Non-perturbative constraints on the quark and ghost propagators
aa r X i v : . [ h e p - t h ] S e p SLAC-PUB-17181
Non-perturbative constraints on the quark and ghostpropagators
Peter Lowdon
SLAC National Accelerator Laboratory, 2575 Sand Hill Rd, Menlo Park, CA 94025, USAE-mail: [email protected]
Abstract
In QCD both the quark and ghost propagators are important for governing the non-perturbativedynamics of the theory. It turns out that the dynamical properties of the quark and ghost fieldsimpose non-perturbative constraints on the analytic structure of these propagators. In thiswork we explicitly derive these constraints. In doing so we establish that the correspondingspectral densities include components which are multiples of discrete mass terms, and that thepropagators are permitted to contain singular contributions involving derivatives of δ ( p ), bothof which are particularly relevant in the context of confinement. Introduction
The non-perturbative behaviour of propagators involving coloured fields enters into many im-portant areas of quantum chromodynamics (QCD), including the dynamics of quark-gluonplasma [1, 2] and the nature of confinement itself [3, 4, 5, 6, 7, 8]. Nevertheless, the overallstructure of these objects remains largely unknown. In order to gain a better understanding ofthe general characteristics of these objects, one requires a framework in which one can probe thenon-perturbative regime. In the literature, many of the new insights into the structure of QCDpropagators have come from non-perturbative numerical approaches [5, 9, 10, 11, 12, 13, 14].Now whilst these approaches provide a powerful way to calculate certain aspects of propagators,they necessarily contain uncertainties due to the approximations that are required in order tocarry out the calculations . In Refs. [8] and [15] an alternative approach was developed in orderto establish the most general structural form of the gluon propagator. This approach inovlvedapplying a local quantum field theory (LQFT) framework, which is constructed via the assertionof a series of physically motivated axioms [4, 16, 17, 18, 19]. Since these axioms are assumed tohold independently of the coupling regime, this enables genuine non-perturbative characteristicsto be derived in a purely analytic manner.An important feature of gauge theories such as QCD is that the gauge symmetry provides anobstacle to the locality of the theory . In order to construct a consistent quantised theory oneis left with two options: either one allows non-local fields, or one preserves locality. A generalfeature of local quantisations is that additional degrees of freedom are introduced into the the-ory, resulting in a space of states with an indefinite inner product. The prototypical example isthe Becchi-Rouet-Stora-Tyutin (BRST) quantisation of QCD, where the space of states V QCD contains negative-norm ghost states. In this case the physical states V phys ⊂ V QCD correspondto those that are annihilated by the BRST charge [4]. Although many of the generic features ofpositive-definite inner product QFTs are preserved in BRST quantised QCD, it turns out thatthe existence of an indefinite inner product can lead to significant changes in the structure ofthe propagators . In particular, in Ref. [8] it was demonstrated that the BRST quantised gluonpropagator can potentially contain singular terms involving derivatives of δ ( p ), a feature whichis related to confinement [7, 20, 21].Since quark and ghost fields are the other degrees of freedom in QCD for which the propagatorsplay a central role in governing the dynamics of the theory, it is also important to determinethe structural properties of the propagators associated with these fields. With this motivationin mind, the aim of this paper is to continue the approach developed in Refs. [8] and [15] forthe gluon propagator, and evaluate the constraints imposed on the quark and ghost propagatorsin BRST quantised QCD. The rest of the paper is organised as follows: in Sec. 2 a local QFTapproach is used to derive the general structural representation of the Lorentz covariant Diracfermion correlator and propagator, and these representations are then used together with thequark Schwinger-Dyson equation to constrain the quark propagator; in Sec. 3 an analogousapproach is applied in order to determine the overall structural form of an anti-commutingghost correlator and propagator, and the subsequent constraints imposed by the Schwinger-Dyson equation on the QCD ghost propagator; and finally in Sec. 4 the main findings aresummarised. In the case of the solutions of the Schwinger-Dyson equations, uncertainties arise for example due to the choiceof truncation scheme employed in order to consistently solve the equations. By locality we mean that the fields in the theory are local fields, and therefore commute or anti-commute(depending on their spin properties) for space-like separations [18]. LQFTs defined with an indefinite inner product space of states can be described using a modified version of thestandard QFT axioms, which are often referred to as the
Pseudo-Wightman axioms . A more in-depth discussion ofthis framework can be found in Ref. [19]. Non-perturbative constraints on the quark propagator
In order to derive the structural form of the quark propagator in QCD one must first determinethe general properties of an arbitrary Dirac fermion correlator and propagator. These propertieswill be discussed in the proceeding sections.
