Embed Size (px)
Citation preview
QUANTUM RELATIVISTIC DYNAMICS
C. BECCHIDipartimento di Fisica, Universita di Genova,
Istituto Nazionale di Fisica Nucleare, Sezione di Genova,via Dodecaneso 33, 16146 Genova (Italy)
AbstractThe aim of these lectures is to describe a construction, as self-contained as possible, of the
dynamics of quantum fields.They are based on a short description of Haag-Ruelle scattering theory and of its relation
with LSZ theory and on an introduction to renormalization theory based on Wilson-Polchinskirenormalization group method which is compared with the subtraction method. The most im-portant results concerning e.g. Wilson operator product expansion and the origin of anomaliesare also briefly described.
1
1Lectures given at Prague in June 2007
1
Contents
1 Introduction 1
2 Difficulties with quantized fundamental fields 2
3 Construction of a relativistic scattering theory 4
4 Properties of the time-ordered functions 13
5 The L.S.Z. reduction formulae 15
6 The functional formalism and the Effective Action 18
7 The construction of the theory, the Euclidean Quantum Field Theory 24
8 The Functional Integral in Euclidean Quantum Field Theory 27
9 The Wilson Effective Action in Euclidean Quantum Field Theory 30
10 The Effective Proper Generator 36
11 The subtraction method 45
12 Bases of local operators in the subtraction scheme and the Wilson operatorproduct expansion 53
13 The Quantum Action Principle 57
A Diagrammatic Expansions 67
A The Wilson Action 71
B Bibliography 72
1 Introduction
Quantum field theory was born from a generalization of QED to other interactions. The maincharacters of the theory where:
1) a one-to-one correspondence between particles and fields, the vector potential for thephoton and the Dirac field for the electron.
2) the Lorentz invariance of a finite number of dynamical field equations with the fieldstransforming according to finite dimensional representation of SL(2C), the universal coveringof the Lorentz group.
3) the locality of the field equations.
1
The most famous extensions were the Fermi theory of weak interactions, where the neutrinofield was introduced and the Yukawa theory of strong interaction where a scalar field wasintroduced assuming the existence of a corresponding spin-less particle identified after thediscovery of the π meson.
This gave origin to a new branch of Physics, that of Elementary Particles. More recently,even though the name survived, the idea of the identification of relativistic dynamics with anelementary particle theory based on assumption (1) above was abandoned due to the discov-ery of a large number of new particles, among which many were unstable. The idea to giveup assumption (1) and formulate a theory involving more elementary fields is rather old (cf.e.g. W.Heisenberg) and eventually found a satisfactory fulfillment within the framework ofnon-abelian gauge theories and their application to strong interactions (QCD). In fact it wasunderstood that, keeping the three points, one is forced to introduce unphysical fields, and theconcept of confinement excluded the identification of strong interacting fundamental fields withparticle fields in the sense of QED.
The multitude of new particles suggested the idea of abandoning a theory based on a finitenumber of field equations and on a limited set of fundamental fields extending the concept ofparticle to any discrete eigenvalue of the mass, thus including the bound states, and introducingon the same ground a local operator for each particle (interpolating field). The hope was thepossibility of reducing the dynamics to direct relations among scattering amplitudes. Thispossibility was strongly supported by the successful formulation of a relativistic scatteringtheory by Haag and many others. This remains one of the basic foundation of relativisticquantum mechanics and will be the subject of next sections.
On the contrary the success of QCD and a better understanding of Renormalization Theorygave new strength to the original scheme where, however point (1) was replaced by the ideathat the dynamics be constructed in terms of fundamental fields and, in general, interpolatingfields be composite operators.
The point of view adopted in these lecture notes is coherent with the above reasoning. Westart from the relativistic scattering theory. Then we come to the dynamical construction basedon renormalized local field theory.
2 Difficulties with quantized fundamental fields
We consider the simplest case of a scalar field φ. Our metric choice is g00 = 1. It turns outthat φ is not an operator. This can be seen using the Lehmann spectral representation for thetwo point vacuum correlator:
< Ω|φ(x)φ(y)|Ω >=< Ω|φ(x− y)φ(0)|Ω >≡ C(x− y) , (1)
due to translation invariance. It is clear that C gives information about the properties ofthe vector state φ(x)|Ω >. Inserting into the correlator a complete set of states labeled bytheir total momentum ~p, their total mass M and further quantum numbers α , excluding thevacuum state since we assume here for simplicity < Ω|φ|Ω >= 0, one gets , without any loss of
2
generality:
C(x) =∑M,α
∫d3p < Ω|φ(x)|~p,M, α >< ~p,M, α|φ(0)|Ω >
=∑M,α
∫d3pe−ipMx| < ~p,M, α|φ(0)|Ω > |2 , (2)
where we have used translation invariance and defined by pM the four vector with time andspace components EM =
√p2 +M2, ~p.
Among the above states we shall distinguish single particle states, corresponding to discreteeigenvalues of the mass and scattering states. The single particle states are assumed to beuniquely identified by their momentum, mass and helicity. The single particle masses satisfy0 < m1 ≤ m2 ≤ ·· ≤ mNp . In the above formulae the sum over the mass value M should beunderstood as the sum over the the single particle state masses and the integral over those ofthe scattering states.
Eq.(2) can be further simplified using the fact that φ is a Lorentz scalar and hence, withthe chosen normalization of vector states, that is < ~p,M, α|~p′,M, α >= δ(~p− ~p′), one has:
< ~p,M, α|φ(0)|Ω >=
√M
EM< ~0,M, α|φ(0)|Ω > . (3)
Introducing the mass density function: ρ(M) = 16π3∑αM | < ~0,M, α|φ(0)|Ω > |2 which is the
sum of a finite number of Dirac deltas corresponding to the single particle, or bound, statesand a continuous part corresponding to the scattering states, we have:
ρ(M) =Np∑a=1
ρaδ(M −ma) + θ(M − 2m1)R(M) , (4)
so that:
C(x) =∫ ∞
0dMρ(M)
∫ d3p
16π3EMe−ipMx ≡
∫ ∞0
dMρ(M)∆(+)M (x) . (5)
Now it is apparent that φ(x)|Ω > has infinite norm, due to the divergence of the momentumintegral, and hence is not a state vector even if there is only a single discrete mass contribution.This difficulty is overcome by noticing that the identification of a single space point needs aninfinite amount of energy and hence the value of a field at a point must be necessarily ill defined.One can therefore consider the field smeared over a finite space region
∫d3r′χ(~r − ~r′)φ(~r′, t) ≡
φχ(~r, t) assuming χ real, supported by a small space region and of C∞ class, that is, infinitelydifferentiable.
In this case one has:
||φχ(~0, 0)|Ω > ||2 =∫ ∞
0dMρ(M)
∫ d3p
EM|χ(~p)|2 . (6)
Now the momentum integral converges since χ(~p) is a fast decreasing function at infinity dueto the smoothness of χ. However there remains the problem of the convergence of the M
3
integral which requires ρ(M) to vanish at infinity since one has asymptotically∫ dM
Mρ(M). As a
matter of fact for purely dimensional-scale-invariance reasons one expects ρ(M)M1−ε to vanishat infinity for any positive ε, since ρ(M) has the dimension of an inverse mass, and hence theabove norm is expected to be finite.
The above result is however not general enough. Had we considered the time derivative ofthe smeared field, which is of course an independent observable, we would have obtained aninfinite norm for φχ(~0, 0)|Ω >, same result, of course, for the D’Alambertian of the field.
Thus one must conclude that local operators must be smeared in space and time. We shallconsider for example: ∫
dt′∫d3r′χ(|~r − ~r′|)χ(t− t′)φ(~r′, t′) ≡ φχ(~r, t) , (7)
where χ can have either compact support, such as an interval, in which case the smearedoperator is called local, or be rapidly vanishing, the resulting operator being called almost-local. Mathematically this means that the basic fields must be considered operator valueddistributions.
As we shall see these smeared operators allow to construct a relativistic scattering theory.However they are of no help for the construction of an interacting theory. Indeed the interactingtheory must be based on strictly local field equations, that is, equations involving strictly localcomposite operators built in terms of products of fundamental fields at the same space-timepoint. We shall see in the following that to overcome this difficulty one must have recourse toa rather technical and sophisticated tool which is called Renormalization Group .
We now sketch the construction of scattering theory.
4
3 Construction of a relativistic scattering theory
We sketch Haag’s construction. Haag considers generalized local, or almost-local, operatorfields of the form:
Q(x) =∫dyf(x− y)φ(y) +
∫dy dzf(x− y, x− z)φ(y)φ(z) + ·· , (8)
where φ is the fundamental field and the coefficient functions are assumed C∞ and, eitherof compact support (denoted by D), or rapidly decreasing at infinity (denoted by S). Thepresence of terms non-linear in the fundamental field may be necessary to deal with boundstates. There is however the possibility, which is consistent with renormalization theory, tointroduce renormalized, strictly local (i.e. distributions) composite operators Φ(x), by analogywith the Wick ordered monomials of free field theory. The latter are monomials of the free fieldsand their derivatives at the same space-time point ordered shifting the annihilation operatorsto the right and the creation operators to the left; in this case the non linear terms in the aboveexpression could be unnecessary. The explicit construction of these operators is postponedafter the construction of Renormalization Theory. In general Φ is, either a fundamental, or acomposite, local field transforming according to a finite dimensional representation under theaction of the Lorentz group.
Having in mind the construction of scattering amplitudes it is convenient to introduce incorrespondence with the a-th single particle a local, and in general composite, field Φa(x) forany 1 ≤ a ≤ Np such that the matrix element < ~p, a|Φa(0)|Ω >≡ 1√
16π3Eaζa does not vanish.
If we assume that the number of discrete mass eigenvalues is finite we can, without loss ofgenerality, refine the choice of the Φa(x)’s by the requirement that their vacuum expectationvalues vanish together with the matrix elements < ~p, a|Φb(0)|Ω > for a 6= b and that |ζa| ≡ 1.Then, if we assume time reversal invariance, ζa ≡ 1. Therefore we have:
< ~p, a|Φb(0)|Ω >≡ 1√16π3Ea
δa,b , < Ω|Φa(0)|Ω >≡ 0 . (9)
Notice that the second condition is trivially satisfied if the field is not scalar. In the case of ascalar field the condition is implemented subtracting a constant, which is a trivial scalar field,from the field operator.
Studying the scattering theory, we limit our discussion to almost local operators of the form:
Qa(x) =∫dyf(x− y)Φa(y) , (10)
where f is a function of class D or S . For simplicity we shall consider only operators associatedwith scalar fields, and we shall deal only with scalar particles and scalar bound states. Howeverour analysis can be extended without major difficulties to particles with any spin. With oursimple choice the Lehmann representation (5) for the two-field vacuum expectation value thatwe call, as it is commonly done, 2-point Wightman functions holds true in general, however theexpected asymptotic behavior of the spectral function ρ(M) depends on the nature of the field.The commutator of two almost local operators is expected to vanish faster than any inversepower of their space distance if the time distance is kept fixed.
5
The physical Hilbert space is assumed to coincide with a Fock space of scattering states.The vacuum state |Ω > is assumed to be an isolated eigenvector of the mass operator, whichmeans that all the particles and bound states involved have positive mass larger than a givenmass gap m.
The core of Haag’s construction is the cluster property that we are going to describe andwhich is a natural consequence of the mass gap.
Les us consider the almost-local operators (10) and consider the long distance, fixed time,behavior of the 2-point Wightman function < Ω|Qa(x)Qb(0)|Ω >. Using an obvious general-ization of (5) we have:
< Ω|Qa(x)Qb(0)|Ω >=∫ ∞
0dMρa,b(M)
∫ d3p
16π3EM
∫dx′dy′fa(x− x′)
fb(−y′)eipM (x′−y′)+ < Ω|Qa(0)|Ω >< Ω|Qb(0)|Ω >
=∫ ∞
0dMρa,b(M)
∫ d3p
16π3EMe−ipMxfa(−pM)fb(pM)
+ < Ω|Qa(0)|Ω >< Ω|Qb(0)|Ω > (11)
where
ρa,b(M) = δa,bδ(M −ma) + θ(M −mNp)16π3M∑α
< Ω|Φa(0)|~0,M, α >< ~0,M, α|Φb(0)|Ω >
(12)is a symmetric matrix if the interpolating fields are Hermitian. We have introduced f(p) ≡∫dxe−ipxf(x) that is assumed to be of class S. It follows from the absence of mass-less particles
that ρa,b(~p,M)fa(−pM)fb(pM)/EM is C∞ in ~p and hence from the Riemann-Lebesgue lemmathe above 2-point function tends to < Ω|Qa(0)|Ω >< Ω|Qb(0)|Ω > faster than any inversepower of r for large r ≡ |~x|.
Notice the we have crucially exploited the mass gap condition. Indeed if the mass sumwould receive contributions from mass-less states and f would not vanish at ~p = 0 the aboveintegrand would not anymore belong to the C∞ class and hence the Riemann-Lebesgue lemmacould not be used.
As a matter of fact the two-point function for space-like distance r of the points vanishesat infinite distance as exp(−m1r).
The above result suggests the introduction of the truncated, or connected, Wightman func-tions. Up to 2-points we define:
< Ω|Qa(x)Qb(0)|Ω >≡< Ω|Qa(x)Qb(0)|Ω >C
+ < Ω|Qa(x)|Ω >C< Ω|Qb(0)|Ω >C
< Ω|Qa(x)|Ω >=< Ω|Qa(0)|Ω >≡< Ω|Qa(0)|Ω >C=< Ω|Qa(x)|Ω >C . (13)
In the second line we have use the translation invariance of the vacuum state. Comparing(11) and (13) we see that the connected 2-point Wightman function vanishes at fixed time andinfinite distance faster that any inverse power of the distance.
This result can be generalized to any n-point Wightman function. Let d be a purely space-like four-vector, d0 = 0 , d2 = −R2 , consider a set of n almost-local operators Qa , for
6
a = 1 · · , n. Let χσi be the characteristic function of a subset σ of the first n integers; χσi = 1if i belongs to σ, χσi = 0 otherwise. If R→∞ the n-point function:
< Ω|n∏a=1
Qa(xa + dχσa)|Ω >→< Ω|∏a∈σ
Qa(xa + d)∏a6∈σ
Qa(xa)|Ω > , (14)
faster that any inverse power of R since the operators at distance R commute up to negligiblecorrections . Notice that the operator products appearing in (14) and in the following formulaeare ordered with the index increasing from the left to the right-hand side. Now we can treatthe right-hand side of (14) as a 2-point function:
< Ω|∏a∈σ
Qa(xa + d)∏a6∈σ
Qa(xa)|Ω >
=∑α
∫ ∞m
dM∫d3p < Ω|
∏a∈σ
Qa(xa + d)|~p,M, α >
< ~p,M, α|∏a6∈σ
Qa(xa)|Ω >=∑α
∫ ∞m
dM∫d3p e−ipMd
< Ω|∏a∈σ
Qa(xa)|~p,M, α >< ~p,M, α|∏a6∈σ
Qa(xa)|Ω > . (15)
Taking into account the C∞ smothness in ~P of the product of matrix elements in the last lineof (15), much in the same way as for (11), we conclude that
< Ω|n∏a=1
Qa(xa + dχσa)|Ω >→< Ω|∏a∈σ
Qa(xa + d)|Ω >< Ω|∏a6∈σ
Qa(xa)|Ω > . (16)
As a matter of fact the study of the ~p dependence of the matrix elements requires an analysismore difficult than a simple use of Lorentz covariance. One must use the fact that in theabsence of mass-less particles the action of the Lorentz group on the Q’s is C∞ in the strongtopology.
Thus generalizing (13) we define implicitly the connected n-point functions by the followingcluster decomposition formula:
< Ω|n∏a=1
Qa(xa)|Ω >≡∑σ1
< Ω|∏a∈σ
Qa(xa)|Ω >C< Ω|∏b 6∈σ1
Qb(xb)|Ω >
=n∑k=1
∑Πj ,j=1,··,k
k∏j=1
< Ω|∏a∈Πj
Qa(xa)|Ω >C (17)
where σ1 runs over all the subsets of the first n integers containing the first one and the indexsets Πj for j = 1, · · · , k are partitions of the first n integers, the second sum above runningover all such partitions. Then the connected n-point functions vanish faster than any inversepower of the maximum space distance among the points if times are kept fixed.
This is the cluster property upon which Haag’s theory is based. For future convenience wecan translate our results into the functional language introducing a tool that will be very usefulin the following.
7
We associate to every almost-local operator Qa(x) a source J(x, θ) of class S where θ is avariable accounting for the position of the operator in the ordered product appearing in theleft-hand side of Eq.(17). Then we can introduce Θ as the θ-ordering operator and define thefunctional generator of the Wightman functions:
W (J) ≡< Ω|Θe∫d4xdθJ(x,θ)Q(x)|Ω >
≡∞∑n=0
< Ω|∫d4x1 · ·d4xn
∫ ∞−∞
dθ1J(x1, θ1)Q(x1)∫ θ1
−∞dθ2J(x2, θ2)Q(x2)
· · ·∫ θn−1
−∞dθnJ(xn, θn)Q(xn)|Ω > . (18)
If the connected n-point function generator W (J)C is defined in a completely analogous way,it is immediate to verify from (17) that:
W (J) = eW (J)C . (19)
Coming back to Haag’s construction we introduce positive/negative frequency solutions ofthe Klein-Gordon equation (∂2 +m2
a)ψa(x) = 0
ψ(s)a (x) =
1
(2π)32
∫ d3p√2Ema
ψ(~p)eispmax (20)
where sk is either + or −, ψ(~p) is of class D and support Sψ.We also introduce the corresponding generalized destruction (s = +)/creation (s = −)
operators;
B(s)ψa
(t) = is∫x0=t
d3x
[ψ(s)a (x)
∂
∂x0Qa(x)−Qa(x)
∂
∂x0ψ(s)a (x)
]≡ is
∫x0=t
d3xψ(s)a (x)∂↔0Qa(x) ,
(21)and Qa(x) =
∫dyfa(x− y)Φa(y) , with fa(Ema , ~p) = 1 on Sψ.
One can study the vacuum expectation value:
< Ω|m∏k=1
B(sk)ψak
(t)|Ω > . (22)
Taking into account that < Ω|B(s)ψb
(t)|Ω >≡ 0 for all b’s, due to our choice of the localfields, it turns out that in the limit |t| → ∞ (22) vanishes unless it contains the same numberof generalized creation and destruction operators. More precisely the asymptotic time limitof (22) reduces to the sum over all the possible pairings of B(+)’s and B(−)’s of the vacuumexpectation values of the pair products.
This result can be proven applying the cluster decomposition (17) to (22) and using thefact that ψ(±)
a (x) in (20) and its time derivative satisfy the following inequalities:
|ψ(±)a (x)| < A|x0|−3/2,
|ψ(±)a (x)| < AN
[|x0|+ |~x|]N, for any N if Sψ 3
|~x||x0|
. (23)
8
The contribution of a k > 2 point connected term in the cluster decomposition of (22) is:(k∏l=1
−sli)∫
x0i≡t
d3x1 · · · d3xk(ψ(s1)a1
(x1)− ψ(s1)a1
(x1)∂
∂x01
) · · ·
(ψ(sk)ak
(xk)− ψ(sk)ak
(xk)∂
∂x0k
) < Ω|Qa1(x1)Qa2(x2) · · ·Qak(xk)|Ω >C (24)
which can be decomposed in the sum of 2k terms of the form:
∫x0i≡t
k∏j=1
(d3xjψ(sj)aj
(xj)) < Ω|Qa1(x1)Qa2(x2) · · · Qak(xk)|Ω >C , (25)
where the functions ψ are solution of the Klein Gordon equation and satisfy (23) and the Qare quasi-local operators satisfying (10).
Now, changing integration variables and exploiting translation invariance, (25) can be writ-ten:
t3k∫ k∏
j=1
(d3vjψ(sj)aj
(~vjt, t)) < Ω|Qa1(~v1t, t)Qa2(~v2t, t) · · · Qak(~vkt, t)|Ω >C
= t3k∫d3v1ψ
(s1)a1
(~v1t, t)k∏j=2
(d3wjψ(sj)aj
((~v1 + ~wj)t, t))
< Ω|Qa1(~v1t, t)Qa2((~v1 + ~w2)t, t) · · · Qak((~v1 + ~wk)t, t)|Ω >C
= t3k∫d3v1ψ
(s1)a1
(~v1t, t)k∏j=2
(d3wjψ(sj)aj
((~v1 + ~wj)t, t))
< Ω|Qa1(~0, 0)Qa2(~w2t, 0) · · · Qak(~wkt, 0)|Ω >C . (26)
Due to the cluster property the connected vacuum expectation value vanishes faster than any
inverse power of the the maximum distance among the points, that is of√t2∑kl=2w
2l , (e.g. as
e−t2∑k
l=2w2l ). Thus, if the supports of ψai overlap the w-integrals give a contribution propor-
tional to |t|−3(k−1) times the absolute value of the product of ψ’s, which, on account of the firstinequality in (23) amounts to |t|−3k/2. Since the integral with respect to ~v1 is limited by (23) toa sphere of radius one, we find that (26) vanishes proportionally to |t|−3(k−2)/2. Notice howeverthat if the supports of ψ’s do not overlap (26) vanishes faster than any inverse power of |t| dueto the second inequality in (23). Notice that this is independent of the si’s, that is of the choiceof generalized creation or destruction operators.
Therefore, in the cluster decomposition of (22) one is left with 2-point connected terms only.Considering the 2-point cluster terms and referring for simplicity to (11) one has instead:
< Ω|B(s1)ψa1
(t)B(s2)ψa2
(t)|Ω >C
= (−s1s2)∫x0i≡t
d3x1d3x2(ψ(s1)
a1(x1)− ψ(s1)
a1(x1)
∂
∂x01
)(ψ(s2)a2
(x2)− ψ(s2)a2
(x2)∂
∂x02
)
9
< Ω|Qa1(x1)Qa2(x2)|Ω >C
= (−s1s2)∫x0i≡t
d3x1d3x2(ψ(s1)
a1(x1)− ψ(s1)
a1(x1)
∂
∂x01
)(ψ(s2)a2
(x2)− ψ(s2)a2
(x2)∂
∂x02
)∫d3p
∫ ∞0
dM
16π3EMρa1,a2(~p,M)fa(−pM)fb(pM)e−ipM (x2−x1)
= (−s1s2)∫d3p
∫ ∞0
dM
EMρa1,a2(~p,M)fa(−pM)fb(pM)
(EM + s1Ema1)(EM − s2Ema2
)
4√Ema1
Ema2
ψa1(−s1~p)ψa2(s2~p)e−i(s1Ema1
+s2Ema2)t . (27)
From the asserted C∞ smoothness of the functions ρa1,a2 (see Eq. (12)) and ψa and from theRiemann-Lebesgue lemma it turns out that (27) vanishes faster than any inverse power of |t|unless ma1 = ma2 and s1 = −s2 and the supports of the ψ’s overlap, in which case (27) it ist-independent.
