1

Renormalization: state-of the-art report with a new concept and new methods

Jan HelmInstitute of Physics, LondonEmail: janhelm@snafu.de

AbstractFirst, we present the state-of-the-art of renormalization with its three main methods: the dimensionalregularization due to ‘t Hooft-Veltmann, the Pauli-Villars cut-off method, and the lattice regularization.Second, we derive a new, mathematically rigorous formulation of the dimensional regularization, analyze thelimit behavior of the lattice regularization numerically, and present relations between the three methods. Theapproach is illustrated by numerical and analytic calculations for the three first-order loop corrections of thequantum electrodynamics (QED).

1 Definition of renormalizabilityRenormalization is a collection of techniques in quantum field theory, that are used to treat infinities arising incalculated quantities by altering values of quantities to compensate for effects of their self-interactions.Mathematically speaking, we get for an Feynman graph expression Γ(κi) a series in ε , where 0 is theregularization parameter:

...)()(

)(),( 11

0

i

iii C

CC , which has a residuum C-1 at 0,

instead of a “normal” complex value )(0 iC , if the expression is non-divergent.

Renormalization is the procedure of finding additional counter-terms in the corresponding lagrangian tocompensate for the residuum-term in (finitely many) divergent Feynman graphs of an interaction.

2 Physical dimension of wave functions and the lagrangianIn the following, we use the usual convention c 1 , i.e. all physical entities have dimension cmn or cm-n ,e.g. [length]=cm, [energy]=cm-1 . If we are working in a space of (geometric) dimension d=4 (normally, ofcourse in Minkowski space of dimension d=4)), then we have the following (physical) dimension

4][ cmL for a lagrangian1][ cm for a vector wave function, e.g. photon, or a Klein-Gordon scalar (lagrangian term 2 )

2/3][ cm , for a spinor, e.g. electron (lagrangian term i )

Now, it can be shown that for a renormalizable interaction, the corresponding term in the lagrangian has the

dimension 4cm , like the lagrangian itself, i.e. the coupling constant is dimensionless.For instance, in QED, the lagrangian is [1]

and the interaction term has the dimension 4cm and the coupling constant 4e is dimensionless(α is the fine-structure constant). For QCD

, where the field tensor is

is the covariant derivative , the coupling constant g is dimensionless, and

is the gluon field with the gluons aA and the the gauge generators )3(SUa

23. Renormalization criteria and methodsThere are four methods for dealing with divergences in the quantum field theory (QFT): power counting,regulation and counterterms [1].Power countingBy simply counting the powers of p in any Feynman graph, we can, for large p, tell whether the integraldiverges by calculating the degree of divergence of that graph:each boson propagator contributes p-2,each fermion propagator contributes p-1,each loop contributes a loop integration with p4,and each vertex with n derivatives contributes at most n powers of p.If the overall power of p; that is, the degree of divergence D, is 0 or positive, then the graph diverges.By simply counting the p-power at infinity, we have a necessary condition for non-renormalizability.Renormalizability criteria:-The degree of divergence D of any graph must be a function only of the number of external legs; that is, itmust remain constant if we add more internal loops.-The number of classes of divergent N-point graphs must be finite.

