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A stagnant gauge for QCD
Nuclear Physics @ North-Holland
B165 (1980) 423-428 Publishing Company
A STAGNANT
GAUGE
C.H. LLEWELLYN
FOR QCD
SMITH
Department of Theoretical Physics, 1 Keble Road, Oxford, UK
Received
9 November
1979
It is shown that the gauge-fixing parameter (Y of standard covariant gauges may legitimately be replaced by an operator a(O). In particular, a may be chosen so that the gluon propagator has a “stagnant” tensor structure proportional to g,, for all momenta. This choice of gauge simplifies explicit calculations and leads to renormalization group equations with no a/k term.
1. Introduction Consider a theory with an unbroken standard gauge-fixing term -&2”)2
gauge symmetry (e.g., QCD). With the
)
(1)
where
the bare propagator
is
q&,+(1-(Y) AU,,(k)+-
(
-&,+F
)I
(k2))’
(for a review of the derivation of the Feynman rules see ref. [l]). Except in the case of Landau gauge (a = 0), the relative proportion of g,, and k,k, terms in the complete propagator AFLyis k* dependent. Therefore, although by choosing Feynand avoid having to integrate over man gauge (a = 1) we can make AL,ag,, k,k,(k2)-’ terms in low orders, these terms inevitably occur in higher orders in the standard formulation. Gauge invariance assures us that the net effect of “intrinsic” k,k, terms in Azy is zero when we calculate S-matrix elements (although Green functions and wavefunction renormalization are, of course, gauge dependent). In this note I show that the net effect of induced k,k, terms is also zero (a result which is relatively obvious 423
C. Llevwllyn
424
Smith
/ Stagnant
gauge for QCD
in QED). Specifically I show that the gauge-fixing parameter (Ycan be replaced by an arbitrary function a(k2) (or an operator ~(0) in the lagrangian). Setting cy= 1 +Cz=i (g*)“(~,,(k~) we can choose the (Y, to cancel induced k,k, terms order by order and ensure that A,,aggy for all k2 to all orders. I call this choice stagnant Feynman gauge or SFG. SFG has two attractive features: (i) it simplifies calculations, an explicit example being provided by leading log calculations in QCD where a fixed tensor structure for A&,, and the absence of (k’)-* terms is a considerable advantage [2]; (ii) in SFG there is no ‘*gauge-changing term” proportional to a/& in the renormalization group equations. This paper is organized as follows. In sect. 2 the rules for the case cy= a(O) are derived in a formal way (in passing I give a compact derivation of the Ward identity k”k”A,, = -ix). In sect. 3 the same conclusions are derived diagrammatically. The renormalization group equations for SFG are discussed in sect. 4.
2. Formalism With the gauge-fixing term (1) the Faddeev-Popov det (a -i/* a”Zb)
determinant
is
)
where
is the covariant derivative. At first sight it might seem that the ghost lagrangian which replaces the FP determinant would depend on the fact that LY= a(O). However, writing det ((Y-I/~ d.D)=det(cu
-1’2) det (a * D) ,
the first term is an irrelevant field-independent constant, which can be ignored, and the ghost lagrangian takes its usual form (we could keep Y”* in Lghost and note that it can be absorbed in a redefinition of the ghost field or cancelled between adjacent vertices and propagators). It follows that the Feynman rules are the usual ones except that (Y=a(k2) in the bare gluon propagator. Is the theory renormalizable and unitary? This requires (a) that the k* dependence of (Ydoes not introduce new divergences and (b) that there are Ward identities. Condition (a) is fulfilled if a depends on powers of In k2, which is the case in SFG. Condition (b) is satisfied provided the action is BRS invariant [3]. Writing L ghost=7]a a.oab&Jb, where n and w are the anticommuting
ghost fields, it is easy to check (using the
C. Llewellyn
Smith
fact that (Ye”’ (0) is a self-adjoint under the BRS transformation
/ Stagnant
gauge for QCD
425
operator) that the action is indeed invariant
6A; = cDihob,
ho” =
-kfabc~h~C,
where the infinitesimal parameter E is an anticommuting object (here and below I ignore fermions for simplicity; they change none of the conclusions). The usual Ward identity, kc”&
=
-iakv,
is valid, from which we infer that
in SFG. In passing I record a particularly simple derivation of this Ward identity which avoids the use of canonical cummutators. Adding source terms to the action S, I define S’ by S’=S+
I
d4x(J”(x)a~D~bob(x)+J’,(x)77c(x))
and a generating
functional
W(J) = 1 DA Dn Dw eiS’ . BRS invariance
I
of the path integral
na(x) d”AfXy) eis DA Dn Dw
then gives
W_‘(O)
62w
mwh(y)
= V’(0)
I
= Wp’(O)$
=$
J=O
qa(x) a”Dhy’w’(y) ea DA Dn Dw
F
$
lAz(x)Ai(y) ”I a
$ $(A:(x)Ahiv))iO, CL y
els DA Dq Dw
426
C. Llewllyn
Smith
/ Stagnant
gauge
for QCD
where LY= ~(0,) if it is not a constant. The first line is easily evaluated by changing variables to n”= n” +J” in the path integral and is equal to -i8”h64(X - y) . Equating this to the last line and Fourier transforming k”k”A,,
gives the Ward identity
= -ia
(from which the identity for kPA,,
follows trivially).