Thus we conclude that the n-point functions of generalized creation/destruction operatorshave non-trivial asymptotic limit only if n is even and the number of creation operators equalsthat of destruction ones. In this case one has:
lim|t|→∞
< Ω|m∏j=1
B(sj)ψaj
(t)|Ω >= δm,2n2n∑i=2
δma1 ,maiδs1,−si
< Ω|B(s1)ψa1
B(si)ψai|Ω > lim
|t|→∞< Ω|
2n∏j=2,j 6=i
B(sj)ψaj
(t)|Ω > . (28)
In the special case in which the destruction operators lie on the left-hand side and their numberequals that of the creation operators, one has:
lim|t|→∞
< Ω|n∏i=1
B(+)ψai
(t)n∏j=1
B(−)ψaj
(t)|Ω >=∑πi
n∏i=1
δai.aπi < Ω|B(+)ψaiB
(−)ψaπi|Ω > , (29)
where πi is a permutation of the corresponding indices.This equation implies that in the state vector
∏mk=1B
(−)ψak
(t)|Ω > and in the limit |t| → ∞the order of the B(−) operators is immaterial. It is indeed immediate to verify that the norm ofthe difference of two such vectors with different ordering of the same operators vanishes in thelimit. If furthermore the supports of the ψ’s do not overlap the norm of the difference vanishesfaster than any inverse power of |t|.
It is possible to get a stronger result with almost local operators of the form:
Qma(x) ≡∫dyfma((x− y)2)Qa(y) , (30)
where fma(p2) ≡
∫dxfma(x
2)e−ipx has compact support contained in the region (p2 −m2a)
2< ε
and ε small enough to include only the four momenta of single particle states with mass ma
10
and satisfies fma(m2a) = 1 , The stronger result is due to the fact that each Qma(x) acting on
the vacuum creates a single particle state of mass ma:
Qma(x)|Ω >=∑M,α
∫d3p |~p,M, α >< ~p,M, α|Φa(0)|Ω > eipMx
f(pM)fma(M2) =
∫d3p
1√16π3Ea
|~p, a > eipmaxf(pma) (31)
and hence from Eq. (12) we get
ρa,b(~p,M) = δa,bδ(M −ma) . (32)
We label the corresponding generalized creation/destruction operators by: C(∓)ψa
(t) . The crucialnew point is that:
d
dtC
(−)ψa
(t)|Ω >= 0 , (33)
as it is easy to verify. This implies that;
d
dtΨψa1 ,··ψam (t) ≡ d
dt
m∏k=1
C(−)ψak
(t)|Ω >→|t|→∞ 0 (34)
since, the derivative of each factor of the operator product can be shifted to the right up toasymptotically vanishing contributions. Furthermore in the present case (28) becomes:
lim|t|→∞
< Ω|m∏j=1
C(sj)ψaj
(t)|Ω >= δm,2n2n∑i=2
δma1 ,maiδs1,+δsi,−
< Ω|C(s1)ψa1
C(si)ψai|Ω > lim
|t|→∞< Ω|
2n∏j=2,j 6=i
C(sj)ψaj
(t)|Ω >
= δm,2n2n∑i=2
δma1 ,maiδs1,+δsi,−
∫d3p ψa1(~p)ψai(~p) lim
|t|→∞< Ω|
2n∏j=2,j 6=i
C(sj)ψaj
(t)|Ω > . (35)
Thuslimt→±∞
Ψψa1 ,··ψam (t) = Ψ(out/in)ψa1 ,··ψam
, (36)
and the convergence rate is faster than any inverse power of |t| if the supports of the ψ’s donot overlap. Otherwise the rate is that of |t|−3/2
From (35) one can easily verify that:
〈Ψ(out/in)ψa1 ,··ψam
|Ψ(out/in)ψ′b1,··ψ′
bn
〉 = δn,m∑
πk,k=1,··,n
n∏j=1
δaj ,bπj
∫d3p ψ∗aj(p)ψ
′bπj
(~p) (37)
11
which implies that the asymptotic states Ψ(out/in)ψa1 ,··ψam
must be interpreted as scattering statesand can be written in the form:
|Ψ(out/in)ψa1 ,··ψam
>=n∏j=1
a(out/in)†ψaj
|Ω > , (38)
where a(out/in)†ψ are the creation operators in the Fock space of the in/out scattering states and
satisfy the commutation rules:[a
(out/in)ψ∗a
, a(out/in)†ψb
]= δa,b
∫d3p ψ∗a(~p)ψ
′b(~p) . (39)
It is a natural hypothesis of relativistic scattering theory that the finite linear combinationsof vectors (38) define a dense subset of the Hilbert space. This condition is called asymptoticcompleteness
What we have found until now leads to a relation between scattering amplitudes and Wight-man functions. However the explicit calculations rely more on the time-ordered functions, thatwe shall introduce in next section, than on Wightman functions. The bridge between time-ordered functions and scattering amplitudes is given by the LSZ reduction formulae. These arebased on a further important result of Haag’s scattering theory which comes from the study ofthe asympotic behavior of the matrix element:
limt→±∞
〈Ψ(out/in)ψ′a1
,··ψ′an|B(s)
ψ”c(t)|Ψ(out/in)
ψa1 ,··ψam〉 . (40)
This matrix element can be decomposed according:
〈Ψ(out/in)ψ′a1
,··ψ′an|B(s)
ψ”c(t)|Ψ(out/in)
ψa1 ,··ψam〉 = 〈Ψ′(out/in)
ψ′a1,··ψ′an
(t)B(s)ψ”c
(t)Ψ(out/in)ψa1 ,··ψam
(t)〉
+〈Ψ′(out/in)ψ′a1
,··ψ′an(t)|B(s)
ψ”c(t)|
[Ψ
(out/in)ψa1 ,··ψam
−Ψ(out/in)ψa1 ,··ψam
(t)]〉
+〈[Ψ′
(out/in)ψ′a1
,··ψ′an−Ψ′
(out/in)ψ′a1
,··ψ′an(t)]|B(s)
ψ”c(t)|Ψ(out/in)
ψa1 ,··ψam〉 . (41)
Now, using again (27), (28) and (29), and on the basis of asymptotic completeness, one can
verify that B(s)ψ”c
(t) is bounded by a positive power of E in the subspace of the Hilbert spacespanned by the scattering states with energy lower than E. It follows that the second and thirdterms in the right-hand side vanish in the asymptotic limit. Thus one finds that the limit (41)coincides with:
limt→±∞
< Ω|n∏j=1
C(+)ψ′aj
(t)B(s)ψ”c
(t)m∏i=1
C(−)ψbi
(t)|Ω > , (42)
From (35) and (29) one sees that this limit is given by:
δn,m+1δs,−∑πj
δc,aπ1< Ω|C(+)
ψ′aπ1
B(−)ψ”c|Ω >
n∏i=2
δaπi ,bi < Ω|C(+)ψ′aπi
C(−)ψbi|Ω >
+δm,n+1δs,+∑πi
δc,bπ1< Ω|B(+)
ψ”cC
(−)ψbπ1|Ω >
m∏i=2
δai,bπi < Ω|C(+)ψ′aiC
(−)ψbπi|Ω > .
(43)
12
Therefore one has:
limt→±∞
〈Ψ′(out/in)ψ′a1
,··ψ′an|B(s)
ψ”c(t)|Ψ(out/in)
ψa1 ,··ψam〉 = δs,−〈Ψ′(out/in)
ψ′a1,··ψ′an
|a(out/in)†ψ”c
|Ψ(out/in)ψa1 ,··ψam
〉
+δs,+〈Ψ′(out/in)ψ′a1
,··ψ′an|a(out/in)ψ”∗c
|Ψ(out/in)ψa1 ,··ψam
〉 . (44)
Taking into account asymptotic completeness we find that we can replace either Ψ′(out/in)ψ′a1
,··ψ′an,
or Ψ(out/in)ψa1 ,··ψam
with a generic vector of a finite energy subspace of the Hilbert space and hencewe have:
limt→∞〈Ψ′(out)ψ′a1
,··ψ′an|B(s)
ψ”c(t)|Ψ〉 = δs,−〈Ψ′(out)ψ′a1
,··ψ′an|a(out)†ψ”c|Ψ〉
+δs,+〈Ψ′(out)ψ′a1,··ψ′an
|a(out)ψ”∗c|Ψ〉 . (45)
and
limt→−∞
〈Ψ′|B(s)ψ”c
(t)|Ψ(in)ψa1 ,··ψam
〉 = δs,−〈Ψ′|a(in)†ψ”c|Ψ(in)
ψa1 ,··ψam〉
+δs,+〈Ψ′|a(in)ψ”∗c|Ψ(in)
ψa1 ,··ψam〉 , (46)
with fast convergence in the case of non-overlapping momentum wave functions.This weak asymptotic limit result is the basis of the L.S.Z. construction of the scattering
amplitudes that we shall describe in next sections.
13
4 Properties of the time-ordered functions
We have thus shown how asymptotic states can be built in field theory. This implicitly allowsthe construction of scattering amplitudes from Wightman functions. It turns out however thatconstructive field theory is better formulated in terms of time-ordered functions, even if, as weshall see in a moment, these functions are more difficult to define than Wightman functions.
An n-point time-ordered function is formally defined in terms of Wightman n-point functionsby the following formula:
< Ω|T (n∏a=1
Φa(xa))|Ω >≡∑
πi,i=1,··,n
n−1∏i=1
θ(x0πi− x0
πi+1) < Ω|Φπ1(xπ1) · ·Φπn(xπn)|Ω > , (47)
where, as above, πi labels the permutations of indices. Since the Wightman functions of strictlylocal operators (fields) are distributions and θ’s are discontinuous, in general the above formuladoes not make sense in general even in the framework of distribution theory. We consider, forexample the case of two points.
τa,b(x) ≡< Ω|T (Φa(x)Φb(0))|Ω >= θ(x0) < Ω|Φa(x)Φb(0)|Ω >
+θ(−x0) < Ω|Φb(0)Φa(x)|Ω >=∫ ∞m
dM∫ d3p
2EM
[e−ipMxθ(x0)ρa,b(M) + eipMxθ(−x0)ρa,b(M)] ≡∫ dq
(2π)4e−iqx τa,b(q
2)
= −i∫ ∞m
dMρa,b(M)∫ dq
(2π)4
e−iqx
M2 − q2 − iε, (48)
where we have used (5) and the fact that ρa,b(M) is a symmetric matrix due to T-invariance.Now, in order the last equation to make sense for distributions, the mass integral must convergeat infinity and this depends on the dimension of the fields. If e.g. they have dimension two itis expected that, for large M , ρ(M) ∼M and hence the mass integral does not converge. Thisimplies that (48) does not define a distribution. A deeper discussion of this point is in orderhere. In order that τa,b(x) be a distribution, the integral τa,b[f ] ≡
∫dxf(x)τa,b(x) with f of
class D should be well defined, this corresponds to the condition that the M -integral in:
τa,b[f ] =∫dqf(q)τa,b(q
2) = −i∫ ∞m
dMρa,b(M)∫dq
f(q)
M2 − q2 − iε, (49)
be absolutely convergent. Notice that the q-integral defines a bounded function of M2 decreas-ing as 1/M2 at infinity.
In the situation under discussion this is not true; however let us multiply f by xµ, we have:
τa,b[xµf ] = −2
∫ ∞m
dMρa,b(M)∫dqqµ
f(q)
(M2 − q2 − iε)2, (50)
which is well defined. Therefore we can conclude that τa,b is ill defined only in the origin.However one can give an alternative definition of the two point function substituting
τa,b(q2) = −i
∫ ∞m
dMρa,b(M)
M2 − q2 − iε
14
with:
τR,a,b(q2) = K − i
∫ ∞m
dMρa,b(M)
M2[
M2
M2 − q2 − iε− 1]
= K − iq2∫ ∞m
dMρa,b(M)
M2
1
M2 − q2 − iε, (51)
where K is an arbitrary constant . Now the mass integral converges and it is easy to verify
that, whenever∫∞m dM
ρa,b(M)
M2 converges the difference between τR and τ is just a constant andhence τa,b and τR,a,b coincide everywhere except in the origin. If we define
τS,a,b(x) ≡ −i∫ dq
(2π)4e−iqx
∫ ∞m
dMρa,b(M)
M2
1
M2 − q2 − iε, (52)
it is apparent that τS,a,b(x) is a distribution and one has:
τR,a,b(x) = Kδ(x)− ∂2τS,a,b(x) , (53)
which is also a distribution.This example shows which are the difficulties related with the definition of T-ordered func-
tions and in particular that these difficulties come from their lack of definition when two ormore point coincide. Finally it shows the need of supplementary conditions to identify theT-ordered functions completely as distributions in R4n. It turns out that the construction ofa consistent set of T-ordered functions in the case in which the fields are identified with theWick monomials of some fundamental free field coincides with the perturbative renormalizationprogram. Postponing any further discussion of the construction of T-functions, we assume tohave a consistent definition of those involving the local fields defined in (9) and we proceed inthe costruction of the scattering amplitudes using the LSZ reduction formulae.
15
5 The L.S.Z. reduction formulae
Given a set of fields Φai we build a corresponding set of almost local operators Qai(x) =∫dyfai(x − y)Φai(y) , where the functions fai(x) belong to the class D and their support is
contained in the slice |x0| < ∆. Then we select a set ψsiai of solutions of the Klein-Gordon
equation as in (20) with ψai(~p) of class D and non-overlapping support and we consider:
Aψ
(s1)a1
,··,ψ(sn)an
≡ in∫ ∏
j
dyj
∫Γ
∏i
(dxiψ(si)ai
(xi)(∂2xi
+m2ai
)fai(xi − yi))
< Ω|T (n∏k
Φak(yk))|Ω > , (54)
where the integration domain Γ is defined by: |x0k| ≤ Tk with Ti − Tl ≥ 2(l − i)∆.
Noticing that:
ψ(si)ai
(xi)(∂2xi
+m2ai
)fai(xi − yi)= ∂xµi
(ψ(si)ai
(xi)∂xµifai(xi − yi)− fai(xi − yi)∂x0iψ(si)ai
(xi)), (55)
one identifies (54) with:
in−∑
sl < Ω|T (n∏k
[B(sk)ψak
(Tk)−B(sk)ψak
(−Tk)])|Ω > , (56)
where the generalized creation and destruction operators B(sk) are defined in (21) and the timeorder refers to the ±Tk variables. Now, using (45) and (46) and taking into account that thevectors
T (n∏k=2
[B(sk)ψak
(Tk)−B(sk)ψak
(−Tk)])|Ω >
and [T (
n∏k=2
[B(sk)ψak
(Tk)−B(sk)ψak
(−Tk)])]†|Ω >
belong to a finite energy subset of the Hilbert space, we have:
limT1→∞
Aψ
(s1)a1
,··,ψ(sn)an
= limT1→∞
in−∑
sl < Ω|B(s1)ψa1
(T1)T (n∏k=2
[B(sk)ψak
(Tk)−B(sk)ψak
(−Tk)])|Ω >
− limT1→∞
in−∑
sl < Ω|T (n∏k=2
[B(sk)ψak
(Tk)−B(sk)ψak
(−Tk)])B(s1)ψa1
(−T1)|Ω >
= δs1,+in−∑
sl < Ω|a(out)ψa1
T (n∏k=2
[B(sk)ψak
(Tk)−B(sk)ψak
(−Tk)])|Ω >
−δs1,−in−∑
sl < Ω|T (n∏k=2
[B(sk)ψak
(Tk)−B(sk)ψak
(−Tk)])a(in)†ψa1|Ω > . (57)
16
Next step is the iteration of this equation with the assumption that the wave packets ψ(+) donot overlap with the wave packets ψ(−) and reminding the above made choice: fai(Ema , ~p) = 1
on the support of the corresponding ψai . In the asymptotic limits for all Tk the B(s)ψ at positive
time commute among themselves since, either they have the same s and hence they have thesame creation/destruction nature, or the corresponding wave packets do not overlap. Thereforeone remains with B(−)’s on the left-hand side and −B(+)’s on the right-hand side in the vacuumexpectation value. Taking into account Eq’s(45) and (46) one gets:
limTn→∞
· · · limT1→∞
Aψs1a1,··,ψsnan
=< Ω|
∏i : si=+
a(out)ψ∗ai
∏j : sj=−
a(in)†ψai
|Ω > , (58)
that is a transition (scattering) amplitude.Now it is apparent from this formula that the order of limits is immaterial. Taking into
account the fast convergence rate we can write the identity:
in∫ ∏
j
dyj
∫ ∏i
(dxiψsiai
(xi)(∂2xi
+m2ai
)fai(xi − yi)) < Ω|T (n∏k
Φak(yk))|Ω >
=< Ω|
∏i : si=+
a(out)ψ∗ai
∏j : sj=−
a(in)†ψai
|Ω > , (59)
where the xi integrals cover the whole R4n.Computing cross sections of a process with 2 particles in the initial and f particles in the
final state, that is n = f + 2, one more comment is in order. Decomposing the time-orderedfunction in the left-hand side of (59) into connected parts one finds a momentum conservationconstraint for each connected part; it follows that there is a single contribution to the transitionamplitude which comes from the connected (f+2)-point function with s1 = s2 = − and sj = +for j ≥ 2. The transition amplitude is given by:
if+2∫ 2∏
j=1
(dyjψ(−)aj
(xj))∏i>2
(dxiψ(+)ai
(xi)) < Ω|T (f+2∏k=1
(∂2xk
+m2ak
)Φak(xk))|Ω >C
≡ A2→f . (60)
The explicit computation of cross sections is made in the limit of perfect resolution which canbe simply defined choosing Gaussian wave packets and forgetting, since irrelevant, the factthat their Fourier transforms are also Gaussian and hence do not meet the compact supportcondition. In this situation one can choose the wave packets
ψ(~p) =1
(√π∆)3/2
e−(~p−~p0)2
2∆2 (61)
whose perfect resolution limit corresponds to ∆→ 0 and:
ψ(~p)→ (√
4π∆)3/2δ(~p−~p0) . (62)
17
A further simplification consists in choosing the same ∆ for all the wave packets. FollowingBecchi and Ridolfi (cited work), one can define the production probability of f final particlesin a certain region of their momentum space through the production probability density whichis defined as the perfect resolution limit
W ≡ lim∆→0
|A2→f |2
(√
4π∆)3f(63)
and one can show that;W = L dσ , (64)
where L is the integrated luminosity of the initial state, which, of course depends on the initialwave packets, and dσ is the differential cross section. As a matter of fact W and L are bothproportional to ∆2
18
6 The functional formalism and the Effective Action
Assuming that all needed time-ordered functions are defined, in analogy with the Wightmanfunction case one can define the functional generator:
Z[J ] ≡∞∑n=0
in
n!
∑a1,··,an
∫ n∏j=1
(dxjJaj(xj)) < Ω|T (n∏k
Φak(xk))|Ω >
≡∞∑n=0
in
n!< Ω|T (
∑a
∫dxJa(x)Φa(x))n|Ω >, (65)
where it is understood that the ordering operator acts before time integration. Notice that,contrary to the Wightman case, in the present case there is no operator ordering problem sincethe time-ordered functions are symmetric functions. The functional generator is a formal powerseries of the J ’s that are chosen of class S and (65) can be formally written;
Z[J ] =< Ω|T (ei∑
a
∫dxJa(x)Φa(x))|Ω > . (66)
In analogy with (19) the cluster decomposition of the time-ordered functions can be describedintroducing the connected generator:
ZC [J ] ≡∞∑n=2
in−1
n!< Ω|T (
∑a
∫dxJa(x)Φa(x))n|Ω >C , (67)
where we have taken into account that the fields have null vacuum expectation value. From(65) we can show that:
Z[J ] = eiZC [J ] . (68)
Indeed, on account of (17), taking a functional derivative of (65) comparing it with that of (66)we have:
δZ[J ]
δJa(x)=∞∑n=0
in+1
n!< Ω|T (Φa(x)(
∑a
∫dxJa(x)Φa(x))n)|Ω >
=∞∑n=0
in+1
n!
n∑m=1
n!
m!(n−m)!< Ω|T (Φa(x)(
∑a
∫dxJa(x)Φa(x))m)|Ω >C
< Ω|T (∑b
∫dyJb(y)Φb(y))n−m|Ω >
=∞∑m=1
im+1
m!< Ω|T (Φa(x)(
∑a
∫dxJa(x)Φa(x))m)|Ω >C
∞∑n=0
in
n!< Ω|T (
∑a
∫dxJa(x)Φa(x))n|Ω >= i
δZC [J ]
δJa(x)Z[J ] . (69)
This is a first order differential equation whose solution with initial conditions Z[0] = 1 , ZC [0] =0 is given by Eq. (68).
19
Given the connected generator, which is also a formal power series in J , one defines itsLegendre transform as follows (See Appendix A). One introduces the functional valued distri-bution:
Φa(x, J) ≡ δZC [J ]
δJa(x). (70)
Then one considers the Fourier transformed connected two-point function:
τ(2)a,b (p2) ≡
∫dxeipx < Ω|T (Φa(x)Φb(0))|Ω >≡≡
∫dxeipxτ
(2)a,b (x) . (71)
If τ(2)−1a,b (p2) is a matrix valued tempered distribution, in the free case it is a polynomial, one
can define a further functional valued distribution Ja(x, ϕ) formal power series in J such that;
Φa(x, J(·, ϕ)) ≡ ϕa(x) . (72)
Indeed from Eq.(70) one has:
Φa(x, J) =∫dyτ
(2)a,b (x− y)Jb(y) +O(J2) , (73)
this is a formal power series relation which implies, on account of the implicit function theorem:
Ja(x, ϕ) =∫dy τ
(2)−1a,b (x− y) ϕb(y) +O(ϕ2) (74)
where the higher order terms in ϕ are recursively defined through Eq. (72).Then one defines the formal power series Legendre transform of ZC ;
Γ[ϕ] ≡ ZC [J(·, ϕ)]−∫dx∑a
ϕa(x)Ja(x, ϕ) . (75)
The new functional is the effective action in the following sense. Taking its functional derivativeone has:
δΓ[ϕ]
δϕa(x)= −Ja(x, ϕ) , (76)
and henceδΓ[Φ(·, J)]
δϕa(x)= −Ja(x) , (77)
thus Φ(x, J) is the solution of the classical field equation induced by Γ and vanishing at J = 0.The inverse equation to (75) is:
ZC [J ] = Γ[Φ(·, J)] +∫dx∑a
Φa(x, J)Ja(x) . (78)
Now the above functional contruction is justified noticing that (60) is equivalent to:
A2→f = i∫ f+2∏
i=1
(dxiψsiai
(xi)(∂2xi
+m2ai
)δ
δJai(xi))ZC [J ]|J=0 . (79)
20
It can be shown (this point is discussed in some details by Becchi and Ridolfi, (cited work))that, defining:
ψ(as)a (x) ≡
f+2∑i=1
δai,aψsiai
(x) , (80)
one can write in the limit of perfect resolution of the wave packets, that is, wave packets withpoint-like support in momentum space:
A2→f = ie∫dx∑f+2
i=1ψsiai
(x)(∂2x+m2
ai) δδJai (x)ZC [J ]|J=0
= ie∫dx∑
aψ
(as)a (x)(∂2
x+m2a) δδJa(x)ZC [J ]|J=0
≡ ieΣZC [J ]|J=0 . (81)
Indeed the terms which are not linear in each ψsiai do not contribute in the perfect resolutionlimit. Let us sketch a proof of this fact.
