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BRS symmetry for Yang-Mills theory with exact renormalization group
ci2i.L --Ii!!
NUCLEAR PHYSICS
B
Nuclear PhysicsB 437 (1995) 163-186
ELSEVIER
BRS symmetry for Yang-Mills theory with exact renormalization group * M. Boninia,
M. D’Attanasio
a. G. Marchesini
b
a Dipartimento di Fisica, Universitci di Parma and INFN, Gruppo Collegato di Parma, Parma, Italy b Dipartimento di Fisica, Universith di Milan0 and INFN? Sezione di Milano, Milan, Italy
Received19 October1994;accepted5 December1994
Abstract
In the exact renormalization-group (RG) flow in the infrared cutoff A one needs boundary conditions. In a previous paper on SU( 2) Yang-Mills theory we proposed to use the nine physical relevant couplings of the effective action as boundary conditions at the physical point A = 0 (these couplings are defined at some non-vanishing subtraction point p # 0). In this paper we show perturbatively that it is possible to appropriately fix these couplings in such a way that the full set of Slavnov-Taylor (ST) identities are satisfied. Three couplings are given by the vector and ghost wave-function normalization and the three-vector coupling at the subtraction point; three of the remaining six are vanishing (e.g. the vector mass) and the others are expressed by irrelevant vertices evaluated at the subtraction point. We follow the method used by Becchi to prove ST identities in the RG framework. There the boundary conditions are given at a non-physical point A = A’ # 0, so that one avoids the need of a non-vanishing subtraction point.
1. Introduction
Renormalization-group (RG) formulation [ 11 provides the most physical way to deal with the ultraviolet (UV) singularities. Following a suggestion of Polchinski [ 21 (see also Refs. [ 3-51) the exact RG formulation has been used recently to give self-contained and simple perturbative rederivations of many properties such as renormalizability [ 2,581, infrared finiteness of massless theories [6,9], operator product expansion [ 101, decoupling theorem [ 111. Some approximations have also been attempted within this context [ 121. * Researchsupportedin part by MURST, Elsevier Science B.V. SSDIO550-3213(94)00569-9
Italy.
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Its application has been recently extended to gauge and chiral gauge theories [ 5,13171. The challenging problem here is the general fact that local gauge symmetry typically conflicts with the presence of a momentum cutoff. For the SU(2) Yang-Mills theory it is shown in Ref. [S] and in the present paper that, in spite of the explicit breaking of gauge symmetry, the Slavnov-Taylor (ST) identities can be implemented perturbatively by appropriately fixing the boundary conditions on the RG flow. Thus the RG formulation provides an alternative to dimensional regularization [ 181 of gauge theories. To describe the situation we briefly recall the exact RG formulation for gauge theories. Consider the wilsonian effective action &E[ 4, y; A] with $ and y the fields and the BRS sources and A an infrared (IR) cutoff. This functional is obtained by path integration over the fields with frequencies above n and up to an UV cutoff Ao. At A = 0 and A0 + 0;) one has performed the full path integral and therefore from the functional &ff[ +, y; A = 0] we can obtain the physical Green functions or the oneparticle irreducible vertices. Actually the vertices of Se~[ 4, y; A = 0] are the amputated connected Green functions. We never write in the functionals the UV cutoff A0 since we always understand Ao -+ 00, which is possible due to perturbative renormalizability. The functional Sen[ #J,y; A] satisfies an evolution equation in the IR cutoff, which is obtained by observing that the path integral over all frequencies can be done first by integrating over the frequencies above A and then below A. The fact that the result does not depend on A gives the mentioned evolution equation. The exact RG equation is non-perturbative but it can be solved perturbatively once the boundary conditions are given at some value of A. An important point concerning the boundary conditions is the distinction between relevant and irrelevant parts of the functional S,E[ 4, y; A] (and of any other functional such as the physical effective action r[ 4, y] ). As usual the vertices with negative mass dimension are called “irrelevant” and contribute to Seff,im[4, y; A]. The remaining part &a,,[ 4, y; A] contains only a finite number of parameters with non-negative dimension (relevant parameters) which, for A # 0, can be defined for instance as the first coefficients of the Taylor expansion of Sen[ 4, y; A] around vanishing momenta. The UV value A = A0 is the obvious value where to fix the boundary conditions for Seff,im[ 4, y; A]. For & 4 00 one expects that all vertices with negative dimension vanish thus one sets Seff,irr[ 4. y; A = Ao] + 0 for & + 00. Finally, the key problem is to fix the boundary conditions for the finite number of relevant parameters in Serrret[ 4, y; A]. Here is where one implements in the theory both the physical parameters (such as masses, couplings, wave-function normalizations) and the symmetry, i.e. ST identities. In Ref. [ 131 we considered the SU(2) Yang-Mills theory which has nine relevant parameters. We proposed to fix these nine parameters at the physical point A = 0. Since the theory involves zero-mass fields, at A = 0 one should define these parameters as the values of some vertices at non-vanishing subtraction points. The reason for selecting A = 0 is that &[ 4, y; A = 0] is related via Legendre transform to the effective action r[ 4, y]. Therefore, some of these parameters are fixed at the physical values of the couplings, masses and wave-function normalizations. We proposed to obtain the remaining parameters by imposing some of the ST identities and showed that this procedure can be implemented perturbatively. Therefore, by construction some of the
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ST identities are satisfied, thus the fundamental question is whether the effective action r[ 4, y] computed by this procedure does indeed satisfy the full set of ST identities. In the paper of Ref. [ 131 we were able to prove this only partially at one-loop level. Recently, Becchi [S] considered the exact RG formulation of the SU(2) YangMills theory and imposed the boundary conditions for the relevant parameters in Sefrrel[ 4, y; A] at a non-physical point A = A’ # 0 so that the relevant parameters can be defined by expanding the vertices around vanishing momenta. In this way the relevant parameters are not directly related to physical couplings in the effective action ~[c#J, y] but the analysis of relevant parts of ST identities becomes easy. By using algebraic methods he was able to prove that the full set of ST identities can indeed be satisfied perturbatively. He identified a functional L&[ 4, y; A] which gives the “defect” to ST identities and then showed that one can obtain perturbatively the nine relevant parameters in &r~~i [ #J,y; A] by solving the fine-tuning equation A efi,rel[A~;Al
=0
for the relevant part of the ST defect. In the present paper we apply the same method to the case in which the boundary conditions are given at the physical point A = 0. In this case the operation of extracting the relevant parameters become rather complex since one has to use non-vanishing subtraction points. We are able to generalize Becchi’s proof to this case and we explicitly prove that the procedure suggested in our previous paper gives indeed an effective action which satisfies perturbatively ST identities. Moreover, we explicitly express the solution of the fine-tuning equation by giving in terms of physical vertices those relevant parameters which are not fixed by the physical couplings and wave function normalizations. This paper is organized as follows. In Section 2 we recall the RG formulation for the SU( 2) Yang-Mills theory given in Ref. [ 131. We introduce the wilsonian effective action &r[+. y; A], define its relevant and irrelevant parts, discuss the role of the boundary conditions and the local symmetry. In Section 3 we study the operator which gives the violation to the ST identities and show that the problem of implementing the symmetry can be reduced to the solution of a finite number of equations, the$rre-tuning equations. Section 4 contains the explicit solution of these fine-tuning equations and in Section 5 we rephrase this solution in the usual algebraic language. Section 6 contains some conclusions.
2. Renormalization-group
flow for W(2) Yang-Mills
The fields and corresponding c/5= {A;,
ca, Z’} ,
j
sources for SU(2) =
theory
Yang-Mills
theory are
, y = {u;, u”} 7 > where ut and v“ are the sources associated to the BRS variation of Al and P, respectively. The generating functional, in the Feynman gauge, is
ZLLrl
=exp(W.Lyl)
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J
D4exp
Physics B 437 (1995) I63-186
i{-$(4P2+)ono
+
(j4>ono
+ Znt[+7Y;
nol} 7
(1)
where the path integral is regularized by assuming a UV cutoff ,40. In general we introduce the cutoff scalar products between fields and sources
- d4p J=J (277)4 ’
P
' ] c"(p) w>,,o-J K,-,o(~) 1 {j;(-p)A;(p)+ [X'"(-p)- ~p,u;(-p) P
+~a(-P)xu(P) I3 where Kin, is the cutoff function which is one for A2 < p* < At and rapidly vanishes outside. The functional Sin, is the UV action involving monomials in the fields, BRS sources and their derivatives which have dimension not larger than four and are Lorentz and SU(2) scalars. There are nine of these independent monomials n
&nt[47
Y; llol
=
d4x
{;A,.
[gpv<
a; + u;J2)
+ u3”c7&] A,
1
+afwr t&A,
+CT,/ + us” (&,A,)
. A,, A A,
A A,) . (A, A A”) + d%+,
. A,) (Afi. 4,)
+a~w,~c~A,+a;u~c~c}, where WE = u: + ga,P and we have introduced the usual scalar and external SU(2) product. The nine couplings cr: depend on A0 and have non-negative dimension (relevant parameters). In order to obtain the physical theory one has to show that the values of the a? can be fixed in such a way that: (i) The no + co limit can be taken by fixing the physical parameters such as the masses, coupling g and wave-function normalization constant at a subtraction point ,x Perturbative renormalizability ensures that this can be done [ 2,5,13]. (ii) In the no t cc limit the Slavnov-Taylor identities must be satisfied. This is the crucial point to be discussed in this paper. According to Wilson one integrates over the fields with frequencies A* < p2 < Ai and obtains (see Appendix A)
exp (WLYI)
J +&dAy; Al},
=Nj,y;Al
v$ expi{--~(4p24)on
+ (jd>o, (2)
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where N[j, y; A] contributes to the quadratic part of W[j, r] logN[j,y;A]
1 KL4o (P> PzKon 0(P)KOA(P)
=-i s
P
X
1
ijE(--p)
j;(p)
[
f”(-p)
- ip&-P)
I
X”(P)
>
The functional See is the wilsonian effective action and is obtained by integrating fields 4’ with the higher modes (p2 > A*)
exp (~{~(~~*~)hb =
s
(3) the
+ &ff[4,r;~lI)
Dq5’ exp (i{--+(~$‘p~&)
AA” + (j’4’>hio
+ hx[4’~~;AOl})
(4)
T
where the source is j’(p)
= P*{ A;(P),
-C’(P)~
(5)
C”(P) 1.
By comparing with the physical generating functional Wj’,x~l
= ~(&*4~).4~~
+ &ff[~,r;~l
as a functional of j’, generates propagators have frequencies in external propagators is cancelled for all values of p. At n + 0 generating functional W[j’,yl
=
W[j’,y;A=Ol
W[j, r] in ( 1) we see that (6)
,
the connected Green functions the range A2 < p* < 4. The by the inverse cutoff so that the and ila --f co, this functional
=
~(~~P'$)oA~
+&ff[&r;A
in which the internal cutoff function in the source j’ can be taken becomes the physical
=Ol.
(7)
Given the relation in (5) between j’ and 4 we have that the vertices of Seg[ 4, y; n = 0] are the amputated connected Green functions. Taking into account that the variable n enters as a cutoff in the internal propagators of connected Green functions, one derives the exact RG equation [2] $
.I
AJAKoA(P>
exp (-&f/fi)
P 1
6
%A;(-&
6
SAC(p)
6
+ Sca( -p)
6
~ l%(p)
exp (iS,&ti)
.
