε→ 0 the formfactor remains complex and is characterized by two real functions, the modulus and the phase

Fπ(s) = |Fπ(s)| ei δ(s) ; (8.77)

213

ii) analyticity relates Re Fπ(s) and Im Fπ(s) by a DR, which may be expressed as a relationbetween modulus and phase δ(s) = arctan(Im Fπ(s)/Re Fπ(s)), known as the Omnes representa-tion [62]

Fπ(s) = P (s) exp

s

π

∫ ∞

4m2π

ds′δ(s′)

s′ (s′ − s)

(8.78)

where P (s) is a polynomial, which determines the behavior at infinity, or, equivalently, the numberand position of the zeros;iii) charge conservation Fπ(0) = 1, which fixes P (0) = 1;iv) Fπ(s) is real below the 2 pion threshold (−∞ < s < 4m2

π), which implies that P (s) must bea polynomial with real coefficients;v) the inelastic threshold is sinel = 16m2

π.

The representation (8.78) tells us that once we know the phase on the cut and the location of thezeros of P (s) the form factor is calculable in the entire s–plane. In the elastic region s ≤ sinelWatson’s theorem43, exploiting unitarity, relates the phase of the form factor to the P wave phaseshift of the ππ scattering amplitude with the same quantum numbers, I = 1, J = 1:

δ(s) = δ11(s) for s ≤ sinel = 16m2π . (8.82)

However, it is an experimental fact that the inelasticity is negligible until the quasi two–bodychannels ωπ, a1π, are open, thus in practice one can take (8.82) as an excellent approximationup to about 1 GeV (while

√sinel ≃ 0.56 GeV). Actually, the phase difference (8.82) satisfies the

bound [63]

sin2(δ(s)− δ11(s)) ≤ 1

2[1−

√1− r2(s) ] , r(s) =

σI=1non−2π

σe+e−→π+π−

and η1 ≤ (1 − r)/(1 + r), provided r < 1, which holds true below 1.13 GeV (below 1 GeVr < 0.143 ± 0.024, or δ − δ11<∼6, strongly decreasing towards lower energies).

The ππ scattering phase shift is due to elastic re-scattering of the pions in the final state (finalstate interaction) as illustrated by Fig. 8.18

The ππ scattering phase shift has been studied recently in the framework of the Roy equations(see [64]), first with no extra input [65], and also exploiting in addition chiral symmetry [66–69].

43The pion isovector form factor is defined by the matrix element

〈out π+(p+)π−(p−)|jµ(0)|0〉 = −i(p+ − p−)µ Fπ(s) , (8.79)

where jµ(x) is the electromagnetic vector current and s = (p+ + p−)2. The π+π− state in this matrix element, in

order not to vanish, must be in a I = 1, J = 1 (P wave) state, J the angular momentum. If we look at the chargedensity j0, time-reversal (T ) invariance tells us that

〈out π+π−|j0(0)|0〉 = 〈in π+π−|j0(0)|0〉∗ , (8.80)

as for fixed J only “in” and “out” get interchanged. The complex conjugation follows from the fact that T mustbe implemented by an anti-unitary transformation. Now, with S the unitary scattering operator, which transformsin and out scattering states according to |X out〉 = S+|X in〉 (X the label of the state) we have

〈out π+π−|j0(0)|0〉 = 〈in π+π−|Sj0(0)|0〉= e2iδππ 〈in π+π−|j0(0)|0〉= e2iδππ 〈out π+π−|j0(0)|0〉∗ (8.81)

which implies Fπ(s) = e2iδππ F ∗π (s). As two pions below the inelastic thresholds may scatter elastically only, by

unitarity the S-matrix must be a pure phase in this case. The factor 2 is a convention, δππ(s) is the ππ–scatteringphase shift.

214

FπIm ⇔

Figure 8.18: Final state interaction due to ππ → ππ scattering

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1E (GeV)

0

20

40

60

80

100

120

140

160

180

δ 11

Figure 8.19: The ππ phase shift δ11 : the values of the phase at the two points (marked by •l) arefree; the Roy equation and chiral symmetry completely fix the solution (from [61])

As a result it turns out that δ11(s)

is constrained to a remarkable degree of accuracy up to about E0 = 0.8 GeV (matching point).The behavior of δ11(s) in the region below the matching point is controlled by three parameters:two S–wave scattering length a00, a

20 and by the boundary value φ ≡ δ11(E0). One may treat φ

as a free parameter and rely on the very accurate predictions for a00, a20 from chiral perturbationtheory. This information may be used to improve the accuracy of the pion form factor and thusto reduce the uncertainty of the hadronic contribution to the muon g-2. Similar approaches,however, not using the constraints from CHPT, have been followed in [70, 71]. Before we canapply the outlined method to reality, we have to take into account the isospin breaking. For thispurpose we separate off an other factor which takes into account the I = 0 contribution due toρ− ω–mixing. Accordingly we now write the form factor in the form [60,61]

Fπ(s) = G1(s) ·Gω(s) ·G2(s) .

• G1(s) is the Omnes-Muskhelishvili factor describing the cut due to the 2π intermediate states:

G1(s) = exp

s

π

∫ ∞

4m2π

ds′δ11(s′)

s′ (s′ − s)

. (8.83)

• Gω(s) accounts for the ω–pole contribution due to ρ− ω–mixing with mixing amplitude ε:

Gω(s) = 1 + εs

sω − s+ . . . sω = (Mω −

1

2iΓω)2 .

215

In order to get it real below the physical thresholds we use an energy dependent width

Γω(s) = Γ(ω → 3π, s) + Γ(ω → π0γ, s) + Γ(ω → 2π, s)

=s

M2ω

Γω

Br(ω → 3π)

F3π(s)

F3π(M2ω)

+Br(ω → π0γ)Fπγ(s)

Fπγ(M2ω)

+Br(ω → 2π)F2π(s)

F2π(M2ω)

where Br(V → X) denotes the branching fraction for the channel X and FX(s) is the phase spacefunction for the corresponding channel normalized such that FX(s)→ const for s→∞. For thetwo-body decays V → P1P2 we have FP1P2(s) = (1− (m1 + m2)

2/s)3/2. The channel V → 3π isdominated by V → ρπ → 3π and this fact is used when calculating F3π(s) [72].

Low energy singularities generated by states with 2 or 3 pions are accounted for by the firsttwo factors of the “master equation” above. The remaining function G2(s) represents the smoothbackground that contains the curvature generated by the remaining singularities. The 4 π channelopens at s = 16 m2

π but phase space strongly suppresses the strength of the corresponding branchpoint singularity of the form (1 − sinel/s)9/2 – a significant inelasticity only manifests itself fors > sin = (Mω +mπ)2. The conformal mapping

z =

√sin − s1 −

√sinel − s√

sinel − s1 +√sin − s

maps the plane cut along s > sinel onto the unit disk in the z–plane. It contains a free parameter s1- the value of s which gets maps into the origin. We find that if s1 is taken negative and sufficientlyfar from the origin the fit becomes rather insensitive to the details of the parametrization. In thefollowing we set s1 = −1.0 GeV2. We approximate G2(s) by a polynomial in z:

G2(s) = 1 +

nP∑

i=1

ci (zi − zi0) ,

where z0 is the image of s = 0. The shift of z by z → z − z0 is required to preserves the chargenormalization condition G2(0) = 1. The form of the branch point singularity (1 − sinel/s)

9/2

imposes four constraints on the polynomial; a non-trivial contribution from G2(s) thus requiresa polynomial of fifth order at least. Varying the degree nP of the polynomial and constrainingthe free coefficients by the data is the only model uncertainty. In this way we let the data fix thelocation of the zeros in the complex s–plane of the Omnes representation (8.78).

Numerical results:The above representation of the vector form factor involves a product of three functions each ofwhich has free parameters which are fitted by the data. The free parameters are the following:

• in G1 we have two free parameters, the values of the phase δ11 at 0.8 GeV and at 1.15 GeV;

• in G2 we have nP −4 parameters, where nP is the degree of the polynomial in the conformalvariable z. Varying nP within a reasonable range nP = 5, · · · , 8 will allow us to estimatethe model uncertainty;

• in Gω we have in principle three free parameters: εω, Mω and Γω – however, since mass andwidth of the ω are rather well known from other experiments, we take their values from thePDG. Since the ω–resonance is very narrow Gω is very sensitive to the energy calibration,which therefore we allow to vary within the experimental systematic uncertainty.

Altogether we have 4+ (nP −4) free parameters, depending on the degree of the polynomial usedto describe the inelastic effects.

For results we refer to [60,61]. This method provides a systematic way to combine experimentalinformation from |Fπ(s)|2 measurements with corresponding information of the phase of the pionform factor.

216

8.15.9 Beyond Chiral Perturbation Theory: Models with Massive Spin 1 Fields

Chiral perturbation theory does not explicitly account for resonances like the ρ mesons, which in away may be considered as ππ bound states. In order to extend the applicability of CHPT towardshigher energies one has to abandon the pure first principles path of low energy QCD and go to aneffective theory which explicitly takes into account vector meson resonances and combines CHPTwith the Vector Dominance Model (VMD) in a consistent manner. Such a framework has beendeveloped to O

(p4)

in [52,53]

L =f2π4

Tr(DµUDµU † + χU † + χ†U)− 1

4FµνF

µν

−1

2Tr(∇λVλµ∇νV νµ − 1

2m2ρVµνV

µν)− 1

2Tr(∇λAλµ∇νAνµ −

1

2m2aAµνA

µν)

+FV

2√

2Tr(Vµνf

µν+ ) +

iGV√2

Tr(Vµνuµuν) +

FA

2√

2Tr(Aµνf

µν− ), (8.84)

where U = exp(i√

2Φ/fπ), Φ describes the SU(3) octet of pseudoscalar mesons, Vµν (Aµν) isantisymmetric field describing the SU(3) octet of polar-vector (axial-vector) mesons, and Fµν =∂µBν − ∂νBµ is the electromagnetic field strength tensor, where the photon field is denoted byBµ. Further, fπ, FV , GV , FA are constants, whose numerical values are specified in [52,53]. Moredetails concerning definitions and notation the reader may find in the original papers. For variantsof resonance Lagrangian approach we refer to [73–76].

The extension to O(p6)

including P odd anomalous terms has been worked out in [77]. It reads

Leff = L(2) + L(4) + L(6)

L(2) = −1

4FµνF

µν + ψ (iD/−m) ψ + e2 C 〈QU+QU〉

+F 20

4

(〈dµU+dµU〉+ 2B0〈M+U + U+M〉

)

L(4) =i e2Nc

24π2εµναβ ∂

µaνAα〈Q2∂βUU+ + Q2U+∂βU

−1

2QUQ∂βU+ +

1

2QU+Q∂βU〉+ · · ·

= − αNc

12πF0εµναβ F

µνAα∂βπ0 + · · ·

L(6) =3 i α2

32π2ψγµγ5ψ

χ1 〈Q2

(U+dµU + dµUU+

)〉

χ2 〈QU+QdµU − QdµU+QU〉

+ · · ·

+α2

4π2F0χ ψγµγ5ψ ∂

µπ0 + · · · (8.85)

whereM = diag(mu,md,ms) is the mass matrix, Q = diag(2/3,−1/3,−1, 3) is the charge matrixand χ = −(χ1 + χ2)/4.

Exercises: Section ??

① Calculate the exact Born cross section for e+e− → µ+µ−, by replacing the hadronic tensorhµν by the leptonic one ℓµν of the muon. What is the difference to e+e− → e+e−?

② Verify that the subtraction changes the DR of the form (8.13), assumed to converge orsupplemented with a cut–off, to (8.14), which we call a subtracted DR.

217

③ Discuss the positivity properties of the functional P (s) defined as the derivative of

P (s) = P∫ ∞

s0

ds′R(s′)

s′(s′ − s) ,

where R(s) is a positive ordinary function. As a distribution (or linear functional) P (s) hasa derivative. Formally,

P (s) =dP (s)

ds= P∫

ds′R(s′)

(s′ − s)2 .

is an integral over an integrand which is almost everywhere positive.

④ Derive the non-relativistic and the narrow width approximation from the field theory versionof the vector–meson resonance cross–section.

218

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[49] A. V. Manohar, Hadrons in the 1/N Expansion, In: At the frontier of Particle Physics, edM. Shifman, (World Scientific, Singapore 2001) Vol. 1, pp 507–568

[50] H. Leutwyler, Nucl. Phys. Proc. Suppl. 64 (1998) 223; R. Kaiser, H. Leutwyler, Eur. Phys.J. C 17 (2000) 623

[51] B. V. Geshkenbein, V. L. Morgunov, arXiv:hep-ph/9410252

[52] G. Ecker, J. Gasser, A. Pich, E. de Rafael, Nucl. Phys. B 321 (1989) 311

[53] G. Ecker, J. Gasser, H. Leutwyler, A. Pich, E. de Rafael, Phys. Lett. B 223 (1989) 425

[54] J. Gasser, H. Leutwyler, Ann. of Phys. (NY) 158 (1984) 142

[55] J. Gasser, H. Leutwyler, Nucl. Phys. B 250 (1985) 517

[56] G. J. Gounaris, J. J. Sakurai, Phys. Rev. Lett. 21 (1968) 244

[57] J. Gasser, U.-G. Meißner, Nucl. Phys. B 357 (1991) 90

[58] G. Colangelo, M. Finkemeier, R. Urech, Phys. Rev. D 54 (1996) 4403

[59] S. R. Amendolia et al. [NA7 Collaboration], Nucl. Phys. B 277 (1986) 168

[60] H. Leutwyler, hep-ph/0212324

[61] G. Colangelo, Nucl. Phys. Proc. Suppl. 131 (2004) 185; ibid. 162 (2006) 256

[62] R. Omnes, Nuovo Cim. 8 (1958) 316

[63] L. Lukaszuk, Phys. Lett. B 47 (1973) 51; S. Eidelman, L. Lukaszuk, Phys. Lett. B 582(2004) 27

[64] B. Hyams et al., Nucl. Phys. B 64 (1973) 134

221

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[67] I. Caprini, G. Colangelo, J. Gasser, H. Leutwyler, Phys. Rev. D 68 (2003) 074006

[68] B. Ananthanarayan, I. Caprini, G. Colangelo, J. Gasser, H. Leutwyler, Phys. Lett. B 602(2004) 218

[69] H. Leutwyler, hep-ph/0612111; hep-ph/0612112

[70] J. A. Casas, C. Lopez, F. J. Yndurain, Phys. Rev. D 32 (1985) 736.

[71] B. V. Geshkenbein, Phys. Rev. D 61 (2000) 033009

[72] F. M. Renard, Nucl. Phys. B 82 (1974) 1; N. N. Achasov, N. M. Budnev, A. A. Kozhevnikov,G. N. Shestakov, Sov. J. Nucl. Phys. 23 (1976) 320; N. N. Achasov, A. A. Kozhevnikov,M. S. Dubrovin, V. N. Ivanchenko, E. V. Pakhtusova, Int. J. Mod. Phys. A 7 (1992) 3187

[73] M. Bando, T. Kugo, S. Uehara, K. Yamawaki, T. Yanagida, Phys. Rev. Lett. 54 (1985) 1215;M. Bando, T. Kugo, K. Yamawaki, Phys. Rept. 164 (1988) 217; M. Harada, K. Yamawaki,Phys. Rept. 381 (2003) 1

[74] U. G. Meissner, Phys. Rept. 161 (1988) 213; see also S. Weinberg, Phys. Rev. 166 (1968)1568

[75] A. Dhar, R. Shankar, S. R. Wadia, Phys. Rev. D 31 (1985) 3256;D. Ebert, H. Reinhardt, Phys. Lett. B 173 (1986) 453

[76] J. Prades, Z. Phys. C63, 491 (1994); Erratum: Eur. Phys. J. C11, 571 (1999).

[77] J. Bijnens, L. Girlanda, P. Talavera, Eur. Phys. J. C 23 (2002) 539

222

9 Quarkonia

Quarkonia name the qq meson states of the heavy quarks q = c and b. Correspondingly we havetwo series of new states the Charmonium and the Bottomonium states. The top quark is tooshort lived under weak decays that it is able to form corresponding toponium bound states. Toclassify charmonium and bottomonium states we remind about the corresponding states in thelight quark sector.

Normal hadrons made of u and d quarks are the pions, the nucleons and the light vector mesonresonances ρ and ω. The latter are JPC = 1−− states:

ρ0 =1√2

(uu− dd) ; ρ+ = ud ; ω =1√2

(uu+ dd) .

Strange hadrons are composed of an s quark together with u, d or s quarks, the Kaons, hyperons

and the vector resonances, which again are JPC = 1−− states:

φ = ss ; K∗+ = us ; K∗0 = ds .

In 1970 Glashow, Illiopoulos and Maiani (GIM) postulated the existence of a heavier weak doubletpartner of the s quark, in order to prevent tree level flavor changing neutral currents, and whichthey named charm c (see Sect. 3.3).

In 1974 Ting [1] and Richter [2] independently discovered a new state called J/ψ in e+e−–annihilation, which actually includes a whole family of resonances the J/ψ spectral family. Thefirst results from SPEAR at SLAC are shown in Fig. 9.1. The plot tells us that the new resonancesmust be very stable. If they would decay easily they would be broad, having a large width. Avery much scaled up version in presented in Fig. 9.2. This second plot clearly demonstrates thatthe hadronic bound states not only decay into hadrons but also directly into lepton pairs. Morerecent substantially improved charm physics results have been obtained with the BES detectorat Beijing.

The new state was interpreted as Charmonium

J/ψ =1√3

∑

a

caca

where a = r(ed), g(reen), b(lue) denotes the 3 color degrees of freedom.

The calculation of hadronic bound states is a key problem of QCD. For light quarks u, d and s per-turbative or semi–perturbative calculation are hopeless, but lattice QCD is making big progresshere with numerical simulations based on the QCD path integral. In contrast, for the relativelyheavy quarks c and b, αs is already relatively small because of the relatively large c.m. energy.Since in addition the constituents are heavy, it is suggestive that methods applied in positroniumphysics or in calculation of hydrogen–atoms could work here as well with appropriate modifi-cations. These methods are based on non–relativistic approximations, i.e., interactions betweenconstituents are approximated by potentials V (r) together with the corresponding Schrodingerequation

[T + V ] Ψ = EΨ , (9.1)

where T is the kinetic energy.

In 1975 Appelquist and Politzer [3] worked out the charmonium spectrum by adapting corre-sponding positronium calculations. Before going into potential model calculations we present abasic phenomenological overview (for a much more detailed review is [4]).

223

Figure 9.1: The total cross section of hadron production vs. center-of-mass energy. As seen fistby Mark I at SLAC (from B. Richter’s Nobel lecture [2]).

9.1 Charmonium

The ground state of charmonium is J/ψ a 3S1 state. The spectroscopic notation is: n2S+1LJ . nis the principal quantum number and J = L+S the total angular momentum, the sum of angularmomentum L and spin S. General properties are

Parity: P = (−1)L+1 = −1

Charge conjugation: C = (−1)L+S = −1

same QN’s as photon

JPC = 1−−,

however, in contrast to the photon they are very heavy: MJ/ψ ∼ 3.1 GeV.

The extraordinary properties of the lowest lying J/ψ states are the large lifetimes! The preciseJ/ψ characteristics are

MJ/ψ = 3096.916 ± 0.011 MeV

Γ(J/ψ)tot = 63.2 ± 2.1 keV

Γ(J/ψ → e+e−) = 5.55± 0.14 ± 0.02 keV

Usually, we would expect such a heavy state to be very instable as there is a lot of energy availableto produce lighter secondary particles. When comparing strong versus electromagnetic decays:

Γ(J/ψ → hadrons) hadronic width (strong interaction) ,

Γ(J/ψ → leptons) leptonic width (elmagn. interaction) ,

we observe a surprisingly small ratio of roughly 10 : 1, between strong and electromagnetic modes,only. The reason for this we will learn to understand in the sequel.

The lowest lying resonance J/ψ(1S) exhibits a number of exited states denoted often by ψ′(2S),ψ

′′, · · ·, where the first exited state has parameters

224

Figure 9.2: Scans of hadron, µ+µ− and e+e− pair production cross section in the regions of theJ/ψ and ψ

′′(from B. Richter’s Nobel lecture [2]).

Mψ′ = 3686.09 ± 0.04 MeV

Γ(ψ′)tot = 317± 9 keV

Γ(ψ′ → e+e−) = 2.38 ± 0.04 keV

About 98% of the decays are hadronic, whereof 57 % of the type ψ′ → J/ψ+anything. Exclusivebranching fraction typically look like:

i) ψ′ → e+e−, µ+µ− (0.75% each) ; ii) ψ′ → (π′s,K ′s) (10%)

iii) ψ′ → J/ψππ (49% ) ; ψ′ → J/ψη (3%)

iv) electromagn. cascades

The electromagnetic cascades proceed via

|−→ χ mesons

ψ′ → · · · + γ|−→ J/ψ + γ

and make possible the transition ψ′ → J/ψ+ γγ while ψ′ → J/ψ+ γ is forbidden by CP parity.In fact, between ψ′ and J/ψ additional states with QN’s 3P0, 3P1, 3P2 with JPC = 0++, 1++, 2++

exist, such that decays

ψ′ →3PJ + γ and 3PJ → J/ψ + γ

are possible.

225

Charm Threshold

γ ψ′ → 3PJ + γ

γ

χJ → J/ψ + γ

J/ψ → ηc + γ

2γ

hc[11P1](3526)

γ

π0

ππ, η

DDη′c[2

1S0](3637)

ηc[11S0](2980)

χ2[3P2](3556)

χ1[3P1](3511)

χ0[3P0](3415)

J/ψ[1 3S1](3097)

ψ′′′

(4421)

ψ′′

(4039)

ψ′

[2 3S1](3686)

ψ(4153)

ψ(3773)

1S0 [0−+] 3S1 [1−−] 1P1 [1+−] 3P0,1,2 [0++, 1++, 2++] 3D1 [1−−]

↑Para

Charmonium

↑Ortho

Charmonium

Figure 9.3: Lowest states of the charmonium spectrum and radiative transitions.

In 1975 the DASP Collaboration at DESY [6] and subsequently SPEAR at SLAC [7, 8] firstobserved the χ states. About 10 years latter the Crystal Ball detector at SPEAR (SLAC) allowedone to perform a very precise spectroscopic investigation of the radative ψ′–decays [5]. A spectrumof this measurement is shown in Fig. 9.4. More recent radiative transition spectra from CLEOare shown in Fig. 9.5. A very rich charmonium spectroscopy emerges. Actual results for massesand widths of the cascade states are:

0++ = χ(3415) : M = 3414.75 ± 0.31 MeV, Γ = 10.2 ± 0.7 MeV Pbr=9.4 %

1++ = χ(3511) : M = 3510.66 ± 0.07 MeV, Γ = 0.89 ± 0.05 MeV Pbr=8.8 %

2++ = χ(3556) : M = 3556.20 ± 0.09 MeV, Γ = 2.03 ± 0.12 MeV Pbr=8.3 % .

By Pbr = Γ(ψ′ → χγ)/Γ(ψ′)tot we denoted the production branching fraction via the radiative

decays ψ′ → χJ + γ.

Exclusive hadronic transitions ψ(2S)→ J/ψ+X are π+π−J/ψ [33.5%], π0π0J/ψ [16.5%], ηJ/ψ[3.3%] and π0J/ψ [0.13%].

In Fig. 9.3 we show the spectrum which has been established experimentally. Later we willdevelop the tools which allow us to understand the observed spectral scenery.

Presently there are 6 J/ψ series 1−− resonances established. The parameters of the one’s notalready given above are:

226

Figure 9.4: Spetrum of ψ′ radiatve charmonium decays as measured by the Crystal Ball Col-laboration. Shown are the number of photon conunts versus the photon energy. Courtesy of H.Kolanoski [5].

State ψ(3770) ψ(4040) ψ(4160) ψ(4415)

Mass MeV 3772.92 ± 0.35 4039± 1 4153± 3 4421± 4

Width MeV 27.3± 1.0 80± 10 103± 8 62± 20

Br(ψ → e+e−) (0.97± 0.07) · 10−5 (1.07± 0.16) · 10−5 (0.81 ± 0.09) · 10−5 (0.94 ± 0.32) · 10−5

Note the much bigger widths of these states, in comparison to the very narrow two lowest statesJ/ψ(1S) and ψ(2S). Actually, the next higher state ψ(3770) decays with 85.3% branching ratioas

ψ(3770) → DD

into a pair of open charm particles (48.7% D0D0, 36.1% D+D−). This tells us that we havepassed the charm threshold! Of the vector mesons, only the two lowest states have negativebinding energy, and this is what makes them so surprisingly stable.

A new state hc[11P1](3526) was discoverd in 2005 bt CLEO. It is produced hadronically via

ψ(2S)→ π0hc with hc → ηcγ. Its precise mass is an important test for qq potental models.

A recent γγ → hadrons spectrum of BABAR showing the pseudoscalars ηc and η′c in the invariantmass distribution of the K0

SK±π∓ channel is shown in Fig. 9.6.

227

Figure 9.5: Inclusive photon spectra from 23S1 decays f cc and bb measured with the CLEOdetector. From Skwarnicki [9].

Figure 9.6: Observation of γγ → ηc(1S) and γγ → ηc(2S) by the BABAR experiment. A peakdue to e+e− → γJ/ψ is also visible. From Skwarnicki [9].

9.2 Charmed Mesons

While charmonium states are of type cc, and thus have net charm zero, charmed mesons haveopen charm and are of type cq and qc, where q = u, d, s.

quark content cu cd cs uc dc sc

pseudo scalars JP = 0− D0 D+ D+s D0 D− D−s

vector mesons JP = 1− D∗0 D∗+ D∗+s D∗0 D∗− D∗−s

The charmed pseudoscalar mesons D0 and D± were established in 1976, the D±s in 1977. Mainproduction mechanism is B(ψ

′′ → DD) ( >∼ 90% ). All decays are parity violating (see theExercises) and hence proceed via weak interactions. The reason is that charm is conserved instrong and electromagnetic interactions. Since the states are heavy there is a plenitude of decaymodes and the main channels are typically at the percent level (some reaching close to 15%):

228

state mass [MeV] mean life τ × 1015 some decays

D0 1864.48 ± 0.17 410.1 ± 1.5 s K−π+[3.89%], K−π+π0[13.9%], K−e+νe[3.58%]

D± 1869.62 ± 0.20 1040 ± 7 s K−π+π+[9.2%], K0Sπ

+π0[6.8%], K0µ+νµ[9.3%]

D±s 1968.49 ± 0.34 500 ± 7 s K+K−π+ [5.5%], K+K−π+π0[5.6%], τ+ντ [6.6%]

Note that the pseudo scalar states only decay via weak processes!

Exercises:

1) Draw the relevant Feynman diagrams (within the SM) for the decays D0 → K−π+, K−e+νeand K−π+π0, based on the quark content of the hadrons.

2) The decay D0 → K−π+π0 is observed with a branching fraction 13.9 %. What is the branchingfraction of the charge conjugate decay D0 → K+π−π0?

3) Explain why D+s → µ+νµ has a branching fraction of 6.2·10−3 % only, while D+

s → τ+ντ hasone of 6.6 %.

The vector states D∗ and D∗s were detected fist in 1977. They are produced in ψ(4.028) → D∗D∗).They are to be considered as excited states, which decay via strong and electromagnetic transitionsinto the lighter charmed D mesons:

state mass [MeV] width decays

D∗0 2006.97 ± 0.19 Γ < 2.1 MeV D0π0 [61.9%], D0γ [38.1%]

D∗± 2010.27 ± 0.17 Γ = 96± 22 keV D0π± [67.7%], D±π0 [30.7%], D±γ [1.6%]

D∗±s 2112.3 ± 0.5 Γ < 1.9 MeV D±s γ [94.2%], D±s π0 [5.8%]

A much more detailed experimental study of this rich field of hadron spectroscopy has beenperformed since and is going on up to day, e.g. with the charm–factory CLEOc at Cornell/USAand a similar c − τ–factory BEPCII/BESIII in Beijing/China. We refer to the Particle DataTables for more up to date information on this program.

What is most important for us are the facts which established the new quantum number charm:

1. Very narrow peaks J/ψ and ψ′

2. Broad peaks above about 4 GeV

3. χ intermediate states

4. D mesons which decay parity violating, i.e., decay via weak interaction. This decay patternestablishes that charm is conserved in strong and electromagnetic processes

5. Exited D∗ states

6. The ratio R = σ(e+e− → hadrons)/σ(e+e− → µ+µ−) increases strongly at about 4 GeV(the charm threshold) (see Fig. 8.7)

229

9.3 The Υ resonances

The first observation of a resonance at 9.4 GeV was made in the dimuon spectrum of 400 GeVproton–nucleon collisions [10] at Fermilab. In 1978, after upgrading the DORIS storage ring atDESY, the PLUTO, DASP and DESY-Heidelberg experiments [11–13] clarified the nature of thenew resonances in e+e−–annihilation: the Υ resonance and the Υ spectral family were to beinterpretation as Bottomonium states

Υ =1√3

∑

a

baba .

Later more precise measuerement were performed with the CUSB and CLEO detectors at theCESR e+e−–ring at Cornell and with the ARGUS and Crystal Ball detectors at DORIS II ringat DESY. Data from CUSB are shown in Fig. 9.7.

The b (bottom) quark was a new quark, which was expected to exist after the discovery of theτ–lepton in 1975, which was the first fermion beyond the complete second lepton–quark family.In the SM the family structure is required by renormalizability which in turn is possible byfamily-wise anomaly cancellation. A third family had been proposed in 1973 by Kobayashy andMaskawa as a natural explanation of CP violation which did not require to introduce a new typeof “superweak” interaction.

