- Email: [email protected]
Abelian Ward identity from the background field dependence of the effective action
.__ BB
s
11 July 1996
c
__
PHYSICS
EJ..SEMER
LETTERS 6
Physics Letters B 380 (1996) 337-340
Abelian Ward identity from the background field dependence of the effective action F. Freire ‘, C. Wetterich 2 htitut $2 Theoretische Physik, Vniversitlit Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany Received 20 January
1996; revised manuscript Editor: PV. Landshoff
received 4 April 1996
Abstract The dependence of the effective action for gauge theories on the background field obeys an exact identity. We show that for Abelian theories the Ward identity follows from the more general background field identity. This observation is relevant for the formulation of effective actions with an infrared cutoff since the solution of exact flow equations must obey an anomalous Ward identity. PACS: 03.70.+k;
11.15-q;
12.20.-m
The effective average action rk is a useful concept for the investigation of infrared problems. In field theory it corresponds to the quantum effective action with an effective infrared cutoff N k for the fluctuations. In statistical physics it is the coarse grained free energy with coarse graining length scale -k-l. Exact non-perturbative flow equations describe the dependence of rk on k [ 11. They are related to the Wilsonian approach to the renormalization group equations [2]. Since non-Abelian gauge theories are plagued in perturbation theory by severe infrared problems the use of the effective average action seems particularly promising here. Writing down exact flow equations for gauge theories poses no additional difficulty. The problem of correct implementation of gauge symmetry arises rather on the level of their solutions. In fact, models with local gauge symmetry correspond to particular trajectories in the space of general solutions of the flow equations. It is crucial to find appropriate identities for rk which enforce a restriction to “gauge-invariant solutions” and guarantee gauge invariance for k ---f 0. So far, two different lines of research have been followed in this respect. The first one [3,4] works in the background field formalism where rk[A, A1 depends on the classical gauge field A, (conjugate to the source) and the background gauge field A, which enters through the gauge fixing and the infrared cutoff. By construction &[A, A] is invariant under simultaneous gauge transformations of A, and A,. This combined gauge invariance is, however, not sufficient to guarantee that the solution lies on a trajectory appropriate for a gauge-invariant theory. An additional exact identity for the background field dependence 8rk/8& was derived
’ E-mail: [email protected]. ’ E-mail: [email protected]. 0370-2693/96/$12.00 Copyright PII SO370-2693(96)00513-S
0 1996 Elsevier Science B.V. All rights reserved.
F. Freire, C. Wetterich/Physics
338
Letters B 380 (1996) 337-340
[ 4] and it is believed that this identity guarantees full gauge invariance of the theory. The second approach [ 561 centers on the Ward-Takahashi or Slavnov-Taylor identities. These identities receive anomalous contributions [ 61 from the presence of the infrared cutoff term N Rk which vanish only for k -+ 0. Obviously, the same type of identities can also be derived within the background field formalism. One would suspect that the background field identity and the Ward identity are not unrelated since both reflect the content of gauge symmetry. Their exact relation has, however, not been revealed in the past. In this letter we indicate how the Ward identity can be derived from the background field dependence of the effective action for Abelian gauge theories. The model we use is scalar electrodynamics (SQED) in arbitrary dimension d. The classical action consists of the usual SQED action plus a gauge-fixing term and a quadratic term implementing an infrared cutoff. For the complex scalar field x(x) the infrared cutoff reads [4]
A:‘)S=
px,&‘*(x)Rk(~[~l)X(-d= /$$ f$$@%‘,P’)x*(P’)x(P).