(8)
This equation is non-perturbative but it can be solved perturbatively once the boundary conditions are given at some A. We have included fi in (8) to show explicitly how the loop expansion is generated. In the r.h.s. there are two contributions. One is quadratic in the action and has a coefficient with the same power of fi as the 1.h.s thus involving Green functions with lower or equal loop order. The second, linear in the action, has a coefficient with an additional factor ti and therefore involves Green functions with a
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lower loop. Thus, starting at zero loop with S$*)
given by the value of Green functions
at the boundary conditions, Eq. (8) allows one to obtain $2 at all loops. The fact that one of the contributions in the r.h.s. in (8) is at the same loop order as the 1.h.s. makes the generation of the loop expansion less transparent and less efficient. Actually, as shown in Refs. [ 6,7], the loop expansion is more easily generated if one integrates the RG flow not for the wilsonian action S,, but for a cutoff effective action r[ 4, y; A] given by the usual Legendre transform of the cutoff functional W [ j, y; A] introduced in (6)
n4,
y; Al = w1.L y; Al - WI07 0; Al -
.I
{ji(-p>A;l,(p)
+ Xa(-p)ca(p)
+ PC-P>x”(P>}
P
This functional of the “classical” fields 4 generates cutoff vertex functions in which the internal propagators have frequencies in the range A2 < p2 < Ao. From (7) we have indeed that at A = 0
UAYI
= Uhy;A
=Ol
(9)
is the physical effective action. One can convert (8) into the RG equation for r[ 4, y; A] which has the following form:
AJ/ifl4
Y; Al = hIF
[d%
Y;
Al ,
(10)
where the functional Zr[ 4,~; A] is given in Ref. [ 131 and depends non-linearly on the vertices of r[ c$,y; A]. Then in the r.h.s. of ( 10) there are vertices at a loop order lower than the l.h.s., so that by solving iteratively this equation one automatically generates the loop expansion. We now discuss the crucial point of the boundary conditions which provide the starting point for the loop expansion. 2.1. Boundary conditions: physical parameters and symmetry There is an obvious value where the boundary conditions should be given which is the UV value A = A,. Here one finds from (4)
thus Sen becomes local and depends only on the nine relevant couplings C: with nonnegative dimension. Since & is the only surviving mass parameter as A0 -+ CC one expects that, for A0 4 co, all vertices with negative dimension vanish. For this reason vertices with negative dimension are called “irrelevant” and their contributions to the wilsonian action will be denoted by Sen,im[#J,y; A]. The remaining part Serrret[ 4, y; Pi (A) ] is a local functional which contains nine relevant parameters c+i(A) with non-negative dimension. The form of Serrret[ 4, y; (+i (A) ] is the same as Si,, in which the UV couplings CT: are replaced by ai( A). Also for
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these parameters we need to fix the boundary conditions at some A = A’. The precise definition of gi( A) in terms of S,ff depends crucially on whether one assumes boundary conditions at A = A’ # 0 (e.g. A’ = Aa) or at the physical point A’ = 0. In the first case the definition of ai is simple since one does not need to impose a subtraction point and can consider the Taylor expansion of the Green functions around vanishing momenta. Therefore, one defines
Y; a.(A) 1 = T4’0’SefiM1,y’ Al 1 seff,rel[+11
A # 0,
where Tie’ is the Taylor expansion around vanishing momenta truncated to the terms with coefficients with non-negative dimension (see for instance Ref. [ 51). Taking advantage of the fact that &[ 4, y; A = 0] and r[ 4, y; A = 0] are physical functionals (see (7) and (9)), we have suggested in Ref. [ 131 to fix the relevant parameters ai at the physical point A = 0 so that Ci( A = 0) are physical couplings, such as for instance the mass, the wave-function normalizations and the coupling constant g at a subtraction point p. At A = 0 the definition of the functional &rrei [ #, y; o-i (A = O)] requires the introduction of a finite subtraction point p. We then define Seff,rel[4JTri(A)]
=Z”/“‘$E[~,Y;AI
3
where the operator TicL) is given in Appendix B for a scalar theory. The extension to the YM theory is simple in principle and Tip) r[ 4, y; A] is defined completely in Appendix B . For the boundary conditions for the RG equation (10) of the cutoff effective action r[ 4, y; A] we follow the same procedure: (i) We assume that the irrelevant parts of r[ 4, y; A = Ao] vanish rim[@,Y;A=Ao] (ii)
= [l -T:“]
r[~$,y;A=Ao]
~0.
We fix the remaining nine relevant parameters at the physical point A = 0. It is useful to rearrange the various monomials and use the following parametrization for the relevant part of the physical effective action r[ 4, r] = r[ 4, y; A = 0]
+dz3 - 1)(&A,) . A, A A, - &J,) +~reh$,r;Pil where Fyv = d,A, - &A,
Ll[4J;
Pi1 =
(11)
7
+ gA, A A,, D,c = a,c + gA, A c and
d4X { +,A,
J +&apAJ +&do.
. (&A,)
. A,
. (&A,)
+ ~3
c A c + $g2ps(A,
wfi +c A A, A A,) . (A, A A,)
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(12) In (11) we have singled out two BRS-invariant monomials, the gauge-fixing contribution and an additional three-vector coupling monomial. We now discuss the values zi and pi that one has to fix in order to have the physical couplings and the local SU(2) gauge symmetry. Some of these couplings can be fixed to give the wave-function normalizations and the three-vector coupling g at a subtraction point ,LL This can be implemented by assuming zi = 22 = ~3 = 1 so that the first part of rret in ( 11) is just the BRS action in the Feynman gauge &as [ 4, r] and we have ~,l[$,y1
= SBRS
+
f&,[#‘~~~~il
.
(13)
Notice that the choice of boundary conditions at n = 0 for the relevant couplings provides a conspicuous advantage with respect to the case in which one takes A = A’ # 0. We have in (13) that the relevant part of the effective action already contains the BRSinvariant classical action which is the starting point of perturbation theory. We have also explicitly written the h-dependence which shows that the other six couplings pi vanish at zero loop. In Ref. [ 131 we proposed to fix the pi’s from the ST identities for the physical effective action r[ 4, r] given by A,-[+,y]
z &-~r’[dtY]
=o,
(14)
where
J SF 6 SF 6 ~ ___ PA;+aAa,(p) a~;(-P) I’I{a~;(-P) SF 6 6s s
T’[&Yl
= nAYI
+ ;
d4Jr ( c&AJ2
and Sr/ is the usual Slavnov operator [ 191
sp =
+suQ(-p) &T”(p) + St”(p) Su”(-p) I . The six couplings pi are involved in all vertices of the functional Al,[ 4, -y] . In Ref. [ 131 we obtained the various pi from the simplest ST identities and found that these couplings are given by some irrelevant part of vertices evaluated at the subtraction point p. We showed that this procedure can be implemented perturbatively. This is due to the fact that, as indicated in ( 13), the couplings pi vanish at zero loop. At higher loops some of the couplings pi are different from zero. The mass of the vector which is given by ~1, as expected, remains zero at all loops. In that paper we were not able to show that this procedure ensures that the ST identities are actually satisfied for all vertices, In the next section we will show that this is the case.
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3. Gauge
symmetry
and fine-tuning
Physics B 437 (1995) 163-186
equation
171
at A = 0
The proof that gauge symmetry can be implemented perturbatively in the RG formulation has been given by Becchi in Ref. [ 51 in the case in which the boundary conditions for the nine parameters a,(n) are given at some non-physical point A = A’ # 0. These parameters are defined by applying the operator Tie’ to the wilsonian effective action. The perturbative analysis of the ST identities is simplified by the fact that T,” commutes with the ST functional derivative at zero loop. Here we follow the same procedure but in the case in which one fixes the boundary conditions at the physical point n = 0. The price for working directly at A = 0 is that the relevant couplings are defined by the operator Tq(y) which does not commute with the zero-loop ST operator. As a result the analysis of the ST identities seems more complex. We are able to show that actually the analysis is easy and one can write a simple fine-tuning equation which allows one to determine the six relevant couplings p,. Following Becchi we consider the generalized BRS transformation
with v a Grassmann parameter. Applying this change of variables to the functional integral (2)) one deduces the following identity:
J
n+exp
SZLLYI =N.Ly;~l x44>y;Al,
+
(i{-~(+P*$)oA
(.@)oA
+&T[~,Y;AI})
where
is the usual ST operator obtained from the variation of the source term (j~$)a,. Notice that the cutoff functions cancel. The functional A arises from the variation of the other terms in the exponent and from the jacobian. It can be divided into two parts, a linear and a quadratic one in the derivatives A=Al
+A*,
Al =i
J{
(16)
with P’A;(P)
I’
&
+ g/YP) CL
~ aA;
6
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and A2 =
6 s &n(P) exp (-is&) & Sua Fep, + Sea(p) Su”(-P) sP CL F x exp (iS,ff) .
Since Z&(P) vanishes for A = 0 we have that at the physical point A[ r$, y; A = 01 = dl[+,y;n=ol. We have to prove that one can select the six relevant couplings pi in such a way that
N~,Y;AI
=O,
(17)
so that the ST identities
szLLr1
=o
are satisfied. Actually we will show that the functional equation (17) can be reduced to a finite number of relations, the$ne-tuning equations. The RG flow for the functional A is given by the following linear evolution equation: AcYnA[q5,y;A]
=M.A[$,y;A]
E M[A;A],
(18)
where the linear operator M depends on Se@
The RG equation ( 18) requires boundary conditions. As before we discuss the boundary conditions for the relevant part of A[ 4, y; A] at A = 0 and for the irrelevant part at the UV point A = &. The definition of the relevant part of A[ 4, y; A] at A = 0 requires a subtraction point. This functional has dimension one and we have
&1[4,y;&l
=~;C”‘AM,y;A=Ol,
(19)
where TiP’ is again defined as in Appendix B. There are 11 relevant parameters 6i (see Appendix C) which are the coefficients of the 11 monomials in the fields, sources and momenta of dimension not greater than five and with ghost number one. Recall that at A = 0 the expression of A simplifies since the non-linear part A2 vanishes due to the cutoff function.
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We can easily prove that the irrelevant part of the functional provided that the UV cutoff is sent to infinity
4,,[4,y;A
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A vanishes at A = A0
for A0 + 00.
= Aol + 0,
(20)
First observe that Al [A = A,] contains only relevant parts. This is due to the fact that at A = A0 we have S,ff[A = Ao] = 5’t”t which is local. For 42[A= Aa] we have that the functional in the integrand is local apart from the p-dependence of the cutoff function Ken,(p) which gives an irrelevant contribution to A2 [ A = Ao] . However, for A0 + 03 we have Kc,(p) = 1 so that A2[ A = A0 4 001 becomes local. Using the boundary condition (19) and (20) we can integrate the RG equation for A and obtain A
A[#,Y;AI
= &l[4,y;&l
+
J
+re,M;
11-
J
O” dA
TM,,[4
11.
(21)
A
0
We now prove the following two perturbative theorems for the functional Ace) [ q5,y; A] at loop f? and for the corresponding vertices AL’) ( . . . ; A) where n denotes the number of fields or sources and the dots denote momenta, internal and Lorentz indices. Theorem 1: if A(“‘) [ 4, y; A] = 0 for all loop C’ < k’, and if Ai? (. . . ; A) = 0 for n’ < n, then Ace) n ( . . ..A)=A$.( Proof
.. :A=O). ,
From the hypothesis the functional
Mce)[A.A],
= -(27r)’
M[ A; A] at loop e is given by
JLA&Kon(~) P2
+ &a(p)
~A’~)bf~,y; Al , where S$) is the tree wilsonian
&a(-p)
- c ++ F
(22)
action in (6). Since the starting point of the perturbative
expansion (see ( 13) ) is I-(‘) [ 4, y; A] = Snns [ 4, ~1, the vertices of S$’ [ 4, y; A] are the amputated connected Green functions at tree level. Moreover, S$ [ 4, y; A] does not contain contributions quadratic in the fields, they are all included in i ($p*~$), see (6). Consider now the RG equations (21) for a vertex Aif) (. . . ; A). From (22) we have that MC”) [A; A] involves only vertices Ai? ( . . . ; I) with n’ < n. They vanish according to the hypothesis thus only the first term in r.h.s. of (21) is left and the theorem is proved. An obvious consequence of this is the second theorem. Theorem 2: if Ace’) [ 4, y; A] = 0 for all loop .!? < ! and if at loop e the relevant part of this functional vanishes at A = 0, namely if
&,[~Y$~)I
=O,
(23)
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then A(‘) [ 4, y; A] = 0. The line-tuning condition (23) will be used to fix the relevant couplings pi of the effective action r[ 4, y] . However, the functional A[ 4, y; A] defined in (16) involves vertices of the wilsonian action Serf[ $, y; A]. It is convenient to express the fine-tuning condition at A = 0 in terms of a functional which involves directly the vertices of r[ 4, y] This can be simply done by taking the Legendre transform of A[ $, y; A] at n = 0. In this way the tine-tuning equation (23) is equivalent to
where Ar,,,l has the same decomposition as A,1 given in Appendix C. Then both Theorems 1 and 2 can be phrased for the functional Ar [ c+h, y] . From now on we consider A = 0 and we show perturbatively the equivalence of (23) and (24). By using the inductive assumption A (“1 = 0 for C’ < e, one has the following results at loop e: (i) For the two-field vertex one has Aj!)(Ac) = A(“)(*@, then the fine-tuning condition A(e)(AC1 = 0 is equivalent to A~?~~lAc)= 0. I mposing this condition the full vertex rel A vanishes. 1. (ii) For vertices with three fields one has A$f)(AAc) = A(e)(AAc) and Af-e)bcc) = A(“)(wcc), since the one-particle reducible terms vanish due to (1). Thus the finetuning conditions A$)‘AAc’ = 0 and A~$e:)(wcc) = 0 are equivalent to A~~~,AAC)= 0 and A~~~lwcc)= 0, respectively. With these conditions these two vertices vanish. (iii)
Similarly for the vertices with four fields one has Ay)(3Ac) = A(‘)(3Ac) and A~)(WACC)= A(e)(wAcc) and the remaining fine-tuning conditions become A~~~13Ac)= 0 and ~(l)(w*Cc) _- 0 and the two functions Ay)(3Ac) and A~)(WACC)vanish. f,rel (iv) By increasing the number of fields one has that the two functionals A(!) and A)? are equal. In the next section we rove by induction on the number of loops that it is possible to fix the six couplings pi ! ) in such a way that si(‘I = 0 so that the fine-tuning equation is solved and from Theorems 1 and 2 the ST identities’hold perturbatively.