RegularizationThere are three regularization techniques: cutoff regularization(Pauli-Villars), dimensional regularization(t’Hooft-Veltman), and lattice regularization.Manipulating divergent integrals is not well-defined , so we must cutoff the integration over d4p in some wayin order to make the integral finite and then proceed to a limit.The cutoff method does it by introducing an additional massive particle with large mass M (a ghost) ,integrating to the cutoff M, and then letting M→∞ .The dimensional method extends the integral to the fractional dimension d=4-ε, where it converges, and letsthen ε →0 .The lattice method calculates the integral over a finite equidistant lattice with lattice distance ε and the upperlimit M(ε) and lets ε →0 , M(ε) →∞ .The first two methods are analytical, i.e. one must be able to calculate the integral as a closed expression, thelattice method is numerical, i.e. it is applicable also for integrals, which cannot be calculated analytically.CountertermsThe method of counterterms, pioneered by Bogoliubov, Parasiuk, Hepp, and Zimmerman (BPHZ), consists ofadding new terms directly to the Lagrangian to subtract off the divergent graphs. The coefficients of thesecounterterms are chosen so that they precisely kill the divergent graphs.In a renormalizable theory, there are only a finite number of counterterms needed to render the theory finite toany order, these counterterms are proportional to terms in the original action. Adding the counterterms to theoriginal action gives us a renormalization of the masses and coupling constants in the action.Multiplicative renormalizationThe method of multiplicative renormalization, pioneered by Dyson and Ward for QED, means in essence tosum formally over an infinite series of Feynman graphs with a fixed number of external lines.The divergent sum is then absorbed into a redefinition of the coupling constants and masses in the theory. Sincethe bare masses and bare coupling constants are unmeasurable, we can assume they are divergent and that theycancel against the divergences of corresponding Feynman graphs, and hence the theory has absorbed alldivergences at that level.Using the formula for the geometric series

one develops the fermion propagator into a series in p2 and sums the integral formally:

...~~11

~1

111)(

2

2

2

2

2

2

2

2222

im

pi

m

p

mi

m

pmimppF

4 Green function, S-matrix and cross-sectionsWe start with the Green function (inverse operator) of the Schrödinger Hamiltonian

3

, where HI the interaction part of the Hamiltonianwe can then describe the evolution of the Green operator under the interaction HI

and for the wavefunction under n interaction steps

From this we can extract the transition probability matrix (S-matrix for short)

From this, we calculate the differential cross section for the transition under the interaction HI

as the number of transitions per unit time per volume divided by the incoming particle fluxV

vvJ 21

dσ= , where Nf is the density of final states

We introduce the transition probability2

fiM in the center-of-mass frame

k p

iffifi

kVE

PPVTMS2

12)(

4422

and with this we get the formula for {1,2}→{3,4,...,N} process for the cross-section

, where the volume V cancels out

fiM is the Feynman integral for the process {1,2}→{3,4,...,N}

)()...()2(

...)2(

1

0

4

4

0

41

4

nn

lk pipipdpd

PI

with integration over internal momenta, where )( npi

is the propagator of the internal process

5 Feynman rules

QED rules

4

general QFT rulesExternal lines radiate from vertices and receive the following factors:(a) For spin-1/2 fermion of momentum p and spin state s• in initial state: u(p, s) on the right,• in final state: ¯u(p, s) on the left;(b) For spin-1/2 antifermion of momentum p and spin state s• in initial state: ¯v(p, s) on the left,• in final state: v(p, s) on the right;(c) For spin-0 boson• in either initial or final state: 1;(d) For spin-1 boson of helicity λ (if massless boson, λ = ±1; if massive boson, λ = 0,±1) • in initial state: εμ(λ), • in final state: ε*μ(λ).

Internal lines

each internal loop: integrate over

closed fermion loop: factor -1closed loop n bosons: factor 1/n!

QCD vertices

5

electroweak quark-boson vertices

electroweak lepton-boson vertices

6 Divergent graphs of QED

degree of divergence: ,where Eψ=number of external electron legs, EA=number of external photon legsddiv(Γ)=D: Γ is divergent with 1/ε(D+1)

divergent diagrams: 6 one-loop

)( p = D=1 , one-loop electron self-energy, momentum p ( pp Dirac dagger)

6

)',( pp = D=0 , one-loop electron-photon vertex, momemtum p, q, p’

)(k = D=2 , photon vacuum polarization, momentum k

D=1 , 3-photon vertex correction one-loop, momentum k, k’, k’’

D=1 , 3-photon vertex correction inverse one-loop, momentum k, k’, k’’

D=0 , 4-photon vertex correction one-loop, momentum k, k’, k’’,k’’’

Actual divergence only Γ1 , Γ2 , Γ3

Γ1 : =O(1/ε) Γ2 : =O(1/ε) Γ3 : =O(1/ε) Γ4 = -Γ5 cancel because of Furry’s theorem: Γ(EA )= -Γ(-EA ) if EA =odd

Γ6 is convergent because of 0kk

Mathematically speaking, we get for a divergent graph Γ a series in ε , where 0 is the regularizationparameter:

...)( 11

0

CC

C , which has a residuum C-1 at 0.