Here Rk is a function that cuts off the modes with momentum smaller than k with &(O) N k2. More precisely, the cutoff distinguishes between eigenvalues of the covariant Laplacian, D[ A] = -( 8, + ig&( x) ) (P + igAp (x) ) , where A,(x) is the background gauge field. The complete set of quadratic cutoff terms includes a term for the gauge sector which can be chosen to be gauge invariant in Abelian gauge theories. If this is the case it does not affect the Ward identity or the background field identity and does not need to be specified for our discussion. We choose the background gauge fixing rgf[A,/i]
= & /ddx
(d,JAp(x)
- A’(x)])~,
(2)
and consider the effective action Ik [ p, A, A] with 9 the classical scalar field related to x. For this gauge fixing it has been shown that the background field dependence of Ik is governed by the identity [4]
(rp) + Rk)-’ where
the arguments
+;
d”d,(A’-
/iv),
(3)
have been omitted. In Eq. (3) Tr stands for integration over the ddp whilst @) + Rk) -’ denotes the inverse of the twoconfiguration or momentum space, e.g. Tr = ( R’ s point connected Green function for the complex scalar field in the presence of the infrared cutoff. In momentum space, where
ry)
(,’
+ ~k)q~p,jp~pl
= &&
-’ =(x*(P’MP))c. >dP’)dP)
+zk
By subtracting
of the functionals
the gauge-fixing
I;[P,A,A]
=rkb.A,Ai
(rk
+ AkS)
3 we have
(4)
term -
rgf[Adi,
(5)
Eq. (3) becomes
(6)
F. Freire, C. Wetterich/Physics
Letters B 380 (1996) 337-340
339
The Ward identity for SQED in the absence of cutoff functions is well known. In the presence of the infrared cutoff term AkS’S it receives an anomalous contribution [6]. For our choice of the gauge-fixing term the anomalous Ward identity reads
A+/$$ 14 g ‘SAP(q)
s
ddp
{+&f4P-d-&GD*~P+d}
ddp’ {Rp)(P+qYP’)
=mm
-g++(P,P’-q)}
(r:2)+%)--(;,)~(p).
(7)
Note that R:(p) (p,p’) is not diagonal in momentum space (c?~ and &(x) do not commute) and depends on A,, Fq. (12). The right-hand side of Eq. (7) vanishes for k = 0 (Rpp) = 0) and the identity reduces to the standard homogeneous linear relation between one-particle irreducible Green functions. In this case the identity is known to encode the transversality of the gauge field propagator and it can be seen as a constraint guaranteeing the absence of non-physical (in this instance longitudinal) degrees of freedom. In the presence of the infrared cutoff the Ward identity is a generalisation of this constraint. Truncated solutions to the exact flow equation should satisfy them at least approximately such that the homogeneous Ward identity is recovered for k=O.
We now want to show that the Ward identity, Eq. (7)) follows from the background field identity, Eq. (3). The invariance of the background field effective action under a simultaneous gauge transformation of 40,A, and /I results in the identity
a-:
1 + ;
Sri
{~dP-q)-~~*(P+q)}=o.
-+/g$ 4’“6A,(q)
(8)
From Eq. (8) the proof that Eq. (7) is contained in Eq. (6) follows immediately if we show
“Rj;:pdp’) =Rf+ (p,p’
j q/1
- q) -
RF) (p
+ q,p’).
(9)
/L
In turn, Eq. (9) is a direct consequence of the invariance of the infrared cutoff A:“‘& Eq. (l), with respect to gauge transformations acting also on the background field (cf. Eq. (8) ) . This completes the proof that the Ward identity is contained in the background field identity. Therefore Eq. (6) embodies all information related to the local gauge invariance of the model. The proof that the background field identity implies the Ward identity can be generalised for all Abelian gauge theories and matter fields with arbitrary spin and charge. This clarifies at least for the Abelian case the relation between the two mentioned approaches to the use of exact flow equations for gauge theories. In particular, the standard covariant gauge fixing is a special case of Eq. (2) for A,(x) = 0. The background field identity, Eq. (6)) can also be used at b,(x) = 0. For general A,(x) Eq. (6) is stronger than the Ward identity since the latter is equivalent to its divergence, or, in momentum space, to a contraction with qp. For practical applications one may isolate the gauge-invariant kernel I!, [ 9, A] = rk [ cp,A, fi = A] and expand the remaining background field dependent part, Irge [cp,A,A] =rj,[p,A,A] -i’k[q,A], in powers of A(x) -A(x) f~ge[p,A,A]
= s
+ $ f$“b,81
ddx
(A,(x)
{
Hf;Lb,Al (A&) -&(x))
(A,(n)
- &Ax)) - &4x>) + . . .}.
