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QUANTUM RELATIVISTIC DYNAMICS
C. BECCHIDipartimento di Fisica, Universita di Genova,
Istituto Nazionale di Fisica Nucleare, Sezione di Genova,via Dodecaneso 33, 16146 Genova (Italy)
Abstract
The 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 itsrelation with LSZ theory and on an introduction to renormalization theory based onWilson-Polchinski renormalization group method which is compared with the subtractionmethod.
1
Contents
1 Introduction 1
2 Difficulties with quantized fundamental fields 2
3 Construction of a relativistic scattering theory 5
4 Properties of the time-ordered functions 13
5 The L.S.Z. reduction formulae 15
6 The functional formalism and the Effective Action 17
7 The construction of the theory, the Euclidean Quantum Field Theory 21
8 The Functional Integral in Euclidean Quantum Field Theory 23
9 The Wilson Effective Action in Euclidean Quantum Field Theory 26
10 The Effective Proper Generator 31
11 The subtraction method 38
12 Bases of local operators in the subtraction scheme and the Wilson operatorproduct expansion 45
13 The Quantum Action Principle 48
A The Wilson Action 57
B Bibliography 58
1 Introduction
Quantum field theory was born from a generalization of QED to other interactions. The maincharacteristics of the theory were:
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.The most famous extensions were the Fermi theory of weak interactions, where the neutrino
field was introduced and the Yukawa theory of strong interaction where a scalar field was
1
introduced assuming the existence of a corresponding spin-less particle identified after thediscovery of the π meson.
This gave rise 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 withan elementary particle theory based on assumption (1) above was abandoned due to the dis-covery of a large number of new particles, many of which 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 to abandon 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 foundations 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 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. It turns out that it is not an operator. Thiscan be seen using the Lehmann spectral representation for the two 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 under the simplifying assumption that < Ω|φ|Ω >= 0, one gets, without any lossof 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)
2
where we have used translation invariance and denoted 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. these states are assumed to be uniquely identified by their momentum,mass and helicity. The single particle masses satisfy m1 ≤ m2 ≤ ·· ≤ mNp , m1 > m > 0.
Eq.(2) can be further simplified using the fact that φ is a Lorentz scalar and hence, withsuitable normalization of vector states:
< ~p,M, α|φ(0)|Ω >=
√M
EM< ~0,M, α|φ(0)|Ω > . (3)
Introducing the mass density function: ρ(M) = 2∑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
2EMe−ipMx ≡
∫ ∞
0dMρ(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 and supported by a small space region and infinitely differentiable, weshall say C∞ as it is commonly written.
In this case one has:
||φχ(~0, 0)|Ω > ||2 =∫ ∞
0dMρ(M)
∫ d3p
2EM|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 Mintegral which requires ρ(M) to vanish at infinity since one has asymptotically
∫ dMMρ(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)
3
where χ can may either have compact support, such as an interval, in which case the smearedoperator is called local, or be rapidly decreasing, the resulting operator being called almost-local. Mathematically this means that the basic fields must be considered as 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 such operators is postponed tillafter the construction of Renormalization Theory. In general Φ is, either a fundamental, or acomposite, local field transforming according to a linear finite dimensional representation underthe action 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 >≡
√ma
Eaζa 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 >≡√ma
Eaδa,b , < Ω|Φa(0)|Ω >≡ 0 . (9)
Notice that the second condition is trivially satisfied if the field is not scalar. In the case ofa scalar field this condition is implemented by subtracting a constant, which is a trivial scalarfield, from the field operator.
Studying the scattering theory, we limit our discussion to almost local operators of the form:
Q(x)a =∫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 >=∫ ∞
mdMρa,b(M)
∫ d3p
2EM
∫dx′dy′fa(x− x′)
fb(−y′)eipM (x′−y′)+ < Ω|Qa(0)|Ω >< Ω|Qb(0)|Ω >
=∫ ∞
mdMρa,b(M)
∫ d3p
2EMeipMxfa(pM)fb(−pM)
+ < Ω|Qa(0)|Ω >< Ω|Qb(0)|Ω >
≡∫d3p
∫ ∞
m
dM
EMσa,b(~p,M)eipMx+ < Ω|Qa(0)|Ω >< Ω|Qb(0)|Ω > (11)
where ρa,b(M) =∑α < Ω|Φa(0)|~0,M, α >< ~0,M, α|Φb(0)|Ω > and we have introduced f(p) ≡∫
dxe−ipxf(x) that is assumed to be of class S. It follows that σa,b(~p,M) is C∞ in ~p and hencefrom the Riemann-Lebesgue lemma the above 2-point function tends to < Ω|Qa(0)|Ω ><Ω|Qb(0)|Ω > faster than any inverse power of r for large r ≡ |~x|. This result suggests theintroduction of the truncated, or connected, Wightman functions. Up to 2-points we define:
< Ω|Qa(x)Qb(0)|Ω >≡< Ω|Qa(x)Qb(0)|Ω >T
+ < Ω|Qa(x)|Ω >T< Ω|Qb(0)|Ω >T
< Ω|Qa(x)|Ω >=< Ω|Qa(0)|Ω >≡< Ω|Qa(0)|Ω >T=< Ω|Qa(x)|Ω >T . (12)
In the second line we have used the translation invariance of the vacuum state. Comparing(11) and (12) 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 , fora = 1 · · , n. Let χσi be the characteristic function of a subset σ of the first n integers; χσa = 1if i belongs to σ, χσa = 0 otherwise. If R→∞ the n-point function:
< Ω|n∏i=1
Qa(xa + dχσa)|Ω >→< Ω|∏a∈σ
Qa(xa + d)∏a 6∈σ
Qa(xa)|Ω > , (13)
faster that any inverse power of R since the operators at distance R commute up to negligiblecorrections . Notice that the operator products appearing in (13) and in the following formulaeare ordered with the index increasing from the left to the right. Now we can treat the right-hand
6
side of (13) as a 2-point function:
< Ω|∏a∈σ
Qa(xa + d)∏a 6∈σ
Qa(xa)|Ω >
=∑α
∫ ∞
mdM
∫d3p < Ω|
∏a∈σ
Qa(xa + d)|~p,M, α >
< ~p,M, α|∏a 6∈σ
Qa(xa)|Ω >=∑α
∫ ∞
mdM
∫d3p e−ipMd
< Ω|∏a∈σ
Qa(xa)|~p,M, α >< ~p,M, α|∏a 6∈σ
Qa(xa)|Ω > . (14)
Taking into account the C∞ smoothness in ~P of the product of matrix elements in the last lineof (14) is a C∞ function of ~P we conclude , 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|∏a 6∈σ
Qa(xa)|Ω > . (15)
Thus generalizing (12) we define implicitly the connected n-point functions according to thefollowing cluster decomposition formula:
< Ω|n∏a=1
Qa(xa)|Ω >≡∑σ1
< Ω|∏a∈σ
Qa(xa)|Ω >T< Ω|n∏
b6∈σ1
Qb(xb)|Ω >
=n∑k=1
∑Πj ,j=1,··,k
k∏j=1
< Ω|∏a∈Πj
Qa(xa)|Ω >T (16)
where σ1 runs over all the subsets of the first n integers containing the first one, as above, andthe index sets Πj for j = 1, · · · , k are partitions of the first n integers, the sum running over allsuch partitions. Then, the connected n-point functions vanish faster than any inverse power ofthe 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.
We associate to every almost-local operator Qa(x) a source J(x, θ) of class S where θ isa variable accounting for the position of the operator in the ordered products. Then we candefine the functional 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 > . (17)
If the connected n-point function generator W (J)C is defined in a completely analogous way,it is immediate to verify from (16) that:
W (J) = eW (J)C . (18)
7
Coming back to Haag’s construction we introduce positive/negative frequency solutions ofthe Klein-Gordon equation (∂2 +m2
a)ψa(x) = 0
ψ(±)a (x) =
1
(2π)32
∫ d3p√2Ema
ψ(~p)e±ipmax (19)
with ψ(~p) of class D and support Sψ and the corresponding generalized destruction/creationoperators;
B(±)ψa
(t) = ±i∫x0=t
d3x
[Qa(x)
∂
∂x0ψ(±)a (x)− ψ(±)
a (x)∂
∂x0Qa(x)
], (20)
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 > , (21)
where sk is either + or −.Taking into account that < Ω|B(±)
ψb(t)|Ω >≡ 0 for all b’s, due to our choice of the local
fields, it turns out that in the limit |t| → ∞ (21) vanishes unless it contains the same numberof generalized creation and destruction operators. More precisely the asymptotic time limitof (21) reduces to the sum over all the possible pairings of B(+)’s and B(−)’s of the vacuumexpectation values of the products.
This result can be proven by applying the cluster decomposition (16) to (21) and using thefact that ψ(±)
a (x) in (19) 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|
. (22)
The contribution of a k > 2 point connected term in the cluster decomposition of (21) is:(k∏l=1
sli
)∫x0
i≡td3x1 · · · 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 (23)
which can be decomposed into the sum of 2k terms of the form:
∫x0
i≡t
k∏j=1
(d3xjψ(sj)aj
(xj)) < Ω|Qa1(x1)Qa2(x2) · · · Qak(xk)|Ω >C , (24)
where the functions ψ are solution of the Klein Gordon equation and satisfy (22) and theoperators Q satisfy (10).
8
Now, changing integration variables (24) can be written:
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 . (25)
Due to the cluster property the connected vacuum expectation value vanishes faster than any
inverse power of the the maximum distance between the points, that is of√t2∑kl=2w
2l , (e.g.
as e−t2∑k
l=2w2
l ). Thus, if the supports of ψaioverlap the w-integrals give a contribution pro-
portional to |t|−3(k−1) times the absolute value of the product of ψ’s, which, on account of thefirst inequality in (22) amounts to |t|−3k/2. Since the integral with respect to ~v1 is limited by(22) to a sphere of radius one, we find that (25) vanishes proportionally to |t|−3(k−2)/2. Noticehowever that if the supports of ψ’s do not overlap (25) vanishes faster than any inverse powerof |t| due to the second inequality in (22). Notice that this is independent of the si’s, that is ofthe choice of generalized creation or destruction operators.
Therefore, in the cluster decomposition of (21) one is left with 2-point connected terms only.Considering the 2-point cluster terms and referring for simplicity to (11) one has instead:
(−s1s2) < Ω|B(s1)ψa1
(t)B(s2)ψa2
(t)|Ω >C
= (−s1s2)∫x0
i≡td3x1d
3x2(ψ(s1)a1
(x1)− ψ(s1)a1
(x1)∂
∂x01
)(ψ(s2)a2
(x2)− ψ(s2)a2
(x2)∂
∂x02
)
< Ω|Qa1(x1)Qa2(x2)|Ω >C
= (−s1s2)∫x0
i≡td3x1d
3x2(ψ(s1)a1
(x1)− ψ(s1)a1
(x1)∂
∂x01
)(ψ(s2)a2
(x2)− ψ(s2)a2
(x2)∂
∂x02
)∫d3p
∫ ∞
0
dM
2EMσa1,a2(~p,M)eipM (x2−x1) = (−s1s2)(2π)3
∫d3p
∫ ∞
0
dM
2EMσa1,a2(~p,M)
(EM + s1Ema1)(EM − s2Ema2
)
2√Ema1
Ema2
ψa1(−s1~p)ψa2(s2~p)ei(s1Ema1
+s2Ema2)t . (26)
From the assumed C∞ smoothness of the functions σa1,a2 and ψa and from the Riemann-Lebesgue lemma it turns out that (26) vanishes faster than any inverse power of |t| unlessma1 = ma2 and s1 = −s2 and the supports of the ψ’s overlap, in which case (26) it t-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 and destructionoperators are equal. 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=iB
(sj)ψaj
(t)|Ω > . (27)
9
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 > , (28)
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) , (29)
where fma(p2) ≡
∫dxfma(x
2)e−ipx satisfies fma(m2a) = 1 , and has compact support contained
in the region in which (p2 −m2a)
2< ε, with ε small enough to include only the four momenta
of single particle states with mass ma. This is due to the fact that each Qma(x) acting on thevacuum 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
√ma
Ea|~p, a > eipmaxf(pma) (30)
We label the corresponding generalized creation/destruction operators by: C(∓)ψa
(t) . The crucialnew point is that:
d
dtC
(−)ψa
(t)|Ω >= 0 , (31)
as it is easy to verify. This implies that;
d
dtΨψa1 ,··ψam
(t) ≡ d
dt
m∏k=1
C(−)ψak
(t)|Ω >→|t|→∞ 0 (32)
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 (27) becomes:
lim|t|→∞
< Ω|m∏j=1
C(sj)ψaj
(t)|Ω >= δm,2n2n∑i=2
δma1 ,maiδs1,+δsi,− < Ω|C(s1)
ψa1C
(si)ψai|Ω >
lim|t|→∞
< Ω|2n∏
j=2,j 6=iC
(sj)ψaj
(t)|Ω >= δm,2n2n∑i=2
δma1 ,maiδs1,+δsi,−
∫d3p ψa1(−~p)ψai
(~p) lim|t|→∞
< Ω|2n∏
j=2,j 6=iC
(sj)ψaj
(t)|Ω > . (33)
10
Thuslimt→±∞
Ψψa1 ,··ψam(t) = Ψ
(out/in)ψa1 ,··ψam
, (34)
and the convergence rate is faster than any inverse power of |t|if the supports of the ψ’s do notoverlap. Otherwise the rate is that of |t|−3/2
From (33) 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) (35)
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 > , (36)
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) . (37)
It is a natural hypothesis of relativistic scattering theory that the finite linear combinationsof vectors (36) 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 rather rely on the time-ordered functions thatwe shall introduce in the 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〉 . (38)
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〉 . (39)
Now, using again (26), (27) and (28), 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 (39)coincides with:
limt→±∞
< Ω|n∏j=1
C(+)ψ′aj
(t)B(s)ψ”c
(t)m∏i=1
C(−)ψbi
(t)|Ω > , (40)
11
From (33) and (28) 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 > .
