STPHY-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) 0 2 1 v 9 Abstract 8 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 / 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