A central feature of local formulations of QFT is that correlators h | φ ( x ) φ ( x ) | i and theirFourier transforms b T (1 , ( p ) are distributions . Due to the Lorentz transformation properties ofthe fields φ and φ it follows that b T (1 , ( p ) can be decomposed in the following manner: b T (1 , ( p ) = N X α =1 Q α ( p ) b T α (1 , ( p ) , (2.1)where b T α (1 , ( p ) are Lorentz invariant distributions, and Q α ( p ) are polynomial functions of p carrying the same Lorentz index structure as φ and φ [19]. The first case of interest in thispaper is where φ = ψ and φ = ψ are Dirac spinor and conjugate spinor fields respectively.In this instance there are two possible Lorentz covariant polynomials: Q ( p ) = I and Q ( p ) = γ µ p µ = /p , where the spinor indices have been suppressed. It follows from Eq. (2.1) that themomentum space fermion correlator can then be written b S ( p ) = F (cid:2) h | ψ ( x ) ψ ( y ) | i (cid:3) = I b S ( p ) + /p b S ( p ) . (2.2)As in the case of the vector correlator [8], the Lorentz invariant distributions b S ( p ) and b S ( p ) arerestricted to have support in the closed forward light cone V + , and therefore have the followingspectral representation [19]: b S i ( p ) = P i ( ∂ ) δ ( p ) + Z ∞ ds θ ( p ) δ ( p − s ) ρ i ( s ) , (2.3)where P i ( ∂ ) is a polynomial of finite order in the d’Alembert operator ∂ = g µν ∂∂p µ ∂∂p ν , and ρ i ( s ) are the corresponding spectral densities . The full fermion correlator therefore takes theform b S ( p ) = Z ∞ ds θ ( p ) δ ( p − s ) (cid:2) ρ ( s ) + /pρ ( s ) (cid:3) + (cid:2) P ( ∂ ) + /pP ( ∂ ) (cid:3) δ ( p ) . (2.4)Taking the inverse Fourier transform of this expression leads to the general representation ofthe position space correlator h | ψ ( x ) ψ ( y ) | i = − Z ∞ ds π (cid:2) ρ ( s ) + ρ ( s ) i /∂ (cid:3) iD ( − ) ( x − y ; s )+ 1(2 π ) h P (cid:2) − ( x − y ) (cid:3) + i /∂ P (cid:2) − ( x − y ) (cid:3) i , (2.5)where D ( − ) ( x − y ; s ) is the negative frequency Pauli-Jordan function [19]. Since P and P are complex polynomials of finite order, one can set: P = P l =0 a l (cid:2) − ( x − y ) (cid:3) l , and P = P m =1 b m (cid:0) − ( x − y ) (cid:1) m where a l , b m ∈ C . The sum in P does not include the m = 0 termbecause this will not contribute due to the derivative in Eq. (2.5). More specifically, they are assumed to belong to the class of tempered distributions S ′ ( R , ) [16]. It turns out that the spectral densities ρ i ( s ) are tempered distributions in the class S ′ ( R + ). .2 The Dirac fermion propagator The fermion propagator involves a time-ordered product of fields, and is defined by h | T { ψ ( x ) ψ ( y ) }| i := θ ( x − y ) h | ψ ( x ) ψ ( y ) | i − θ ( y − x ) h | ψ ( y ) ψ ( x ) | i . (2.6)In order to determine the spectral representation of this propagator one must first establish thespectral representation for the correlator h | ψ ( y ) ψ ( x ) | i . Since the CPT operator Θ transformsDirac spinor fields as: Θ ψ ( x )Θ − = iγ ψ † ( − x ), and the vacuum state is invariant under theaction of Θ, one has the following