(10)
Here Hi are gauge-invariant Lorentz-tensors which may depend on C+Y( x) and A,(x) but not on A,(x) . For any given i?k[ p, A] all Hi can be computed from appropriate functional derivatives of Eq. (6) with Il, replaced by
F. Freire, C. Wetterich/Physics Letters B 380 (19961337-340
340 fgauge.
For example,
a photon mass term for 40 = 0 is contained
fkauge to the photon two-point k
function
&2+w-
&+w
- q’) = -
AGp”“(q)8(q
where
[. . .I0 means evaluated
+
completely
aA,(q)iAv(-q’)
sA,(q)hd-q’)
at A = A = p = 0. The photon mass term is extracted
and one finds from Eq. (6)
from
82$wge
+ [ ~&(q)~~v(-q’)
m~2P’)
= rni 8‘” + . . . . The correction
in Hy
reads
fixed in terms of l?k by the background
for q = 0 (AC’-‘” (0) =
for k + 0 as rni N g2k2. We conclude
that it vanishes
field identity Eq. (6) -at
(11)
I o ’
that f”,“““” is
least as long as the part fyge
(2)
in lYj2) on the right-hand side can be treated iteratively. An explicit computation of the terms on the right-hand side of Eq. (11) can be done by using functional derivatives of the background field identity Eq. (6). This involves vertices for the background field A, which follow from an expansion of Ry’ (p, p’) in quadratic order in A, @+(PvP’)
+g2
x
= Rk(P2>&P
J a w
ddQ’
-g
s
A?Q)A”(Q’)
A’“(Q)
(2,rr>d ddQ
&P
+
&(P2> - &(PJ2) p2 - p’2
(P + P’)P
Q + Q’
S(P
+
Q
-P’>
-P’) 2
Rk(p2) - &(P’2> p2-p’2
Rk((p’ -
ddQ
mm
-P’>
- Q’)2) (p’
_
(zp -
+
Q)&Jp’
_
Q’)u
Rk(p2)
-
sRk(Pt2)
p2 - p’2
(P+Q)~-P~
(” -,,f)2R&‘2)
Q’)2
!p’2
+ o(g3A3>
9
(12)
&(p2) equals Rk(-a2) in a momentum space representation and 8(p) = (27~)~8(p). One can verify that Eq. ( 12) is consistent with the identity (9). A background field identity has also been derived for non-Abelian gauge theories [3] and up to now has been little exploited. It would be very interesting to understand its relation with the anomalous Slavnov-Taylor identities [6]. In view of the findings of this letter one may suspect that the non-Abel&t background field identity contains much relevant information beyond the Slavnov-Taylor identity. We thank the referee for remarks which contributed
substantially
to the final form of this note.
References [I] [2]
[3] [4] [ 51 [6]
C. Wetterich, Z. Phys. C 57 (1993) 451; C 60 (1993) 461; Phys. Lett. B 301 (1993) 90; M. Bonini, M. D’Attanasio and G. Marchesini, Nucl. Phys. B 409 ( 1993) 441. E Wegner and A.Houghton, Phys. Rev. A 8 (1973) 401; K. G. Wilson and I. G. Kogut, Phys. Rep. 12 (1974) 75; S. Weinberg, Critical Phenomena for Field Theorists, Erice Subnucl. Phys. (1976) 1; J. Polchinski, Nucl. Phys. B 231 (1984) 269. M. Reuter and C. Wetterich, Nucl. Phys. B 391 (1993) 147; B 408 (1993) 91; B 417 (1994) 181; preprint HD-THEP-94-39. M. Reuter and C. Wetterich, Nucl. Phys. B 427 ( 1994) 291. M. Bonini, M. D’ Attanasio and G. Marchesini, Nucl. Phys. B 418 ( 1994) 8 1; B 421 (1994) 429; B 437 (1995) 163; Phys. Lett. B 346 (1995) 87. U. Ellwanger, Phys. L&t. B 335 (1994) 364; U. Ellwanger, M. Hirsch and A. Weber, preprint LPTHE Orsay 95-39, to appear in Z. Phys. C.
