Table Of ContentSTPHY-Th/95-1
Quantizing SU(N) gauge theories without gauge fixing
G B Tupper and F G Scholtz
Institute of Theoretical Physics
University of Stellenbosch
5
9
9 7600 Stellenbosch
1
n South Africa
a
J
(January 17, 1995)
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2
1
v
9 Abstract
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0
1
0 We generalize and extend the quantization procedure of [1] which is designed
5
9 to quantize SU(N) gauge theories in the continuum without fixing the gauge
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h
t and thereby avoid the Gribov problem. In particular we discuss the BRS
-
p
e symmetry underlying the effective action. We proceed to use this BRS sym-
h
:
v metry to discuss the perturbative renormalization of the theory and show
i
X
that perturbatively the procedure is equivalent to Landau gauge fixing. This
r
a
generalizes the result of [1] to the non-abelian case and confirms the widely
held believe that the Gribov problem manifests itself on the non-perturbative
level, while not affecting the perturbative results. A relation between the
gluon mass and gluon condensate in QCD is obtained which yields a gluon
mass consistent with other estimates for values of the gluon condensate ob-
tained from QCD sum rules.
PACS numbers 11.15.–q, 11.10.Gh
Typeset using REVTEX
1
I. INTRODUCTION
A major obstacle in the quantization of non-abelian gauge theories is the
Gribovproblem[2],especiallyasformulatedbySinger[3]: consideracompact,
semisimple non-abelian gauge theory in Euclidean space-time with boundary
conditions at infinity implying the identification of space-time with S4, then
due to a topological obstruction no global continuous gauge fixing is possible.
This excludes a very general class of gauge fixing conditions and in particular
all the practically implementable ones.
It seems to be generally accepted that the Gribov problem is not impor-
tant in the perturbative domain, but that it may play an important role in
understandingthenon-perturbativeaspectsofagaugetheory[4]. Ourpresent
results confirmthat the Gribov problem is perturbatively unimportant. How-
ever, the situation regarding the non-perturbative aspects is much less clear.
Indeed,Fujikawa [5] argued thatnon-perturbatively the usualBRS symmetry
[6] is spontaneously broken when a Gribov problem is present, invalidating
the associated Slavnov-Taylor identities. Thus, while perturbatively innocu-
ous, the Gribov problem casts doubt on e.g. the program of solving the
Schwinger-Dyson equations non-perturbatively via the gauge technique [7].
It is therefore of paramount importance that the Gribov problem should be
brought under control before reliable investigations into the non-perturbative
aspects of gauge theories can be launched.
Within the continuum limit there has been two recent proposals for quan-
tizing non-abelian gauge theories without gauge fixing and thereby avoiding
theGribovproblem. Thefirstofthese,’softgaugefixing’,duetoZwanziger[8]
andJona-Lasinio[9]amountstoanimplementation ofPopov’ssuggestion [10]
that the Faddeev-Popov trick should be generalized to 1 = [dU]F[UG]/I[G]
with I[G] = [dU]F[UG] and F a non-gauge invariant funcRtion of the gauge
R
2
field G such that I[G] exists. For a particular choice of F[G] this method has
been shown by Fachin [11] to reproduce the usual perturbative result for the
renormalized Landau-gauge propogator in a suitable limit.
The second approach is due to the present authors [1]. Here the point
of departure is a supersymmetry-like resolution of the identity in terms of
bosonic and fermionic auxiliary fields combined with a U-gauge transforma-
tion to give the gauge field a mass. Since the starting identity has a BRS
symmetry one also expects this to be the case for the effective action result-
ing from this quantization scheme. This is indeed the case as is shown below.
There are, however, several outstanding problems connected to the pro-
cedure of [1]. Firstly it has not yet been demonstrated how the usual pertur-
bative results can be recovered from this procedure for a non-abelian theory.
Indeed one may appreciate that since the program invokes a massive U-gauge
like non-abelian gauge field the ordinary perturbation theory and renormal-
ization analysis are problematic. Secondly it can be shown (see section 6)
that the original proposition of breaking the gauge symmetry at the tree level
is perturbatively incompatible with the BRS invariance as it would imply the
spontaneous breaking of the BRS symmetry. Thirdly the sequential scheme
originally proposed to deal with SU(N>2) is clumsy and obscures the BRS
symmetry underlying the quantization procedure.
In this paper we will show how these problems are overcome. We be-
gin in section 2 by reformulating our procedure so as to effectively deal with
any SU(N>2) theory without resorting to the sequential scheme of [1]. This
is done by imbedding in a U(N) gauge theory with the identity realized in
terms of local U(N) global U(N) fields. In section 3 we discuss the BRS
L R
⊗
symmetry, which becomes very transparent in the present formulation, for a
SU(N) theory. In the remainingsections the BRS symmetry and the pinching
technique of Cornwall [12] are used to perform the perturbative renormal-
3
ization, which becomes much more tractable in the present formulation, of
the effective theory. This culminates in our main result, namely, that in the
perturbative domain the present quantization procedure corresponds to Lan-
dau gauge fixing for non-abelian theories. The normal perturbative results
are therefore recovered, showing that the Gribov problem has no effect in the
perturbative domain. Some technical results are collected in two appendices.