For a generic choice of particle average momenta, ~p0 in Eq. (61), translation invariance,implying global energy-momentum conservation, selects in Eq. ( 81) contributions of the samedegree m in all the f +2 wave packets ψsiai . Indeed, for a generic choice of the particle momentain the 2→ f process, each f + 2-particle subset has vanishing average total energy-momentumwhile it does not contain any subset with vanishing average total energy-momentum. Them(f + 2)-particle contributions correspond to matrix elements for transition processes with 2minitial states to mf final states . Among the m× 2→ m× f amplitudes one has contributionscorresponding to processes which factorize into m factors corresponding to 2 → f particlestransitions times m− 1 factors corresponding to forward elastic scattering between initial andfinal particles of different subsets (the diagrammatic expansions of the Feynman amplitudesare discussed in Appendix A). In this situations the average energy-momentum of forwardscattering particles are on the mass-shell and hence there are 2(m− 1) mass-shell singularitiesdue to vanishing denominators in 2(m− 1) propagators. For example consider the case shownin the figure below in which one has two 2→ 2 particle processes and one final particle of thefirst process forward scatters a final particle of the second process.
&%'$
@@I
@@I
~p1 ~p2
~p3 ~p4
&%'$
@@I
@@@I
~p1 ~p2
~p3 ~p4
@I ~p3 ~p4
The corresponding diagrams contain two lines connecting each 2→ 2 process sub-diagramto the forward scattering sub-diagram, the small circle in the figure. The average energy-momenta carried by these two lines lie on the corresponding particle mass-shell. However thisdoes not mean that the amplitude diverges since the initial and final momenta are integratedover the support of the wave functions. This means in particular that the singularity associatedwith one line is smeared by the integration over the angle between the average line momentum
21
and its fluctuations which are of order ∆. Hence one has for each line an integral analogous to∫ 1−1 d cos θ/(a∆ cos θ + b+ iε) ∝ ∆−1 .
Taking into account this consideration one concludes that the dominant contributions in the∆ → 0 limit to the m× 2 → m× f amplitudes come from the just mentioned diagrams sincethese are those with the maximum number of lines with average momentum on the particlemass-shell. Therefore a generic m× 2→ m× f amplitude scales in the limit proportionally to∆(2+f)3m/2−2m = ∆3mf/2+m. It is therefore clear that the term with m = 1 dominates in theperfect resolution limit.
Thus we have reached the following conclusion, if we define:
Φa(x) ≡ eΣΦa(x, J)|J=0 , (82)
using Eq.(78) we can write the 2→ f amplitude (81) in the form:
A2→f = i[Γ[Φ] +∫dx∑a
ψ(as)a (x)(∂2
x +m2a)Φa(x)] . (83)
In order to understand the meaning of Eq. (83) let us consider Φ more closely.
eΣ δΓ[Φ(·, J)]
δϕa(x)|J=0 =
δΓ[Φ]
δϕa(x)= −eΣJa(x)|J=0
= −∫dy∑a
ψ(as)a (y)(∂2
y +m2a)δ(x− y) =
∑a
(∂2x +m2
a)ψ(as)a (x) = 0 . (84)
Thus Φ satisfies the field equation induced by the effective action Γ. Therefore Eq. (83)identifies the transition amplitude with the value taken by the modified effective action Γ[ϕ] +∫dx∑a ψ
(as)a (x)(∂2
x+m2a)ϕa(x) on the solution Φ of the effective field equation. This solution is
uniquely identified once its asymptotic properties, that is, the boundary condition to Eq.(84),are given. It remains to discuss these asymptotic properties of Φ.
Since Φ is a distribution we should better discuss the asymptotic properties of Qa(x) =∫dyfa(x − y)Φa(y) however selecting, as above, fa(p) = 1 on the intersection of the particle
mass-shells with the support of∫d3xψ(as)
a (x) exp(isap · x)|x0=0 .Taking into account the reduction formulae we have:
Qa(x) = eΣ∫dyfa(x− y)
δ
δJa(y)ZC |J=0
=2∑q=0
f∑m=0
∑k1<··<km=3,··,f+2
∑l1<··<lp=1,2
< Ω|m∏j=1
a(out)ψ∗akj
Qa(x)p∏i=1
(a(in)ψai
)†]|Ω >C
+Ra(x) , (85)
where Ra(x) accounts for the terms at least quadratic in one of the ψa’s. Considering theasymptotic limit of the first term in the right-hand side of (85) we notice that this is the sum ofa finite number of matrix elements of Qa(x) between out-going and in-going scattering states.
22
Let us use < out|Qa(x)|in > as a symbol for the generic matrix element. It is shown in Eq.(44) that:
limt→±∞
is∫ d3x
(2π)3/2
∫ d3p√2Ea(~p)
f(~p)eispa·x∂↔0 < out|Qa(x)|in >C
= δs,+ < out|a(out/in)f∗,a |in >C +δs,− < out|(a(out/in)
f,a )†|in >C . (86)
Therefore, selecting s = − in the limit t → ∞ one finds that asymptotically the right-handside of Eq. (85) converges to:
2∑q=0
f∑m=0
∑k1<··<km=3,··,f+2
∑l1<··<lp=1,2
< Ω|m∏j=1
a(out)ψ∗akj
(a(out)f,a )†
p∏i=1
(a(in)ψai
)†]|Ω >C . (87)
This is related to a sum of scattering amplitudes between a q-particle initial states and am−1-particle final states corresponding to all the possible choices of q initial and m final singleparticle wave functions. On account of energy-momentum conservation it is immediate to verifythat, if the 2 → f transition is allowed and initial and final single particle wave functions donot overlap, the only possibly non-vanishing matrix elements are those with q = 0 and m = 1.That is, one is left with:
f+2∑k=3
< Ω|a(out)ψ∗ak
(a(out)f,a )†|Ω >C=
f+2∑k=3
δa, ak
∫d3pf(~p)ψak(~p) . (88)
In much the same way if we select s = + in the limit t→ −∞ the first term in the right-handside of Eq. (85) converges to:
2∑q=0
f∑m=0
∑k1<··<km=3,··,f+2
∑l1<··<lp=1,2
< Ω|m∏j=1
a(out)ψ∗akj
a(in)f∗,a
p∏i=1
(a(in)ψai
)†]|Ω >C . (89)
Once again, taking onto account energy-momentum conservation, one is left with:
2∑l=1
< Ω|a(in)f∗,a(a
(in)ψal
)†]|Ω >C=2∑l=1
δa,al
∫d3pf(~p)ψal(~p) . (90)
Therefore we have shown that
is∫ d3x
(2π)3/2
∫ d3p√2Ea(~p)
f(~p)eispa·x∂↔0Qa(x) (91)
converges for s = − in the limit t→∞ and s = + in the limit t→ −∞ to
is∫ d3x
(2π)3/2
∫ d3p√2Ea(~p)
f(~p)eispa·x∂↔0ψ(as)a (x) (92)
23
for any f(~p) of class C∞ and with compact support. These boundary conditions are completelyequivalent to those satisfied by the classical fields in the semi-classical approximation (see againBecchi and Ridolfi (cited work))
Therefore the asymptotic behavior of first term in the right-hand side of Eq. (85) is thatof a linear combination of asymptotic wave packets, that is ∝ ∆3/2 . The second term Ra(x) isby definition non-linear in some wave packets and hence if one deals with it in much the sameway as with the first term one finds results vanishing faster than the first term than ∝ ∆3/2 .Thus these terms can be disregarded. It is clear that also in the analysis of Ra(x) one couldencounter mass singularities, however, repeating the above analysis one finds that the masssingularities cannot compensate the vanishing degree of the further wave packet factors.
In conclusion, through (85), (88) and (90) we have shown that, up to terms not contribut-ing to the transition amplitude in the perfect resolution limit, Φa(x) satisfies the classical fieldequation corresponding to the effective action Γ and the same boundary conditions . If fur-thermore we consider the expression of the transition amplitude given by (83) in terms of Γ weconclude that the fully quantized scattering theory coincides with its semi-classical counterpartprovided one replaces of the action with the effective action.
24
7 The construction of the theory, the Euclidean Quan-
tum Field Theory
Having so concluded the presentation of Haag-Ruelle relativistic scattering theory we shall de-scribe in the rest of these notes a construction procedure for the connected functional generator(67). Let us first of all consider a free scalar field theory; it is an elementary exercise to provethat:
ZC [J ] =1
2
∫dxdy∆F (x− y)J(x)J(y) , (93)
with
∆F (x) =∫ dx
(2π)4
e−ipx
m2 − p2 − i0+(94)
coincides with the solution of the differential equation:
(m2 + ∂2)δZC [J ]
δJ(x)= J(x) , (95)
with the condition that ZC vanishes at J = 0 and that positive/negative frequency components
of δZC [J ]δJ(x)
vanish in the limit x0 → ∓∞.It was noticed by Symanzik that the above equation, and hence its generalization to non
free theories is highly simplified if one turns to its Euclidean version. In the present situationthis is obtained through the analytic continuation of the Fourier transformed 2-point function∆;
∆(p) ≡∫dxeipx∆(x) =
1
m2 − p2 − i0+
=1
(Em(~p)− i0+ − p0)(Em(~p)− i0+ + p0), (96)
to the domain p0 = eiθp4 with 0 ≤ θ < π and in particular to the Euclidean points whereθ = π/2. The Fourier transform of the analytic continuation to the Euclidean space:
S(x) =∫ d4p
(2π)4
ei∑4
j=1xjpj
m2 +∑4j=1 p
2j
(97)
is called Schwinger 2-point function S(x1 − x2).Defining
FC [J ] =1
2
∫dxdy S(x− y)J(x)J(y) , (98)
one sees immediately that the equation corresponding to (95) is
(m2 − ∂2)δFC [J ]
δJ(x)= J(x) , (99)
and that it identifies FC [J ] with the solution of (95) vanishing at J = 0 and at infinity of R4.
25
It is clear from (96) that the analytic continuation to imaginary energies of the Fouriertransformed 2-point, time ordered, function is allowed due to the fact that it has singularitiesin the points ±(Em(~p)−i0+) and, if the theory is massive, Em(~p) ≥ m > 0 for some m. It turnsout that this is a general property of the energy singularities of the Fourier transformed n-pointtime-ordered functions of interacting theories, and hence one can relate the n-point Schwingerto time-ordered functions in much the same way as the two-point functions. This is a veryimportant point since the analytic continuation corresponds to the equivalence of Minkowskianand Euclidean (massive) field theories and, as we shall see in a moment, the construction ofthe Euclidean theory appears simpler than that of the Minkowskian theory.
A clear, however naively formal, way of understanding the possibility of the analytic con-tinuation starts from the decomposition of the time-ordered functions given in (47) that wediscuss, for simplicity, in the case of a single scalar field. Computing the Fourier transform of(47) we get:
τ(p1, · · ·, pn)|∑ pi=0 =∫ n−1∏
i=1
dxiei∑n−1
l=1pl·(xl−xn) < Ω|T (Φ(x1) · ·Φ(xn))|Ω > (100)
=∑
πi,i=1,··,n
∫ n−1∏i=1
(dxπiθ(x0πi− x0
πi+1))e
i∑n−1
k=1
(∑k
j=1pπj ·(xπk−xπk+1
)
)< Ω|Φ(xπ1) · ·Φ(xπn)|Ω >
where it is apparent that the contribution from every permutation does not depend on xπnsince the Wightman functions are translation invariant. Introducing the 3-dimensional Fouriertransformed Wightman functions:∫ n−1∏
i=1
d~xie−i∑n−1
k=1~pk·(~xk−~xn) < Ω|Φ(x1) · ·Φ(xn)|Ω >=
∫ ∞m
n−1∏i=1
dEi ρn(~p, E)e−i∑n−1
j=1Ej(x
0j−x
0j+1)
(101)which generalizes to the n-point case the spectral properties given in (11) for the two-pointfunctions, we find:
τ(p1, · · ·, pn)|∑ pi=0
=∑
πi,i=1,··,n
∫ ∞m
n−1∏i=1
dEπiρn(~pπ, Eπ)∫ n−1∏
i=1
dx0i e−i∑n−1
k=1(x0k−x
0k+1)(
∑k
j=1p0πj−Eπk )
= in−1∑
πi,i=1,··,n
∫ ∞m
n−1∏i=1
dEπiρn(~pπ, Eπ)∏n−1
l=1 (∑lj=1 p0
πj− Eπl + i0+)
. (102)
This equation clearly shows the positions of the singularities in the energy variables p0i and
hence the possibility of an analytic continuation to the domain p0i = eiθp4
i with 0 ≤ θ < π.It turns also out that for a general interacting theory the connected n-point Schwinger
functions Sn are also obtained, up to contact terms, that is, terms supported at coincidingpoints, by the analytic continuation of n-point Wightman functions to imaginary times. Thatis:
Sn(x1, ··, xn) = Wn(x′1, ··, x′n) ≡< Ω|Φ(x′1) · ·Φ(x′n)|Ω >
for x41 > x4
2 > ·· > x4n and ~xi ≡ ~x′i , −ix4
i ≡ x′0i . (103)
26
Notice that Sn is a symmetric function of its arguments while Wn is not.In the case of the free theory two-point functions one has, on account of (97) and (5) and
for x4 > 0:
S(~x, x4) =∫ d~p
(2π)4ei~p·~x
∫dp4
eip4x4
E2 + p24
=∫ d~p
(2π)4ei~p·~x
2πi
2iEe−Ex
4
= W (~x,−ix4) . (104)
It turn out furthermore that a local Poincare invariant theory goes into an Euclidean in-variant one.
A non-trivial advantage of the Euclidean theory is that one can choose the smearing func-tions appearing in (8) and in (10) Euclidean invariant as e.g. the Gaussian
Λ40
(2π)2e−Λ2
0(x−y)2
2 ≡ gΛ0(x− y) . (105)
Notice that whenever treating Euclidean functions we shall use the Euclidean scalar productas in (97).
Osterwalder and Schrader have described a complete set of conditions (axioms) guarantee-ing the possibility of constructing a consistent Wightman theory from a given set of Schwingerfunctions. A clear account of their results is presented in Haag’s book quoted in the bibli-ography. In Feynman perturbation theory these axioms are generally satisfied up to infraredproblems that are not discussed in these notes. Therefore in the following we shall consider theconstruction problem in the framework of Euclidean field theory.
27
8 The Functional Integral in Euclidean Quantum Field
Theory
We now consider how interacting Euclidean Quantum Field Theory can be built starting fromthe field equations.
In principle one should start from the equations of the fundamental fields. We shall first,for simplicity, limit our discussion to the case of a single scalar field φ. The field equationsshould relate a certain combination of derivatives of φ to a suitable local composite operatorJ (φ(x), ∂φ(x), ··) ≡ J [φ(x)]. For instance we can choose
−∂2φ(x) = J [φ](x) . (106)
As a matter of fact the above choice is the simplest, non trivial possible. It is worth noticing,however, that the order of partial differential operator appearing in this equation can be reducedadding more field components. In spite of its simplicity (106) is seriously sick since J [φ] doesnot make sense being φ a distribution. Translating (106) into a functional differential equationone should heuristically replace that of the free theory (99) with :
−∂2 δ exp(FC [J ])
δJ(x)= J(x) exp(FC [J ]) + J [
δ
δJ(x)] exp(FC [J ]) , , (107)
where the last term should insert the operator J into the Schwinger functional. As above thisdoes not make sense since Schwinger functions are distributions.
Therefore we see that the construction of a non-trivial quantum field theory must be basedon the definition of local composite operators. Notice that these operators play a crucialrole, not only in the construction of the theory through the field equations, but also in thedefinition of physical observables, e.g. interpolating fields, currents, energy-momentum tensorsand many others. Even if in the construction the fundamental fields, which are identifiedwith the dynamical variables, have a dominant role, the final goal is that of computing localoperator correlation functions. A standard formal mean to insert into the theory local compositeoperators is that of adding to the interaction terms coupling local operators to classical externalfields, and hence to generate the operators through functional derivatives of the action withrespect to the corresponding external fields. This method will be used systematically whenwe shall be interested in local operators and hence the dependence of the effective action onexternal fields will be always understood.
In order to define composite operators overcoming the difficulties due to the distributioncharacter of the fields we should introduce the smeared field
qΛ0(x) ≡∫dy gΛ0(x− y)φ(y) ≡ (gΛ0 ∗ φ)(x) , (108)
where ∗ is the convolution symbol and we define J as a function of q. After this substitution(107) should be written
−∂2 δ exp(FC [J ])
δJ(x)= J(x) exp(FC [J ]) + J [(gΛ0 ∗
δ
δJ(y))(x)] exp(FC [J ]) , , (109)
28
However replacing as above J (φ(x), ∂φ(x), ··) with J (q(x), ∂q(x), ··) ≡ JΛ0(x) destroys thelocal character of (106) and hence one should find a method to recover the locality of thetheory. This is the Renormalization Group (RG) method.
The rough idea is that if one limits the range of momenta to a suitably bounded region itshould be possible to find an interaction built with the qΛ0 operators which, however compli-cated, has the same effect as a local operator.
In other and more precise words, if one considers test functions J whose Fourier transformJ(p) has support bounded by p < Λ Λ0 it should be possible to choose non-local, Λ0-dependent JΛ0 and define a suitable FC [J,Λ0], regular in the limit Λ0 →∞ (that we shall callthe UV limit), such that (109) be true independently of Λ0 and JΛ0 become local, however notnecessarily regular, in this limit. In these conditions JΛ0 plays the role of a bare interactionterm in the field equation. Strictly speaking one should choose smearing functions with compactsupport in p, which is not true with the Gaussian in (105) . We claim that we can use Gaussianswithout any loss of generality, at least in perturbation theory.
Proceeding further we consider a sufficiently wide class of positive functionals IΛ0 [φ], that wecall bare interaction, with a range of non-locality of the order of 1/Λ0 and choose JΛ0 [φ](x) =−∫dygΛ0(x− y)(δIΛ0 [φ]/δφ(x)). The restriction of the bare interaction term term in the field
equation to the functional derivative of some bare interaction is ”a priori” arbitrary in contrastwith the scattering effective action whose existence is guaranteed by the reduction formulae.However to my knowledge this restriction is a price to pay for renormalization.
With the mentioned choice it is apparent that (106) can be written:
δ
δφ(x)(∫dy
(∂φ)2
2+ IΛ0 [qΛ0 ]) = J(x) , (110)
and (109) reads:
−∂2 δ exp(FC [J,Λ0])
δJ(x)+ (gΛ0 ∗
δIΛ0 [gΛ0 ∗ δδJ
]
δφ)(x) exp(FC [J,Λ0]) = J(x) exp(FC [J,Λ0]) , (111)
whose solution, equal to one for vanishing J and positive for J real, as it is required by theaxioms, is:
eFC [J,Λ0] =∫dµ[φ]e
∫dy(J(y)φ(y)− (∂φ)2
2)−IΛ0
[qΛ0]) , (112)
where dµ[φ] is a positive and translation invariant probability measure normalized according:∫dµ[φ] exp(−
∫dy(∂φ)2/2) = 1. It is an exercise left to the reader to verify this.
The ”solution” (112) deserves a long list of comments that we are forced to shorten quitedrastically. Let us only mention that the condition at J = 0 fixes the field independent termin IΛ0 . Concerning the appearance of the functional integral, consider the ordinary differentialequation:
d
dxf(x) + a(
d
dx)2n+1f(x) = x , (113)
with a real and positive, its solution:
f(x) =∫ ∞−∞
dy e−[ y2
2+a y
2n+2
2n+2+c+xy] , (114)
29
with c fixed by the condition f(0) = 1 is unique if we ask f to be real and positive. Equation(112) is the functional version of (114). An alternative form to (114), which is equivalent inthe sense of formal power series in a is:
f(x) = e−[ a
2n+2d2n+2
(dx)2n+2 +c]∫ ∞−∞
dye−[ y2
2+xy] = e
−[ a2n+2
d2n+2
(dx)2n+2 +c′]ex2
2 . (115)
It is apparent that this form is obtained from (112) by partially replacing the variable y underintegral sign with the x-derivative and exchanging the result with the integral sign. It is alsoeasy to see that computing (115) by expanding it in power series of a leads to a power seriesin x which does not converge, at least, absolutely. It is an asymptotic series. This justifies thecaveat about the interpretation as formal power series.
In much the same way an alternative form to (112), which is equivalent in the sense offormal power series in IΛ0 is:
eFC [J,Λ0] = e−IΛ0[gΛ0∗ δδJ
])e12
∫dx J(S∗J) = eFC [J,Λ0] = e−IΛ0
[gΛ0∗ δδJ
])e12
(JS∗J) , (116)
where we have further simplified our notation inserting (fg) for∫dxf(x)g(x) and
S(x) =∫ dp
(2π)4
eipx
p2=
1
(2π)2x2, (117)
The formal expression given in (116) is very important since it generates the Feynmandiagram expansion. Indeed Feynman series is obtained expanding both exponentials in (116)and performing the functional derivatives in IΛ0 which corresponds to the vertex generator one
12
(JS∗J) which plays the role of line generator. The analysis of FC as the generator of connectedFeynman diagrams is given in Appendix A.
Now the RG condition we want to require is that FC has a regular limit for Λ0 →∞ and thefield equation becomes local in this ultra-violet (UV) limit. This is, of course, a condition onIΛ0 . However this does not require IΛ0 to remain regular in the limit. The requirement is that
FC has a regular limit andδIΛ0
δφ, however singular, becomes local. This means that
δ2IΛ0[qΛ0
]
δφ(x)δφ(y)→ 0
when Λ0 →∞ for (x− y)2 = d2 > 0 fixed.
30
9 The Wilson Effective Action in Euclidean Quantum
Field Theory
When one tries to construct the Schwinger functional FC in the UV limit satisfying the abovestated RG condition and using e.g. (116) one faces an obvious difficulty since FC appearsexplicitly built starting from IΛ0 which is expected to be singular in this limit. Wilson’s solutionto this problem exploits the above mentioned possibility of replacing IΛ0 [gΛ0 ∗ (δ/δJ)] with adifferent, non local functional IΛ,Λ0 [gΛ ∗ (δ/δJ)], which is called the Wilson effective action,when the support of J lies in the region p Λ Λ0. The idea is that, keeping Λ fixed in theUV limit, IΛ,Λ0 [gΛ ∗ (δ/δJ)] should have a regular IΛ,∞[gΛ ∗ (δ/δJ)] limit, since FC is expectedto be regular in this limit. What remains to be required is IΛ,∞[gΛ ∗ (δ/δJ)] to become local,however singular, in the Λ→∞ limit.