s
11 July 1996
c
__
PHYSICS
EJ..SEMER
LETTERS 6
Physics Letters B 380 (1996) 337-340
Abelian Ward identity from the background field dependence of the effective action F. Freire ‘, C. Wetterich 2 htitut $2 Theoretische Physik, Vniversitlit Heidelberg, Philosophenweg 16, D-69120 Heidelberg, Germany Received 20 January
1996; revised manuscript Editor: PV. Landshoff
received 4 April 1996
Abstract The dependence of the effective action for gauge theories on the background field obeys an exact identity. We show that for Abelian theories the Ward identity follows from the more general background field identity. This observation is relevant for the formulation of effective actions with an infrared cutoff since the solution of exact flow equations must obey an anomalous Ward identity. PACS: 03.70.+k;
11.15-q;
12.20.-m
The effective average action rk is a useful concept for the investigation of infrared problems. In field theory it corresponds to the quantum effective action with an effective infrared cutoff N k for the fluctuations. In statistical physics it is the coarse grained free energy with coarse graining length scale -k-l. Exact non-perturbative flow equations describe the dependence of rk on k [ 11. They are related to the Wilsonian approach to the renormalization group equations [2]. Since non-Abelian gauge theories are plagued in perturbation theory by severe infrared problems the use of the effective average action seems particularly promising here. Writing down exact flow equations for gauge theories poses no additional difficulty. The problem of correct implementation of gauge symmetry arises rather on the level of their solutions. In fact, models with local gauge symmetry correspond to particular trajectories in the space of general solutions of the flow equations. It is crucial to find appropriate identities for rk which enforce a restriction to “gauge-invariant solutions” and guarantee gauge invariance for k ---f 0. So far, two different lines of research have been followed in this respect. The first one [3,4] works in the background field formalism where rk[A, A1 depends on the classical gauge field A, (conjugate to the source) and the background gauge field A, which enters through the gauge fixing and the infrared cutoff. By construction &[A, A] is invariant under simultaneous gauge transformations of A, and A,. This combined gauge invariance is, however, not sufficient to guarantee that the solution lies on a trajectory appropriate for a gauge-invariant theory. An additional exact identity for the background field dependence 8rk/8& was derived
’ E-mail: [email protected]. ’ E-mail: [email protected]. 0370-2693/96/$12.00 Copyright PII SO370-2693(96)00513-S
0 1996 Elsevier Science B.V. All rights reserved.
F. Freire, C. Wetterich/Physics
338
Letters B 380 (1996) 337-340
[ 4] and it is believed that this identity guarantees full gauge invariance of the theory. The second approach [ 561 centers on the Ward-Takahashi or Slavnov-Taylor identities. These identities receive anomalous contributions [ 61 from the presence of the infrared cutoff term N Rk which vanish only for k -+ 0. Obviously, the same type of identities can also be derived within the background field formalism. One would suspect that the background field identity and the Ward identity are not unrelated since both reflect the content of gauge symmetry. Their exact relation has, however, not been revealed in the past. In this letter we indicate how the Ward identity can be derived from the background field dependence of the effective action for Abelian gauge theories. The model we use is scalar electrodynamics (SQED) in arbitrary dimension d. The classical action consists of the usual SQED action plus a gauge-fixing term and a quadratic term implementing an infrared cutoff. For the complex scalar field x(x) the infrared cutoff reads [4]
A:‘)S=
px,&‘*(x)Rk(~[~l)X(-d= /$$ f$$@%‘,P’)x*(P’)x(P).