7 Renormalization of QEDFor the 3 divergent graphs Γ1 , Γ2 , Γ3 we introduce 4 infinite expressions Z2, Z1, Z3 , δm and the 4 fundamentalparameters ψ, Aμ, e, m are renormalized, i.e. made finite, with unrenormalized (infinite) parametersψ0, Aμ0, e0, m0 .We develop )( p around mp

, where )(m infinite and 1)(' m and set

)('1

12

mZ

)(mm

7

)(~

)( 3 kZk

)',(~

)',( 1 ppZpp , where the ~expressions are the finite parts.

Now we make a transformation from renormalized to unrenormalized parameters

and insert them into the lagrangian, making it unrenormalized

are the counterterms

yields finite graphs 321

~,

~,

~ , that is, )(

~0 jj C

The procedure of finding the appropriate counterterms , i.e. the appropriate transformation to theunrenormalized parameters using the infinite terms Z2, Z1, Z3 , δm is the essence of renormalization.Ward-Takahashi identitiesFrom the simple identity in γ-matrices

one can prove that

ppp

p,

)(

(Ward-Takahashi identity)

and from this followsZ1=Z2 , i.e. the renormalization factors of electron-photon vertex )',( pp and electron self-energy )( p are

equal.

8 Types of regularizationThere are basically 3 types of regularization, i.e. forming a series in ε for Feynman graphs: Pauli-Villars cut-off regularization, t’Hooft-Veltman dimensional regularization and lattice regularization.Pauli-Villars cutoff regularizationThis was the first widely used regularization scheme.For the high-energy limit (ultraviolet divergence) we cutoff the integrals by assuming the existence of a virtualparticle (ghost) of large mass M. The propagator is then modified by:

The propagator now behaves as 1/p4 , which renders all graphs finite. Then we let M→∞ , making thetransition to the original operator. This method preserves the gauge invariance and the Ward-Takahashiidentities, enabling cancellation of terms due to gauge symmetry.As an example, the inner photon contribution

becomes

))(()2(2)(

222

02

22

0

0

4

402

iMpimp

Mmpdipi

M

8

If we set

m

Mlog

1

, we get a series in ε identical to the dimensional regularization.

For the low-energy limit (infrared divergence) we modify the photon propagator by a cut-off mass μ

222

1),(

1)(

kkP

kkP and let 0 , the resulting divergence is of the form

mlog

1in

analogy to the high-energy limit.Dimensional regularizationThis method is widely used now, as it is analytically simpler than the cutoff method. It also preserves the gaugeinvariance and the Ward-Takahashi identities, enabling cancellation of terms due to gauge symmetry.We calculate n-dimensional integrals in spherical coordinates {r,φ1,..., φn-1}n-dim spatial angle

)2/(

2 2/

n

nn

n-dim integral in spherical coordinates

121

2

0

1

0

3

22

2

1

0 0

11 ,...,,...sinsin

nn

nnRr

r

n rfdddrdr

Now, we will find the rigorous mathematical definition for xd n

First, cartesian xi in spherical coordinates φ n=3: {x1, x2, x3}= {sin[φ2]cos[φ1], sin[φ2]sin[φ1], cos[φ2]}ݎn=4: {x1, x2, x3, x4}= ,sin[32]sin[φ2]cos[φ1]}ݎ sin[32]sin[φ2]sin[φ1], sin[φ3]cos[φ2], cos[φ3]}With the function

)1(2

2/)1()1(2

0

)1(2

sin)2/2()2/3(2

)2(sin

sin

1

)3(

)4(),(

d

fomega

)1(2sin)0),,(lim(

fomega

we can formulate the integral xd n

for n=4-ε correctly (so that it provides an analytic expression, which gives

the correct value for n=3 and n=4 )