(41)
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
〉 . (42)
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〉 . (43)
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〉 , (44)
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.
12
4 Properties of the time-ordered functions
We have thus shown how asymptotic states can be built in field theory. This implicity 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
∏i<j
θ(x0πi− x0
πj) < Ω|Φπ1(xπ1) · ·Φπn(xπn)|Ω > , (45)
where, as above, πi labels a permutation of indices. Since the Wightman functions of strictlylocal operators (fields) are distributions and the θ’s are discontinuous, the above formula doesnot make sense in general even within 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 >=∫ ∞
mdM
∫ d3p
2EM
[e−ipMxθ(x0)ρa,b(M) + eipMxθ(−x0)ρa,b(M)] ≡∫ dq
(2π)4e−iqx τa,b(q
2)
= −i∫ ∞
mdMρa,b(M)
∫ dq
(2π)4
e−iqx
M2 − q2 − iε, (46)
where we have used (5) and the fact that ρa,b(M) is a symmetric matrix due to T-invariance.Now, in order for the last equation to make sense for distributions, the mass integral mustconverge at infinity and this depends on the dimension of the fields. If e.g. they have dimensiontwo it is expected that, for large M , ρ(M) ∼M and hence the mass integral does not converge.This implies that (46) does not define a distribution. A deeper discussion of this point is inorder here. 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∫ ∞
mdMρa,b(M)
∫dq
f(q)
M2 − q2 − iε, (47)
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
∫ ∞
mdMρa,b(M)
∫dqqµ
f(q)
(M2 − q2 − iε)2, (48)
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(q
2) =
13
−i∫∞m dM
ρa,b(M)
M2−q2−iε with:
τR,a,b(q2) = K − i
∫ ∞
mdM
ρa,b(M)
M2[
M2
M2 − q2 − iε− 1]
= K − iq2∫ ∞
mdM
ρa,b(M)
M2
1
M2 − q2 − iε, (49)
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
∫ ∞
mdM
ρa,b(M)
M2
1
M2 − q2 − iε, (50)
it is apparent that τS,a,b(x) is a distribution and one has:
τR,a,b(x) = Kδ(x)− ∂2τS,a,b(x) , (51)
and hence it is also a distribution.This example shows what 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. We postpone any further discussion of the construction of T-functions, we assumewe have a consistent definition of those involving the local fields defined in (9) and we proceedwith the construction of the scattering amplitudes using the LSZ reduction formulae.
14
5 The L.S.Z. reduction formulae
Given a set of fields Φaiwe 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 iscontained in the slice |x0| < ∆. Then we select a set ψsi
aiof solutions of the Klein-Gordon
equation as in (19) with ψai(~p) of class D and non-overlapping supports and we consider:
Aψs1a1,··,ψsn
an≡ in
∫ ∏j
dyj
∫Γ
∏i
(dxiψsiai
(xi)(∂2xi
+m2ai
)fai(xi − yi))
< Ω|T (n∏k
Φak(yk))|Ω > , (52)
where the integration domain Γ is defined by: |x0k| ≤ Tk with Ti − Tl ≥ 2(l − i)∆. Noticing
that:
ψsiai
(xi)(∂2xi
+m2ai
)fai(xi − yi)
= ∂x0i
(ψsiai
(xi)∂x0ifai
(xi − yi)− fai(xi − yi)∂x0
iψsiai
(xi)), (53)
one identifies (52) with:
(−)n−∑
sl2 < Ω|T (
n∏k
[B(sk)ψak
(Tk)−B(sk)ψak
(−Tk)])|Ω > , (54)
where the generalized creation and destruction operators B(sk) are defined in (20) and the timeorder refers to the ±Tk variables. Now, using (43) and (44) 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:
15
limT1→∞
Aψs1a1,··,ψsn
an
= limT1→∞
(−)n−∑
sl2 < Ω|B(s1)
ψa1(T1)T (
n∏k=2
[B(sk)ψak
(Tk)−B(sk)ψak
(−Tk)])|Ω >
− limT1→∞
(−)n−∑
sl2 < Ω|T (
n∏k=2
[B(sk)ψak
(Tk)−B(sk)ψak
(−Tk)])B(s1)ψa1
(−T1)|Ω >
= δs1,+(−)n−∑
sl2 < Ω|a(out)
ψa1T (
n∏k=2
[B(sk)ψak
(Tk)−B(sk)ψak
(−Tk)])|Ω >
−δs1,−(−)n−∑
sl2 < Ω|T (
n∏k=2
[B(sk)ψak
(Tk)−B(sk)ψak
(−Tk)])a(in)†ψa1
|Ω > . (55)
Taking the asymptotic limits for all Tk we get:
limTn→∞
· · · limT1→∞
Aψs1a1,··,ψsn
an
=< Ω|
∏i : si=+
a(out)ψ∗ai
∏j : sj=−
a(in)†ψai
|Ω > , (56)
that is a transition 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 > , (57)
where the xi integrals cover the whole R4n. Finally we notice that, if ,as chosen above,fai
(Ema , ~p) = 1 on the support of the corresponding ψaione can get rid of the fai
’s in (57) andreplace them by Dirac deltas.
Computing cross sections of a process with f particles in the final state, that is n = f + 2,one more comment is in order. Decomposing the time-ordered function in the left-hand sideof (57) into connected parts one finds a momentum conservation constraint for each connectedpart; it follows that the only contribution to a cross section comes from the connected (f + 2)-point function with s1 = s2 = − and sj = + for j ≥ 2. In this case the transition amplitude isgiven 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 . (58)
16
6 The functional formalism and the Effective Action
Assuming that all needed time-ordered functions are defined in analogy with the case of theWightman functions 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 >, (59)
where it is understood that the ordering operator acts under integral sign. Notice that inthe present case there is no operator ordering problem since the time-ordered functions aresymmetric functions. The functional generator is a formal power series of the J ’s that arechosen of class S and (59) can be formally written;
Z[J ] =< Ω|T (ei∑
a
∫dxJa(x)Φa(x))|Ω > . (60)
In analogy with (18) the cluster decomposition of the time-ordered functions can be describedin terms of the connected generator:
Z[J ]C ≡∞∑n=2
in−1
n!< Ω|T (
∑a
∫dxJa(x)Φa(x))
n|Ω >C , (61)
where we have taken into account that the fields have null vacuum expectation value. From(59) we can show that:
Z[J ] = eiZC [J ] . (62)
Indeed, on account of (16), taking a functional derivative of (59) and comparing it with thatof (60) we have:
δZC [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 >= iδZC [J ]
δJa(x)Z[J ] . (63)
This is a first order differential equation whose solution with initial conditions Z[0] = 1 , ZC [0] =0 is given by (62).
Given the connected generator, which is a formal power series too, one defines its Legendretransform as follows. One defines the distribution valued functional:
Φa(x, J) ≡ δZC [J ]
δJa(x). (64)
17
Then one considers the Fourier transformed connected two-point function:
τ(2)a,b (p
2) ≡∫dxeipx < Ω|T (Φa(x)Φb(0))|Ω > . (65)
If τ(2)−1a,b (p2) is a matrix valued tempered distribution one can define a further distribution
valued functional Ja(x, ϕ) such that;
Φa(x, J(·, ϕ)) ≡ ϕa(x) , (66)
and the formal power series Legendre transform of ZC ;
Γ[ϕ] ≡ ZC [J(·, ϕ)]−∫dx∑a
ϕa(x)Ja(x, ϕ) . (67)
The new functional is the effective action in the following sense. Taking its functional derivativeone has:
δΓ[ϕ]
δϕa(x)= −Ja(x, ϕ) , (68)
and henceδΓ[ϕ(·, J)]
δϕa(x)= −Ja(x) , (69)
thus Φ(x, J) is the solution of the classical field equation induced by Γ vanishing at J = 0.The inverse equation to (67) is:
ZC [J ] = Γ[Φ(·, J)] +∫dx∑a
Φa(x, J)Ja(x) . (70)
It is worth noticing now that (58) is equivalent to:
A2→f = i∫ f+2∏
i=1
(dxiψsiai
(xi)(∂2xi
+m2ai
)δ
δJai(xi)
)ZC [J ]|J=0 . (71)
It can be shown that in the limit of perfect resolution of the wave packets, defining:
ψ(as)a (x) ≡
f+2∑i=1
δai,aψsiai
(x) , (72)
one can write:
A2→f = ie∫dx∑f+2
i=1ψ
siai
(x)(∂2x+m2
ai) δ
δJai (x)ZC [J ]|J=0
= ie∫dx∑
aψ
(as)a (x)(∂2
x+m2a) δ
δJa(x)ZC [J ]|J=0
≡ ieΣZC [J ]|J=0 , (73)
since the terms which are not linear in all ψsiai
do not contribute in the perfect resolution limit.If we define:
Φa(x) ≡ eΣΦa(x, J)|J=0 , (74)
18
we can write (73) in the form:
A2→f = i[Γ[Φ] +∫dx∑a
ψ(as)a (x)(∂2
x +m2a)Φa(x)] . (75)
Let us consider ϕ more closely.
eΣδΓ[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 . (76)
Thus ϕ satisfies the field equation induced by the effective action Γ.It remains to discuss the asymptotic properties of ϕ that determine it. Since ϕ is a dis-
tribution we should better discuss the asymptotic properties of Qa(x) =∫dyfa(x − y)ϕa(y) .
Taking into account the reduction formulae we have:
Qa(x) =< Ω|[1 +f∑
m=1
i−m∑
k1<··<km=3,··,f+2
∏j
a(out)ψ∗akj
]
Qa(x)[1− i2∑i=1
a(in†)ψai
− a(in)†ψa1
a(in†)ψa2
]|Ω >C
+Ra(x) , (77)
where Ra(x) accounts for the terms at least quadratic in one of the ψa’s and hence not con-tributing to A2→f in the limit of perfect resolution. Considering the asymptotic limit of thefirst term of (77) we notice that Eq. (42) says that in this limit Qa contributes as an asymptoticcreation or destruction operator. In particular its contribution to the negative frequency partin the t→∞ limit corresponds to a a(out)† operator, while that to the positive frequency partin the t→∞ limit corresponds to a a(in) operator.
