Completing Continuum Coulomb Gauge in the Functional Formalism
aa r X i v : . [ h e p - t h ] N ov Completing Continuum Coulomb Gauge in the Functional Formalism
P. Watson and H. Reinhardt
Institut f¨ur Theoretische Physik, Universit¨at T¨ubingen,Auf der Morgenstelle 14, D-72076 T¨ubingen, Deutschland
It is argued that within the continuum functional formalism, there is no need to supply a further(spatially independent) gauge constraint to complete the Coulomb gauge of Yang-Mills theory. Itis shown explicitly that a natural completion of the gauge-fixing leads to a contradiction with theperturbative renormalizability of the theory.
PACS numbers: 11.15.-q,12.38.-t
Consider Yang-Mills theory, invariant under the fol-lowing (local) gauge transform characterized by the in-finitesimal parameter θ ax ( σ = A ): ~A ax → ~A θax = ~A ax + 1 g ~ ∇ x θ ax − f acb ~A cx θ bx , (1) σ ax → σ θax = σ ax − g ∂ x θ ax − f acb σ cx θ bx . (2)We are interested in Coulomb gauge, defined as the con-dition ~ ∇ x · ~A ax = 0. At the classical level it is clear that thiscondition restricts θ ax to be independent of spatial argu-ment ~x but which can be time-dependent or global. Leav-ing aside the issue of global gauge fixing, the questionaddressed here is whether or not it is necessary to com-pletely fix the local gauge; i.e., must we specify a time-dependent gauge condition in addition to the Coulombgauge condition? It is shown that trying to completethe gauge-fixing leads to a contradiction and thereforethat the system has in a sense ‘chosen’ its own nontrivialcompletion of the gauge.In the quantum, functional formulation of the theory,the central object of interest is the functional integral Z = Z D Φ exp { ı S} (3)(Φ denotes the collection of fields). Because the action, S , (see [1] for our notation and conventions) is invariantunder gauge transforms, we have to isolate the zero modeof the integral (generated by integration over the gaugegroup) and if we fix to Coulomb gauge using the Faddeev-Popov technique, then we have Z = Z D Φ exp { ı S + ı S fp } , S fp = Z d x h − λ a ~ ∇· ~A a − c a ~ ∇· (cid:16) δ ab ~ ∇ − gf acb ~A c (cid:17) c b i , (4)where λ a is a Lagrange-multiplier field introduced to en-force the gauge condition, c a and c b are Grassmann-valued ghost fields. The functional integral above stillcontains zero modes corresponding to time-dependentgauge transforms (we do not consider the Gribov copieshere). If a further gauge-fixing condition is required to eliminate these zero modes, it must be spatially indepen-dent (so as not to interfere with Coulomb gauge itself)and we desire that it is local in the fields, such that func-tional techniques can be applied. An obvious choice is F [ σ ] = Z d~xσ a ( x , ~x ) = 0 . (5)The same condition exists when one considers theCoulomb gauge limit of interpolating (Landau-Coulomb)gauges in a finite volume and with periodic boundaryconditions [2]. Note that if we consider the (weaker) con-straint ∂ x Z d~xσ a ( x , ~x ) = 0 , (6)then since we have a single temporal dimension, this im-plies that Z d~xσ a ( x , ~x ) = C (7)where C is a constant. However, under time-reversal, σ a ( − x , ~x ) = − σ a ( x , ~x ), which forces C = 0 and thecondition, Eq. (5), above. The form of the gauge-fixingcondition, Eq. (5), has an immediate consequence in thefunctional formalism – the Faddeev-Popov determinantgenerated is independent of the fields on the gauge-fixedhypersurface and is thus trivial. To be specific, usingthe Faddeev-Popov technique we isolate the integrationover the time-dependent gauge group by inserting thefollowing identity into the functional integral: Z D θδ ( F [ σ ])det (cid:2) M ba ( y , x ) (cid:3) , (8)where M ba ( y , x ) = δF [ σ aθ ( x , ~x )] δθ b ( y ) (cid:12)(cid:12)(cid:12)(cid:12) F =0 = (cid:20) − g δ ba ∂ x δ ( y − x ) Z d~x − f acb δ ( y − x ) Z