121

2

0

3

0

221

0 0

13 ,...,,sin),(

n

Rr

r

rfddfomegadrdr

Lattice regularizationIn the lattice method a four dimensional hypercube lattice in space-time is introduced, and the integral iscalculated as a sum over lattice points. This method is numerical and can be also used for non-perturbative (notwith Feynman diagram series) calculations, but it violates gauge invariance and the Ward-Takahashi identities,cancellation of terms because of gauge symmetry is only approximative, and an integral can become thereforenumerically divergent of a higher degree, e.g. the photon vacuum polarization )(k is originally

quadratically divergent with D=2, but becomes logarithmically divergent with D=0 because of termcancellation).

We choose the lattice equidistant with upper limit M=Mvr m/ε , and number of intervals ]1

[

n , where [] is

the next higher integer and Mvr is the step adaptation factor. The lattice step constant is then mMn

Md vr

independent of ε , and four-dimensional lattice-hypercube has4

nN cells with cell volume4

4

n

Mv

9

We approximate the integral by

N

i ikfam

MSkfkd1vr

vr4 )()M()M

,()( by the sum over the lattice

points ki , where a(Mvr) is the integral adaptation factor .The Taylor-series of f(k) around a lattice point ki gives then the error formula

dfkfkfkdMSerN

i i )()()()),((1

4 for M=const., where ξ is a point within the lattice range.

That means for the ε-series of Sε(ε ,Mvr m/ ε) that both terms are shifted by a constant:

C-1(Sε(ε ,Mvr m/ ε))= C-1( )(4 kfkd )+c-1

C0(Sε(ε ,Mvr m/ ε))= C0( )(4 kfkd )+c0 .

The term C-1(Sε) can be calculated numerically from

),()

2/,2/()(1

mMS

mMSSC vrvr , the constant

term of the integral C0( )(4 kfkd ) from )0),(),(())(( 14

0 SCMSLimkfkdC numerically for

small ε with M kept constant.

9 Divergent one-loops in QED9.1 Electron self-energyWe discuss here the first divergent QED graph

Γ1= )( p = D=1 , one-loop electron self-energy, momentum p

Dimensional regularization

expressed as kd d ,

)( p =

or mathematically correct

22

02

0

2

0

3

0

22

)1(2

1

0 0

14

32

0)(

)(sinsin

16)(

kmkp

mkpddd

kdkeip

then kd d transformed into integral 1

0

dx (Feynman trick) and with substitution q=k- p x

= =and with d=4-ε ,

=which yields the ε-series coefficients

C-1(Γ1)= )4(8

02

2

0 mpe

C0(Γ1)/e02 ( with pd= , p2=p2)

=

10

which is practically =const*m0 for =m0, p2= m02 :

2 4 6 8 10

0.2

0.4

0.6

0.8

The renormalization factor becomes

1

21

1

81

2

2

02

eZ , Z1=Z2 because of Ward-Takashi identity.

Pauli-Villars cut-offWe introduce the cutoff at M by the replacement

222

02

22

222

022

02

111

Mpmp

Mm

Mpmpmp

and get for the cross section

2222

02

22

004

42

0)()(

))()((

16)(

kMkpmkp

Mmmkpkdep

If we set

0

log1

m

M

, we get the same ε-series as for the dimensional regularization, i.e. the same

coefficients C-1(Γ1) and C0(Γ1) .Lattice regularization

We get for )( p for ε=π/60 nε=20, i.e.4

nN , Mvr =0.1 ,with a CPU-time per lattice point tε=0.0108s ,

the following values:a(Mvr)=214.4 the integral adaptation factor , ),( vrMk =0.913 the numerical ε-power (should be 1).

Physical effectsThe electron self-energy renormalization factor contributes to the density distribution of the electron

200 Z , e.g. the effective mass of the electron in matter is [11]

obsm , where Ecut-off is the limit energy of the interaction.

The measured energy values in matter are then

, where the correction factor

C=This leads to the famous Lamb-shift (difference in energy between two energy levels 2S1/2 and 2P1/2 of thehydrogen atom)

, where nj is transition energy between the two levels.