Therefore, taking into account the energy-momentum conservation constraint, we find thatthe negative frequency part in the t→∞ limit is given by
f∑k=3
< Ω|a(out)ψ∗ak
Qa(x)|Ω >→t→∞
f∑k=3
δa,ak
∫dyfa(x− y)ψ(−)
ak(y) , (78)
since all other terms in the first term in (77) which asymptotically correspond to a n→ m− 1transition amplitude violates energy momentum conservation. The positive frequency partin the t → ∞ limit corresponds to a n − 1 → m transition amplitude which violate energymomentum conservation with the exception of
< Ω|Qa(x)[a(in)†ψa1
+ a(in)†ψa2
]|Ω >→t→−∞
2∑k=1
δa,ak
∫dyfa(x− y)ψ(+)
ak(y) . (79)
Thus, through (77), (78) and (79) we have shown that, up to terms not contributing to thetransition amplitude in the perfect resolution limit, ϕa(x) satisfies the classical field equation
19
corresponding to the effective action Γ and satisfies the same boundary conditions as the clas-sical field in the semi-classical approximation. If furthermore we consider the expression of thetransition amplitude given by (75) in terms of Γ we conclude that the fully quantized scatteringtheory coincides with its semi-classical counterpart provided one replaces the action with theeffective action.
20
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 givein the rest of these notes a construction procedure for the connected functional generator (61).Let us first of all consider a scalar field theory; it is an elementary exercise to prove that:
ZC [J ] =1
2
∫dxdy∆F (x− y)J(x)J(y) , (80)
with
∆F (x) =∫ dx
(2π)4
e−ipx
m2 − p2 − i0+(81)
coincides with the solution of the differential equation:
(m2 + ∂2)δZC [J ]
δJ(x)= J(x) , (82)
with the condition that ZC should vanish 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. This is obtained extendinganalytically the Fourier transformed 2-point function ∆;
∆(p) ≡∫dxeipx∆(x) =
1
m2 − p2 − i0+
=1
(Em(~p)− i0+ − p0)(Em(~p)− i0+ + p0), (83)
to the domain p0 = eiθp4 with 0 ≤ θ < π and in particular to the Euclidean points whereθ = π/2. If we call Schwinger 2-point function S(x1−x2) the Fourier transform of the analyticcontinuation to the Euclidean space:
S(x) =∫ d4p
(2π)4
ei∑4
j=1xjpj
m2 +∑4j=1 p
2j
(84)
and
FC [J ] =1
2
∫dxdy S(x− y)J(x)J(y) , (85)
the equation corresponding to (82) is
(m2 − ∂2)δFC [J ]
δJ(x)= J(x) , (86)
identifying FC [J ] with its solution vanishing at J = 0 and at infinity of R4.It turns also out that, for a general interacting theory, the connected Euclidean n-point
functions Sn obtained by the same analytical continuation from the Fourier transforms of the
21
time-ordered functions are also obtained, up to contact terms, that is terms supported atcoinciding points, by analytical continuation of n-point Wightman functions Wn at imaginarytimes. that is:
Sn(x1, ··, xn) = W (x′1, ··, x′n) for x41 > x4
2 > ·· > x4n
and ~xi ≡ ~x′i , −ix4i ≡ x
′0i . (87)
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 (84) 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) . (88)
It turn out furthermore that a local Poincare invariant theory goes into a Euclidean invariantone.
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) . (89)
Notice that whenever we deal with Euclidean functions we shall use the Euclidean scalar productas in (84).
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 actions 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.
22
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) . (90)
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 (90) is seriously sick since J [φ] does not make sense being φ adistribution. Translating (90) into a functional differential equation one should heuristicallyreplace that of the free theory (86) with :
[−∂2 δ
δJ(x)− J [
δ
δJ(x)]] expFC [J ] = J(x) expFC [J ] , (91)
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 crucial role,not only in the construction of the theory through the field equations, but also in the defini-tion of physical observables, e.g. interpolating fields, currents, energy-momentum tensors andmany others. Even if in the construction the fundamental fields, which are identified with thedynamical variables, have a dominant role, the final goal is that of computing local operatorcorrelation functions. A standard formal mean to insert into the theory local composite oper-ators is that of adding to the interaction coupling terms of 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 we should rather introduce the smeared field
qΛ0(x) ≡∫dy gΛ0(x− y)φ(y) ≡ (gΛ0 ∗ φ)(x) , (92)
where ∗ is the convolution symbol, and define J as a function of q. After this substitution (91)should be written
−∂2 δ
δJ(x)expFC [J ] = [J(x) + J [(gΛ0 ∗
δ
δJ(y))(x)]] expFC [J ] , (93)
23
However replacing as above J (φ(x), ∂φ(x), ··) with J (q(x), ∂q(x), ··) ≡ JΛ0(x) destroys thelocal character of (90) and hence one should find a method to recover the locality of the theory.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 compute a suitable FC [J,Λ0], regular in the limit Λ0 → ∞ (that we shallcall the UV limit), such that (93) be true independently of Λ0 and JΛ0 become local, howevernot necessarily regular, in this limit. In these conditions JΛ0 plays the role of an effective termin the field equation. Strictly speaking one should choose smearing functions with compactsupport in p, which is not true with the Gaussian in (89). We claim that we can use toGaussians without any loss of generality, at least in perturbation theory.
Proceeding further we consider a suitable class of positive functionals IΛ0 [φ], that we calleffective interaction, with a range of non-locality of the order of 1
Λ0and choose JΛ0(x) =
−∫dygΛ0(x− y)(δIΛ0 [φ]/ δφ(x)). The restriction of the effective term in the field equation to
the functional derivative of some effective interaction is ”a priori” arbitrary in contrast with thescattering effective action whose existence is guaranteed by the reduction formulae. Howeverto my knowledge this restriction is a price to pay for renormalization.
With the mentioned choice it is apparent that (90) can be written:
δ
δφ(x)(∫dy
(∂φ)2
2+ IΛ0 [qΛ0 ]) = J(x) , (94)
and (93) reads:
−∂2 δ
δJ(x)expFC [J,Λ0] + (gΛ0 ∗
δIΛ0 [gΛ0 ∗ δδJ
]
δφ)(x) expFC [J,Λ0] = J(x) expFC [J,Λ0] , (95)
whose solution is 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]) , (96)
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” (96) 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 , (97)
with a real and positive, its solution:
f(x) =∫ ∞
−∞dy e−[ y2
2+a y2n+2
2n+2+c+xy] , (98)
24
with c fixed by the condition f(0) = 1 is unique if we ask f to be real and positive. Equation(96) is the functional version of (98).
An alternative form to (98), which is equivalent in the sense of formal power series in a is:
f(x) = e−[ a
2n+2d2n+2
(dx)2n+2 +c]∫ ∞
−∞dy e−[ y2
2+xy] = e
−[ a2n+2
d2n+2
(dx)2n+2 +c′]e
x2
2 . (99)
It is apparent that this form is obtained from (98) 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 (99) by expanding it in power series of a leads to a power series inx 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 (96), which is equivalent in the sense of formalpower series in IΛ0 is:
eFC [J,Λ0] = e−[IΛ0[gΛ0
∗ δδJ
]+KΛ0 ]e12
∫dx J(S∗J) ≡ e−[IΛ0
[gΛ0∗ δ
δJ]+KΛ0 ] e
12(JS∗J) , (100)
where we have further simplified our notation using(fg) for∫dxf(x)g(x) and
S(x) =∫ dp
(2π)4
eipx
p2=
1
(2π)2x2, (101)
It is a further exercise left to the reader to verify by (100) that FC is the generator of connectedFeynman diagrams corresponding to the interaction IΛ0 and to the propagator (101).
Now the RG condition we want to require is that FC has a regular limit for Λ0 → ∞ andthat field equation becomes local in this ultra-violet (UV) limit. This is, of course, a conditionon IΛ0 . However this does not require IΛ0 to remain regular in the limit. The requirement isthat FC has a regular limit and (δIΛ0/δφ), however singular, becomes local. This means that(δ2IΛ0 [qΛ0 ]/δφ(x)δφ(y)) → 0 when Λ0 →∞ for (x− y)2 = d2 > 0 fixed.
25
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. (100) 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)] witha different, non local functional IΛ,Λ0 [gΛ ∗ (δ/δJ)] when the support of J lies in the regionp Λ Λ0. The idea is that, keeping Λ fixed in the UV limit, IΛ,Λ0 [gΛ ∗ (δ/δJ)] should havea regular IΛ,∞[gΛ ∗ (δ/δJ)] limit, since FC is expected to do. What remains to be required isthat IΛ,∞[gΛ ∗ (δ/δJ)] 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[gΛ∗ δ
δJ+ g∗S∗J ]) : e
12(JS∗J) , (102)
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. The proof of (102) isgiven in Appendix A. It is clear that the functional built up in Appendix A it what we werelooking for, indeed, if the support of J lies in the region p Λ Λ0, g ∗ S ∗ J vanishes and(102) coincides with (100) after the substitution IΛ0 ↔ IΛ,Λ0 .
A further exercise left to the reader is to prove that −IΛ,Λ0 is the functional generatorof connected amputated Schwinger functions corresponding to the interaction IΛ0 and to thepropagator:
S ≡ gΛ0 ∗ S ∗ gΛ0 − gΛ ∗ S ∗ gΛ . (103)
Notice that, using the equivalence (100)-(102), one has:
eFC [J ] =∫dµ[φ]e
∫dy(J(y)φ(y)− (∂φ)2
2)−IΛ,Λ0
[gΛ∗φ+ g∗S∗J ]) , (104)
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 ]) , (105)
which must be Λ independent. Notice that in (105) g ∗ f ≡ f − gΛ ∗ f .One can derive the field equation requiring, at the first order in ε, the invariance of FC
under 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∗J
e∫dy(J(y)φ(y)− (∂φ)2
2)−IΛ,∞[gΛ∗φ+ g∗S∗J ]) . (106)
26
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 (106) tends to (ε(δIΛ/δϕ)[(δ/δJ)]) exp(FC [J ]) sinceg ∗ J vanishes in the limit. Therefore we have obtained an equation strictly analogous to(95). Furthermore, due to the Λ independence of (106), 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 crucialpoint is that exp(FC [J ]) does not depend on Λ therefore, taking the derivative of (102) withrespect of Λ, that we label by a dot, that is writing:
F (Λ) ≡ Λ2 δF (Λ)
δΛ2, (107)
one has:
: (−IΛ,Λ0 [φ]−(δIΛ,Λ0
δφgΛ ∗ [
δ
δJ− S ∗ J ]
)e−IΛ,Λ0
[φ]|φ=gΛ0∗ δ
δJ+ g∗S∗J :
e12
∫dxdy(J S∗J) = 0 (108)
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 at afirst sight the two terms in brackets should sum to zero. This is however not true because theproduct S ∗ J induced by (δ/δJ) lies at the right of the ordered expression and hence, beforecoming to any conclusion, we have to shift it to the left. What remains of the two contributionsis thus what comes in re-ordering S ∗ J to the left of the J functional derivative appearing inIΛ,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)] . (109)
Thus, asking for the resulting ordered expression to vanish, which is a sufficient condition for(108) to 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)] . (110)
Considering the Fourier transform of (103), that is, switching to momentum space, in whichthe propagator (101) is S = (1/p2) and choosing the Gaussian smearing (89) one has:
˜S(p) =
e− p2
Λ20 − e−
p2
Λ2
p2(111)
and
˙S(p) = −e
− p2
Λ2
Λ2, (112)
27
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)] . (113)
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 derivative is a functional derivative. For example the operator that, in the limitΛ → Λ0 is identified with (gΛ0 ∗ φ)4/4! corresponds to the functional derivative of the effectiveaction with respect to some bare coupling constant considered space-time dependent and aconserved current is usually defined by means of the functional derivative with respect to someexternal (background) vector field. At the effective theory level the operator coupled to theexternal field ω is identified by
ΩΛ.Λ0 [φ] ≡ δIΛ.Λ0 [φ]/δω (114)
whose evolution equation is linear:
ΩΛ.Λ0 [φ] =1
2
∫ dp
(2π)4
e−p2
Λ2
Λ2[δIΛ,Λ0
δφ(p)
δΩΛ,Λ0
δφ(−p)+δΩΛ,Λ0
δφ(p)
δIΛ,Λ0
δφ(−p)− δ2ΩΛ,Λ0
δφ(p)δφ(−p)] . (115)
The analysis of the solutions of this operator evolution equation is identical to that of theeffective action which follows.