The main properties of the experimentally confirmed Υ states are:

MΥ(1S) = 9460.30 ± 0.26 MeV ΓΥ(1S) = 54.02 ± 1.25 keV ΓΥ(1S)e+e− = 1.340 ± 0.018 keV



MΥ(4S) = 10579.4 ± 1.2 MeV ΓΥ(4S) = 20.5 ± 2.5 MeV ΓΥ(4S)e+e− = 0.272 ± 0.029 keV

MΥ(5) = 10865 ± 8 MeV ΓΥ(5) = 110± 13 MeV ΓΥ(5)e+e− = 0.31± 0.07 keV

MΥ(6) = 11019 ± 8 MeV ΓΥ(6) = 97± 16 MeV ΓΥ(6)e+e− = 0.13± 0.03 keV

The branching fractions into the cascade states are:

Υ(2S) → γχb2(1P ) (7.15 ± 0.35)% Υ(3S) → γχb2(2P ) (13.1 ± 1.6)%

→ γχb1(1P ) (6.9 ± 0.4)% → γχb1(2P ) (12.6 ± 1.2)%

→ γχb0(1P ) (3.8 ± 0.4)% → γχb0(2P ) (5.9 ± 0.6)%

Other major modes are Υ(2S)→ Υ(1S)π+π−(18.8 ± 0.6)% ; Υ(1S)π0π0(9.0 ± 0.8)%, Υ(3S)→Υ(2S)+anything(10.6±0.8)% [Υ(2S)π+π−(2.8±0.6)% ; Υ(2S)π0π0(2.0±0.3)% ; Υ(2S)γγ(5.0±0.7)%]. Furthermore, Υ(3S)→ Υ(1S)π+π−(4.5 ± 0.2)% ; Υ(1S)π0π0(2.1 ± 0.3)%.

Two series of cascade states the 1P and 2P states are well establised:

230

Figure 9.7: A scan of the first 3 Υ resonances from the CUSB detector at the CESR ring atCornell.

Mχb0(1P )= 9859.44 ± 0.42 ± 0.31 χb0(1P ) → γΥ(1S) < 6%

Mχb1(1P )= 9892.78 ± 0.26 ± 0.31 χb1(1P ) → γΥ(1S) (35 ± 8)%

Mχb2(1P )= 9912.21 ± 0.26 ± 0.31 χb2(1P ) → γΥ(1S) (22 ± 4)%

Mχb0(2P )= 10232.50 ± 0.40 ± 0.50 χb0(2P ) → γΥ(2S) (4.6 ± 2.1)% , γΥ(1S) (0.009 ± 0.006)%

Mχb1(2P )= 10255.46 ± 0.22 ± 0.50 χb1(2P ) → γΥ(2S) (21 ± 4)% , γΥ(1S) (8.5 ± 1.3)%

Mχb2(2P )= 10268.65 ± 0.22 ± 0.50 χb2(2P ) → γΥ(2S) (16.2 ± 2.4)% , γΥ(1S) (7.1 ± 1.0)%

While many aspects of the Bottomonium are very similar to the properties of the Charmonium,the bottomonium states, being much heavier than the charmonium states, decay in many morechannels, in particular also into charmonium and charmed meson states. Famous is the the Υ(4S)state as it is just above the BB threshold of B anti-B meson production. Running an e+e−–annihilation machine at the Υ(4S) is a B–factory. B–factories have been relaized at SLAC withthe BaBar detector and at KEK Tsukuba/Japan with the Belle detector.

231

BB Threshold

γE1

γM1

2γ

hb[2P ]

hb[1P ]ππ

ππ

ππ

BB

ηb[3S]

ηb[2S]

ηb[1S]

χb[2P ]

χb[1P ]

Υ(11020)

Υ(10860)

Υ(4S)

Υ(3S)

Υ(2S)

Υ(1S)

Ψ(32D)

Ψ(31D)

1S0 [0−+] 3S1 [1−−] 1P1 [1+−] 3P0,1,2 [0++, 1++, 2++] 3D1 [1−−]

↑Para

Bottomonium

↑Ortho

Bottomonium

Figure 9.8: Experimentally establisehd states of the bottomonium spectrum and radiative (red)and some hadronic (blue) transitions in full lines. Dashed lines: states and transitons expectedfrom potential model predictions. See also Fig. 9.3 for other possible transitions.

9.4 Bottom Mesons

The b–flavored mesons, usually called B–mesons, are composed of the 3rd family b–quark, thebottom partner of the top quark in the weak iso–doublet, an one of the lighter quarks u, d, s, orc:

quark content ub db sb bu bd bs

pseudo scalars JP = 0− B+ B0 B0s B− B0 B0

s

vector mesons JP = 1− B∗+ B∗0 B∗0s B∗− B∗0 B∗0s

and the charmed B mesons, we obtain be replacing the u by the c quark in the table.

The bottom pseudoscalar mesons B0 and B± were established in 1976, the B±s in 1977. Mainproduction mechanism is Υ(4S) → BB ( >∼ 96% [B+B− : 51.6 ± 0.6 %, B0B0 : 48.4 ± 0.6 %]).All decays are parity violating and hence proceed via weak interactions. The main properties of

232

the normal B mesons are

MB± = 5279.15 ± 0.31 MeV ; τB± = (1.638 ± 0.011) · 10−12 s

MB0 = 5279.53 ± 0.33 MeV ; τB0 = (1.530 ± 0.009) · 10−12 s

Note that MB0 −MB± = 0.37± 024 MeV and τB±/τB0 = 1.071 ± 0.009.

Like Kaon physics, B physics plays a key role in the flavor structure of the SM. Both show CPviolating X0 ↔ X0 oscillations. In 1973 Kobayashi and Maskawa observed that incorporatingthe observed CP violation of the K–system in a natural way is possible only in a SM with 3quark–lepton families. Given the CP pattern in the Kaon sector the three family SM predictedmany aspects of B physics and in particular the O(1) CP voilation in the B sector, which in 2001was fully confirmed at the B–factories by Belle and BaBar.

9.5 Non-Relativistic Potential Models

In spite of the relatively high energy and the related relatively small effective strong coupling,quarkonium resonances are a non–perturbative phenomenon and pQCD is of limitted value inpredicting their precise properties. Nevertheless, it is possible to give QCD inspired heuristcarguments which allow us to model quite realistically the physics of quarkonia. The argumentsare as follows: at distances >∼ 1 GeV−1 we know quarks must be confined, whereas deep inelasticelectron–proton scattering (see later) tells us that for Q2 >∼ 1 GeV2 quarks in a proton are “free”,as a consequence of asymptotic freedom. For a meson M = (qq) with a confinement radius RMthe uncertaintity relation tells us that the quark’s momentum is p ∼ R−1M . For light quarksmq

<∼R−1M we are in a relativistic situation as it applies for the light hadrons. For quarkonia,we are confronted with hadronic bound states of heavy quarks mc,mb ≫ R−1M . For charmoniumtypically: Mcc ∼ 3 − 4 GeV and binding energy ∆Ecc ∼ 100 − 600 MeV ≪ Mcc. Thereforea non–relativistic describtion seen not unrealistic, which suggests a potential model description.How could a qq–potential look like?1) At large distances confinement requires a linearly rising potential as known from the Reggebehavior of excited hadrons: V (r) ∝ r. Such behavior is supported by strong coupling expansionsas well as by numerical simulations in lattice regulated QCD.2) At short distances, by virtue of asymptotic freedom, we expect a Coulomb like potentialmediated by the massless gluon: V (r) ∝ −1/r. These two limits suggest a model:

V (r) = −ar

+ V0 + σ r . (9.2)

One important observation is the approximate congruence of cc and bb meson spectral as shownin Fig. 9.9. which would be exact for a logarithmic potential V (r) = C ln(r/r0) +C0, and whichin some sence is an interpolation between the ∝ −1/r and the ∝ r behavior expected in QCD(Quigg and Rosner 1977). Similarly, a power potential V (r) = A+B rα with α ≃ 0.1 leads to asuccesful description of the cc and the bb spectroscopies (Martin 1981).

Note, there does not exist a “derivation” of a potential model! What is used are QCD motivatedansatze. Also, a potential does not account for the full dynamics as it assumes interactions tobe instantaneous. Therefore, be aware of the limitations of potential models, which neverthelessusually work rather well. Let us look at this a little bit more specific:

a) The mas spectrum of a heavy quark qq bound state is of the form

Mn = 2mq + En(mq, V ) ,

233

4S?

3S

2S

1S

1P

2P?

Charmonium

Mass

GeV

3.00

3.25

3.50

3.75

4.00

4.25 4S

3S

2S

1S

1P

2P

Bottomonium

9.50

9.75

10.00

10.25

10.50

Figure 9.9: Congruence (=equal level spacings) of charmonium and bottomoium spectra.

where mq is an appropriately defined version of the quark mass, En is the energy eigenvalueof a non-relativistic Schrodinger equation and V (r) a suitable Schrodinger potential, whichis flavor independent, because the string interaction are flavor independent besides theflavordependent quark mass effects. For the potential we first look at the spin–independentpart and discuss spin effects later.

b) The asymptotic behavior of the two–body qq- potential should satisfy

1) V (r) ≃ −43αs(Λ r)

r + corrections ; r→ 0 ,

where the corrections are obtained by the low velocity v/c–expansion of the one–gluonexchange potential (Λ = ΛQCD).

2) V (r) ≃ σ r + corrections ; r →∞ ,

imposing manifest confinement, whith σ > 0 a spin and flavor independent string tensionconstant44.

Point 1) requiring a Coulomb like potential at short distances suggests the application of thecorresponding technique applied for the positronium (or hydrogen–atom) in QED, which yieldsthe ground state splitting of the 0−+ (Para-) and 1−− (Ortho-) ground states via spin–spin

44If a heavy quark antiquark pair is separated the color electric field is concentrated into a string of collimatedflux lines. The shape of the string can be characterized by the chromo–electric field energy density

E(x) ∼< qq|Tr ~E 2(x)|qq > − < qq|qq >< 0|Tr ~E 2(x)|0 > .

Quantitatively the string picture manifests itself by the ln r–behavior of the thickness of the “chromoelectric fluxtube” (Luscher, Munster, Weisz 1980 [14]):

σ2 ≡∫

d2x⊥x2⊥ E(x)

∫

d2x⊥ E ∼ ln r .

This behaviour is confirmed by corresponding lattice QCD simulations (see e.g. [15]).

234

gQ

q q

q q

Figure 9.10: Quark–antiquark scattering models the short distance tail of the heavy quark qq(two–body) potential.

interaction ((↑↓) vs. (↑↑)), as well as the fine– and hyperfine–splitting by other spin dependenteffects (see below).

The strategy for “deriving” a “realistic” potential attempts to use all information we have fromQCD. The short distance tail shoud be reliably described by pQCD. At lowest order the potentialis given by the one–gluon exchange as shown in Fig. 9.10.

V (Q2) = −C2(R)g2sQ2

where Q is the space–like four momentum transfer vector and C2(R) = 4/3 is the SU(3)c Casimir.This factor is due to the non–Abelian structure of QCD: the two quark–gluon vertices exhibitfactors Ti and Tk, respectively, and the gluon propagator is diagonal (∝ δik) such that we obtain

(TRi T

Ri

)ab

=

r∑

i=1

d(R)∑

c=1

(TRi)ac

(TRi)cb

= δab C2(R) .

The strong coupling constant gs is to be identified with the effective running QCD couplinggoverned by the RG (see Sect. 8). We will adopt the convention introduced earlier within the

context of effective QCD in Sect. 8.10. Thus αs = g2s4π and a = αs

π and the β–function normalized

by β(a) = Q2 ∂∂Q2 a(Q2) (see (8.15)). The β–function should satisfy the following properties:

① By asymptotic freedom we have

β(a) = −a2(β0 + β1 a+ β2 a

2)

+O(a3) ; a→ 0 .

Asymptotically, for large Q2, exanding in addition with respect to leading logs, we have

a(Q2) =1

β0 lnQ2/Λ2− β1β30

ln lnQ2/Λ2

(lnQ2/Λ2)+O(

1

(lnQ2/Λ2)3) ; Q→∞ .

Note that this in fact means that the potential at short distances is logarithmically modified

V (r) ∼ 1

r ln(

1Λ2r2

) ; r → 0 . (9.3)

② Confinement suggests a β–function which has no IR fixed point (no further zero beyond thetrivial one at a = 0). In this case the strong coupling constant grows to a(Q2)→∞ as Q2 → 0.Such statements are to be taken with caution, however. At low Q2 the quark gluon coupling loosesits meaning, as large numbers of quarks and gluons get strongly coupled and behave collectively.In other words, using perturbative concepts like αs in the non–perturbative regime is highlyambiguous, one could say extremely scheme dependent. Nevertheless, we use this concept as it

235

Figure 9.11: Model β–function adopted for modelling the quarkonium potential. Parameters:ℓ = 24 and αs = 0.24.

has been introduced in the original papers on the subject (Richardson, Buchmuller and Tye andmany others [16–21]). In fact it will turn out that the calculations of quarkonium spectra, locatedin the ranges about 2mq± at most a few GeV, do not depend very much on the detailed treamentof this long range asymptotics. Inspired by V (r)→ σ r as r→∞ and the fact that perturbatvelyV (r) ∝ a(Λr) the string like behavior of the potential may be obtained by assuming45

a(Q2) =K

Q2[1 + o(1)] ; Q2 → 0 ,

such that

β(a) = −a (1 + o(1)) ; a→∞ .

Matching the V (r) ∼ σ r string behavior, which in turn is related to the phenomenologicallyknown Regge slope (α′)−1 ≃ 1 GeV−2, one finds

(α′)−1 = 2π σ = 4π2 C2(R)K , (9.4)

which fixes σ ∼ 0.159 GeV−2 as well as K ≃ 1.9 · 10−2 GeV−2. Today one can obtain the stringconstant reliably from QCD on a lattice, a recent value obtained in [22] is

σ ≃ 0.2162 GeV−2 , (9.5)

see (9.13) below, which amounts to an increase by a factor 1.36, relative to the one obtained fromthe Regge slope.

We now may choose a(Q2) such that ① and ② are the boundary conditions. The general ansatzis

β(a) = −β0 a2 − β1 a3 + σ(a) ,

45The symbols O(x) and o(x) have the usual meaning defined in mathematics of asymptotic expansions: “f(x) =O(g(x)) as x→ ∞” means, there exist real positive numbers c and x0 such that |f(x)| ≤ c |g(x)| for all x > x0. Incontrast, “f(x) = o(g(x)) as x→ ∞” means that g(x) grows faster than f(x), i.e., f(x)/g(x) → 0 as x→ ∞.

236

where σ(a) must be choosen such that we satisfy ②. With the appropriate choice of ΛQCD, wehave

lnQ2

Λ′2 =

a(Λ′2)∫

a(Q2)

dx

β(x),

and adopting the strategy introduced in Sect. 8.5, when discussing the “higher order” effects, wemay define Λ by ①

lnQ2

Λ2=

1

β0 a+β1β20

ln(β0 a) +

a∫

0

dx

(1

β0x2− β1β20

1

x+

1

β(x)

)

for a(Λ2) > a(Q2) and ②

lnK

Q2= ln a+

∞∫

a

dx

(1

x+

1

β(x)

)

for a(K) < a(Q2). The consistency condition of ① and ② requires:

lnK

Λ2=

1

β0 a+β1β20

ln(β0 a) +

a∫

0

dx

(1

β0x2− β1β20

1

x+

1

β(x)

)+ ln a+

∞∫

a

dx

(1

x+

1

β(x)

)

independent of Q2 and therefore of a. A model β–function which satisfies both conditions is givenby the explicit function

1β(a) = − 1

β0 a2(

1−exp(− 1β0 a

)) + β1

β20

1a exp(−ℓ a) . (9.6)

One easily checks that it satisfies the boundary conditions:

β(a) = −β0 a2 − β1 a3 −β1 (β1 − ℓ β0)

β0a4 +O(a5) ; a→ 0

β(a) = −a+1

2β0+O(1/a) ; a→∞ .

However, the second condition is satisfied only when ℓ > 0 such that the second term of the ansatzis damped at large a. On the other hand, the asymptotic form for a small seems to determine ℓuniquely by β2 (up to a sign the coefficient of the a4–term, given in (8.15)). The result we obtainis

ℓ =β21 − β0β2β0β1

≃ −11.8[−48.1] for Nf = 3[0] ,

a nonsensical value, because of the wrong sign. Note that the model ansatz only makes sense forpositive ℓ. It is not very surprising that this does not work. It is a typical example of abuse ofthe perturbative concept of the running coupling in the infrared. We conclude that the potential(9.2) is a reasonable ansatz if we treat the three parameters as what they are: uncorrelated fitparameters with the exeption of the first term which we know in QCD has a well predicted form

237

Figure 9.12: A comparison of various quarkonia potentials. Labels 1: Martin [24], 2: Buchmuller,Grunberg and Tye [23], 3: Bhanot and Rudaz [25] 4: Cornell group [16]. Also shown are somemean–square radii for Υ and J/ψ states. (taken from Ref. [18])

(9.3), plus higher order corrections. With a fitted value for ℓ Eq. (9.6), works rather well. In [23]specifically, the QCD correction to two–loops are incorporated:

V (r) ∼ −4

3

1

r

4π

β0 ln(

1/r2Λ2MS

) ×

1− β1

β20

ln[ln(

1/r2Λ2MS

)]

ln(

1/r2Λ2MS

) + c1

ln(

1/r2Λ2MS

) +O

1

ln2(

1/r2Λ2MS

)

, (9.7)

where c = 1360

(31− 10

3 Nf

)+ 2γE and γE = 0.5772 · · · is Euler’s constant. The RG coefficient are

as given in (8.5).

A set of successful potentials are shown in Fig. 9.12. Note that the radii of the states areintermediate and neither probe the short distance nor the long distance asymptotic regimes. Thevarious potentials have been shifted by a constant such that they coincide at r0 = 0.5 fm, andthe error bars indicate the uncertainty in this normalization. Typical predictions are given inTables 1 and 2. They have been obtained based on the spin independent model: Coulomb–termsupplemented with the β–function model (9.6), which at long distances matches a linear potentialwith appropriate string constant. The parameter ℓ = 24 has been fixed such that experimetalresult for the mass difference MΥ′−MΥ is reproduced correctly. Note that in the spin independentapproximation pseudoscalar (0−+) vs. vector states (1−−) splitting is neglected. Similarly for thehc vs. χc2 etc. Remarkably, this crude model reflects the spectrum quite well.

In general one can say that the non-relativistic description for charmonium is quite crude, and asexpected, works much better for bottomonium. For much more recent and more detailed reviewswe refer to the Refs. [20, 21,26,27].

238

Table 1: cc spectrum: masses, ratios of widths, velocities, and mean–square radii. The mass ofthe ground state is input; it determines the c–quark mass mc = 1.48 GeV [18].

State Mass (GeV) Γee/Γee(1S) 〈v2/c2〉〈r2〉1/2 (fm)

J/ψ(1S) 3.10 1 0.23 0.42

χc2(1P ) 3.52 0.25 0.67

ψ′(2S) 3.70 0.46 0.29 0.85

ψ(1D) 3.81 0.29 0.87

χ′c2(2P ) 3.79 0.32 1.05

ψ′′(3S) 4.12 0.32 0.36 1.20

ψ(2D) 4.19 0.36 1.22

ψ′′′(4S) 4.48 0.25 0.44 1.48

Table 2: bb spectrum: masses, ratios of widths, velocities, and mean–square radii. The mass ofthe ground state is input; it determines the b–quark mass mb = 4.87 GeV [18].

State Mass (GeV) Γee/Γee(1S) 〈v2/c2〉〈r2〉1/2 (fm)

Υ(1S) 9.46 1 0.077 0.23

χb2(1P ) 9.89 0.069 0.39

Υ(2S) 10.02 0.44 0.075 0.50

Υ(1D) 10.14 0.072 0.53

χ′b2(2P ) 10.25 0.078 0.65

Υ(3S) 10.35 0.32 0.085 0.75

Υ(2D) 10.43 0.083 0.77

χ′′b2(3P ) 10.53 0.090 0.87

Υ(4S) 10.62 0.26 0.098 0.95

Υ(3D) 10.68 0.098 0.97

Υ(10860) 10.86 0.25 0.130 1.10

239

Table 3: cc spectrum: masses in MeV from lattice QCD in the quenched approximation

JPC State CP-PACS [28] Columbia [29] QCD-TARO [30] experiment

0−+ ηc 3013(1) 3014(4) 3010(4) 2980(1)

η′c 3739(46) 3707(20) 3654(10)

1−− J/ψ 3085(1) 3084(4) 3087(4) 3097

ψ(2S) 3777(40) 3780(43) 3686

1+− hc 3474(10) 3774(20) 3528(25) 3525

h′c 4053(95) 3886(92) −0++ χc0 3408 3413(10) 3474(15) 3415(1)

1++ χc1 3472(9) 3462(15) 3524(16) 3511

2++ χc2 3503(24) 3488(11) 3556

Today it is possible to calculate the charmonium spectra in lattice QCD, although there arestill some limitations, like quenched approximation (no sea quark fluctuations), extrapolations tophysical parameters, corrections for lattice artefacts etc. Table 3 shows a set of typical results forCharmonium. When simulating heavy quarks of mass m on a lattice of spacing a one inevitablyhas to cope with ma or (ma)2 corrections, i.e. the condition for ma≪ 1 must be satisfied. Whilethis can be reached for charmonium, bottomonium is still at the limit of the possibilities. Onecan expect that reliable simulations will be possible in not to far future.

9.5.1 More on the heavy quark potential

The leading heavy quark potential (9.2) has a non-perturbative string like IR piece, which actuallycan be calcuated by lattice QCD, and a perturbative Coulombic UV piece which recieves higherorder perturbative QCD corrections [31,32].

The static potential is defined in a manifestly gauge invariant way in terms of the vacuum expec-tation value of a Wilson loop [33,34]

V (r) = − limT→∞

1

Tln⟨WΓ

⟩, WΓ = TrP exp

(i g

∮

ΓdxµAµ

). (9.8)

The path Γ is taken to be rectangular loop with time extension T and spatial extension r (seeFig. 9.13). The prescription P indicates that the gauge fields Aµ are path-ordered along the loop.

The color trace is normalized according to Tr(..) = Tr(..)/Tr11 .

r

T

Figure 9.13: Wilson loops (closed path Γ) used in the definition of the static potential.

240

=

Figure 9.14: The one–loop diagrams contributing to the static potential. Double, wiggly, dottedand solid lines denote source, gluon, ghost and (light) fermion propagators, respectively. A blobon a gluon line stands for one–loop self–energy corrections.

It can be shown that contributions to Eq. (9.8) containing connections to the spatial componentsof the gauge fields Ai(~r,±T/2) vanish in the limit of large time extension T in perturbation theory(at least up to two loops). Therefore, Eq. (9.8) can be simplified to

Vpert = − limT→∞

1

Tln

⟨TrT exp

(∫

xJaµA

aµ

)⟩, (9.9)

where T now means time ordering and the static sources separated by the distance r = |~r − ~r ′|are given by

Jaµ(x) = ig δµ0 Ta[δ(~x− ~r)− δ(~x− ~r ′)

],

where T a are the generators in the fundamental representation.

Expanding the expression in Eq. (9.9) perturbatively, one encounters in addition to the usualFeynman rules the source–gluon vertex i gδµ0T

a, with an additional minus sign for the antisource.Furthermore, the time–ordering prescription generates step functions, which can be viewed assource propagators, analogous to the heavy–quark effective theory (HQET).

The perturbative corrections have been calculated to two loops. The relevant diagrams at one–loop are displayed in Fig. 9.14 and have been calculated in [33,35]. Shown are the bare diagramswhich have to be supplemented as usual by the relevant counterterms. The two–loop correctionsstem from diagrams as shown in Fig. 9.15. The renormalization counterterms have to be includedas usual. The calculations have been performed in [36–38]. More recently also the three–loop [39]and four–loop [40,41] leading log correction have been calculated.

Important checks of the calculation include

• gauge independence of appropriate classes of diagrams,

• confirmation of cancelation of infrared divergences,

• correct renormalization properties.

The result for the static potential in coordinate space including the complete two–loop as well asthe logarithically enhanced three–loop and four–loop terms is given by

V (r) = −CFαV (1/r)

r, (9.10)

241

a1 a2 a3 a4 a5 a6 b1 b2

b3 c1 c2 c3 c4 c5 c6 d1

d2 e1 e2 e3 f1 f2 f3 g1

g2 h1 h1g h1f h2 h3 i1

(2)

i2

i1

(2) =

i1.a i1.ag1 i1.ag2 i1.af1 i1.af2 i1.b

i1.bg i1.bf i1.c i1.d i1.e i1.f

Figure 9.15: Classes of two–loop diagrams contributing to the static potential. Double, wiggly,dotted and solid lines denote source, gluon, ghost and (light) fermion propagators, respectively.A blob on a gluon line stands for one–loop self–energy corrections (from [36]).

with

αV (1/r) = αs(1/r)

1 +

(αs(1/r)

4π

)[a1 + 2γEβ0]

+

(αs(1/r)

4π

)2 [a2 +

(π2

3+ 4γ2E

)β20 + 2γE (β1 + 2β0a1)

]+

(αs(1/r)

4π

)3 [aL3 ln

CAαs(1/r)

2+ a3

]

+

(αs(1/r)

4π

)4 [aL24 ln2 CAαs(1/r)

2+ aL4 ln

CAαs(1/r)

2+ a4

]+ · · ·

. (9.11)

Here, αs(1/r) = αMS

(µ = 1/r) and γE is Euler’s constant. The first two terms of the beta function

needed here are given by β0 = 113 CA − 4

3 TFNf and β1 = 343 C

2A − 4CFTFNf − 20

3 CATFNf . Theexpansion coefficients are given by

a1 =31

9CA −

20

9TFNf ,

242

0.6 0.8 1 1.2 1.4 1.6

1.4

1.6

1.8

2

2.2

2.4

integration of the force-3 loops

bosonic string

Figure 9.16: The force and the static potential. The dashed line represents the bosonic stringmodel and the solid line the prediction of perturbation theory as detailed in the text. From [22].

a2 =

(4343

162+ 4π2 − π4

4+

22

3ζ(3)

)C2A −

(55

3− 16ζ(3)

)CFTFNf

+400

81T 2FN

2f −

(1798

81+

56

3ζ(3)

)CATFNf ,

aL3 =16π2

3C3A (9.12)

aL24 =16π2

3C3A

(−11

3CA +

4

3TF Nf

)

aL4 = 16π2C3A

[a1 + 2γeβ0 + TFNf

(−40

27+

8

9ln 2

)+ CA

(149

27− 22

9ln 2 +

4

9π2)]

.

This establishes the “constant” a of Eq. (9.2) with remarkable accuracy. Mass effects to two–loops have been calculated in [38, 42]. The three and four loop results are incomplete: theconstant pieces a3 and a4 are not yet known. We have not included the results for the “ultrasoft”contributions to a3 and a4 which have been given in [43–46]. Since corrections are potentiallylarge and very much scheme dependent, a complete calculation of a3 is highly needed.

As already mentioned, lattice QCD allows one to compute the string constant σ of Eq. (9.2). Acomputation which explored the static potential in the infrared and down to perturbative shortdistance scales has been performed in [22]. For large values of r, the force F (r) is expected to begiven by a constant, the string tension σ. The correction term is proportional to 1/r2 and hasa universal coefficient π/12 (Luscher term) [47]. This yields the parameter free bosonic stringmodel,

F (r) =dV (r)

dr= σ +

π

12 r2, σr20 = 1.65 − π/12 , r0 = 0.5 fm , (9.13)

which is in excellent agreement with the Nf = 0 lattice result Fig. 9.16 for a surprisingly widerange down to r ≥ 0.8 r0. The agreement at lower r at first sight looks accidental. In any caseone would expect corrections to this formula to be negligible only for much larger r. However,in a high precision computation [48] the coefficient of the subeading term proportional to r−2

243

was shown to agree with the one in the string model with high accuracy. Also the next term inthe expansion has been determined recently [49]. In any case eq. (9.13) is a very good effectivedescription of F (r) for 0.8 r0 ≤ r ≤ 1.6 r0.

The reference length scale r0 (Sommer scale) [50] used in this analysis is defined in terms of theforce F (r) between static quarks by the implicit equation

r2F (r)|r=r(c) = c , r0 = r(1.65) .

In QCD, r0 has a value of about 0.5 fm. At short distances the force may be obtained by anintegration of the perturbative renormalization group [for Nf = 0],

F (r) = CF g2qq(r)/(4πr2) , CF = 4/3 , (9.14)

−r d

drgqq = β(gqq) = −

2∑

ν=0

bν g2ν+3qq , (9.15)

b0 =11

16π2, b1 =

102

(16π2)2, b2 =

1

(4π)6

(−3470 + 2519

π2

3− 99

π4

4+ 726ζ(3)

)

Here the 3-loop coefficient, b2, could be extracted from [36,37,42]. Inserting the result

ΛMS r0 = 0.602(48) (9.16)

from [51], as well as the known relation between ΛMS and Λqq [33, 35] to fix the integrationconstant, one has a parameter free perturbative prediction. For short distances, r < 0.3 r0,the perturbative prediction V (r) = V (0.3 rc) +

∫ r0.3 rc

dyF (y) agrees rather well with the non-perturbative results. However, the convergence of the pQCD result very much depends on therenormalization scheme utilized, this is discussed in detail in [52]. A corresponding lattice QCDcalculation including dynamical light quarks (Nf = 3) at present does not exist, but would be veryimportant. For more recent investgations of the heavy quark potential see also [53] and [43–46].

244

9.6 A digression: positronium in QED

Positronium is an e+e− bound state, a state very much like a hydrogen atom, except that the twopartners are of equal mass and the positron is the antiparticle of the electron. The theory of thepositronium is a classical topic in QED textbooks and there are a number of excellent expositions(see e.g. Landau and Lifshitz Course of Theoretical Physics Vol 4 ).

Starting point is the e+e− → e+e− scattering amplitude at tree level:

+

e−, p1

e+, p2

e−, p′1

e+, p′2

q↓p→

.

The amplitude is given by

T e+e−fi = e2

u(p′1, r

′1) γµ u(p1, r1) v(p2, r2) γν v(p′2, r

′2)Dµν(q)

−v(p2, r2) γµ u(p1, r1) u(p′1, r

′1) γν v(p′2, r

′2)Dµν(p)

.