d~xσ c ( x , ~x ) (cid:21) F =0 = − g δ ba ∂ x δ ( y − x ) Z d~x. (9)In the above, the spatial integral of σ is a number andvanishes due to the gauge condition F = 0. The remain-ing part of det( M ) is thus a pure number, independent ofthe fields and which can be incorporated into the normal-ization of the functional integral. Clearly there will be noadditional temporal Gribov ambiguity. Our completelygauge-fixed functional integral now reads: Z = Z D Φ δ (cid:18)Z d~xσ ax (cid:19) exp { ı S + ı S fp } . (10)It is to be emphasized that the interaction content of thetheory has not been modified by the extra gauge-fixing.This means that the Dyson–Schwinger equations will notchange their form; what will change are the propagatorsand the effects will be seen at tree-level. Thus, we maydiscard the interaction content of the theory from thediscussion, save for one-loop integrals which will be con-sidered later.Let us then consider the generating functional of thetheory by including source terms and restricting to atmost quadratic terms in the action. For definiteness, weexpress the δ -functional constraint as an integral over anew time-dependent Lagrange multiplier field, χ a ( x )[7].We have Z [ J ] = Z D Φ exp { ı S + ı S s } , S = Z d x (cid:26) − A ai (cid:2) δ ij ∂ − δ ij ∇ + ∇ i ∇ j (cid:3) A aj − λ a ∇ i A ai − c a ∇ c a − χ a ( x ) σ a − A ai ∂ ∇ i σ a − σ a ∇ σ a (cid:27) , S s = Z d x { ρ a σ a + J ai A ai + c a η a + η a c a + ξ a λ a + κ ax χ a ( x ) } . (11)The generating functional of connected Green’s functionsis W [ J ], where Z = e W (in the context here, J denotesa generic source). Also defining the classical fields Φ α = δW [ J ] /δıJ α we can construct the effective action, Γ, asthe Legendre transform of W :Γ[Φ] = W [ J ] − ıJ α Φ α (12)(condensed index notation implies summation over alldiscrete indices and integration over all continuous ar-guments). For notational convenience, we introduce abracket notation to denote the functional derivatives ofboth W and Γ: <ıJ α > = δWδıJ α , <ı Φ α > = δ Γ δı Φ α . (13)We can now write down our tree-level equations for bothproper and connected two-point functions using the tech-niques of [1, 3]. In the case of the proper functions, this ismore or less trivial – for example, the Lagrange multiplierfield λ gives rise to the equation <ıλ ax > = −∇ ix A aix (14) and all the further functional derivatives can be writtendown without ambiguity in either configuration or mo-mentum space. The case for the connected (propagator)two-point functions is far less clear. The full set of equa-tions reads: ρ ax = ∂ x ∇ ix <ıJ aix > + ∇ x <ıρ ax > + <ıκ ax >, (15) J aix = (cid:2) δ ij ∂ x − δ ij ∇ x + ∇ ix ∇ jx (cid:3) <ıJ ajx > + ∂ x ∇ ix <ıρ ax > −∇ ix <ıξ ax >, (16) ξ ax = ∇ ix <ıJ aix >, (17) Z d~xκ ax = Z d~x <ıρ ax >, (18) η ax = ∇ x <ıη ax > . (19)Let us begin with Eq. (17). The only non-zero func-tional derivative of the left-hand side is that with respectto ıξ by , leading to − ıδ ba δ ( y − x ) = ∇ ix <ıξ by ıJ aix > . (20)The solution to this is written as <ıξ by ıJ aix > = δ ba Z ¯ d k e − ık · ( y − x ) k i ~k (21)where we recognize that when sources are set to zero,the function must be translationally invariant and beodd under the parity transform (it is a spatial vector).This latter constraint necessarily precludes the possibil-ity that there may be other (spatially independent) so-lutions to the homogeneous equation. Indeed, the func-tional derivative of Eq. (17) with respect to ıρ by is justsuch a homogeneous equation:0 = ∇ ix <ıρ by ıJ aix > . (22)Since < ıρ by ıJ aix > is a spatial vector, the function van-ishes (as is clear if we Fourier transform into momentumspace). The same holds for <ıκ by ıJ aix > : i.e., <ıρ by ıJ aix > = <ıκ by ıJ aix > = 0 . (23)The spatial gluon propagator, <ıJ bjy ıJ aix > = Z ¯ d k e − ık · ( y − x ) D baAAji ( k , ~k ) , (24)is derived from the corresponding functional derivativesof equations (16) and (17):0 = Z ¯ d k e − ık · ( y − x ) k i D baAAji ( k , ~k ) , Z ¯ d k e − ık · ( y − x ) × h ( k − ~k ) D baAAji ( k , ~k ) − ıδ ba t ji ( ~k ) i (25)( t ji ( ~k ) = δ ji − k j k i /~k is the transverse projector). Thesolution to this is D baAAji ( k , ~k ) = δ ba t ji ( ~k ) ı ( k − ~k ) . (26)Now let us examine the ghost propagator stemmingfrom Eq. (19). Since the ghost field is Grassmann-valued whereas the propagators must be scalar, the ghostfields/sources must come in pairs. Because the ghostfields anticommute, in the absence of sources the quan-tity < ıη by ıη ax > must vanish – we cannot construct anyantisymmetric, color diagonal, scalar, translationally in-variant function of invariants ( x − y ) and ( ~x − ~y ) .Thus, the only functional derivative of Eq. (19) that is ofinterest is ∇ x <ıη ax ıη by > = ıδ ab δ ( x − y ) (27)and the solution is D abc ( x − y , ~x − ~y ) = − δ ab Z ¯ d k e − ık · ( x − y ) ı~k . (28)In principle, we could add a homogeneous term ∼ δ ( ~k ) D ( k ) to the integrand above. However, because theghost propagator is connected to a ghost-gluon vertexwith the factor k i in any loop integral [1], the δ -functionguarantees the situation whereby this term never appearsin a calculation and we can disregard it.Let us now turn to the remaining scalar propagators.Since the λ and χ fields are Lagrange-multiplier fields,propagators involving them will not contribute to anyloop integral (they have no interaction term) and onlythe temporal gluon propagator, < ıρ by ıρ ax > , is of conse-quence. We notice that Eq. (18) is integrated over ~x (adirect consequence of the fact that we must have a spa-tially independent second gauge condition) and will onlydetermine the functions in momentum space at ~k = 0.Indeed, we have that D σσ ( k , ~k = 0) = 0 . (29)This applies for all values of k , including the limit k → ∞ . We are now led to a contradiction. Since D σσ is the only propagator that can cancel the well-known en-ergy divergence of the ghost loop (see [1] for an explicitrealization of this cancellation), it must have a finite partas k → ∞ , just as the ghost propagator, in order to ef-fect the cancellation (we have explicitly shown that themixed propagator, D Aσ , is zero and the spatial gluonpropagator, D AA , vanishes in this limit). However, ondimensional grounds the above constraint, Eq. (29), tellsus that D σσ must vanish – in the absence of any externalscale it can only behave as k − ( ~k /k ) µ for some positivepower µ in this limit. Moreover, if we try to determine D σσ by solving the simultaneous set of equations gen-erated from Eqs. (15) and (16) then we are led to thefollowing (with the common color factors omitted): k − ı k ı~k − ~k D σσ ( k , ~k ) k D λσ ( k , ~k ) D χσ ( k , ~k ) k D χλ ( k , ~k ) = ık . (30) The matrix in the above has a zero determinant (forall momenta) and the solution is determined only up toan unknown scalar function. This function is only con-strained by the requirement that it vanishes as ~k = 0 soas to agree with Eq. (29). The (physically meaningful)temporal gluon propagator is not determined, even attree-level! Thus, we have a situation whereby completelyfixing the gauge has resulted in the energy divergencesof the ghost loops not being canceled and the propagatorcontent of the theory being ill-defined.In summary, we have attempted to completely fix theCoulomb gauge by adding a further spatially independentgauge condition. Whilst this extra condition seems justi-fied (and even necessary) to deal with the zero modes inthe functional formalism, its implementation has led toan explicit contradiction in defining the tree-level propa-gators of the theory. There could be one of two reasonsfor this. Since one is not familiar with such spatially inde-pendent