4. Perturbative
solution
of the fine-tuning
equation
The starting point of the proof by induction over the number of loops is L!~‘O) [ 4, y] = 0 which is valid since r(‘=O) = &as. We suppose that the fine-tuning equation is solved at e’ < e SO that Al?” [ 4, y] = 0. We want to show that it is possible to fix the six couplings pIc) in such a way that Af-e’[ 4, y] = 0. Notice that there are six parameters pj’) and 11 equations 6:“’ = 0. Therefore, the solvability of these equations requires that there must be five relations connecting the various Sj”‘. They are provided by the so-called consistency conditions, a set of equations which must be identically satisfied by the functional A,- due to its definition (14). From the anticommuting character of S~J we have S~IA,. = S$r’ = 0, where Sr., is defined in (15). By using again the
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Physics B 437 (1995) 163-186
inductive hypothesis A,(I” - 0 for !’ < e, the consistency condition loop 4 one finds
175
simplifies
S r; A(“) r = 0,
and at
(25)
where Sr; is defined in ( 15) with r given by Snns. In order to obtain from this equation the desired relations connecting the various S,“‘, we have to extract its relevant part by applying the operator TiFL). Notice that this operator mixes relevant and irrelevant parts of Ar. This in general makes some of the consistency conditions complicated due to the presence of irrelevant contributions. We analyse in detail the fine-tuning equations and consistency conditions for the various vertices of Ar. We start form the vertices with two and three fields and sources. We then analyse the remaining vertices. In the following all vertices will be considered at loop !. (i) The vertices with two and three fields are obtained from the definition of Ar in (14) AjlA,“(p) A?$?(~,q,k)
=T’;*)(p)I-y)(p) =I-y)(k)rz)(p,q,k) +l+p
Ajlw,‘(p,q,
,
(sY~~*)(qJw4
k) =r~)(p)Z+=)(p,q, +T;“C)(k)TEA)(p,q,k).
(26) - f’~?p)~$‘%,
kq)
1 k) + T;WC)(q)T~f(p,
(27) k,q) (28)
The relevant parameters of these vertices are the five 6i with i = I, 2,3,4,5 (see Appendix C) while the relevant couplings of the r vertices are the four pi with i = 1,2,3,4 (see ( 11) and Appendix B). Then we need one relation among these 6i. Indeed if we consider the consistency condition (25) for the vertex of the ACC fields and take its relevant part, we obtain ig& = 83 + & .
(29)
It is a nice fact that this relation involves only relevant parameters. This is not a general fact since the operator TiPL) applied to SrAr typically extracts irrelevant vertices of Ar evaluated at the subtraction point. We are now able to satisfy the fine-tuning equation for the vertices in (26)(28). From (26) we obtain
and we have 61 = 82 = 0 by fixing p1 =p2=0. From (27) we obtain
(30)
M. Bonini el al. /Nuclear
176
Physics B 437 (I 995) 163-186
and we have 6s = Sd = 0 by fixing (31)
p3=0.
Finally, from (28) we obtain -$p&vi$Y+“(p, P
q, k)
I
3SP >
,
and we have Ss = 0 by fixing +p,k.F$Y’)(p, P
q, k)
1
.
(32)
3SP
From Theorem 2 of the previous section we have that, since the relevant parts of the vertices in (26)-( 28) vanish, the complete vertices vanish. Moreover, from Theorem 1 of previous section the four-field vertices of A,- could be only relevant. (ii) We COLZ now to analyse the vertices of A,- involving four fields. Consider first the vertex Ay/)abCd (p, q, k, h) , which contains the relevant parameter Se (see Appendix C). From the definition of A=[ 4, r] in ( 14) one easily sees that this vertex involves only couplings pi already fixed thus one should be able to show that the fine-tuning condition Se = 0 is satisfied automatically. Indeed there is a consistency condition which, after having imposed (30)) (3 1) and (32)) reduces to S6 = 0. This is due to Theorem 1 which implies that irrelevant parts of the vertices of Ar with four fields vanish. The last vertex of A,- we have to consider is
= r~wc)(h)r~~~b”bcd(p,q,k,h)
+ l-‘:A,A1(p)l-~;AA)adbc(p,h,q,k)
+~‘;~)(q)~~~AA)bduc(q,h,p,k) +~eda~ebc~~~A)(q+
k,h,p)I$$‘(p
~~~~~~~~~~~~~~~ + k, h,q)T;$(q +eedceebar;~*)
(q + p, h, k) Z$;’
+l+~‘(k)l-~v+‘++d”b(k,h,p,q) + h,q, k) + h,p, k) (k + h, q, p) .
(33)
This vertex contains the five relevant parameters Si with i = 7,. . . , 11 while the rvertices involved here contain only the two relevant couplings not yet fixed, ps and p6. The consistency condition (25) gives the three relations &=S9=S,,,
S8 = SlfJ.
(34)
Notice that these relations involve only relevant parameters. As stated before this is a consequence of the fact that we have already set to zero the vertices of Ar with two and three fields and, according to Theorem 1, the vertices with four fields do not have any irrelevant parts. From (34) we have that it is possible to have Si = 0, i = 7,. . . , 11, simply setting to zero two independent Si (e.g. ST and Ss) by fixing p5 and p6. The easiest way
M. Bonini et al./Nuclear
Physics B 437 (1995) 163-186
II?
of obtaining & and 8s is to contract in two different ways the vertex in Eq. (33) by external momenta and summing over internal indices. Since this vertex does not have any irrelevant part, these contractions are proportional to two different linear combinations of 87 and 8s. Using for simplicity the symmetric point we find PPqvkP &Ac)aabb(p, WfP P4 = --$(Zi7
q, k, h) 4SP
+ S,) P&‘kPhfl~C4A)onhb(g, PF P4
q, k, h)
1
4SP
and A(3Ac)aabb
PIJquhP
@UP
P4 = 4(2&
4SP
- 76s)
ig(y%s +‘F(h,
(~7 4, k h)
i p,qvhphu p4 g
- $P6) -
(P, q, k, h)
+ ~kp)T~~AA)aabb(k,h,p,q)
+12- p~qvhp~~wcA)(q p4 UP The fine-tuning
$,,;?bb
1
$ k, h,p)I$;;‘(p
+ h,q, k) 4sp.
equations 87 = 0 and 8s = 0 give k, h)
1
(35)
4SP
and ’ 3i i pcLqvhph, 1 r$;b”abb(p, p4 L’S= gP6 - 16g ; 1 +‘F(hp,
q, k h)
ikp)l$;AA)aabb(k,h,p,q)
+12PPq”hPr(wcA) r,F P4
(q+k,h,p)r~~~)(p+h,q,k)
1
.
(36)
4SP
This completes the proof that we can set to zero the 11 parameters 6i by fixing, at every loop, the six couplings pi according to (30), (31), (32), (36) and (35).
’ In Ref. [ 131 some contributions
to (36) were missing.
178
M. Bonini et al./Nuclear
5. Algebraic
Physics B 437 (1995) 163-186
formulation
We want to connect the perturbative solution we have discussed with the algebraic formulation given in Ref. [ 51. This is based on the fact that if AI;~~I[ $, y; Si] can be written in the following form:
&,,,I [ 4, Y; &I = Spm rre~[ $7 Y; Pi 1 ,
(37)
then it is possible to find a perturbative solution of the fine-tuning equation by appropriately fixing the pi. This property is discussed in Ref. [5] and recalled in Appendix D. Eq. (37) is valid provided the 11 parameters in Ar,,el fulfill the following five relations: ig82
(38)
= S3 + 64 ,
8~=-ig&,
&=S9=S*,,
S8 =s10.
(39)
In our case, in which we have a non-vanishing subtraction point p # 0, the first relation (38) holds (see (29) ) , while the relations (39) are not valid in general since there are contributions from irrelevant parts. Still also for ,u + 0 we can use the above theorem as follows. First observe that the relation (38) holds also for p # 0 and involves only vertices with two and three fields, i.e. AL*‘), Ay) and A$..“) . Therefore, the form (37) is valid for these vertices and from Appendix D we deduce that we can solve the fine-tuning equations Si = 0 for the five relevant parameters with i = 1,. . . ,5. This is obtained by fixing the four couplings pt, ~2, p3 and ~4, which enter in the vertices of r with two and three fields. From the previous section the solution is pr = p2 = p3 = 0 and p4 given in (32). After solving these five fine-tuning conditions, we have from Theorem 2 that the complete vertices ApI, AjfiAc) and Ay) vanish and from Theorem 1 that the vertices with four fields are only relevant. Therefore, the relations in (39) are now valid and all vertices of Allrel [ 4, y; Si] can be expressed as in (37). From Appendix D we can perturbatively solve the remaining tine-tuning equations and from previous section the solution is ps in (36) and p6 in (35).
6. Conclusions
In this paper we have fixed the renormalization conditions for the SU(2) Yang-Mills theory, i.e. the nine couplings (zi and pi, see ( 11) ), which enter in the relevant part of the effective action r[ 4, r] as boundary conditions at n = 0 for the exact RG flow. Three of these couplings are fixed to give, at the subtraction point p # 0, the vector and ghost wave functions and the three-vector coupling (~1 = ~2 = ~3 = 1). In this way one fixes a contribution of the effective action to be the BRS classical action (for instance in the Feynman gauge, see (13)). The other six couplings pi in (12) are absent at tree level. They are at our disposal in order to implement the gauge symmetry for the physical effective action, i.e. the ST identities Ar [ 4, r] = 0 in ( 14). The fact that the three couplings zi = 1 are not affected by loop corrections is one of the advantages of working at the physical point n = 0 rather than at A = A’ # 0. (We shall recall
M. Bonini
et ul./Nuclear
Physics B 437 (1995) 163-186
179
later the difficulties arising from the non-vanishing subtraction point.) We have shown perturbatively the following two results. The first result (see Section 3) is that it is possible to satisfy the ST identities A,.[ 4,r] = 0 if one is able to solve the fine-tuning equations A,-,,,,[ qb,y; Si] = 0. It is a consequence of the RG equation (18) (see Theorems 1 and 2) and of the fact that the starting point of the loop expansion is the classical BRS action which satisfies the ST identities Ajf=“’ = 0. The second result (see Sections 4 and 5) is that it is possible to solve (perturbatively) the 11 fine-tuning equations 6i = 0 by fixing the six pi couplings. This is possible since, due to the consistency condition (25)) there are only six independent 6i. Moreover, we have constructed the solution and found that the relevant part of the effective action is
rrel[4t’Y;fil
=SBRS
+
d4x { $4~
fi
. c A c +
$&&4p
A A,)
. (A,
A A,)
J
+$g2p6(A,-A,)(Ap~AAv)}.
The only non-vanishing couplings ~4, ps and p,j are given in (32), (36) and (35), respectively, in terms of irrelevant vertices of r evaluated at the subtraction point. This form allows one to deduce the perturbative expansion since irrelevant vertices at loop ! involve relevant couplings at lower loops P < !. We have followed the method used by Becchi [5] in which the boundary conditions are taken at the non-physical point n = A’ # 0. In that case one can define the relevant parameters by expanding around vanishing momenta thus avoiding irrelevant contributions in the consistency conditions. Since we work at A = 0 we have needed non-vanishing subtraction points and introduced the operator Tk(” which defines the relevant parts of the various functionals. This makes the analysis of the ST identities more difficult since the operator Tk(‘) mixes relevant and irrelevant parts when applied to the product of two functionals. As discussed in Section 5 the fine-tuning equation Ar [ 4, y; Si] = 0 can be perturbatively solved provided that the local functional A,tret can be parametrized as in (37). This is valid if the relations (38) and (39) hold. Actually the relations (39) are not valid in general if ,u + 0 due to the presence of the irrelevant contributions generated by the TCLL’operator. However, we have shown that one can use the parametrization in (37) by p;oceeding in two steps. First one uses the fact that Eq. (38) is valid also for p f: 0. This allows one to solve the fine-tuning equations for the vertices of A,. with two and three fields. Once these equations are solved the remaining relations (39) hold since the irrelevant contributions vanish. In this way one can use (37) and solve the remaining fine-tuning equations. The method is general. As shown in Ref. [ 171 it can be applied for instance to SU(2) gauge theory with fermions. The application to the case of chiral gauge theory without anomalies should be also possible along the same lines.
M. Bonini et aLlNuclear
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Physics B 437 (1995) 163-186
Acknowledgements We have benefited greatly from discussions with C. Becchi and M. Tonin.