relation h | ψ ( y ) ψ ( x ) | i = − γ h | ψ ( − x ) ψ ( − y ) | i γ . (2.7)Using the spectral representation of the fermion correlator in Eq. (2.4), the propagator can thenbe written h | T { ψ ( x ) ψ ( y ) }| i = θ ( x − y ) Z ∞ ds Z d p (2 π ) e − ip ( x − y ) θ ( p ) δ ( p − s ) (cid:2) ρ ( s ) + /pρ ( s ) (cid:3) + θ ( x − y ) Z d p (2 π ) e − ip ( x − y ) (cid:2) P ( ∂ ) + /pP ( ∂ ) (cid:3) δ ( p )+ θ ( y − x ) Z ∞ ds Z d p (2 π ) e ip ( x − y ) θ ( p ) δ ( p − s ) (cid:2) ρ ( s ) − /pρ ( s ) (cid:3) + θ ( y − x ) Z d p (2 π ) e ip ( x − y ) (cid:2) P ( ∂ ) − /pP ( ∂ ) (cid:3) δ ( p ) . (2.8)In order to simplify this expression one can use the relation i /∂ h θ ( x − y ) e − ip ( x − y ) + θ ( y − x ) e ip ( x − y ) i = /p h θ ( x − y ) e − ip ( x − y ) − θ ( y − x ) e ip ( x − y ) i + iγ δ ( x − y ) h e − ip ( x − y ) − e ip ( x − y ) i , which upon substitution into Eq. (2.8) implies that the Dirac fermion propagator has the fol-lowing general structure h | T { ψ ( x ) ψ ( y ) }| i = − Z ∞ ds π (cid:2) ρ ( s ) + ρ ( s ) i /∂ (cid:3) i ∆ F ( x − y ; s )+ 1(2 π ) P (cid:2) − ( x − y ) (cid:3) + i (2 π ) /∂P (cid:2) − ( x − y ) (cid:3) , (2.9)where ∆ F ( x − y ; s ) is the Green’s function of the Klein-Gordon equation. The momentum spacepropagator b S F ( p ) therefore has the form b S F ( p ) = i Z ∞ ds π (cid:2) ρ ( s ) + /pρ ( s ) (cid:3) p − s + iǫ + (cid:2) P ( ∂ ) + /pP ( ∂ ) (cid:3) δ ( p ) . (2.10)The representations in Eqs. (2.9) and (2.10) follow only from the assumption that the momentumspace correlators are Lorentz covariant distributions with support in the closed forward lightcone. Since this assumption is a generic feature of any QFT, these representations are thereforemodel independent. Since the general spectral properties of a Dirac fermion propagator have been outlined in theprevious section, one can now use the dynamical information in BRST quantised QCD to derivethe model-dependent constraints on the structure of the quark propagator. .3.1 General structure In BRST quantised QCD the renormalised quark field ψ i satisfies the equation of motion( iγ µ ∂ µ − m ) ψ i = − gγ µ A aµ ( x )( t a ψ ) i = K i , (2.11)where t a is the colour group generator in the fundamental representation, i is the colour index,and g, m are the renormalised coupling and mass parameters. The quark fields also satisfy theequal-time anti-commutation relation (cid:8) ψ i ( x ) , ψ j ( y ) (cid:9) x = y = δ ij Z − γ δ ( x − y ) , (2.12)where Z is the quark field renormalisation constant. Taking the vacuum expectation value ofEq. (2.12), and applying Eq. (2.5) together with Eq. (2.7) gives h |{ ψ i ( x ) , ψ j ( y ) }| i x = y = − (cid:20)Z ∞ ds π h ρ ij ( s ) + ρ ij ( s ) i /∂ i iD ( − ) ( x − y ; s ) (cid:21) x = y + (cid:20)Z ∞ ds π h ρ ij ( s ) + ρ ij ( s ) i /∂ i iD ( − ) ( y − x ; s ) (cid:21) x = y = − (cid:20)Z ∞ ds π h ρ ij ( s ) + ρ ij ( s ) (cid:0) iγ ∂ + iγ j ∂ j (cid:1)i iD ( x − y ; s ) (cid:21) x = y (2.13)Using the initial conditions: D ( x − y ; s ) x = y = 0 and ˙ D ( x − y ; s ) x = y = δ ( x − y ), and comparingwith