In this case nj =1.041GHz, which was measured by W.E. Lamb and first calculated by Hans Bethe in 1947.

9.2 Electron-photon vertex

Γ2= )',( pp = D=0 , one-loop electron-photon vertex, momemtum p, q, p’

11Dimensional regularization

expressed as kd d ,

)',( pp =

or mathematically correct

22

022

02

002

0

3

0

22

)1(2

1

0 0

14

32

0)'()(

)()'(sinsin

16)',(

kmkpmkp

mkpmkpddd

kdkeipp

then kd d transformed into double integral x

dydx1

0

1

0

(double Feynman trick) and with substitution

k→k- p x-p’ y

)',( pp =

and with d=4-ε the result is up to O(ε)

2

1

12

''

3

1

22 2

0

22

m

ppppand therefore

2

1

12

''

3

1

228)',( 2

0

22

2

2

0

m

ppppepp

and the ε-series is

E

m

ppppepp

64

''

12

11

8)',( 2

0

22

2

2

0 , where γE=0.557 is the Euler-Mascheroni-number.

The renormalization factor becomes

1

21

1

81

2

2

01

eZ , Z1=Z2 because of Ward-Takashi identity.

Pauli-Villars cut-off

If we set

0

log1

m

M

, we get the same ε-series as for the dimensional regularization, i.e. the same

coefficients C-1(Γ2) and C0(Γ2) .Lattice regularization

We get for )',( pp for ε=π/60 nε=20, i.e.4

nN , Mvr =0.1 ,with a CPU-time per lattice point

tε=0.00371s , the following values:a(Mvr)=62.6 the integral adaptation factor , ),( vrMk =0.868 the numerical ε-power (should be 1).

Physical effectsThe electron-photon vertex correction contributes to the gyromagnetic factor of the electron:

0011614.1*22

12

S

Mg s

s , which is verified experimentally up to 10-10, making the magnetic

moment of the electron the most accurately verified prediction in the history of physics .

9.3 Photon vacuum polarization

Γ3= )(k = D=2 , photon vacuum polarization, momentum k

Dimensional regularization

expressed as kd d ,

12

)(k =

or mathematically correct

2

022

02

002

0

3

0

22

)1(2

1

0 0

14

32

0)(

))((sinsin

16)(

mkpmp

mkpmpTrddd

pdpeik

then kd d transformed into integral 1

0

dx (Feynman trick) and with substitution

p→q- k x ,

=and after cancellation of the first and the third term under the integral

)(k =-4

and with d=4-ε the result is up to O(ε)

)(k =

setting μ=m0 we get the ε-series:

C-1(Γ3)= )(6

2

2

2

0 kgkke

and with k2=k2

C0(Γ3)= )( 22

0 kgkke

C0(Γ3) in dependence on m0:

The renormalization factor becomes

1

3

21

1

61

2

2

03

eZ

Pauli-Villars cut-off

If we set

0

log1

m

M

, we get the same ε-series as for the dimensional regularization, i.e. the same

coefficients C-1(Γ3) and C0(Γ3) .Lattice regularizationWith the lattice regularization the cancellation of the first and third term under the integral above happens onlyin the limit, so we use the formula with cancellation, which is only logarithmically divergent:

13

2220

2

21

0

4

42

0))1((

))(1(2

24)(

xxkmq

kgkkxxqddxeik

We get for )(k for ε=π/80 nε=26, i.e.4

nN , Mvr =0.1 ,with a CPU-time per lattice point tε=0.00131s ,

the following values:a(Mvr)=204.4 the integral adaptation factor , ),( vrMk =0.99 the numerical ε-power (should be 1).

Physical effectsThe photon vacuum polarization correction contributes to the electrostatic potential of the electron [11]

, where the reduced Compton wavelength

This was confirmed experimentally at the TRISTAN experiment in Japan in 1997.

The effects of vacuum polarization become significant when the external field approaches:

10. Infrared divergence: vertex correction

A general interaction graph can have an infrared divergence (right above) [5].