Appendix
A
The generating functional
( 1) is equivalent
to
(A.11
KOA,(P) = Ken(P) + KIAO(P> and the coefficient N is given in EQ. (3). This can be easily seen by making in (A.l) the change of variables 4, = 4’ - 4, which gives
x
I
~+exp(i{-~($p24)
OA- ~(4p’dhn, + (j14>,,o + (j4)0,}) >
(A.2)
where the source jl (p) is h(p)
=p2(AE(p),
-C’“(P),
C’“(P) ).
By performing the integral over the field 4, which is gaussian, one obtains ( 1). On the other hand, by integrating over the field 4’ in (A.2) and using (4) one obtains (2).
Appendix
B
We explicitly give the form of the operator TifiL), which extracts the relevant part of a functional of a multicomponent massless scalar field k. Due to the masslessness of the field, the Taylor expansion around vanishing momenta is affected by infrared singularities when we consider the four-dimensional terms in the fields and momenta. Thus one is forced to introduce a non-vanishing subtraction point. For the two-field components this point is assumed at p2 = p2, while for the N field components it is assumed at the symmetric point NSP defined by - PiP,=&(NSij-l),
N=3,4
,....
In order to have a more compact notation, we also introduce the Fourier transform Ji of the field +i. The form of T4(‘) is then
M. Bonini et al./Nuclear
Physics B 437 (1995) 163-186
181
In the following we explicitly give the relevant part of the effective action T[+,y] for the SU( 2) Yang-Mills theory. We first isolate the vertices of r[ 4, r] which contain the relevant couplings
P
PUP +GE ISrc3*)(p,q,r)A~(p>A~(q)A~(r) 1
abc
P
9
+AJJJ
k, h)A;(p)A:(q)A;(k)A:(W
I’z:bcd(p,q,
P
q
k
J
+ q%)w;(-P)cYP) P
+cnbc
JJ JJ
rEA)(p,q,r)w;(p)cb(q)A;(r)
P ++abc
9
T(““)(p,
P
q, r)u”(p)cb(q)cc(r)
+ ... ,
03.1)
4
where r = -p - q, h = -p - q - k and the dots stand for all the remaining terms which contain only irrelevant vertices, since they are coefficients of monomials in the fields and sources with dimension higher than four. The six vertices in (B. 1) contain nine relevant couplings which are defined as follows:
182
M. Bonini et al. /Nuclear
Physics B 437 (1995) 163-186
r;*)(P) =gpv[~m* +p2c-1 +t,v(p)
[gA
r(3A)(pt 4, r) = [(P - 4)&v WP X[c3A rggbCd(p,
[ u;A
rcwcA) PJJ
where
t,, p,;“,
+ (4 - r),&J + (1. - P>&pl 9, r> 3 + 7; j$)
(P, 9. r) 9
q> k h) 1
+ $4A)(P,
4, k h) 1
(p, q, k, h) ,
=~[-i+(Twc+P)(p)l,
(p, 4, r> = &v
r(vcc)(p,
+ 2L(P>l 13
+ $4A)(p,
+Fj$gbcd
q@(p)
+ ST(p)
+ 2 (3A)(p,
q, k, h) =+$&4A +tg&a
+ca)
q, r) = -1
[ VwcA + 2 (wcA) (p, q, r>
1 + Fp)
+ (T,,,
,
+ ~(vcc)(p,
q, I)
(p,
q, r) ,
(P 1 = p2 gcLp - P,P~, = (Ealazcp3~4
is the four-vector
SU(2)
abed
t2;llvpg = ( rbs”d
_
&wucp2a3
) &L,p3&2w + (2 4-b 3) + (3 - 4)
structure appearing in the BRS action and
+ 6”” abd + lYd fibc) @u&l,
+ g,&u
+ &?,,g, 1 .
The relevant couplings are defined by the conditions XL(O)
zT(p)
= 0,
= 0,
.x(3A)(p,q,r)
= 0,
pG$
~~4A’(P,q,k,h)l
3SP
= 0,
$?p,q,W)
= 0,
4SP 2’“c’(p)
4SP = 0,
J&p2 JGvcc’(p,
q,
r)
-CwC*)(p,
q, r)
= 0 ) 3SP
= 0. 3SP
From these conditions2 we can factorize in the vertices 2~ a dimensional function of p. Thus Z; are “irrelevant” and contribute to the irrelevant part of the functional r[ r$, ~1. Similarly the vertices Ti are irrelevant since their Lorentz structure is (partially in the case of r(4A)) given by external momenta. We recall that the ghost propagator and the &A vertex are given in terms of the vertices ry) and I-p) by ’ Notice that in order to follow the general rule of extracting the relevant part of a functional given by the operator TiFL), the couplings CT* and (TwC are defined in a slightly different way with respect to Ref. [ 131.
M. Bonini et al. /Nuclear
F)(p) PA)(p, ”
=p2 + ip2[aw,
q, t-) = -igp,TEA’
Physics B 437 (1995) 163-186
+ /.P”‘(p)]
183
,
(p, q, i-)
In Subsection 2.1 in order to better identify the physical couplings in the effective action and study the gauge symmetry we have made a different definition of the relevant part of the effective action (see Eq. (1 I)), introducing the couplings zi and pi. The relation of these couplings with the gi is the following: z]=l
-(TA-(Tar
z2
PI =a,,
9
p4=uycc
+ iflT,,,
z3 = 2 - ‘,,A g
= 1 + iuwc ,
pz = @a 7
- U,L, - ua,
p3 = 1 + iuwc + flwcA,
3 p5 = 1 - (TA - (T, + -$A
+ +;A)
,
p6 = 5--&7'4A.
In particular when the physical boundary conditions ~1 = ~2 = ~3 = 1 are imposed one finds a,-/-aA
u WC--0,
=o,
UrnA = PI 1 o;cc
= P4 9
ua
= P2 1
u4A
= g2(-1
u3A uv,,A +PS
-
$6,
>
u'4A
= -ig, = -1
+p3,
= ;g2p6,
which show that these boundary conditions fix the vector and ghost wave-function normalization and the three-vector coupling. The other couplings in T[+,y] are given in terms of the pi and determined by the symmetry as in Section 4. Appendix
C
We now extract the relevant part of the most general one-dimensional fields and sources with ghost number one. We call this generic functional we isolate the vertices of A which contain the relevant couplings A=
JI'
Ajt\“‘(p)A;(
functional of A. First of all
-p)c”(p)
A~?'(P, 4, ~M;(p>A,bhWW JJ I,4 +I&JJ JJJ JJJ + I Eubc
2
A~CC’tp,q,r)~~(p)cb(q)cC(~)
2
P Y
+;
A@“ACC)“bCd(p,q,k,h)~;(p)A~(q)cc(k)cd(h) P‘
+t
A(3Ac,abcd(,,q,k,h)A;(p)A;(q)A;(k)cd(h) VP
I’
4
k
+. . . ,
184
M. Bonini et al./Nuclear
Physics B 437 (1995) 163-186
where r = -p - q, h = -p - q - k and the dots stand for the remaining terms which are all irrelevant. Then we define the 11 relevant parameters as follows: AfYP)
=pp[&
Ar’(p,q,r)
+p*s*
+
A’*“‘(p)1
,
-q2>C~3+~~AAc~~~,q,~~l
=g,,(p*
+(~,~~-q,q~)[&+A:AAc)(~~q~r)l +ip)(p,
Ap)(p,q,r) A(WACC)=bCd(p,q, P”
9, r) ,
=p,[G
+ A(WCC)(p,q,r)l,
k, h) =g#==Sbd
- Fdc+)
+$wAcc)abcd(p, A(3Ac)ubcd(p,q, WP
k, h) = (Gu&dp,g,
[S, + A@‘*@(p,q,
k, h)]
q, k, h) , + . . .) [&
+ Ai3*@(p,q,
k, h)]
+(6=d~bcp,g,+...)[c%+A;3Ac)(p,q,k,h)l
+WbScdp,g,,+
. ..)[a.
+~:3Ac’(~,q,kh)l
+(GQC@dp,g,p+ . . .>iho + A:3Ac)(p, q, k, h) 1
+Wd@‘cp,g,p + . . .>i&l + A:3Ac)(p, q, k, h) 1 +i63Ac)abcd WP
(p, q, k, h) ,
where the dots stand for permutation over the gluon momenta and Lorentz and colour indices. The conditions defining the relevant parameters are A(*“‘(O)
= --&A’nc’(p)
=o,
Ajy’
p24
A(wCC) =o, 3SP
Ai3Ac)
=o, 4SP
=o, 3SP
A(W*CC)
=o.
4SP
Due these conditions one can isolate in these vertices a dimensional function of p, thus they are irrelevant. Similarly the vertices ii have the Lorentz indices carried by momenta in a different way with respect to their relevant parts and are irrelevant.
Appendix
D
In this appendix we prove that the fine-tuning equations (24) can be solved if satisfies
Af,rel[ 4, Y; Sil = S~~CW r’ rel[$9Y;Pil.
Ar,,,=l
(D.1)
An explicit calculation shows that Eq. (D.l) implies the relations (38) and (39) among the 6i. The parameters pi are given by the following functions of Si:
M. Bonini et al. /Nuclear
81 = -p,, 65 = +
-Ps)
1
Physics B 437 (1995) 163-186
82
= -;p2,
67
= MPS -
63 i-=j3
+
P4)
185
= P5 3
6, = -2ig(&
,
-
.
/53)
(D.2) Now we show that it is possible to select the six parameters pje’ in r,,r[~#~r; pje’] such a way that six fine-tuning equations, 6ie) = 0, are satisfied. From ( 14) we have e-i Ay) [ 4, y] = 2S,-,w I+(‘) + c S,,ca r’(e-k) k=l By applying T:‘),
in
.
we obtain the relevant part
&I[ 4, y; 6;“) 1 = 2S,-co,I-;, [$, y; pje’ I + de) [qb,yl , where E-l
The crucial observation now is that fi (0 does not depend on the relevant parameters pie). This is obvious, since the product of two relevant vertices is a relevant vertex. This implies T;pc”’S,vco,T,(IL) = Sr,co,T4(pcL) . As a consequence (Z’:P)Sy,~~~ - S,.,C~,T4(p’)r:e, = 0 and ace) does not receive contribution from the couplings pje). Thus the last term in (D.3) involves only I$) [ d,r] which is given in terms of P!~‘) at lower loops P < C. Eq. (D.l) implies that L?ct) must be of the form n(“) = Spa, r;,, [ 4, y; pry)] )
pfie) = jjp
- 2py
.
As previously observed, p’je) are known from the calculation Therefore, by fixing #)
= +j”)
of pjr”
,
(D.4)
we have pi = 0 and from (D.2) we have the final result Ay) [ q5,y] = 0. References [ 1I K.G. Wilson, Phys. Rev. B 4 (1971) 3174,3184; K.G. Wilson and J.G. Kogut, Phys. Rep. 12 (1974)
at loop C’ < e.
75.
186
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163-186
[2] J. Polchinski, Nucl. Phys. B 231 (1984) 269. [ 31 G. Gallavotti, Rev. Mod. Phys. S7 ( 1985) 471. [4] C. Becchi, Lectures on the renormalization of gauge theories, in Relativity, groups and topology, Vol. 2, Les Houches 1983, eds. B.S. Dewitt and R. Stora (North-Holland, Amsterdam, 1984). [ 5 ] C. Becchi, On the construction of renormalized quantum field theory using renormalization-group techniques, in Elementary particles, field theory and statistical mechanics, eds. M. Bonini, G. Marchesini and E. Onofri (Parma University, 1993). [6] M. Bonini, M. D’Attanasio and G. Marchesini, Nucl. Phys. B 409 (1993) 441. [7] C. Wetterich, Phys. Len B 301 (1993) 90. [ 81 G. Keller, C. Kopper and M. Salmhofer, Helv. Phys. Acta 65 (1992) 32; T. Morris, Int. J. Mod. Phys. A 9 (1994) 2411; R.D. Ball and R.S. Thome, Renormalizability of effective scalar field theory, CERN-TH 7067193 preprint. [9] M. Bonini, M. D’Attanasio and G. Marchesini, Perturbative infrared finiteness of a massless scalar theory in d = 4, UPRF 94-396 University of Parma preprint; G. Keller and C. Kopper, Perturbative renormalization of massless @$ with flow equations, preprint UniGoe-ThPhy 4-93; R.D. Ball and R.S. Thorne, Infrared and ultraviolet behaviour of effective scalar field theory, CERN-TH 7233194 preprint. [lo] J. Hughes, Nucl. Phys. B 312 (1989) 125. ] 111 L. Girardello and A. Zaffaroni, Nucl. Phys. B 424 ( 1994) 219; R.D. Ball and R.S. Thome, The decoupling theorem in effective scalar field theory, CERN-TH 7233-94 preprint; C. Kim, Decoupling, factorization and irrelevant operators in effective field theory: RG-flow equation approach, SNUTP 94-20 preprint. ] 121 N. Tetradis and C. Wetterich, Nucl. Phys. B 422 ( 1994) 541; T.R. Morris, Phys. Lett. B 329 (1994) 241, Phys. Len. B 334 (1994) 335; U. Ellwanger, Z. Phys. 62 (1994) 503; M. Alford, Phys. Len B 336 (1994) 237. [ 131 M. Bonini, M. D’Attanasio and G. Marchesini, Nucl. Phys. B 421 (1994) 429. 1141 B.J. Warr, Ann. Phys. (NY) 183 (1988) 1, 89; G. Keller and C. Kopper, Phys. Lett. B 273 (1991) 323. [ 151 M. Reuter and C. Wetterich, Nucl. Phys. B 417 (1994) 181. [ 161 U. Ellwanger, Phys. Lett. B 335 (1994) 364. [ 171 M. Bonini, M. D’Attanasio and G. Marchesini, Nucl. Phys. B 418 (1994) 81; Phys. Lett. B 329 (1994) 249. ] 181 G. ‘t Hooft and M. Veltman, Nucl. Phys. B 44 (1972) 189; K. Wilson, Phys. Rev. D 7 (1973) 2911; G. Bollini and J. Giambiagi, Nuovo Cimento 12 (1972) 20; Acta Phys. Austr. 38 ( 1973) 211; L. Ashmore, Nuovo Cimento Lett. 4 (1972) 289; E.M. Cicuta and E. Montaldi, Nuovo Cimento Lett. 4 (1972) 392; P Butera, E.M. Cicuta and E. Montaldi, Nuovo Cimento 19 (1974) 513; D.A. Akiempong and R. Delburgo, Nuovo Cimento 17 (1973) 578; 18 (1973) 94; 19 (1973) 219; T. Marinucci and M. Tonin, Nuovo Cimento 31 (1976) 381; G. ‘t Hooft, Nucl. Phys. B 61 (1973) 455; E. Speer, J. Math. Phys. 15 ( 1974) 1; l? Breitenlohner and D. Maison, Commun. Math. Phys. 52 (1977) 11, 39, 55. ] 191 C. Becchi, A. Rouet and R. Stora, Commun. Math. Phys. 42 (1975) 127; Ann. Phys. (NY) 98 (1976) 287.