We will justify below that the non-relativistic limit is well justified as a leading approximation.A systematic expansion in small velocities v/c may be worked out in a straight forward mannerif one writes quantities in natural units, with the appropriate factors in c, in particular. Thespinors then read

u(p, r) =1√c

√p0 +mc

U(r)

~σ·~pp0+mc U(r)

v(p, r) =1√c

√p0 +mc

~σ·~pp0+mc

V (r)

V (r)

and the photon propagator

Dµν = −(gµν − (1− ξ) qµqν

q2

)1

q2 + iε

= DCoulombµν (q) +Dc µν(q) , with

DCoulombµν (q) : D00 =

1

~q 2; D0i = 0 ; Dik =

1

q2 + iε

(δik −

qiqk~q 2

),

Dc µν(q) = O(1/c) .

The Schrodinger wave functions for the scattering states then are given by

U(r) = U−(r) for the electron ,

V (r) = iσ2 U+(r) for the positron .

We then obtain

T e+e−fi = −2m1 2m2 U

+′− U+′

+ U(~p1, ~p2, ~q) U+ U− ,

245

with the following total interaction operator (Breit potential) for e+e−–scattering:

U(~p1, ~p2, ~q) = −e2

1

~q 2− 1

8m21c

2− 1

8m22c

2+ i

~σ− (~q × ~p1)4m2

1c2~q 2

− i~σ+ (~q × ~p2)

4m22c

2~q 2

+(~p1 · ~q)(~p2 · ~q)m1m2c2~q 4

− (~p1 · ~p2)m1m2c2~q 2

+ i(~q × ~p1) · ~σ+2m1m2c2~q 2

− i(~q × ~p2) · ~σ−2m1m2c2~q 2

+(~σ− · ~q) (~σ+ · ~q)

4m1m2c2~q 2− ~σ+~σ−

4m1m2c2− 3 + ~σ+~σ−

8m2m2c2

.

We have given the result of the expansion up to O((1/c)3)for different masses, for our purposewe set m1 = m2 = m. It is useful to write the corresponding Hamiltonian in configuration spacein the c.m. system:

H =~p 2

m− α

r+ V1 + V2 + V3 ,

V1 = − ~p 4

4m3c2+ 4π µ20 δ(~r)−

α

2m2c21

r

[~p 2 +

~r (~r ~p) ~p

r2

],

V2 = 6µ201

r3~ℓ · ~S ,

V3 = 6µ201

r3

[(~S · ~r)(~S · ~r)

r2− 1

3~S 2

]+ 4πµ20 (

7

3~S 2 − 2) δ(~r) , (9.17)

where

µ0 = e~2mc Bohr magneton,

~~ℓ = ~r × ~p orbital angular momentum,

~S = ~σ++~σ−2 total spin (~S 2 = 1

2 (3 + ~σ+~σ−)).

Using this Hamiltonian we are able to treat the positronium bound state problem perturbatively.The undisturbed Hamiltonian is given by

H =~p 2

m− α

r,

which can be solved, and yields the hydrogen spectrum with the replacement m→ m/2:

En = − mα2

4~2n2; n the principal quantum number.

The perturbations Vi (i = 1, 2, 3) yield the fine structure splitting, with the classification of

the levels according to the angular momentum. Since only ~S = ~σ++~σ−2 , not however ~µS =

µ0 (~σ+ − ~σ−), the magnetic spin moment, are present in the Hamiltonian, we have

[H, ~S 2] = 0 ,

which means that the total spin is a good quantum number(holds exactly):

S = 0 : para–positronium,

S = 1 : ortho–positronium.

246

Furthermore, parity P , time reversal T and charge conjugation C provide constraints. For aparticle-antiparticle pair the phases of P , T and C are given by

ηP ηP = (−1)2j ; ηT ηT = ηC ηC = 1 ,

with j the spin. The total parity of the e+e−–system therefore is

P = (−1)ℓ+1 .

The wave function of the e+e−–system is therefore odd under exchange of the positions, the spinand charge conjugation:

(−1)ℓ · (−1)S+1 · C = −1 ,

such that

C = (−1)S+ℓ ; PC = (−1)S+1 ,

which are symmetries of the electro-magnetic interactions. For the positronium system this meansthat the spin is conserved. If j is given, we thus have the two possibilities:

1S0 : S = 0 ; ℓ = j ,

3S1 : S = 1 ; ℓ = j, j ± 1 .

Rather than going into a discussion of level splitting by the relativistic corrections here, we areinterested in the possible positronium decay channels.

9.6.1 e+e−–annihilation into photons: e+e− → 2γ, 3γ

The differential rate is given by the standard decay rate formula

dσnγ =(2π)4 δ(4)(Pf − Pi)

2√λ(s,me,me

1

4

∑

pol

|Tfi|2dµ(k1) · · · dµ(kn) .

The leading processes are

Tfi : 2γ : + 1 permutation

3γ : + 5 permutation .

The total cross–sections in the non–relativistic approximation are given by

σ2γ = π

(e2

mc2

)2c

vrel,

σ3γ =4 (π2 − 9)

3α

(e2

mc2

)2c

vrel, (9.18)

247

where vrel is the relative velocity between electron and positron.

The transition probability enters the cross–section definition as

dσ =dw

j,

where dw is the process probability and j the incoming particle current, in the c.m. system givenby

j =v1 + v2V

.

Thus

dw = j dσ ,

and for the total rates averaged over spin. Thus

wnγ = j σnγ

are the probability rates for e+e− → nγ. In the c.m. system we have

j =vrelV

for a free e+e−-pair. For an e+e−-pair bound in the state ψ(r)

j = vrel |ψ(0)|2 ,

in the non–relativistic approximation. For positronium in the ground state, we have

ψ(r) =1√πa3

e−r/a ; a =2~2

me2= 2 r0

where r0 is the Bohr radius of the H-atom. Positronium may be considered as a H-atom withreduced mass m→ m/2.

We are now able to consider the decay of positronium.

Positronium annihilation: we consider the decay of the ground state. Energy–momentum con-servation requires at least 2 γ’s. As P = (−1)ℓ+1 and CP = (−1)S+1 charge conjugation isC = (−1)ℓ+S . Therefore we have the correspondence

1S0 : S = 0 para–positronium: C = 1→ 2n γ′s ,

3S1 : S = 1 ortho–positronium: C = −1→ 2n + 1 γ′s ,

where n = 1, 2, · · ·, and the number of possible spin states is 1 for para and 3 for ortho, i.e., in totalfour. Note that our ground state wave function ψ(r) is normalized to describe four degeneratestates. Switching on the spin interaction responsible for the ground state splitting, para describes1, ortho 3 spin states out of the 4. The corresponding properly normalized wave functions arethen ψ1S0

(r) = 41 ψ(r) and ψ3S1

(r) = 43 ψ(r), respectively.

The probability for para–positronium decaying into 2 γ’s is

w0 = 4w2γ = 4 |ψ(0)|2 (v σ2γ)∣∣v→0

=mc2α5

2~, (9.19)

248

with α = e2/~c .

The life time is then given by

τ0 = w−10 =2~

mc2α5≃ 1.23 × 10−10sec . (9.20)

The corresponding width is

Γ0 = ~/τ0 =mc2α5

2≪ |E0| =

mc2α2

4,

and we can conclude that positronium is a quasi stationary system, as required for a non–relativistic treatment. The same exercise we can do for ortho–positronium.

Similarly, the probability for ortho–positronium decaying into 3 γ’s is

w1 =4

3w3γ =

4

3|ψ(0)|2 (v σ3γ)

∣∣∣∣v→0

=2 (π2 − 9)

9π

mc2α6

~. (9.21)

The life time is then

τ1 = w−11 =9π

2 (π2 − 9)

~

mc2α6≃ 1.4 × 10−7sec . (9.22)

We note that τ1 ≫ τ0 and we have the hierarchy: Γ1 ≪ Γ0 ≪ |E0|. Ortho-positronium is muchmore stable than para-positronium, in spite of the fact that ortho-positronium lies higher inenergy:

∆E0 = E(3S1)− E(1S0) =7

2α2 me

4

~2≃ 8.2× 10−4 eV .

The reason for the enhanced stability is that it cannot decay into 2 γ’s, and the decay into 3 γ’srequires one additional power in α:

w1 = w04 (π2 − 9)

9πα .

In addition, we observe a cancellation of two large terms, an irrational π2 minus the rational 9,such that the factor π2 − 9 ≃ 0.87 is an order smaller than the individual terms.

End of the Digression

For the application to Quarkonia systems we summarize a few relevant results from positronium(we go back to the unit system ~ = c = 1 : e2 → α here) in the following. The decay rates are

Γ(1S0 → 2γ) = 4πα2

m2

∣∣ψ1S0(0)∣∣2

Γ(3S1 → 3γ) =4

3

4

3(π2 − 9)

α3

m2

∣∣ψ3S1(0)∣∣2 , (9.23)

where ψ1S0(r) and ψ3S1

(r) are the ground state wave functions for the 1S0 spin singlet and the 3S1spin triplet,respectively. In the digression above ψ(r) denoted the ground state wave function forthe mixed state of all 4 spins. Therefore, the decay of the normalized state ψ(r) proceeds with1/4 propabiity to 2 γ and with 3/4 probability to 3 γ, and ψ1S0

(r) = 41 ψ(r) and ψ3S1

(r) = 43 ψ(r)

in terms of the wave function normalized for the degenerate ground state.

249

The energy splitting between 1S0 and 3S1 states is explained by the spin dependent part of thepotential. Since in the ground state the orbital angular momentum is zero, ∆H = V3 only,contributes. Since the ground state is spherically symmetric, we have

〈 1

r3

((~S · ~r)(~S · ~r)

r2− 1

3~S 2

)〉 = 0 .

As

~S 2 =

0 ; S = 0 ,

S (S + 1) = 2 ; S = 1 ,

we obtain

∆E0 = 〈∆HSS〉 =

∫d3r ψ∗(r)V3(~r)ψ(r) = 4π µ20 · 2 |ψ(0)|2 =

14

3πα

m2|ψ(0)|2 ,

with µ0 = e~2mc .

An important conceptual difference concerns the role of the annihilation diagramms

e+

e−

e+

e−

γ

q

q

q

q

g

in QED on the one hand and in QCD on the other hand. In QED at the order discussed abovethe annihilation diagram only contributes a term

δH =πα

m2~S 2 δ(~r)

to the Hamiltonian. In QCD in place of (e+e−) we have a color singlet 1√3

∑c(qcqc) and since

the gluon carries color the annihilation diagram contribution vanishes (because of∑

c (Ti)cc =TrTi = 0) for any colorless state. This means that in QCD only the scattering diagram

1√3

∑

c

g

qc

qc

contributes to the qq–potential. This leads to a modified spin-spin interaction term

δH =4

3

πα

m2~S 2 δ(~r) ,

with coefficient 4/3 replacing 7/3. The contribution of the QED scattering diagram to the groundstate splitting is

∆E(exchange)0 =

8

3

πα

m2|ψ(0)|2 . (9.24)

250

9.7 Quarkonia Decays

The first question asked after the discovery of the J/ψ and later the Υ was: why are these heavystates so stable?

The answer is given by the Okubo-Zweig-Iizuka (OZI) rule (1964): decays of heavy resonances,which require disconnected quark diagrams are suppressed. In QCD language: processes whichproceed via exchange of intermediate gluons only are suppressed. Thereby it is important to notethat physical processes cannot take place via single gluon exchange, since this violates the colorsinglet condition of physical states. In fact the lowest quarkonia states have negative bindingenergy and lie below the qq thershold. Since QCD preserves quark flavors a decay into lighterquark flavors is possible only via intermediate gluons.

We first consider the J/ψ and Υ, which are the lowest lying JPC = 1−− vector mesons, andwhich correspond to ortho-positronium, i.e., they can decay only via 2n+ 1 gluons (n ≥ 1). It isinteresting to compare directly the hadronic with the radiative decays:

q

q ψ

gTi

g

g

g

u, d, s

;q

q ψ

eQq

γ

γ

γ

The respective rates read

Γ(1−− → 3g) =10

81α3s 4 (π2 − 9)

|ψ(0)|2m2q

; Γ(1−− → 3γ) =4

3α3Q6

q 4 (π2 − 9)|ψ(0)|2m2q

(9.25)

The result may be obtained by using the parton cross–section

σ3γ =4 (π2 − 9)

3

α3Q6q

m2qv

,

which determines the width of the bound qq system according to

Γ = τ−1 = limv→0

v |ψ(0)|2 σ(v) .

We can then calculate the ratio between hadronic and radiative dacays: with Ti = λi2 the gener-

ators of SU(3) in the fundamental representation we find

Γ(1−− → 3g)

Γ(1−− → 3γ)=

α3s

Q6q α

3

∑

i,j,k

(Tr (λiλjλk)symmetrized

8 · 3

)2

=α3s

Q6q α

3

∑

i,j,k

(dijk12

)2

=5

54

α3s

Q6q α

3. (9.26)

We have used λiλj + λjλi = 43 δij + 2 dijk λk and

∑ijk (dijk)

2 = 403 . There is also a mixed single

photon radiative decay

with a rate given by

Γ(1−− → γ3g) =8

9αQ2

q α2s 4 (π2 − 9)

|ψ(0)|2m2q

. (9.27)

251

q

q ψ

gTi

eQq γ

g

g

u, d, s

q

q ψ

gTi

g

g

u, d, s

;q

q ψ

eQq

γ

γ

The pseudoscalar ηc as a JPC = O−+ state decays to an even number of gluons or photons,preferably to 2g or 2γ,

with rates given by

Γ(0−+ → 2g) =8

3αs π

|ψ(0)|2m2q

; Γ(0−+ → 2γ) = 12α2Q4q π|ψ(0)|2m2q

. (9.28)

Again, the result may be obtained starting from the parton annihilation cross–section

σ2γ = πα2Q4

q

m2qv

,

from which we get width of the bound qq system via

Γ = τ−1 = limv→0

v |ψ(0)|2 σ(v) .

Here the ratio between hadronic and radiative decay widths is given by

Γ(0−+ → 2g)

Γ(0−+ → 2γ)=

α2s

Q4q α

2

∑

i,j

(Tr (λiλj)

4 · (√

3)2

)2

=α2s

Q4q α

2

∑

i,j

(δij6

)2

=2

9

α2s

Q4q α

2. (9.29)

Note the color factors: color singlet states have a wave function ∝ 1√3δcc′ and

∑color

1√3δcc′ =

√3,

which in rates yields a factor (√

3)2 = 3, as we learned earlier.

For the 1−− states leptonic decays play a central role as they proceede via the direct coupling tothe photon. Thereby the VMD picture confronts the qq bound state picture:

V

iγVM2V

γe+

e−;

q

q ψ

eQq

γe+

e−

The latter, characterized by the second diagram, is encoded by the van Royen-Weisskopf formula(see below)

Γ(V → e+e−) = 16π α2Q2q

|ψ(0)|2M2V

, (9.30)

where in our case V = J/ψ, · · · ,Υ, · · ·.

252

For completeness let us calculate the given decay rate: the basic matrixelement for the transition qq → γ∗ → e+e−

is obtained with the transition operator

Top = −4π αQq

q2jeµ j

µq ,

where jeµ = −e γµ e and jqµ = q γµ q. We have used that from the photon propagator Dµν = −gµν 1q2+ gauge terms,

only the first term contributes by gauge invariance. In vector meson decay q2 = M2V . Then

〈e+e−|Top|V 〉 = −4π αQq

M2V

〈e+e−|jeµ(0)|0〉lep〈0|jµq(0)|V 〉had ,

where 〈e+e−|jeµ(0)|0〉lep = −u(p−, r−) γµ v(p+, r+) and 〈0|jµq(0)|V 〉had = γV M2V ε

µ(λ). The width then followsfrom the standard 2–body decay formula

Γ =1

2M

|T |28π

√

1− 4m2e

M2,

with |T |2 is the spin averaged amplitude 13

∑

λ |T |2. The factor 18π

√

1− 4m2e/M

2V ≃ 1

8πcomes from the phase space

intergal (2π)4∫

dµ(p+) dµ(p−) δ(4)(pV − p+ − p−). In the terminology of Sect. ??, the matrix elements squared

read∑

spins

v(p+, r+)γµu(p−, r−)u(p−, r−)γνv(p+, r+) = 4ℓµν

and∑

spins

〈0|jµ(0)|V (pV , λ)〉〈V (pV , λ)|jν(0)|0〉 =∑

λ

εµ(pV , λ)ε∗ν(pV , λ)M

4V γ

2V = hµν

with

Xµν =∑

λ

εµ(pV , λ)ε∗(pV , λ) = −gµν +

pV µpV ν

M2V

; pV ≡ q .

In the rest frame of the vector meson V we have pV µ = (MV ,~0 ) and hence

Xµν =

0 ; (µν) = (0, 0)

δik ; (µν) = (i, k) .

This means that we need the spacial part of ℓµν only:

4ℓik = 2

M2V δik − 4 pipk

.

The momenta are pV = p+ + p− = (MV ,~0 ), thus ~p+ = −~p− = ~p and E+ = E− = E = MV /2. With ~p+ − ~p− =2 ~p ; E2 −m2

e = ~p 2 we arrive at

(

M4V γ

2V

)−14ℓµν h

µν = 4 ℓik δik = 4∑

i

ℓii = 2(

3M2V − 4 ~p 2) = 4M2

V

(

1 +2m2

e

M2V

)

.

Finally, with

∑

|T |2 =(4παQq)

2

M2V

· 4M2V

(

1 +2m2

e

M2V

)

γ2VM

4V ,

we arrive at the decay rate

Γ =1

3

16π2

2MV

α2Q2q

M2V

4

8π

(

1 +2m2

e

M2V

)

√

1− 4m2e

M2V

γ2V M

4V ,

and using the van Royen–Weisskopf formula (Nuovo Cimento 40 (1967))

γ2V.= Q2

q γ2V = 4Nc 〈Qq〉2 |ψ(0)|2/M3

V

and the approximation me ≪MV we find

Γ = 16 π α2 〈Qq〉2 |ψ(0)|2M2

V

=4

3π α2 γ2

V MV .

Note that dimψ = 3/2 and like in positronium, where |ψ(0)|2 = 1πa3 = m3α3

8π.

253

9.8 Matrix elements of mesonic bound states

Here we reproduce the derivation of the Van Royen - Weisskopf formula in detail. This exerciseat the same time may be considered as a repetitorium of basic QFT concepts.We first consider a quark field, single quark states and their properties:

ψαa(x) =∑

r

∫dµ(p)

uα(p, r) aa(~p, r) e−ipx + vα(p, r) b+a (~p, r) eipx

with creation and annihilation operators satisfying the normalization

[aa(~p, r), a

+a′(~p

′, r′)]+

= δaa′ δrr′ (2π)3 2ω~p ′ δ(3)(~p− ~p ′)etc.

The single quark states are then represented by

|~p, r, a〉 = a+a (~p, r) |0〉 ,

and normalized according to

〈~p ′, r′, a′|~p, r, a〉 = δaa′ δrr′ (2π)3 2ω~p ′ δ(3)(~p − ~p ′) .

The integral measure

dµ(p) =d3p

~3 (2π)31

2ωp; ωp =

√~p 2 +m2 ,

denotes the relativistically invariant on–mass shell element equivalent to Θ(p0) δ(p2 − m2) d4p,which is normalized to one particle per unit volume element:

∑

a′,r′

∫dµ(p) 〈~p ′, r′, a′|~p, r, a〉 = 1 .

The particle density operator is thus given by

~na(~p, r) =1

(2π)3 2ωpa+a (~p, r) aa(~p, r) ,

and the particle number operator

Na =∑

r

∫d3p~na(~p, r) =

∑

r

∫dµ(p) a+a (~p, r) aa(~p, r)

satisfies the comutation relation

[Na, a

±(~p, r)]

= ± δab a±(~p, r) .

The one–particle wave functions are represented by the Dirac spinors u(p, r) for quarks and v(p, r)for antiquarks, which are solutions of the momentum space Dirac equation and in our notationsatisfy

∑

r

uα(p, r)uβ(p, r) = (p/+m)αβ etc.

The free wave pakets may be replaced by more general solutions of the Klein-Gordon equation

e−ipx → f~p (x) .

254

They are independent of any quantum numbers. We then have the single particle matrix elements

〈0|ψαb(x)|~p, r, a〉 =∑

r′

∫dµ(p′)

[ab(~p

′, r′), a+a (~p, r)]+× uα(p′, r′) f~p ′ (x)

= uα(p, r) f~p ′ (x) ,

which represents the complete one particle wave function.

In the stationary case we may write

f~p (x) = e−iωpx0 g~p (~x) ; ωp = ω(| ~p |)g∗~p (~x) = g−~p (~x) .

We are interested in mesonic bound states of quarks. Interpolating fields for these states withthe correct tranformation properties and the appropriate quantum numbers can be constructedwith the help of the free quark fields as “composite” operators:

MΓik(x) =1√3

∑

c

ψαci(x) (Γ)αβ ψβck(x) , (9.31)

where c is the color index and i, k are flavor indices. The field M(x) in any case is a color singletand depending on Γ we have

Γ = 1 i γ5 γµ γµγ5 σµν

scalar pseudoscalar vector axialvector tensor

defining the transformation property under Lorentz–transformations.

Consider a meson, a quark–antiquark color neutral bound state. Depending on the spin andparity combination, the meson may have spin 0, 1 or 2 with parities

Γ = 1 i γ5 γµ γµγ5 σµν

spin 0 0 1 1 2

parity + − − + +

The Fourier–transform of the composite field is

MΓik(P ) =

∫d4x eiPxMΓik , (9.32)

where P is the four momentum in the center of mass system of the meson.

We consider the “meson” in the c.m. frame at rest (~P = 0) at time t = x0 = 0:

|M(~P = 0)〉 =

∫d3x

1√3

∑

c

: ψαci(x) (Γ)αβ ψβck(x) :∣∣∣x0=0

|0〉 .

By the Wick ordering we achieve that when experessed in terms of creation and annihilationoperators only the term with two creation operators in non-vanishing. This term creates the

255

desired quark-antiquark color singlet:

|M(~P = 0)〉 =1√3

∑

c

∑

r,r′

∫dµ(p)dµ(p′) uα(p, r) (Γ)αβ vβ(p′, r′)×

×∫

d3x ei (p′+p)x

∣∣∣x0=0︸︷︷︸

(2π)3 δ(3)(~p ′+~p)

a+ci(~p, r) b+ck(~p

′, r′) |0〉

=∑

r,r′

∫dµ(p)

1

2ωpuα(~p, r) (Γ)αβ vβ(−~p, r′) · 1√

3

∑

c

a+ci(~p, r) b+ck(−~p, r′) |0〉 ,

where the ~x integration yielded a δ–function, which allowed us to perform the ~p ′ integrationtrivially. As expected, quark and antiquark have opposite momenta, such that ~P = 0.

Of course the quarks in a bound state are not free, such that the free wave functions u(p, r) e−i px

etc. have to be replaced by corresponding, generally unknown, wave function of the bound state.The wave function of the bound state also determines the spin and flavor properties of the boundsystem. A creation operator for a mesonic bound state in the rest system, formally has the form:

C+M (~P = 0, λ) = N ·

∑

cc′,ii′

∑

r,r′

∫dµ(p) f(~p)ϕcc′;ii′(λ; r, r′) · a+ci(~p, r) b+c′i′(−~p, r′) ,

where f(~p) is the momentum distribution of the valence quarks and the coefficients ϕcc′;ii′(λ; r, r′)have to be determined in such a way that the meson creation operator C+

M has the desired quantumnumbers. The creation operators a+ and b+ of course do not create free quarks, but the so calledvalence quarks, which characterize the macroscopic properties of the sysyem. Besides the valencequarks there are the virtual quarks, called sea–quarks, and the gluons. N is a normalizationfactor. The state

|M(~P = 0, λ)〉 = C+M (~P = 0, λ)|0〉 (9.33)

describes a meson at rest. In order to fix the normalization we need a propagating state. Ourstandard normalization is then given by

〈M(~P , λ′)|M(~0, λ)〉 = δλλ′ 2P 0 (2π)3 δ(3)(~P ) ,

where P 0 = M is the mass of the meosn. Note that with this normalization only, our standardformulae for T–matrix elements, cross sections and decay widths remain valid.

Formally, a moving quark–antiquark state at time t = 0 is given by

|M(~P , λ)〉 =

∫d3x ei

~P ~x 1√3

∑

c

: ψαci(x) (Γ)αβ ψβck(x) : |0〉

=1√3

∑

c

∑

r,r′

∫dµ(p)dµ(p′) uα(p, r) (Γ)αβ vβ(p′, r′)×

×∫

d3x ei (~P−~p ′−~p)~x

︸︷︷︸(2π)3 δ(3)(~P−~p ′−~p)

a+ci(~p, r) b+ck(~p

′, r′) |0〉

=∑

r,r′

∫dµ(p)

1

2ωP−puα(~p, r) (Γ)αβ vβ(~P − ~p, r′) · 1√

3

∑

c

a+ci(~p, r) b+ck(

~P − ~p, r′) |0〉 .

256

We thus can calculate the normalization

〈0|CM (~P , λ′)C+M (~0, λ)|0〉

=∑

s,s′

∫dµ(q)

1

2ω~P−~qv(~P − ~q, s′) Γλ′ u(~q, s)×

∑

r,r′

∫dµ(p)

1

2ω~pu(~p, r) Γλ v(−~p, r′)

×∑

c,c′

(1√3

)2

δcc′ δii′ δcc′ δkk′ δrs δr′s′ (2π)6 2ω~p 2ω~P−~q × δ(3)(~q − ~p) δ(3)(~P − ~q + ~p)

= 3 δii′δkk′∑

r,r′

∫dµ(p)

1

2ω~pv(−~p, r′) Γλ′ u(~p, r) u(~p, r) Γλ v(−~p, r′)× (2π)3 δ(3)(~P ) .

We now have to replace the free wave functions by the ones of the bound state. This is achievedby the substitution

∑

ii′

∑

cc′

∑

rr′

1√3

1

2ω~pu(p, r) Γλ v(−~p, r′) δcc′ · · · ⇒

∑

ii′

∑

cc′

∑

rr′

N f(~p)ϕcc′;ii′(λ, r, r′) · · ·

and results in

〈M(~P , λ′)|M(~0, λ)〉 =∑

cc′

∑

ii′

∑

cc′

∑

ii′

∑

ss′

∑

rr′

ϕ∗cc′ ;ii′(λ′, s, s′)ϕcc′;ii′(λ, r, r

′)

× δcc δc′c′ δii δi′i′ δrs δr′s′ ×∫

dµ(p) |f(~p)|2 2ω~p (2π)3 δ(3)(~P ) ·N2 .

Our normalization thus requires the bound state wave function factors to have the normalizations:

1.∑

cc′

∑

ii′

∑

r,r′

ϕ∗cc′;ii′(λ, r, r′)ϕcc′;ii′(λ, r, r

′) = δλ′λ ,

2.

∫d3p

(2π)3|f(~p)|2 = 1 ,

3. N =√

2M . (9.34)

Herewith we are equipped to calculate simple matrix elements. We are interested here in thematrix element

〈0|Jµ(0)|M(~P = 0.λ)〉 (9.35)

of the electromagnetic current

Jµ(x) =∑

c,i

Qi : ψci(x) γµ ψci(x) : (9.36)

Only the term of Jµ exhibiting two annihilation operators yields a contribution,

〈0| : Jµ(0) :=∑

c,f

Qf∑

ss′

∫dµ(q) dµ(q′) vα(q′, s′) (γ)αβ uβ(q, s) · 〈0|bc,f (~q ′, s′) ac,f (~q, s) ,

and we obtain

〈0|Jµ(0)C+M (~P = 0, λ)|0〉

=√M∑

c,f

Qf∑

s,s′

∑

c,c′

∑

i,i′

∑

r,r′

∫dµ(q) dµ(q′)

∫dµ(p) f(~p )× vα(q′, s′) γµαβ uβ(q, s)

257

×ϕcc′;ii′(λ; r, r′) δcc δfi δsr (2π)3 2ω~p δ(3)(~p− ~q) δcc′ δfi′ δs′r′ (2π)3 δ(3)(−~p− ~q ′)

=√M∑

c,f

Qf∑

s,s′

∫dµ(p) f(~p ) vα(−~p, s′) γµαβ uβ(~p, s)ϕcc;ff (λ; s, s′)

=√M∑

c,f

Qf∑

s,s′

v(0, s′) γµ u(0, s)

∫dµ(p) f(~p )

2m

ω~p +mϕcc;ff (λ; s, s′)

≃√M∑

c,f

Qf∑

s,s′

v(0, s′) γµ u(0, s)ϕcc;ff (λ; s, s′)ψ(0) ,

where in the last step we assumed the non-relativistic limit to be appropriate. Let us work outthe non-relativistic limit for the electromagnetic current explicitly:

v(−~p, s′) γµ u(~p, s) = v+(−~p, s′) γ0 γµ u(~p, s)

= (p0 +m)

(V +(s′)

~σ · ~pp0 +m

,V +(s′)

)γ0 γµ

U(r)

~σ·~pp0+m U(r)

=

0 ; µ = 0

2mp0+m

V +(s′)σi U(s) ; µ = i

=2m

p0 +mv(0, s′) γµ u(0, s) .

We have been using ~σ · ~p σi ~σ · ~p = ~p 2 σi and p0 = ω~p =√~p 2 +m2. Furthermore, taking the

non-reletivistic approximation we have∫

dµ(p)2m

ω~p +mf(~p) =

∫d3p

(2π)32m

ω~p (ω~p +m)f(~p) ≃ 1

m

∫d3p

(2π)3f(~p) .

In fact, the non-relativistic wave function ψ(~r) of the bound state is given by

ψ(~r) =

∫d3p

(2π)3ei ~p~rf(~p)

and hence∫

d3p

(2π)3f(~p) = ψ(0) .

This last relation we have been using above, already.

Color, flavor and spin:Color ,flavor and spin are independent and hence factorize

ϕcc;ff (λ; s, s′) = Φ(c)cc′ · Φ

(F )ff ′ · χspin(λ; s, s′) .

1. Color: we consider color first: the mesons are color singlet

Φ(c)cc′ =

1√3δcc′ .