Here Rk is a function that cuts off the modes with momentum smaller than k with &(O) N k2. More precisely, the cutoff distinguishes between eigenvalues of the covariant Laplacian, D[ A] = -( 8, + ig&( x) ) (P + igAp (x) ) , where A,(x) is the background gauge field. The complete set of quadratic cutoff terms includes a term for the gauge sector which can be chosen to be gauge invariant in Abelian gauge theories. If this is the case it does not affect the Ward identity or the background field identity and does not need to be specified for our discussion. We choose the background gauge fixing rgf[A,/i]
= & /ddx
(d,JAp(x)
- A’(x)])~,
(2)
and consider the effective action Ik [ p, A, A] with 9 the classical scalar field related to x. For this gauge fixing it has been shown that the background field dependence of Ik is governed by the identity [4]
(rp) + Rk)-’ where
the arguments
+;
d”d,(A’-
/iv),
(3)
have been omitted. In Eq. (3) Tr stands for integration over the ddp whilst @) + Rk) -’ denotes the inverse of the twoconfiguration or momentum space, e.g. Tr = ( R’ s point connected Green function for the complex scalar field in the presence of the infrared cutoff. In momentum space, where
ry)
(,’
+ ~k)q~p,jp~pl
= &&
-’ =(x*(P’MP))c. >dP’)dP)
+zk
By subtracting
of the functionals
the gauge-fixing
I;[P,A,A]
=rkb.A,Ai
(rk
+ AkS)
3 we have
(4)
term -
rgf[Adi,
(5)
Eq. (3) becomes
(6)
F. Freire, C. Wetterich/Physics
Letters B 380 (1996) 337-340
339
The Ward identity for SQED in the absence of cutoff functions is well known. In the presence of the infrared cutoff term AkS’S it receives an anomalous contribution [6]. For our choice of the gauge-fixing term the anomalous Ward identity reads
A+/$$ 14 g ‘SAP(q)
s
ddp
{+&f4P-d-&GD*~P+d}
ddp’ {Rp)(P+qYP’)
=mm
-g++(P,P’-q)}
(r:2)+%)--(;,)~(p).
(7)
Note that R:(p) (p,p’) is not diagonal in momentum space (c?~ and &(x) do not commute) and depends on A,, Fq. (12). The right-hand side of Eq. (7) vanishes for k = 0 (Rpp) = 0) and the identity reduces to the standard homogeneous linear relation between one-particle irreducible Green functions. In this case the identity is known to encode the transversality of the gauge field propagator and it can be seen as a constraint guaranteeing the absence of non-physical (in this instance longitudinal) degrees of freedom. In the presence of the infrared cutoff the Ward identity is a generalisation of this constraint. Truncated solutions to the exact flow equation should satisfy them at least approximately such that the homogeneous Ward identity is recovered for k=O.
We now want to show that the Ward identity, Eq. (7)) follows from the background field identity, Eq. (3). The invariance of the background field effective action under a simultaneous gauge transformation of 40,A, and /I results in the identity
a-:
1 + ;
Sri
{~dP-q)-~~*(P+q)}=o.
-+/g$ 4’“6A,(q)
(8)
From Eq. (8) the proof that Eq. (7) is contained in Eq. (6) follows immediately if we show
“Rj;:pdp’) =Rf+ (p,p’
j q/1
- q) -
RF) (p
+ q,p’).
(9)
/L
In turn, Eq. (9) is a direct consequence of the invariance of the infrared cutoff A:“‘& Eq. (l), with respect to gauge transformations acting also on the background field (cf. Eq. (8) ) . This completes the proof that the Ward identity is contained in the background field identity. Therefore Eq. (6) embodies all information related to the local gauge invariance of the model. The proof that the background field identity implies the Ward identity can be generalised for all Abelian gauge theories and matter fields with arbitrary spin and charge. This clarifies at least for the Abelian case the relation between the two mentioned approaches to the use of exact flow equations for gauge theories. In particular, the standard covariant gauge fixing is a special case of Eq. (2) for A,(x) = 0. The background field identity, Eq. (6)) can also be used at b,(x) = 0. For general A,(x) Eq. (6) is stronger than the Ward identity since the latter is equivalent to its divergence, or, in momentum space, to a contraction with qp. For practical applications one may isolate the gauge-invariant kernel I!, [ 9, A] = rk [ cp,A, fi = A] and expand the remaining background field dependent part, Irge [cp,A,A] =rj,[p,A,A] -i’k[q,A], in powers of A(x) -A(x) f~ge[p,A,A]
= s
+ $ f$“b,81
ddx
(A,(x)
{
Hf;Lb,Al (A&) -&(x))
(A,(n)
- &Ax)) - &4x>) + . . .}.