II. THE U(N) FORMALISM
In[1]theauxiliarybosonicandfermionicfieldsweretakenasvectorvalued
in the fundamental representation of SU(N). To integrate the gauge degrees
of freedom out in this setting one has to resort to a sequential procedure in
which the gauge symmetry is broken down according to SU(N) SU(N-1)....
⊃
This procedure obscures many aspects of the theory and leads to technical
complications.
Here we avoid this sequential procedure by taking the auxiliary bosonic
and fermionic fields to be matrix valued in the fundamental representation of
U(N).Weexploit, asamatterofconvenience, thefactthatSU(N) U(N).Our
⊂
procedure is as follows: consider a pure Yang–Mills type theory (for notation
see appendix A)
1
L = tr(G2 )
YM −2 µν
(1)
G = ∂ G ∂ G +ig[G ,G ] , G = Gat
µν µ ν − ν µ µ ν µ µ a
This lagrangian is invariant under the gauge transformations
i
G G′ = UD U† , U U(N) (2)
µ → µ −g µ ∈
where
D = ∂ +igG (3)
µ µ µ
4
is the gauge covariant derivative in the fundamental representation. Now for
some set of gauge invariant functionals O[G] = O[G′] define
< O > = Z−1 [dG]O[G]exp (i d4xL ) ,
YM
Z Z
(4)
Z = [dG]exp (i d4xL )
YM
Z Z
For the restricted class of functionals, O, which depend on the SU(N)
gauge field
G = Gat (5)
µ µ a
only, it follows from U(N) SU(N) U(1) that
≃ ⊗
< O > = Z−1 [dG]O[G]exp (i d4xL ),
YM
Z Z
(6)
Z = [dG]exp (i d4xL )
YM
Z Z
which is canonical. Thus quantization of an SU(N) gauge theory proceeds
from (4) applied to O.
While (4) and (6) are well defined on the lattice, in the continuum the
measure
[dG] Π dGa(x) (7)
∝ x,µ,a µ
is itself gauge invariant, [dG′] = [dG], so < O > / which is ill defined.
∼ ∞ ∞
To factor the volume of the gauge group while eschewing the Faddeev–Popov
ansatz together with its associated Gribov problem consider the (minimal)
identity
1 = [dΦ][dΦ†][dζ][dζ†]exp (i d4xL ) ,
aux
Z Z
(8)
L =tr(Φ†D†DµΦ+ζ†D†Dµζ).
aux µ µ
Hereintheauxiliaryfieldsarematrixvaluedinthefundamentalrepresentation
of U(N)
5
Φ = Φ t , Φ† = Φ†t and ζ = ζ t , ζ† = ζ†t (9)
a a a a a a a a
representingN2 complex(2N2 real)scalardegreesoffreedomandN2 complex
Grassmann valued scalar degrees of freedom (ghosts), respectively. Further-
more D† acts to the left
µ
←
Dµ† =∂µ −igGµ. (10)
The measure on the auxiliary fields is
dΦ (x)dΦ†(x)
[dΦ][dΦ†][dζ][dζ†]= Π a a dζ (x)dζ†(x). (11)
x,a 2πi a a
By construction L has local U(N) invariance, under the transforma-
aux L
tions
G iU D U† , Φ U Φ , Φ† Φ†U† ,
µ −→ −g L µ L −→ L −→ L
(12)
ζ U ζ , ζ† ζ†U† , U = exp (iθLt ) U(N)
−→ L −→ L L a a ∈ L
as well as an independent global U(N) symmetry:
R
G G , Φ ΦU† , Φ† = U Φ† ,
µ −→ µ −→ R R
(13)
ζ ζU† , ζ† U ζ† , U = exp (iθRt ) U(N) .