Equation (113) can be further elaborated introducing the scaled field variables:
φ(p) ≡ Λ−3ϕ(p
Λ) , (116)
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) . (117)
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) . (118)
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), (119)
28
and hence the evolution equation (113) 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)
]. (120)
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] . (121)
which looks very much like a diffusion equation in the functional space with the time variableidentified with ln(Λ0/Λ). A very simple way to understand the possible consequences of (121)is to consider its form at zero space-time dimensions. In this ultra-simplified form the generalsolution of this equation is easily written and it is also easy to understand its implications fromthe point of view of the renormalization program. It is obvious that the zero dimensional fieldis a real variable that we identify with x after multiplication by suitable constant factor. Thuswe write (121) in the form:
Λ2∂Λ2e−IΛ,Λ0(x) = −1
2
∂2
(∂x)2e−IΛ,Λ0
(x) . (122)
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) , (123)
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 tells us that our scalar field theory is necessarily free.
It is interesting to see how this triviality property appears if we choose F (x, 0) = exp(−g0x4/4!)
and we develop F (x, t) in power series of g0. We find:
F (x, 2 lnΛ0
Λ) =
∞∑n=0
(4n)!
n!(−g0 ln2 Λ0
Λ
4!)n
2n∑l=0
( x2
lnΛ0Λ
)l
(2l)!(2n− l)!. (124)
Needless to say, this power series is not absolutely convergent: consider, e.g. the series corre-sponding to F (0, λ). Considering (124) as an asymptotic power series in g0 and limiting theexpansion to the second order one has:
lnF (x, 2 lnΛ0
Λ) =
g0 ln2 Λ0
Λ
2+
4
3(g0 ln2 Λ0
Λ)2 −
[g0 ln
Λ0
Λ− 16
3g20 ln3 Λ0
Λ
]x2
2
−[g0 −20
3g20 ln2 Λ0
Λ]x4
4!+O(g3
0, x6) . (125)
29
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
ΛR
1±√√√√
1−80gR ln2 Λ0
ΛR
3
. (126)
This means that there is no g0 corresponding to the wanted coupling constant as soon as Λ0/ΛR
becomes large enough. Since program is to take the limit Λ0/ΛR → ∞ we see that it worksonly in the trivial case gR = 0. Notice that our conclusion should be that a sensible, non-trivialscalar field theory cannot be built in this way.
A more formal approach which is however less critically dependent of the particular natureof the field and of the structure of the propagator S is based on the iterative construction ofthe effective interaction which defines perturbative renormalization.
Next step of our analysis consists in the search for a solution of the evolution equation sat-isfying the locality constraint in the UV limit and a suitable set of normalization conditions (asa matter of fact, initial conditions) at Λ = ΛR. The nature of these conditions and the value ofΛR remain to be specified. With this aim one could perfectly well continue the analysis study-ing the solutions to (120), however many applications are simplified if one considers anotherfunctional which coincides with IΛ,Λ0 in the Λ → Λ0 limit. We shall 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 (127)
will represent the connected n point functions and we shall replace φ , φ by φ ,˜φ if the
corresponding legs are amputated (multiplied by S−1). We shall also use
<˜φ(p1) · · · ·˜φ(pn−1)φ(0)|Λ >1−PI (128)
for the n-point 1-PI functions. For example the field functional derivatives of IΛ∞[φ] are thefunctions:
< φ(x1) · · · ·φ(xn)|Λ >C (129)
30
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 convenient to introduce the effectiveconnected generator:
ZΛ,Λ0 [J ] ≡ 1
2(JS ∗ J)− IΛ,Λ0 [S ∗ J ] , (130)
this is the generator of the connected diagrams corresponding to the interaction IΛ0 and to thepropagator S, that is, in the UV limit,
< φ(x1) · · · φ(xn)|Λ >C . (131)
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 (67) .
Assuming, in order to simplify the formulae, that IΛ,Λ0 [φ] does not contain a linear term inits functional variable φ, we put
φ[J ] ≡ S ∗ (J − δIΛ,Λ0
δφ[S ∗ J ]) . (132)
The inverse functional J [φ] is defined as the formal power series in φ solution of:
J = Cφ+δIΛ,Λ0
δφ
[S ∗ J [φ]
], (133)
where
C(p) ≡ (S)−1(p) =p2
e− p2
Λ20 − e−
p2
Λ2
, (134)
and φ is chosen of class D in momentum space, since we would like (φ C ∗φ) to be finite. Thenwe define:
VΛ,Λ0 [φ] ≡ (φJ [φ])−ZΛ,Λ0 [J [φ]] = (φJ [φ])− 1
2(J [φ]S ∗ J [φ]) + IΛ,Λ0 [S ∗ J [φ]] , (135)
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(xn)|Λ >1−PI . (136)
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 (137)
31
Furthermore from (110) 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
])
, (138)
where we have introduced: Tr(AB) ≡∫dxdy A(x, y)B(y, x) . Notice that on the basis of the
first equation in (137) one can see that this trace is well defined in spite of the singular behaviorof C at infinite momentum.
Now it is possible to define Effective Proper Generator VΛ,Λ0 by:
VΛ,Λ0 [φ] ≡ 1
2(φC ∗ φ) + VΛ,Λ0 [φ] . (139)
With the chosen sign convention VΛ,Λ0 is positive. 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) , (140)
and the evolution equation for V [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+ ··)
)≡ R[φ] . (141)
Taking into account that IΛ,Λ0 is the generator of amputated connected diagrams whose skeletonvertices are generated by VΛ,Λ0 we can reformulate our renormalization program in terms ofV . We have to analyze the iterative solution of (141), discussing in particular under whichconditions the solution has regular UV limit and this limit has local Λ →∞ limit.
32
Once again it is convenient to use, as above (116), 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
(dpiϕ(pi))δ(n∑j=1
pj)vn(p1, ··, pn,Λ,Λ0) . (142)
Notice that in the notation introduced above
Vn(p1, ··, pn,Λ,∞) =<˜φ(p1) · · · ·˜φ(pn−1)φ(0)|Λ >1−PI . (143)
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) , (144)
where we have set χ = Λ2/Λ20. Notice that if the |vn|’s are bounded and ϕ is of class D
all integrals in the right-hand side of (144) are absolutely convergent uniformly in χ due tomomentum conservation which limits the range of the integration variables (e.g. p − q in thesecond term in brackets). Thus the absolute value of rn is also bounded uniformly in χ.
Now we discuss the solutions of (144) noticing, first of all, that the coefficients vn and rn inthe corresponding expansion of R are analytic functions of the momenta due to the presenceof the cut-off Λ which plays the role of an infra-red cut-off. Then we startconsidering theevolution equation of the coefficients of the Taylor expansion of v2 and r2 up to the secondorder in the momenta, that is v2(0, 0,Λ,Λ0) ≡ v2(0) and ∂q2v
′2(q,−q,Λ,Λ0))|q=0 ≡ v2(0) and
analogous coefficients for r2. Next we study the evolution of the value of v4 and r4 at zeromomenta. Finally we complete the analysis studying ∂2
q2v2(q,−q,Λ,Λ0), the first derivatives∂qv4(q1, ··, q4,Λ,Λ0) and vn for n ≥ 6.
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(0, 0,Λ,Λ0) = r2(0, 0,Λ,Λ0)
Λ∂Λv′2(0, 0,Λ,Λ0) = r′2(0, 0,Λ,Λ0)
Λ∂Λv4(0, 0, 0, 0,Λ,Λ0) = r2(0, 0, 0, 0,Λ,Λ0) , (145)
33
which are solved by:
v0(Λ,Λ0) =(
Λ
ΛR
)−4c0 +
∫ ( ΛΛR
)1
dλ λ3r0 (ΛRλ,Λ0)
v2(Λ,Λ0) =
(Λ
ΛR
)−2c2 +
∫ ( ΛΛR
)1
dλ 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)
, (146)
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 alow 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 (146). If the coefficients depend on Λ0 this is dueto the r’s, and hence to the Λ0-dependence of the other vn’s. Had one chosen initial conditionsat Λ = Λ0 there would appear an explicit Λ0-dependence which would influence the UV limit.
In order to complete 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)of vn, one has evolution equations completely analogous to (144); that is :
[Λ∂Λ + 4− n− k − q∂q] ∂kq vn(q1, ..., q2n,Λ,Λ0) = ∂kq rn(q1, .., q2n,Λ,Λ0) (147)
In the cases we are studying, in which n+ k > 5, 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
)= Λn+k−4
∫ Λ
Λ0
dx x3−n−k∂kq rn
(Λq
x, x,Λ0
). (148)
Indeed the rightmost expression in (148) clearly shows that (147) has a UV regular solution if∂kq rn remains uniformly and absolutely bounded in the limit. Therefore one can identify the
coefficients of VΛ,∞[ϕ] with the solutions of the system of integral equations (146) and of
∂kq vn(q,Λ,∞) = Λn+k−4∫ Λ
∞dx x3−n−k∂kq rn
(Λq
x, x,∞
), (149)
with rn computed from (144). In the following we shall systematically omit the label∞ from theUV limits of the coefficients and the UV limit will be understood for all functions of interestwhenever Λo will not explicitly appear. The solution of these equations puts outstandingconvergence problems, since it is an infinite system of integral equations, therefore we shall
34
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 the system (146)). Takinginto account (148) and (146) one finds coefficient functions vn(q,Λ) that are power series of theinput 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
))(150)
where Pn,k,ν is a suitable polynomial of degree lower than or equal to ν. A completely analogousinequality holds true for sup |∂kq r(ν)
n (q,Λ)|. We conclude that the UV limit can be taken onthe iterative solutions and it appears regular since the x-integral in (148) and those on (144)converge uniformly in the limit.
The last point which remains to be discussed is the locality condition for limΛ→∞ V [φ,Λ]which guarantees that our field theory be local, in particular, that field equations be local. Ona very superficial level this is trivially verified taking into account (149) , (150) and recallingthat:
Vn(p,Λ) = Λ4−nvn(p
Λ,Λ). (151)
Indeed combining these equations it is easy to find that ∂kq V(ν)n (q,Λ) with k + n > 4 vanish in
the UV limit. This is however not enough, indeed one should verify that e.g. V4 has a sensibleUV limit. Unfortunately this is not true since it changes sign for Λ large enough. This is afurther sign that the scalar theory built in this way is sick.
The result that we have reached is based on the construction of the solution in power seriesof c2, c
′2 and c4, As a matter of fact it is not a problem to sum the series expansion in c2 that
plays the role of mass parameter and c′2 that fixes the scale of the field, for simplicity one canchose both constants vanishing and remain with c4 playing the role of a coupling constant g.This must be positive, otherwise the theory becomes unstable and the functional integral (96)does not make sense. One gets in this way a perturbation theory which should be accurate forsmall values of c4 ≡ g.
There is however a different and, potentially, more interesting construction strategy which isbased on the smallness of V4(Λ) for some value of Λ rather than g. In particular this strategy iscrucial for theories with V4(Λ) vanishing at infinite Λ, the limit we are interested in to implementlocality of the theory. Indeed, in this situation that is called asymptotic freedom, one would beconfident in the consistency of the theory independently of the possibility of producing accurateresults at low energy.
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) this implies a strong correction tov2 that, by the way, should apply to the Higgs particle mass and hence is the basic reason toinvoke super-symmetric extensions of the Standard Model.
However the crucial point of our argument concerns the evolution equation of v4 itself.Considering the evolution equation of v4 at vanishing momenta and up to the first non-trivialorder in v4 one has from (144) and (145):
Λ∂Λv4(0,Λ) =4!