3. Diagrammatics The considerations in sect. 2 are rather formal but the same conclusions can be reached by straightforward diagrammatic means. In fact the diagrammatic proof that the S-matrix is independent of LYwhen a is fixed also applies if (Y= a(k’) because it consists of cancelling a-dependent terms in the expressions for Green functions and wave-function renormalization before integrating over virtual momenta. Explicitly, we start in a standard covariant gauge (a =const) for which the calculational rules are well-known and well-defined. We then add a term AMU) a”A;)’ to the lagrangian and show that the S-matrix is unchanged order by order in A. The demonstration follows ‘t Hooft and Veltman’s proof of the gauge invariance of the S-matrix [4] (simplified by using BRS invariance rather than brute force to derive their Slavnov-Taylor-Ward identity; we continue to ignore fermions which cause no difficulties). For A = 0, a = constant, BRS invariance gives S(O/T($(x) =+a.
a. A”(y)A;,(x,)
. .. AL:,(.r,))10)= 0
A”(x) a. A”(y)A>,
-1 (O]T(na a . A”(y)A:‘,
A
.. . Af;n)lO)
. .. D$&,)
... A>m)(0),
B Fig. 1
C
C. Llewellyn
Smith
/ Stagnant
gauge for QCD
427
Fig. 2
corresponding to the diagrams in fig. 1. Now let q = k, multiply by Af2(k2) and let ky +O. Integrating over k, the first term is just the change in the n-point Green function AG, due to treating the perturbation to order A. Assuming that the onshell legs are attached to physical sources J” (which satisfy _P(ghost (k)~D,o~O)ak”.Tp = 0) only fig. 2 survives from figs. lb, c. Fig. 2 gives nXG, where X is a constant, so the equation depicted in fig. 1 reads AG,-nXG,
=O.
Applying this equation to the two-point function, it follows that AZ-2xz=o, where Z is the residue at the pole as k* +O. Hence the S-matrix, which is given by &=&G,,
(J-v
satisfies AS, = 0 and is independent of A to first order. It is straightforward to continue the proof to higher orders and obvious that it must work since it is certainly true for f= constant and the fact that f# constant plays no role.
4. The renormalization group equation (RGE) The lagrangian can be written in terms of either bare or renormalized In the case of fixed (Yin 4 - E dimensions L = L(A,B, gB, aB1 = L(&,
g~-~“~, ~1 +L(A,
quantities.
g, a, E, ~1,
where the counter terms L, could be determined either by the minimal subtraction prescription or by imposing normalization conditions on superficially divergent Green functions at momenta characterized by p. The RGE express the fact that the lagrangian is invariant under the transformation
428
C. Llewellyn Smith / Stagnant guage for QCD
(Y-fa
(
1+s-
ScL) P
>
for suitable /3, y and 8. Now consider SFG in which a and aB are fixed uniquely by the requirement that A,,ccgCLV for all k2. It follows that if the kinetic and interaction terms in L are invariant when p, g and the scale of A, are changed suitably, the (a 9A)* term will automatically be invariant, i.e., aB(O, gB, E) = @(CLCL,g, E)Z&, /4 6) will not change. Therefore the RGE take the form
without the usual a/& term in SFG. However, beyond leading order, p and y are not the same as in any fixed (Ygauge and must be calculated in SFG. I am grateful to John C. Taylor for useful discussions.
References [l]
ES. Abers and B.W. Lee, Phys. Reports 9 (1973) 1; J.C. Taylor, Gauge theories of weak interactions (Cambridge University [2] A.B. Carter and C.H. Llewellyn Smith, Nucl. Phys. B162 (1980) 397. [3] C. Becchi, A. Rouet and R. Stora, Ann. of Phys. 98 (1976) 287. [4] G. ‘t Hooft and M. Veltman, Nucl. Phys. BSO (1972) 318.