NUCLEAR PHYSICS
B
Nuclear PhysicsB 437 (1995) 163-186
ELSEVIER
BRS symmetry for Yang-Mills theory with exact renormalization group * M. Boninia,
M. D’Attanasio
a. G. Marchesini
b
a Dipartimento di Fisica, Universitci di Parma and INFN, Gruppo Collegato di Parma, Parma, Italy b Dipartimento di Fisica, Universith di Milan0 and INFN? Sezione di Milano, Milan, Italy
Received19 October1994;accepted5 December1994
Abstract
In the exact renormalization-group (RG) flow in the infrared cutoff A one needs boundary conditions. In a previous paper on SU( 2) Yang-Mills theory we proposed to use the nine physical relevant couplings of the effective action as boundary conditions at the physical point A = 0 (these couplings are defined at some non-vanishing subtraction point p # 0). In this paper we show perturbatively that it is possible to appropriately fix these couplings in such a way that the full set of Slavnov-Taylor (ST) identities are satisfied. Three couplings are given by the vector and ghost wave-function normalization and the three-vector coupling at the subtraction point; three of the remaining six are vanishing (e.g. the vector mass) and the others are expressed by irrelevant vertices evaluated at the subtraction point. We follow the method used by Becchi to prove ST identities in the RG framework. There the boundary conditions are given at a non-physical point A = A’ # 0, so that one avoids the need of a non-vanishing subtraction point.
1. Introduction
Renormalization-group (RG) formulation [ 11 provides the most physical way to deal with the ultraviolet (UV) singularities. Following a suggestion of Polchinski [ 21 (see also Refs. [ 3-51) the exact RG formulation has been used recently to give self-contained and simple perturbative rederivations of many properties such as renormalizability [ 2,581, infrared finiteness of massless theories [6,9], operator product expansion [ 101, decoupling theorem [ 111. Some approximations have also been attempted within this context [ 121. * Researchsupportedin part by MURST, Elsevier Science B.V. SSDIO550-3213(94)00569-9
Italy.
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Its application has been recently extended to gauge and chiral gauge theories [ 5,13171. The challenging problem here is the general fact that local gauge symmetry typically conflicts with the presence of a momentum cutoff. For the SU(2) Yang-Mills theory it is shown in Ref. [S] and in the present paper that, in spite of the explicit breaking of gauge symmetry, the Slavnov-Taylor (ST) identities can be implemented perturbatively by appropriately fixing the boundary conditions on the RG flow. Thus the RG formulation provides an alternative to dimensional regularization [ 181 of gauge theories. To describe the situation we briefly recall the exact RG formulation for gauge theories. Consider the wilsonian effective action &E[ 4, y; A] with $ and y the fields and the BRS sources and A an infrared (IR) cutoff. This functional is obtained by path integration over the fields with frequencies above n and up to an UV cutoff Ao. At A = 0 and A0 + 0;) one has performed the full path integral and therefore from the functional &ff[ +, y; A = 0] we can obtain the physical Green functions or the oneparticle irreducible vertices. Actually the vertices of Se~[ 4, y; A = 0] are the amputated connected Green functions. We never write in the functionals the UV cutoff A0 since we always understand Ao -+ 00, which is possible due to perturbative renormalizability. The functional Sen[ #J,y; A] satisfies an evolution equation in the IR cutoff, which is obtained by observing that the path integral over all frequencies can be done first by integrating over the frequencies above A and then below A. The fact that the result does not depend on A gives the mentioned evolution equation. The exact RG equation is non-perturbative but it can be solved perturbatively once the boundary conditions are given at some value of A. An important point concerning the boundary conditions is the distinction between relevant and irrelevant parts of the functional S,E[ 4, y; A] (and of any other functional such as the physical effective action r[ 4, y] ). As usual the vertices with negative mass dimension are called “irrelevant” and contribute to Seff,im[4, y; A]. The remaining part &a,,[ 4, y; A] contains only a finite number of parameters with non-negative dimension (relevant parameters) which, for A # 0, can be defined for instance as the first coefficients of the Taylor expansion of Sen[ 4, y; A] around vanishing momenta. The UV value A = A0 is the obvious value where to fix the boundary conditions for Seff,im[ 4, y; A]. For & 4 00 one expects that all vertices with negative dimension vanish thus one sets Seff,irr[ 4. y; A = Ao] + 0 for & + 00. Finally, the key problem is to fix the boundary conditions for the finite number of relevant parameters in Serrret[ 4, y; A]. Here is where one implements in the theory both the physical parameters (such as masses, couplings, wave-function normalizations) and the symmetry, i.e. ST identities. In Ref. [ 131 we considered the SU(2) Yang-Mills theory which has nine relevant parameters. We proposed to fix these nine parameters at the physical point A = 0. Since the theory involves zero-mass fields, at A = 0 one should define these parameters as the values of some vertices at non-vanishing subtraction points. The reason for selecting A = 0 is that &[ 4, y; A = 0] is related via Legendre transform to the effective action r[ 4, y]. Therefore, some of these parameters are fixed at the physical values of the couplings, masses and wave-function normalizations. We proposed to obtain the remaining parameters by imposing some of the ST identities and showed that this procedure can be implemented perturbatively. Therefore, by construction some of the
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165
ST identities are satisfied, thus the fundamental question is whether the effective action r[ 4, y] computed by this procedure does indeed satisfy the full set of ST identities. In the paper of Ref. [ 131 we were able to prove this only partially at one-loop level. Recently, Becchi [S] considered the exact RG formulation of the SU(2) YangMills theory and imposed the boundary conditions for the relevant parameters in Sefrrel[ 4, y; A] at a non-physical point A = A’ # 0 so that the relevant parameters can be defined by expanding the vertices around vanishing momenta. In this way the relevant parameters are not directly related to physical couplings in the effective action ~[c#J, y] but the analysis of relevant parts of ST identities becomes easy. By using algebraic methods he was able to prove that the full set of ST identities can indeed be satisfied perturbatively. He identified a functional L&[ 4, y; A] which gives the “defect” to ST identities and then showed that one can obtain perturbatively the nine relevant parameters in &r~~i [ #J,y; A] by solving the fine-tuning equation A efi,rel[A~;Al
=0
for the relevant part of the ST defect. In the present paper we apply the same method to the case in which the boundary conditions are given at the physical point A = 0. In this case the operation of extracting the relevant parameters become rather complex since one has to use non-vanishing subtraction points. We are able to generalize Becchi’s proof to this case and we explicitly prove that the procedure suggested in our previous paper gives indeed an effective action which satisfies perturbatively ST identities. Moreover, we explicitly express the solution of the fine-tuning equation by giving in terms of physical vertices those relevant parameters which are not fixed by the physical couplings and wave function normalizations. This paper is organized as follows. In Section 2 we recall the RG formulation for the SU( 2) Yang-Mills theory given in Ref. [ 131. We introduce the wilsonian effective action &r[+. y; A], define its relevant and irrelevant parts, discuss the role of the boundary conditions and the local symmetry. In Section 3 we study the operator which gives the violation to the ST identities and show that the problem of implementing the symmetry can be reduced to the solution of a finite number of equations, the$rre-tuning equations. Section 4 contains the explicit solution of these fine-tuning equations and in Section 5 we rephrase this solution in the usual algebraic language. Section 6 contains some conclusions.
2. Renormalization-group
flow for W(2) Yang-Mills
The fields and corresponding c/5= {A;,
ca, Z’} ,
j
sources for SU(2) =
theory
Yang-Mills
theory are
, y = {u;, u”} 7 > where ut and v“ are the sources associated to the BRS variation of Al and P, respectively. The generating functional, in the Feynman gauge, is
ZLLrl
=exp(W.Lyl)
166
M. Bonini et (11./Nuclear =
J
D4exp
Physics B 437 (1995) I63-186
i{-$(4P2+)ono
+
(j4>ono
+ Znt[+7Y;
nol} 7
(1)
where the path integral is regularized by assuming a UV cutoff ,40. In general we introduce the cutoff scalar products between fields and sources
- d4p J=J (277)4 ’
P
' ] c"(p) w>,,o-J K,-,o(~) 1 {j;(-p)A;(p)+ [X'"(-p)- ~p,u;(-p) P
+~a(-P)xu(P) I3 where Kin, is the cutoff function which is one for A2 < p* < At and rapidly vanishes outside. The functional Sin, is the UV action involving monomials in the fields, BRS sources and their derivatives which have dimension not larger than four and are Lorentz and SU(2) scalars. There are nine of these independent monomials n
&nt[47
Y; llol
=
d4x
{;A,.
[gpv<
a; + u;J2)
+ u3”c7&] A,
1
+afwr t&A,
+CT,/ + us” (&,A,)
. A,, A A,
A A,) . (A, A A”) + d%+,
. A,) (Afi. 4,)
+a~w,~c~A,+a;u~c~c}, where WE = u: + ga,P and we have introduced the usual scalar and external SU(2) product. The nine couplings cr: depend on A0 and have non-negative dimension (relevant parameters). In order to obtain the physical theory one has to show that the values of the a? can be fixed in such a way that: (i) The no + co limit can be taken by fixing the physical parameters such as the masses, coupling g and wave-function normalization constant at a subtraction point ,x Perturbative renormalizability ensures that this can be done [ 2,5,13]. (ii) In the no t cc limit the Slavnov-Taylor identities must be satisfied. This is the crucial point to be discussed in this paper. According to Wilson one integrates over the fields with frequencies A* < p2 < Ai and obtains (see Appendix A)
exp (WLYI)
J +&dAy; Al},
=Nj,y;Al
v$ expi{--~(4p24)on
+ (jd>o, (2)
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where N[j, y; A] contributes to the quadratic part of W[j, r] logN[j,y;A]
1 KL4o (P> PzKon 0(P)KOA(P)
=-i s
P
X
1
ijE(--p)
j;(p)
[
f”(-p)
- ip&-P)
I
X”(P)
>
The functional See is the wilsonian effective action and is obtained by integrating fields 4’ with the higher modes (p2 > A*)
exp (~{~(~~*~)hb =
s
(3) the
+ &ff[4,r;~lI)
Dq5’ exp (i{--+(~$‘p~&)
AA” + (j’4’>hio
+ hx[4’~~;AOl})
(4)
T
where the source is j’(p)
= P*{ A;(P),
-C’(P)~
(5)
C”(P) 1.
By comparing with the physical generating functional Wj’,x~l
= ~(&*4~).4~~
+ &ff[~,r;~l
as a functional of j’, generates propagators have frequencies in external propagators is cancelled for all values of p. At n + 0 generating functional W[j’,yl
=
W[j’,y;A=Ol
W[j, r] in ( 1) we see that (6)
,
the connected Green functions the range A2 < p* < 4. The by the inverse cutoff so that the and ila --f co, this functional
=
~(~~P'$)oA~
+&ff[&r;A
in which the internal cutoff function in the source j’ can be taken becomes the physical
=Ol.