(2!)2
∫ dp
(2π)4e−p
2 1− e−p2
p2v2
4(0,Λ)
35
=6π2
(2π)4
∫ ∞
0dx(e−x − e−2x)v2
4(0,Λ) =3
(4π)2v2
4(0,Λ) , (152)
where we have disregarded the term in v6 which is O((v4)3) and any momentum dependence of
v4 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 ΛΛR
. (153)
This result is apparently deceiving since, due to the sign in the rightmost term of (152) 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 that the only consistent scalar theory we can build is thefree 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 (152) end 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 a linear evolution equation(115). It is clear from this and from the above analysis that every operator is defined givinga certain number of low-energy data analogous to c2, c
′2 and c4 in (146). The question is how
these parameters identified. The answer depends on the properties of the operator, that is: itsdimension dΩ , Lorentz transformation (tensor) properties, and possibly charges, in case thetheory has conserved charges. We encourage the reader to verify on the basis of (146) thatthe low-energy parameters are in one-to-one correspondence with the monomials in the fieldsand their derivatives, that have the same Lorentz transformation properties and charges ofthe considered operator and dimension not exceeding dΩ. As a matter of fact an operator isidentified by the initial value of the effective operator ΩΛR.Λ0 [φ] defined in (114). Just to give asimple example, consider, in the framework of neutral scalar field theory invariant under reflec-tion φ→ −φ a dimension dΩ ≤ 4 local operator 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 identifying a particular choice ofeach operator concern the 1-PI vertex with one ω leg and 4 φ legs at zero momenta and thecoefficients of the Taylor expansion up to degree 2 in the momenta of the scalar legs of thevertex with one ω and 2 φ legs. Both vertices must be computed at Λ = ΛR. Let a1 be thevalue of the 4-leg vertex. Due to the chosen Lorentz covariance and Bose symmetry the Taylorexpansion of the 2-φ vertex can be written as: a2 + a3p1 · p2 + a4(p
21 + p2
2). Notice that weunderstand, except in case of explicit warning, the normalization condition that the vacuumexpectation values of operators vanish.
Since the operators are elements of a linear space it is convenient to chose a basis for thisspace which is 4 dimensional in the case under discussion. This basis can be e.g. chosen settingone of the ai, i = 1, ··, 4 equal to 1 and the rest equal to zero. It is clear that the different choicesof i identify 4 independent local operators. A more systematic construction of these bases and
36
a deeper analysis of the properties of renormalized local operators is more easily done using thesubtraction method. Thus we postpone the analysis to a dedicated section.
In this section we have tried to give a general idea of the renormalization group basedconstruction of a quantum field theory transforming the differential evolution equations for therelevant functional generators into integral equations to be solved iteratively.
The discussion has concerned a mass-less scalar field with a particular cut-off; it should beclear that a number of generalizations are possible, introducing e.g. masses in the propaga-tor (101) or changing the Lorentz covariance of the field. In next section we shall present aperturbative solution of the evolution equations, in particular of (141) based on the Feynmandiagram decomposition and on a slight variant of Zimmermann’s subtraction formula. Thisstates a connection between the present RG construction and the well known BPHZ one.
37
11 The subtraction method
We are going to show that the subtraction method gives a solution to the evolution equation(141) which also satisfies the corresponding integral equations (146) and (149). In order toleave open the possibility of fixing the initial values at ΛR = 0 we modify the propagator (141)introducing a mass:
˜S(p) =
e− p2
Λ20 − e−
p2
Λ2
p2 +M2(154)
and hence:
˙S(p) = − p2e−
p2
Λ2
Λ2(p2 +M2)p2 , (155)
and:˜C(p) ≡ (S)−1(p) =
p2 +M2
e− p2
Λ20 − e−
p2
Λ2
. (156)
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, (157)
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 by Sn(p,Λ), we assume the reader to be familiar with the construction ofthe, possibly divergent, Feynman integrals corresponding to the diagrams contributing to itwith vertices corresponding to the gφ4/4! interaction (and possibly to two-leg vertices that willbe induced by the subtraction procedure) and with the propagator (157). S2n appears as a
formal power series in g of the form: S2n =∑∞L=1 g
n+m−1S(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 one of these diagrams, that isdiagram 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) , (158)
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) = cΓ2m+2n−2∏
l=1
S(pl + kl) ≡ cΓ∏
l∈L(Γ)
S(pl + kl) , (159)
38
where l label 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 polynomial factors in the momenta appearing whenever one hasfermions 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 (159) more precise, let us remind that a diagram is a set V (Γ) of verticesjoined by a set L(Γ) of lines, a trivial diagram has a single vertex and no lines, we have alreadyspecified what we mean by 1-particle irreducible (1-PI) diagram. A sub-diagram of a diagramcorresponds to a subset of vertices and a subset of lines joining them. A sub-diagram is, ofcourse contained in the original diagram, two sub-diagrams of the same diagram that have linesand/or vertices in common, either are contained into one another, or intersect, one says overlap.The presence of divergent overlapping diagrams corresponds to that of overlapping divergences.
Let us now consider how the p’s and k’s above can be assigned. As an example we considerthe diagram in figure,
&%'$
A B
1
3
2
(160)
it has three lines 1, 2, 3 two vertices A,B and an external legs attached to each vertex, thusit 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 starts 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 and pV is thetotal momentum flowing into V through the external legs and, in the case of a sub-diagram,the lines not belonging to the diagram. This is momentum conservation at the vertex. Thesecond law refers to any loop L and 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.The ρl’s are resistances associated with the lines which can also vanish or diverge with thecondition that loops formed by lines with resistance identically zero or infinite are forbidden.In the example, assigning the same resistance to the three lines one has pl ≡ p/3. Therefore the
39
external 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) . (161)
The subtraction procedure requires that a momentum routing be also assigned to each 1-PI sub-diagram considering as external the momentum flowing into its vertices through the externallegs and the lines not belonging to the sub-diagram and keeping the same resistances. 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 idea offorest of sub-diagrams, that is a set of non overlapping non-trivial sub-diagrams of Γ includingpossibly Γ itself. The set of all forests FΓ of our example has eight elements, the empty forest,4 forests with a single element, be it the whole diagram or its sub-diagrams mentioned above,and three forests with two elements: the whole diagram and one of its sub-diagrams. For eachdiagram/sub-diagram γ we introduce the operator Sγ which replaces the momentum routingof the original diagram with that of γ and the Taylor operator tdγ (d integer) whose action onIγ(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 of Iγ(p, k), considered as function of p(γ) and k(γ) upto degree d if d is non-negative
• tdγ gives zero if d is negative.
Finally we set dγ equal to the superficial divergence of γ, i. e. in our simple example,dγ = 4−nγ where nγ is the number of external legs/lines of the diagram/sub-diagram γ. Thatis the number of external legs and the lines not belonging to the sub-diagram attached to thevertices 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 (158)IΓ(p, k) with:
RΓ(p, k) ≡ SΓ
∑F∈FΓ
∏γ∈F
(−tdγγ Sγ)IΓ(p, k) . (162)
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(k2/2− k1/2)
−S(k2/2− k1)S(+k1 − k2/2)S(p/3 + k2)], (163)
here t2p takes the Taylor expansion in p up to the second degree.
40
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 right-hand 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 every 4-dimensionalhyperplane as it is easy to verify setting k1 = αk , k2 = (1 − α)k and computing the degreein k of the bracket. However on the whole R4m we see that the content of the bracket in (163)vanishes at infinity as d−6 which is clearly not enough. Now it is clear that asymptoticallymasses and exponentials in (157) do not play any role and hence the content of the bracketin (163) tends to a homogeneous function of the momenta p and k of degree −6. It is alsoclear that the operator 1 − t2p in front of the bracket selects terms of degree higher than 2 inp and hence, asymptotically, terms of degree lower than −6 − 2 in a generic direction in R8
thus implementing the absolute convergence criterion on R8. This should give an idea of howthe subtraction method works. It should be also clear that the exponential in (157) does notplay any role in the UV regime and hence the general convergence proof given by Zimmermannworks also in our case.
Therefore we see that the subtraction method systematically applied to the expansion of thecoefficient functions Vn(p ,Λ) in (142) (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 (141). In order to prove thislet us take the Λ derivative of a generic subtracted Feynman integral corresponding to a 1-PI diagram and hence contributing to VΛ, due to the absolute convergence of the momentumintegral we are allowed to commute derivative with momentum integration and hence we cometo the k-integral of:
∂ΛRΓ(p, k) = ∂Λ
SΓ
∑F∈FΓ
∏γ∈F
(−tdγγ Sγ)IΓ(p, k)
= SΓ
∑F∈FΓ
∏γ∈F
∂Λ
[(−tdγ
γ Sγ)IΓ(p, k)]. (164)
Now some more diagrammatic analysis is needed. A 1-PI diagram Γ is primitively divergent ifits superficial divergence dΓ is non-negative. In the case of a primitively divergent diagram onehas
RΓ(p, k) = (1− tdΓΓ )RΓ(p, k) (165)
whereRΓ(p, k) ≡ SΓ
∑F∈F ′
Γ
∏γ∈F
(−tdγγ Sγ)IΓ(p, k) (166)
and F ′Γ is the family of forests not containing Γ as an a element. In other words, computing
RΓ(p, k) one excludes the subtraction of the whole diagram. This distinction is in order since,the subtraction being made at Λ = ΛR one has
∂ΛRΓ(p, k) = ∂ΛRΓ(p, k) , (167)
41
which is equivalent to limiting the sum over the forests in (164) to F ′Γ. For any forest F
appearing in the sum in (162) one can define the sub-forest F as the maximal set of elements ofF that are not contained into other elements of F , and the reduced diagram Γ/
∏γ∈F γ which
is built by the lines and vertices of Γ not belonging to any element of F and by a further setof vertices each corresponding to an element of F shrunk to a point-like vertex. Then one hasfor any forest in F ′
Γ:
∏γ∈F
(−tdγγ Sγ)IΓ(p, k) =
∏γ∈F
(−tdγγ Sγ)Rγ(p, k)
IΓ/∏γ∈F
γ(p, k) , (168)
in this expression gives the same contribution as a reduced diagram in which the verticescorresponding to the elements γ of F carry a factor equal to (−tdγ
γ Sγ)Rγ(p, k). This factor,i.e. the bracket above, is, of course, Λ independent. Therefore, inserting (168) into (164) andtaking into account what noticed above about the restriction to F ′
Γ, one has
∂ΛRΓ(p, k) = SΓ
∑F∈F ′
Γ
∏γ∈F
(−tdγγ Sγ)Rγ(p, k)
∑l∈L(Γ)
∂ΛIΓ/∏
γ∈Fγ(p, k)
= 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)
(169)
where we have taken into account the fact that the Λ dependence comes from the propagatorand Γ/
∏γ∈F γ ∪ l means the reduced diagram Γ/
∏γ∈F γ deprived of the line l. Exchanging the
order of summations over the forests and over l we get:
∂ΛRΓ(p, k) = SΓ
∑l∈L(Γ)
˙S(pl + kl)
∑F∈FΓ/l
∏γ∈F
(−tdγγ Sγ)IΓ/l(p, k) (170)
where we have extended the idea of forest to diagrams, such as Γ/l which are connected butnot necessarily 1PI. Notice that in (170) the second sum runs over all the forests in FΓ/l sinceF ′
Γ contains FΓ/l. Let us now consider the possibility of Γ/l not being 1PI. Γ/l is howeverconnected and hence it decomposes according to its skeleton structure, a chain 1PI sub-diagramsconnected by lines, into a product of line and 1PI factors, one of the end points of the line lis attached to the first 1PI sub-diagram of the chain the other one to the last. Labelling thesesub-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) (171)
in case Γ/l be 1PI the product above reduce to a single factor. Now a forest F in Γ/l appearsas the union of, possibly trivial, forests in the 1PI sub-diagrams, therefore the sum over theforests in FΓ/l decomposes into the product of the sums over the Fαi
’s and hence we have:
42
∂ΛRΓ(p, k) = SΓ
∑l∈L(Γ)
˙S(pl + kl)
∑F∈FΓ/l
∏γ∈F
(−tdγγ Sγ)IΓ/l(p, k) = SΓ
∑l∈L(Γ)
˙S(pl + kl)
Sα0
∑F∈Fα0
∏γ∈F
(−tdγγ Sγ)Iα0(p, k)
n∏i=1
S(pi + ki)
Sαi
∑F∈Fαi
∏γ∈F
(−tdγγ Sγ)Iαi
(p, k)
= SΓ
∑l∈L(Γ)
˙S(pl + kl)Rα0(p, k)
n∏i=1
S(pi + ki)Rαi(p, k) . (172)
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 ofthe effective proper generator VΛ[φ] and, of course, to that of its coefficient functions givenin (141) whose first two terms are shown, after scaling of the fields, in (144). Indeed one
finds a chain of 1PI amplitudes connected by propagators S and closed by˙S. The only point
that remains to verify is the correct counting of diagrams. In general this is automaticallyguaranteed by the recourse to the functional method upon which (141) is based. Just to clarifythis point with an example, let us consider the three line two leg diagram discussed in some detailabove. 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 (141) this diagramshould appear only once. This is however a wrong argument since it forgets the combinatorialfactors of the diagrams. A diagram with N sets of ni , i = 1, ··, N indistinguishable linescarries 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 we get the resulting contributionto the evolution equation with weight 1/2 which is exactly the combinatorial factor of thecorresponding diagram with two identical lines.