Note that∑

c

Φ(c)cc′ =

√3

yields the color factors in matrix elements.

2. Flavor: Qf = 23 ; f = u, c, t and Qf = −1

3 ; f = d, s, b. Our main interest here are thevector mesons JPC = 1−−:

258

ρ0 = 1√2(uu− dd) : Φρ0

uu = 1√2, Φρ0

dd = − 1√2,∑

f Qf Φρ0

ff = 1√2(23 + 1

3) = 1√2

ω = 1√2(uu+ dd) : Φω

uu = 1√2, Φω

dd = 1√2,∑

f Qf Φωff = 1√

2(23 − 1

3) = 13√2

φ = ss : Φφff 6=s = 0, Φφ

ss = 1,∑

f Qf Φφff = −1

3

J/ψ = cc : ΦJ/ψff 6=c = 0, Φ

J/ψcc = 1,

∑f Qf Φ

J/ψff = 2

3

Υ = bb : ΦΥff 6=b = 0, ΦΥ

bb = 1,∑

f Qf ΦΥff = −1

3 .

In general color– and flavor–factors take to form

∑

c

∑

f

Φ(c)cc Φ

(F )ff =

√3 〈Q〉 =

√3

1√2,

1

3√

2, − 1

3,

2

3, − 1

3, · · ·

(9.37)

ρ0 ω φ J/ψ Υ

3. Spin: χspin(λ; s, s′)We again consider a massive vector boson (spin 1). The possible spin states are

|↑↑〉 , 1√2|↑↓ + ↓↑〉 , |↓↓〉

J3 : +1 , 0 , − 1

The spin dependence of

ϕcc′;ii′(λ; s, s′) = Φ(c)cc Φ

(F )ff χ(λ; s, s′)

is characterized by the quark–antiquark wave function

u(0, s) γµ v(0, s′) =

0 ; µ = 0

−U+(s)σi V (s′) ; µ = i = 1, 2, 3,

which couples in an obvious manner to the electromagnetic current:

Jµ(0) = · · · v(0, s′) γµ u(0, s) ,

which means

Jµ(0) u(0, s) γµ v(0, s′) = J0(0) u(0, s) γ0 v(0, s′)− J i(0) u(0, s) γi v(0, s′)

= J i(0)(−U+(s)σi V (s′)

),

where the minus sign comes from the metric γi = −γi. We now look for states with fixed J3: wehave

U(↑) =

(1

0

), U(↓) =

(0

1

); V (↑) =

(0

1

), V (↓) = −

(1

0

),

and with σ1 = σ+ + σ− ; σ2 = −i (σ+ − σ−), i.e.,

σ+ =σ1 + iσ2

2=

0 1

0 0

; σ− =

σ1 − iσ22

=

0 0

1 0

,

the relationship between the normalized J3 eigenstates and −U+(s)σi V (s′) is fixed:

δs↑ δs′↑ = U+(s)σ+ V (s′) : λ = +

δs↓ δs′↓ = −U+(s)σ− V (s′) : λ = −1√2

(δs↑ δs′↓ + δs↓ δs′↑

)= − 1√

2U+(s)σ3 V (s′) : λ = 0 .

259

We thus have

−U+(s)σi V (s′) = −δik U+(s)σk V (s′)

= δi1 U+(s) (−σ+ − σ−)V (s′) + δi2 U

+(s) (iσ+ − iσ−)V (s′) + δi3 U+(s) (−σ3)V (s′)

= (−δi1 + i δi2) δs↑ δs′↑ + (δi1 + i δi2) δs↓ δs′↓ +√

2 δi31√2

(δs↑ δs′↓ + δs↓ δs′↑

)

=√

2 δiλ χ(λ; s, s′) .

The coefficients χ(λ; s, s′) are just the Clebsch-Gordon coefficients of the reduction 1/2 ⊗ 1/2 =0⊕ 1. Non-vanishing components are

χ(+; ↑↑) = 1 ; χ(−; ↓↓) = 1

χ(0; ↑↓) =1√2

; χ(0; ↓↑) =1√2,

and we have defined

δiλ : δi+.=−δi1 + iδi2√

2, δi−

.=δi1 + iδi2√

2, δi0

.= δi3 .

The wave function of the massive spin 1 boson are given by the polarization vector ε(P, λ), withthe properties:

ε0(P, λ) = 0 ; ~P · ~ε(P, λ) = 0 ,

εµ(P, λ) ε∗µ(P, λ′) = −δλλ′∑

λ=0,±εµ(P, λ) ε∗ν(P, λ) = −gµν +

PµPνM2

~P=0=

0 ; (µ, ν) 6= (i, k)

δik ; (µ, ν) = (i, k).

In the rest system of the meson, in the kartesian basis, we thus have ei(λ) = δiλ for i = 1, 2, 3.This translates into εi(±) = 1√

2(∓ei(1) − i ei(2)) ; εi(0) = ei(3), or

εi(+) =1√2

−1

−i

0

; εi(−) =

1√2

1

−i

0

; εi(0) =

0

0

1

.

Back to the matrix element

〈0|Jµ(0)|M(~P = 0, λ)〉 ,we know it must be proportional to the above polarization vectors:

∑

s,s′

v(0, s′) γµ u(0, s)︸︷︷︸

=

0

V +(s′)σiU(s)

χ(λ; s, s′) ∝ εµ(λ) ,

whereby

λ = + : V +(s′)σiU(s) = (0, 1)σi

(1

0

)= (1, i, 0) = −

√2 εi(+)

λ = − : V +(s′)σiU(s) = −(1, 0)σi

(0

1

)= (−1, i, 0) = −

√2 εi(−)

λ = 0 : V +(s′)σiU(s) = −(1, 0)σi

(1

0

)+ (0, 1)σi

(0

1

)= (0, 0,−2) = −2 εi(0) .

260

Table 4: Properties of leading JPC = 1−− vector mesons

V ρ ω φ J/ψ Υ

Mass [MeV] 775.49 ± 0.34 782.65 ± 0.12 1019.455 ± 0.020 3096.916 ± 0.011 9460.30 ± 0.26

Γ [MeV] 149.4 ± 3.0 8.49 ± 0.08 4.26 ± 0.04 0.0932 ± 0.0021 0.05402 ± 0.00125

Γe+e− [keV] 7.04± 0.06 0.60 ± 0.02 1.26 ± 0.02 5.55± 0.14 1.340 ± 0.018

〈Qf 〉 1/2 1/18 1/9 4/9 1/9

λV = e γV 0.0611 0.0178 0.0225 0.0271 0.0076

|ψ(0)|2 6.327 44.487 39.627 100.674 3629.101

Taking into account εi(λ) = −εi(λ) (i = 1, 2, 3) and the Clebsch-Gordon coefficients we obtainindeed:

∑

s,s′

v(0, s′) γµ u(0, s)χ(spin)(λ; s, s′) =√

2 εµ(λ)

with εµ(λ) = εµ(P, λ)| ~P=0. Alltogether, we have obtained

〈0|Jµ(0)|M(~P = 0, λ)〉 = 2√M√Nc · 〈Qf 〉M εµ(~P = 0, λ) · ψ(0) . (9.38)

We finally may compare this result with the matrix element of an elementary vector meson V inthe VDM

〈0|Jµ(0)|MV (~P = 0, λ)〉 = γV M2V 〈0|V µ(0)|MV (~P = 0, λ)〉︸︷︷︸

εµ(~P=0,λ)

.

The VDM ansatz JµV (x).= γV M

2V V

µ(x) implies JµV (x)Aµ(x) = γV M2V V

µ(x)Aµ(x) as an effec-tive coupling to the photon. Note, Jµ has dim J=3 while V µ has dimV=1. We thus have therelationship

γV M2V = 2

√3√MV 〈Qf 〉V ψ(0)

between the phenomenological VDM ansatz, which defines the phenomenological V − γ couplingparameter γV , and the bound state wave function at the origin ψ(0), which appears as the onlyas yet undetermined quantity in the above bound state calculation, in the non–relativistic limit.Other notation frequently used are: fV = γV M

2V and gV = 1/γV etc. In our notation

γ2V = 4Nc 〈Qf 〉2|ψ(0)|2M3V

=3ΓV→e+e−4πα2MV

, (9.39)

incorporates the compositeness of the vector meson. Like 〈Qf 〉 which is the effectve charge of thevector meson, which is composed of constituents of different charges Qf .

In Table 4 we have collected the parameters of the known leading vector mesons. The parameterλV we have defined as

λV = e γV =

(3ΓV→e+e−αMV

)1/2

.

Lit.:R. van Royen and U. Weisskopf, Nuovo Cim. 40 (1967), 617;H. Ito, Prog. Theor. Phys. 77 (1987), 681

261

9.9 Applications

9.9.1 Determination of αs

In our withs predictions typically depend on the wave function ψ(0). The latter non–perturbativequantity is generally not well known, theoretical predictions in general are model dependent. Itis therefore interesting to consider ration which are independent of ψ(0). Such a ratio is

BΥh/ℓ =

Γ(Υ3g→ hadrons)

Γ(Υ→ e+e−)=

10 (π2 − 9)

81πα2Q2b

α3s(MΥ) . (9.40)

This is a very interesting result as it is ∝ α3s in leading order, and hence potentally an ideal

observable the determine αs. The price to pay is that the both widthes are not known with thesame precision because the leptonic one is one order of magnitude smaller. Nevertheless it is aninteresting observalbe for the determination of αs. As usual in pQCD, higher order correctionsare important and they have been calculated to O(α2

s) relative to the leading term. We considerhere the leading corrections:

γ+

γ

3g + 3g

which yields corrections

Γe+e− = Γe+e−,0

(1− 16

3

αsπ

)

Γhad = Γhad,0

(1 + c

αsπ

)(9.41)

with c = 3.8± 0.5 (Mackenzie and Lepage 1981), such that

BΥh/ℓ = BΥ

h/ℓ,0

(1 + (1.1± 0.5)

αsπ

)(9.42)

Note that the inclusive decays via 3g can be extracted from observable quantities as follows

Γ(Υ→ 3g → hadrons) ≃ ΓΥtot − 3 ΓΥ→e+e− −R(µ = 10 GeV) ΓΥ→e+e− ≃ 34.8 ± 9.6 keV .

Since me,mµ,mτ ≪MΥ, we have used

ΓΥ→e+e− + ΓΥ→µ+µ− + ΓΥ→τ−τ− ≃ 3 ΓΥ→e+e− ; ΓΥ→e+e− = 1.22± 0.05 keV ,

and

RΓΥ→e+e− =σ(e+e− → hadrons)

σ(e+e− → µ+µ−)· ΓΥ→µ+µ− ; R ≃ 3.7 ,

262

These are the hadronic channels Υ→ γ → hadrons which do not proceede via the 3g intermediatestate. Using the experimental branching fraction BΥ = 28.5 ± 7.9 we obtain

αs(MΥ) =

(BΥh/ℓ ·

81πα2 19

10 (π2 − 9)

1

1 + (1.1± 0.5)αsπ

)1/3

= 0.167 ± 0.015 .

With this value we may estimate

R(s)(udsc) = 3.5(

1 +αsπ

)≃ 3.69 ± 0.02 .

The approximation works the better the heavier the quarks. However we als expect that we cantreat the J/ψ decays in the same way. At MJ/ψ we have R(s)(uds) ≃ 2.1, Γ(J/ψ → had) =51.86 ± 1.28 keV and Γ(J/ψ → e+e−) = 4.7 ± 0.3 keV, and subtracting R · Γe+e− yields

Γ(J/ψ → 3g → hadrons) ≃ 42.0 ± 1.4 keV ,

such that

BJ/ψh/ℓ =

Γ(J/ψ → hadrons)

Γ(J/ψ → e+e−)= 8.9± 0.6 .

From the ratio

BJ/ψh/ℓ

BΥh/ℓ

=α3s(MJ/ψ)

α3s(MΥ)

· Q2b

Q2c

=1

4

α3s(MJ/ψ)

α3s(MΥ)

Using 4BJ/ψh/ℓ ≃ 35.6 > BΥ

h/ℓ ≃ 28.4 we may compute αs(MJ/ψ):

αs(MJ/ψ) = αs(MΥ) ·

4B

J/ψh/ℓ

BΥh/ℓ

1/3

= αs(MΥ)× (1.08 ± 0.042)

= 0.18 ± 0.07 ,

which implies that αs is running! Indeed, the RG with αs(MΥ) as an input yields

αs(MJ/ψ) = 0.210 ± 0.028 ,

in reasonable agreement.

In a similar way one may use the ratio

Bγh =Γ(V → γgg → γ + hadrons)

Γ(V → ggg → hadrons)=

36

5〈Qf 〉2V

α

αs(MV )

(1 + (2.2 ± 0.6)

αsπ

)

to determine αs(MV ).

Here we presented very simple estimates based on NLO corrections. For uptodate determinationsincluding higher order effects I refer to Brambilla et al. 2007 [54] and references therein. Thisrecent determination of αs from Γ(Υ(1S)→ Xγ)/Γ(Υ(1S)→ X) with CLEO data by taking intoaccount color octet contributions and avoiding any model dependence in the extraction, yieldsαs(MΥ(1S)) = 0.184+0.015

−0.014, which corresponds to αs(MZ) = 0.119+0.006−0.005.

Pre-Coulombic behavior: hyperfine splitting In the region where the one gluon exchange diagramdetermines the qq–potential and the the extent that a potential si suitable to describe the leadingphysics effects, the energy level splitting of the ground state is given by

∆E0 = M(13S1)−M(11S0) =32π

9

α− s(M)

m2|ψ(0)|2 ,

as in positronium with the replacement α→ 43 αs. The states in question here are

263

13S1 : Υ or J/ψ : JPC = 1−−

11S0 : ηb or ηc : JPC = 0−+

and |ψ(0)|2 can be determined experimentally from Γ(13S1 → e+e−) or calculated within quarko-nia potential models. Using van Royen–Weisskopf relationship

|ψ(0)|2 =M2V ΓV→e+e−

16π α2Q2q

we obtain

∆E0 =2

9Q2q α

2

M2V

m2q

αs(MV ) Γ(V → e+e−) . (9.43)

For the Υ–system then

MΥ −Mηb ≃2

α2

M2Υ

m2b

αs(MΥ) · Γ(Υ→ e+e−) ≃ 31 MeV

where we have used mb(mb) = 4.7 ± 0.1 GeV, MΥ ≃ 9.46 GeV and αs ≃ 0.167. For the J/ψ–system similarly

MJ/ψ −Mηc ≃2

α2

M2J/ψ

m2c

αs(MJ/ψ) · Γ(J/ψ → e+e−) ≃ 66 MeV

where we have used mc(mc) = 1.27 ± 0.05 GeV, MJ/ψ ≃ 3.01 GeV and αs ≃ 0.25. The experi-mental value is MJ/ψ −Mηc = 116± 9 MeV. Charmonium models yield ≃ 99 MeV, which showsthat nonperturbative effects are substantial.

As a last example we illistrate the valididy and meanung of the OZI-rule: the stability of thestates J/ψ and ψ′ is the result of the fact that they are OZI forbidden! Hadronically, they onlydecay via gluons into light hadrons

J/ψ, ψ′ u, d, s

because their masses are below the charm threshold. This is different for ψ′′

which predominantlydecays into a pair of the charmed D mesons (see Fig. 9.17)

ψ′′ → DD .

The D mesons into which ψ′′

decay at the 90% level are the JP = 0− staes D0, D+, D0 andD−, with quark content cu, cd, cu and cd, respectively. The decays are OZI allowed, such that

+ψ

′′

D0

D0

D−

D+

c

c

u

u

c

c

d

d

Figure 9.17: Charm conserving ψ′′

decay modes (with c and c as spectator quarks).

the process proceeds witout purely gluonic intermediate states and the original quarks which

264

made up the ψ′′

are preserved and appear in the final states hadrons. Such quarks are calledspectator quarks. The condition for the decay ψ → DD into charmed mesons obviously is

Mψ > 2MD ≃ 3726.6 MeV

which is fulfilled for ψ′′

not however for the very slightly lighter ψ′. For the D meson masses wehave M(D0) = 1863.3±0.9 MeV and M(D±) = 1868.3±0.9 MeV . If we compare the parametersof ψ

′′and ψ′

M(ψ′′) = 3770.0 ± 3 MeV M(ψ′) = 3686 ± 0.10 MeV

Γ(ψ′′) = 25± 3 MeV Γ(ψ

′) = 0.215 ± 0.040 MeV

Γ(ψ′′ → e+e−) = 0.26 ± 0.05 keV Γ(ψ

′ → e+e−) = 2.05 ± 0.2 keV

we indeed observe that ψ′′

is much less stable then the ψ′.

Further reading: much more details may be found in the collection of expert contribution in [19]and the updates you may find in the Particle Data Tables [55]. All about present and futurequarkonium the reader may find in the CERN Yellow report [56].


① Draw the relevant Feynman diagrams (within the SM) for the decays D0 → K−π+, K−e+νeand K−π+π0, based on the quark content of the hadrons.

② The decay D0 → K−π+π0 is observed with a branching fraction 13.9 %. What is thebranching fraction of the charge conjugate decay D0 → K+π−π0?

③ Explain why D+s → µ+νµ has a branching fraction of 6.2·10−3 % only, while D+

s → τ+ντhas one of 6.6 %.

④ The top quark decays via the weak process t→ Wb, with subsequent decay of the W andthe b (e.g. W → eν and b → b–jet of hadrons). Above threshold for real W production(with mt ≃ 172 GeV and MW ≃ 80.1 GeV well satisfied), in the approximation mb ≪ mt,the tree level QPM top width is given by

Γt =

√2GF m

3t

32π2 (1− y)2 (1 + 2 y)

where y = (MW/mt)2. Compare the lifetime of the top quark obtained from this formula

with the actual lifetime of the top and with the lifetimes of Υ and J/ψ resonances. Try toexplain why no toponium resonances are formed.

265

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268

10 Jets

In this lecture we introduce in a more quantitative manner the concept of hadronic jets. We startwith a brief summary of the outline given in Sect. ??, already.

For highly energetic processes (“hard kicks”) asymptotic freedom suggests that perturbative QCDshould work, with the provisos discussed earlier. This manifests itself in the correct prediction ofthe total hadronic cross section in electron–positron annihilation σtot(e

+e− → γ∗ → hadrons) innon–resonant energy ranges. The consequences of the applicability of pQCD go far beyond that.According to pQCD, the production of hadrons in e+e−–annihilation proceeds via the primarycreation of a quark–antiquark pair (up to effects suppressed by one power in α) whereby thequarks hadronize into jets (see Fig. 8.11).

Ve+

e−

q

q

γ e+ e−

q

q

θ

Figure 10.1: Quark–pair production as a leading primary process in hadron production in e+e−–annihilation. At high energies the two highly energetic quarks are produced at fixed productionangle θ.

The elementary process e+e− → qq implies that in a highly energetic collision of e+ and e− inthe c.m. system q and q are produced with high energy in opposite directions as suggested byFig. 10.1. In pQCD the leading contribution is given in this case by the quark–parton model(strong interaction switched off), and the corresponding lowest order differential cross section isgiven by

dσ

dΩ(e+e− → qq) =

4

3

α2

s

∑

f

Q2f

(1 + cos2 θ

), (10.1)

with s = E2cm and θ the production angle. This is an angular distribution typical for the produc-

tion of a pair of spin 1/2 particles, up to a color factor identical with the cross section for muonpair production. Indeed one observes that the two quarks hadronize each by itself by formingjets of hadrons. At high enough energies, these bundles of hadrons show up in relatively narrowangular cones:

q

q

The quarks are not observable, however their production angle can be identified with the jet–axis,which, in principle, is reconstructible from the hadrons in the jets 46. The angular distribution is

46For first attempts to find and analyze jet events at the SPEAR ring at SLAC see [1].

269

γ

e+

e−

q

q

g

γ

e+

e−

q

q

g

g

Figure 10.2: Feynman diagrams for the production of up to 4 hard partons resulting into 2– 3–and 4–jets. With 4 partons in the final state for the first time the triple gluon self-interaction(non-Abelian gauge structure) comes into play (in the last diagram).

thus observable as a distribution of the jet axis. While color and fractional charge are balancedvia infrared effects (sea–quarks and gluons) between the jets, in such a way that only colorsinglet hadrons of integer charge show up in the final state, the momentum has not time enoughto distribute evenly. The point is that the primary quarks supply themselves with sea–quarksfrom the vacuum to form hadrons, which allows to balance color and charge without the need todistribute the momentum isotropically. As a rule: in highly energetic kicks perturbation theorymay be utilized if one identifies ”hard partons (quarks, antiquarks and gluons)” with “hadronjets”.

The key points are that, in the e+e− c.m. system (laboratory system of the electron positronstorage ring), 2–jet events are back-to-back, while 3–jets are coplanar (in a common plane) as itfollows from the primary parton kinematics.

The discovery of the gluon at the PETRA e+e− storage ring at DESY marked the beginning ofa new epoch in QCD and strong interaction physics. First clear evidence was reported by theTASSO collaboration in 1979 [6] immediately followed by JADE, MARK J and PLUTO [7]. Itmade possible to determine the spin of the gluon, which proved to be a vector particle as requiredby QCD. Also the “string–effect”, the hadronization of quarks and gluons via the formation ofcolor strings, could be established. Last but not least jet physics is an important tool for theprecise measurement the strong coupling constant as the ratio of rates 3–jet/2–jet is directlyproportional to αs modulo higher order effects.

In 1989 the Large Electron Positron (LEP) collider went into operation at CERN and due tothe very large event samples QCD tests, αs determinations and jet analyses did become possibleat much higher precision. For example, it was possible now to investigate 4–jet signals in anunambiguous way, which allowed one to determine the group structure of QCD. Types of Feynmandiagrams for perturbative jet cross-section calculations in QCD are shown in Fig. 10.2. Onlytypical cases shown, gluons are to be attached to quarks in all possible ways. Final states areqq (2 jets), qqg (3 jets) and qq, q′q′ (4 jets), qqg g (4 jets). A nice 4–jet event is shown in Fig. 10.4.

Ratio of gluon jets/quark jets ∝ CA/CF . The Casimir coefficients are obtained in squaring the

270

sum of amplitudes represented by the Feynman diagrams (10.2), where typically

∣∣∣∣∣

∣∣∣∣∣

2

∝ αs · CF ,

∣∣∣∣∣

∣∣∣∣∣

2

∝ αs · CA ,

∣∣∣∣∣

∣∣∣∣∣

2

∝ αs · TF Nf

σ4−jet ∼ CF · σA(

+ · · ·)

+ CA · σB(

+ · · ·)

+ TF · σC(

+ · · ·)

+ interference terms

Including all terms yields a gauge invariant result for any unbroken local gauge group, whichimplies that at the given perturbative order the terms proportional to the different Casimirsmust be gauge invariant separately. Therefore the Casimirs are quasi observales, which may beextracted from experimental jet cross–sections with the help of theory (pCD). In NLO QCD theprediction for an observable like R is

R = Aαs + (BCFCF +BCACA +BTFTF Nf )CF α2s .

Usually, experiments fit αs, CA/CF and (TF Nf )/CF , from which the Casimirs may be calculated.Theory values

Theory CF CA TF

QCD 4/3 3 1/2

Abelian 1 0 3

vary widely depending on the gauge group.

Results from 4–jet analysis at LEP typically are:

Experiment αs(MZ) CA CF

ALEPH [8] 0.119 ± 0.027 2.93 ± 0.60 1.35 ± 0.27

OPAL [9] 0.120 ± 0.023 3.02 ± 0.55 1.34 ± 0.26

Ref. [13] 0.119 ± 0.010 2.84 ± 0.24 1.29 ± 0.18

where the last entry was obtained from an event shape analysis. Other results for the gaugestructure determinations are

271

Experiment CA/CF TF/CF

DELPHI [11] 2.25 ± 0.06± 0.12 0.34 ± 0.14

ALEPH 2.24 ± 0.40 0.58 ± 0.29

OPAL [10] 2.23 ± 0.01± 0.14 0.40 ± 0.11 ± 0.14

DELPHI [12] 2.26 ± 0.16

The last result was extracted form observed scaling violations in quark and gluon jets [12].

For the average values one finds [15]

αs = 0.1211 ± 0.0010(exp.)± 0.0018(theo.) ,

CA = 2.89 ± 0.01(stat.)± 0.21 (syst.) ,

CF = 1.30 ± 0.01(stat.)± 0.09 (syst.) ,

in excellent agreement with structure constants of QCD, ruling out an Abelian vector gluon modelas well as many other possible gauge field structures. Results are depicted in Fig. 10.3, showsthe relative n–jet contributions, and which shows how much SU(3)c is favored by experimentaljet data from the LEP experiments.

0

0.5

1

1.5

2

2.5

0 1 2 3 4 5 6

U(1)3

SU(1)

SU(2)

SU(4)

SU(5)Combined result

SU(3) QCD

ALEPH 4-jet

OPAL 4-jet

Event Shape

OPAL Ngg

DELPHI FF

CF

CA

86% CL error ellipses

Figure 10.3: Experimental results showing n–jet cross-sections and the status of measurementsof the gauge group Casimirs.

Typical LEP jet events are shown in Fig. 10.4. Figure ?? presents another plot of the experimentaldetermination of the gauge structure of QCD.

What precisely is a jet? The original approach is that of Sterman and Weinberg [2] and allowsto define this concept to all orders of perturbation theory. Introducing jet resolution parameterstakes care of non–perturbative effects by averaging them out. We will discus this, as an example,below in detail. In fact it turned out soon that other concepts of characterizing hadronic finalstates are more practical and also are calculable in pQCD and after appropriate smearing ofsoft and collinear effects can be compared directly with the data. The key requirement is thatthe observables must be infrared and collinear safe. What this means we will investigate bysimple analytic calculations below. Infrared safe means that the quantity, in order to be an

272

Figure 10.4: A 3–jet event from OPAL and a 4–jet event from the ALEPH detectors at LEP.

true observable, should not change discontinuously if one adds a soft particle to the final state.Similarly, collinear safe means that the quantity should not change abruptly if one splits one finalstate particle into two particles with equal momentum.

In general, the properties of hadronic events may be analyzed using a set of event shape observ-ables. These may be used to classify the distribution of particles and the event topology (planar,spherical etc.). The characteristic features can be computed either using the measured chargedparticle tracks and calorimeter clusters, or using the true hadrons or partons in simulated events.

Historically, one of the first tools for the analysis of jet events was the sphericity tensor. Unfortu-nately the latter is not collinear safe and would not be used in practice any more. Nevertheless,we discuss it here as a pedagogical example, as it was used in some of the pioneering experiments.Conceptually similar save observables we will considered as a next step below.

If in a e+e− → qq event r hadrons with momenta ~pa (a = 1, · · · , r) are produced, one can constructthe sphericity tensor

Mij =

∑a piapja∑a piapia

and performs a transformation to principal axes. The corresponding eigenvalues are λ1, λ2, λ3which we assume ordered according to 0 ≤ λ1 ≤ λ2 ≤ λ3 and normalized to λ1 + λ2 + λ3 = 1.They determine the sphericity S = 3

2 (λ1 + λ2) and the acoplanarity A = 32 λ1. Extremal cases

are1) isotropic distribution: S = 1, A = 1

22) “ideal jet”: S = 0, A = 0Denoting n1. n2 and n3 the directions of the corresponding main axes, the eigenvalues have tofollowing interpretation:

λ1 = minn1

∑

a (~pa·n1)∑

a ~pa2 ”flatness” of event

λ2 = minn2⊥n1

∑

a (~pa·n2)∑

a ~pa2 ”width” of event

λ3 = maxn3

∑

a (~pa·n3)∑

a ~pa2 ”length” of event

Events with λ1 ≪ λ2 are called “coplanar”, events with λ2 ≪ λ3 are called “collinear”.

273

One may use a triangular plot Y =√32 (λ2−λ1) as a function of S = 3

2(λ2 +λ1) to separate 2–jet,3–jet and non-coplanar events, as shown in Fig. 10.5.

A=

3

2

λ1

coplan

ar(3

Jets)

noncoplanar

(2 Jets)0 → S = 3

2(λ1 + λ2)

Y =√

3

2(λ2 − λ1)

Figure 10.5: Triangular plot which allows to separate 2–jets from 3–jets and noncoplanar events.

Figure 10.6: TASSO distributions of sphericity versus aplanarity at W = 14, 34 and 41.5 GeV.

As expected, data show that preferably 2–jet events are produced. If one investigates the observedfrequency of 2–jet events, which are produced at an angle θ of the jet-axis n3, one indeed observesclearly the perturbative 1 + cos2 θ dependence typical for the production of a pair of spin 1/2particles. For scalar quarks one would get a sin2 θ dependence instead. The spin 1/2 nature ofthe hadronic constituents is thus experimentally established.

274

Figure 10.7: The angular distribution of the jet axis determined by sphericity. The curve showsthe 1 + cos2 θ shape.

As mentioned, sphericity

S =3

2

∑a |~paT |2∑a |~pa|2

,

with ~pa the four momentum of particle a and ~paT is taken relative to the axis which minimizes∑a |~paT |2, is not a good physical observable, because its not collinear safe. It can be replaced by

• Spherocity S: defined as

S =4

πmin

(∑a |~paT |∑a |~pa|

)2

, (10.2)

which is both infrared and collinear safe. Other frequently utilized event shapes are the following(see e.g. [15] and references therein):

• Thrust T : is defined by the expression

T = max~n

(∑i |~pi · ~n|∑i |pi|

). (10.3)

The thrust axis ~nT is the direction ~n which maximizes the expression in parentheses. A planethrough the origin and perpendicular to ~nT divides the event into two hemispheres H1 and H2.

• C-parameter: The linearized momentum tensor Θαβ is defined by

Θαβ =

∑i(p

αi p

βi )/|~pi|∑

i |~pi|, α, β = 1, 2, 3 .

The three eigenvalues λj of this tensor define C through

C = 3(λ1λ2 + λ2λ3 + λ3λ1) .

275

• Heavy Jet Mass MH:

The hemisphere invariant masses are calculated using the particles in the two hemispheres H1

and H2. Then MH is defined as the heavier mass, divided by√s.