(10)
Here Hi are gauge-invariant Lorentz-tensors which may depend on C+Y( x) and A,(x) but not on A,(x) . For any given i?k[ p, A] all Hi can be computed from appropriate functional derivatives of Eq. (6) with Il, replaced by
F. Freire, C. Wetterich/Physics Letters B 380 (19961337-340
340 fgauge.
For example,
a photon mass term for 40 = 0 is contained
fkauge to the photon two-point k
function
&2+w-
&+w
- q’) = -
AGp”“(q)8(q
where
[. . .I0 means evaluated
+
completely
aA,(q)iAv(-q’)
sA,(q)hd-q’)
at A = A = p = 0. The photon mass term is extracted
and one finds from Eq. (6)
from
82$wge
+ [ ~&(q)~~v(-q’)
m~2P’)
= rni 8‘” + . . . . The correction
in Hy
reads
fixed in terms of l?k by the background
for q = 0 (AC’-‘” (0) =
for k + 0 as rni N g2k2. We conclude
that it vanishes
field identity Eq. (6) -at
(11)
I o ’
that f”,“““” is
least as long as the part fyge
(2)
in lYj2) on the right-hand side can be treated iteratively. An explicit computation of the terms on the right-hand side of Eq. (11) can be done by using functional derivatives of the background field identity Eq. (6). This involves vertices for the background field A, which follow from an expansion of Ry’ (p, p’) in quadratic order in A, @+(PvP’)
+g2
x
= Rk(P2>&P
J a w
ddQ’
-g
s
A?Q)A”(Q’)
A’“(Q)
(2,rr>d ddQ
&P
+
&(P2> - &(PJ2) p2 - p’2
(P + P’)P
Q + Q’
S(P
+
Q
-P’>
-P’) 2
Rk(p2) - &(P’2> p2-p’2
Rk((p’ -
ddQ
mm
-P’>
- Q’)2) (p’
_
(zp -
+
Q)&Jp’
_
Q’)u
Rk(p2)
-
sRk(Pt2)
p2 - p’2
(P+Q)~-P~
(” -,,f)2R&‘2)
Q’)2
!p’2
+ o(g3A3>
9
(12)
&(p2) equals Rk(-a2) in a momentum space representation and 8(p) = (27~)~8(p). One can verify that Eq. ( 12) is consistent with the identity (9). A background field identity has also been derived for non-Abelian gauge theories [3] and up to now has been little exploited. It would be very interesting to understand its relation with the anomalous Slavnov-Taylor identities [6]. In view of the findings of this letter one may suspect that the non-Abel&t background field identity contains much relevant information beyond the Slavnov-Taylor identity. We thank the referee for remarks which contributed
substantially
to the final form of this note.
References [I] [2]
[3] [4] [ 51 [6]
C. Wetterich, Z. Phys. C 57 (1993) 451; C 60 (1993) 461; Phys. Lett. B 301 (1993) 90; M. Bonini, M. D’Attanasio and G. Marchesini, Nucl. Phys. B 409 ( 1993) 441. E Wegner and A.Houghton, Phys. Rev. A 8 (1973) 401; K. G. Wilson and I. G. Kogut, Phys. Rep. 12 (1974) 75; S. Weinberg, Critical Phenomena for Field Theorists, Erice Subnucl. Phys. (1976) 1; J. Polchinski, Nucl. Phys. B 231 (1984) 269. M. Reuter and C. Wetterich, Nucl. Phys. B 391 (1993) 147; B 408 (1993) 91; B 417 (1994) 181; preprint HD-THEP-94-39. M. Reuter and C. Wetterich, Nucl. Phys. B 427 ( 1994) 291. M. Bonini, M. D’ Attanasio and G. Marchesini, Nucl. Phys. B 418 ( 1994) 8 1; B 421 (1994) 429; B 437 (1995) 163; Phys. Lett. B 346 (1995) 87. U. Ellwanger, Phys. L&t. B 335 (1994) 364; U. Ellwanger, M. Hirsch and A. Weber, preprint LPTHE Orsay 95-39, to appear in Z. Phys. C.