In conclusion we have shown that adapting the subtraction method to the Feynman diagramsbuilt with the propagator S(p) given in (157), and possibly of its spinor, or gauge field variants,yields to a diagrammatic construction of VΛ[φ] solving the RG evolution equation (141). Thusthe question comes: what is the difference between the RG and the subtraction approach? Thechoice of the infra-red cut-off put apart, we can forget it inserting the mass as above and settingΛR = 0, from the UV point of view the subtraction approach deals with one diagram at a time,the forest decomposition solving the problem of overlapping divergences. On the contrary theRG approach groups the contributions of many diagrams together and overlapping divergencesare disentangled taking the Λ-derivative and integrating the evolution equation. What is notclear in the RG approach is how the diagrams should be selected. In an asymptotically freetheory one would select iteratively the asymptotically dominant contributions, this is howeverstill to be done. Much less ambitious is the criterion of selecting all the diagrams with thesame number of loops, which corresponds to the order in h, thus building an iterative solutionto (141) ordered in h; we have just shown that in this case what one gets is identical to whatone obtains applying the subtraction method to a loop ordered diagrammatic expansion. Theproblem is that very often the need of the above selection is overlooked and hence people studyiterative solutions to (141) which are based on arbitrary selections of diagrams and hence haveno physical meaning.
43
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) ≡
∫ 1Λ2
0dα e−αp
2
(173)
and extending the resulting Gaussian momentum integral in d dimensions through:∫ddp e−Xp
2
=(π
X
) d2
. (174)
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. Extending the theory tod Euclidean dimensions also changes the physical dimensions of the parameters. Let µ be thereference mass scale. In d dimensions the fundamental scalar field φ has the mass dimensionof 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
∫ 1Λ2
0dαdβe−(αk2+β(p+k)2)
=g2µ8−2d
2(4π)d2
∫ 1Λ2
0
dαdβ
(α+ β)d2
e−αβ
α+βp2 = g2µ4−d (Λ/µ)d−4
2(4π)d2
∫ 1
0
dxdy
(x+ y)d2
e−xy
x+yp2
Λ2 , (175)
its minimally subtracted version is
g2µ4−d[(Λ/µ)d−4
2(4π)d2
∫ 1
0
dxdy
(x+ y)d2
e−xy
x+yp2
Λ2 − 1
(4π)2(4− d)
]. (176)
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)md2
(Λ/µ)m(d−4)+4−2n∫ 1
0
∏2(m−1)+ni=1 dxi
Dd2 (x)
e−
∑2n−1
a,b=1Na,b(x)pa·pb
Λ2D(x) (177)
where the pa’s span a basis of external momenta and D and Nab are homogeneous polynomialsof degreem andm+1 respectively. In the hypercube xi ≤ 1 Nab ≤ D and hence the singularitiesof the integral come from the vanishing of D in the origin of sectors of the x-space. A systematicdescription of the subtraction procedure is given by Breitenlohner and Maison.
It is clear that repeating the analysis of the subtracted theory in the present case would leadto completely analogous results provided the coefficient of the poles be chosen Λ independentas above.
44
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→ −φ weconsider a basis of dΩ ≤ 4 local operators transforming as an even under reflection scalar field.The basis has 4 elements which have been characterized in the framework of the RG constructionspecifying the corresponding initial conditions at Λ = ΛR. These conditions concern the value,a1, of the 1-PI vertex with one ω leg and 4 φ-legs at zero momenta and the coefficients of theTaylor expansion up to degree 2 in the momenta of the φ-legs of the vertex with one ω leg and2 φ-legs, this expansion is identified by three coefficients according to a2 +a3p1 ·p2 +a4(p
21 +p2
2).Now it is apparent that three elements of the basis, that is those corresponding to the choicesai = δi,j with j = 1, 3, 4 are φ4/24 (∂φ)2/2 and φ∂2φ. Indeed the 1-PI vertices involving oneof these operators and with n < 5 legs are superficially divergent and hence, according to (165)the corresponding functions have the structure:∫
dk(1− tdΓ)RΓ(P, k) (178)
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 thewanted initial condition is φ2/2, however this has dimension two and hence the degree of thesubtraction, d in the above formula, is reduced by 2. Thus the only subtracted 1-PI vertex isthat with two legs and hence for this operator none of the ai’s vanishes, in particular a2 = 1.The situation would be different if the subtraction degree d were systematically increased bytwo for the vertices containing this operator as an internal vertex. This is what is calledextra-subtraction. It corresponds to consider the operator which has in fact dimension 2 as adimension 4 operator. In Zimmemann’s notation this is N4[φ
2/2] which, of course, differs fromthe naive one, that in the same notation is N2[φ
2/2].Zimmemann’s notation for a generic local operator is: Nδ[M ] and specifies a polynomial M
in the fields and their derivatives, which identifies the structure of the Feynman diagram vertexassociated with the operator, and an index δ from which the subtraction degrees of the 1-PIvertices is computed. In the scalar field case the subtraction degree dγ for a diagram γ with nexternal legs is δ minus the number of external legs (d = δ − n). In the case of local operatorsδ must be larger or equal to the dimension of M in order the Feynman integrals be absolutelyconvergent.
As a matter of fact one could have used many other ways of increasing the subtractiondegrees 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 the
45
number and choice of these lines. In this case one speaks of anisotropic subtractions, we shallnot any more consider anisotropic subtractions.
Coming back to our example and having found two different bases, the first one with ele-ments N4[φ
4/24], N4[(∂φ)2/2], N4[φ∂2φ] and N4[φ
2/2], while in the second one the last elementis replaced by N2[φ
2/2], it is clear that the elements of the second basis can be written as linearcombinations of the elements of the first one. In particular this holds true for N2[φ
2/2] whichcan be written in the form:
N2[φ2/2] = e1N4[φ
4/24] + e2N4[φ2/2] + e3N4[(∂φ)2/2] + e4N4[φ∂
2φ] (179)
In order to make this linear relation explicit we must compute the coefficients ei. With thispurpose we introduce the notation < N2[φ
2/2(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 (179). In particular the functions obtainedusing the right-hand side of (179) are easily computed taking into account the definition ofN4[M ] operators. Indeed, for example, choosing the amplitude < N2[φ
2/2(0)](φ(0))4|ΛR >1−PIone has immediately that the contribution of the right-hand side is only due to N4[φ
4/24] andhence this gives the value of e1. The complete decomposition is given by:
N2[φ2/2] = N4[φ
2/2]+ < N2[φ2/2(0)](φ(0))4|ΛR >1−PI N4[φ
4/24]
+1
4∂p1 · ∂p2 < N2[φ
2/2(0)]φ(p1)φ(p2)|ΛR >1−PI |pi≡0N4[(∂φ)2/2]
+1
8∂2p1< N2[φ
2/2(0)]φ(p1)φ(p2)|ΛR >1−PI |pi≡0N4[φ∂2φ] (180)
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, how the coefficients should be computed.
Consider the operator φ(x)φ(0), this is an operator whose Feynman amplitudes have alreadybeen defined together with the renormalized theory. What is ill-defined is the x→ 0 limit of theoperator. 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 of dimension 2. This is a new operator which is expected to have regular x→ 0limit since N2[φ
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 Feynman integrals. This allows theexchange of the x→ 0 limit with the momentum integral.
Let us now consider the difference:
∆(x) ≡ φ(x)φ(0)−N2[φ(x)φ(0)] . (181)
It is clear from the above definition that this difference originates from the extra-subtractionsthat we have inserted into the definition of N2[φ(x)φ(0)]. Let us consider the set of subtracted,
46
but not necessarily connected, Feynman diagrams involving the product φ(x)φ(0). These di-agrams are clearly not 1-PI, however one can still define the set of their forests FΓ and onefinds for the subtracted n-point functions expressions completely analogous to the integral of(162). On the contrary, if the two-point operator is N2[φ(x)φ(0)] the un-subtracted integrandIΓ(x, p, k) is exactly the same, while, whenever both points belong to the same connected partof the whole diagram, the set of forests F ′
Γ is larger than FΓ and contains it. Indeed, if oneconsiders the two-point operator as a single vertex, all the connected sub-diagrams containingboth fields give rise to 1-PI sub-diagrams which were not included in any forest of FΓ. Amongthese diagrams are considered divergent those which are disconnected from the rest of the dia-gram, which thus factorizes into the product of the two-point function < φ(x)φ(0)|ΛR >C andthe rest, and those which are connected to the rest of the diagram by two lines. The diagramscontributing to the first set are just disregarded since the two point factor is considered as thevacuum expectation value of N2[φ(x)φ(0)] which must be subtracted.
The sub-diagrams connected to the rest through two lines, let us call them γ2, are subtractedat vanishing momenta of the two lines. Thus, following the same line of reasoning that hasled to (168), the subtraction term of a forest in F ′
Γ containing some γ2 appears as the product
of the contribution of the smallest γ2 in the forest, let it be γ2, to < φ(x)φ(0)(˜φ(0))2|ΛR >C
times the subtracted Feynman integral of the reduced diagram Γ/γ2 in which γ2 is replaced bya N2[φ
2(0)/2] vertex. The N2 prescription accounts for the subtractions associated with theelements of the forest containing γ2.
Therefore considering ∆(x) defined above, we have that the function with a ∆(x) vertexjust 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 thiscan be expressed in the operator relation:
∆(x) =< φ(x)φ(0)|ΛR >C I+ < φ(x)φ(0)˜φ
2
(0)|ΛR >C N2[φ2(0)/2] , (182)
where I represents the identity operator whose insertion into a diagram reproduces the samediagram. Last equation can be written according:
φ(x)φ(0) =< φ(x)φ(0)|ΛR >C I+ < φ(x)φ(0)˜φ
2
(0)|ΛR >C N2[φ2(0)/2] +N2[φ(x)φ(0)] , (183)
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, the term ' x−2, while the second term contributes as ln x2 the lastterm being regular. This is Wilson OPE in the case of the product φ(x)φ(0). It can easily begeneralized to the case in which the fields are replaced by renormalized composite operators.If needed the expansion can also be pushed further, developing N2[φ(x)φ(0)] in Taylor series ofx.