Therefore what one has to do is, first, to identify IΛ,Λ0 , then to see how it depends on Λand how it is related to IΛ0 , this will lead us to the RG evolution equation with the initialcondition IΛ0,Λ0 = IΛ0 . Using this evolution equation there remains to understand under whichconditions IΛ,∞ is regular and it has a local, however singular, in the Λ→∞ limit.
Let us begin with the first step. Introducing g = gΛ0 − gΛ, one can show that there exists afunctional IΛ,Λ0 such that
eFC [J,Λ0] =: e−IΛ,Λ0[gΛ0∗ δδJ
+ g∗S∗J ]) : e12
(JS∗J) , (118)
where the symbol : X[J, δ/δJ ] : implements the ordering prescription according to whichfunctional derivatives should be placed on the right-hand side of J ’s and we have shortened∫dxJ(x)(gΛ0 ∗ J)(x) into (JS ∗ J). The proof of (118) is given in Appendix A. It is clear that
the functional built up in Appendix B it what we were looking for. Indeed, if the support ofJ lies in the region p Λ Λ0, g ∗ S ∗ J vanishes and (118) coincides with (116) after thesubstitutionIΛ0 ↔ IΛ,Λ0 .
A further exercise left to the reader is to prove that −IΛ,Λ0 is the functional generator ofconnected amputated diagrams corresponding to the interaction IΛ0 and to the propagator:
S ≡ gΛ0 ∗ S ∗ gΛ0 − gΛ ∗ S ∗ gΛ . (119)
Notice that, using the equivalence (116)-(118), one has:
eFC [J ] =∫dµ[φ]e
∫dy(J(y)φ(y)− (∂φ)2
2)−IΛ,Λ0
[gΛ∗φ+ g∗S∗J ]) , (120)
Having introduced IΛ,Λ0 we notice that, if this functional has a regular UV limit (Λ0 →∞)IΛ,∞ (this is the point that will take few pages to be verified), one has
eFC [J ] =∫dµ[φ]e
∫dy(J(y)φ(y)− (∂φ)2
2)−IΛ,∞[gΛ∗φ+ g∗S∗J ]) , (121)
which must be Λ independent. Notice that in (121) g ∗ f = f − gΛ ∗ f .
31
One can derive the field equation requiring, at the first order in ε, the invariance of FCunder the change of functional integration variables φ→ φ+ ε:
(ε(J + ∂2 δ
δJ))eFC [J ]
=∫dµ[φ](ε
δIΛ,∞
δϕ)|ϕ=gΛ∗φ+ g∗S∗Je
∫dy(J(y)φ(y)− (∂φ)2
2)−IΛ,∞[gΛ∗φ+ g∗S∗J ]) . (122)
If the support of the Fourier transform J(p) of J is kept bounded and the limit Λ → ∞ istaken, it is clear that the right-hand side of (122) tends to (ε(δIΛ,∞/δϕ)[(δ/δJ)]) exp(FC [J ])since g ∗ J vanishes in the limit. Therefore we have obtained an equation strictly analogous to(111). Furthermore, due to the Λ independence of (122), the right-hand side of this equationcorresponds to a local operator if IΛ,∞ tends to a local Λ→∞ limit.
Next step in our program is to deduce the RG evolution equation for IΛ,Λ0 . The crucial pointis that exp(FC [J ]) does not depend on Λ; therefore, taking the derivative of the right-hand sideof (118) with respect of ln(Λ2), that we label by a dot, that is writing:
F (Λ) ≡ Λ2 δF (Λ)
δΛ2, (123)
one has:
: (−IΛ,Λ0 [φ]−(δIΛ,Λ0
δφgΛ ∗ [
δ
δJ− S ∗ J ]
)e−IΛ,Λ0
[φ]|φ=gΛ0∗ δδJ
+ g∗S∗J :
e12
∫dxdy(J S∗J) = 0 (124)
With the chosen ordering prescription the functional derivative (δ/δJ) in the second term ofthis expression lies on the right-hand side and hence it can be replaced by S ∗ J . Thus ata first sight the two terms in brackets should sum to zero. This is however not true becausethe product S ∗ J induced by (δ/δJ) lies at the right of the ordered expression and hence,before coming to any conclusion, we have to shift them to the left. What remains of the twocontributions is thus what comes re-ordering S ∗ J to the left of the J functional derivativeappearing in IΛ,Λ0 . This is given by:
∫dxdy(gΛ ∗ S ∗ gΛ)(x− y)[
δIΛ,Λ0
δφ(x)
δIΛ,Λ0
δφ(y)− δ2IΛ,Λ0
δφ(x)δφ(y)] . (125)
Thus, asking for the resulting ordered expression to vanish, which is a sufficient condition for(124) be satisfied, one has the evolution equation for the effective interaction IΛ.Λ0 :
IΛ.Λ0 [φ] =∫dxdy(gΛ ∗ S ∗ gΛ)(x− y)[
δIΛ,Λ0
δφ(x)
δIΛ,Λ0
δφ(y)− δ2IΛ,Λ0
δφ(x)δφ(y)]
=1
2
∫dxdy
˙S(x− y)[
δ2IΛ,Λ0
δφ(x)δφ(y)− δIΛ,Λ0
δφ(x)
δIΛ,Λ0
δφ(y)] . (126)
32
This equation can be easily translated in terms of Fourier transformed fields. For this we needthe Fourier transform of (119), in which the propagator (117) is S = (1/p2). Choosing theGaussian smearing (105) we have:
˜S(p) =
e− p2
Λ20 − e−
p2
Λ2
p2(127)
and
˙S(p) = −e
− p2
Λ2
Λ2. (128)
Then the evolution equation reads:
IΛ.Λ0 [φ] =1
2
∫ dp
(2π)4
e−p2
Λ2
Λ2[δIΛ,Λ0
δφ(p)
δIΛ,Λ0
δφ(−p)− δ2IΛ,Λ0
δφ(p)δφ(−p)] . (129)
It is important to notice here that this evolution equation for the effective action induces acorresponding evolution equations for operators. Indeed a generic operator of the theory isidentified by the derivative of the effective action with respect to a parameter, which, in thecase of local operators is a space-time dependent function, that we have called external field,and hence the above derivative is a functional derivative. For example the operator that, inthe limit Λ→ Λ0 is identified with (gΛ0 ∗ φ)4/4! corresponds to the functional derivative of theeffective action with respect to some bare coupling constant considered space-time dependentand a conserved current is usually defined by means of the functional derivative with respectto some external (background) vector field.
In conclusion, at the effective theory level, the operator coupled to the external field ω isidentified by
ΩΛ,Λ0 [φ] ≡ δIΛ,Λ0 [φ]/δω (130)
whose evolution equation is linear:
ΩΛ.Λ0 [φ] =1
2
∫ dp
(2π)4
e−p2
Λ2
Λ2[2δIΛ,Λ0
δφ(p)
δΩΛ,Λ0
δφ(−p)− δ2ΩΛ,Λ0
δφ(p)δφ(−p)] . (131)
The analysis of the solutions of this operator evolution equation is identical to that of theeffective action which follows.
Equation (129) can be further elaborated introducing the scaled field variables
φ(p) ≡ Λ−3ϕ(p
Λ) (132)
and the series expansion of IΛ,Λ0 in powers of the field;
IΛ,Λ0 [φ] =∞∑n=0
1
n!
∫ n∏i=1
(dpiφ(pi))δ(n∑j=1
pj)In(Λ,Λ0, p) . (133)
33
Setting In(Λ,Λ0, p) ≡ Λ4−nin(Λ,Λ0,pΛ
) one can write IΛ,Λ0 as a functional of ϕ through:
IΛ,Λ0 [φ] = IΛ,Λ0 [ϕ] =∞∑n=0
1
n!
∫ n∏i=1
(dpiϕ(pi))δ(n∑j=1
pj)in(Λ,Λ0, p) . (134)
One has furthermore:
δIΛ,Λ0 [φ]
δφ(p)=
1
Λ
δIΛ,Λ0 [ϕ]
δϕ( pΛ
)
IΛ,Λ0 [φ] =˙IΛ,Λ0 [ϕ]− 1
2
∫dp(3ϕ(p)) + p · ∂pϕ(p))
δIΛ,Λ0 [ϕ]
δϕ(p), (135)
and hence the evolution equation (129) becomes:
˙IΛ,Λ0 [ϕ]− 1
2
∫dp(3ϕ(p)) + p · ∂pϕ(p))
δIΛ,Λ0 [ϕ]
δϕ(p)
=1
2
∫ dp
(2π)4e−p
2
[δIΛ,Λ0
δϕ(p)
δIΛ,Λ0
δϕ(−p)− δ2IΛ,Λ0
δϕ(p)δϕ(−p)
]. (136)
and the equation for exp(−IΛ,Λ0 [ϕ]) is:
[Λ2∂Λ2 − 1
2
∫dp(3ϕ(p)) + p · ∂pϕ(p))
δ
δϕ(p)]e−IΛ,Λ0
[ϕ]
= −1
2
∫ dp
(2π)4e−p
2 δ2
δϕ(p)δϕ(−p)e−IΛ,Λ0
[ϕ] . (137)
which looks very much like a diffusion equation in the functional space with the time variableidentified with ln(Λ0/Λ). A very simple way of understanding some possible consequencesof (137) is to consider its form at zero space-time dimensions. In this ultra-simplified formthe general solution of this equation is easily written and it is also easy to understand itsimplications from the point of view of the renormalization program. It is obvious that thezero dimensional field is a real variable that we identify with x after multiplication by suitableconstant factor. Thus we write (137) in the form:
Λ2∂Λ2 exp(−IΛ,Λ0(x)) = − ∂2
(∂x)2exp(−IΛ,Λ0(x)) . (138)
Setting t = ln(Λ0/Λ)2 and F (x, t) ≡ exp(−IΛ,Λ0(x)) it is easy to show that:
F (x, t) =∫ ∞−∞
dy√2πt
e−(x−y)2
2t F (y, 0) , (139)
and it is apparent that F (x, t) tends to a Gaussian for t → ∞, for any integrable choice ofF (x, 0). This is a triviality property since a Gaussian corresponds to a free theory and hencethe general solution seems to suggest that our scalar field theory is necessarily free. This result
34
is however far from established, e.g. it is fake in two-dimensional filed theory, since nowhereit is said that the solution should only depend on the ratio Λ/Λ0. Being interested in theconstruction of a non trivial effective theory at Λ in the limit Λ0 →∞ we remain free to insertan explicit Λ0-dependence into F (x, 0). In this situation one would have the solution of theevolution equation F (x, t,Λ0) and would be interested in the limit t→∞ of F (x, t,Λ exp(−t/2))whose triviality is not guaranteed. It is however clear that, in order to avoid a trivial result, itis necessary, and not sufficient, that F (x, 0,Λ0) be singular in the Λ0 →∞ limit.
The simplicity of the equations allows to see how the solution (139) appears in perturbationtheory giving an idea of what can happen in more realistic situations. With F (x, 0,Λ0) =exp(−g0(Λ0)x4/4!) and expanding F (x, t,Λ0) in power series of g0 we find:
F (x, 2 ln(Λ0/Λ)) =∞∑n=0
(4n)!
n!(−g0 ln(Λ0/Λ)
12)n
2n∑l=0
(x2
ln(Λ0/Λ)
)l(2l)!(2n− l)!
. (140)
Needless to say, this power series is not absolutely convergent as it appears e.g. looking atthe series corresponding to F (0, λ). Considering (140) as an asymptotic power series in g0 andlimiting the expansion to the second order one has:
F (x, 2 ln(Λ0/Λ)) =g0 ln2(Λ0/Λ)
2+
4
3(g0 ln2(Λ0/Λ))2 − [g0 ln(Λ0/Λ)− 16
3g2
0 ln3(Λ0/Λ)]x2
2
−[g0 −20
3g2
0 ln2(Λ0/Λ)]x4
4!+O(g3
0, x6) . (141)
Identifying the coefficient of x4/4! at Λ = ΛR with the renormalized low energy coupling con-stant gR and solving in terms of g0 we find:
g0 =3
40 ln2(Λ0/Λ)
1±
√1− 80gR ln2(Λ0/Λ)
3
. (142)
This means that there is no g0 corresponding to the wanted coupling constant as soon as Λ0/ΛR
becomes large enough. Of course it is not possible to further speculate on this result since wehave chosen a particular initial condition at Λ0. We shall see that in general this is the wrongstrategy.
Next step of our analysis consists in the search for a solution of the evolution equationsatisfying the locality constraint in the UV limit and a suitable set of normalization conditions(as a matter of fact, initial conditions) at Λ = ΛR, the nature of these conditions and the valueof ΛR remaining to be specified. With this aim one could perfectly well continue the analysisstudying the solutions to (136), however many applications are simplified if one considers another functional which coincides with IΛ,Λ0 in the Λ → ∞ limit. We call this new functionalthe effective proper generator
When needed we shall use the following notations to indicate the Schwinger functions withinfrared cut-off Λ in the UV limit:
< φ(p1) · · · φ(pn−1)φ(0)|Λ >C (143)
35
will represent the connected n point functions and we shall replace φ , φ by φ , ˆφ , if thecorresponding legs are amputated (multiplied by S−1). We shall also use
< ˆφ(p1) · · · ˆφ(pn−1)φ(0)|Λ >1P I (144)
for the n-point 1-PI functions. For example the field functional derivatives of IΛ,∞[φ] are thefunctions:
< φ(x1) · · · φ(pn)|Λ >C . (145)
36
10 The Effective Proper Generator
In this section we introduce the effective proper generator, we study its evolution equation andwe discuss its iterative solution. In order to define the effective proper generator it is convenientto introduce the corresponding effective connected generator:
ZΛ,Λ0 [J ] ≡ 1
2(JS ∗ J)− IΛ,Λ0 [S ∗ J ] , (146)
that is the generator of the connected amplitudes corresponding to the interaction IΛ0 and tothe propagator S. In the UV limit these amplitudes are:
< φ(x1) · · · φ(pn)|Λ >C . (147)
The effective proper generator V is defined as the formal power series Legendre transform ofZ, which is built much in the same way as the Effective Action in section 6 (75) .
Assuming, in order to simplify the formulae, that IΛ,Λ0 [φ] does not contain a linear term inits functional variable φ, we put
φ ≡ S ∗ (J − δIΛ,Λ0
δφ[S ∗ J ]) , (148)
The inverse functional J [φ] is defined as the formal power series in φ solution of:
J = Cφ+δIΛ,Λ0
δφ[S ∗ J ] , (149)
where
C(p) ≡ (S)−1(p) =p2
e− p2
Λ20 − e−
p2
Λ2
. (150)
and φ is chosen of class D in momentum space, since this would guarantee (φC ∗ φ) to befinite.Then we define:
VΛ,Λ0 [φ] ≡ (φJ [φ])−ZΛ,Λ0 [J [φ]] = (φJ [φ])− 1
2(J [φ]S ∗ J [φ]) + IΛ,Λ0 [J [φ]] , (151)
It follows from this definition that V [φ] is the functional generator of the one-particle irreducibleparts of the connected functions generated by Z[J ] , that is:
< φ(x1) · · · φ(pn)|Λ >1−IP . (152)
As already mentioned 1-PI means diagrams that cannot be broken in two disconnected partsby cutting a single line.
It is a direct consequence of the above definitions that:
δ2ZΛ,Λ0
δJ(x)δJ(y)=
(δ2VΛ,Λ0
δφ(x)δφ(y)
)−1
= S(x− y)− (S ∗ δ2IΛ,Λ0
δφ2∗ S)(x, y)
VΛ,Λ0 = −ZΛ,Λ0 ,δVΛ,Λ0
δφ= J (153)
37
Furthermore from (126) one has:
ZΛ,Λ0 =1
2(J
˙S ∗ J)− IΛΛ0 − (J
˙S ∗ δIΛΛ0
δφ)
=1
2
[((J − δIΛΛ0
δφ)
˙S ∗ (J − δIΛΛ0
δφ)
)− (
δ
δφ˙S ∗ δ
δφ)IΛΛ0
]= −VΛ,Λ0
=1
2
−(φ˙C ∗ φ) + Tr(
˙C[S −
(δ2VΛ,Λ0
δφ2
)−1
])
, (154)
where we have introduced: Tr(AB) ≡∫dxdy A(x, y)B(y, x) . Notice that on the basis of the
first equation in (153) one can see that the above trace is well defined in spite of the singularbehavior of C at infinite momentum.
Now it is possible to define the Effective Proper Generator VΛ,Λ0 by :
VΛ,Λ0 [φ] ≡ 1
2(φC ∗ φ) + VΛ,Λ0 [φ] . (155)
Then one has: (δ2VΛ,Λ0
δφ2
)−1
=
(C +
δ2VΛ,Λ0
δφ2
)−1
=
((1 +
δ2VΛ,Λ0
δφ2∗ S) ∗ C
)−1
= S ∗ (∞∑n=0
(− ∗ δ2VΛ,Λ0
δφ2∗ S)n) , (156)
and the evolution equation for VΛ,Λ0 [φ]:
VΛ,Λ0 [φ] = −1
2(φ
˙C ∗ φ) + VΛ,Λ0 [φ]
= −1
2Tr(
˙C[S −
(δ2VΛ,Λ0
δφ2
)−1
])
=1
2Tr(
˙CS
∞∑n=1
(− ∗ δ2VΛ,Λ0
δφ2∗ S)n)
=1
2Tr
(˙Sδ2VΛ,Λ0
δφ2
∞∑n=0
(− ∗ S ∗ δ2VΛ,Λ0
δφ2)n)
=1
2Tr
(˙S(δ2VΛ,Λ0
δφ2− δ2VΛ,Λ0
δφ2∗ S ∗ δ
2VΛ,Λ0
δφ2+ ··)
)
≡ 1
2R[φ] . (157)
This equation is better represented by the following figure:
38
Λ∂Λ&%'$
=X&%'$
−X
+ X
+ · · ·
where double lines correspond to the propagator S and the crossed double one to ˆS whilebubbles correspond to the 1-PI parts generated by V .
Taking into account that IΛ,Λ0 is the generator of amputated connected diagrams whoseskeleton vertices are generated by VΛ,Λ0 we can reformulate our renormalization program interms of V . We have to analyze the iterative solution of (157), discussing in particular underwhich conditions the solution has regular UV limit and this limit has local Λ→∞ limit.
Once again it is convenient to use, as above( in (132)), the scaled fields setting:
VΛ,Λ0 [φ] =∞∑n=0
1
n!
∫ n∏i=1
(dpiφ(pi))δ(n∑j=1
pj)Vn(p1, ··, pn,Λ,Λ0)
= VΛ,Λ0 [ϕ] =∞∑n=0
1
n!
∫ n∏i=1
(dqiϕ(qi))δ(n∑j=1
qj)vn(q1, ··, qn,Λ,Λ0) . (158)
Notice that in the notation introduced above
Vn(p1, ··, pn,Λ,∞) = Λ4−nvn(p1/Λ, ··, pn/Λ,Λ,∞) =< ˆφ(p1) · · · ·ˆφ(pn−1)φ(0)|Λ >1−PI . (159)
Then one finds for the coefficient functions the evolution equations:∫ n∏i=1
(dqiϕ(qi))δ(n∑j=1
qj) [Λ∂Λ + 4− n− q∂q] vn(q1, ..., qn,Λ,Λ0)
= −∫ n∏
i=1
(dqiϕ(qi))δ(n∑j=1
qj)
[∫ dp
(2π)4e−p
2
vn+2(q1, ..., qn, p,−p,Λ,Λ0)
+∫ dp
(2π)4e−p
2∫dqe−χq
2 − e−q2
q2δ(
m∑j=1
qj + p− q)n∑
m=1
n!
m!(n−m)!
vm+2(q1, ..., qm, p,−q,Λ,Λ0)vn−m+2(qm+1, ..., qn, q,−p,Λ,Λ0) + ··]
≡∫ n∏
i=1
(dqiϕ(qi))δ(n∑j=1
qj)rn(q1, .., q2n,Λ,Λ0) , (160)
where we have set χ = Λ2/Λ20 and we have explicitly given the first two terms of the expansion
in (157).This equation justifies the recourse to the scaled filed variables. Indeed if the |vn|’s are
bounded and ϕ is of class D it is apparent that all integrals in the right-hand side of (160)
are absolutely convergent uniformly in χ ≥ 0 due to the the cut-off factors in˙S and S and to
momentum conservation which limits the range of the integration variables (e.g. p − q in the
39
second term in brackets) and hence there is no problem in taking the limit χ→ 0 in the aboveequation.
Now we transform (160) into a system of integral equations which will be suitable fordiscussing the Λ0 → ∞ limit. We start considering the evolution equation of v0. Then weconsider that of the coefficients of the Taylor expansion of v2(p) up to the second order in p,that is v2(0, 0,Λ,Λ0) ≡ v2 and ∂q2v2(q,−q,Λ,Λ0))|q=0 ≡ v′2. Next we study the evolution of thevalue of v4 at zero momenta. Finally we complete the set of evolution equations consideringthose of the k-th momentum derivatives ∂kq vn(q,Λ,Λ0) for n+ k > 4.
The Λ dependence of the q-independent coefficients is controlled by the evolution equations
[Λ∂Λ + 4] v0(Λ,Λ0) = r0(Λ,Λ0)
[Λ∂Λ + 2] v2(Λ,Λ0) = r2(Λ,Λ0)
Λ∂Λv′2(Λ,Λ0) = r′2(Λ,Λ0)
Λ∂Λv4(0Λ,Λ0) = r4(Λ,Λ0) , (161)
which are solved by:
v0(Λ,Λ0) =(
Λ
ΛR
)−4c0 +
∫ ( ΛΛR
)1
dλ λ3r0 (ΛRλ,Λ0)
v2(Λ,Λ0) =
(Λ
ΛR
)−2c2 +
∫ ( ΛΛR
)1
λdλ r2 (ΛRλ,Λ0)
v′2(Λ,Λ0) =
c′2 +∫ ( Λ
ΛR
)1
dλ
λr′2 (ΛRλ,Λ0)
v4(Λ,Λ0) =
c4 +∫ ( Λ
ΛR
)1
dλ
λr4 (ΛRλ,Λ0)
, (162)
for some ΛR. These equations clearly show a strategy which is different from that discussedat the end of the last section where we have computed the effective action corresponding to agiven IΛ0 . Indeed the present strategy, aiming at the construction of a VΛ,Λ0 [ϕ] with regularΛ0 →∞ (UV) limit, is based on the choice of the initial values of the evolution equation at afixed scale ΛR, at least for the first coefficients v0 , v2 , v
′2 and v4. The advantage of this choice
is to eliminate any explicit Λ0-dependence from (162). If the coefficients depend on Λ0 this isdue to the r’s. Had one chosen initial conditions at Λ = Λ0 there would appear an explicitΛ0-dependence which would influence the UV limit.