If we modify the graph by a vertex correction with μ-modified photon propagator22

1),(

kkP , the

infrared divergence cancels out (left above).

We calculate below the μ-modified vertex correction ),',( ppc with the corresponding infrared divergence

and show, that it cancels out in the case of bremsstrahlung as the interaction graph above [1].

To show this, we will begin our calculation with the one-loop vertex correction:

),',( pp =

Our goal is to write this expression in the form:

sandwiched between u(p') and u(p), where F1 and F2 are the form factors that measure the deviation from the

simple , vertex. We will calculate explicit forms for these two form factors. We will find that F1 cancels

against the infrared divergence found in the bremsstrahlung calculation, giving us a finite result.

With and

14

we get

with

follows

where represents the Pauli-Villars contribution, where μ is replaced by Λ.

With this insertion, the integral converges for finite but large Λ. To perform this integration, we must do an

analytic continuation of the previous equation.

With

,

the result for F2 is

and

This gives us the correction to the magnetic moment of the electron

The result for F1 is:

15

compare

E

m

pppppp

64''

12

11

2)',(

2

0

22

with

4

2

0e ,

0

log1

m

M

, where M is the

high-energy cutoff

radiated energy

so

))'((log

2164

''

12

1log

2),',( 20

2

0

22

0

ppm

m

pppp

m

Mpp E

c

finally we get the correction factor

with

we recall that the bremsstrahlung amplitude was given by:

while the vertex correction graph yields for :

Adding these two amplitudes are added together, we find a finite, convergent result independent of μ2, as

desired. The cancellation of infrared divergences is thus shown for the bremsstrahlung. This can be shown for a

general interaction graph from above.

1611 The renormalization groupThe renormalization group equations are based on the plausible assumption that the physical theory cannotdepend on the mass μ which we use as the point of regularization. We have a multiplicative relation between

the vertex functions of the unrenormalized theory and the vertex functions of the renormalized theory

. But the unrenormalized vertex function is independent of the regularization point μ.

Let R represent some renormalization scheme. If Γ0 is an unrenormalized quantity and ΓR is the quantityrenormalized by R, then:

where Z(R) is the renormalization factor under R.For a different renormalization scheme R' we have

Then the relationship between these two renormalized quantities is given by:

Trivially, this satisfies a group multiplication law:Z(R", R')Z(R', R) = Z(R", R)where the identity element is given by:Z(R, R) = 1

The renormalization group equation

Let us make the following definitions (where we now take the limit as ε0):

we derive from the defining equationthe renormalization group equation

where g is the coupling constant, n is the order of g in β(g) , m is the mass, Zφ is thenormalization factor of the fermion wave function.We can solve this equation for β(g)and get for QED

β(g) , β(g)>0 , so the coupling constant increases with the energyFor QCD we obtain the expression

β(g) , where Cad =3 is the number of charges and 4Cf =6 is the number offermions in the interaction, so the coupling constant decreases with the energy, the theory is asymptoticallyfree.

17

References[1] M.Kaku, Quantum Field Theory, Oxford University Press 1993[2] J.D.Bjorken, S.D.Drell, Relativistic Quantum Mechanics, McGraw-Hill 1964[3] Quang Ho-Kim, Xuan-Yem Phan, Elementary particles and their interactions[4] Einan Gardi, Modern Quantum Field Theory, University of Edinburgh, Lecture 2015[5] Douglas Ross, Quantum Field Theory 3, University of Southampton, Lecture 2018[6] Joao Magueijo, Quantum Electrodynamics, Imperial College London, Lecture 2015[7] Andrey Grozin, Lectures on QED and QCD, [arxiv hep-ph/0508242], 2005[8] Robbert Rietkerk, One-loop amplitudes in perturbative quantum field theory, master thesis Utrechtuniversity, 2012[9] Yasumichi Aoki, Non-perturbative renormalization in lattice QCD, [arxiv hep-lat/1005.2339], 2010[10] Jim Branson, Quantum Physics, University of California San Diego, Lecture 2013[11] Wikipedia, Vacuum polarization, 2018