−→ R −→ R R a a ∈ R
For infinitesimal transformations
δ Φ = 2itr(t t t )θLΦ = iθL(T ) Φ (14)
L a a c b c b c c ab b
whereas
δ Φ = iθR( T∗) Φ (15)
R a c − c ab b
which, together with(A14), identifiesT and T∗ as generators of theadjoint
a − a
representationofU(N) andU(N) respectively;ourauxiliaryfieldstransform
L R
as the basis for the adjoint representation of chiral U(N). Note that this is
in contrast to [1] where the auxiliary fields transform as the fundamental
representation. Writing
6
1 1
d4xL = d4x d4y Φ†(x)M (x,y)Φ (y)+ ζ†(x)M (x,y)ζ(y) ,
aux 2 a ab b 2 a ab
Z Z Z (cid:18) (cid:19)
(16)
M (x,y) = DaeDµδ(4)(x y)
ab − µ eb −
with
Dab = δ ∂ +igGe(T ) (17)
µ ab µ µ e ab
the gauge covariant derivative in the adjoint representation of U(N) , the
L
proof of the identity is immediate:
det (M)
[dΦ][dΦ†][dζ][dζ†]exp (i d4xL ) = = 1. (18)
aux
det (M)
Z Z
Injecting (8) into (4)
< O > = Z−1 [dΦ][dΦ†][dζ][dζ†][dG]O[G]exp (i d4x(L +L )) ,
YM aux
Z Z
(19)
Z = [dΦ][dΦ†][dζ][dζ†][dG]exp (i d4x(L +L ))
YM aux
Z Z
we make the change of variables
Φ = U(π)φ , Φ† = φU†(π)
− −
(20)
U(π) = exp (iπ t )
a a
followed by
i
G G′ = U(π)D U†(π) , ζ ζ′ = U(π)ζ , ζ† ζ′† = ζ†U†(π) (21)
µ −→ µ −g µ −→ −→
which is the form of a (unitary) gauge transformation. Using the result (B9)
< O > = Z−1 [dU][det (J(φ))dφ][dζ][dζ†][dG]O[G]exp (i d4xL )
eff
Z Z
(22)
Z = [dU][det (J(φ))dφ][dζ][dζ†][dG]exp (i d4xL )
eff
Z Z
where
7
L = L +tr(φD†Dµφ+ζ†D†Dµζ) (23)
eff YM µ µ
and
J (φ) = D φ (24)
ab abc c
The integrand being independent of U(π) the volume of the gauge group,
[dU], factors and cancels in the normalization, leaving
Z
< O > = Z−1 [det (J(φ))dφ][dζ][dζ†][dG]O[G]exp (i d4xL )
eff
Z Z
(25)
Z = [det (J(φ))dφ][dζ][dζ†][dG]exp (i d4xL )
eff
Z Z
Comparingourformulation to that of [8,9] one sees thatthe latter constitutes
a non-linear realization of the chiral symmetry with the ghosts represented
by pseudofermions.
III. BRS SYMMETRY
The essential content of the identity (8) is that G appears as an ex-
µ
ternal source field, expanding in powers of Ga one observes that for each
µ
closed Φ loop there is a closed ζ loop with a relative minus sign from ghost
− −
statistics so they exactly cancel – including vacuum graphs – much as in a
supersymmetry. In turn this suggests that our procedure possesses a type of
BRS symmetry, and such is indeed the case as we now proceed to show. For
the representation of the auxiliary fields used here the relevant (anti) BRS
transformations generated by (S¯)S are
8
SΦ† = ζ† S¯Φ† = 0
SΦ = 0 S¯Φ = ζ
−
Sζ† = 0 S¯ζ† = Φ† (26)
Sζ = Φ S¯ζ = 0
SG = 0 S¯G = 0
µ µ
Clearly S and S¯ are nilpotent but do not anticommute, rather
(SS¯+S¯S)(Φ† , Φ , ζ† , ζ)= (Φ† , Φ, ζ† , ζ) (27)
− −
Using
S(XY)= (SX)Y X(SY) , S¯(XY) = (S¯X)Y X(S¯Y) (28)
± ±
where the + ( ) sign applies if X is ghost even (odd) one sees that
−
L = SW = S¯W∗ (29)
aux
with
W = tr(Φ†D†Dµζ), W∗ = tr(ζ†D†DµΦ). (30)
µ µ
Thus from nilpotency L is (anti) BRS invariant and moreover a simple
aux
calculation shows the BRS invariance of the measure in (8). Then, for F any
functional of the auxiliary fields and G
µ
[dΦ][dΦ†][dζ][dζ†]Fexp (i d4xL )=
aux
Z Z
[dΦ][dΦ†][dζ][dζ†](F +χSF)exp (i d4xL ) (31)
aux
Z Z
where we have performed a BRS transformation with χ a global Grassmann
variable. It follows that
0 = [dΦ][dΦ†][dζ][dζ†](SF)exp (i d4xL ) (32)
aux
Z Z
9
From (32) we have an alternative proof of (8): let
I[G] = [dΦ][dΦ†][dζ][dζ†]exp (i d4xL ). (33)
aux
Z Z
By (29), differentiation of I[G] with respect to G produces a quantity of
µ
the form (32), i.e. δI[G]/δGa = 0 which is to say I[G] is a constant that may
µ
benormalized to unity. Note from(29) that L correspondsto atopological
aux
field theory of the Witten type [13].
Next we need to establish what becomes of this BRS symmetry under the
change of variables (20), (21). We begin by defining the ‘BRS current’.
C = U†(π)SU(π)
(34)
SU(π) = U(π)C , SU†(π) = CU†(π)
−
Then
S( φU†(π)) = (Sφ)U†(π)+φCU†(π) = ζ†U†(π) (35)
− −
or
Sφ= ζ†+φC (36)
−
while
S(U(π)φ) = U(π)Cφ+U(π)(Sφ) (37)
so
Sφ = Cφ. (38)
−
Similarly
S(ζ†U†(π)) = (Sζ†)U†(π)+ζ†CU†(π)) = 0
S(U(π)ζ) = U(π)Cζ +U(π)(Sζ)) = U(π)φ
−
10