In order to continue our analysis of the UV limit we now come to the evolution of theremaining coefficients vn. Notice that, taking a generic k-th momentum derivative ∂kq vn(q,Λ,Λ0)of vn, one has evolution equations completely analogous to (160); that is : :
[Λ∂Λ + 4− n− k − q∂q] ∂kq vn(q1, ..., q2n,Λ,Λ0) = ∂kq rn(q1, .., q2n,Λ,Λ0) (163)
In the cases we are studying, in which n+ k > 4 it is convenient to consider the solution:
∂kq vn(q,Λ,Λ0) =(
Λ
Λ0
)n+k−4 ∫ ( ΛΛ0
)1
dλ λ3−n−k∂kq rn
(Λq
Λ0λ,Λ0λ,Λ0
)
40
= Λn+k−4∫ Λ
Λ0
dx x3−n−k∂kq rn
(Λq
x, x,Λ0
). (164)
Indeed the rightmost expression in (164) clearly shows that (163) has a UV regular solution if∂kq rn is UV regular and absolutely bounded by polynomials in ln Λ uniformly in Λ0.
Now it appears clearly in our integral equations that the crucial step of the analysis concernsthe properties of the rn’s. It should be apparent from (160) that the momentum integralsdefining rn in terms of the vn’s are limited by the exponential cut-off factor to a domaincontained into a R4 spherical shell of radii of order 1. Thus if the coefficients vn where absolutelylimited every integral would lead to an absolutely limited result. This is not enough to claim thatthe rn’s are absolutely limited since the number of terms contributing to each rn is potentiallyinfinite due to the presence of v2. As a matter of fact summing all the possible insertions of
v2 is equivalent to replace the propagator˜S(q) that, after rescaling and for χ = 1, is
˜S(q) =
(1− exp(−q2))/q2 by:
˜S(q) =˜S(q)
∞∑n=0
(−v2(q)˜S(q))n
=˜S(q)
q2
q2 + v2(q)(1− exp(−q2))≡ ˜S(q)Z(q) . (165)
For the same reason the Λ-derivative of the propagator gets multiplied by Z2(q). Now theproblem is that Z is not necessarily absolutely limited in Λ.
Indeed, summing over the v2(p,−p,Λ,Λ0) insertions, the evolution equation of v2(p,−p,Λ,Λ0)given in (160) becomes:
[Λ∂Λ + 2− q∂q] v2(q,−q,Λ,Λ0) = −∫ dp
(2π)4e−p
2
Z2(p)v4(q,−q, p,−p,Λ,Λ0) ≡ r2(q,−q,Λ,Λ0)
(166)and it is apparent that r2 in our scalar theory is negative since v4 must be positive. Therefore,limiting our analysis to v2 ≡ v2(0, 0,Λ,Λ0), and assuming a weak, logarithmic, Λ-dependenceof r2 we find v2 ' (ΛR/Λ)2c2 − P2(ln Λ), that is: for large enough Λ, v2 is negative and itsabsolute value increases as a power of ln Λ, and hence Z diverges for some Λ. Notice that v2
changes sign since we have chosen a finite (null) value of v2 at the initial point ΛR. Given thevalue of V2(Λ) ≡ Λ2v2(Λ) at Λ0 one has V2(Λ) = V2(Λ0) + (Λ2
0P2(ln Λ0) − Λ2P2(ln Λ)) with P2
positive, hence in order to have a small value of V2 at ΛR one has to choose V2(Λ0) negative.If, on the contrary, one keeps V2(Λ0) positive V2(ΛR) diverges quadratically in the U.V. limit.2
It is worth recalling here that V2 is a mass and this is an aspect of the famous naturalnessproblem with the scalar field masses, in particular with Higgs field one. There are theoriesin which the masses of the fundamental fields are protected by a symmetry, this is the caseof super-symmetry, and, much more important, that of non-abelian gauge theories where theproblem we have just discussed does not exist. In the present case the mass problem forbids aconsistent construction of our scalar theory except if we limit our study to a formal perturbative
2The situation is different in super-renormalizable theories, such as e.g. P (φ) in two dimensions in whichr2 ' −m2/Λ2 up to log’s, and hence v2 has a similar behavior.
41
expansion in the initial data c2 , c′2 and c4. Indeed in a perturbative expansion the right-hand
side of (160) receives a finite number of contributions at any finite order.We shall thus limit our future considerations to the iterative solution in powers of c2 , c
′2
and c4 (c0 is just a normalization constant which only appears in the first equation of thesystem (162)). Taking into account (164) and (162) one finds coefficient functions vn(q,Λ) thatare power series of the input data c2, c
′2 and c4 and the terms of order ν in c′2 and c4 satisfy,
uniformly in Λ,ΛR and q, the inequality:
sup |∂kq v(ν)n (q,Λ)| ≤ Pn,k,ν
(log
(Λ
ΛR
))(167)
where Pn,k,ν is a suitable polynomial of degree lower or equal to ν. A completely analogousinequality holds true for sup |∂kq r(ν)
n (q,Λ)|. Therefore the UV limit can be taken on the foundsolutions and it appears regular since the x-integral in (164) and those in (160) converge uni-formly in the limit. The inequality (167) is obtained identifying the coefficients of VΛ,∞ withthe solutions of the system of integral equations (162) and taking into account that, from (164),one has:
∂kq vn(q,Λ,∞) = Λn+k−4∫ Λ
∞dx x3−n−k∂kq rn
(Λq
x, x,∞
), (168)
with rn computed from (160). In the following we shall often omit the label ∞ from theUV limits of the coefficients and the UV limit will be understood for all functions of interestwhenever Λ0 will not explicitly appear.
The last point which remains to be discussed is the locality of limΛ→∞ V [φ,Λ] which guaran-tees that our field theory and, in particular, that field equations be local. On a very superficiallevel this is trivially verified taking into account (168) , (167) and recalling that:
Vn(p,Λ,Λ0) = Λ4−nvn(p
Λ,Λ,Λ0) . (169)
Indeed combining these equations it is easy to find that ∂kqV(ν)n (q,Λ) with k+n > 4 vanishes in
the UV limit. This is however not enough, indeed one should verify that e.g. V4 has a sensibleUV limit. This is apparently true in our perturbative expansion.
The result we have reached so far is based on the construction of the solution of the RGequations in power series of c2, c
′2 and c4. As a matter of fact we have already discussed what
happens when one sums the series in c2 and it is not a problem to sum the series in c′2 that fixesthe scale of the field, for simplicity one can chose c2 and c′2 vanishing and remain with c4 playingthe role of a coupling constant g. This must be positive, otherwise the theory becomes unstableand the functional integral (112) does not make sense. One gets in this way a perturbationtheory which should be accurate for small values of c4 ≡ g.
There is however a different and, potentially, more interesting construction strategy whichis based on the smallness of V4(Λ), rather than g, for some value of Λ > ΛR. In particular thisstrategy is crucial for theories with V4(Λ) vanishing at infinite Λ, the limit we are interested inin order to implement locality of the theory. Indeed, if this situation that is called asymptoticfreedom, were actual, one would have a guarantee of the consistency of the theory, independentlyof the possibility of producing accurate results at low energy.
42
In order to give a more precise idea of this possibility let us analyze our scalar model. Ifv4 is small it is not difficult to verify that r2n ∼ O((v4)n). Thus, considering the evolutionequation of v4 at vanishing momenta and up to the first non-trivial order in v4 one has from(160) and (161):
Λ∂Λv4(0,Λ) =4!
(2!)2
∫ dp
(2π)4e−p
2 1− e−p2
p2v2
4(0,Λ)
=6π2
(2π)4
∫ ∞0
dx(e−x − e−2x)v24(0,Λ) =
3
(4π)2v2
4(0,Λ) , (170)
where we have disregarded the term in v6 which is O((v4)3) and any momentum dependence ofv4 itself which would induce further higher order corrections in v4. Integrating this differentialequation between ΛR and Λ with v4(0,ΛR) = g > 0 one has:
v4(0,Λ) =g
1− 3(4π)2 g ln Λ
ΛR
. (171)
This result is apparently deceiving since, due to the sign in the rightmost term of (170), v4
increases with Λ reaching a singular point in Λ = exp((4π)2/3g)ΛR beyond which it changessign. This is called the Landau singularity.
It is clear that this result confirms the belief that the only consistent scalar theory we canbuild is the free one.
It has been a surprise the result found about 20 year ago that the same calculation in thecase of the Yang-Mills gauge theory gives the opposite sign in (170) and hence a v4 vanishingin the Λ→∞ limit.
A further comment is necessary concerning renormalized operators that appear in the con-struction of the theory, as specified in section 8, through functional derivatives of the effectiveaction with respect to external fields, ω in the following, and satisfy the linear evolution equa-tion (131). It is clear from this and from the above analysis that every operator is definedinserting a certain number of initial parameters, analogous to c2, c
′2 and c4, into the solutions of
the evolution equation for the coefficients Ωn(p,Λ) of the field expansion of the effective operatorΩΛ[φ](x). These solutions are strictly analogous to (162). Of course, in order to complete theconstruction, one also has to extend to the local operators the equations for the derivatives ofthe higher order coefficients ∂kqΩn(p,Λ), for n+k > dΩ, that turn out to be analogous to (168).The need of a complete identification of the operator relates the number of initial parametersto dΩ. Indeed completeness of initial conditions requires that initial values be specified for allthe coefficient of ΩΛ[φ](x) in its expansion in powers of fields and momenta up to the totaldegree dΩ; while (168) is equivalent to the prescription that the remaining coefficients havenull values at infinity. This means that the initial conditions that are not automatically trivialcorrespond to the coefficients of a local polynomial in the fields and partial derivatives whosetotal degree is dΩ. Thus one has to specify the terms of mass dimension up to dΩ of the localoperator at Λ = ΛR. Considering as above an iterative solution of the evolution equations onesees that, at the first iterative order, ΩΛ[φ](x) is identified with an operator of mass dimension
d(m)Ω ≤ dΩ. We shall call d
(m)Ω naive dimension. A further comparison with the results for the
43
effective proper generator shows that dΩ identifies the behavior of the coefficients Ωn(p,Λ) forlarge Λ and hence momenta. This turns out to be given, up to logs, by Ωn(p,Λ) = O(ΛdΩ−n).Thus we call dΩ power counting dimension.
The question is how these parameters and dΩ are chosen. The answer depends on theproperties of the operator, that is: its naive dimension, dΩ, Lorentz transformation (tensor)properties, and possibly charges, in case the theory has conserved charges. Just to give a simpleexample, consider, in the framework of neutral scalar field theory invariant under reflectionφ → −φ, the dimension dΩ = 4 local operators transforming as a scalar field and even underreflection. There are 4 monomials in the fields and their derivatives with the desired properties,these are φ2 , (∂φ)2 , φ ∂2φ and φ4, while the initial conditions for the proper effective actionidentifying a particular choice of an operator with the desired properties concern the 1-PI vertexwith one ω leg and 4 φ legs at zero momenta and the coefficients of the Taylor expansion up todegree 2 in the momenta of the scalar legs of the vertex with one ω and 2 φ legs. Both verticesmust be computed at Λ = ΛR. Let a1 be the value of the 4-leg vertex. Due to the chosenLorentz covariance and Bose symmetry the Taylor expansion of the 2-φ vertex can be writtenas: a2 + a3 p1 · p2 + a4(p2
1 + p22)). Notice that we understand, except in case of explicit warning,
the normalization condition that the vacuum expectation values of operators vanish.It can be helpful to represent the initial values of the evolution equations in the functional
formalism, indeed in this formalism the initial value of the effective proper generator is givenby:
VΛR [φ] =∫dx[c′2(∂φ)2(x)/2 + c2φ
2(x)/2 + c4φ4(x)/4!] + V
(D>4)ΛR
[φ] , (172)
where, expanding V(D>4)
ΛR[φ] in series of φ as in (142) and recalling that the coefficient functions
V (D>4)n (p1, · · ·, pn,ΛR) are analytic functions of the momenta in the origin, one has V (D>4)
n (p1, · ··, pn,ΛR) = O(p4−n) for n ≤ 4.
In much the same way for the effective operator corresponding to the generic 4 dimensionaloperator discussed above one has:
ΩΛR [φ](x) ≡ δVΛR [φ]
δω(x)= a1φ
4(x)/4! + a2φ2(x)/2− a3(∂φ)2(x)/2− a4φ(x)∂2φ(x) + ΩD>4
ΛR[φ](x) .
(173)Now, due to the linearity of the evolution equation, ΩD>4
ΛR[φ](x) is a linear function of the ai’s.
Since the operators are elements of a linear space it is convenient to chose a basis for this spacewhich is 4 dimensional in the case under discussion. This basis can be e.g. chosen setting one ofthe coefficients ai , i = 1, .., 4 equal to 1 and the rest equal to zero. It is clear that the differentchoices of i identify 4 independent local operators Ωi.
At this point a further comment is in order. Suppose we are interested in the operatorswith the same properties of the above operator (173) but with dΩ = 2, we should have found,instead of the right-hand side of (173) bφ2(x)/2 + ΩD>2
ΛR[φ](x). The linearity of the operator
space implies that this operator corresponds to a linear combination of the elements of the basisdiscussed above, that is, that it can be written in the form (173) for a special choice of the ai’s.
A more systematic construction of these bases and a deeper analysis of the properties ofrenormalized local operators can be done using the subtraction method. Thus we postponethe analysis to a dedicated section. In this section we have tried to give a general idea of the
44
renormalization group based construction of a quantum field theory transforming the differentialevolution equations for the relevant functional generators into integral equations to be solvediteratively. The discussion has concerned a mass-less scalar field with a particular cut-off; itshould be clear that a number of generalizations are possible, introducing e.g. masses in thepropagator (117) or changing the Lorentz covariance of the field. In next section we shallpresent a perturbative solution of the evolution equations, in particular of (157), based on theFeynman diagram decomposition and on a slight variant of Zimmermann’s subtraction formula.This states a connection between the present RG construction and the well known BPHZ one.
45
11 The subtraction method
We are going to show that the subtraction method gives a solution to the evolution equation(157) satisfying the integral equations (162) and (168). Leaving open the possibility of fixingthe initial values at ΛR = 0 we can modify the propagator (157) introducing a mass:
˜S(p) =
e− p2
Λ20 − e−
p2
Λ2
p2 +M2(174)
and hence:
˙S(p) = − p2e−
p2
Λ2
Λ2(p2 +M2)(175)
and˜C(p) ≡
(˜S)−1
(p) =p2 +M2
e− p2
Λ20 − e−
p2
Λ2
. (176)
The details of Zimmermann’s construction can be found in the lectures notes by Lowensteinand Zimmermann quoted in the bibliography. Owing to the fact that the subtracted Feynmanintegrals are absolutely convergent also in the absence of our exponential cut-off factors, andin agreement with our conclusions about the evolution equations in the UV ( Λ0 → ∞) limit,we shall understand this limit setting in the following formulae:
˜S(p) =
1− e−p2
Λ2
p2 +M2, (177)
for simplicity we shall also choose once and for all c2 = c′2 = 0 and c4 = g.Considering now the generic n-point function in the UV limit: < φ(p1) · ·φ(pn−1)φ(0) >
that we shorten to Sn(p,Λ), we assume the reader to be familiar with the construction of the,possibly divergent, Feynman integrals. These are associated with to a certain set of diagramswith vertices corresponding to the gφ4/4! interaction (and possibly to two-leg vertices thatwill be induced by the subtraction procedure) and with the propagator (177). S2n appears
as a formal power series in g of the form: S2n =∑∞L=1 S
(m)2n where S
(m)2n gives the sum of the
contributions from the diagrams with 2n external legs and m loops. These are finite in number.The subtraction formula refers to the integral corresponding to each of these diagrams, let usconsider the diagram Γ.
The unsubtracted, and hence possibly divergent, Feynman integral contributing to S(m)2n and
corresponding to Γ has the form:
SΓ(p) =∫ d4mk
(2π)4mIΓ(p, k) , (178)
where k ≡ k1, ...., km is a basis of internal momenta of the diagram, to be better specified, andp ≡ p1, ...., p2n−1 a basis of external momenta. The integrand is given by:
IΓ(p, k) = cΓ
2(m+n−1)∏l=1
S(pl + kl) ≡ cΓ
∏l∈L(Γ)
S(pl + kl) , (179)
46
where l labels the lines of the diagram and pl and kl are linear combinations of the p’s and ofthe k’s respectively giving the momentum flow through the line l. cΓ is a combinatorial factor.Notice that we limit ourselves considering a very special and simple class of diagrams which,e.g. excludes the presence of the polynomial factors in the momenta that appear whenever onehas fermions and/or operators depending on the field derivatives. As a matter of fact two-legvertices carrying factors linear in the leg squared momenta are induced by the subtraction,these complicate the form of IΓ without changing our conclusions. We believe that the examplewe are presenting are sufficient to give a clear enough idea of the subtraction procedure and ofits correspondence with the result of the RG construction. In any case students interested inthe most general situation can find every needed information in the cited references.
In order to make (179) more precise, let us remind that a diagram is identified by a set V (Γ)of vertices joined by a set L(Γ) of lines, a trivial diagram has a single vertex and no lines, wehave already specified what we mean by 1-particle irreducible (1-PI) diagram. A sub-diagram ofa diagram corresponds to a subset of vertices and a subset of lines joining them. A sub-diagramis, of course contained in the original diagram, two sub-diagrams of the same diagram that havelines and/or vertices in common, either are contained into one another, or intersect, one saysthey overlap. The presence of divergent overlapping divergent diagrams corresponds to that ofoverlapping divergences. Let us now consider how the p’s and k’s above can be assigned. Asan example we consider the diagram in the following figure:
&%'$
A B
1
2
3
(180)
it has three lines 1, 2, 3, two vertices A,B and a single external legs attached to each vertex,thus it contributes to V
(2)2 . It is a 1-PI diagram that we label (A,B/1, 2, 3) indicating the sets
of vertices and lines, its 1-PI sub-diagrams are (A,B/1, 2), (A,B/1, 3) and (A,B/2, 3). Theyoverlap
The momentum assignment begins from the k’s whose basis is given considering two loops,say (1, 2) and (2, 3), and two momenta: k1 flowing clockwise around the first loop and k2 flowingclockwise around the second. In this way we have k1 = −k1, k2 = k1 − k2 and k3 = k2. Wehave oriented the positive line momentum flow from B to A. Now we consider the flow ofthe external momentum p entering into B. This is determined by Kirchhoff’s law of circuittheory. The first law is
∑lV ±plV + pV = 0 for every vertex, where the sum runs over the
lines attached to the vertex with the sign taken positive if the line momentum flows in andnegative if it flows out, and pV is the total momentum flowing into V through the externallegs and, in the case of a sub-diagram, the lines not belonging to the sub-diagram. This lawcorresponds to momentum conservation at the vertex. The second law refers to any loop Land prescribes that
∑lL ±plLρlL = 0 where the sum runs over the lines forming the loop with
the sign taken positive if the line momentum flows clockwise and negative in the opposite case.The ρl’s are resistances associated with the lines, they can possibly vanish or diverge with thecondition that loops formed by lines with resistance identically zero or infinite are forbidden.
47
In the example, assigning the same resistance to the three lines one has pl ≡ p/3. Therefore theexternal momentum flow through the lines of (A,B/1, 2, 3) is p1 = p2 = p3 = p/3 and henceone has in the case of our example:
IΓ(p, k) =1
6S(p/3− k1)S(p/3 + k1 − k2)S(p/3 + k2) . (181)
The subtraction procedure requires that a momentum routing be also assigned to each 1-PIsubdiagram keeping the same resistances and considering as external the momenta flowing intoits vertices through the external legs and the lines not belonging to the sub-diagram. Thereforeconsidering e.g. γ = (A,B/1, 3) one has p
(γ)1 = p
(γ)2 = (p+k2−k1)/2 and k
(γ)1 = −kγ, k(γ)
2 = kγ.Comparing this with the momentum routing of the whole diagram one finds that the newinternal momentum kγ = (k1 + k2)/2 depends only on the k’s. This is a general consequence ofthe momentum assignment procedure.
Now we come to the construction of the subtraction scheme. This is based on the introduc-tion of the forests of sub-diagrams, these are sets of non overlapping non-trivial sub-diagrams ofΓ including possibly Γ itself. We call FΓ the set of all forests of Γ. In our example FΓ containseight forests, the empty forest, 4 forests with a single element, be it the whole diagram or itssub-diagrams mentioned above, and three forests with two elements: the whole diagram andone of its sub-diagrams.
For each diagram/sub-diagram γ we introduce the operator Sγ which replaces the momen-tum routing of the original diagram with that of γ and the Taylor operator tdγ (d integer) whoseaction on Iγ(p, k), the factor of IΓ(p, k) corresponding to the lines of γ is threefold:
• tdγ replaces Λ with ΛR in the propagators
• tdγ takes the p(γ) Taylor expansion up to degree dγ, of Iγ(p, k), considered as function of
p(γ) and k(γ)
• tdγ gives zero if dγ is negative.
Finally we set dγ equal to the superficial divergence of γ, i. e. in our simple example, dγ = 4−nγwhere nγ is the number of external legs/lines of the diagram/sub-diagram γ. That is the numberof external legs and the lines not belonging to the sub-diagram attached to the vertices of γ.In the example dΓ = 2 and dγ ≡ 0 for the three sub-diagrams.
Having set this apparatus the subtracted Feynman integral is obtained replacing into (178)IΓ(p, k) with:
RΓ(p, k) ≡ SΓ
∑F∈FΓ
∏γ∈F
(−tdγSγ)IΓ(p.k) . (182)
In the case of our example:
RΓ(p, k) =1
6(1− t2p)
[S(p/3− k1)S(p/3 + k1 − k2)S(p/3 + k2)
−S(p/3− k1)S(k1/2− k2)S(k2 − k1/2)
−S(−k1/2− k2/2)S(p/3 + k1 − k2)S(k1/2 + k2/2)
−S(k2/2− k1)S(k1 − k2/2)S(p/3 + k2)], (183)
48
here t2p takes the Taylor expansion in p up to the second degree.In order to prove the absolute convergence of the Feynman integral
∫d4mkRΓ(p, k) one has
to verify that, on every hyperplane of dimension 4a with a ≤ m in R4m, |RΓ(p, k)| vanishes atinfinity faster than d−4a where d is the distance from the origin. Considering our example ande.g. the hyperplane spanned by k2 one sees immediately that the first two terms in the righ-thand side, each of which would give a contribution d−4, in fact combine to give a term goingas d−5, while the remaining terms vanish as d−6. The same holds true for the 4-dimensionalhyperplanes spanned by k1 and k1 = k2. However on the whole R4m we see that the contentof the bracket in (183) vanishes at infinity as d−6 which is clearly not enough. Now it is clearthat asymptotically the mass and the exponential in (177) do not play any role and hence thecontent of the bracket in (183) tends to a homogeneous function of the momenta p and k ofdegree −6. It is also clear that the operator 1 − t2p in front of the bracket selects terms ofdegree higher than 2 in p and hence, asymptotically, terms of degree lower than −8 in a genericdirection in R8 thus implementing the absolute convergence criterion on R8. This should givean idea of how the subtraction method works. It should be also clear that the exponential in(177) does not play any role in the UV regime and hence the general convergence proof givenby Zimmermann works also in our case.