(7)
Given the relation in (5) between j’ and 4 we have that the vertices of Seg[ 4, y; n = 0] are the amputated connected Green functions. Taking into account that the variable n enters as a cutoff in the internal propagators of connected Green functions, one derives the exact RG equation [2] $
.I
AJAKoA(P>
exp (-&f/fi)
P 1
6
%A;(-&
6
SAC(p)
6
+ Sca( -p)
6
~ l%(p)
exp (iS,&ti)
.
(8)
This equation is non-perturbative but it can be solved perturbatively once the boundary conditions are given at some A. We have included fi in (8) to show explicitly how the loop expansion is generated. In the r.h.s. there are two contributions. One is quadratic in the action and has a coefficient with the same power of fi as the 1.h.s thus involving Green functions with lower or equal loop order. The second, linear in the action, has a coefficient with an additional factor ti and therefore involves Green functions with a
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Physics B 437 (1995) 163-186
lower loop. Thus, starting at zero loop with S$*)
given by the value of Green functions
at the boundary conditions, Eq. (8) allows one to obtain $2 at all loops. The fact that one of the contributions in the r.h.s. in (8) is at the same loop order as the 1.h.s. makes the generation of the loop expansion less transparent and less efficient. Actually, as shown in Refs. [ 6,7], the loop expansion is more easily generated if one integrates the RG flow not for the wilsonian action S,, but for a cutoff effective action r[ 4, y; A] given by the usual Legendre transform of the cutoff functional W [ j, y; A] introduced in (6)
n4,
y; Al = w1.L y; Al - WI07 0; Al -
.I
{ji(-p>A;l,(p)
+ Xa(-p)ca(p)
+ PC-P>x”(P>}
P
This functional of the “classical” fields 4 generates cutoff vertex functions in which the internal propagators have frequencies in the range A2 < p2 < Ao. From (7) we have indeed that at A = 0
UAYI
= Uhy;A
=Ol
(9)
is the physical effective action. One can convert (8) into the RG equation for r[ 4, y; A] which has the following form:
AJ/ifl4
Y; Al = hIF
[d%
Y;
Al ,
(10)
where the functional Zr[ 4,~; A] is given in Ref. [ 131 and depends non-linearly on the vertices of r[ c$,y; A]. Then in the r.h.s. of ( 10) there are vertices at a loop order lower than the l.h.s., so that by solving iteratively this equation one automatically generates the loop expansion. We now discuss the crucial point of the boundary conditions which provide the starting point for the loop expansion. 2.1. Boundary conditions: physical parameters and symmetry There is an obvious value where the boundary conditions should be given which is the UV value A = A,. Here one finds from (4)
thus Sen becomes local and depends only on the nine relevant couplings C: with nonnegative dimension. Since & is the only surviving mass parameter as A0 -+ CC one expects that, for A0 4 co, all vertices with negative dimension vanish. For this reason vertices with negative dimension are called “irrelevant” and their contributions to the wilsonian action will be denoted by Sen,im[#J,y; A]. The remaining part Serrret[ 4, y; Pi (A) ] is a local functional which contains nine relevant parameters c+i(A) with non-negative dimension. The form of Serrret[ 4, y; (+i (A) ] is the same as Si,, in which the UV couplings CT: are replaced by ai( A). Also for
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Physics B 437 (1995) 163-186
169
these parameters we need to fix the boundary conditions at some A = A’. The precise definition of gi( A) in terms of S,ff depends crucially on whether one assumes boundary conditions at A = A’ # 0 (e.g. A’ = Aa) or at the physical point A’ = 0. In the first case the definition of ai is simple since one does not need to impose a subtraction point and can consider the Taylor expansion of the Green functions around vanishing momenta. Therefore, one defines
Y; a.(A) 1 = T4’0’SefiM1,y’ Al 1 seff,rel[+11
A # 0,
where Tie’ is the Taylor expansion around vanishing momenta truncated to the terms with coefficients with non-negative dimension (see for instance Ref. [ 51). Taking advantage of the fact that &[ 4, y; A = 0] and r[ 4, y; A = 0] are physical functionals (see (7) and (9)), we have suggested in Ref. [ 131 to fix the relevant parameters ai at the physical point A = 0 so that Ci( A = 0) are physical couplings, such as for instance the mass, the wave-function normalizations and the coupling constant g at a subtraction point p. At A = 0 the definition of the functional &rrei [ #, y; o-i (A = O)] requires the introduction of a finite subtraction point p. We then define Seff,rel[4JTri(A)]
=Z”/“‘$E[~,Y;AI
3
where the operator TicL) is given in Appendix B for a scalar theory. The extension to the YM theory is simple in principle and Tip) r[ 4, y; A] is defined completely in Appendix B . For the boundary conditions for the RG equation (10) of the cutoff effective action r[ 4, y; A] we follow the same procedure: (i) We assume that the irrelevant parts of r[ 4, y; A = Ao] vanish rim[@,Y;A=Ao] (ii)
= [l -T:“]
r[~$,y;A=Ao]
~0.
We fix the remaining nine relevant parameters at the physical point A = 0. It is useful to rearrange the various monomials and use the following parametrization for the relevant part of the physical effective action r[ 4, r] = r[ 4, y; A = 0]
+dz3 - 1)(&A,) . A, A A, - &J,) +~reh$,r;Pil where Fyv = d,A, - &A,
Ll[4J;
Pi1 =
(11)
7
+ gA, A A,, D,c = a,c + gA, A c and
d4X { +,A,
J +&apAJ +&do.
. (&A,)
. A,
. (&A,)
+ ~3
c A c + $g2ps(A,
wfi +c A A, A A,) . (A, A A,)
M. Bonini et al. /Nucleur
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Physics B 437 (1995) 163-186
(12) In (11) we have singled out two BRS-invariant monomials, the gauge-fixing contribution and an additional three-vector coupling monomial. We now discuss the values zi and pi that one has to fix in order to have the physical couplings and the local SU(2) gauge symmetry. Some of these couplings can be fixed to give the wave-function normalizations and the three-vector coupling g at a subtraction point ,LL This can be implemented by assuming zi = 22 = ~3 = 1 so that the first part of rret in ( 11) is just the BRS action in the Feynman gauge &as [ 4, r] and we have ~,l[$,y1
= SBRS
+
f&,[#‘~~~~il
.
(13)
Notice that the choice of boundary conditions at n = 0 for the relevant couplings provides a conspicuous advantage with respect to the case in which one takes A = A’ # 0. We have in (13) that the relevant part of the effective action already contains the BRSinvariant classical action which is the starting point of perturbation theory. We have also explicitly written the h-dependence which shows that the other six couplings pi vanish at zero loop. In Ref. [ 131 we proposed to fix the pi’s from the ST identities for the physical effective action r[ 4, r] given by A,-[+,y]
z &-~r’[dtY]
=o,
(14)
where
J SF 6 SF 6 ~ ___ PA;+aAa,(p) a~;(-P) I’I{a~;(-P) SF 6 6s s
T’[&Yl
= nAYI
+ ;
d4Jr ( c&AJ2
and Sr/ is the usual Slavnov operator [ 191
sp =
+suQ(-p) &T”(p) + St”(p) Su”(-p) I . The six couplings pi are involved in all vertices of the functional Al,[ 4, -y] . In Ref. [ 131 we obtained the various pi from the simplest ST identities and found that these couplings are given by some irrelevant part of vertices evaluated at the subtraction point p. We showed that this procedure can be implemented perturbatively. This is due to the fact that, as indicated in ( 13), the couplings pi vanish at zero loop. At higher loops some of the couplings pi are different from zero. The mass of the vector which is given by ~1, as expected, remains zero at all loops. In that paper we were not able to show that this procedure ensures that the ST identities are actually satisfied for all vertices, In the next section we will show that this is the case.
M. Bonini et al. /Nuclear
3. Gauge
symmetry
and fine-tuning
Physics B 437 (1995) 163-186
equation
171
at A = 0
The proof that gauge symmetry can be implemented perturbatively in the RG formulation has been given by Becchi in Ref. [ 51 in the case in which the boundary conditions for the nine parameters a,(n) are given at some non-physical point A = A’ # 0. These parameters are defined by applying the operator Tie’ to the wilsonian effective action. The perturbative analysis of the ST identities is simplified by the fact that T,” commutes with the ST functional derivative at zero loop. Here we follow the same procedure but in the case in which one fixes the boundary conditions at the physical point n = 0. The price for working directly at A = 0 is that the relevant couplings are defined by the operator Tq(y) which does not commute with the zero-loop ST operator. As a result the analysis of the ST identities seems more complex. We are able to show that actually the analysis is easy and one can write a simple fine-tuning equation which allows one to determine the six relevant couplings p,. Following Becchi we consider the generalized BRS transformation
with v a Grassmann parameter. Applying this change of variables to the functional integral (2)) one deduces the following identity:
J
n+exp
SZLLYI =N.Ly;~l x44>y;Al,
+
(i{-~(+P*$)oA
(.@)oA
+&T[~,Y;AI})
where
is the usual ST operator obtained from the variation of the source term (j~$)a,. Notice that the cutoff functions cancel. The functional A arises from the variation of the other terms in the exponent and from the jacobian. It can be divided into two parts, a linear and a quadratic one in the derivatives A=Al
+A*,
Al =i
J{
(16)
with P’A;(P)
I’
&
+ g/YP) CL
~ aA;
6
M. Bonini et al./Nuclear
172
Physics B 437 (1995) 163-186
and A2 =
6 s &n(P) exp (-is&) & Sua Fep, + Sea(p) Su”(-P) sP CL F x exp (iS,ff) .
Since Z&(P) vanishes for A = 0 we have that at the physical point A[ r$, y; A = 01 = dl[+,y;n=ol. We have to prove that one can select the six relevant couplings pi in such a way that
N~,Y;AI
=O,
(17)
so that the ST identities
szLLr1
=o
are satisfied. Actually we will show that the functional equation (17) can be reduced to a finite number of relations, the$ne-tuning equations. The RG flow for the functional A is given by the following linear evolution equation: AcYnA[q5,y;A]
=M.A[$,y;A]
E M[A;A],
(18)
where the linear operator M depends on Se@
The RG equation ( 18) requires boundary conditions. As before we discuss the boundary conditions for the relevant part of A[ 4, y; A] at A = 0 and for the irrelevant part at the UV point A = &. The definition of the relevant part of A[ 4, y; A] at A = 0 requires a subtraction point. This functional has dimension one and we have
&1[4,y;&l
=~;C”‘AM,y;A=Ol,
(19)
where TiP’ is again defined as in Appendix B. There are 11 relevant parameters 6i (see Appendix C) which are the coefficients of the 11 monomials in the fields, sources and momenta of dimension not greater than five and with ghost number one. Recall that at A = 0 the expression of A simplifies since the non-linear part A2 vanishes due to the cutoff function.
M. Bonini et al. /Nuclear
We can easily prove that the irrelevant part of the functional provided that the UV cutoff is sent to infinity
4,,[4,y;A
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Physics B 437 (1995) 163-186
A vanishes at A = A0
for A0 + 00.
= Aol + 0,
(20)
First observe that Al [A = A,] contains only relevant parts. This is due to the fact that at A = A0 we have S,ff[A = Ao] = 5’t”t which is local. For 42[A= Aa] we have that the functional in the integrand is local apart from the p-dependence of the cutoff function Ken,(p) which gives an irrelevant contribution to A2 [ A = Ao] . However, for A0 + 03 we have Kc,(p) = 1 so that A2[ A = A0 4 001 becomes local. Using the boundary condition (19) and (20) we can integrate the RG equation for A and obtain A
A[#,Y;AI
= &l[4,y;&l
+
J
+re,M;
11-
J
O” dA
TM,,[4
11.