47
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] , (184)
into the functional integral (96) and we have at the first order in ε:∫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 (185)
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 σΛ0 into the effective actionthrough the source ω:
expFC [J, ω] =∫dµ[φ] e−IΛ0
+(ω σΛ0)[gΛ0
∗φ]e−12(φC∗φ)+(φJ) , (186)
getting ∫dµ[φ]
((gΛ0 ∗
δ
δφIΛ0 + (ω σΛ0) [gΛ0 ∗ φ] + C ∗ gΛ0 ∗ φ− gΛ0 ∗
δ
δφ
)σΛ0 [gΛ0 ∗ φ]
)
exp(φJ)− (φC ∗ φ)
2− IΛ0 [gΛ0 ∗ φ]+ (J gΛ0 ∗
δ
δω) expFC [J, ω] = 0 . (187)
Now let IΛ,Λ0 [φ, ω] be the solution of the evolution equation corresponding to the bare, local,operator IΛ0 + (ω σΛ0)[φ]. We have (see (134) ):
expFC [J, ω] =∫dφ e−IΛ,Λ0
[gΛ∗φ+g∗S∗J,ω]e−12(φC∗φ)+(φJ) , (188)
performing the change of variable:
φ→ φ+ εgΛ ∗δ
δωIΛ,Λ0 [gΛ ∗ φ+ g ∗ S ∗ J, ω] , (189)
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−IΛ,Λ0 [gΛ ∗ φ+ g ∗ S ∗ J, ω, ] exp−1
2(φCφ) + (φJ)
=∫dφ BΛ[gΛ ∗ φ+ g ∗ S ∗ J, ω] exp−IΛ,Λ0 [gΛ ∗ φ+ g ∗ S ∗ J, ω] exp−1
2(φCφ) + (φJ)
−(J gΛ0 ∗δ
δω) expFC [J, ω] = 0 , (190)
48
where, of course:
BΛ[ϕ, ω] =
((gΛ ∗
δ
δφIΛ,Λ0 [φ, ω] + C ∗ φ− gΛ ∗
δ
δφ
)gΛ ∗
δ
δωIΛ,Λ0 [φ, ω]
), (191)
is the solution of the evolution equation corresponding to the bare operator:
BΛ0 [ϕ, ω] =
((gΛ0 ∗
δ
δφIΛ0 + (ωσΛ0) [φ, ω] + C ∗ φ− gΛ0 ∗
δ
δφ
)gΛ0 ∗ σΛ0 [φ, ω]
), (192)
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 (184). Equation (190)represents what is usually called the quantum action principle.
From the RG point of view one could wonder if in some special situation the theory couldbe fine tuned at Λ = ΛR so that BΛ, that is, its initial value vanishes. As a matter of fact, inthe Λ0 →∞ limit, due to the linearity of the operator evolution equations, if the initial valueof the symmetry breaking term BΛ vanishes, BΛ identically vanishes and the theory satisfiesthe Ward-like identity:
−(Jδ
δω) expFC [J, ω] = 0 . (193)
Notice that it is apparent from (191) that the vanishing condition for BΛ 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 in the sum of the elements of the suitable operator basis and(190) and its vanishing condition can be analized using the so called algebraic method that onecan find presented in some details in the book by Piguet and Sorella quoted in the bibliography.
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 + ψγ5A/ψ + iωγ5αψ − iψγ5αω + l0[A] , (194)
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 (96) is:
eFC [A,η,η,ω,ω,Λ0] =∫dµ[ψ]dµ[ψ] exp[−i(ψ∂/ψ) + (ψ gΛ0 ∗ γ5A/gΛ0 ∗ ψ) + l0[A]]
exp[+i(ωγ5αgΛ0 ∗ ψ)− i(ψ gΛ0 ∗ γ5αω) + (ηψ)− (ψη)] (195)
where we have understood space-time integrals. Considering the spinor field change:
ψ → ψ + igΛ0 ∗ γ5αgΛ0 ∗ ψψ → ψ + iψ ∗ gΛ0γ
5α ∗ gΛ0 , (196)
49
and that of A:A→ A+ ∂α (197)
at the first order in α we have∫dµ[ψ]dµ[ψ]i(ψ[ gΛ0 ∗ α gΛ0 ∗ gΛ0 ∗ A/gΛ0 ∗ −gΛ0 ∗ A/ gΛ0 ∗ gΛ0 ∗ α gΛ0∗]ψ)
exp[−i(ψ∂/ψ) + (ψ gΛ0 ∗ γ5A/gΛ0 ∗ ψ) + i(ωγ5αgΛ0 ∗ ψ)− i(ψ gΛ0 ∗ γ5αω) + δAl0[A]]
+[(η gΛ0 ∗δ
δω)− (
δ
δωgΛ0 ∗ η)] expFC [A, η, η, ω, ω,Λ0]
= (∂αδ
δA) expFC [A, η, η, ω, ω,Λ0] (198)
where one does not find contributions from the Jacobian of the functional measure since γ5
is traceless. We have used δA ≡ (∂α δ/δA). Notice that, if the l0[A] term only depends on∂µAν − ∂νAµ ≡ Fµν , the bare breaking term reduces to
i(ψ[ gΛ0 ∗ α gΛ0 ∗ gΛ0 ∗ A/gΛ0 ∗ −gΛ0 ∗ A/ gΛ0 ∗ gΛ0 ∗ α gΛ0 ∗ ψ)
≡ i(ψ gΛ0
[α(gΛ0∗)2A/− (gΛ0∗)2αA/
]gΛ0 ∗ ψ) (199)
and tends in the Λ0 →∞ to the commutator [α , A] which, of course, vanishes. Thus naivelyone would say that in the UV limit one recovers the exact Ward identity:
[(ηδ
δω)− (
δ
δωη)] expFC [A, η, η, ω, ω,Λ0] = (∂α
δ
δA) expFC [A, η, η, ω, ω,Λ0]
≡ δA expFC [A, η, η, ω, ω,Λ0] , (200)
this result is however not guaranteed since the naive functional integral does not make sense.To verify the validity of (200) one has to first consider the effective theory and then to go tothe UV limit.
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:
ΣΛΛ0(p) ≡g2Λ0
(p)− g2Λ(p)
p/, (201)
and the effective action:
IΛ,Λ0 [A,ψ, ψ, ω, ω] = −ψΓΛΛ0ψ − iωγ5α (1− ΣΛΛ0 ∗ ΓΛΛ0)ψ
+iψ (1− ΓΛΛ0 ∗ ΣΛΛ0)αγ5ω −WΛΛ0 [A] + II [Fµν ] (202)
where:
WΛΛ0 [A] =∞∑n=1
(−1)n−1
nTr((γ5A/ΣΛΛ0∗)n) + V [A] . (203)
V [A] accounts for possible terms introduced by the initial conditions of the evolution equationand hence is a space-time integrated local operator of dimension 4, and ΓΛΛ0 satisfies
ΓΛΛ0 = γ5A/ (1− ΣΛΛ0 ∗ ΓΛΛ0) = (1− ΓΛΛ0 ∗ ΣΛΛ0) γ5A/ . (204)
50
Now from (197) one has:
δAΓΛΛ0 = γ5[∂/, α] (1− ΣΛΛ0 ∗ ΓΛΛ0)− γ5A/ΣΛΛ0 ∗ δAΓΛΛ0
= (1− ΓΛΛ0 ∗ ΣΛΛ0) γ5[∂/, α] (1− ΣΛΛ0 ∗ ΓΛΛ0)
= i (1− ΓΛΛ0 ∗ ΣΛΛ0) γ5[p/, α] (1− ΣΛΛ0 ∗ ΓΛΛ0) (205)
where we have used: (1 + γ5A/ ΣΛΛ0∗
)(1− ΓΛΛ0 ∗ ΣΛΛ0) = 1 . (206)
One also has:
δAWΛΛ0 [A]− δAV [A] =∞∑n=1
(−1)nTr(γ5[∂/, α]ΣΛΛ0 ∗ (γ5A/ΣΛΛ0∗)n)
= −iT r(γ5[p/, α] ΣΛΛ0 ∗ ΓΛΛ0 ∗ ΣΛΛ0)
= −iT r(α[((gΛ0∗)2 − (gΛ∗)2) ∗ ΓΛΛ0 ∗ γ5ΣΛΛ0)
−ΣΛΛ0 ∗ ΓΛΛ0γ5 ∗ ((gΛ0∗)2 − (gΛ∗)2)∗
]. (207)
Now we apply (190) 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)− (ψη)] (208)
where we have set:
Ψ = gΛψ +1
p/g ∗ η , Ψ = ψgΛ − η
1
p/∗ g . (209)
First of all we evaluate the contribution in the functional integral (190) of the Jacobian of thefield transformations:
ψ → ψ + igΛ ∗ γ5α (1− ΣΛΛ0 ∗ ΓΛΛ0) ∗Ψ
ψ → ψ + iΨ (1− ΓΛΛ0 ∗ ΣΛΛ0) γ5α ∗ gΛ , (210)
this is given by
δJΛΛ0 [A] ≡ Tr(δ
δψgΛ ∗
δ
δωIΛ,Λ0)− Tr(
δ
δψgΛ ∗
δ
δωIΛ,Λ0)
= −iT r(gΛ ∗[αγ5 (1− ΣΛΛ0 ∗ ΓΛΛ0) + (1− ΓΛΛ0 ∗ ΣΛΛ0)αγ
5]∗ gΛ)
= −iT r([ΣΛΛ0 ∗ α (gΛ∗)2 − (gΛ∗)2αΣΛΛ0∗]ΓΛΛ0 ∗ γ5) (211)
once again we have taken into account that γ5 is traceless. Thus we can write (198) according:
(∂αδ
δA) expFC [A, η, η, ω, ω,Λ0] =
∫dµ[ψ]dµ[ψ] [δAJΛΛ0 [A] + δAWΛΛ0 [A]
+i(Ψ[−iδAΓΛΛ0 + (1− ΓΛΛ0 ∗ ΣΛΛ0)αγ
5( gΛ∗)2ΓΛΛ0 ∗+ΓΛΛ0 ∗ (gΛ∗)2αγ5 (1− ΣΛΛ0 ∗ ΓΛΛ0)]Ψ)
−(Ψ (1− ΓΛΛ0 ∗ ΣΛΛ0)αγ
5gΛ ∗ (p/ψ − iη))
+((ψp/+ iη)gΛ ∗ αγ5 (1− ΣΛΛ0 ∗ ΓΛΛ0∗) Ψ
)]
51
exp[−i(ψ∂/ψ)− IΛ,Λ0 [A,Ψ, Ψ, ω, ω] + (ηψ)− (ψη)]
=∫dµ[ψ]dµ[ψ]
[δAJΛΛ0 [A] + δAWΛΛ0 [A] + i
(ΨBΛΛ0 [A]Ψ
)+ i(ηgΛ0 ∗ αγ5 (1− ΣΛΛ0 ∗ ΓΛΛ0) Ψ))
−i(Ψ (1− ΓΛΛ0 ∗ ΣΛΛ0)αγ5gΛ0 ∗ η)
]exp[−i(ψ∂/ψ)− IΛ,Λ0 [A,Ψ, Ψ, ω, ω] + (ηψ)− (ψη)]
=∫dµ[ψ]dµ[ψ]
[δAJΛΛ0 [A] + δAWΛΛ0 [A] + i
(ΨBΛΛ0 [A]Ψ
)]exp[−i(ψ∂/ψ)− IΛ,Λ0 [A,Ψ, Ψ, ω, ω] + (ηψ)− (ψη)]
+[(η gΛ0 ∗δ
δω)− (
δ
δωgΛ0 ∗ η)] expFC [A, η, η, ω, ω,Λ0] (212)
where we have set:
BΛΛ0 [A] = −iδAΓΛΛ0 + (1− ΓΛΛ0 ∗ ΣΛΛ0)αγ5( gΛ∗)2ΓΛΛ0 ∗
+ΓΛΛ0 ∗ (gΛ∗)2αγ5 (1− ΣΛΛ0 ∗ ΓΛΛ0) + p/γ5α (1− ΣΛΛ0 ∗ ΓΛΛ0)
+ (1− ΓΛΛ0 ∗ ΣΛΛ0) γ5αp/ = (1− ΓΛΛ0 ∗ ΣΛΛ0) γ
5[p/, α] (1− ΣΛΛ0 ∗ ΓΛΛ0)
+ (1− ΓΛΛ0 ∗ ΣΛΛ0)αγ5( gΛ∗)2ΓΛΛ0 ∗+ΓΛΛ0 ∗ (gΛ∗)2αγ5 (1− ΣΛΛ0 ∗ ΓΛΛ0)
+p/γ5α (1− ΣΛΛ0 ∗ ΓΛΛ0) + (1− ΓΛΛ0 ∗ ΣΛΛ0) γ5αp/
= (1− ΓΛΛ0 ∗ ΣΛΛ0) γ5α(gΛ0∗)2ΓΛΛ0 + (gΛ0∗)2γ5α (1− ΣΛΛ0 ∗ ΓΛΛ0) . (213)
Now, taking into account (204) one finds:
BΛΛ0 [A] = (1− ΓΛΛ0 ∗ ΣΛΛ0)[α(gΛ0∗)2A/− A/(gΛ0∗)2α
](1− ΣΛΛ0 ∗ ΓΛΛ0) . (214)
In order to evaluate the contribution of this term let us notice that, in the UV limit, the vertexcorresponding to the square bracket contributes by a factor α(q)A(k)[exp(−(p + q)2/Λ2
0) −exp(−(p + k)2/Λ2
0)] where p is the momentum entering into the vertex from the left. We arekeeping the Fourier transform conventions of the first chapter (11), that is:
f(x) =∫ dp
(2π)4eip·xf(p) . (215)
If A, η, and ˜η are fast decreasing functions at high momenta the momenta p, q and k remainlimited uniformly in Λ0 and hence the vertex factor vanishes in the UV limit.