• Jet Broadening observables BT and BW:

These are defined by computing the quantity

Bk =

(∑i∈Hk

|~pi × ~nT |2∑

i |~pi|

)

for each of the two event hemispheres, Hk, defined above. The two observables are defined by

BT = B1 +B2 , and BW = max(B1, B2)

where BT is the total and BW is the wide jet broadening.

• Transition value between 2 and 3 jets y23:

Jet algorithms are applied to cluster the large number of particles of an hadronic event into asmall number of jets, reflecting the parton structure of the event. Defining each particle initiallyto be a jet, a resolution variable yij is calculated for each pair of jets i and j:

yij =2min(E2

i , E2j )

E2vis

(1− cos θij),

where Ei and Ej are the energies, cos θij is the angle between the two jets and Evis is the sumof the energies of all visible particles in the event (or the partons in a theoretical calculation).If the smallest value of yij is less than a predefined value ycut, the pair is replaced by a jet withfour momentum pµij = pµi + pµj , and the clustering starts again with pµij instead of the momentapµi and pµj . Clustering ends when the smallest value of yij is larger than ycut. The remaining jetsare then counted. The value of ycut at which for an event the transition between a 2-jet and a3-jet assignment occurs is called y23.

For applications of these concepts we refer to [15].

Exercise:1) Show that sphericity is infrared safe but not collinear safe.2) Show that spherocity is both infrared and ultraviolet save.

Safe observables have a very useful property: they can be defined in calorimetric terms. In thespherodicity definition a can be used to label the calorimetric cells and ~pa is the energy detectedin cell a times the unit vector pointing to it. One can show that for sufficiently small cells andsufficiently high energy the two definitions coincide, provided hadronization effects are powersuppressed.

10.1 QCD corrections O(αs) to the 2–jet cross–section

The lowest order 2–jet cross–section is given by (10.1). The corrections O(αs) in pQCD maybe calculated exactly like corresponding photon emission corrections in QED. Not surprisingly,in pQCD the same type of problems show up, like infrared and collinear singularities. Goingfrom QED [U(1)em] to QCD [SU(3)c] the difference is the Casimir operator C2(R) = 4

3 , whichamount to replace α by 4

3 αs. Strong interaction corrections affect only the final state (quarks).We therefore need only consider the sub–process γ∗ → qq in the following.

276

10.1.1 Virtual corrections

The virtual corrections

+ +

for massless gluons and on-shell quarks (i.e., after on-shell renormalization, which because of itsgauge invariance is advocated here) is infrared divergent.

***

Remember: the origin of the troubles are the Coulomb like forces between colored quarks, which make perturbativelydefined quark scattering states illdefined. Looking at q(p)q(p) → q′(p′)q′(p′) in leading order,

+

q, p

q, p

q, p′

q, p′

k↓P→

the scattering diagram has a singularity in the physical region, the forward peak. The scattering amplitude is

∝ 1

k2=

1

(p− p′)2= − 1

2~p 2 (1− cos θ)

θ

~p

~p ′

and therefore dσd cos θ

is singular in forward direction. As a consequence the total cross section σtot =∫ +1

−1d cos θ dσ

d cos θ

is inexistent, while σcut(δ) =∫ 1−δ

−1d cos θ dσ

d cos θincreases unboundedly as δ → 0. This carries over to loops, like

which is infrared singular iff quarks are on–shell. On shell quarks are unphysical due to steady emission and re-

absorption of soft gluons, which causes “physical” quarks to be never realy on–shell. In pQCD of quarks we thus

encounter exactly the same problems as in QED with the charged leptons.

***

The required calculations we have performed in Sect. 7. As advocated there, in the case of one–loop corrections, we may use and infinitesimal gluon mass mg to regularize the IR singularitiesat intermediate steps. In observable quantities the limit mg → 0 exists and will be taken. Inaddition we consider the energy regions where the active quarks, in the sense of Nf–flavor effectiveQCD, have negligible mass: mq ≪

√s. Only then the perturbative treatment is adequate. The

result then may be worked out easily from the results presented in Sects. 7.1 and 7.4. The resultis given by

(dσ

dΩ

)

virtual

=

(dσ

dΩ

)

parton

· Cvirtual (10.4)

with

Cvirtual =4

3

αs2π

4

(ln

s

m2q

− 1

)lnmg√s

+ ln2 s

m2q

+ lns

m2q

− 4− 2π2

3

.

277

Like in QED, an IR finite result can only be obtained by looking for a truly observable quantity. Infact in quantum field theories with Coulomb type long range interactions, with the steady emissionand absorption of soft massless quanta, any measurement beyond the finite energy resolution ωof any detector, cannot distinguish between virtual and soft quanta of energy Eg ≤ ω. Thusone has to include soft real emission on the same footing to obtain a realistic measurable result.In our case, we thus have to take into account the soft real gluon emission (Bloch–Nordsieckprescription!) in order to get a sensible result.

10.1.2 Real soft gluons

Taking into account soft real gluons of energy Eg ≤ ω = ε√s, we have to calculate the contribu-

tions from the diagrams

+

Since the singular part comes from the very low energetic gluons we may consider the approxima-tion where the gluon four momentum k can be neglected in the energy balance. The correspondingcorrection is the proportional to the Born term, such that the real soft gluon corrections factorize.This factorization in fact is a necessary condition for the singularities to cancel between virtualand real soft contributions:

(dσ

dΩ

)

soft bremsstrahlung

=

(dσ

dΩ

)

parton

· Csoft bremsstrahlung (10.5)

with

Csoftbremsstrahlung =4

3

αs2π2

∫

|~k|≤ω

d3k

2ωk

2 (p1p2)

(kp1)(kp2)− m2

q

(kp1)2− m2

q

(kp2)2

=4

3

αs2π

4

(ln

s

m2q

− 1

)ln

2ω

mg− ln2 s

m2q

+ 2 lns

m2q

− 2π2

3

.

This result again considered as a separate contribution is IR singular and it depend on the energyresolution ω. Again we have taken an infinitesimally small gluon mass mg. Actually, in the

mg → 0 limit we need only the keep the gluon mass in the phase-space integral measure d3k2ωk

with

ωk =√~k 2 +m2

g.

The origin of singular behavior can easily be traced back to the behavior of the virtual quarkpropagators in the bremsstrahlung diagrams: with

1

k/ + p/i −mq + iε=

k/ + p/i −mq

(k + pi)2 −m2q + iε

singular behavior comes from the zeros in the denominator of the last expression when gluons aremassless k2 = 0 and quarks are on–shell p2i = m2

q. Then,

1

(k + pi)2 −m2q + iε

=1

k2 + 2 kpi + p2i −m2q + iε

→ 1

2 kpi + iε.

278

Now,

kpi = k0Ei − ~k ~pi = |~k|√~p 2i +m2

q − |~k||~pi| cos θ = |~k|(√

~p 2i +m2

q − |~pi| cos θ)

mq=0= |~k| |~pi| (1− cos θ)

In the bremsstrahlung cross–section we thus get by performing the phase space integral

d3k = d|~k | |~k |2 dΩ ; dΩ = sin θ dθ dϕ = −d(cos θ) dϕ ,

with mg = 0,

∫

|~k|≥|~k|min

d|~k | |~k |22|~k|

1

|~k| 2∼ log |~k|min

which is logarithmically singular as |~k|min → 0. Alternatively, we may keep an infinitesimally

small gluon mass in the phase space integration whereupon, with ωk =√m2g + ~k 2,

∫

|~k|≥0

d|~k | |~k |22ωk

1

|~k| 2∼ logmg .

In the limit mq = 0 also the intergral∫ +1−1 d(cos θ) 1

(1−cos θ) is singular (collinear singularity), andeither a finit mq or an appropriate angulat cut 1 − cos θ ≥ δ is required in order to get a finiteanswer.

Exercise:a) Show that the hard gluon spectrum, after performing the angular intergrations, may be writtenin the form

dσ

dk= σ0(s)C2(R)

αsπ

(ln

s′

m2q

− 1

)(1 +

(s′

s

)2)

1

k

where k = |~k | is the gluon energy and s′ = s(

1− kEb

)and Eb =

√s/2. Note that s′ is the

invariat mass square of the qq–system after radiation of the gluon, which carried away a fractionk/Eb of the energy.

b) In contrast in the soft gluon limit (factorization) may be represented as

σ(s) = σ0(s) +C2(R)αsπ

∫

k≤ωds′ P (s′)σ0(s)

with P (s′) radiator function

P (s′) =

(p1kp1− p2kp2

)2

integrated over cos θ. The azimutal integration over ϕ yield a factor 2π. Note ds′ = −2√s dk.

Work out P (s′).

279

10.1.3 Virtual plus real soft gluons

Both the virtual and the soft real gluon emission are proportional to the lowest order cross–sectionand both are O(αs) corrections such that they can, and actually must, be added

(dσ

dΩ

)

qqjets

=

(dσ

dΩ

)

parton

(1 + C<ω) (10.6)

with

C<ω = Cvirtual + Csoft bremsstrahlung

=4

3

αs2π

4

(ln

s

m2q

− 1

)ln

2ω√s

+ 3 lns

m2q

− 4− 4π2

3

.

The following points are remarkable about this result:

1) the infrared singular terms (the ones ∝ lnmg) dropped out; thus the sum of virtual plusreal soft corrections is IR finite,

2) the big double logarithms proportional to ln2 sm2

q(Sudakov Logarithms) also canceled,

3) however, a limit mq → 0 does not exist!, these are the so called collinear mass singularitiesdue to the degeneracy of a massless quark traveling together with the massless gluon alongthe same direction.

Because of point 3) the result for QCD is physically not satisfactory. We can remedy this bytaking into account the finite angular resolution of identifying the jets. Hard gluons which travelalmost parallel to the quark or antiquark cannot be resolved experimentally. Each detector notonly has a finite energy detection threshold, but also a limited angular resolution. Before wediscuss the angular cuts, let us first look at the result we get if we integrate over all the real gluon(full phase–space integration)

(dσ

dΩ

)(e+e− → qq, qqg) =

(dσ

dΩ

)

parton

(1 +Call) (10.7)

with

Call =4

3

αs2π

3

2,

a completely regular perturbative sensible result, all kinds of possible singularities are absent.The result tells us that singularities are caused by unphysical “cuts”. Physically sensible cuts arethose which admit to separate unambiguously 2–jet events from 3–jet events. At the given orderat most 3 jets are possible. At the next order also 4 jets are possible, and actually up to 4–jetevents have been investigated with remarkable precision at LEP (see e.g. Fig. 10.4). Here weare going to investigate 3–jet structure in more detail. The experimental identification of 3–jetevents, as we mentioned earlier, is a direct for the existence of the gluon and an experimentalconfirmation of the qqg–gauge coupling.

280

e+ e−

q

q

gluon

δ

θ

u d

π−

uu

π0 ud

π+π0

Figure 10.8: Left: three jets separated by cones; δ is the half opening angle of the cone and θ theangle between two cone axes. Right: possible low multiplicity hadronization mode

10.2 3–Jet Events

In the process γ∗ → qqg one can get 3–jet events, provided q, q and g all have large momenta indifferent directions, as illustrated in the Fig. 10.8

In order to understand the gross features it is sufficient to work in the approximation mq ≃ 0:we use the following notation:

eq =Eq√s

=1

2

(1− eg

(1− cos θ

)),

eq =Eq√s

=1

2

(1− eg

(1 + cos θ

)),

eg =Eg√s

; θ : <)(q, g) in the qq cm system . (10.8)

One then has the relations

eq = −eq + (1− eg) ,

eq = eq1− cos θ

1 + cos θ

cos θ

1 + cos θ. (10.9)

In the laboratory system, which coincides with the e+e− c.m. system, the production angle isdenoted by θ :<)(q, g) and we have the relations

cos θ =1− v σ1 + v σ

with v.=

1− cos θ

1 + cos θ; σ

.=

(p1 + p2)2

s(10.10)

A 3–jet event is then defied by the cuts

eq, eq, eg > ε and θqq, θqg, θqg > δ . (10.11)

Of all events a fraction f(√s, ε, δ) are 2–jet events:

1− f =4

3

αsπQ(√s; ε, δ) (10.12)

has first been calculated by Sterman and Weinberg 1977, in the approximation

1≫ δ ≫ 2mq

Eand ε≪ 1

281

eg

1/2

1/2 eq

eq

eg = 12

eg = 0

δδ

δε

ε

ε

Figure 10.9: Three jet event defined by soft (ε) and collinear (δ) kinematical cuts.

, with the result:

Q(ε, δ) ≃ (4 ln 2ε+ 3) lnδ

2+π2

3− 7

4. (10.13)

A little thinking tells us that this approximation is not adequate in practice of jet physics, becausesuch sharp cuts are not realizable in hadronic processes. What we need is the result for the abovedefinition of a 3–jet event for arbitrary cuts in δ and ε. Such results were given in [3] for radiativelepton pair production γ∗ → ℓ+ℓ−γ. The necessary integrations are straight forward but quitenon-trivial, if one attempts to do them analytically. In the following we just give the resultsadapted for hadronic jets.

The result for the fraction f of 2–jet final states is given by

1− f =4

3

αsπQ(ε, δ) +O(α2

s) +O(αsm2q

s, (10.14)

where Q consists of different pieces which we now present one by one.

1.) Configurations of hard non–collinear gluons defined by cuts

Eg/E > ε , θqg, θqg > δ

Here

QI(ε, δ) =C>ω,>δ0α/π

where C>ω,>δ0 is given in Eq. (11) of Ref. [3] with λ = 2ε. Thus

Q(mq=0)I (ε, δ) =

1

2

(4 ln 2ε+ (1− 2ε)(3 − 2ε)) ln

1− ρ1 + ρ

− 41 + v

v2ln(1 + v(1 − 3ε) +

4

v(1− 2ε)

(1− 2ε)(3 − 2ε) ln(1− 2ε)2

3π2 − 4Sp (2ε) − 3

2(1− 2ε)2

(10.15)

where ρ = cos δ and v = 1−ρ1+ρ . One easily checks that in the limit (1 ≫ δ ≫ 2

mq

E , ε ≪ 1) weobtain Sterman and Weinbergs result.

282

δε

I IIδ

δ

ε

III

Figure 10.10: The three integration regions I, II and III, referred to in the text.

2.) Configurations with quark and antiquark in the same jet

θqq < δ

∆QII(δ) = Sp (∆) +1

2ln2(1−∆) +

1

2

1 + 2∆

∆2ln(1−∆) +

1

2δ+

5

4(10.16)

with ∆ =1−ρqq

2 , and ρqq = cos δqq.

3.) Configurations with soft q and/or q at angle

<) (q, g) > δ , (q, g) > δ and (q, q) > δ

Here one obtains

∆QIII(ε, δ) = 2 (ln(1− 2ε) + ε (1 + ε)) ln v + ln2(1− 2ε) + 2 ε (1 + ε) ln(1− 2ε) − 8/, ε

+1− 4 δ

δ2ln(1− 2ε∆) +

2ε

∆, (10.17)

where δ = δqq = δqg = δqg, v = 1−ρ1+ρ , ∆ = 1−ρ

2 and ρ = cos δ. In any case δ < π2 and ε ≤ 1

4 isassumed. Altogether, we find

Q(ε, δ) = QI(ε, δ) −QII(δ) +QIII(ε, δ) (10.18)

Results presented in Tab. 1 illustrate the strong dependence of the rates on the cuts. Plots may befound in Ref. [3]. Especially for narrow cuts one probes the soft (ε→ 0) and the collinear (δ → 0)singularities. In reality the latter are smoothed by the quark masses, however at high energiesthe collinear logarithms get huge and may cause the breakdown of the perturbative expansion.The soft logarithms exponentiate, upon which the soft limit gets regularized (screening).

10.3 Summary and Discussion

We have presented results at O(αs) with all terms O(αsm2q/s) neglected, which is a good approx-

imation sufficiently far above thresholds. The results may be summarized as follows:

1.) The differential quark pair production cross section

283

Table 1: Values for the function Q(ε, δ) for various values of the parameters ε and δ.

δ \ ε .05 .10 .15 .20 .25

5.0 21.36 12.92 7.92 4.15 .83

10.0 16.92 10.32 6.40 3.43 .83

15.0 14.31 8.79 5.50 3.01 .82

20.0 12.44 7.69 4.86 2.71 .81

25.0 10.98 6.83 4.35 2.47 .80

30.0 9.77 6.12 3.93 2.26 .79

35.0 8.73 5.51 3.57 2.09 .77

40.0 7.81 4.97 3.25 1.93 .75

(dσ

dΩ

)(e+e− → qq, qqg) =

4

3

α2

s

∑

q:4m2q≪s

Q2q

(1 + cos2 θ

)×

1 +αsπ

in non-resonant regions agrees with the experiments in magnitude as well as in angular distribution(the number of 2–jet events at given angle θ of the jet axes as shown in Fig. 10.7). The virtual γ∗

exchange process can be measured directly below about√s ∼ 30 GeV before γ−Z mixing comes

into play. Hadron production at the Z–resonance as very precisely studied at LEP/SLC is a topicto be discussed in a separate chapter. The QCD corrections, known to O(α4

s) clearly improve theagreement between theory and experiment. Mass effect are known at O(α2

s) and matter as oneapproaches thresholds from above (see Fig. 8.7). The strong interaction constant αs has beendetermined in an up-to-date reanalysis [14, 15] from the total cross section at PETRA to 20%αs(42.4 GeV) = 0.144±0.029 and from jet event shapes at the 6% level αs(MZ) = 0.1286±0.0072(total error).

2.) Neglecting n–jet (n ≥ 4) events, i.e. O(α2s) corrections,

and denoting by f(√s, ε, δ) the fraction of 2–jet event at given energy cut ε and given angular

cut δ, the fraction of 3–jet events is of leading order O(αs), given by

4

3

αsπQ(√s; ε, δ) = 1− f(

√s, ε, δ) .

Because of asymptotic freedom, for s → ∞ we have f(√s, ε, δ) → 1, i.e., 2-jet events dominate

more and more the higher the energy. The 3–jet events directly probe the existence of the gluonand of the gauge coupling qqg, i.e., quarks only interact with each other via gluon exchange.

The decrease of the number of 3–jet events with increasing energy for fixed cut parameters ε andδ means that events more and more populate the boundaries of phase space. This means thatjets narrow with increasing energy.

To illustrate jet narrowing we consider the Sterman-Weinberg approximation ε≪ 1 and 2mq

E ≪δ ≪ 1 and αs in leading approximation αs = 4π

β0 ln s/Λ2 :

1− f =4

3

αsπQ(ε, δ)

Q(ε, δ) = (4 ln 2ε + 3) lnδ

2+π2

3− 7

4.

284

3 Jets

Figure 10.11: 2 jet events (red hatched) at high energies concentrate close to the triangular phasespace boundary.

We then ask the question, how do we have to change δ as a function of s such that the fraction of3–jet events remains the same. To this end we just have to solve the equation f(

√s; ε, δ) =contant

for δ, with the result

δ = 2 exp

(74 − π2

3

4 ln 2ε+ 3

)· exp

(3β016

1− f4 ln 2ε+ 3

lns

Λ2

)

︸︷︷︸(√

sΛ

)

3β08

1−f4 ln 2ε+3

and hence

δ ∝(√

s

Λ

)−dwith d > 0 .

Since ε≪ 1 we have 4 ln 2ε+3 < 0, such that QCD jets narrow powerlike with increasing energy.An example: Nf = 3, ε = 0.1, 1 − f = 0.1 yields d = 0.098.

From the observed ratio σ(3 jets)/σ(2 jets), which at leading order is proportional to αs one candetermine the strong coupling constant. A serious problem is the fact that the cuts ε and δ, whichhave been imposed for the partons!, do not simply translate into experimental cuts on hadronicevents, since the hadronization, also called fragmentation, is modifying the perturbative cuts ina non–perturbative manner. In particular, the angular cut δ is broadening by hadronizatio. Thisproblem is dealt with using phenomenologically tuned fragmentation models. As a typical resultat PETRA energies is the one from the JADE Collaboration mentoned under item 1.) above.As mentioned earlier, LEP experiments at the Z peak allowed one to perform QCD studies withhuge event rates with millions of events. There, including the appropriate higher order effects, onecould neatly separate also 4 jet events, and the jet analyses were able to experimentally determinethe SU(3)c gauge structure be measuring the characteristic Casimir coefficients as shown oncemore in Fig. 10.12.

285

0

0.5

1

1 2 3

masslessgluinos

SU(3) QCD

ALEPH

OPAL

DELPHI 3Jts.

β-function

Sp(2N)

SU(N)

SO(N)

SU(2),Sp(2)

SU(4)

SU(4)´

G

Sp(4)

Sp(6)

SO(3) ,E

SO(4),Sp(6)´

SO(5),F ,Sp(4)´

E

E

8

2

7

6

CA/CF

TF/

CF

Figure 10.12: Determination of the gauge structure from jet analyses.

286


① Show that sphericity is infrared safe but not collinear safe.

② Show that spherocity is both infrared and ultraviolet save.

③ Exercise:a) Show that the hard gluon spectrum, after performing the angular intergrations, may bewritten in the form

dσ

dk= σ0(s)C2(R)

αsπ

(ln

s′

m2q

− 1

)(1 +

(s′

s

)2)

1

k

where k = |~k | is the gluon energy and s′ = s(

1− kEb

)and Eb =

√s/2. Note that s′ is

the invariat mass square of the qq–system after radiation of the gluon, which carried awaya fraction k/Eb of the energy.

b) In contrast in the soft gluon limit (factorization) may be represented as

σ(s) = σ0(s) + C2(R)αsπ

∫

k≤ωds′ P (s′)σ0(s)

with P (s′) radiator function

P (s′) =

(p1kp1− p2kp2

)2

integrated over cos θ. The azimutal integration over ϕ yield a factor 2π. Note ds′ =−2√s dk. Work out P (s′).

④

287

References

[1] G. Hanson et al., Phys. Rev. Lett. 35 (1975) 1609

[2] G. Sterman, S. Weinberg, Phys. Rev. Lett. 39 (1977) 1436.

[3] J. Fleischer and F. Jegerlehner, Z. Phys. C 26 (1985) 629.

[4] J. R. Ellis, M. K. Gaillard and G. G. Ross, Nucl. Phys. B 111 (1976) 253 [Erratum-ibid. B130 (1977) 516].

[5] G. Kramer, G. Schierholz and J. Willrodt, Phys. Lett. B 79 (1978) 249 [Erratum-ibid. B80 (1979) 433].

[6] R. Brandelik et al. [TASSO Collaboration], Phys. Lett. B 86 (1979) 243; Phys. Lett. B 97(1980) 453.

[7] JADE Collaboration 1980 Phys. Lett.B91 142.JADE Collaboration 1981 Phys. Lett.B101 129.MARK J Collaboration 1979 Phys. Rev. Lett.43 830.MARK J Collaboration 1983 Phys. Rev. Lett.50 2051.PLUTO Collaboration 1979 Phys. Lett.B86 418.PLUTO Collaboration 1985 Z. Phys.C28 365.

[8] ALEPH Coll. Heister A et al 2003 Eur. Phys. J. C 27 1 17

[9] OPAL Coll. Abbiendi G et al 2001 Eur. Phys. J. C 20 601 615

[10] OPAL Coll. Abbiendi G et al 2002 Eur. Phys. J. C 23 597 613

[11] DELPHI Coll. Abreu P et al 1999 Phys. Lett. B 449 383

[12] DELPHI Coll. Abreu P et al 2000 Eur. Phys. J. C 13 573

[13] Kluth S et al 2001 Eur. Phys. J. C 21 199 210

[14] C. Pahl, S. Kluth, S. Bethke, P. A. Movilla Fernandez and J. Schieck [JADE Collaboration],arXiv:hep-ex/0408123.

[15] S. Kluth, Rept. Prog. Phys. 69 (2006) 1771 [arXiv:hep-ex/0603011].

288

A Solved Problems

A.1 Exercises: Section 1

① Show that 8⊗ 8 = 1⊕ 8⊕ 8⊕ 10⊕ 10∗ ⊕ 27.

8⊗ 8 = × = × a

b

a

= a

a b

⊕ a

b

a

⊕ a

a

b

⊕ a a

b

⊕ a

a b ⊕ a a

b

In order to append to the first octet the second one in all admissible ways which respect the(anti-) symmetrization, we replace the second one by letters with identical letters in the rows(symmetrized). The different elements of a column if they appear in the same column of theenlarged tableau must appear in the old order (anti-symmetrized). Hence append the elementsfrom the first row in all possible ways to the first tableau, then to the such enlarged ones theelements of the second row etc. For our case one easily reads off the result.

Note that the tableau

a

a

is not allowed. Also

a a b

is not allowed, since baa is not an admissible sequence of letters.

② Isospin symmetry was introduced by Heisenberg in 1932 right after the discovery of theneutron (Chadwick 1932). It is obvious that isospin symmetry is violated by electromag-netism: the iso-doublet members (p,n) have different charge (1, 0). The nucleon system isdescribed by a iso-spinor Dirac field

Ψ =

ψp

ψn

.

Write down the charges Q, B and Ti (i = 1, 2, 3) in terms of the nucleon field operator andshow that B commutes with isospin, but Q does not. Write down the T = 1 and T = 0 twonucleon states.

The generators of the different symmetry operations are given by the operators

Q =

∫d3x ψpγ

0ψp =

∫d3x Ψγ0

1 + τ32

Ψ

289

B =

∫d3x

(ψpγ

0ψp + ψnγ0ψn

)=

∫d3x Ψγ0Ψ

T3 =1

2

∫d3x

(ψpγ

0ψp − ψnγ0ψn)

= Ψγ0τ32

Ψ

Ti =

∫d3x Ψγ0

τi2

Ψ

The operator version of the Lie algebra follows directly from the SU(2) generator matrices τi/2(τi the Pauli matrices):

[Ti, Tj ] = i εijkTk .

We furthermore have Q = T3 + B2 and since B corresponds the the unity matrix in isospin space

such that [B,Ti] = 0 we formally obtain

[Q,Ti] = [T3, Ti] 6= 0 ; i = 1, 2

the manifestation of isospin breaking by electromagnetism. The two-nucleon states of fixed isospinfollow the usual SU(2) spin rules:

T = 1 : ψ(1)p ψ(2)

p ,1√2

(ψ(1)p ψ(2)

n + ψ(1)n ψ(2)

p

), ψ(1)

n ψ(2)n ,

T = 0 :1√2

(ψ(1)p ψ(2)

n − ψ(1)n ψ(2)

p

).

③ Discuss the iso-spin properties of the triplet of pions (π+, π0, π−) .

The iso-spin symmetry of the scattering operator S not only leads to relations betweenmatrix elements but also to selection rules: Suppose

(a) T is a generator of a symmetry transformation such that [T, S] = 0 ,

(b) | α > and | β > are eigenstates of T i.e. T | α >= tα | α >, T | β >= tβ | β >

What does this imply for the S-matrix elements

Sβα =< β | S | α > ?

Find a few examples.

The pions form an isospin triplet, for SU(2) the representation is equivalent to the adjoint rep-resentation (which has dimension 3) : (ta)ij = −i εija thus

t1 =

0 0 0

0 0 −i

0 i 0

; t2 =

0 0 i

0 0 0

−i 0 0

; t3 =

0 −i 0

i 0 0

0 0 0

,

By a change of basis we may arrange t3 to be diagonal

t′3 =

1 0 0

0 0 0

0 0 −1

; t′1 =

1√2

0 1 0

1 0 1

0 1 0

; t′2 =

1√2

0 −i 0

i 0 −i

0 i 0

,

290

and we may use the Lie algebra relations in the form T± = T1 ± iT2, [T3, T±] = ±T± and[T+, T−] = 2T3 to find t′1 and t′2 for given t′3. The states describing π+, π0, π− are

1

0

0

;

0

1

0

;

0

0

1

,

with T3 = +1, 0,−1.

Two-pion states decompose into T = 0, 1 and 2 irreducible parts:

3⊗ 3 = × = ⊕ ⊕= ⊕ ⊕ = 1⊕ 3⊕ 5 .

The tensor product space is 9 dimensional (π+π+, π+π0, π+π−, π0π+, π0π0, π0π−, π−π+, π−π0, π−π−)and decomposes into a singlet, a triplet and a quintet. In strong interactions a two pion state hasdefinite total isospin (analogous to angular momentum) ~T = ~t (1) + ~t (2) with

~T 2 = T (T + 1) = (~t (1) + ~t (2))2 = (~t (1))2 + 2~t (1) · ~t (2) + (~t (2))2

such that with (~t (i))2 = t (t + 1) = 2 we find

~t (1) · ~t (2) =1

2T (T + 1)− 2 ,

which may be used to construct a basis of two–pion states of isospin T = 2, 1 and 0. Obviously

~t (1) · ~t (2) =

−2 , T=0

−1 , T=1

+1 , T=2

such that the combinations

2 + ~t (1) · ~t (2) = 0, 1, 3 for T = 0, 1, 2

1 + ~t (1) · ~t (2) = −1, 0, 2 for T = 0, 1, 2

1− ~t (1) · ~t (2) = 3, 2, 0 for T = 0, 1, 2

project to zero the T = 0, 1, 2 part, respectively. Normalized isospin projection operators thenread

Π2 =1

6(2 + ~t (1) · ~t (2)) (1 + ~t (1) · ~t (2)) ,

Π1 =1

2(2 + ~t (1) · ~t (2)) (1− ~t (1) · ~t (2)) ,

Π0 =1

3(~t (1) · ~t (2) − 1) (1 + ~t (1) · ~t (2)) .

A particular property of the pion triplet is G-parity conservation. G–parity is defined by

G = C eiπT3

Let φ denote the triplet of pion fields. Under charge conjugation

CφC−1 = φ∗

291

We may write φ0 = φ3 and φ∓ = 1√2

(φ1 ± iφ2), in terms of 3 real Cartesian fields. Then

C

φ1

φ2

φ3

C−1 =

φ1

−φ2φ3

and an isospin rotation by angle π about the 3-axis implies

eiπT3

φ1

φ2

φ3

e−iπT3 =

−φ1φ2

−φ3

,

which tells us that

GφiG−1 = −φi ; i = 1, 2, 3 .