(21)
A
0
We now prove the following two perturbative theorems for the functional Ace) [ q5,y; A] at loop f? and for the corresponding vertices AL’) ( . . . ; A) where n denotes the number of fields or sources and the dots denote momenta, internal and Lorentz indices. Theorem 1: if A(“‘) [ 4, y; A] = 0 for all loop C’ < k’, and if Ai? (. . . ; A) = 0 for n’ < n, then Ace) n ( . . ..A)=A$.( Proof
.. :A=O). ,
From the hypothesis the functional
Mce)[A.A],
= -(27r)’
M[ A; A] at loop e is given by
JLA&Kon(~) P2
+ &a(p)
~A’~)bf~,y; Al , where S$) is the tree wilsonian
&a(-p)
- c ++ F
(22)
action in (6). Since the starting point of the perturbative
expansion (see ( 13) ) is I-(‘) [ 4, y; A] = Snns [ 4, ~1, the vertices of S$’ [ 4, y; A] are the amputated connected Green functions at tree level. Moreover, S$ [ 4, y; A] does not contain contributions quadratic in the fields, they are all included in i ($p*~$), see (6). Consider now the RG equations (21) for a vertex Aif) (. . . ; A). From (22) we have that MC”) [A; A] involves only vertices Ai? ( . . . ; I) with n’ < n. They vanish according to the hypothesis thus only the first term in r.h.s. of (21) is left and the theorem is proved. An obvious consequence of this is the second theorem. Theorem 2: if Ace’) [ 4, y; A] = 0 for all loop .!? < ! and if at loop e the relevant part of this functional vanishes at A = 0, namely if
&,[~Y$~)I
=O,
(23)
174
M. Bonini et al./Nuclear
Physics B 437 (1995) 163-186
then A(‘) [ 4, y; A] = 0. The line-tuning condition (23) will be used to fix the relevant couplings pi of the effective action r[ 4, y] . However, the functional A[ 4, y; A] defined in (16) involves vertices of the wilsonian action Serf[ $, y; A]. It is convenient to express the fine-tuning condition at A = 0 in terms of a functional which involves directly the vertices of r[ 4, y] This can be simply done by taking the Legendre transform of A[ $, y; A] at n = 0. In this way the tine-tuning equation (23) is equivalent to
where Ar,,,l has the same decomposition as A,1 given in Appendix C. Then both Theorems 1 and 2 can be phrased for the functional Ar [ c+h, y] . From now on we consider A = 0 and we show perturbatively the equivalence of (23) and (24). By using the inductive assumption A (“1 = 0 for C’ < e, one has the following results at loop e: (i) For the two-field vertex one has Aj!)(Ac) = A(“)(*@, then the fine-tuning condition A(e)(AC1 = 0 is equivalent to A~?~~lAc)= 0. I mposing this condition the full vertex rel A vanishes. 1. (ii) For vertices with three fields one has A$f)(AAc) = A(e)(AAc) and Af-e)bcc) = A(“)(wcc), since the one-particle reducible terms vanish due to (1). Thus the finetuning conditions A$)‘AAc’ = 0 and A~$e:)(wcc) = 0 are equivalent to A~~~,AAC)= 0 and A~~~lwcc)= 0, respectively. With these conditions these two vertices vanish. (iii)
Similarly for the vertices with four fields one has Ay)(3Ac) = A(‘)(3Ac) and A~)(WACC)= A(e)(wAcc) and the remaining fine-tuning conditions become A~~~13Ac)= 0 and ~(l)(w*Cc) _- 0 and the two functions Ay)(3Ac) and A~)(WACC)vanish. f,rel (iv) By increasing the number of fields one has that the two functionals A(!) and A)? are equal. In the next section we rove by induction on the number of loops that it is possible to fix the six couplings pi ! ) in such a way that si(‘I = 0 so that the fine-tuning equation is solved and from Theorems 1 and 2 the ST identities’hold perturbatively.
4. Perturbative
solution
of the fine-tuning
equation
The starting point of the proof by induction over the number of loops is L!~‘O) [ 4, y] = 0 which is valid since r(‘=O) = &as. We suppose that the fine-tuning equation is solved at e’ < e SO that Al?” [ 4, y] = 0. We want to show that it is possible to fix the six couplings pIc) in such a way that Af-e’[ 4, y] = 0. Notice that there are six parameters pj’) and 11 equations 6:“’ = 0. Therefore, the solvability of these equations requires that there must be five relations connecting the various Sj”‘. They are provided by the so-called consistency conditions, a set of equations which must be identically satisfied by the functional A,- due to its definition (14). From the anticommuting character of S~J we have S~IA,. = S$r’ = 0, where Sr., is defined in (15). By using again the
M. Bonini et al./Nuclear
Physics B 437 (1995) 163-186
inductive hypothesis A,(I” - 0 for !’ < e, the consistency condition loop 4 one finds
175
simplifies
S r; A(“) r = 0,
and at
(25)
where Sr; is defined in ( 15) with r given by Snns. In order to obtain from this equation the desired relations connecting the various S,“‘, we have to extract its relevant part by applying the operator TiFL). Notice that this operator mixes relevant and irrelevant parts of Ar. This in general makes some of the consistency conditions complicated due to the presence of irrelevant contributions. We analyse in detail the fine-tuning equations and consistency conditions for the various vertices of Ar. We start form the vertices with two and three fields and sources. We then analyse the remaining vertices. In the following all vertices will be considered at loop !. (i) The vertices with two and three fields are obtained from the definition of Ar in (14) AjlA,“(p) A?$?(~,q,k)
=T’;*)(p)I-y)(p) =I-y)(k)rz)(p,q,k) +l+p
Ajlw,‘(p,q,
,
(sY~~*)(qJw4
k) =r~)(p)Z+=)(p,q, +T;“C)(k)TEA)(p,q,k).
(26) - f’~?p)~$‘%,
kq)
1 k) + T;WC)(q)T~f(p,
(27) k,q) (28)
The relevant parameters of these vertices are the five 6i with i = I, 2,3,4,5 (see Appendix C) while the relevant couplings of the r vertices are the four pi with i = 1,2,3,4 (see ( 11) and Appendix B). Then we need one relation among these 6i. Indeed if we consider the consistency condition (25) for the vertex of the ACC fields and take its relevant part, we obtain ig& = 83 + & .
(29)
It is a nice fact that this relation involves only relevant parameters. This is not a general fact since the operator TiPL) applied to SrAr typically extracts irrelevant vertices of Ar evaluated at the subtraction point. We are now able to satisfy the fine-tuning equation for the vertices in (26)(28). From (26) we obtain
and we have 61 = 82 = 0 by fixing p1 =p2=0. From (27) we obtain
(30)
M. Bonini el al. /Nuclear
176
Physics B 437 (I 995) 163-186
and we have 6s = Sd = 0 by fixing (31)
p3=0.
Finally, from (28) we obtain -$p&vi$Y+“(p, P
q, k)
I
3SP >
,
and we have Ss = 0 by fixing +p,k.F$Y’)(p, P
q, k)
1
.
(32)
3SP
From Theorem 2 of the previous section we have that, since the relevant parts of the vertices in (26)-( 28) vanish, the complete vertices vanish. Moreover, from Theorem 1 of previous section the four-field vertices of A,- could be only relevant. (ii) We COLZ now to analyse the vertices of A,- involving four fields. Consider first the vertex Ay/)abCd (p, q, k, h) , which contains the relevant parameter Se (see Appendix C). From the definition of A=[ 4, r] in ( 14) one easily sees that this vertex involves only couplings pi already fixed thus one should be able to show that the fine-tuning condition Se = 0 is satisfied automatically. Indeed there is a consistency condition which, after having imposed (30)) (3 1) and (32)) reduces to S6 = 0. This is due to Theorem 1 which implies that irrelevant parts of the vertices of Ar with four fields vanish. The last vertex of A,- we have to consider is
= r~wc)(h)r~~~b”bcd(p,q,k,h)
+ l-‘:A,A1(p)l-~;AA)adbc(p,h,q,k)
+~‘;~)(q)~~~AA)bduc(q,h,p,k) +~eda~ebc~~~A)(q+
k,h,p)I$$‘(p
~~~~~~~~~~~~~~~ + k, h,q)T;$(q +eedceebar;~*)
(q + p, h, k) Z$;’
+l+~‘(k)l-~v+‘++d”b(k,h,p,q) + h,q, k) + h,p, k) (k + h, q, p) .
(33)
This vertex contains the five relevant parameters Si with i = 7,. . . , 11 while the rvertices involved here contain only the two relevant couplings not yet fixed, ps and p6. The consistency condition (25) gives the three relations &=S9=S,,,
S8 = SlfJ.
(34)
Notice that these relations involve only relevant parameters. As stated before this is a consequence of the fact that we have already set to zero the vertices of Ar with two and three fields and, according to Theorem 1, the vertices with four fields do not have any irrelevant parts. From (34) we have that it is possible to have Si = 0, i = 7,. . . , 11, simply setting to zero two independent Si (e.g. ST and Ss) by fixing p5 and p6. The easiest way
M. Bonini et al./Nuclear
Physics B 437 (1995) 163-186
II?
of obtaining & and 8s is to contract in two different ways the vertex in Eq. (33) by external momenta and summing over internal indices. Since this vertex does not have any irrelevant part, these contractions are proportional to two different linear combinations of 87 and 8s. Using for simplicity the symmetric point we find PPqvkP &Ac)aabb(p, WfP P4 = --$(Zi7
q, k, h) 4SP
+ S,) P&‘kPhfl~C4A)onhb(g, PF P4
q, k, h)
1
4SP
and A(3Ac)aabb
PIJquhP
@UP
P4 = 4(2&
4SP
- 76s)
ig(y%s +‘F(h,
(~7 4, k h)
i p,qvhphu p4 g
- $P6) -
(P, q, k, h)
+ ~kp)T~~AA)aabb(k,h,p,q)
+12- p~qvhp~~wcA)(q p4 UP The fine-tuning
$,,;?bb
1
$ k, h,p)I$;;‘(p
+ h,q, k) 4sp.
equations 87 = 0 and 8s = 0 give k, h)
1
(35)
4SP
and ’ 3i i pcLqvhph, 1 r$;b”abb(p, p4 L’S= gP6 - 16g ; 1 +‘F(hp,
q, k h)
ikp)l$;AA)aabb(k,h,p,q)
+12PPq”hPr(wcA) r,F P4
(q+k,h,p)r~~~)(p+h,q,k)
1
.
(36)
4SP
This completes the proof that we can set to zero the 11 parameters 6i by fixing, at every loop, the six couplings pi according to (30), (31), (32), (36) and (35).
’ In Ref. [ 131 some contributions
to (36) were missing.
178
M. Bonini et al./Nuclear
5. Algebraic
Physics B 437 (1995) 163-186
formulation
We want to connect the perturbative solution we have discussed with the algebraic formulation given in Ref. [ 51. This is based on the fact that if AI;~~I[ $, y; Si] can be written in the following form:
&,,,I [ 4, Y; &I = Spm rre~[ $7 Y; Pi 1 ,
(37)
then it is possible to find a perturbative solution of the fine-tuning equation by appropriately fixing the pi. This property is discussed in Ref. [5] and recalled in Appendix D. Eq. (37) is valid provided the 11 parameters in Ar,,el fulfill the following five relations: ig82
(38)
= S3 + 64 ,
8~=-ig&,
&=S9=S*,,
S8 =s10.
(39)
In our case, in which we have a non-vanishing subtraction point p # 0, the first relation (38) holds (see (29) ) , while the relations (39) are not valid in general since there are contributions from irrelevant parts. Still also for ,u + 0 we can use the above theorem as follows. First observe that the relation (38) holds also for p # 0 and involves only vertices with two and three fields, i.e. AL*‘), Ay) and A$..“) . Therefore, the form (37) is valid for these vertices and from Appendix D we deduce that we can solve the fine-tuning equations Si = 0 for the five relevant parameters with i = 1,. . . ,5. This is obtained by fixing the four couplings pt, ~2, p3 and ~4, which enter in the vertices of r with two and three fields. From the previous section the solution is pr = p2 = p3 = 0 and p4 given in (32). After solving these five fine-tuning conditions, we have from Theorem 2 that the complete vertices ApI, AjfiAc) and Ay) vanish and from Theorem 1 that the vertices with four fields are only relevant. Therefore, the relations in (39) are now valid and all vertices of Allrel [ 4, y; Si] can be expressed as in (37). From Appendix D we can perturbatively solve the remaining tine-tuning equations and from previous section the solution is ps in (36) and p6 in (35).
6. Conclusions
In this paper we have fixed the renormalization conditions for the SU(2) Yang-Mills theory, i.e. the nine couplings (zi and pi, see ( 11) ), which enter in the relevant part of the effective action r[ 4, r] as boundary conditions at n = 0 for the exact RG flow. Three of these couplings are fixed to give, at the subtraction point p # 0, the vector and ghost wave functions and the three-vector coupling (~1 = ~2 = ~3 = 1). In this way one fixes a contribution of the effective action to be the BRS classical action (for instance in the Feynman gauge, see (13)). The other six couplings pi in (12) are absent at tree level. They are at our disposal in order to implement the gauge symmetry for the physical effective action, i.e. the ST identities Ar [ 4, r] = 0 in ( 14). The fact that the three couplings zi = 1 are not affected by loop corrections is one of the advantages of working at the physical point n = 0 rather than at A = A’ # 0. (We shall recall
M. Bonini
et ul./Nuclear
Physics B 437 (1995) 163-186
179
later the difficulties arising from the non-vanishing subtraction point.) We have shown perturbatively the following two results. The first result (see Section 3) is that it is possible to satisfy the ST identities A,.[ 4,r] = 0 if one is able to solve the fine-tuning equations A,-,,,,[ qb,y; Si] = 0. It is a consequence of the RG equation (18) (see Theorems 1 and 2) and of the fact that the starting point of the loop expansion is the classical BRS action which satisfies the ST identities Ajf=“’ = 0. The second result (see Sections 4 and 5) is that it is possible to solve (perturbatively) the 11 fine-tuning equations 6i = 0 by fixing the six pi couplings. This is possible since, due to the consistency condition (25)) there are only six independent 6i. Moreover, we have constructed the solution and found that the relevant part of the effective action is
rrel[4t’Y;fil
=SBRS
+
d4x { $4~
fi
. c A c +
$&&4p
A A,)
. (A,
A A,)
J
+$g2p6(A,-A,)(Ap~AAv)}.