constraints, the implementation here may be de-ficient in some way, although it is not clear how – the con-dition Eq. (29) certainly appears a robust consequence ofthe gauge-fixing and the requirement of canceling the en-ergy divergence of the ghost-loop is not ambiguous. Themore likely explanation is that the gauge-fixing conditioncontradicts the dynamics of the renormalizable quantumtheory and in particular, Gauß’ law. In the functionalapproach, Gauß’ law appears as the dynamical equationof motion for the σ -field and it is primarily this equationthat leads to definition of the temporal gluon propagator, D σσ . We are thus led to the conclusion that the construc-tion of the functional formalism in Coulomb gauge im-plicitly ‘chooses’ the remaining temporal gauge conditionwith the requirement of perturbative renormalizability sothat a further constraint such as Eq. (5) is not necessary.Because in principle we should be able to choose any (rea-sonable) gauge-fixing condition, we can further say thatif the condition given by Eq. (5) is not allowed, then nei-ther is any other condition, except that implicit condition‘chosen’ by the system itself.Whilst we have used the second order formalism here,the same arguments will apply to Coulomb gauge in thefirst order formalism. Formally, within the first orderformalism, the system can be reduced to physical (spa-tially transverse gluon) degrees of freedom [3, 4] and, onreflection, this would indeed seem to imply that there isno need for a further, spatially independent gauge con-straint and in agreement with the conclusions here. Fur-ther, the Gribov-Zwanziger confinement scenario [4, 5]in Coulomb gauge alludes to an infrared divergent tem-poral gluon propagator (from which a confining poten-tial can be constructed) in contradistinction to conditionEq. (29).Given that the temporal gauge condition, Eq. (5),occurs in the interpolating gauge [2], one might betempted to infer from the results here that the Coulombgauge end-point of interpolating gauge does not exist, orleads to different physical mechanisms when comparedto Coulomb gauge. This is not necessarily true and notour conclusion. It is seen in the interpolating gauge lat-tice calculations of Ref. [6] that as one approaches theCoulomb gauge limit, an increasingly infrared (but stillfinite | ~k | ) enhanced, yet ~k = 0 vanishing temporal propa-gator emerges whilst the Coulomb gauge temporal prop-agator itself is infrared divergent. Thus, it should bekept in mind that in discussing the perturbative propa-gators here, what matters physically are the integrals andcombinations of the tree-level factors that form, for ex-ample, the nonperturbative propagators. That the tree-level propagators, canceled energy divergences etc., havea different form in Coulomb gauge in distinction to inter-polating gauges is not in itself either a drawback or a sur-prise – quite tautologically, many different integrals havethe same value. Indeed, one may regard the differences between the internal constructions of Coulomb gauge andthe interpolating gauge and how they still should result inthe same observable physics as another fascinating pieceof the puzzle to study. Acknowledgments
The authors would like to thank R. Alkofer for acritical reading of the manuscript. They would alsolike to thank M. Quandt and G. Burgio for use-ful discussions. This work has been supported bythe Deutsche Forschungsgemeinschaft (DFG) under con-tracts no. DFG-Re856/6-1 and DFG-Re856/6-2. [1] P. Watson and H. Reinhardt, arXiv:0709.3963 [hep-th].[2] L. Baulieu and D. Zwanziger, Nucl. Phys. B , 527(1999) [arXiv:hep-th/9807024].[3] P. Watson and H. Reinhardt, Phys. Rev. D , 045021(2007) [arXiv:hep-th/0612114].[4] D. Zwanziger, Nucl. Phys. B (1998) 237. [5] V. N. Gribov, Nucl. Phys. B (1978) 1.[6] A. Cucchieri, A. Maas and T. Mendes, Mod. Phys. Lett.A22