Among other things this tells us that in the isospin symmetry limit a decay like

η 6→ π+π−π0 ???

is strictly forbidden. Note that η0 → 2γ requires T = 0 for the η, furthermore Cη0C−1 = η0 andhence Gη0G−1 = η0. In fact the decay η → π+π−π0 is a ∆I = 1 transition (the pions are in aT=1 state) mediated in leading order (up the electromagnetic isospin violating corrections) bythe ∆I = 1 part of the QCD Hamiltonian

HQCD(x) =1

2(mu −md) (uu− dd)(x) ,

due to isospin breaking by the md −mu mass difference. Thus the decay is chirally suppressedand would be forbidden in the chiral limit up to even smaller electromagnetic isospin breakingeffects:

HQED(x) = −1

2e2∫

dyDµν(x− y)Tjµem(x) jνem(y)

where Dµν(x − y) is the photon propagator and jµem = jµ3 + 12jµY is the electromagnetic current,

consisting of an isovector and an isoscalar part. Electromagnetic effects are suppressed by thefact that only the interference part has the right G-parity. The η–decay branching fractions are

η → 2γ 3π0 π+π−π0 π+π−γ other

Br 39.31% 32.56 % 22.73 % 4.6 % 2.8 %

int QED QCD QCD QED

6I,G 6I, 6G 6I, 6G 6I,G

With T |α〉 = tα|α〉 and Sβα = 〈β|S|α〉 we find

〈β|[T, S]|α〉 = 〈β|TS|α〉 − 〈β|ST |α〉 = (tβ − tα) 〈β|S|α〉 = 0 .

Thereforeeither tβ = tα or 〈β|S|α〉 = 0 .

292

Such a circumstance is called a selection rule. If T is an absolutely conserved quantity likethe charge operator Q or Baryon number B we call it a superselection rule: e.g. each Hilbertspace of states of a fixed charge is totally orthogonal to any other Hilbert space of states of adifferent charge.

In the limit of vanishing neutrino masses in the SM lepton number Lℓ is separately conservedfor ℓ = e, µ and τ :

particle e− e+ µ− µ+ τ− τ+ γ other

Le 1 −1 0 0 0 0 0 0

Lµ 0 0 1 −1 0 0 0 0

Lτ 0 0 0 0 1 −1 0 0

As an example, we may consider the process µ− → e−γ. In fact [Le, S] = 0 implies that〈µ−|S|e−γ〉 = 0: 〈µ−|[Le, S]|e−γ〉 = 〈µ−|LeS|e−γ〉 − 〈µ−|SLe|e−γ〉 = (0 − 1) 〈µ−|S|e−γ〉 = 0from which the assertion follows. While leptonic flavor transitions are practically forbidden, thecorresponding transitions in the quark sector are allowed by quark flavor mixing, as the ∆S = 1transition Λ0 → nγ, Σ+ → pγ, etc. with branching fractions of order ∼ 10−3 (see below).

Back to isospin: in pionic systems (no hypercharge or baryon number) we always have Q = T3and isospin selection rules coincide with the one of charge conservation. As we discussed above,for the η G-parity seems to provide a good selection rule. Indeed, the η does not decay by stronginteractions as it violates isospin. That’s why it is so stable, with a width Γη = 1.30 ± 0.07 keVonly. Further interesting cases are ρ0 → π+π− [∼ 100% ] versus ρ0 → π+π−π0 [Br=(1.01+0.64

−0.50) ·10−4] in comparison to and ω → π+π−π0 [89.2 %] versus ω → π+π− [(1.53+0.11

−0.13) %]. These arestrong interaction processes taking place in absence of electromagnetic and weak interactions.If isospin is conserved ρ0 → π+π−π0 as well as ω → π+π− are forbidden by G-parity. Thereoccurrence is due to isospin breaking by the quark mass difference mu−md, which leads to ρ−ω–mixing (briefly mentioned in Sec. 1.4). Thus the physical ρ and ω are not pure SU(2) states andtherefore in fact can decay via the wrong G-parity modes.

What about 〈γ∗|π+π−〉 and 〈γ∗|π0π0〉 ?Another example is proton decay p → e+γ, e+π0, π+π0, · · · , which is forbidden by B conserva-tion. We know that the matter–antimatter asymmetry in the universe requires B to by violatedat scales above about MX ∼ 1016 GeV.

④ Use the Young tableaus to construct the meson states in

3⊗ 3∗


3⊗ 3⊗ 3 .

The states in the pseudo-scalar meson octet of flavor SU(3) are characterized by the 3rd

component of iso-spin and by hypercharge Y = B+S (B baryon number B = 0 for mesons,S strangeness S = 0 for pions). Display the weight diagram (I3 − Y plot) of the mesonstates. How are they composed of u, d and s quarks in the SU(3)flavor quark model ?

293

⑤ The structure constants cikl of a Lie-algebra [Ti, Tk] = i ciklTl satisfy the Jacobi identity.

cikncnlm+ terms cyclic in (ikl) = 0

Use this to show that (Ti)kl = −i cikl also satisfies the Lie-algebra (adjoint representation).

Cyclic sum is: “(ikl)+(lik)+(kli)”= 0 and thus Jacobi reads: cikncnlm+clincnkm+cklncnim = 0 .We take the 1st term as the r.h.s of the Lie algebra. In the 2nd and 3rd term: since n is summedover it should be the index of matrix multiplication of two successive T ’s, i.e no Tn should appear,which may be achieved by antisymmetric permutation of the index n away from its first positionin the cnkm etc. and m as a dummy index is good to be kept in the last position. Also only Tiand Tk should appear i.e. I is the other dummy to the left: therefore we may write cikncnlm =i cikn (Tn)lm, clincnkm = cilncknm = − (Ti)ln (Tk)nm, cklncnim = −cklncinm = (Tk)ln (Ti)nm. Thus

i cikn (Tn)lm − (Ti)ln (Tk)nm + (Tk)ln (Ti)nm = 0 ,

which is the algebra

([Ti, Tk])lm = i cikn (Tn)lm .

q.e.d.

⑥ Discuss the electromagnetic decays of π0 and η0 into photons. Use the fact that the Cparity of the photon is ηγC = −1. What are the observed decays? What do they imply forthe C parities of the neutral pseudoscalar mesons π0 and η0 ?. Which decays are strictlyforbidden? Which Feynman diagram is responsible for these decays?

C parities of π0 and η0:

Since the photon has C–parity ηγC = −1 as mentioned earlier, the observed C–conserving electro-magnetic decays

π0 → γγ , η0 → γγ

tell us that

ηπ0

C = ηη0

C = +1 .

The conservation of C–parity on the other hand implies that

π0 6→ γγγ , η0 6→ π0γ or γγγ

are forbidden decays.

294

As the π0 is a qq bound state the decay is possible via the coupling of the photon to the chargedquarks: the leading diagram is the 1–loop diagram

: Aπ0γγq−loop .

γ

γu, dπ0 = (uu− dd)/

√2

Via PCAC this transition amplitude is related to the axial vector current anomaly, which maybe written as an effective Lagrangian (Wess-Zumino Lagrangian see below)

⑦ Which experiments (processes) allow us to determine the intrinsic parity (ηP ) of the pions?

⑧ Lepton number Le is another additive quantum number which is strictly conserved. Le(e−) =

1 by convention. Determine Le for the other particles from the observed reactions:

1. Le(e+) = −1, Le(γ) = 0 :

p+ e → p+ e+ γ

γ∗ → e+ + e−

2. Le(π0) = Le(π

±) = 0 :

π0 → 2γ, γ + e+ + e−

p+ π− → n+ π0

p+ π0 → n+ π+

3. Le(νe) = −1, Le(νe) = 1 :

π− → e− + νe

π+ → e+ + νe

From the last two reactions we learn the important result νe 6= νe !

⑨ Baryon number conservation is responsible for the stability of the proton. By conventionB(p) = 1, B(e−) = 0 . Determine the baryon numbers of particles from the observation of

295

the following reactions:

a.) Baryons and mesons:

1. B(π0) = 0 :

p+ p → p+ p+ π0

2. B(n) = B(p), B(π±) = B(π0) = 0 :

p+ p → p+ n+ π+

π− + p → n+ π0

3. B(K±) = B(K0) = 0 :

K± → π± + π0

K0 → π+ + π−, π+ + π− + π0

4. B(Λ), B(Σ) = 1 :

π− + p → Λ0 +K0, Σ− +K+

π+ + p → Σ+ +K+, Σ0 + Λ0

5. B(Ξ), B(Ω−) = 1 :

K− + p → Ξ− +K+, Ξ0 +K0, Ω− +K+ +K0

b.) Antibaryons:

6. B(p) = −1 :

p+ p → p+ p+ p+ p

7. B(B) = −1 :

p+ p → n+ n, Λ0 + Λ0, Σ0 + Σ0, Σ± + Σ∓, Ξ+ + Ξ−

c.) Photon:

8. B(γ) = 0 :

p → p+ γ

d.) Leptons: All leptons are produced in pairs, B(e−) = 0 by convention.

9. B(e) = B(µ) = 0 :

γ∗ → e+ + e−, µ+ + µ−

10. B(νe) = B(νµ) = 0 :

n → p+ e− + νe

µ− → e− + νe + νµ

µ+ → e+ + νe + ν−µ

π− → µ− + νµ

π+ → µ+ + νµ

296


① Discuss the symmetry breaking of SU(2)flavor (Isospin) and SU(3)flavor (Isospin andStrangeness) in the spin 1/2 baryon octet. Comment the decays Σ0 → Λγ and Σ+ → pγand compare them to the strong decays Σ− → Λπ−(?) and Σ− → nπ−. Use the quarkmodel schema for the discussion.

Note the following parallelism between meson and baryon states

Y T Baryons Mesons

+1 12 p, n K+,K0

0 1 Σ+,Σ0,Σ− π+, π0, π−

-1 12 Ξ0,Ξ− K0,K−

0 0 Λ0 η0

Note the hierarchy mu < md << ms and its influence on the hadron masses. They are theprimary source of symmetry breaking and in first place are reflected in the masses.

Spin 1/2 Baryon masses:

Y T Baryon masses in MeV

+1 12 mp = 937.27203 ± 0.00008 mn = 939.56536 ± 0.00008

0 1 mΣ+ = 1189.37 ± 0.07 mΣ0 = 1192.642 ± 0.024 mΣ− = 1197.449 ± 0.030

-1 12 mΞ0 = 1314.86 ± 0.20 mΞ− = 1321.71 ± 0.07

0 0 mΛ0 = 1115.683 ± 0.006

A key role for the decay patterns play the mass differences, which determine the available decayphase space.

Mass shifts in MeV from replacing quark flavors:

u→ d d→ s

mn −mp = 1.2933317 ± 0.0000005 mΣ+ −mp = 252.1 ± 0.07

mΣ− −mΣ0 = 4.807 ± 0.035 mΣ0 −mn = 253.1 ± 0.02

mΣ0 −mΣ+ = 3.27± 0.07 mΞ0 −mΣ0 = 122.2 ± 0.20

mΞ− −mΞ0 = 6.85± 0.21 mΞ− −mΣ− = 124.3 ± 0.08

mΣ− −mΛ0 = 81.77 ± 0.03 mΛ0 −mn = 176.1 ± 0.01

mΛ0 −mΣ+ = −73.69 ± 0.07 mΞ0 −mΛ0 = 199.2 ± 0.20

297

x

x

x

x

x

x

x

x

x

x

x

x

d→ u s→ ds→ u

t t tc c cu u u

b b bs s sd d d

Figure A.13: The relevant CKM transitions for weak Baryon Decays. The horizontal lines cor-respond to Flavor Changing Neutral Currents (FCNC) which are strictly forbidden (x) in theSM.

We note a surprisingly large mass shiftmΣ0 −mΛ0 = 76.959 ± 0.023 MeV

between the singlet T=0 and the octet T=1 states with the same conserved quantum numbersT3 = 0, Y = 0.Mass differences (phase space) and the different types flavor transitions Fig. A.13 essentiallydetermine the lifetimes. Of course Baryon number must be conserved. For phase space reasonsat most one additional hadrons, a pion, can be produced in ∆S = 1 transitions s → u, d. SinceFCNC (flavor changing neutral currents) in the SM are forbidden at tree level, most (all exceptone) decays are second order weak CC (charged current) transitions

LCC =g√2

(J+µ W

µ− + J−µWµ+)

J−µ = (u, c, t) γµ (1− γ5) UCKM

d

s

b

with J+µ = (J−µ )+ and unitary 3× 3 matrix

UCKM =

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

.

Fig. A.13 shows the possible transitions for the u, d s subspace. The relevant decay typesare depicted in Fig. A.15. ∆S = 0 transitions d → u only are possible semileptonically viad→ ue− νe [ diagram type a)]. Flavor changing radiative decays B → B′γ [diagram type b)] aresuppressed by an additional factor α. All decays involving charged particle (one the quark level all)may include one or more additional photons, usually called electromagnetic radiative corrections(em RC). Processes where physical charged pions are produced via virtual W ’s [diagram typef)], W± → π± have direct semileptonic counterparts W− → e−νe, µ−νµ or W+ → e+νe, µ

+νµ,as mµ < mπ also decays into muons show up, unlike in normal d → u β–decays, where decaysinto muons are absent because mµ >> md −mu. The neutron is quasi stable due to the smallphase space mn − mp available. All quarks in a hadron are strongly interacting via gluons.Strong interactions like the electromagnetic ones are flavor preserving. In a decay a quark whichis inactive with respect to weak and electromagnetic transitions is called a spectator quark.Decays which just differ by different spectator quarks have similar properties, except from massand phase space effects.

Because of the specific structure of the SM quark flavor mixing, a number of transitions areforbidden at order O(G2

F ), while possible in general at order O(G4F ). Since the latter are dramat-

ically suppressed they usually are called forbidden transitions. In semileptonic hyperon decays

298

proceeding via Fig. A.15 a) type diagrams we obviously must have ∆S = ∆Q [where ∆S refers tothe hadronic current and ∆Q to the leptonic one]. Since FCNC are forbidden a s→ q transitiononly is possible by a W− exchange. Accordingly Σ+ → ne+νe is forbidden [while Σ− → ne−νe isallowed], another forbidden decay is Ξ0 → Σ−e+νe, here also Ξ0 → Σ−π+ is absent by kinematics.Note that Σ+ → nπ+ is allowed as well as a subsequent decay π+ → e+νe, however, overall itis a strongly suppressed O(G4

F ) process. Also ∆S = 2 transitions are forbidden Ξ0 → pπ− andΞ− → nπ−.

Baryon lifetimes:

Y T Baryon lifetimes in seconds

+1 12 τp > 2.1 · 1029 years τn = 885.7 ± 0.8

0 1 τΣ+ = (0.802 ± 0.003) · 10−10 τΣ0 = (7.4 ± 0.7) · 10−20 τΣ− = (1.479 ± 0.011) · 10−10

-1 12 τΞ0 = (2.90 ± 0.09) · 10−10 τΞ− = (1.639 ± 0.015) · 10−10

0 0 τ0Λ = (2.631 ± 0.020) · 10−10

Note: the neutral spin 1 meson ρ0 has a width of about 125 MeV which corresponds to a lifetimeτρ = 5.266 · 10−25 sec, which is a typical strong decay lifetime. In fact the decay ρ0 → π+π− is aflavor conserving strong decay and no weak or electromagnetic transition is needed to mediate it(see Fig. A.14). Main decay modes (see also Figs. A.13, A.15 and A.16) [q] denotes the inactivespectator quark:

299

Decay Branching fraction transition diagram Fig. A.15

p → stable – – –

n → pe−νe 100 % βdu a)

n → pe−νeγ Br= (3.13 ± 0.35) · 10−3 em RC a) +γ’s

Λ0 → nπ0 35.8 % s→ d[d] c), d)

Λ0 → pπ− 63.9 % s→ u[u] e), f)

Λ0 → nγ Br= (1.75 ± 0.15) · 10−3 s→ d rad b)

Λ0 → pe−νe Br= (8.32 ± 0.14) · 10−4 βsu a)

Σ+ → pπ0 51.57 % s→ d[u] c), d)

Σ+ → nπ+ 48.31 % s→ d[u] c), d)

Σ+ → pγ Br= (1.23 ± 0.05) · 10−3 s→ d rad b)

Σ+ → nπ+γ Br= (4.5± 0.5) · 10−4 em RC c), d) +γ’s

Σ+ → Λ0e+νe Br= (2.0± 0.5) · 10−5 β∗du

Σ0 → Λγ 100 % elmag [M1] “q → qγ”

Σ− → nπ− 99.848 ± 0.005 % s→ u[d] e), f)

Σ− → ne−νe Br= (1.02 ± 0.03) · 10−3 βsu a)

Σ− → nπ−γ Br= (4.6± 0.6) · 10−4 em RC e), f) +γ’s

Σ− → Λ0e−νe Br= (5.73 ± 0.27) · 10−5 βdu a)

Ξ0 → Λ0π0 99.925 ± 0.012 % s→ u[s] e)

Ξ0 → Λ0γ Br= (1.17 ± 0.07) · 10−3 s→ d rad b)

Ξ0 → Σ0γ Br= (3.33 ± 0.10) · 10−3 s→ d rad b)

Ξ0 → Σ+e−νe Br= (2.53 ± 0.08) · 10−4 βsu a)

Ξ− → Λ0π− 99.887 ± 0.035 % s→ u[s] e), f)

Ξ− → Σ−γ Br= (1.27 ± 0.23) · 10−4 s→ d rad b)

Ξ− → Λe−νe Br= (5.36 ± 0.31) · 10−4 βsu a)

Ξ− → Ξ0e−νe Br< 2.3 · 10−3 βdu a)

u

u

ρ0 π−

d

d

u

uπ+

g

d

d

ρ0 π−

u

u

d

dπ+

g

sdu

Σ0 Λ0

sdu

γ

Figure A.14: Examples on non-weak flavor preserving hadron decays. Top: strong decay of theρ–meson. Bottom: electromagnetic M1 transition.

300

νe

e−

udd

n pudu

W

a)

uds

udd

u, c, t

Λ n

γ

W

b)

ds

u

Λ n

π0

dud

dd

W

c)

du

s

ddu

uu

Λ n

π0

W

d)

ds

u

Λ p

π−

duu

ud

W

e)

uds

udu

Λ p

du

π−

W

f)

Figure A.15: Various Baryon Decay Schemes a) β–decay [d, s → ue−νe], b) Penguin radiativedecay [s→ dγ], c), d) and e) B → B′ + π [s→ d], and f) B → B′ + π [s→ u].

Note on Σ0 → Λ0γ: this is an electromagnetic transition of M1 type (as the hadrons have no netcharge) (see Fig. A.14). The amplitude of this transition is defined by

M(Σ0 → Λ0γ) =√

4πα e∗µ(k, λ)〈Λ0(kΣ0 , σΣ0)|Jµ(0)|Σ0(kΛ0 , σΛ0)〉, (A.1)

where Jµ(0) is the electromagnetic hadronic current, e∗µ(k, λ) is a polarization vector of a photonand α = 1/137.036 is the fine–structure constant. Due to the M1 transition the matrix elementof the electromagnetic hadronic current is

〈Λ0(kΣ0 , σΣ0)|Jµ(0)|Σ0(kΛ0 , σΛ0)〉 = − gΣ0Λ0 uΛ0(kΛ0 , σΛ0)σµνkνuΣ0(kΣ0 , σΣ0), (A.2)

where σµν = (γµγν − γνγµ)/2, gΣ0Λ0 is the effective coupling constant of the transition Σ0 → Λ0,caused by strong low–energy interactions, and k = kΣ0−kΛ0 , and uΛ0(kΛ0 , σΛ0) and uΣ0(kΣ0 , σΣ0)are the bispinors of the Λ0 and Σ0 hyperons, normalized by uY (kY , σY )uY (kY , σY ) = 2mY forY = Σ0 and Λ0.

The width of the Σ0 → Λ0γ decay for unpolarized baryons is proportional to

1

2

∑

σΛ0=±1/2

∑

σΣ0=±1/2

∑

λ=±1|M(Σ0 → Λ0γ)|2 =

= 2παg2Σ0Λ0

∑

λ=±1e∗µ(k, λ)eα(k, λ)kνkβ tr(mΣ0 + kΣ0)σµν(mΛ0 + kΛ0)σαβ =

= 2παg2Σ0Λ0

∑

λ=±1e∗µ(k, λ)eα(k, λ)kνkβ trkΣ0σµν kΛ0σαβ = 32πα g2Σ0Λ0(k · kΣ0)(k · kΛ0) =

= 32πα g2Σ0Λ0 m2Σ0ω

20 , (A.3)

301

x

x

xx x

d→ ue−νe s→ d γ

s→ d s→ u

∆S 6= ∆Q

∆S = 2

Figure A.16: Types of Transitions in the Baryon Octet. Directions are driven by the quarkmass hierarchy ms > md > mu. Only s–decay transitions have enough phase space allowingto produce real pions [blue arrows] (Fig. A.15 diagrams c) to f)). In d → u generally onlysemileptonic decays may occur [green arrows] (diagram a)). Flavor changing electromagnetictransition must be second order weak [red arrows] (diagram b)). The flavor neutral Σ→ Λγ in apure electromagnetic M1 (uncharged particles must couple via magnetic moment) decay.

su

s

Ξ0 Σ−

π+∗

sdd

d

u

νe

e+

WW

a)

ss

u

Ξ0 p

π−

W

u

udu

ud

WW

b)

Figure A.17: “Forbidden” [=not allowed at O(G2F )] Baryon Decays (see text): a) ∆S = ∆Q

violating, b) ∆S = 2 decay. Being suppressed by an additional factor G2F ∼ 10−10 these modes

in fact are not observable by present standards.

where ω0 = (m2Σ0 −m2

Λ0)/2mΛ0 = 74.5 MeV is the photon energy in the rest frame of the Σ0 –hyperon.

The width of the Σ0 → Λ0γ decay is equal to

Γ(Σ0 → Λ0γ) =α

π

g2Σ0Λ0

mΣ0

∫δ(4)(kΛ0 + k − kΣ0) (k · kΣ0)2

d3kΛ0

EΛ0(~kΛ0)

d3k

ω(~k)=

= 4αg2Σ0Λ0 ω30. (A.4)

Using the experimental value for the width of the Σ0 → Λ0γ decay Γexp(Σ0 → Λ0γ) = (8.9 ±0.8) × 10−3 MeV we get gΣ0Λ0 = (8.6 ± 0.4) × 10−4 MeV−1. The effective coupling constantgΣ0Λ0 can be expressed in terms of the transition magnetic moment gΣ0Λ0 = µΣ0Λ0/2mN , where|µΣ0Λ0 | = 1.61 ± 0.08. Data from PDG.

② In the Goldstone model of spontaneous symmetry breaking two real fields ϕ =(ϕ1

ϕ2

)interact

by the Lagrangian L = 12 (∂µϕ)2 + µ2

2 ϕ2 − λ

4! ϕ4. For µ2 > 0 let the ground state be given

302

by ϕ0 =(0v

), v > 0. Show that the curvature of the potential ∂2V

∂ϕi∂ϕkat the ground state ϕ0

represents the mass matrix of the system.

The claim is

Mik =∂2V

∂ϕi∂ϕk

∣∣∣∣ϕ=ϕ0

.

Note that in the symmetric phase V is already diagonal in the fields, depending on ϕ2 only. Alsowe have ϕ2

0 = v2. Therefore

∂V

∂ϕi=

∂V

∂ϕ2

∂ϕ2

∂ϕi= (

λ

6ϕ2 − µ2)ϕi

which must vanish at the minimum and this means that the first factor must vanish, because oneof the fields ϕi is a non-vanishing constant v at the minimum. Thus µ2 must be equal to λv2/6.On the other hand

Mik =∂2V

∂ϕi∂ϕk

∣∣∣∣ϕ=ϕ0

= 2λ

6ϕkϕi

∣∣∣∣ϕ1=0,ϕ2=v

=

λv2

3 = m22 ; ϕi = ϕk = ϕ2

0 ; otherwise

which proves the assertion.

③ Write down the Feynman rules for the Goldstone model (see Fig. 2.2).

We denote the shifted fields by ϕ1 and ϕ2, where, after the shift, both fields have zero vacuumexpectation values as expected for normal quantized fields. The vacuum action density must havebeen subtracted i.e. 〈0|L(x)|0〉 = 0. The tadpole constant c1 is zero at tree level, but must beadjusted to guarantee 〈0|ϕ2(x)|0〉 = 0 in higher orders:

〈0|ϕ2(x)|0〉 = + = 0 .

Note that for µ2 > 0 a “quantization” of the fields for before performing a shift on the classicallevel, would lead to an ill defined perturbative expansion (negative mass square fields [tachyonfields]). The Feynman rules are depicted in Fig. A.18. Note that factorial normalization factors1n! coming from permutations of identical fields in monomials ϕni are not included in the Feynmanrule couplings, since diagrams do not differ by taking different permutations, and hence arecounted once only.

④ Calculate the Noether currents for the Goldstone model, with respect to the unbroken globalsymmetry. Discuss how it is possible that a current couples to the vacuum, i.e. why it canbee that for a conserved current we have : jµi (x) : |0〉 6= 0.

Under a variation of the field the Lagrangian varies by

δL =∂L∂∂µϕ

δ∂µϕ+∂L∂ϕ

δϕ ,

303

Propagators :

ϕ1 ( ) ϕ1i

p2+iǫ

ϕ2 ( +m22) ϕ2

ip2−m2

2+iǫ

+ interaction vertices :

−i λ v3

−i λ v

−i λ

−i λ

−i λ3

−i c1

ϕ1

ϕ2

ϕ1

ϕ2

ϕ1

ϕ2

ϕ2

ϕ2

ϕ1ϕ1

ϕ1ϕ1

ϕ2ϕ2

ϕ2ϕ2

ϕ2ϕ1

ϕ2ϕ1

ϕ2

Figure A.18: Feynman rules for the O(2) Goldstone model

and by the Euler-Lagrange equation of motion

∂L∂ϕ

= ∂µ∂L∂∂µϕ

,

and for global variations δ∂µϕ = ∂µδϕ we obtain

δL =∂L∂∂µϕ

∂µδϕ+

(∂µ

∂L∂∂µϕ

)δϕ

= ∂µ

(∂L∂∂µϕ

δϕ

)= ∂µ (∂µϕ δϕ) .

An infinitesimal ϕ rotation (with cos θ → 1, sin thetatoθ (θ ≪ 1) we may write

δϕ = θ

0 1

-1 0

ϕ = θ

ϕ2

−ϕ1

304

and thus

δL = θ ∂µ jµ

with

jµ =(∂µϕ1, ∂µϕ2

) ϕ2

−ϕ1

= (∂µϕ1)ϕ2 − (∂µϕ2)ϕ1 ≡ ϕ2

↔∂µ ϕ1

the requested Noether current. Since L is invariant under rotations we actually have aconserved current

∂µ jµ = 0 .

Spontaneous symmetry breaking (for µ2 > 0) of the O(2) field rotations requires us toadjust the fields ϕ2 = ϕ′2 + v while ϕ1 = ϕ′1, which affects the current in a very interestingway, namely it acquires a linear term:

jµ = ϕ′2↔∂µ ϕ′1 + v ∂µϕ1

The quantized current

jµ =: ϕ′2↔∂µ ϕ′1 : +v ∂µϕ1

obviously creates ϕ1 quanta, i.e. Nambu-Goldstone bosons, from the ground state

: jµ : |0〉 = i v

∫d3p pµ

2|~p| (2π)3eipx |~p〉1

i.e. : jµ : |0〉 6= 0, while in the symmetric phase (µ2 < 0) , without a linear term in thecurrent, we would have

: jµ : |0〉 = 0 .

Actually, for Q|0〉 =∫

d3x : j0 : |0〉 we obtain

Q|0〉 = i v

∫d3p |~p|

2|~p| (2π)3ei|~p|x

0∫

d3xei~p~x |~p〉1

= i v1

2|~0〉1

which is a projection to a one Nambu-Goldstone boson state of zero momentum.

A.3 Exercises: Section 3.2

① In the SM the leptonic hypercharge current has the form

jµY =1

2

(ℓLγ


)+ ℓRγ

µℓR .

Write it in terms of vector plus axialvector contributions.

305

② Calculate the trace and the Feynman integral (3.5) and verify (3.8).


① The left (L)- and right (R)-handed fields are given by

ψL =1− γ5

2ψ , ψR =

1 + γ52

ψ .

Show that a mass term is given by

ψψ = ψLψR + ψRψL .

Discuss helicity mixing for the terms

S = ψψ , P = ψγ5ψ , V µ = ψγµψ , Aµ = ψγµγ5ψ , T µν = ψσµνψ

The chiral fields ψR and ψL are obtained from a Dirac field by chiral projection ψR = Π+ψ andψL = Π−ψ (see footnote p. 42). For the adjoint field ψ

.= ψ+γ0 due to γ0, γ5 = 0 we have

ψR = ψΠ− and ψL = ψΠ+. Therefore, ψRψR = 0 and ψLψL = 0. Furthermore, the Π± areeigenoperators of γ5: γ5Π± = ±Π± and Π±γµ = γµΠ∓. Using these properties and the fact thatthe Π± are hermitian projection operators one easily works out

ψΓiψ =(ψR + ψL

)Γi (ψR + ψL)

with Γi appropriate for i = S,P, V µ, Aµ and T µν given above. The results read:

S =(ψR + ψL

)(ψR + ψL) = ψRψL + ψLψR ,

P =(ψR + ψL

)γ5 (ψR + ψL) = −ψRψL + ψLψR ,

V µ =(ψR + ψL

)γµ (ψR + ψL) = ψRγ

µψR + ψLγµψL ,

Aµ =(ψR + ψL

)γµγ5 (ψR + ψL) = ψRγ

µψR − ψLγµψL ,

T µν =(ψR + ψL

)σµν (ψR + ψL) = ψRσ

µνψL + ψLσµνψR .

Thus S,P and T are helicity flip operators, while V and A are helicity preserving; all gaugeinteractions are necessarily of the latter type.

② Consider two free fields ψ1 and ψ2 with different masses m1 and m2 . Calculate the diver-gences of the vector and axial vector currents

V µ = ψ1γµψ2 and Aµ = ψ1γ

µγ5ψ2

∂µVµ =? ∂µA

µ =?