The only non-vanishing couplings ~4, ps and p,j are given in (32), (36) and (35), respectively, in terms of irrelevant vertices of r evaluated at the subtraction point. This form allows one to deduce the perturbative expansion since irrelevant vertices at loop ! involve relevant couplings at lower loops P < !. We have followed the method used by Becchi [5] in which the boundary conditions are taken at the non-physical point n = A’ # 0. In that case one can define the relevant parameters by expanding around vanishing momenta thus avoiding irrelevant contributions in the consistency conditions. Since we work at A = 0 we have needed non-vanishing subtraction points and introduced the operator Tk(” which defines the relevant parts of the various functionals. This makes the analysis of the ST identities more difficult since the operator Tk(‘) mixes relevant and irrelevant parts when applied to the product of two functionals. As discussed in Section 5 the fine-tuning equation Ar [ 4, y; Si] = 0 can be perturbatively solved provided that the local functional A,tret can be parametrized as in (37). This is valid if the relations (38) and (39) hold. Actually the relations (39) are not valid in general if ,u + 0 due to the presence of the irrelevant contributions generated by the TCLL’operator. However, we have shown that one can use the parametrization in (37) by p;oceeding in two steps. First one uses the fact that Eq. (38) is valid also for p f: 0. This allows one to solve the fine-tuning equations for the vertices of A,. with two and three fields. Once these equations are solved the remaining relations (39) hold since the irrelevant contributions vanish. In this way one can use (37) and solve the remaining fine-tuning equations. The method is general. As shown in Ref. [ 171 it can be applied for instance to SU(2) gauge theory with fermions. The application to the case of chiral gauge theory without anomalies should be also possible along the same lines.
M. Bonini et aLlNuclear
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Physics B 437 (1995) 163-186
Acknowledgements We have benefited greatly from discussions with C. Becchi and M. Tonin.
Appendix
A
The generating functional
( 1) is equivalent
to
(A.11
KOA,(P) = Ken(P) + KIAO(P> and the coefficient N is given in EQ. (3). This can be easily seen by making in (A.l) the change of variables 4, = 4’ - 4, which gives
x
I
~+exp(i{-~($p24)
OA- ~(4p’dhn, + (j14>,,o + (j4)0,}) >
(A.2)
where the source jl (p) is h(p)
=p2(AE(p),
-C’“(P),
C’“(P) ).
By performing the integral over the field 4, which is gaussian, one obtains ( 1). On the other hand, by integrating over the field 4’ in (A.2) and using (4) one obtains (2).
Appendix
B
We explicitly give the form of the operator TifiL), which extracts the relevant part of a functional of a multicomponent massless scalar field k. Due to the masslessness of the field, the Taylor expansion around vanishing momenta is affected by infrared singularities when we consider the four-dimensional terms in the fields and momenta. Thus one is forced to introduce a non-vanishing subtraction point. For the two-field components this point is assumed at p2 = p2, while for the N field components it is assumed at the symmetric point NSP defined by - PiP,=&(NSij-l),
N=3,4
,....
In order to have a more compact notation, we also introduce the Fourier transform Ji of the field +i. The form of T4(‘) is then
M. Bonini et al./Nuclear
Physics B 437 (1995) 163-186
181
In the following we explicitly give the relevant part of the effective action T[+,y] for the SU( 2) Yang-Mills theory. We first isolate the vertices of r[ 4, r] which contain the relevant couplings
P
PUP +GE ISrc3*)(p,q,r)A~(p>A~(q)A~(r) 1
abc
P
9
+AJJJ
k, h)A;(p)A:(q)A;(k)A:(W
I’z:bcd(p,q,
P
q
k
J
+ q%)w;(-P)cYP) P
+cnbc
JJ JJ
rEA)(p,q,r)w;(p)cb(q)A;(r)
P ++abc
9
T(““)(p,
P
q, r)u”(p)cb(q)cc(r)
+ ... ,
03.1)
4
where r = -p - q, h = -p - q - k and the dots stand for all the remaining terms which contain only irrelevant vertices, since they are coefficients of monomials in the fields and sources with dimension higher than four. The six vertices in (B. 1) contain nine relevant couplings which are defined as follows:
182
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Physics B 437 (1995) 163-186
r;*)(P) =gpv[~m* +p2c-1 +t,v(p)
[gA
r(3A)(pt 4, r) = [(P - 4)&v WP X[c3A rggbCd(p,
[ u;A
rcwcA) PJJ
where
t,, p,;“,
+ (4 - r),&J + (1. - P>&pl 9, r> 3 + 7; j$)
(P, 9. r) 9
q> k h) 1
+ $4A)(P,
4, k h) 1
(p, q, k, h) ,
=~[-i+(Twc+P)(p)l,
(p, 4, r> = &v
r(vcc)(p,
+ 2L(P>l 13
+ $4A)(p,
+Fj$gbcd
q@(p)
+ ST(p)
+ 2 (3A)(p,
q, k, h) =+$&4A +tg&a
+ca)
q, r) = -1
[ VwcA + 2 (wcA) (p, q, r>
1 + Fp)
+ (T,,,
,
+ ~(vcc)(p,
q, I)
(p,
q, r) ,
(P 1 = p2 gcLp - P,P~, = (Ealazcp3~4
is the four-vector
SU(2)
abed
t2;llvpg = ( rbs”d
_
&wucp2a3
) &L,p3&2w + (2 4-b 3) + (3 - 4)
structure appearing in the BRS action and
+ 6”” abd + lYd fibc) @u&l,
+ g,&u
+ &?,,g, 1 .
The relevant couplings are defined by the conditions XL(O)
zT(p)
= 0,
= 0,
.x(3A)(p,q,r)
= 0,
pG$
~~4A’(P,q,k,h)l
3SP
= 0,
$?p,q,W)
= 0,
4SP 2’“c’(p)
4SP = 0,
J&p2 JGvcc’(p,
q,
r)
-CwC*)(p,
q, r)
= 0 ) 3SP
= 0. 3SP
From these conditions2 we can factorize in the vertices 2~ a dimensional function of p. Thus Z; are “irrelevant” and contribute to the irrelevant part of the functional r[ r$, ~1. Similarly the vertices Ti are irrelevant since their Lorentz structure is (partially in the case of r(4A)) given by external momenta. We recall that the ghost propagator and the &A vertex are given in terms of the vertices ry) and I-p) by ’ Notice that in order to follow the general rule of extracting the relevant part of a functional given by the operator TiFL), the couplings CT* and (TwC are defined in a slightly different way with respect to Ref. [ 131.
M. Bonini et al. /Nuclear
F)(p) PA)(p, ”
=p2 + ip2[aw,
q, t-) = -igp,TEA’
Physics B 437 (1995) 163-186
+ /.P”‘(p)]
183
,
(p, q, i-)
In Subsection 2.1 in order to better identify the physical couplings in the effective action and study the gauge symmetry we have made a different definition of the relevant part of the effective action (see Eq. (1 I)), introducing the couplings zi and pi. The relation of these couplings with the gi is the following: z]=l
-(TA-(Tar
z2
PI =a,,
9
p4=uycc
+ iflT,,,
z3 = 2 - ‘,,A g
= 1 + iuwc ,
pz = @a 7
- U,L, - ua,
p3 = 1 + iuwc + flwcA,
3 p5 = 1 - (TA - (T, + -$A
+ +;A)
,
p6 = 5--&7'4A.
In particular when the physical boundary conditions ~1 = ~2 = ~3 = 1 are imposed one finds a,-/-aA
u WC--0,
=o,
UrnA = PI 1 o;cc
= P4 9
ua
= P2 1
u4A
= g2(-1
u3A uv,,A +PS
-
$6,
>
u'4A
= -ig, = -1
+p3,
= ;g2p6,
which show that these boundary conditions fix the vector and ghost wave-function normalization and the three-vector coupling. The other couplings in T[+,y] are given in terms of the pi and determined by the symmetry as in Section 4. Appendix
C
We now extract the relevant part of the most general one-dimensional fields and sources with ghost number one. We call this generic functional we isolate the vertices of A which contain the relevant couplings A=
JI'
Ajt\“‘(p)A;(
functional of A. First of all
-p)c”(p)
A~?'(P, 4, ~M;(p>A,bhWW JJ I,4 +I&JJ JJJ JJJ + I Eubc
2
A~CC’tp,q,r)~~(p)cb(q)cC(~)
2
P Y
+;
A@“ACC)“bCd(p,q,k,h)~;(p)A~(q)cc(k)cd(h) P‘
+t
A(3Ac,abcd(,,q,k,h)A;(p)A;(q)A;(k)cd(h) VP
I’
4
k
+. . . ,
184
M. Bonini et al./Nuclear
Physics B 437 (1995) 163-186
where r = -p - q, h = -p - q - k and the dots stand for the remaining terms which are all irrelevant. Then we define the 11 relevant parameters as follows: AfYP)
=pp[&
Ar’(p,q,r)
+p*s*
+
A’*“‘(p)1
,
-q2>C~3+~~AAc~~~,q,~~l
=g,,(p*
+(~,~~-q,q~)[&+A:AAc)(~~q~r)l +ip)(p,
Ap)(p,q,r) A(WACC)=bCd(p,q, P”
9, r) ,
=p,[G
+ A(WCC)(p,q,r)l,
k, h) =g#==Sbd
- Fdc+)
+$wAcc)abcd(p, A(3Ac)ubcd(p,q, WP
k, h) = (Gu&dp,g,
[S, + A@‘*@(p,q,
k, h)]
q, k, h) , + . . .) [&
+ Ai3*@(p,q,
k, h)]
+(6=d~bcp,g,+...)[c%+A;3Ac)(p,q,k,h)l
+WbScdp,g,,+
. ..)[a.
+~:3Ac’(~,q,kh)l
+(GQC@dp,g,p+ . . .>iho + A:3Ac)(p, q, k, h) 1
+Wd@‘cp,g,p + . . .>i&l + A:3Ac)(p, q, k, h) 1 +i63Ac)abcd WP
(p, q, k, h) ,
where the dots stand for permutation over the gluon momenta and Lorentz and colour indices. The conditions defining the relevant parameters are A(*“‘(O)
= --&A’nc’(p)
=o,
Ajy’
p24
A(wCC) =o, 3SP
Ai3Ac)
=o, 4SP
=o, 3SP
A(W*CC)
=o.
4SP
Due these conditions one can isolate in these vertices a dimensional function of p, thus they are irrelevant. Similarly the vertices ii have the Lorentz indices carried by momenta in a different way with respect to their relevant parts and are irrelevant.
Appendix
D
In this appendix we prove that the fine-tuning equations (24) can be solved if satisfies
Af,rel[ 4, Y; Sil = S~~CW r’ rel[$9Y;Pil.
Ar,,,=l
(D.1)
An explicit calculation shows that Eq. (D.l) implies the relations (38) and (39) among the 6i. The parameters pi are given by the following functions of Si:
M. Bonini et al. /Nuclear
81 = -p,, 65 = +
-Ps)
1
Physics B 437 (1995) 163-186
82
= -;p2,
67
= MPS -
63 i-=j3
+
P4)
185
= P5 3
6, = -2ig(&
,
-
.
/53)
(D.2) Now we show that it is possible to select the six parameters pje’ in r,,r[~#~r; pje’] such a way that six fine-tuning equations, 6ie) = 0, are satisfied. From ( 14) we have e-i Ay) [ 4, y] = 2S,-,w I+(‘) + c S,,ca r’(e-k) k=l By applying T:‘),
in
.
we obtain the relevant part
&I[ 4, y; 6;“) 1 = 2S,-co,I-;, [$, y; pje’ I + de) [qb,yl , where E-l
The crucial observation now is that fi (0 does not depend on the relevant parameters pie). This is obvious, since the product of two relevant vertices is a relevant vertex. This implies T;pc”’S,vco,T,(IL) = Sr,co,T4(pcL) . As a consequence (Z’:P)Sy,~~~ - S,.,C~,T4(p’)r:e, = 0 and ace) does not receive contribution from the couplings pje). Thus the last term in (D.3) involves only I$) [ d,r] which is given in terms of P!~‘) at lower loops P < C. Eq. (D.l) implies that L?ct) must be of the form n(“) = Spa, r;,, [ 4, y; pry)] )
pfie) = jjp
- 2py
.
As previously observed, p’je) are known from the calculation Therefore, by fixing #)
= +j”)
of pjr”
,
(D.4)
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