Therefore we see that the subtraction method systematically applied to the expansion ofthe coefficient functions Vn(p,Λ) in (158) (we have omitted the∞ label) generates formal seriesexpansions in g of these coefficients such that V4(0,ΛR) = g and V2(p,ΛR) = O(p4) and hencecorresponding to the choice c2 = c′2 = 0 and c4 = g.
It remains to prove that the effective proper generator VΛ[φ] corresponding to the coefficientfunctions Vn(p,Λ) so obtained satisfies the evolution equation (157) with the chosen initialconditions. In order to prove this let us take the Λ-derivative of a generic subtracted Feynmanintegral corresponding to a 1-PI diagram and hence contributing to VΛ. Due to the absoluteconvergence of the momentum integral we are allowed to commute derivative with momentumintegration and hence we come to the k-integral of ∂ΛRΓ(p, k). The crucial point in this proofis that an un-subtracted Feynman integrand depends on Λ only through the propagators S andthat the sub-diagram subtraction terms generated by the Taylor operators tdγγ are Λ-independentsince they are computed at Λ = ΛR. Thus we have to consider in (182) the contributions ofthe propagators of un-subtracted sub-diagrams.
For a generic 1-PI diagram Γ one has:
RΓ(p, k) = (1− tdΓΓ )RΓ(p, k) (184)
whereRΓ(p, k) = SΓ
∑F∈F ′Γ
∏γ∈F
(−tdγSγ)IΓ(p.k) (185)
and F ′Γ is the set of forests non containing Γ as an element. In other words, computing RΓ(p, k)one excludes the subtraction of the whole diagram. This distinction is in order since, thepossible subtraction being made at Λ = ΛR one has:
∂ΛRΓ(p, k) = ∂ΛRΓ(p, k) , (186)
which is equivalent to restricting the sum over the forests in (186) to F ′Γ.
49
Now some more diagrammatic analysis is needed. For every forest F in F ′Γ, γ ∈ F is amaximal element of F if it is not contained into other elements of F . We define the maximalsub-forest F of F , the set of maximal elements of F and we label by F ′Γ the set of maximalsub-forests in F ′Γ. Notice that F ′Γ coincides with the set of forests made of mutually disjointsub diagrams of Γ.
It is clear that any forest F in F ′Γ is equal to the union of non empty forests Fγ in theelements γ of F , that is:
F ≡ ∪γ∈FFγ|Fγ 6=∅ . (187)
Therefore we can write (185) as:
RΓ(p, k) = SΓ
∑F∈F ′Γ
∏γ∈F
∑Fγ∈Fγ ,Fγ 6=∅
∏γ∈Fγ
(−tdγSγ)IΓ(p.k) . (188)
Given a maximal sub-forest F of Γ we define the reduced diagram Γ/(∏γ∈F γ) which is built
with the lines and vertices of Γ not belonging to any element of F and of a further set ofvertices corresponding to the elements γ of F shrunk to point vertices. The reduced diagramΓ/(
∏γ∈F γ) is relevant to our discussion since the corresponding integrand identifies the part
of IΓ(p.k) which is not changed by the subtraction operation corresponding to the forest F .Indeed one can write:
RΓ(p, k) = SΓ
∑F∈F ′Γ
∏γ∈F
((−tdγγ Sγ)Rγ(p, k))
IΓ/(∏γ∈F γ)(p, k) (189)
This expression is identical to that associated with a diagram coinciding with the reduceddiagram in which the vertices corresponding to the elements γ of F carry a factor equal to(−tdγγ Sγ)Rγ(p, k). This factor, i.e. the bracket above, is, of course, Λ-independent. Therefore,inserting (189) into (186) one has:
Λ2∂Λ2RΓ(p, k) = SΓ
∑F∈F ′Γ
∏γ∈F
((−tdγγ Sγ)Rγ(p, k))
Λ2∂Λ2IΓ/(∏γ∈F γ)(p, k) (190)
= SΓ
∑F∈F ′Γ
∏γ∈F
((−tdγγ Sγ)Rγ(p, k))
∑l∈L(Γ/(
∏γ∈F γ))
˙S(pl + kl)IΓ/(
∏γ∈F γ∪l)
(p, k)
where we have also used the fact that the Λ-dependence comes from the propagators andΓ/(
∏γ∈F γ ∪ l) means the reduced diagram Γ/(
∏γ∈F γ) deprived of the line l.
Now we interchange the summation over the forests with that over the line l upon whichthe Λ-derivative acts. This is possible since every line contributes to the above sum in corre-spondence with the forests F in F ′Γ whose elements do not contain l. If we extend the idea offorest to diagrams, such as Γ/l which are connected but not necessarily 1-PI, the set of forestswe are speaking of is FΓ/l which is, of course, contained in F ′Γ. Thus we get:
Λ2∂Λ2RΓ(p, k) = SΓ
∑l∈L(Γ)
˙S(pl + kl)
∑F∈FΓ/l
∏γ∈F
(−tdγSγ)IΓ/l(p, k) , (191)
50
Let us now consider the possibility of Γ/l not being 1-PI. Γ/l is however connected and hence itdecomposes according to its skeleton structure. In the present situation, in which the diagramis obtained cutting a line from a 1-PI diagram, Γ/l is a chain 1-PI sub-diagrams pairwiseconnected by lines. Therefore IΓ/l(p, k) factorizes into a product of line and 1-PI factors, oneof the end points of the line l above being attached to the first 1-PI sub-diagram of the chainthe other one to the last. Labelling these sub-diagrams by αi , i = 0, .., n we can write:
IΓ/l(p, k) = Iα0(p, k)n∏i=1
S(pi + ki))Iαi(p, k) . (192)
If Γ/l is 1-PI, the product above reduce to a single factor. Now a forest F in Γ/l appears as theunion of, possibly trivial, forests in the 1-PI sub-diagrams, therefore the sum over the forestsin FΓ/l decomposes into the product of the sums over the Fαi ’s and hence we have:
Λ2∂Λ2RΓ(p, k) = SΓ
∑l∈L(Γ)
˙S(pl + kl)
∑F∈FΓ/l
∏γ∈F
(−tdγSγ)IΓ/l(p, k) = SΓ
∑l∈L(Γ)
˙S(pl + kl)
Sα0
∑F∈Fα0
∏γ∈F
(−tdγγ )Iα0(p, k)
n∏i=1
S(pi + ki))
Sαi ∑F ′∈Fαi
∏γ′∈F ′
(−t′d′γ
γ )Iαi(p, k)
= SΓ
∑l∈L(Γ)
˙S(pl + kl)Rα0(p, k)
n∏i=1
[S(pi + ki))Rαi(p, k)
]. (193)
Summing over all the possible diagrams it clearly appears that the structure of the rightmostterm of this equation coincides with that of the right-hand side of the evolution equation of theeffective proper generator VΛ[φ] (157) and, of course, to that of its coefficient functions whosefirst two terms are shown, after scaling of the fields, in (160). Indeed one finds a chain of 1-PI
amplitudes connected by propagators S and closed by˙S.
The only point that remains to verify is the correct counting of diagrams. In generalthis is automatically guaranteed by the recourse to the functional method upon which (157)is based. Just to clarify this point with an example, let us consider the three line two legdiagram discussed in some detail above. This diagram seems to violate what just claimed,indeed it has three indistinguishable lines, and hence its Λ-derivative gives three identical
contributions in which˙S is linked to a single diagram with two identical lines; in a diagrammatic
expansion of (157) this diagram should appear only once. This is however a wrong argumentsince it forgets the combinatorial factors of the diagrams. A diagram with N sets of ni, i =1, ..., N indistinguishable lines carries a combinatorial factor equal to 1/(
∏Ni=1 ni!) that is 1/6
in the example. Combining the three identical contributions from the three lines together weget the resulting contribution to the evolution equation with weight 1/2 which is exactly thecombinatorial factor of the corresponding diagram with two identical lines.
In conclusion we have shown that, adapting the subtraction method to the Feynman di-agrams built of the propagator S given in (177), and possibly of its spinor, or gauge fieldvariants, yields to a diagrammatic construction of VΛ[φ] solving the RG evolution equation(157). Furthermore, considering the functional generator of the sum of all the subtracted 1-PIdiagrams at Λ = ΛR and comparing it with (172) we see, as already noticed, that the part
51
with dimension up to four, corresponding to the integral in (172), only receives contributionsfrom the trivial diagrams, since tdΓ
Γ RΓ = tdΓΓ (1 − tdΓ
Γ )RΓ = 0. Thus in the present case thesubtracted functional generator satisfies (172) with c2 = c′2 = 0 and c4 = g. While, concern-
ing V(D>4)
ΛRand recalling that the coefficients of its field expansion correspond to superficially
convergent diagrams, one easily sees that it vanishes in the Λ→∞ limit using the inequality:(1 − exp(−p2/Λ2))/p2 ≤ 2/(p2 + Λ2), and pure scale arguments. Therefore we conclude thatthe construction of the effective proper generator VΛ[φ] by the subtraction method leads to thesame evolution equation and the same boundary condition as the Wilson construction, thus itleads to the same result.
At this point it is natural to ask what is the difference between the RG and the subtractionapproach? Let us disregard the choice of the infra-red cut-off, we can skip it inserting themass as above and setting ΛR = 0. The main difference, from the UV point of view, isthat the subtraction approach deals with one diagram at a time, the forest decompositionsolving the problem of overlapping divergences. On the contrary the RG approach groups thecontributions of many diagrams together and overlapping divergences are disentangled takingthe Λ-derivative and integrating the evolution equation. What is not clear in the RG approachis how the diagrams should be selected. In an asymptotically free theory one would selectiteratively the asymptotically dominant contributions, this is however still to be done. Muchless ambitious is the criterion of selecting all the diagrams with the same number of loops,which corresponds to the order in h, thus building an iterative solution to (157) ordered in h;we have just shown that in this case what one gets is identical to what one obtains applyingthe subtraction method to a loop ordered diagrammatic expansion. The problem is that veryoften the need of the above selection is overlooked and hence people study arbitrarily selectediterative solutions to (157).
Before concluding this section we have to mention the existence of further versions of thesubtraction method among which by far the most used is based on dimensional regularizationand follows the lines given by Breitenlohner and Maison in the papers quoted in the bibliog-raphy. In dimensional regularization one transforms the momentum integral associated with aFeynman diagram into a parametric integral writing the (massless) propagator according:
˜S(p) ≡
∫ 1/Λ2
0dα e−αp
2
(194)
and extending the resulting Gaussian momentum integral in d dimensions through:
∫ddpe−Xp
2
=(π
X
) d2
.
The resulting expression for a Feynman integral, computed for Euclidean independent momentawithout vanishing partial sums, that is in the non-exceptional situation, corresponds to a mero-morphic function in d and the subtraction procedure consists in subtracting the pole terms ind = 4 coming from the α-integral in the neighborhood of the origin.
A technical aspect of this procedure is related to the change of the physical dimensionsof the parameters due to the extension of the theory to d Euclidean dimensions. Let µ be areference mass scale. In d dimensions the fundamental scalar field φ has the mass dimension
52
of µ(d−2)/2 and hence the φ4 coupling constant g has the mass dimension of µ4−d, this meansthat, in order to extend the theory to d dimensions, one has to replace g by gµ4−d. Considerthe simple example of a diagram with two identical lines joining two vertices, the rest of ourformer example once a line has been taken away, the corresponding Feynman integral is:
g2µ8−2d
2
∫ ddk
(2π)d1
k2(p+ k)2=g2µ8−2d
2
∫ ddk
(2π)d
∫ 1/Λ2
0dαdβ e−(αk2+β(p+k)2)
=g2µ8−2d
2(4π)d/2
∫ 1/Λ2
0
dαdβ
(α + β)d/2e−
αβα+β
p2
= g2µ4−d (Λ/µ)d−4
2(4π)d/2
∫ 1
0
dxdy
(x+ y)d/2e−
xyx+y
p2
Λ2 , (195)
its minimally subtracted version is:
g2µ4−d[
(Λ/µ)d−4
2(4π)d/2
∫ 1
0
dxdy
(x+ y)d/2e−
xyx+y
p2
Λ2 − 1
(4π)2(4− d)
]. (196)
The general structure of the unsubtracted Feynman integral corresponding to a generic 1-PIdiagram with m loops and 2n external legs of a scalar field theory with only φ4 couplings is:
cΓgn+m−1µd−n(d−2)
(4π)md/2(Λ/µ)m(d−4)+4−2n
∫ 1
0
∏2(m−1)+ni=1 dxiDd/2(x)
e−
∑2n−1
a,b=1Na,b(x)pa·pb
Λ2D(x) , (197)
where the indices a and b label n+m−2 of the n+m−1 vertices of the diagram, pa and pb arethe external momenta entering into the corresponding vertices, D and Na,b are homogeneouspolynomials of degree m and m + 1 respectively. It can be shown that in the hypercubexi ≤ 1 , i = 1, .., 2(m− 1) + n, Na,b ≤ D and hence the singularities of the integral come fromthe vanishing of D in the origin of the x-space. A systematic description of the subtractionprocedure is given by Breitenlohner and Maison. It is clear that repeating the analysis of thesubtracted theory in the present case would lead to completely analogous evolution equationsprovided the coefficient of the subtracted terms be chosen Λ-independent as above.
53
12 Bases of local operators in the subtraction scheme
and the Wilson operator product expansion
Let us now consider how the bases of independent local operators discussed in the previoussection appear in the subtraction scheme. We consider in particular the example discussed atthe end of section [10], that is: in a neutral scalar theory invariant under reflection φ → −φwe consider a basis of dΩ = 4 local operators transforming as an, even under reflection, scalarfield. The basis has 4 elements which have been characterized in the framework of the RGconstruction specifying the corresponding initial conditions at Λ = ΛR. These conditionsconcern the value, a1, of the 1-PI vertex with one ω leg and 4φ-legs at zero momenta andthe coefficients of the Taylor expansion up to degree 2 in the momenta of the φ-legs of thevertex with one ω-leg and 2 φ-legs, this expansion is identified by three coefficients accordingto a2 + a3p1 · p2 + a4(p2
1 + p22). Now it is apparent that three elements of the basis, that is those
corresponding to the choices ai = δi,j with j = 1, 3, 4 correspond, after subtractions, to theoperators φ4/4!, (∂φ)2/2 and φ∂2φ . Indeed the 1-PI vertices involving one of these operatorsand with n < 5 legs are superficially divergent and hence, according to (184) the correspondingfunctions have the structure: ∫
dk(1− tdΓ)RΓ(p, k) (198)
where d = 4−n. Therefore one has zero at Λ = ΛR and zero momenta for the vertices with 2 and4 legs together with their momentum derivatives of degree d. The only contributions which arenot subtracted are those corresponding to the trivial, single vertex zero loop, diagrams whichgive the chosen values of the ai’s.
Things are quite different if we choose ai = δi,2 since the operator that would give the wantedinitial condition is φ2/2, which has dimension two and hence the degree of the subtraction, din the above formula, is reduced by 2. Thus the only subtracted 1-PI vertex is that with twolegs and hence for this operator none of the ai’s vanishes, in particular a2 = 1. The situationwould be different if the subtraction degree d were systematically increased by two for thevertices containing this operator as an internal vertex. This is what is called extra-subtraction.It corresponds to considering the operator which has in fact dimension 2 as a dimension 4operator. In Zimmemann’s notation this is N4[φ2/2] which, of course, differs from the normallysubtracted one, that in the same notation is N2[φ2/2].
Zimmemann’s notation for a generic local operator is: Nδ[M ] and specifies a polynomialM in the fields and their derivatives, which identifies the structure of the Feynman diagramvertex associated with the operator, and an index δ from which the subtraction degrees of the1-PI vertices is computed. In the scalar field case the subtraction degree dγ for a diagram γwith n external legs is δ minus the number of external legs (dγ = δ − n). In the case of localoperators δ must be larger or equal to the dimension of M in order the Feynman integrals beabsolutely convergent. Comparing this discussion to the analysis in section [10] we see that δ
coincides with dΩ while d(m)Ω is the naive mass dimension of the operator. We also see that M
identifies the initial conditions of the operator evolution at Λ = ΛR. Thus extra subtractionsappear whenever d
(m)Ω < dΩ.
As a matter of fact one could have introduced many other ways of increasing the subtraction
54
degrees of sub-diagrams containing an operator as a vertex. For example one could have chosena subtraction degree dependent on how the operator vertex is connected to the rest of thediagram, in particular in the cases in which some of the lines attached to the operator vertexare external lines of the sub-diagram one could choose a subtraction degree dependent on thenumber and choice of these lines. In this case one speaks of anisotropic subtractions, they lieoutside the scope of this text.
Coming back to our example we notice that we have found two different bases, the firstone with elements N4[φ4/24], N4[(∂φ)2/2], N4[φ∂2φ] and N4[φ2/2], while in the second one thelast element is replaced by N2[φ2/2]. It is clear that the elements of the second basis can bewritten as linear combinations of the elements of the first one. In particular this holds true forN2[φ2/2] which can be written in the form:
N2[φ2/2] = e1N4[φ4/24] + e2N4[φ2/2] + e3N4[(∂φ)2/2] + e4N4[φ∂2φ] . (199)
In order to make this linear relation explicit we must compute the coefficients ei. With thispurpose we introduce the notation < N2[φ2/2(0)]φ(p1) · · · φ(pn)|Λ >1−PI , for the sum of theperturbative contributions to the 1-PI function containing a vertex corresponding to the oper-ator φ2/2 with the subtraction degree corresponding to the N2 prescription and we notice thatthese functions can be computed using both sides of (199). In particular the functions obtainedusing the right-hand side of (199) are easily computed taking into account the definition ofN4[M ] operators. Indeed, for example, choosing the amplitude < N2[φ2/2(0)](φ(0))4|Λ >1−PIone has immediately that the contribution of the right-hand side is only due to N4[φ4/24] andhence this gives the value of e1. Thus the complete decomposition is given by:
N2[φ2/2] = N4[φ2/2]+ < N2[φ2/2(0)](φ(0))4|Λ >1−PI N4[φ4/24]
+1
4∂p1 · ∂p2 < N2[φ2/2(0)]φ(p1)φ(p2)|Λ >1−PI |pi≡0 N4[(∂φ)2/2]
+1
8∂2p1< N2[φ2/2(0)]φ(p1)φ(p2)|Λ >1−PI |pi≡0 N4[φ∂2φ] . (200)
This is Zimmermann’s reduction formula applied to the present example.The use of operator bases that has led us to the above reduction formula can be extended
to analyze properties of products of operators. This can lead to the formal proof of the renownWilson operator product expansion (OPE) which has been given by Zimmermann. We shalldiscuss the simplest possible example of OPE with the aim of giving an idea of the method ofthe general proof and, at the same time, showing how the coefficients should be computed.
Consider the operator φ(x)φ(0), this is an operator whose Feynman amplitudes have alreadybeen defined in the framework of our renormalized theory. What is ill-defined is its x →0 limit. Instead of φ(x)φ(0) let us introduce N2[φ(x)φ(0)] applying our modified version ofZimmermann’s subtraction prescriptions to the 1-PI diagrams containing φ(x)φ(0) consideredas a single vertex (operator) of dimension dΩ = 2. This is a new operator which is expectedto have regular x → 0 limit since N2[φ2/2(0)] is a well defined operator and the subtracted 1-PI functions containing it, or else N2[φ(x)φ(0)], correspond to absolutely convergent Feynmanintegrals uniformly in x. It is therefore allowed the exchange of the x → 0 limit with themomentum integral.
55
Let us now consider the difference:
∆(x) ≡ φ(x)φ(0)−N2[φ(x)φ(0)] . (201)
It is clear from the above comments that this difference does not vanish due to the extra-subtractions that we have inserted into the definition of N2[φ(x)φ(0)]. Let us consider theset of subtracted, but not necessarily connected, Feynman diagrams involving the productφ(x)φ(0). These diagrams are clearly not 1-PI, however one can still define the set of theirforests FΓ and one finds for the subtracted n-point functions expressions completely analogousto the integral of (182).
On the contrary, if the two-point operator is N2[φ(x)φ(0)], and thus is considered as asingle vertex, the un-subtracted integrand IΓ(x, p, k) does not change, while, whenever bothpoints belong to the same connected part of the whole diagram, the set of forests FΓ is largerthan FΓ and contains it. Indeed, if one considers the two-point operator as a single vertex,all the connected sub-diagrams containing both fields give rise to 1-PI sub-diagrams whichwere not included in any forest of FΓ. Among these diagrams are considered divergent thosewhich are disconnected from the rest of the diagram, which thus factorizes into the productof the two-point function < φ(x)φ(0)|ΛR >C and the rest, and those which are connectedto the rest of the diagram by two lines. The diagrams contributing to the first set are justdisregarded, if the two-point operator is N2[φ(x)φ(0)], since the two point factor is consideredas the vacuum expectation value of N2[φ(x)φ(0)] which must be subtracted. If instead the splitvertex N2[φ(x)φ(0)] is contained in sub-diagrams connected to the rest through two lines, thesesub-diagrams are subtracted at vanishing momenta of the two lines while this subtraction doesnot appear for the field product.
Therefore the Green functions containing ∆ receive contributions from two kinds of dia-grams, the first kind corresponds to diagrams which factorize into a diagram with only twoexternal legs corresponding to φ(x) and φ(0) and the rest, while in the diagrams of the secondkind the field product is connected to the rest by two external lines. Of course one excludesfrom the second kind diagrams which belong to the first one.
Considering diagrams of the second kind let us call FΓ,2 the sub-set of FΓ whose elementsare forests containing at least one 1-PI sub-diagram including the split vertex and connectedto the rest by two lines, let us call γ2 the sub-diagrams satisfying this condition. It is clear thatthe contribution of a diagram of the second kind to a Green functions containing ∆ correspondsto the subtracted integrand:
∆RΓ(x, p, k) = SΓ
∑F∈FΓ,2
∏γ∈F
(−tdγSγ)IΓ(p.k) . (202)
Now let γ(F )2 be the smallest γ2 in the forest F in FΓ,2, let F< be the sub-forest of F strictlycontained in γ(F )2 and F> = F (F<⊕ γ(F )2) , let furthermore IΓ/γ(F )2(p.k) be the integrandof the reduced diagram Γ/γ(F )2 we have:
∆RΓ(x, p, k) = SΓ
∑F∈FΓ2
∏γ∈F>
(−tdγSγ)IΓ/γ(F )2(p, k) (t0γ(F )2Sγ(F )2)
∏γ′∈F<
(−tdγ′Sγ′)Iγ(F )2(x, p, k) .