306

What is the consequence of this result for the weak currents ? (Hint: Use the Diracequations). Comment on the hadronic charged weak current responsible for β–decay bycomparing the nucleon current with the quark current.

From the Dirac equation (iγµ∂µ −m)ψ = 0, we have γµ∂µ ψ = −imψ and taking the adjoint,we have ∂µψ

+γµ+γ0 = imψ+γ0 or, with γ0γµ+γ0 = γµ implying γµ+γ0 = γ0γµ we get ∂µψγµ =

imψ. Together with γµ, γν = 0 we easily find

∂µVµ = i (m1 −m2) ψ1ψ2 ; ∂µA

µ = i (m1 +m2) ψ1γ5ψ2

and the vector current is conserved only if ψ1 = ψ2 (flavor neutral) the axial current is conservedonly in the chiral limit of massless fermions.The charged weak (V-A) hadronic nucleon current which seems to mediate neutron β–decayn→ p+ e− + νe for the hadrons is (1st family only, CKM mixing discarded))

Jµnucleon = ψp γµ (1− γ5)ψn

(thats what it was thought to be before the discovery of the quarks). The true SM weak hadroniccurrent is

Jµhadronic = ψu γµ (1− γ5)ψd .

Note that for the vector part

∂µVµnucleon = i (mp −mn) ψpψn

and

∂µVµhadronic = i (mu −md) ψuψd

the symmetry braking is of the same order of magnitude: mn −mp ≃ 1.23 MeV vs. md −mu ≃1.95 MeV (using mu ∼ 2.35 MeV,md ∼ 4.25 MeV). In contrast for the axial current

∂µAµnucleon = i (mp +mn) ψpγ5ψn

compares with

∂µAµhadronic = i (mu +md) ψuγ5ψd

a mismatch by a factor about 1434. The plain absence axial current conservation on the nucleonlevel lead Nambu in 1960 to the discovery of spontaneous symmetry breaking and the interpre-tation of the pions as quasi Nambu-Goldstone bosons.

③ Find the form of the quark currents which couple to W and Z and the photon. As a startingpoint use the SU(2)L ⊗ U(1)Y covariant derivative form

Lq = bRiγµ(∂µ + i

1

3g′Bµ

)bR + tRiγµ

(∂µ − i

2

3g′Bµ

)tR

+Lqiγµ

(∂µ − i

1

3

g′

2Bµ − ig

τa2Wµa

)Lq

Check the correctness of the covariant derivatives with the quantum numbers assigned tothe quarks. We have denoted by Lq =

(tb

)L

the left–handed doublet. tR and bR are thecorresponding right–handed singlets. Quark mixing should be ignored.

307

Consider the charged current first. In terms of the real Cartesian fields Wi (i = 1, 2) one obtainsthe charged fields which correspond to the ladder operators of SU(2)L: W

± = 1√2

(W1 ∓ iW2)

such that

g

2

∑

a=1,2

τaWµa =g√2

0 W−

W+ 0

,

and hence

LCC =g√2

(tLγ

µbLW−µ + bLγ

µtLW+µ

).

The neutral gauge fields mix via a rotation

Z

A

=

cw −sw

sw cw

W3

B

;

W3

B

=

cw sw

−sw cw

Z

A

with cw = cos θW = g√g2+g′2

and sw = sin θW = g′√g2+g′2

. sin2 θW is called weak mixing parame-

ter. Note that√g2 + g′2 = g/cw and

√g2 + g′2swcw = gsw = e the electric charge unit. For the

neutral current we first have (note hypercharges are Y = 4/3,−2/3, 1/3, 1/3 for tR, bR, tL, bL andY = 2 (Q− T3))

LNC+em =1

2tRγ

µ[2Qt g

′Bµ]tR +

1

2bRγ

µ[2Qb g

′Bµ]bR

+1

2tLγ

µ[(2Qt − 1) g′Bµ + gWµ3

]tL +

1

2bLγ

µ[(2Qb + 1) g′Bµ − gWµ3

]bL

= Qt tγµg′Bµt+Qb bγ

µg′Bµb−1

2tLγ

µ[g′Bµ − gWµ3

]tL +

1

2bLγ

µ[g′Bµ − gWµ3

]

and with gW3 − g′B = g/cw Z and g′B = −s2w g/cw Z + swcw q/cw A we find

LNC+em = e∑

q=t,b

Qq qγµqAµ +

g

cw

∑

q=t,b

−Qq s2w qγµqZµ +∑

q=t,b

Tq qγµ (1− γ5)

2qZµ

.

Thus

Lq,neutralmatter, int = ejµemAµ +g

cos θwJµZZµ

where

jµem, had =∑

q

Qqψqγµψq

is the hadronic electromagnetic current and

JµZ = Jµ3 − sin2 θW jµem

weak neutral current.

308

④ Give numerical values for the Higgs couplings to the various particles. Calculate the decaywidth of the Higgs into a fermion pair and estimate the branching ratios for the differentflavors. Discuss the dependence of the Higgs mass for the range 0 < mH < 2MW . Comparethe Higgs width with the width of known particles.

In the unitary gauge the SM Yukawa Lagrangian takes the “trivial” from

LYukawa = −∑

f

mf ψfψf

(1 +

H

v

)

where mf is the fermion mass and v the Higgs vacuum expectation value (with√

2GF = 1/v2).The Higgs decay matrix element is given by

=M[H → f(q1, r1)f(q2, r2)] =mf

vu(q1, r1) v(q2, r2) .

The transition rate is then

∑

r1,r2

|M[H → f(q1, r1)f(q2, r2)]|2 =√

2GFm2f

∑

r1,r2

u(q1, r1)α v(q2, r2)α v(q2, r2)β u(q1, r1)β

=√

2GFm2f (q/1 +mf )αβ (q/2 −mf )βα =

√2GFm

2f 4 ((q1 · q2)−m2

f )

The decay rate is

ΓHff =1

16πmH

√

1−4m2

f

m2H

∑|T12|2

and with q1q2 = 12(m2

H − 2m2f ) we have 4 (q1q2 −m2

f ) = 2 (m2H − 4m2

f ) and hence

ΓHff =

√2GFm

2f

8π

(1−

4m2f

m2H

) 32

mH

The direct Higgs mass limit from LEP is mH>∼114 GeV. If mH < 2MW ∼ 160 GeV the Higgsdecays into all SM leptons and quarks, except for the top (as mt ∼ 173 GeV), preferably into theheaviest particle accessible, which then is the b quark.

The couplings are given by the lepton and quark masses in units of v = 246.221(1) GeV: thusthe strengths are ordered according to the masses (in GeV) me = 0.000511,mu = 0.003,md =0.006,ms = 0.105,mµ = 0.1057,mc = 1.25,mτ = 1.777,mb = 4.25 . We have

∑Ncfm

2f ≃

62.08 GeV2 [with threshold corrections 61.66 GeV2] and with GF = Gµ = 1.16637(1)·10−5 GeV−2

we obtain, neglecting the threshold factors which are very close to unity for all light fermions,

ΓHff/mH = 4.05 · 10−5

or with mH = 120[160] GeV a width ΓHff ≃ 4.9[6.5] MeV which is much smaller than e.g. the ρ

meson width Γρ ≃ 145 MeV. The branching fractions are Bre = 4.2 ·10−9,Bru = 4.4 ·10−7,Brd =1.8 · 10−6,Brs = 5.4 · 10−4,Brµ = 1.8 · 10−4,Brc = 0.0760,Brτ = 0.0511,Brb = 0.872 .

Addendum: a relatively light Higgs of mass below 2MW mainly decays into 2 b hadron jets. Atthe LHC such decays suffer from huge background which is expected to bury all hadronic Higgs

309

decay events. The most promising decay channel for the Higgs discovery at a hadron collider isH → γγ, which has a unique signature but a very low rate. The decay rate is

ΓHγγ =

√2GF α

2m3H

256π3

∣∣∣∣∣∣

∑

f

Ncfe2f AHγγf−loop +AHγγW−loop

∣∣∣∣∣∣

2

,

with 1–loop amplitudes

: AHγγf−loop = 2x [1 + (1− x) f(x)] ,

+ : AHγγW−loop = −x [3 + 2/x + 3 (2 − x) f(x)] ,

where x = 4m/m2H with m the mass of the particle in the loop, m = mf or MW , and

f(x = 1/τ) =

arcsin2√τ for τ ≤ 1

−14

[ln√τ+√τ−1√

τ−√τ−1 − iπ

]2for τ > 1 .

For mH = 120[160] GeV one obtains ΓHγγ = 1.7 · 10−4[5.1 · 10−2] MeV . The branching fractionis 2.7 · 10−5[7.8 · 10−3] only, which compares to the τ–pair branching fraction of 5.1% . Note thatHγγ branching fraction increases strongly with the Higgs mass and close to theW–pair thresholdreaches almost 1% .


① Write the PCAC relation in terms of physical parameters (masses and decay constants ofhadrons) and try to understand the Goldberger-Treiman (GT) relation [Phys. Rev. 110(1958) 1178, ibid. 111 (1958) 354]

gA (mp +mn)/2 = Fπ gπpn

where mp and mn are the proton and neutron mass, respectively, and gπpn the effective pionnucleon coupling. The other couplings show up in the matrix elements for neutron β–decay

M(n→ p+ e− + νe) =GF√

2

[up (fV γλ − gAγλγ5)un Lλ

]

and π–decay

M(π− → µ− + νµ) =GF√

2

(−iFπpλ L

λ).

By Lλ = uℓγλ (1− γ5) vνℓ we denoted the leptonic “current”, p is the pion momentum and

GF is the Fermi constant.

For the hadron states we use the notation introduced in Sect. 1.5. The GT relation attempts torelate the nucleon current matrix element 〈p|J+λ(0)|n〉 to the hadronic pion decay matrix element〈0|J+λ(0)|π〉: thus

up (fV γλ − gAγλγ5)un ⇔ − iFπ p

λ

310

with the pion decay constant Fπ. The vector and axialvector couplings fV and gA may be takenas separate parameters here. We first consider the divergence of the weak nucleon current

∂λ ψp (fV γλ − gV γλγ5)ψn = +i fV (mp −mn) ψpψn − i gA (mp +mn) ψpγ5ψn

≃ −i gA (mp +mn) ψpγ5ψn in the isospin limit mn = mp = mN ,

which is dominated by the divergence of the axial current, and which we may compare with thePCAC relation47

∂λJ+λ5 (x) = Fπm

2π ϕ

+(x)

which relates the divergence of the axial nucleon current

J+λ5 (x) =

1√2ψp γ

λγ5 ψn(x)

to the pion field ϕ+(x). The latter is a composite field with the appropriate quantum numbersas an interpolating field for the pion state. In the spirit of the LSZ reduction formalism we mayreduce the pion field by relating it to its source jπ

+

5 [this means amputation of the pion leg] via

jπ+

5 (x).=( +m2

π

)ϕ+(x) .

The matrix element ( the factor√

2 is a convention)

〈p(p′)|jπ+

5 (0)|n(p)〉 = i√

2 g(q2) up(p′) γ5 un(p)

np

π−

with q = p − p′ defines a pion nucleon form factor, where gπpn = g(m2π) is the experimentally

determined effective coupling [see e.g. Arndt et al., Phys.Rev.C74:045205,2006]

g2πpn4π≃ 13.76 ± 0.01 .

Alternatively, applying PCAC with ∂µ → iqµ, we may write

〈p(p′)|jπ+

5 (0)|n(p)〉 = (m2π − q2) 〈p(p′)|ϕ+(0)|n(p)〉

= i qµ(m2

π − q2)Fπm2

π

〈p(p′)|J+µ5 (0)|n(p)〉 .

The latter matrix element is given by

〈p(p′)|J+µ5 (0)|n(p)〉 =

1√2up(p

′)γµγ5 aA(q2) + qµ γ5hA(q2)

un(p) ,

for the divergence

〈p(p′)|∂µJ+µ5 (0)|n(p)〉 =

i√2up(p

′)q/γ5 aA(q2) + q2 γ5hA(q2)

un(p) ,

47We use the notation introduced in Sect.3.2.2. In the PCAC relation

∂µAaµ(x) = Fπm

2π ϕ

a(x) .

we use the convention in which Fπ = 92.4 MeV. Experimental papers often use fπ =√2Fπ ∼ 132 MeV. The

charged fields we obtain as

A+µ =

(

A1µ + iA2

µ

)

/√2 ; ϕ+ =

(

ϕ1 + iϕ2) /√2 .

311

such that, applying the Dirac equation,

〈p(p′)|∂µJ+µ5 (0)|n(p)〉 =

i√2up(p

′)

(mp +mn) γ5 aA(q2) + q2 γ5hA(q2)un(p) .

thus

〈p(p′)|jπ+

5 (0)|n(p)〉 = i(m2

π − q2)√2Fπm2

π

(mp +mn) gA(q2) + q2 hA(q2)

up(p

′) γ5 un(p) .

and, using the definition of the effective pion nucleon coupling, obtain the identity

√2 g(q2) =

m2π − q2√

2Fπm2π

2mN gA(q2) + q2 hA(q2)

,

where mN is the nucleon mass mN = (mp + mn)/2. Assuming form factors to have a smallvariation in the range 0 ≤ q2 ≤ m2

π, such that we may take as an approximation

gπpn = g(m2π) ≃ g(0) , and gA = gA(0) ,

one obtainsFπ gπpn = mN gA

which is the Goldberger-Treiman relation. Numerically we have gA/gV = −1.2695±0.0029, gπpn =13.150 ± 0.005, Fπ = 92.21 ± 0.20 MeV [fπ = 130.4 ± 0.04 ± 0.2 PDG 2008] and mN =(mp + mn)/2 = 938.92 MeV which yields Fπ gπpn(mN gA) ≃ 1.017. Thus the GT relation issatisfied at the 2% level, quite good as a leading approximation.

In fact the smoothness assumption made by Goldberger-Treiman is not evident, because theinduced pseudoscalar term hA actually may be expected to have a pole. According to Nambu’sspontaneous chiral symmetry breaking scenario, where the chiral quark flavor group SU(2)V ⊗SU(2)A is broken down to SU(2)V , the proper hadronic current is the quark current

J+λ5 (x) =

1√2ψu γ

λγ5 ψd(x)

which in the chiral limit of massless quarks is conserved. Indeed, for m2π → 0 we realize axial

current conservation

∂λJ+λ5 (x) = 0 ,

by the PCAC relation. The latter for mπ finite requires a pion pole in the nucleon matrix element:

〈p|∂λJ+λ5 (0)|n〉 = C m2

π 〈p|φ+(0)|n〉

= Cm2π

(pp − pn)2 −m2π

〈p|j+5 (0)|n〉

np

φ

where 〈p|j+5 (0)|n〉 is given above. By comparison of the two evaluations of the PCAC relation wefind

This requires

2mN gA(q2) + q2hA(q2) = 0 .

312

i.e. hA(q2) must exhibit a pole in q2, which corresponds to the pseudoscalar Nambu-Goldstonepion pole:

limq2→0

hA(q2) = −2Fπgπpn(0)

q2,

which in the limit mπ → 0 again yields the GT relation, but this time based on the spontaneouschiral symmetry breaking pattern.

Models proposed for inferring the proper chiral structure of the hadronic currents and implement-ing the CVC and the PCAC relations are the σ model and the Nambu–Jona-Lasinio model.

The σ–model of Gell-Mann–Levy: Ψ is the nucleon doublet field Ψ =(ψp

ψn

), ~φ is the pseudoscalar

pion triplet field and σ a scalar auxiliary field:

L(σ) = Ψ (iγµ∂µ − g0 (σ + i~τ · ~φ γ5)) Ψ +1

2

(∂µ~φ

)2+

1

2(∂µσ)2

−1

2µ20

(~φ 2 + σ2

)− λ0

(~φ 2 + σ2 − 1

4f20

)2

− µ202f0

σ

Jµ5 =1

2Ψ γµγ5~τ Ψ +

(σ∂µ~φ− ~φ∂µσ

)

∂µJµ5 =

µ202f0

~φ

The Nambu–Jona-Lasinio model:

L(NJL) = Ψ0 (iγµ∂µ −M0) Ψ0 + g0[Ψ0Ψ0Ψ0Ψ0 − Ψ0γ5~τΨ0 · Ψ0γ5~τΨ0

]

For M0 = 0 the NJL Lagrangian is chirally symmetric under SU(2)L ⊗SU(2)R, the symmetry isspontaneously broken and 〈0|Ψ0Ψ0|0〉 6= 0. By the shift

Ψ0Ψ0 = ΨΨ + 〈0|Ψ0Ψ0|0〉a nucleon field satisfying 〈0|ΨΨ|0〉 = 0 may be achieved, whereby the shifted field acquires thenucleon mass

M = −g0 〈0|Ψ0Ψ0|0〉entirely due to the interactions of the nucleons. The spontaneous breakdown of the chiral sym-metry SU(2)L ⊗ SU(2)R → SU(2)V implies the existence of the triplet of pseudoscalar pions(π+, π0, π−). For M0 = 0 they are true Nambu-Goldstone bosons. The non-vanishing pion massrequires a non-vanishing M0 of the order of a few MeV.

Addendum:In pion physics G–parity plays an important role. it allows to single out so called “first–class”and “second–class” currents, among the hypercharge conserving weak currents. The G–parityoperator is given by (C charge conjugation, I2 isospin generator)

G .= C ei π I2

and one easily checks that currents may be odd [first–class] or even [second–class] under GGJµ5 G−1 = −Jµ5 first− class current

GJµ5 G−1 = +Jµ5 second − class current

313

② Calculate the decay rate for π0 → γγ using the effective O(p4) coupling

LWZW =α

π

Nc

12Fπ

(π0 +

1√3η8 + 2

√2

3η0

)FµνF

µν ,

which is the Wess-Zumino-Witten Lagrangian. The latter reproduces the ABJ anomalyon the level of the hadrons. π0 is the neutral pion field, Fπ the pion decay constant(Fπ = 92.4 MeV). The pseudoscalars η8, η0 are mixing into the physical states η, η′.

Let us first write down the Feynman rule for the π0γ(q1, µ)γ(q2, ν) vertex: first we note thatFµν = 1

2εµναβFαβ and Fµν = ∂µAν − ∂νAµ. In momentum space ∂µ → iqµ. We have FµνF

µν =12 εµνρσF

µνF ρσ = 12 × 2 × 2 × εµνρσ ∂

µAν ∂ρAσ = −2εµναβ ∂αAµ ∂βAν as the other terms of the

field strength tensors just yield a factor 2 each, by relabeling of contracted indices. In momentumspace the coefficient of the vertex (a factor 1

2 is commonly included for Bose symmetry of the two

photons) 12 π

0(p) A(q1, µ) A(q2, ν) is απ

Nc12Fπ

× (i)2 (−2) εµναβqα1 q

β2 × 2 (the last factor compensates

for the 12 included in the field monomial). We thus have the Feynman rule:

=α

π

Nc

12Fπ4εµναβq

α1 q

β2

π0Aµ(q1)

Aν(q2)

In the matrix element the photon field will be represented by the polarization vector of an outgoingphoton Aµ → εµ∗(q, λ). Thus the matrix element 〈γ(q1, λ1), γ(q2, λ2)|π0(p)〉 of the WZW vertexin momentum space reads (for antisymmetrically contracted Fµν use Fµν = 2∂µAν a factor twofor Bose symmetry)

2α

π

Nc

12Fπ

1

2εµναβ × −4 qµ1 ε

ν∗(1)qα2 εβ∗(2) =

α

π

Nc

12Fπ4εµναβ ε∗µ(1) ε∗ν(2) q1αq2β ,

in agreement with the above Feynman rule. It is customary to define a π0γ∗γ∗ form factor F by

Aµν(π0 → γ∗γ∗) = i

∫d4x eiq·x〈 0|Tjµ(x)jν(0)|π0(p)〉

= εµναβ qαpβ Fπ0γ∗γ∗(m2

π, q2, (p − q)2) ,

with jµ(x) the electromagnetic current. The pion is on-shell, but the photons are in generaloff-shell. Then

=M[π0 → γ(q1, λ1)γ(q2, λ2)] = e2Aµν(π0 → γγ) ε∗µ(q1, λ1) ε∗ν(q2, λ2)

= e2 εµναβq1αq2βFπ0γ∗γ∗(m2π, 0, 0) ε∗µ(1) ε∗ν(2) .

The WZW fromfactor (a constant) corresponds to the formfactor in the chiral limit m2π → 0 and

for real photons:

MWZWπ0γγ = e2 Fπ0γ∗γ∗(0, 0, 0) =

e2Nc

12π2Fπ=

α

πFπ≈ 0.025 GeV−1

where we used Fπ ∼ 92.4 MeV and the color factor is Nc = 3. The transition rate into photonsof arbitrary polarization is

∑

λ1λ2

|Mπ0γγ |2 = e4 gµρgνσεµναβqα1 q

β2 ερσα′β′qα

′1 q

β′

2 |F|2

= e4 ×−2(q21q

22 − (q1 · q2)2

)|F|2 =

m4π

2|e2F|2 .

314

where we utilized, as usual, the completeness relation∑

λ εµ∗(q, λ)ερ(q, λ) = −gµρ + qµfρ +

qρfµ with f some arbitrary vector which must drop out from the transition amplitude (gauge

invariance). We furthermore used εµναβερσα′β′

= −2 (δα′

α δβ′

β − δβ′α δα

′β ), (q1 · q2) = 1

2 (p2 − q21 − q22)

and p2 = m2π, q

21 = 0, q22 = 0.

The total 2–body decay width into two massless particles is (a factor 1/2 is required for identicalparticles)

Γπ0γγ =1

32πmπ

∑

λ1λ2

|Mπ0γγ |2

and using the WZW amplitude, we obtain

ΓWZWπ0γγ =

1

64πm3π |MWZW

π0γγ |2 =α2m3

π

64π3 F 2π

,

where |Mπ0γγ | was given above. Using the experimental π0 → 2γ width, we may calculate

|M expπ0γγ| =

√64πΓπ0γγ/m

3π = 0.025 ± 0.001 GeV−1

in excellent agreement with the WZW prediction.


① Show that the Maxwell equation ∂µFµν = 0 as a field equation for the vector potential

takes the form

Aν(x)− ∂ν (∂µAµ(x)) = 0 .

Show that Aν(x) is not determined by this equation because the operator gµν − ∂µ∂ν hasno inverse. Hint.: ϕµ = ∂µα(x), α(x) an arbitrary scalar function, is a solution of theabove equation with eigenvalue 0.

We have ∂µ (∂µAν − ∂νAµ) = Aν−∂ν (∂µAµ) = (gµν − ∂ν∂µ)Aµ = 0 and (gµν − ∂µ∂ν) ∂να(x) =

∂µα(x) − ∂µα(x) ≡ 0 whatever α(x) is (but assumed to be differentiable). As claimed theoperator gµν − ∂µ∂ν has no inverse, because it has a zero eigenvalue.

We may try to solve for the propagator in momentum space directly. The kernel is Kµν(p) =

p2gµν − pµpν while the propagator should be of the form Dνρ(p) =(Agνρ +B

pνpρp2

)1p2, and a

solution of

Kµν(p)Dνρ(p) = δµρ .

Kµν(p)Dνρ(p) = Aδµρ −Apµpρp2

+Bpµpρp2−B pµpρ

p2

= Aδµρ −Apµpρp2

?= δµρ ,

which has no solution, as it should satisfy A = 1 and A = 0 at the same, a contradiction.

315

② Show that for a massive spin 1 field the Proca equation

( +m2

)Aν(x)− ∂ν (∂µA

µ(x)) = 0

implies ∂µAµ(x) ≡ 0 automatically. Comment on the number of degrees of freedom ! Show

that the Proca equation is the Euler-Lagrange equation of the Lagrangian

L = −1

4FµνF

µν +m2

2AµA

µ ; Fµν = ∂µAν − ∂νAµ .

Discuss the invariance properties of L under gauge transformations.

Taking the derivative ∂ν of the Proca equation as stated above, we obtain

( +m2

)(∂νA

ν(x))− (∂µAµ(x)) = 0

which means

m2 (∂νAν(x)) = 0

and since m 6= 0 we must have ∂νAν(x) = 0 as claimed. It means there is automatically no

scalar component in Aµ which solves the Proca equation. As a massive spin 1 field has 3 physicaldegrees of freedom the 2 transversal plus one longitudinal, the Proca field does not exhibit anyunphysical ghost degrees of freedom.

The Euler-Lagrange equation reading

∂µ∂L

∂∂µAν=

∂L∂Aν

yields

∂L∂∂µAν

=∂L∂Fρσ

∂Fρσ∂∂µAν

= −1

42F ρσ (δµρ δ

νσ − δνρδµσ ) = −1

2(Fµν − F νµ) = −Fµν

while

∂L∂Aν

=1

2m2 2Aν = m2Aν

and hence

−∂µFµν = m2Aν

and inserting Fµν = ∂µAν − ∂νAµ directly yields the Proca equation.

While under a gauge transformation Aµ(x) → Aµ(x) − ∂µα(x) as in QED Fµν and hence thekinetic term of the Proca Lagrangian is invariant, the mass term AµAµ is obviously not gaugeinvariant.

③ Prove that under local gauge transformations

ψ → e−ieα(x)ψ , Aµ(x)→ Aµ(x)− ∂µα(x)

the covariant derivative Dµ = ∂µ − ieAµ has the property: Dµψ transforms identical to ψand ψΓDµψ is gauge invariant provided Dµ commutes with the 4 by 4 matrix Γ.

316

If ψ → e−ieα(x)ψ and Aµ(x)→ Aµ(x)− ∂µα(x) then

Dµψ → (∂µ − ieAµ + ie (∂µα(x))) e−ieα(x)ψ

= e−ieα(x) (∂µ − i e(∂µα(x)) − ieAµ + ie(∂µα(x))) ψ

= e−ieα(x) (∂µ − ieAµ) ψ = e−ieα(x)Dµψ

q.e.d.

Invariance of ψΓDµψ requires eieα(x)Γe−ieα(x) = Γ, which for an Abelian transformation is triv-ial. For chiral transformations this becomes non-trivial. Gauge invariance is obtained only if[γ0Γ, γ5] = 0. For non-Abelian transformations U = U(x) gauge invariance requires [Γ, Ti] = 0for all generators Ti. Indeed with Dµ = ∂µ − i g Vµ and Vµ =

∑i TiViµ we have [Γ,Dµ] =

−i g∑

i Viµ[Γ, Ti] and thus [Γ,Dµ] = 0 implies [Γ, Ti] = 0, which implies [Γ, U ] = 0.


① If Vµ = igU−1(x) (∂µUx) the field is called a “ pure gauge” field. Show that in this case

Gµν ≡ 0 .

The non-Abelian field strength tensor has been obtained from calculating the commutator of twocovariant derivatives:

[Dµ,Dν ]il = −ig∑

j

(Tj)il (∂µVjν − ∂νVjµ)

−g2∑

j′, j′′

[T j′ , Tj′′

]ilVj′µVj′′ν

= −ig∑

j

(Tj)ilGjµν = −ig (Gµν)il .

This directly translates into a relation for elements of the Lie algebra (sums∑

i TiXi). We easilyread off that the field strength tensor can be written in as:

Gµν = ∂µ Vν − ∂ν Vµ − i g [Vµ,Vν ]

Inserting the “pure gauge” field, we have

i gGµν =

−∂µ[U−1x (∂νUx)] + ∂ν [U−1x (∂µUx)]−

[U−1x (∂µUx) , U−1x (∂νUx)

]

= −(∂µU−1x )(∂νUx)− U−1x (∂µ∂νUx) + (∂νU

−1x )(∂µUx) + U−1x (∂ν∂µUx)

−U−1x (∂µUx)U−1x (∂νUx) + U−1x (∂νUx)U−1x (∂µUx) .

From ∂µ(U−1x Ux) = 0 we find (∂µU−1x )Ux = −U−1x (∂µUx) or (∂µU

−1x ) = −U−1x (∂µUx)U−1x . In-

serting this and using that ∂µ and ∂ν commute, we easily find that Gµν vanishes for any U . Notethat because of the non-linearity of the problem the signs of Vµ = i

gU−1(x) (∂µUx) is crucial. The

opposite sine would not yield Gµν ≡ 0 . Also note that the “pure gauge” field can be representedas a gauge transformation of the zero field.

317

② Show that a mass term M2

2

∑i ViµV

µi cannot be locally gauge invariant.

In order to check gauge invariance in general , i.e. beyond infinitesimal transformations, it issimplest to write the fields in terms of elements of the Lie-algebra. Using the orthogonalityTrTiTj = 1

2 δij of the generators we obviously may write the mass term as

M2TrVµVµ

A general gauge transformation reads

Vµ → Vµ′ = U(x)

(Vµ −

i

gU−1(∂µU)

)U−1(x) .

and hence

TrVµVµ

→ TrUx

(Vµ −

i

gU−1(∂µU)

)U−1x Ux

(Vµ − i

gU−1(∂µU)

)U−1x

= Tr

(Vµ −

i

gU−1(∂µU)

)(Vµ − i

gU−1(∂µU)

),

which is not gauge invariant.

③ Show that LYM can be written in the form of a trace

LYM = −1

2Tr (GµνG

µν)

By the normalization adopted for the generators: TrTiTj = 12 δij we have

Tr (GµνGµν) = Tr

∑

ij

TiTjGµν iGµνj =

1

2

∑

ij

δijGµν iGµνj =

1

2

∑

i

Gµν iGµνi .

Therefore,

LYM = −1

2Tr (GµνG

µν) = −1

4Gµν iG

µνi

the usual representation.

④ Prove the validity of the “homogeneous Maxwell equation” or “Bianchi-identity”:

DρGµν + terms cyclic in (ρµν) ≡ 0

318


① Spin 1 propagators are symmetric second rank Lorentz tensors proportional to gµν andkµkν , where k is the four momentum of the spin 1 boson. Show that

T µν = gµν − kµkν

k2; Lµν =

kµkν

k2

are projection tensors to the transversal and the longitudinal field components. Verifythe projector property of T and L. Use the decomposition

Kµν = AT µν +B Lµν

for the bilinear part of the L0G = GµKµνGν of the free part of the Lagrangian (what are

A and B in case of the gluon), calculate the propagator (inverse of the kernel K) using theprojection technique. Formulate the condition under which the inverse exists.