Press,
1976).
B165 (1980) 423-428 Publishing Company
A STAGNANT
GAUGE
C.H. LLEWELLYN
FOR QCD
SMITH
Department of Theoretical Physics, 1 Keble Road, Oxford, UK
Received
9 November
1979
It is shown that the gauge-fixing parameter (Y of standard covariant gauges may legitimately be replaced by an operator a(O). In particular, a may be chosen so that the gluon propagator has a “stagnant” tensor structure proportional to g,, for all momenta. This choice of gauge simplifies explicit calculations and leads to renormalization group equations with no a/k term.
1. Introduction Consider a theory with an unbroken standard gauge-fixing term -&2”)2
gauge symmetry (e.g., QCD). With the
)
(1)
where
the bare propagator
is
q&,+(1-(Y) AU,,(k)+-
(
-&,+F
)I
(k2))’
(for a review of the derivation of the Feynman rules see ref. [l]). Except in the case of Landau gauge (a = 0), the relative proportion of g,, and k,k, terms in the complete propagator AFLyis k* dependent. Therefore, although by choosing Feynand avoid having to integrate over man gauge (a = 1) we can make AL,ag,, k,k,(k2)-’ terms in low orders, these terms inevitably occur in higher orders in the standard formulation. Gauge invariance assures us that the net effect of “intrinsic” k,k, terms in Azy is zero when we calculate S-matrix elements (although Green functions and wavefunction renormalization are, of course, gauge dependent). In this note I show that the net effect of induced k,k, terms is also zero (a result which is relatively obvious 423
C. Llevwllyn
424
Smith
/ Stagnant
gauge for QCD
in QED). Specifically I show that the gauge-fixing parameter (Ycan be replaced by an arbitrary function a(k2) (or an operator ~(0) in the lagrangian). Setting cy= 1 +Cz=i (g*)“(~,,(k~) we can choose the (Y, to cancel induced k,k, terms order by order and ensure that A,,aggy for all k2 to all orders. I call this choice stagnant Feynman gauge or SFG. SFG has two attractive features: (i) it simplifies calculations, an explicit example being provided by leading log calculations in QCD where a fixed tensor structure for A&,, and the absence of (k’)-* terms is a considerable advantage [2]; (ii) in SFG there is no ‘*gauge-changing term” proportional to a/& in the renormalization group equations. This paper is organized as follows. In sect. 2 the rules for the case cy= a(O) are derived in a formal way (in passing I give a compact derivation of the Ward identity k”k”A,, = -ix). In sect. 3 the same conclusions are derived diagrammatically. The renormalization group equations for SFG are discussed in sect. 4.
2. Formalism With the gauge-fixing term (1) the Faddeev-Popov det (a -i/* a”Zb)
determinant
is
)
where
is the covariant derivative. At first sight it might seem that the ghost lagrangian which replaces the FP determinant would depend on the fact that LY= a(O). However, writing det ((Y-I/~ d.D)=det(cu
-1’2) det (a * D) ,
the first term is an irrelevant field-independent constant, which can be ignored, and the ghost lagrangian takes its usual form (we could keep Y”* in Lghost and note that it can be absorbed in a redefinition of the ghost field or cancelled between adjacent vertices and propagators). It follows that the Feynman rules are the usual ones except that (Y=a(k2) in the bare gluon propagator. Is the theory renormalizable and unitary? This requires (a) that the k* dependence of (Ydoes not introduce new divergences and (b) that there are Ward identities. Condition (a) is fulfilled if a depends on powers of In k2, which is the case in SFG. Condition (b) is satisfied provided the action is BRS invariant [3]. Writing L ghost=7]a a.oab&Jb, where n and w are the anticommuting
ghost fields, it is easy to check (using the
C. Llewellyn
Smith
fact that (Ye”’ (0) is a self-adjoint under the BRS transformation
/ Stagnant
gauge for QCD
425
operator) that the action is indeed invariant
6A; = cDihob,
ho” =
-kfabc~h~C,
where the infinitesimal parameter E is an anticommuting object (here and below I ignore fermions for simplicity; they change none of the conclusions). The usual Ward identity, kc”&
=
-iakv,
is valid, from which we infer that
in SFG. In passing I record a particularly simple derivation of this Ward identity which avoids the use of canonical cummutators. Adding source terms to the action S, I define S’ by S’=S+
I
d4x(J”(x)a~D~bob(x)+J’,(x)77c(x))
and a generating
functional
W(J) = 1 DA Dn Dw eiS’ . BRS invariance
I
of the path integral
na(x) d”AfXy) eis DA Dn Dw
then gives
W_‘(O)
62w
mwh(y)
= V’(0)
I
= Wp’(O)$
=$
J=O
qa(x) a”Dhy’w’(y) ea DA Dn Dw
F
$
lAz(x)Ai(y) ”I a
$ $(A:(x)Ahiv))iO, CL y
els DA Dq Dw
426
C. Llewllyn
Smith
/ Stagnant
gauge
for QCD
where LY= ~(0,) if it is not a constant. The first line is easily evaluated by changing variables to n”= n” +J” in the path integral and is equal to -i8”h64(X - y) . Equating this to the last line and Fourier transforming k”k”A,,
gives the Ward identity
= -ia
(from which the identity for kPA,,
follows trivially).