(203)
56
Once again we can exchange the sum over the forests in FΓ,2 with a sum over the sub-diagramsγ2 combined with the sum over the forests in Γ/γ2, that is in FΓ/γ2 , and the sum over theforests in F ′γ2
where, keeping the notation used in section [11], the apex ′ excludes the forestscontaining γ2 as an element. It is clear that there is a one-to-one correspondence between theelements of FΓ/γ2 and the F> appearing in (203).
The two summation procedures are equivalent and using the second procedure we get:
∆RΓ(x, p, k) = SΓ
∑γ2
∑F∈FΓ/γ2
(−tdγSγ)IΓ/γ2(p, k) (t0γ2Sγ2)
∑F ′∈F ′γ2
∏γ′∈F ′
(−tdγ′Sγ′)Iγ2(x, p, k) . (204)
Now it is apparent that (t0γ2Sγ2)
∑F ′∈F ′γ2
∏γ′∈F ′(−tdγ′Sγ′)Iγ2(x, p, k) is the subtracted integrand of
the contribution of γ2 to < φ(x)φ(0)(ˆφ(0))2|ΛR >C , indeed the selection of forests correspondsto considering the field product as it is and not as a single vertex since F ′ ∈ F ′γ2
does notcontain any element containing the split vertex. On the contrary the reduced diagram Γ/γ2 issubtracted with the rules corresponding to the identification of the vertex corresponding to γ2
with a N2[φ2] operator.Therefore considering ∆(x) defined above, we have that a Green function with a ∆(x) vertex
just corresponds to the contributions coming from the subtractions due to the N2[φ(x)φ(0)]vertex considered before. Summing over all possible diagrams containing the split vertex thisresult can be expressed by the operator relation:
∆(x) =< φ(x)φ(0)|ΛR >C I+ < φ(x)φ(0)(ˆφ(0))2|ΛR >C N2[φ2(0)/2] , (205)
where I represents the identity operator whose insertion into a diagram reproduces the samediagram. Recalling (201) this last equation can be written according:
φ(x)φ(0) =< φ(x)φ(0)|ΛR >C I+ < φ(x)φ(0)(ˆφ(0))2|ΛR >C N2[φ2(0)/2] +N2[φ(x)φ(0)] ,(206)
where the first term of the right-hand side gives the most divergent part of the operator productφ(x)φ(0) in the x → 0 limit, this term diverges since < φ(x)φ(0)|ΛR >C' x−2. The secondterm contributes as lnx2 and the last term is regular.
This is Wilson’s OPE applied to the product φ(x)φ(0). This procedure can easily be gen-eralized to the case in which the fields are replaced by renormalized composite operators. Ifneeded the expansion can also be pushed further, by replacing N2[φ(x)φ(0)] with Nd[φ(x)φ(0)]with d > 2 and using Zimmermann’s reduction procedure.
57
13 The Quantum Action Principle
An important result which is easily reached in the functional approach is the quantum actionprinciple. The basic idea is to exploit the invariance of the functional integral under change ofintegration variables. We consider in particular the change of variable:
φ→ φ+ εgΛ0 ∗ σΛ0 [gΛ0 ∗ φ] , (207)
into the functional (112). Here, in the limit Λ0 → ∞, σΛ0 [φ] is a polynomial in the fields andtheir derivatives and g0 is the smearing Gaussian introduced in (105). At the first order in ε,using the short notation introduced in Eq. (116) we get:∫
dµ[φ]
((gΛ0 ∗
δ
δφIΛ0 [gΛ0 ∗ φ] + C ∗ gΛ0 ∗ φ− gΛ0 ∗ (J +
δ
δφ)
)σΛ0 [gΛ0 ∗ φ]
)
exp(φJ)− (φC ∗ φ)
2− IΛ0 [gΛ0 ∗ φ] = 0 . (208)
Notice that the term containing the φ functional derivative of σΛ0 corresponds to the first orderin ε variation of the measure, that is of the Jacobian of the transformation. This equationcan be translated into a functional differential equation inserting into the effective action σΛ0
coupled to the source ω:
expFC [J, ω] =∫dµ[φ] exp−[IΛ0 + (ωσΛ0)][gΛ0 ∗ φ] exp−1
2(φC ∗ φ) + (Jφ) , (209)
getting: ∫dµ[φ]
((gΛ0 ∗
δ
δφ[IΛ0 + (ωσΛ0)][gΛ0 ∗ φ] + C ∗ gΛ0 ∗ φ− gΛ0 ∗
δ
δφ
)σΛ0 [gΛ0 ∗ φ]
)
exp(φJ)− (φC ∗ φ)
2− IΛ0 [gΛ0 ∗ φ]+ (JgΛ0 ∗
δ
δω) expFC [J, ω] = 0 . (210)
Now let IΛ,Λ0 [φ, ω] be the solution of the evolution equation corresponding to the bare, local,operator[IΛ0 + (ωσΛ0)][φ]. We have (see (120) ):
expFC [J, ω] =∫dµ[φ] exp−IΛ,Λ0 [gΛ ∗ φ+ g ∗ S ∗ J, ω] exp−1
2(φC ∗ φ) + (Jφ) , (211)
performing the change of variable:
φ→ φ+ εgΛ ∗δ
δωIΛ,Λ0 [gΛ ∗ φ+ g ∗ S ∗ J, ω] , (212)
at the first order in ε, we get:∫dµ[φ]
((gΛ ∗
δ
δφIΛ,Λ0 [gΛ ∗ φ+ g ∗ S ∗ J, ω] + C ∗ gΛ ∗ φ− gΛ ∗ (J +
δ
δφ)
)δ
δωIΛ,Λ0 [gΛ ∗ φ+ g ∗ S ∗ J, ω]
)exp(φJ)− (φC ∗ φ)
2− IΛ,Λ0 [gΛ ∗ φ+ g ∗ S ∗ J, ω]
≡∫dµ[φ]BΛ,Λ0 [gΛ ∗ φ+ g ∗ S ∗ J, ω] exp(φJ)− (φC ∗ φ)
2− IΛ,Λ0 [gΛ ∗ φ+ g ∗ S ∗ J, ω]
+(JgΛ0 ∗δ
δω) expFC [J, ω] = 0 . (213)
58
where, of course:
BΛ,Λ0 [φ, ω] =
((gΛ ∗
δ
δφIΛ,Λ0 [φ, ω] + C ∗ gΛ ∗ φ− gΛ ∗
δ
δφ
)δ
δωIΛ,Λ0 [φ, ω]
), (214)
is the solution of the evolution equation corresponding to the bare operator:
BΛ0 [φ, ω] =
((gΛ0 ∗
δ
δφ[IΛ0 + (ωσΛ0)][φ] + C ∗ φ− gΛ0 ∗
δ
δφ
)σΛ0 [φ]
). (215)
The most important property of the symmetry breaking operator BΛ is that it appears as anintegrated operator which is local in the Λ0 → ∞ limit and whose dimension is equal to thatof the naive variation of a generic bare action under the transformation (207). Equation (213)represents what is usually called the quantum action principle. From the RG point of view onecould wonder if in some special situation the theory could be fine tuned at Λ = ΛR so thatBΛ, that is, its initial value, vanishes. As a matter of fact, in the Λ0 → ∞ limit, due to thelinearity of the operator evolution equations, if the initial value of the symmetry breaking termBΛR vanishes, BΛ identically vanishes and the theory satisfies the Ward-like identity:
(Jδ
δω)) expFC [J, ω] = 0 . (216)
Notice that it is apparent from (214) that the vanishing condition for BΛR is in fact a conditionon the effective action and hence on its initial value. BΛ vanishes in many physically interestingsituations, while in other cases it turns out that, however vanishing in the semiclassical limit,the breaking BΛ appears at the quantum correction level and cannot be fine-tuned to zero.This is the case of anomalies.
The above comments hold clearly true in the subtraction scheme in which the breakingoperator can be decomposed into the sum of the elements of the suitable operator basis andits vanishing condition can be analized using the so called algebraic method that one can findpresented in some details in the book by Piguet and Sorella quoted in the bibliography. Weshall come back to this method at the end of this section.
It is interesting to apply the above analysis to the case of free fermion current algebra; inthe simplest possible non-trivial case one considers a massless fermion field ψ in an abelianaxial vector field background Aµ. The bare Lagrangian is:
L0 = −iψ∂/ψ + ψA/ψ + iωγ5αψ − iψγ5αω + l0[A] , (217)
where the terms in α define the field gauge transformation and l0[A] represents terms whichonly depend on the background field Aµ.
In this theory the cut-off Feynman functional (112) is:
eFC [A,η,η,ω,ω,Λ0] =∫dµ[ψ]dµ[ψ] exp
[−i(ψ∂/ψ) + (ψ gΛ0 ∗ γ5A/ gΛ0 ∗ ψ) + l0[A]
+i(ωγ5α gΛ0 ∗ ψ)− i(ψ gΛ0 ∗ γ5αω) + (ηψ)− (ψω)], (218)
59
where we have understood space-time integrals and l0 =∫dx l0. Considering the spinor field
change:
ψ → ψ + igΛ0 ∗ γ5α gΛ0 ∗ ψψ → ψ + iψ gΛ0 ∗ γ5α ∗ gΛ0 , (219)
and that of A:A→ A+ ∂α , (220)
at the first order in α we have:∫dµ[ψ]dµ[ψ]
(δA l0[A] + i(ψ gΛ0 ∗ [α gΛ0 ∗ gΛ0 ∗ A/− A/ gΛ0 ∗ gΛ0 ∗ α] gΛ0 ∗ ψ)
)exp
[−i(ψ∂/ψ) + (ψ gΛ0 ∗ γ5A/ gΛ0 ∗ ψ) + l0[A] + (ηψ)− (ψω)
]+
[(η gΛ0 ∗
δ
δω)− (
δ
δωgΛ0 ∗ η)
]expFC [A, η, η, ω, ω,Λ0]
= (∂αδ
δA) expFC [A, η, η, ω, ω,Λ0] ≡ δA expFC [A, η, η, ω, ω,Λ0] , (221)
where one does not find contributions from the Jacobian of the functional measure since γ5 istraceless. Notice that, if l0[A] only depends on ∂µAν − ∂νAµ ≡ Fµν , the bare breaking termreduces to:
i(ψ gΛ0 ∗ [α (gΛ0∗)2 A/− A/ (gΛ0∗)2 α] gΛ0 ∗ ψ) , (222)
and tends in the Λ0 → ∞ limit to the commutator [α,A] which, of course, vanishes. Thusnaively one would say that in the UV limit one recovers the exact Ward identity:[
(ηδ
δω)− (
δ
δωη)
]expFC [A, η, η, ω, ω] = δA expFC [A, η, η, ω, ω] , (223)
this result is however not guaranteed since the naive functional integral does not make sense.To verify the validity of (223) one has to discuss the UV limit of the effective theory.
The effective action of this theory can be computed directly taking into account its in-terpretation as the functional generator of connected-amputated functions corresponding to asuitable cut-off propagator. In the present case the propagator is:
ΣΛ,Λ0(p) ≡g2
Λ0− g2
Λ
p/, (224)
and the effective action:
IΛ,Λ0 [A, ψ, ψ, ω, ω] = −(ψΓΛ,Λ0ψ)− i(ωγ5α(1− ΣΛ,Λ0 ∗ ΓΛ,Λ0)ψ)
+i(ψ(1− ΓΛ,Λ0 ∗ ΣΛ,Λ0)αγ5ω)−WΛ,Λ0 [A] + II [Fµν ] , (225)
where:
WΛ,Λ0 [A] =∞∑n=1
(−1)n−1
nTr((γ5A/ΣΛ,Λ0∗)n) + VΛR [A] . (226)
60
The functional VΛR [A] accounts for possible terms introduced through the initial conditions ofthe evolution equation and hence it is a space-time integrated local operator of dimension 4.ΓΛ,Λ0 satisfies:
ΓΛ,Λ0 = γ5A/(1− ΣΛ,Λ0 ∗ ΓΛ,Λ0) = (1− ΓΛ,Λ0 ∗ ΣΛ,Λ0)γ5A/ . (227)
Now from (220) one has:
δAΓΛ,Λ0 = γ5[∂/, α](1− ΣΛ,Λ0 ∗ ΓΛ,Λ0)− γ5A/ΣΛ,Λ0 ∗ δAΓΛ,Λ0
= (1− ΓΛ,Λ0 ∗ ΣΛ,Λ0)γ5[∂/, α](1− ΣΛ,Λ0 ∗ ΓΛ,Λ0)
= i(1− ΓΛ,Λ0 ∗ ΣΛ,Λ0)γ5[p/, α](1− ΣΛ,Λ0 ∗ ΓΛ,Λ0) , (228)
where we have used:(1 + γ5A/ ΣΛ,Λ0∗)(1− ΓΛ,Λ0 ∗ ΣΛ,Λ0) = 1 , (229)
which follows from (227). One also has:
δAWΛ,Λ0 [A]− δAVΛR [A] =∞∑n=1
(−1)nTr(γ5[∂/, α]ΣΛ,Λ0 ∗ (γ5A/ΣΛ,Λ0∗)n
)= −i T r(γ5[p/, α]ΣΛ,Λ0 ∗ ΓΛ,Λ0 ∗ ΣΛ,Λ0) (230)
= −i T r(((gΛ0∗)2 − (gΛ∗)2)∗
[ΓΛ,Λ0γ5 ∗ ΣΛ,Λ0α− αΣΛ,Λ0 ∗ ΓΛ,Λ0γ
5]).
Now we apply (213) to our model starting from:
eFC [A,η,η,ω,ω,Λ0] =∫dµ[ψ]dµ[ψ] exp
[−i(ψ∂/ψ)− IΛ,Λ0 [A, Ψ,Ψ, ω, ω] + (ηψ)− (ψω)
], (231)
where we have set:
Ψ = gΛψ +1
p/g ∗ η , Ψ = ψgΛ − η
1
p/∗ g . (232)
First of all we evaluate the contribution in the functional integral (213) of the Jacobian of thefield transformations:
ψ → ψ + igΛ ∗ γ5α(1− ΣΛ,Λ0 ∗ ΓΛ,Λ0) ∗Ψ
ψ → Ψ (1− ΓΛ,Λ0 ∗ ΣΛ,Λ0)γ5α ∗ gΛ , (233)
this is given by:
δJΛ,Λ0 [A] ≡ Tr(δ
δψgΛ ∗
δ
δωIΛ,Λ0)− Tr( δ
δψgΛ ∗
δ
δωIΛ,Λ0)
= −i T r(gΛ ∗ [γ5α(1− ΣΛ,Λ0 ∗ ΓΛ,Λ0) + (1− ΓΛ,Λ0 ∗ ΣΛ,Λ0)γ5α] ∗ gΛ
)= −iT r
([ΣΛ,Λ0 ∗ α(gΛ∗)2 − (gΛ∗)2αΣΛ,Λ0∗]ΓΛ,Λ0 ∗ γ5
), (234)
61
once again we have taken into account that γ5 is traceless. Thus we can write (221) according:
δA expFC [A, η, η, ω, ω,Λ0] =∫dµ[ψ]dµ[ψ]
[δAJΛ,Λ0 [A] + δAWΛ,Λ0 [A] +
(ΨδAΓΛ,Λ0Ψ
)+i(Ψ[(1− ΓΛ,Λ0 ∗ ΣΛ,Λ0)γ5α(gΛ∗)2ΓΛ,Λ0 + ΓΛ,Λ0 ∗ (gΛ∗)2αγ5(1− ΣΛ,Λ0 ∗ ΓΛ,Λ0)]Ψ
)−(Ψ(1− ΓΛ,Λ0 ∗ ΣΛ,Λ0)γ5αgΛ ∗ (p/ψ − iη)
)+((ψp/+ iη)gΛ ∗ αγ5(1− ΣΛ,Λ0 ∗ ΓΛ,Λ0)Ψ
)]exp
[−i(ψ∂/ψ)− IΛ,Λ0 [A, Ψ,Ψ, ω, ω] + (ηψ)− (ψη)
]≡∫dµ[ψ]dµ[ψ]
[δAJΛ,Λ0 [A] + δAWΛ,Λ0 [A] + i
(ΨBΛ,Λ0 [A]Ψ
)+i(ηgΛ0 ∗ αγ5(1− ΣΛ,Λ0 ∗ ΓΛ,Λ0)Ψ
)− i
(Ψ(1− ΓΛ,Λ0 ∗ ΣΛ,Λ0)γ5αgΛ0 ∗ η
)]exp
[−i(ψ∂/ψ)− IΛ,Λ0 [A, Ψ,Ψ, ω, ω] + (ηψ)− (ψω)
]=∫dµ[ψ]dµ[ψ]
[δAJΛ,Λ0 [A] + δAWΛ,Λ0 [A] + i
(ΨBΛ,Λ0 [A]Ψ
)]exp
[−i(ψ∂/ψ)− IΛ,Λ0 [A, Ψ,Ψ, ω, ω] + (ηψ)− (ψω)
]+
[(η gΛ0 ∗
δ
δω)− (
δ
δωgΛ0 ∗ η)
]expFC [A, η, η, ω, ω,Λ0] , (235)
where we have set:
BΛ,Λ0 [A] = −iδAΓΛ,Λ0 + (1− ΓΛ,Λ0 ∗ ΣΛ,Λ0)γ5α(gΛ∗)2ΓΛ,Λ0 + p/γ5α(1− ΣΛ,Λ0 ∗ ΓΛ,Λ0)
+ΓΛ,Λ0 ∗ (gΛ∗)2αγ5(1− ΣΛ,Λ0 ∗ ΓΛ,Λ0) + (1− ΓΛ,Λ0 ∗ ΣΛ,Λ0)αγ5p/
= (1− ΓΛ,Λ0 ∗ ΣΛ,Λ0)γ5[p/, α](1− ΣΛ,Λ0 ∗ ΓΛ,Λ0)
+(1− ΓΛ,Λ0 ∗ ΣΛ,Λ0)γ5α(gΛ∗)2ΓΛ,Λ0 + ΓΛ,Λ0 ∗ (gΛ∗)2αγ5(1− ΣΛ,Λ0 ∗ ΓΛ,Λ0)
+p/αγ5(1− ΣΛ,Λ0 ∗ ΓΛ,Λ0) + (1− ΓΛ,Λ0 ∗ ΣΛ,Λ0)γ5αp/
= (1− ΓΛ,Λ0 ∗ ΣΛ,Λ0)γ5α(gΛ∗)2ΓΛ,Λ0 + ΓΛ,Λ0 ∗ (gΛ∗)2αγ5(1− ΣΛ,Λ0 ∗ ΓΛ,Λ0) .(236)
Now, taking into account (227) one finds:
BΛ,Λ0 [A] = (1− ΓΛ,Λ0 ∗ ΣΛ,Λ0)[α(gΛ0∗)2A/− A/(gΛ0∗)2α
](1− ΣΛ,Λ0 ∗ ΓΛ,Λ0) . (237)
In order to evaluate the contribution of this term to the Ward identity (235) in the UV limit, letus notice that the operator corresponding to the square bracket contributes by a vertex factorα(q)A/(k)[exp(−(p+ q)2/Λ2
0)− exp(−(p+ k)2/Λ20)] where p is the momentum entering into the
vertex from the left. We are keeping the Fourier transform conventions of the first chapter (11),that is:
f(x) =∫ dp
(2π)4eip·xf(p) . (238)
If A, and η and ˜η are fast decreasing functions at high momenta, the momenta p, q and kremain limited uniformly in Λ0 and hence we can take the U.V. limit directly on the vertexfactor getting a vanishing result.
62
It remains to compute: δAJΛ,Λ0 [A] + δAWΛ,Λ0 [A]. Using (231) and (234) and, once again,(227), one gets:
δAJΛ,Λ0 [A] + δAWΛ,Λ0 [A] = δAVΛR [A]− iT r[ΣΛ,Λ0 ∗ α(gΛ0∗)2 − (gΛ0∗)2αΣΛ,Λ0∗]ΓΛ,Λ0γ5
= δAVΛR [A] + iT r[α(gΛ0∗)2A/− A/(gΛ0∗)2α]ΣΛ,Λ0 ∗ (1− ΓΛ,Λ0 ∗ ΣΛ,Λ0) . (239)
The crucial point of this analysis is that, even if the content of the square bracket in the rightmost term of this equation seems to vanish in the UV limit, the trace does not vanish, as wecan see in a moment, since it involves the difference of momentum integrals that are divergentin the UV limit.
Notice that the same term in brackets appears in (237), (239) and (222). This is by no meanssurprising since the term breaking the Ward identity at the effective theory level must be thatgenerated by the evolution of (222). In other words until now we have just verified that theoperator breaking the Ward identity (235), which is the sum of (237) and (239) is the effectiveoperator corresponding the the bare operator (222), i.e. that the effective breaking operator isthe functional generator of connected and amputated diagrams containing the vertex (222) andbuilt with the propagator (224). Thus what we have done until now is just a lengthy exerciseverifying Wilson effective theory.
Now we have to analyze what remains of the breaking in the Λ0 → ∞ limit taking intoaccount that it is contained in (239) and depends on the choice of VΛR [A]. One has to keep inmind that VΛR [A], is the initial value (at Λ = ΛR) of the background field part of the effectiveaction. It remains to be determined in particular asking, if possible, that the Ward identityremains unbroken.
We expand the trace in the right-hand side of (239) in powers of A and compute one termat a time. Let us start considering Tr[α(gΛ0∗)2A/− A/(gΛ0∗)2α]ΣΛ,Λ0. Putting:
∆Λ,Λ0(p) ≡ (exp(−p2/Λ20)− exp(−p2/Λ2))/p2 , (240)
we find:
Tr[α(gΛ0∗)2A/− A/(gΛ0∗)2α]ΣΛ,Λ0
= 4i∫ dpdq
(2π)8α(q p · A(−q)[exp(−(p+ q)2/Λ2
0)− exp(−(p− q)2/Λ20)]∆Λ,Λ0(p2)
= i∫ dq
(2π)4α(q q · A(−q)[aΛ2
0 + bq2 +O(1/Λ20)] =
∫dx[aΛ2
0A · ∂α− b∂2A · ∂α]
+O(1/Λ20) = δA
∫dx[aΛ2
0A2/2− b A · ∂2A/2] +O(1/Λ2
0) . (241)
This result shows that the considered contribution to the breaking can be fine-tuned to zeroinserting the term
∫dx[−aΛ2
0A2/2 + b A ·∂2A/2] into VΛR [A]. Indeed, after this insertion, what
remains linear in A in δAJΛ,Λ0 [A] + δAWΛ,Λ0 [A] is just O(1/Λ20).