It remains to compute: δJΛΛ0 [A]+δAWΛΛ0 [A]. Using (207) and (211) and, once again, (204)one gets:
δJΛΛ0 [A] + δAWΛΛ0 [A] = δAV [A]− iT r[ΣΛΛ0 ∗ α(gΛ0∗)2 − (gΛ0∗)2αΣΛΛ0∗
]ΓΛΛ0γ
5
= δAV [A] + iT r[α(gΛ0∗)2A/− (gΛ0∗)2αA/
]ΣΛΛ0 ∗ (1− ΓΛΛ0 ∗ ΣΛΛ0) . (216)
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 we cansee in a moment, since it involves the difference of momentum integrals each of which divergesin the UV limit.
Comparing (214) , (216) and (199) we find the same term in brackets. This is by no meanssurprising since the term breaking the Ward identity at the effective theory level must be that
52
generated by the evolution of the bare breaking (199). This operator corresponds to the sum ofconnected and amputated diagrams containing the vertex (199) and built with the propagator(201). Thus what we have done until now is just a lengthy exercise verifying Wilson effectivetheory.
Now we have to analyze what remains of the breaking in the Λ0 → ∞ limit taking intoaccount that it is contained in (216) and depends on the choice of V [A]. One has to keep inmind that V [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, the Ward identity toremain unbroken.
Let us start considering Tr[α(gΛ0∗)2A/− (gΛ0∗)2αA/] ΣΛΛ0.Putting: ∆ΛΛ0(p
2) ≡ (exp(−p2/Λ20)− (exp(−p2/Λ2))/p2 we find:
4i∫ dpdq
(2π)8α(q)p · A(−q)[exp(−(p+ q)2/Λ2
0)− exp(−p2/Λ20)]∆ΛΛ0(p
2)
≡ 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− bA · ∂2A/2] +O(1/Λ2
0) . (217)
This result shows that the considered contribution to the breaking can be fine-tuned to zeroinserting the term
∫dx[−aΛ2
0A2/2 + bA · ∂2A/2] into V [A]. Indeed, after this insertion, what
remains linear in A in δJΛΛ0 [A] + δAWΛΛ0 [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/− (gΛ0∗)2αA/
]ΣΛΛ0 ∗ (ΓΛΛ0 ∗ ΣΛΛ0)
n. (218)
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.
We consider these contributions separately. We begin with n = 2 getting:
i∫ dpdq1 . . · dq3
(2π)16α(−
∑qi)[exp
(p−∑qi)
2
Λ20
− expp2
Λ20
]
TrA/(q1)ΣΛΛ0(p−∑
qi + q1)A/(q2)ΣΛΛ0(p− q3)A/(q3)ΣΛΛ0(p) . (219)
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
Λ20(2π)16
exp−p2
Λ20
∆3ΛΛ0
(p)[4pµpνpρ − 3pµδνρp2]p ·∑
qiα(−∑
qi)Aµ(q1)A
ν(q2)Aρ(q3)
= −2i∫ dpdq1 . . · dq3
p2Λ20(2π)16
exp−p2
Λ20
(exp−p2
Λ20
− exp−p2
Λ2)3α(−
∑qi)∑
qi · A(q1)A(q2) .A(q3)
= 2∫ dp
p2Λ20(2π)4
exp−p2
Λ20
(exp−p2
Λ20
− exp−p2
Λ2)3∫dx∂α(x) · A(x)A2(x)
' 1
2(4π2)
∫dx ∂α(x) · A(x)A2(x) (220)
53
Once again we find a term which can be written as the variation under (197) of an integratedlocal functional of the background field A. Indeed we have found δA
18(4π2)
∫dx(A2(x))2. There-
fore, in much 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 V [A].
Finally we consider the n = 1 term in (218):
i∫ dpdq1dq2
(2π)12α(−q1 − q2)[exp
−(p−∑qi)
2
Λ20
− exp−p2
Λ20
]TrA/(q1)ΣΛΛ0(p− q2)γ5A/(q2)ΣΛΛ0(p)
−4i∫ dpdq1dq2
(2π)12α(−q1 − q2)[exp
−(p−∑qi)
2
Λ20
− exp−p2
Λ20
]
εµνρσAµ(q1)qν2A
ρ(q2)pσ∆ΛΛ0(p− q2)∆ΛΛ0(p) , (221)
selecting under the integral sign the terms of order Λ−40 for p ' Λ0 we get:
−8i∫ dpdq1dq2
Λ20p
4(2π)12α(−q1 − q2)(p ·
∑qi) exp
−3p2
Λ20
εµνρσAµ(q1)qν2A
ρ(q2)pσ
= −2i∫ dpdq1dq2
Λ20p
2(2π)12exp
−3p2
Λ20
α(−q1 − q2)εµνρσAµ(q1)q
ν2A
ρ(q2)qσ1
= −2i∫ dp
Λ20p
2(2π)4exp
−3p2
Λ20
∫dxα(x)εµνρσ∂µAν(x)∂ρAσ(x)
= − i
24π2
∫dxα(x)εµνρσ∂µAν(x)∂ρAσ(x) ≡ A[α,A] . (222)
This last term cannot be written as the variation under (197) 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 the ABJanomaly. 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,∞] . (223)
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 example isparticularly simple since what remains at the effective level of the bare breaking is restrictedto terms which only depend of the background field A. The fine-tuning process appears asa compensation between the effective breaking induced by the radiative corrections and thegauge variation of V [A], the initial value of the effective action (more precisely on the purebackground part of it.) This is in principle possible since the effective breaking and δAV [A]have the same dimension: 4 in the example. In order that the fine-tuning be complete, oneneeds a one-to-one correspondence between the effective breaking terms and the independentones in δAV [A]. This appears to be almost true in our example, the only exception beingthe last term which has originated the anomaly. One could wonder about the reason of this
54
quasi-complete correspondence. The answer to this question turns out to be quite simple dueto the pure background character of the breaking. Indeed one is in fact limiting the study to(200) for vanishing η , η , ω , ω which, for a generic choice of the low energy normalizationconditions can be written:
(∂αδ
δA) FC [A,∞] = B[α ,A] , (224)
where B[α ,A] represents the generic breaking. In principle B[α ,A] should depend on Λ andone should limit the discussion to Λ = ΛR, however in the present case it does not depend, sincethe left-hand side of (224) is Λ independent. Now (224) 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 onehas:
(∂αδ
δA)B[β ,A]− (∂β
δ
δA)B[α ,A] = 0 . (225)
Let us now consider how B[α ,A] is constrained by dimensional, covariance and parity con-straints (due to parity conservation it must have positive parity being α a pseudo-scalar and Aa 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] .
(226)and hence
(∂βδ
δA)B[α ,A] =
∫dxα(x)
[c1∂
2β + c2(∂2)2β + c3(2A · ∂β∂ · A+ A2∂2β)
+c4(∂β · ∂A2 + 2A · ∂(A · ∂β))](x) + (∂β
δ
δA)(∂α
δ
δA)V [A] (227)
which must be antisymmetric in α and β. We leave as an exercise to the reader to verify thatthis expression is antisymmetric in α and β if and only if c3 = c4. Under this condition onehas:
B[α ,A] = (∂αδ
δA)∫dx[−c1A2 + c2(∂ · A)2 − c3
2(A2)2](x) + V [A]
+c5
∫dxα(x)[εµνρσ∂µAν∂ρAσ](x) (228)
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/24π2.
This is a simple example of a non-trivial application of the already mentioned algebraicmethod. Let us try to put into evidence the aspects which have general validity. I think thatthe main points to be accounted for are the following.
A local quantized-external field transformation analogous to (196) and (197) 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[V ] of a generic choice of the initial value of the effective
55
action V . As a matter of fact BΛR= Br + δ[V ] where Br accounts for the radiative correction
terms.In our example δ is a linear functional differential operator, however in general it is not.
In particular it turn out to be non linear whenever the field are transformed non-linearly.BRS transformations are a typical example of non linear transformations. However in theframework of loop ordered perturbation theory, if the classical (zero loop) value V0 of V isuniquely identified by the invariance constraint δ[V0] = 0, the n-th order contribution of δ[V ]nis linear in Vn, let it be δV0Vn and hence one has δ[V ]n = δV0Vn+ρ[V ]n where ρ[V ]n is independentof Vn. Thus in this situation the recursive symmetry condition is
BΛR,n = Br,n + δV0Vn + ρ[V ]n = 0 . (229)
In complete analogy with our example one can deduce directly from (190) a consistencycondition for BΛR,n which is recursively written as a linear functional differential equation
δV0 ∧BΛR,n = 0 , (230)
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 perturbationtheory whenever the general solution to the consistency condition 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.
56
A The Wilson Action
To prove (102) we introduce a simplified notation understanding space-time integrals and vari-ables and introducing the symbol ∂1 for gΛ ∗ (δ/δJ) and ∂2 for gΛ ∗ (δ/δJ). It is then apparentthat
gΛ0 ∗δ
δJ= gΛ ∗
δ
δJ+ g ∗ δ
δJ≡ ∂1 + ∂2 . (231)
Thus we write (100) 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 . (232)
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 , (233)
where we have introduced (SJ)2 ≡ g ∗ S ∗ J , and S2,2 ≡ g ∗ S ∗ g and by[n/2] we mean theinteger part of n/2. Now from (232) and (233) 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∗ δ
δJ+ g∗S∗J ]) : e
12(JS∗J) , (234)
where we have introduced S1,2 ≡ gΛ ∗ S ∗ g.
57
B Bibliography
The major reference for the first part on the scattering theory is:R. Haag Local Quantum Physics. Berlin: Springer-Verlag 1992Further 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 1987and for what concerns the calculation of cross-sections:C. Becchi and G. Ridolfi An introduction to relativistic processes and the Standard Model
of electro-weak interactions Berlin: Springer-Verlag 2005An introduction to the Euclidean Theory can be found in:K. Symanzik Euclidean Quantum Field Theory. In Local Quantum Theory. Scuola Internaz.
di Fisica ”Enrico Fermi”, Corso 45. R. Jost Ed. New York: Academic Press 1969For the Renormalization Group Method the general 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 given
by:C.Becchi On the construction of renormalized quantum field theory using renormalization
group techniques. In Elementary Particles, Quantum Fields and Statistical Mechanics. Semi-nario Nazionale di Fisica Teorica M.Bonini, G. Marchesini, E. Onofri Eds. Parma 1993 (hep-th/9607188 )
A general reference for basic quantum field theory isC. 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.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.About the BPHZ subtraction method the basic references are:J. H. Lowenstein BPHZ renormalization. andW. Zimmermann The power counting theorem for Feynman integrals with massless propa-
gators.in Renormalization Theory. Proceedings of the NATO Advanced Study Institute held at the
International School of Mathematical Physics at the ’Ettore Majorana’ Center for ScientificCulture in Erice (Italy) G. Velo and A. Wightman Ed.s - D.Reidel Publishing Company ,Boston 1976.
P.Breitenlohner and D.Maison, Commun. Math. Phys. 52(1977)11, ibid. pg. 39, ibid pg.55.
An account of the algebraic method can be found in:O. Piguet, S. Sorella Algebraic renormalization: Perturbative renormalization, symmetries
and anomalies. in Lect.Notes Phys.M28 Berlin: Springer-Verlag 1995?
58
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