3. Diagrammatics The considerations in sect. 2 are rather formal but the same conclusions can be reached by straightforward diagrammatic means. In fact the diagrammatic proof that the S-matrix is independent of LYwhen a is fixed also applies if (Y= a(k’) because it consists of cancelling a-dependent terms in the expressions for Green functions and wave-function renormalization before integrating over virtual momenta. Explicitly, we start in a standard covariant gauge (a =const) for which the calculational rules are well-known and well-defined. We then add a term AMU) a”A;)’ to the lagrangian and show that the S-matrix is unchanged order by order in A. The demonstration follows ‘t Hooft and Veltman’s proof of the gauge invariance of the S-matrix [4] (simplified by using BRS invariance rather than brute force to derive their Slavnov-Taylor-Ward identity; we continue to ignore fermions which cause no difficulties). For A = 0, a = constant, BRS invariance gives S(O/T($(x) =+a.
a. A”(y)A;,(x,)
. .. AL:,(.r,))10)= 0
A”(x) a. A”(y)A>,
-1 (O]T(na a . A”(y)A:‘,
A
.. . Af;n)lO)
. .. D$&,)
... A>m)(0),
B Fig. 1
C
C. Llewellyn
Smith
/ Stagnant
gauge for QCD
427
Fig. 2
corresponding to the diagrams in fig. 1. Now let q = k, multiply by Af2(k2) and let ky +O. Integrating over k, the first term is just the change in the n-point Green function AG, due to treating the perturbation to order A. Assuming that the onshell legs are attached to physical sources J” (which satisfy _P(ghost (k)~D,o~O)ak”.Tp = 0) only fig. 2 survives from figs. lb, c. Fig. 2 gives nXG, where X is a constant, so the equation depicted in fig. 1 reads AG,-nXG,
=O.
Applying this equation to the two-point function, it follows that AZ-2xz=o, where Z is the residue at the pole as k* +O. Hence the S-matrix, which is given by &=&G,,
(J-v
satisfies AS, = 0 and is independent of A to first order. It is straightforward to continue the proof to higher orders and obvious that it must work since it is certainly true for f= constant and the fact that f# constant plays no role.
4. The renormalization group equation (RGE) The lagrangian can be written in terms of either bare or renormalized In the case of fixed (Yin 4 - E dimensions L = L(A,B, gB, aB1 = L(&,
g~-~“~, ~1 +L(A,
quantities.
g, a, E, ~1,
where the counter terms L, could be determined either by the minimal subtraction prescription or by imposing normalization conditions on superficially divergent Green functions at momenta characterized by p. The RGE express the fact that the lagrangian is invariant under the transformation
428
C. Llewellyn Smith / Stagnant guage for QCD
(Y-fa
(
1+s-
ScL) P
>
for suitable /3, y and 8. Now consider SFG in which a and aB are fixed uniquely by the requirement that A,,ccgCLV for all k2. It follows that if the kinetic and interaction terms in L are invariant when p, g and the scale of A, are changed suitably, the (a 9A)* term will automatically be invariant, i.e., aB(O, gB, E) = @(CLCL,g, E)Z&, /4 6) will not change. Therefore the RGE take the form
without the usual a/& term in SFG. However, beyond leading order, p and y are not the same as in any fixed (Ygauge and must be calculated in SFG. I am grateful to John C. Taylor for useful discussions.
References [l]
ES. Abers and B.W. Lee, Phys. Reports 9 (1973) 1; J.C. Taylor, Gauge theories of weak interactions (Cambridge University [2] A.B. Carter and C.H. Llewellyn Smith, Nucl. Phys. B162 (1980) 397. [3] C. Becchi, A. Rouet and R. Stora, Ann. of Phys. 98 (1976) 287. [4] G. ‘t Hooft and M. Veltman, Nucl. Phys. BSO (1972) 318.
Press,
1976).