Let us now study the terms of higher degree in A. These terms are contained in:
i∞∑n=1
(−1)nTr[α(gΛ0∗)2A/− A/(gΛ0∗)2α]ΣΛ,Λ0 ∗ (γ5A/ ΣΛ,Λ0)n . (242)
63
Taking into account that, due to the factor in square brackets, in the UV limit only the mo-mentum region where p ' Λ0 gives non-vanishing contributions to the trace integral, we seethat we can limit our analysis to the terms with n = 1 and 2 in the above sum since the higherorder terms give contributions vanishing in the U.V. limit. We begin with the terms with n = 2getting:
i∫ dpdq1 · · · dq3
(2π)16α(−
∑qi)[exp(−(p−
∑qi)
2/Λ20)− exp(−(p+ q1)2/Λ2
0)]
TrA/(q1)ΣΛ,Λ0(p+ q1 −∑
qi)A/(q2)ΣΛ,Λ0(p− q3)A/(q3)ΣΛ,Λ0(p) . (243)
In this expression we have to select the terms with non-vanishing UV limit. Under the integralsign these correspond to terms of order Λ−4
0 for p ' Λ0. Thus we get:
8i∫ dpdq1 · · · dq3
(2π)16Λ20
exp−p2
Λ20
∆3Λ,Λ0
(p)[4pµpνpρ − (pµδνρ + pνδµρ + pρδµν)p2]
p · (q1 +∑
qi)α(−∑
qi)Aµ(q1)Aν(q2)Aρ(q3)
= −2
3i∫ dpdq1 · · · dq3
(2π)16p2Λ20
exp−4p2
Λ20
α(−∑
qi)(q1 +∑
qi) · A(q1)A(q2) · A(q3)
=4
3
∫ dp
(2π)4p2Λ20
exp−p2
Λ20
(exp−p2
Λ20
− exp−p2
Λ2)∫dx ∂ · α(x)A2(x)
=(
1
12π2+O(Λ2/Λ2
0)) ∫
dx ∂ · α(x)A2(x) . (244)
Once again we find a term which can be written as the variation under (220) of an integratedlocal functional of the background field A. Indeed we have found δA[1/(32π2)
∫dx(A2(x))2].
Therefore, much in the same way as the already found terms linear in A, we can fine-tune thecorresponding breaking terms to zero by a suitable choice of VΛR [A]. Finally we consider then = 1 term in (242):∫ dpdq1dq2
(2π)12α(−q1 − q2)[exp(−(p− q1 − q2)2/Λ2
0)− exp(−(p+ q1)2/Λ20)]
TrA/(q1)ΣΛ,Λ0(p− q2)γ2A/(q2)ΣΛ,Λ0(p)
= −4i∫ dpdq1dq2
(2π)12α(−q1 − q2)[exp(−(p− q1 − q2)2/Λ2
0)− exp(−(p+ q1)2/Λ20)]
εµνρσAµ(q1)q2νAρ(q2)∂σ∆Λ,Λ0(p− q2)∆Λ,Λ0(p) , (245)
selecting under the integral sign the terms of order Λ−40 for p ' Λ0, we get:
−8i∫ dpdq1dq2
(2π)12Λ20p
4α(−q1 − q2)(p · (2q1 + q2)) exp(−3p2/Λ2
0)εµνρσ)Aµ(q1)q2νAρ(q2)pσ
= −4i∫ dpdq1dq2
(2π)12Λ20p
2α(−q1 − q2) exp(−3p2/Λ2
0)εµνρσ)Aµ(q1)q2νAρ(q2)q1σ
= −4i∫ dp
(2π)4Λ20p
2exp(−3p2/Λ2
0)∫dxα(x)εµνρσ)∂µAν(x)∂ρAσ(x)
= − i
12π2
∫dxα(x)εµνρσ)∂µAν(x)∂ρAσ(x) ≡ A[α,A] . (246)
64
This last term cannot be written as the variation under (220) of an integrated local functionalof the background field A and hence it cannot be fine tuned to zero. It is an unavoidablebreaking term in the Ward identity, that is an anomaly analogous, but not identical, to theABJ anomaly. The final form of the Ward identity is:
(∂αδ
δA) expFC [A, η, η, ω, ω]−
[(η
δ
δω)− (
δ
δωη)
]expFC [A, η, η, ω, ω]
= A[α,A] expFC [A, η, η, ω, ω] , (247)
The lengthy exercise we have now concluded contains a remarkable amount of informationuseful to study the general case.
However, before considering the general case let us spend some comment on the fine-tuningthat we have repeatedly appealed to in order to recover the wanted identity. Our exampleis particularly suitable for a simple discussion since what remains at the effective level of thebare breaking is restricted to terms which only depend of the background field A. Let us,first of all, remark that we could have considered the UV limit of our model before trying toimplement the Ward identity. Indeed there is no problem with the UV limit since the modelis power counting renormalizable, however for a generic choice of the effective action initialconditions, and in particular of VΛR [A], we should have obtained a broken identity. What wehave tried to do is to search for a VΛR [A] for which the breaking vanishes. From this point ofview the fine-tuning process appears as a compensation between the effective breaking inducedby the radiative corrections and the gauge variation of VΛR [A], the initial value of the effectiveaction (more precisely on the pure background part of it.) This is in principle possible since theeffective breaking and have the same dimension: 4 in the example. In order that the fine-tuningbe complete, one needs a one-to-one correspondence between the effective breaking terms andthe independent ones in δAVΛR [A]. This appears to be almost true in our example, the onlyexception being the last term which has originated the anomaly. One could wonder about thereason of this quasi-complete correspondence. The answer to this question turns out to be quitesimple due to the pure background character of the breaking. Indeed one is in fact limiting thestudy of (221) to vanishing η, η, ω, ω. With this choice for a generic choice of the low energynormalization conditions this equation can be written:
(∂αδ
δA)FC [A] = B[α,A] , (248)
where B[α,A] represents the breaking part. In principle B[α,A] should depend on Λ and oneshould limit the discussion to Λ = ΛR, however in the present case it does not depend, sincethe left-hand side of (248) is Λ-independent. Now (248) is a functional differential equationwhich admits a consistency condition for B[α,A]. Indeed the generators of two independentinfinitesimal gauge transformation (∂α δ
δA) and (∂β δ
δA) obviously commute, thus one has:
(∂αδ
δA)B[β,A] = (∂β
δ
δA)B[α,A] . (249)
Let us now consider how B[α,A] is constrained by dimensional, covariance and parity constraints
65
(due to parity conservation it must have positive parity being α a pseudo-scalar and A a pseudo-vector). Its general form is:
B[α,A] =∫dxα(x)[c1∂ ·A+ c2∂
2∂ ·A+ c3A2∂ ·A+ c4A ·∂ A2 + c5ε
µνρσ∂µAν∂ρAσ](x)+ δAV [A] .
(250)and hence:
(∂βδ
δA)B[α,A] =
∫dxα(x)
[c1∂
2β + c2(∂2)2 + c3(2∂ · AA · ∂β + A2 + ∂2β)
+c4(∂β · ∂ A2 + 2A · ∂(A · ∂β))]
(x) + (∂βδ
δA)(∂α
δ
δA)V [A] , (251)
which must be symmetric in α and β. We leave as an exercise to the reader to verify that thisexpression is symmetric in α and β if and only if c3 = c4. Under this condition one has:
B[α,A] = (∂αδ
δA)∫dx[−c1∂ · A+ c2(∂ · A)2 − c3
2(A2)2](x) + V [A]
+c5
∫dxα(x)[εµνρσ∂µAν∂ρAσ](x) , (252)
and it is clear that the first term in the right-hand side can be fine-tuned to zero by a suitablechoice of V [A] while the last cannot. Our former calculations add to this result the actual valueof c5 = −i/12π2.
I think that this is the most convenient point for a crucial remark on Wilson approach. Itappears in the literature the idea that the presence of the infra-red cut-off induces symmetryviolations and breakings of gauge invariance. We have just shown that this is a completelywrong idea. If the bare theory were invariant, that is the bare breaking operator in (221) werenull, also the effective breaking would be null and the Ward identity would have an unbrokenUV limit. As a matter of fact, the cut-off and other non-local effects appearing in the effectiveaction, would be completely compensated by the variation of the field transformation laws,much as it partially appears in the present case.
We have discussed a simple example of a non-trivial application of the already mentionedalgebraic method. Let us try to put into evidence the aspects which have general validity. Ithink that the main points to be accounted for are the following.
A local quantized/external field transformation analogous to (219) and (220) which naivelyleaves invariant the bare action induces at the effective level a symmetry breaking BΛ whosevalue at Λ = ΛR has the same general properties, that is, Lorentz covariance, dimension andsymmetry properties, of the variation δ[VΛR ] of a generic choice of the initial value of theeffective action V . As a matter of fact BΛR = Br + δ[VΛR ] where Br accounts for the radiativecorrection terms.
In our example δ is a linear functional differential operator, however in general it is not. Inparticular it turn out to be non linear whenever the field are transformed non-linearly. BRStransformations are a typical example of non linear transformations. In the framework of loopordered perturbation theory, if the classical (zero loop) value V0 of V is uniquely identified bythe invariance constraint δ[V0] = 0, the n-th order contribution of δ[V ]n is linear in Vn, let itbe δV0Vn and hence one has δ[V ]n = δV0Vn + ρ[V ]n where ρ[V ]n is independent of Vn.
66
Thus in this situation the recursive symmetry condition is:
BΛR,n = Br,n + δV0Vn + ρ[V ]n = 0 . (253)
In complete analogy with our example one can deduce directly from (213) a consistency condi-tion for BΛR,n which is recursively written as a linear functional differential equation:
δV0 ∧BΛR,n = 0 , (254)
with δV0 ∧ δV0 = 0.The algebraic method is directly based on the analysis of the solutions of this equation and
guarantees the possibility of extending the wanted invariance to all orders of perturbation theorywhenever the general solution to the consistency condition (254) has the form BΛR,n = δV0Cn.
The advantage of this method is that the possibility of recovering the wanted symmetryfine-tuning the effective action initial values can be proven without computing Feynman di-agrams of increasing complexity, and hence the method allows rigorous proofs of all orderrenormalizability.
67
A Diagrammatic Expansions
We have shown in section 8 that the generator of Schwinger functions can be computed by thefunctional integral (112):
eFC [J,Λ0] =∫dµ[φ]e
∫dy(J(y)φ(y)− (∂φ)2
2)−IΛ0
[qΛ0]) , (255)
whose Feynman expansion is generated by (116):
eFC [J,Λ0] = e−IΛ0[gΛ0∗ δδJ
])e12
∫dx J(S∗J) = eFC [J,Λ0] = e−IΛ0
[gΛ0∗ δδJ
])e12
(JS∗J) . (256)
Notice that here and in the following we use the shortened notation introduced in section 8.We want to show in this appendix that FC [J,Λ0] is the generator of connected Feynman
diagrams, that is, the coefficient of its J-expansions can be expanded in Feynman integralscorresponding to connected diagrams, and that the Legendre transform of FC is the generatorof the one-particle irreducible diagrams, that is, of the diagrams which cannot be divided intwo disconnected parts cutting a single line.
It turns out that it is convenient to start form the field equation (111):
−∂2 δ exp(FC [J,Λ0])
δJ(x)+ gΛ0 ∗
δIΛ0 [gΛ0 ∗ δδJ
]
δφ(x) exp(FC [J,Λ0]) = J(x) exp(FC [J,Λ0]) , (257)
that we convert into integral form:
δ exp(FC [J,Λ0])
δJ(x)=∫dyS(x− y)[J(y)− (gΛ0 ∗
δIΛ0 [gΛ0 ∗ δδJ
]
δφ)(x)] exp(FC [J,Λ0]) , (258)
and simplify omitting the smearing functions and choosing IΛ0 [φ] = λ0/4!∫dxφ4(x)
δ exp(FC [J ])
δJ(x)=∫dyS(x− y)[J(y)− λ0
3!(δ
δJ)3(x)] exp(FC [J ]) , (259)
that is, in terms of FC [J ]:
δ
δJ(x)FC [J ] =
∫dyS(x− y)
J(y)− λ0
3!
(δFC [J ]
δJ(y)
)3
+iλ0
2
(δFC [J ]
δJ(y)
)δ2FC [J ]
δj2(y)+λ0
3!
δ3FC [J ]
δJ3(y)
]. (260)
We can translate this equation into graphical form defining the following symbols:
δnFC [J ]
δJ(x1), .., δJ(xn)≡ @@
y1
n
, S(x− y) ≡ (261)
68
(260) becomes:
y = J − λ0
3!
@@@
yyy +iλ0
2y&$y+
λ0
3!y
(262)It is apparent that the iterative solution of (262) naturally generates Feynman diagrams
with four line vertices corresponding to the interaction IΛ0 and two kind of lines: those joiningvertices that we call internal and those ending in a J , external. A sketch of the terms of FC [j]produced by the first iterations is given in the following figure:
FC [j] =1
2J J− λ0
4!J J
J
J
− λ0
4J Ji
+λ2
0
72J J
J J
J J
+ · · · , (263)
from which it clearly appears that the terms of the J and λ0 expansion of FC [J ] correspondto connected diagrams and hence exponentially vanish when the end points of the externallines are taken apart. This is, of course, consistent with the cluster decomposition of the time-ordered functions discussed in section 6 and justifies the definition of FC [J ] as generator of theconnected Schwinger functions.
Now we consider the Legendre transform of FC [J ].Let f(x) be a convex function, F (y) ≡ Infx(yx − f(x)) also is convex and is called the
Legendre transform of f . If f(x) has continuous first derivative f ′(x) the equation f ′(x) = yhas a unique solution x(y) and one has: F (y) = x(y)y − f(x(y)) and F ′(y(x)) ≡ x. This istrivially extended to any number of variables and hence to the functionals.
In thermodynamics the internal energy U(V ), considered as a function of the volume andminus the free energy −F (V ) are Legendre transforms of one another and ∂U/∂V = −p. Inthe case of phase equilibrium U can fail to be convex, consider e.g. U(V ) = λ2V 4−µ2V 2, usingthe first definition one gets a convex −F (−p) with however discontinuous first derivative inthe origin. A second Legendre transform generates a different U(V ) which is convex and, as inthe example, flat in the region in which U(V ) is not convex, that is, in the mixed phase regionwhere the Maxwell criterion is automatically fulfilled.
Going back to field theory, FC [J ] could be convex, it is in the free situation, and hence it is inthe sense of formal power series in J . In the same sense it has continuous functional derivative,while this is not guaranteed in the general situation, as it is apparent in the thermodynamicexample. Therefore, at least in the sense of formal power series, we can define the inversefunctional J [φ](x) such that:
φ(x) ≡ δ
δJ(x)FC [J [φ]]− δ
δJ(x)FC [0] , (264)
and the Legendre transform:
U [φ] ≡∫dx(φ(x) +
δ
δJ(x)FC [0])J [φ](x)− FC [J [φ]] . (265)
69
It is apparent that:δU [φ]
φ(x)= J [φ](x) . (266)
Notice that in the case we are now studying δδJ(x)
FC [0] = 0 and that the same condition canbe implemented in any scalar field theory. In the case of non scalar fields this condition followsfrom Euclidean invariance.
Now we want to study the functional U [φ] in the framework of the diagrammatic expansionof FC . In this framework we notice that we can single out the first term of the expansionsketched in (263)
FC [J ] =1
2(J, S ∗ J) + FC [J ] , (267)
thus writing:
J [φ](x) = −∂2(φ(x)− δFC [J [φ]]
δJ(x)) , (268)
from which one gets:
U [φ] = (φ(−∂2(φ− δFC [J [φ]]
δJ))) +
1
2((φ− δFC [J [φ]]
δJ)∂2(φ− δFC [J [φ]]
δJ))− FC [J [φ]]
= −1
2(φ∂2φ)− FC [J [φ]] +
1
2(δFC [J [φ]]
δJ∂2 δFC [J [φ]]
δJ) . (269)
In the last expression it appears rather clearly that the third term subtracts from the secondone the contributions from the diagrams that are one particle reducible, i.e. that can be dividedinto two disconnected parts cutting a single line. To verify this result let us have again recourseto a graphical representation. We decompose U = −1
2(φ∂2φ) + U [φ]. This allows us to write
(266) according:δU [φ]
φ(x)= J [φ](x) = −∂2φ(x) +
δU [φ]
δφ(x), (270)
and hence
φ[J(x)] = S ∗ (J(x)− δU [φ[J ]]
δφ(x)) , (271)
In analogy with (268) we define:
δnU [J ]
δφ(x1), .., δφ(xn)≡ @
@
1
.
.
n
.
. φ[J ](x) ≡ y S(x− y) ≡ (272)
and we translate (271) into the form:
y = J −∑∞n=0
@
y2n + 1
y1y.y.y.
@
(273)
70
whose iterative solution begins with:
y = J − J − J +
J
J
J + · · · ·
J
J
(274)The point to be put into evidence is that following the iteration procedure one gets a completeset of tree skeleton diagrams representing a complete Feynman expansion of the whole set ofconnected diagrams. Now the tree skeleton decomposition of a connected Feynman diagram isunique and identifies the vertices of the skeleton diagram with one-particle irreducible parts.Thus U is the generator of one-particle irreducible diagrams.
A further comment is in order: the coefficient δ2Uδφ(p)δφ(0)
|φ=0 is the inverse of δ2FCδJ(p)δJ(0)
|J=0,
indeed the above iterative solution (273) gives:
δ2FC
δJ(p)δJ(0)|J=0 = 1/p2[
∞∑n=0
(− δ2U
δφ(p)δφ(0)|φ=0/p
2
)n] =
[p2 +
δ2U
δφ(p)δφ(0)|φ=0
]−1
. (275)
71
A The Wilson Action
To prove (118) we introduce a simplified notation understanding space-time integrals and vari-ables and introducing the symbol ∂1 for gΛ ∗ δ
δJand ∂2 for gΛ ∗ δ
δJ. It is then apparent that
gΛ0 ∗δ
δJ= gΛ ∗
δ
δJ+ g ∗ δ
δJ≡ ∂1 + ∂2 . (276)
Thus we write (116) as:
eFC [J,Λ0] = e−IΛ0[∂1+∂2])e
(JS∗J)2 ≡ F [∂1 + ∂2]e
(JS∗J)2 ≡
∞∑n,m=0
Fn+m
n!m!∂n1 ∂
m2 e
(JS∗J)2 . (277)
Then we notice that:
∂m2 e(JS∗J)
2 = e(JS∗J)
2
m∑k=0
m!
k!(m− k)!∂k2 (S ∗ J)m−k2
= e(JS∗J)
2
[m2
]∑k=0
m!
k!(m− 2k)!(SJ∗)m−2k
2 Sk2,2 , (278)
where we have introduced (SJ)2 ≡ g ∗S ∗J , and S2,2 ≡ g ∗S ∗ g and by[n2] we mean the integer
part of n2. Now from (277) and (278) we get:
eFC [J,Λ0] =∞∑
n,m=0
[n2
]∑k=0
Fn+m
k!m!(n− 2k)!∂m1 e
(JS∗J)2 (S ∗ J)n−2k
2 Sk2,2
=∞∑
n,m,k=0
Fn+m+2kSk2,2
k!m!n!∂m1 (S ∗ J)n2e
(JS∗J)2
=∞∑
n,m,k,q=0
Fn+m+2q+2kSk2,2S
q1,2
q!k!m!n!(S ∗ J)n2∂
m1 e
(JS∗J)2
≡: e−IΛ,Λ0[gΛ∗ δδJ+ g∗S∗J ]) : e
12
(JS∗J) , (279)
where we have introduced S1,2 ≡ gΛ ∗ S ∗ g.
72
B Bibliography
• The major reference for the first part on the scattering theory is:
R. Haag Local Quantum Physics. Berlin: Springer-Verlag 1992
• Further information, in particular concerning reduction formulae can be found in:
J.Glimm and R. Jaffe Quantum Physics. A functional integral point of view. Berlin:Springer-Verlag 1987
• and for what concerns the calculation of cross-sections:
C. Becchi and G. Ridolfi An introduction to relativistic processes and the Standard Modelof electro-weak interactions Berlin: Springer-Verlag 2005
• an introduction to the effective action is given in:
C. Becchi Lectures on the renormalization of gauge theories. In Les Houches 1983 -Relativity, groups and topology B. S. De Witt and R. Stora Eds. Amsterdam: NorthHolland 1984.
• An introduction to the Euclidean Theory can be found in:
K. Symanzik Euclidean Quantum Field Theory. In Local Quantum Theory. Scuola In-ternaz. di Fisica ”Enrico Fermi”, Corso 45. R. Jost Ed. New York: Academic Press1969
• For the Renormalization Group Method the basic reference is: K. Wilson and J. Kogut,Phys. Reports 12 (1974) 75.
• An account of the Wilson-Polchinski approach with application to gauge theories is givenby:
C.Becchi On the construction of renormalized quantum field theory using renormalizationgroup techniques. In Elementary Particles, Quantum Fields and Statistical Mechanics.Seminario Nazionale di Fisica Teorica M.Bonini, G. Marchesini, E. Onofri Eds. Parma1993 (hep-th/9607188 )
• The evolution equation for the effective proper generator was first obtained by: M. Bonini,M. D’Attanasio, G. Marchesini, Nucl.Phys.B409 (1993) 441.
• A general reference for basic quantum field theory is
C. Itzykson and J.-B. Zuber: Quantum Field Theory. New-York: Mc Graw-Hill 1980.
• More specialized seminal contributions are:
R.Haag, Phys.Rev.112 (1958) 669.
P.Ruelle, Helv. Phys. Acta 35 (1962) 147.
H. Araki, R. Haag. Commun. Math. Phys. 4 (1967) 77.
73
B. Simon. The P (φ)2 Euclidean (Quantum) Field Theory. Princeton U.P. 1973.
K. Gawedzki, A. Kupiainen, in Les Houches 84 Ed’s Osterwalder and Stora.
D.J. Gross, F. Wilczek, Phys.Rev.D9 (1974) 980.
G. Gallavotti, Rev. Mod. Phys. 57 (1985) 471.
• On the BPHZ subtraction method the basic references are:
J. H. Lowenstein BPHZ renormalization. and
W. Zimmermann The power counting theorem for Feynman integrals with massless prop-agators.
in Renormalization Theory. Proceedings of the NATO Advanced Study Institute held atthe International School of Mathematical Physics at the ’Ettore Majorana’ Center forScientific Culture in Erice (Italy) G. Velo and A. Wightman Ed.s - D.Reidel PublishingCompany , Boston 1976.
• The basic reference for dimensional regularization is:
P.Breitenlohner and D.Maison, Commun. Math. Phys. 52(1977)11, ibid. pg. 39, ibidpg. 55.
• An account of the algebraic method can be found in:
O. Piguet, S. Sorella Algebraic renormalization: Perturbative renormalization, symmetriesand anomalies. in Lect.Notes Phys.M28 Berlin: Springer-Verlag 1995.
74