② Given the Yang-Mills Lagrangian in coordinate space derive the triple gluon coupling

−igscijk gµν (p2 − p1)ρ + gµρ (p1 − p3)ν + gνρ (p3 − p2)µand the quartic gluon coupling

−g2s cnijcnkl (gµρgνσ − gµσgνρ) + cnikcnjl (gµνgρσ − gµσgνρ) + cnilcnjk (gµνgρσ − gµρgνσ)in momentum space.

③ According to Faddeev and Popov the fatal violation of gauge invariance by the gauge fixingterm can be avoided by taking into account the functional determinant obtained in thefunctional integral under a gauge transformation of the fields (integration variables). If wedefine the functional integral as follows, with a Faddeev-Popov determinant,

∫DGµaDet

(δCaδωb

)ei∫

(

Linv− 12ξC2

a

)

d4x

one easily checks that now the functional integral is independent on the specific choice ofthe gauge function Ca. Check this!

By introducing anticommuting scalars, the FP ghost fields ηa and ηa, we may represent theFP-determinant as a Berezin integral over Grassmann variables (algebra of anticommutingc-numbers)

Det

(δCaδωb

)=

∫DηDη ei

∫

LFPd4x

with

LFP = ηaMabηb ; Mab.=δCaδωb

.

Check that this agrees with the expression adopted above.

319

④ Becchi-Rouet-Stora (BRS) symmetry: given the “quasi invariant” effective Lagrangian

Leff = Linv + LGF + LFP ,

the local gauge invariance of the functional integral (path integral quantization)∫DGDηDη ei

∫

Leffd4x (A.5)

yields relations between Green functions, the Slavnov-Taylor (ST) identities. They gener-alize the Ward-Takahashi (WT) identities, relations between Green functions which derivefrom global symmetries. In contrast to canonical quantization, which relies on the nongauge-invariant splitting into a free and an interacting part of the Lagrangian, the pathintegral formulation work with the complete effective Lagrangian.

The ST-identities provide the tool needed for proofs of

i) gauge invariance

ii) unitarity

iii) renormalizability

of the S-matrix. ST-identities may be obtained from the BRS-symmetry of Leff .

Convince yourself that the BRS procedure (described in the following) to control gauge in-variance indeed works. The idea behind BRS-symmetry is to dispose of the as yet undefinedtransformation properties of the FP-ghost fields η and η such that

δBRSLeff = 0 . (A.6)

In order to achieve this it is natural to demand the following relations to hold:

i) δLinv = 0

ii) δLGF = −1ξCaMabωb

iii) δLFP = δηMabηb + ηaδ(Mabηb)

= −δLGF

iv) DGDηDη invariant .

A solution for this set of conditions may be obtained as follows:1) Introduce anticommuting global c−number variables δλ, δλ anticommuting with η and η,and identify ωb = ηbδλ. Thus

• δGa = Dabηbδλ

where Ga can be a gauge field, a scalar or a Fermi field with Dabηb given by

δGaµδωb

ηb.= Dµabηb

in case of the gauge field.

2) Assume η to transform according to the regular representation, thus

320

• δηa = −12gfabcηbηcδλ

where a permutation symmetry factor 1/2 (antisymmetry of fabc and anticommutativity ofthe η’s) has been taken into account. 1) and 2) imply δ(Mabηb) = 0 . We thus take thefreedom to choose: 3) The field η transforms as

• δηa = −1ξCaδλ

such that conditions i) to iii) are satisfied. One can show iv) to be true for the above set oftransformations which define the BRS-transformation.ST-identities:The BRS invariance of Leff allows a simple derivation of the ST-identities. Performing achange of integration variables in the functional integral does not change the value of theintegral. See my TASI Lectures Renormalizing the Standard Model for more details.


① Prove the relations (6.4) and (6.5) for the d dimensional Dirac algebra.

The γ’s–contractions can all be worked out using the Dirac algebra: 1) γαγα = gαβ γ

βγα =gαβ

12

(γβγα + γαγβ

)= gαβ

12 2 gαβ 1 = d1

where besides the Dirac algebra we have used the symmetry of the metric tensor for a sym-metrization and gαα = d in d dimensions.2) γαγ

µγα = −γµγαγα + 2 δµαγα == −d γµ + 2 γµ

3) γαγµγνγα = γµγνγαγ

α + 2 δµαγνγα − 2 δναγµγα = γµγν d+ 2 γνγµ − 2 γµγν

4) γαγµγνγργα = −γµγνγργαγα + 2 γνγρ︸︷︷︸γ

µ − 2 γµ γργν︸︷︷︸+2 γµγνγρ = (4− d) γµγνγρ − 2 γργνγµ

applying the Dirac algebra to the underbraced terms yields the given result, in the given (non-unique) form.etc.The traces are obtained using the Dirac algebra in conjunction with the cyclicity property:1) In d = 2n (n=integer) dimensions the representation of the Dirac algebra is composed of atensor product of n = d/2 Pauli matrices σi1 ⊗ σi2 ⊗ · · · ⊗ σin such that the dimension of therepresentation is Tr (1) = 2n = 2d/2, which provides a natural extension to arbitrary dimensions.2) Tr γµ = 0 is a basic property of the γ–matrices, being hermitian and traceless.3) Tr γµγν = 1

2 (Tr γµγν + Tr γνγµ) = 12 Tr γ

µ, γν = gµν Tr(1)4) Tr γµγνγργσ = Tr γνγργσγµ = −Tr γµγνγργσ + 2 gµσ Tr γνγρ− 2 gµρ Tr γνγσ + 2 gµν Tr γργσ

thereforeTr γµγνγργσ = (gµσgνρ − gµρgνσ + gµνgρσ) Tr(1)as claimed.5) γ–odd traces: Tr γµ1γµ2 · · · γµ2n−1 = Tr γµ1γµ2 · · · γµ2n−1γ5γ5 = Tr γ5γ

µ1γµ2 · · · γµ2n−1γ5 an-ticommuting the first γ5 through all γµi ’s to the right we end up with Tr γµ1γµ2 · · · γµ2n−1 =(−)2n−1 Tr γµ1γµ2 · · · γµ2n−1 which means Tr γµ1γµ2 · · · γµ2n−1 = 0 for all integer n ≥ 1. Thisargument is based strictly on the anti–commutativity of γ5: γ5, γµ = 0, which is granted in anycase in d = 4. In dimensional regularization, one eventually has to give up the anti–commutativityof γ5 (’t Hooft, Veltman 1972 [1], see also F. Jegerlehner 2001 [2]). See next exercise.

321

② Show that the γ5–odd traces are not cyclic in the d dimensional Dirac algebra. Note thatfor a mathematically well defined set of matrices a trace trivially is cyclic. In particularprove the claim (6.6). Hint: calculate Tr γαγ

µγνγργσγαγ5 and join the connected γ’s suchthat one may use γαγ

α = d. Note that there are two ways to join these γ’s.

Xµνρσ = Tr γαγµγνγργσγαγ5 may be computed in two ways: 1) anticommute γα with γ5 to the

right and use cyclicity of the trace we obtain Xµνρσ = −Tr γαγαγµγνγργσγ5 = −dEµνρσ ; 2)anticommute γα through γµγνγργσ to the right until it meets γα, which yields 2 δµαγνγργσγα −2 δναγ

µγργσγα + 2 δραγµγνγσγα − 2 δσαγµγνγργα + γµγνγργσγαγ

α = 2 γνγργσγµ − 2 γµγργσγν +2 γµγνγσγρ − 2 γµγνγργσ + d γµγνγργσ. Finally, we are anticommuting the γ–matrices into thecommon order. Note that at this stage the gµν terms from the Dirac algebra do not contributeto the trace as all γ5 odd traces with less than 4 γ’s vanish. Thus we obtain

−dEµνρσ = −8Eµνρσ + dEµνρσ

or

(4− d)Tr γµγνγργσγ5 = 0

and hence

Tr γµγνγργσγ5 = 0 if d 6= 4

in case we assume γ5 to be anticommuting when d 6= 4. Note that the fact of the existence ofthe ABJ anomaly must be understood as a proof that no regularization exists which satisfiesall symmetries which seem to be there at the classical level. This is a no–go theorem: it is notpossible to find a regulatization which preserves all symmetries (gauge and chiral) we would liketo preserve in first place (in this context we should mention the Nielsen and Ninomiya theorem1981 concerning lattice regularizations [3]).

③ Prove the relations (6.39) and derive the corresponding relations for the tensor integralsB1, B21, B22 as well as for the three point amplitudes C1i and C2i. Verify (6.46).

④ Calculate B0(m1,m2; p2) performing the 1-dimensional integral (6.36) in terms of elemen-

tary functions.

Loop integrals typically lead to logarithms and at higher orders to polylogarithms (diloga-rithm[=Spence function], trilogarithm, ... ). These are multi–valued analytic functions andone of the major complications is the proper specification of the sheet which is appropriate for agiven physical application. The unique answer in physics is dictated by the iε–prescription in theFeynman propagators. In Feynman diagram calculations one commonly chooses the logarithm tohave the cut along the negative axis. This implies that

ln(ab) = ln a+ ln b+ η(a, b)

322

with

η(a, b) = 2π i Θ(−Im a)Θ(−Im b)Θ(Im ab)−Θ(Im a)Θ(Im b)Θ(Im − ab)The following simple rules apply:

ln ab = ln a+ ln b if Im a and Im b have different sign ,

ln a/b = ln a− ln b if Im a and Im b have the same sign .

If A and B are real we have

ln(AB − iε) = ln(A− iε′) + ln(B − iε/A)

where ε′ has the same sign as ǫ.

The scalar two–point function is given by Eq.(??):

B0(m1,m2; p2)− Reg = −1∫

0

dx ln(xm21 + (1− x)m2

2 − x (1− x) p2 − iε) .

The argument of the logarithm is a quadratic form in the integration variable x and we maydecompose it in its roots. From q(x) = x2 p2 + x (m2

1 −m22 − p2) +m2

2 − iε which has the zeros

x± = −(m21 −m2

2 − p2)/(2p2)±√

((m21 −m2

2 − p2)2 − 4m22p

2)/(4p4)

Note that the roots have imaginary parts of different signs such that we may write

ln[p2 (x− x+) (x− x−)

]= ln(p2 − iε) + ln(x− x+) + ln(x− x−)

The integral is then easily obtained. The primitive of ln(x− a) is (x− a) ln(x− a)− x and thus∫ 1

0dx ln q(x) =

[x ln(p2 − iε) + (x− x+) ln(x− x+) + (x− x−) ln(x− x−)− 2x

]∣∣10

= ln(p2 − iε) + (1− x+) ln(1− x+) + x+ ln(−x+)

+(1− x−) ln(1− x−) + x− ln(−x−)− 2

= ln(p2 − iε) +∑

i=±

[ln(1− xi)− xi ln

xi − 1

xi− 1

].

Note: in the form −s x (1−x)+m21 x+m2

2 (1−x) the iε–prescription form the Feynman propagatoris m2

i → m2i − iε for s translates to s → s + iε. Thus when m1 = m2 = 0 we may write

q(x) = −x (1−x) p2− iε = (−p2 + iε)x (1−x) (since x (1−x) is positive in the integration rangeiε in q(x) is equivalent to x (1− x) iε). The integral reads

ln(−p2 + iε) + [x lnx− x− (1− x) ln(1− x)− x]10 = ln(−p2 + iε) − 2 .

If p2 = 0 q(x) = x (m21 −m2

2) +m22 − iε has one root only Special cases:

B0(0,m; s) = 1−A0(m)/m2 +m2 − ss

ln(

1− s

m2

)

B0(m,m, s) =

B0(0, 0; s) = Reg + 2− ln(−s− iε)

B0(0,m; 0) = = −A0(m)/m2

B0(m,m; 0) = = −[1 +A0(m)/m2

]

B0(m1,m2; 0) = = − [A0(m1)−A0(m2)] /(m21 −m2

2)

323


① Calculate the group invariants C2(G) and C2(R) using the known properties of the SU(n)groups.

1) General consideration: for any irreducible representation R of dimension d(R) we have thequadratic Casimir invariants T (R) and C2(R) (i.e. an operator commuting with all the generatorsTRi (i = 1, · · · , r) of the group, r = d(G) the order or dimension of the group G [r = n2 − 1 forSU(n)]). T (R) is defined by

TrTRi TRk =

d(R)∑

a,b=1

(TRi)ab

(TRk)ba

= δik T (R) ,

and C2(R) by

(TRi T

Ri

)ab

=

r∑

i=1

d(R)∑

c=1

(TRi)ac

(TRi)cb

= δab C2(R) .

Taking the trace of the second form using the first and the fact that for square matrices Tr (A+B) = TrA+ TrB, we have

d(G)T (R) = d(R)C2(R) or C2(R) =d(G)

d(R)T (R) .

2) For G = SU(n) and R = F the fundamental representation using the last relation we imme-diately obtain

CF = C2(F ) =n2 − 1

nTF ,

where TF = T (F ) is fixed by the standard normalization

TrTiTk = δik T (R) ; T (R) =1

2

of the SU(n) generators. Hence

C2(R) = CF =n2 − 1

2n

in the fundamental representation.

3) C2(G) corresponds to C2(R) for the adjoint (regular) representation R = A of dimensiond(R) = d(G) = r and is defined correspondingly by

fijl fjlk = δik C2(G) ,

which we may write, with fijl = i (Ti)jl and the antisymmetry of fijl, as

i(Ti

)jl· (−i)

(Tk

)lj

= Tr TiTk = δik C2(G) .

324

Again this trace is diagonal, i.e., proportional to δik, and C2(G) provides the normalization ofthe generators in the adjoint representation. In other words

C2(G) = CA = TA as d(R) = d(G) for R = A .

Again a simple consequence of the general relation given under 1). However, we do not yet knowthe normalization TA of the adjoint generators. Once the normalization of the generators in thefundamental representation is fixed, the normalization of the adjoint generators is fixed by theLie algebra, actually TA = g TF . But what is g (in group theory called Coxeter number, n g isthe number of roots).

In fact since we know (as inferred from the normalization equation) that all generators havethe same normalization, we need compute only one, and we may choose the simplest possibility:calculate frkl where r = n2 − 1 and Tr = 1

2 λr with λr the last of the diagonal λ–matricesEq. (1.1) in the fundamental representation. The commutator of Tr vanishes in the upper leftn − 1–dimensional sub-matrix where it just is the unit matrix. Thus the only non-vanishingelements are those which have a 1 or −i in the n-th column. These elements have the form

[Tr, Tk] =1

2

√2

n(n− 1)(1− (−(n− 1))) iTl

k= i frkl

kTl

k,

where, if Tk is chosen to be the generator having a 1 in the n-th column, then Tlkis fixed to be

the generator which has a −i in the same position. Thus we have n−1 possible pairings [kl] withvalues

fr[kl] =

√n

2(n − 1).

The same we obtain for the n− 1 possibilities when we replace 1 by −i in the n-th column on theleft-hand side. This we have to take in quadrature, and we have to sum over all 2(n− 1) possiblenon-vanishing terms: thus

∑

[kl]

f2r[kl] = 2(n − 1)n

2(n− 1)= n .

Therefore we obtain the known result

CA = TA = n

q.e.d.

② Derive from our one-loop QCD results the corresponding ones for QED. Discuss main dif-ferences.

③ Calculate the integral C0(mg,m,m;m2, k2,m2) in the regime m2g ≪ m2, |k2| using the

representation (6.38). Discuss typical properties of the result.

325

We may use a cyclic permutation of masses and momenta and consider C0(m,m,√δ; s,m2,m2)

with δ = m2g infinitesimal and s = k2. Then a = m2, b = s, c = −s, d = −δ, e = 0 and f = δ.

Starting from the representation

C0(m1,m2,m3; s1, s2, s3) =

1∫

0

dx

x∫

0

dy1

D(x, y)

we may rescale the variable as y = xy′ to obtain

C0(m1,m2,m3; s1, s2, s3) =

1∫

0

dx

1∫

0

dy′x

D(x, y = xy′).

Thus the bi-quadratic form D(x, y′) form in the denominator of the last integral then reads

D(x, y) = Ax2 + δ (1− x) with A = s y2 − s y +m2 ,

where y here and in the following denotes y′. The integration over x may now be performed. Tothis end we write, using a partial fraction decomposition,

x

Ax2 + δ (1− x)=

1

A

x

(x− x1)(x− x2)=

1

A

1

x1 − x2

(x

x− x1− x

x− x2

)

=1

A

1

x1 − x2

(x1

x− x1− x2x− x2

)

with x1 + x2 = x1 x2 = δ/A. For the roots we find

x1,2 = ±i

√δ

A+O

(δ

A

)

Up to a factor 1/A the integral is then given by

x1x1 − x2

ln1− x1x1

− x2x1 − x2

ln1− x2x2

=1

2ln

(1− x1)(1 − x2)x1x2

+O

(√− δA

)= −1

2lnδ

A+O

(√− δA

).

As a result of the x integration we have

1∫

0

x

Ax2 + δ (1− x)=−1

2Alnδ

A+O

(√− δA

).

We now may perform the remaining y integration. With appropriate function A(y), this resultjust given also holds if we let the two non-zero masses be different. The function A then reads

A(y) = −s y (1− y) +m21 (1− y) +m2

2 y − iε = s (y − y1) (y − y2) .

As the coefficient of −s is real positive in the integration range we may give s a small imaginarypart s→ s+ iε, which is equivalent to m2

i → m2i − iε. For space–like s, −s = Q2 > 0 A is positive

and has no real roots. In any case the two roots yi have opposite signs such that we may write

lnA(y) = ln(s+ iε) + ln(y − y1) + ln(y − y2)

326

and use the partial fraction decompositions

1

(y − y1)(y − y2)=

1

y1 − y2

(1

y − y1− 1

y − y2

).

The integral to be calculated is then

C0 =

1∫

0

dy−1

2Alnδ

A

=−1

2 s (y1 − y2)

1∫

0

dy

(1

y − y1− 1

y − y2

) (ln

δ

s+ iε− ln(y − y1)− ln(y − y2)

).

The different types of integrals needed are

1∫

0

dy1

y − yi= ln

yi − 1

yi,

1∫

0

dy1

y − yiln(y − yi) =

1−yi∫

−yi

dy1

yln(y)

=1

2ln2 y

∣∣∣∣1−yi

−yi=

1

2

(ln2(1− yi)− ln2(−yi)

),

and for i 6= k

1∫

0

dy1

y − yiln(y − yk)

=

1∫

0

dy1

y − yi

ln

(y − ykyi − yk

)+ ln(yi − yk)

=

1−yi∫

−yi

dy1

y

ln

(y + yi − ykyi − yk

)+ ln(yi − yk)

=

1−yi∫

−yi

dy1

yln

(1 +

y

yi − yk

)+ ln

yi − 1

yiln(yi − yk)

The integral we write as

1−yi∫

−yi

dy · · · =1−yi∫

0

dy · · · −−yi∫

0

dy · · ·

and rescale the “upper” limit to one: y = (1 − yi) y′ and y = −yi y′, repectively. Thereby dy/yremains invariant. This way we obtain

1−yi∫

0

dy1

yln

(1 +

y

yi − yk

)=

1∫

0

dy1

yln

(1 +

1− yiyi − yk

y

)= −Sp

(− 1− yiyi − yk

)

327

and

−yi∫

0

dy1

yln

(1 +

y

yi − yk

)=

1∫

0

dy1

yln

(1 +

−yiyi − yk

y

)= −Sp

(yi

yi − yk

).

Finally, adding up all terms we arrive at the somewhat clumsy expression

C0(m1,m2,mg; s,m22,m

21) =

−1

2 s (y1 − y2)

ln

m2g

s+ iε

(lny1 − 1

y1− ln

y2 − 1

y2

)

−1

2

(ln2(1− y1)− ln2(−y1)− ln2(1− y2) + ln2(−y2)

)

− ln(y1 − y2) lny1 − 1

y1+ ln(y2 − y1) ln

y2 − 1

y2

+Sp(y1 − 1

y1 − y2)− Sp(

y1y1 − y2

)

−Sp(y2 − 1

y2 − y1) + Sp(

y2y2 − y1

)

For m1 = m2 = m (as in QED or QCD) we obtain

C0(m,m,mg; s,m2,m2) =

1

s

[F1(y) ln

m2g

s+ F2(y)

]

where, with y = 4m2/s and r =√

1− y we have

F1(y) =1

rlnr + 1

r − 1

F2(y) =1

2r

− ln

r + 1

r − 1

[ln−r + 1

2+ ln

r − 1

2+ 2 ln r

]

+2 Sp

(r + 1

2r

)− 2 Sp

(r − 1

2r

)

④ Elaborate the following statement [from ’t Hooft and Veltman’s Diagrammar].Statement: Order-by-order renormalization is not equivalent to “throwing away ε–poles andtheir residues” of the unrenormalized S–matrix.

Illustrate this for the following example of a double lepton–loop contribution to the photonself–energy:

Up to a factor p2 gµν − pµpν, the irreducible self–energy bubble may be written in the form

:= f(p2) =1

d− 4f1(p

2) + f2(p2) + (d− 4) f3(p3) .

What is the correct two–loop contribution from the factorizable double–loop above?

328

Without performing a subtraction, the contribution from the two–loop diagram is

(f(p2)

)2=

(1

d− 4

)2

f21 + 2f1f2d− 4

+ f22 + 2 f1f3 +O(d− 4) .

If we now throw away the pole parts (MS–scheme) the result for d = 4 is

f22 + 2 f1f3 .

This result is wrong because it violates unitarity.

Suppose now the correct order–by–order subtraction. After the one–loop treatment one has

+ := f2 + (d− 4) f3 .

The physical result in d = 4 is just f2. Including the proper counterterms at two loops, we find

+ + + := f22 +O(d− 4) .

This result can be shown to be consistent with unitarity.

Exercises: Section ??

⑤ Calculate the exact Born cross section for e+e− → µ+µ−, by replacing the hadronic tensorhµν by the leptonic one ℓµν of the muon. What is the difference to e+e− → e+e−?

⑥ Verify that the subtraction changes the DR of the form (8.13), assumed to converge orsupplemented with a cut–off, to (8.14), which we call a subtracted DR.

Consider the DR

Π′γ(q2) =1

π

Λ2∫

s0

dsIm Π′γ(s)

s− q2 − iε,

with s0 > 0, which does not have a limit as Λ→∞ for the photon vacuum polarization, becausein QCD Im Π′γ(s) = R(s)/12π) and R(s) → 5 as s → ∞. Thus Im Π′γ(s) → constant as s → ∞and the integral is logarithmically divergent: (assuming q2 < s0 for simplicity)

Π′γ(q2) ∼ R(∞)

12π2ln

Λ2 − q2s0 − q2

.

Since the divergence is logarithmic one subtraction suffices to render the intergal finite. Weperform a subtraction at q2 = 0 (charge renormalization in the classical low energy limit; a

329

subtraction at any other fixed q2 like q2 = m2 or the like would remove the UV singularity aswell). At q2 = 0 we have

Π′γ(0) =1

π

Λ2∫

s0

dsIm Π′γ(s)

s− iε,

here the iε prescription has no effect because there is no pole in s > s0. The required subtractionyields

Π′γ(q2)− Π′γ(0) =1

π

Λ2∫

s0

ds Im Π′γ(s)

1

s− q2 − iε− 1

s

=q2

π

Λ2∫

s0

dsIm Π′γ(s)

s (s− q2 − iε).

The last version of the integral is absolutely concergent above the pole at s = q2, and the limitΛ →∞ exists and can be taken in the integral. This proves the finitenes of the subtracted DR.q.e.d.

⑦ Discuss the positivity properties of the functional P (s) defined as the derivative of

P (s) = P∫ ∞

s0

ds′R(s′)

s′(s′ − s) ,

where R(s) is a positive ordinary function. As a distribution (or linear functional) P (s) hasa derivative. Formally,

P (s) =dP (s)

ds= P∫

ds′R(s′)

(s′ − s)2 .

is an integral over an integrand which is almost everywhere positive.

References

[1] G. ’t Hooft, M. J. G. Veltman, Nucl. Phys. B 44 (1972) 189.

[2] F. Jegerlehner, Eur. Phys. J. C 18 (2001) 673 [arXiv:hep-th/0005255].

[3] H. B. Nielsen, M. Ninomiya, Phys. Lett. B 105 (1981) 219; Nucl. Phys. B 185 (1981) 20;Erratum: Nucl. Phys. B 195 (1982) 541; Nucl. Phys. B 193 (1981) 173.

330

Index

B–mesons, 232B–violation, 16U(1)-axial current, 42U(n)V ⊗ U(n)A, 45Υ–resonances, 230σ-model, 30

Abelian gauge field, 68adjoint representation, 4Adler–function, 194, 199Adler-Bell-Jackiw anomaly, 89analyticity, 98anomaly cancellation, 45, 47anti–screening, 89asymptotic freedom, 18, 89, 138, 188asymptotic symmetry, 19axial-vector anomaly, 42, 44

baryon asymmetry, 16baryon number

conservation, 89bottomonium, 223broken symmetry, 19

Casimir invariants, 271causality, 98charmed mesons, 228charmonium, 223chiral

currents, 61perturbation theory, 61–65, 89, 204symmetry, 89symmetry breaking, 61

chiral fields, 42chiral group, 18, 19, 45chiral limit, 31chiral symmetry, 38, 42chiral transformations, 42, 43CKM–matrix, 51color, 38, 87color confinement, 38color singlets, 38confinement, 18, 85, 89conformal invariance, 196conjugate representation, 5covariant derivative, 72, 77, 79, 87, 201current

conserved, 61

partially conserved, 61current quarks, 61current–field identity, 203currents, 82curvature tensor, 82CVC, 61

decoupling by hand, 164dilatation current, 196dimension of a representation, 6dimensional transmutation, 152dispersion relation, 182duality

quark–hadron, 188dynamical symmetries, 84dynamical symmetry breaking, 31Dyson

series, 116, 125summation, 183

electroweak theory, 74e+e− cross–section

in pQCD, 189–193equations of motion, 82equivalence principle, 77equivalent representation, 5Euclidean field theory, 98exponentiation

Coulomb singularity, 191

Faddeev-Popovghosts, 88term, 88

Feynman propagator, 98Feynman rules

QCD, 88sQED, 201

field strength tensornon–Abelian, 87

fine structure constanteffective, 185

first–class currents, 313flavor mixing, 51flavor mixing pattern, 57

G–parity, 291, 313gauge

coupling, 87fixing, 87

331

group, 87parameter, 87

gauge copies, 70gauge fields, 42gauge fixing, 69gauge fixing term, 70gauge invariance, 68gauge orbits, 70gauge parameter, 70gauge potential, 73gauge principles, 68generators of a group, 1glueballs, 19gluon radiation, 278gluons, 18, 87, 189

jet, 194Goldstone bosons, 26, 28, 64Goldstone model, 34Goldstone theorem, 26GOR relation, 64

hadronization, 193hadrons, 38Heisenberg model, 26Higgs

mechanism, 89

imaginary time, 98, 99integral

contour, 98, 181form factor, 101self–energy, 101tadpole, 101

interactionfinal state, 214strong, 87

internal symmetry, 1invariance

scale, 196irreducible representation, 5

Jacobi identity, 2jet shape analysis, 273jets, 194, 269

gluon jet, 194

ladder operators, 2lattice QCD, 89lepton number, 293lepton-quark duality, 45Lie algebra, 2

local gauge invariance, 68

Majorana neutrino, 18mass renormalization

quarks, 114matter fields, 39minimal

substitution, 201minimal couplings, 78minimal substitution, 73, 79

gauge potential, 73minimal subtraction, 94, 95, 114

scheme, 101

Nambu-Jona-Lasinio model, 30neutrino–oscillations, 17non–perturbative effects, 195non-Abelian field strength tensor, 80non-Abelian gauge fields, 79

one loop integralsscalar, 100, 101

one–particle irreducible, 117OPE, 64, 195operator product expansion , see OPE 64order of a group, 1order parameter, 26OZI–rule, 251

parallel displacement, 77particles

unstable, 205partons, 194PCAC, 61permutations, 5perturbation theory

chiral (CHPT), 61potential models, 233production threshold, 102pseudo Goldstone boson, 20pure Yang-Mills theory, 81

QCD, 5, 16, 74, 85gauge structure, 270renormalization group, 146

QEDNoether theorem, 71

Quantum Chromodynamics, 87quantum chromodynamics, 18quantum flavordynamics (QFD), 74quantum numbers: color, 18quark

332

condensates, 65, 195quark flavor mixing, 38quark mixing, 51quark parton model, 189quarkonia

decays, 251quarkonium, 223

hyperfine splitting, 263quarks, 87, 189

radiative corrections, 111rank of a group, 2reducible representation, 5renormalizability, 87renormalization

scale, 101wave function, 183

representation of a group, 4resonance, 194

Breit-Wigner, 209narrow width, 209

ρ− ω mixing, 209Roy equation, 214running charge, 183

Schwarz reflection principle, 181second–class currents, 313selection rule, 293self–energy

quark, 111SM, 74space–like, 185spin-statistics crisis, 38spontaneous symmetry breaking, 26Standard Model, 38, 74Sterman-Weinberg jets, 272symmetry class, 5

tensordecomposition, 94energy momentum, 196integral, 103vacuum polarization, 117

tensor product of representations, 5theorem

Cauchy’s, 181optical, 179, 202

Thomson limit, 183threshold, 194triangle anomaly, 47

unimodular group, 1unitary group, 1unitary groups, 1

Van Royen - Weisskopf formula, 252–254VMD model, 201

wave function renormalization, 115weak hypercharge, 40weak interaction

quantum numbers, 40weak isospin, 39weight diagram, 2Wess-Zumino-Witten Lagrangian, 65Wess-Zumino-Witten Lagrangian, 314Wick

rotation, 98width

finite, 205

Yang-Mills fields, 79Young tableau, 5Yukawa interaction, 51Yukawa sector, 51

333

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