Eq. (2.12), it follows from Eq. (2.13) that ρ ij ( s ) satisfies the spectral density constraint Z ∞ ds ρ ij ( s ) = 2 πδ ij Z − . (2.14)In contrast to the gluon propagator case [8], the equal-time anti-commutation relation imposesan integral constraint on one of the spectral densities, not both.BRST quantised QCD has a space of states with an indefinite inner product. Among other thingsthis implies that not all correlators are guaranteed to define positive-definite distributions [19].In certain cases, such as correlators constructed from gauge-invariant fields, one can demonstratethough that correlators do indeed possess this property. However, since the interacting quarkcorrelator h | ψ ( x ) i ψ j ( y ) | i itself is not composed of gauge-invariant fields, nor is it related toa gauge-invariant correlator which consists of the quark field or its derivatives (like the photoncorrelator in QED [8]), neither the state space structure nor the dynamical equations [Eqs. (2.11)and (2.12)] are sufficient to rule out the possibility of terms involving derivatives of δ ( p ). Inparticular, this implies that the corresponding (momentum space) polynomial terms P ij ( ∂ ) = P l a ijl ( ∂ ) l and P ij ( ∂ ) = P m b ijm ( ∂ ) m for the quark correlator may be non-vanishing, andhence the quark propagator has the general form b S ijF ( p ) = i Z ∞ ds π h ρ ij ( s ) + /pρ ij ( s ) i p − s + iǫ + h P ij ( ∂ ) + /pP ij ( ∂ ) i δ ( p ) . (2.15)Although the overall analytic structure of the quark propagator has been discussed many times inthe literature [5, 9, 22], the possibility of singular terms in the quark propagator is a feature thathas generally not been emphasised before, and yet could potentially be important in the contextof QCD confinement. In Ref. [7] it was established that the appearance of non-measure-definingterms in correlators, which includes derivatives of δ ( p ), can cause the correlation strength be-tween the states created by the fields in these correlators to increase with the separation of thestates, a violation of the so-called cluster decomposition property [20, 21]. If one could demon-strate that this occurs for any correlator involving fields which create coloured states, this wouldimply that the corresponding states could not be measured independently of one another, whichis a sufficient condition for confinement [4, 23]. .3.2 Schwinger-Dyson equation constraints Now that the general structure of the quark propagator has been outlined, one can evaluatethe further constraints that the equation of motion [Eq. (2.11)] imposes. As demonstrated inRef. [15] for the gluon propagator, a direct way to determine these constraints is to derive thecorresponding Schwinger-Dyson equation, and then use this to separately constrain the singularand non-singular terms in the propagator. Combining Eq. (2.11) and Eq. (2.12), together withthe definition of the Dirac fermion propagator in Eq. (2.6), one obtains the coordinate spacequark Schwinger-Dyson equation( iγ µ ∂ µ − m ) h | T { ψ i ( x ) ψ j ( y ) }| i = iδ ij Z − δ ( x − y ) + h | T {K i ( x ) ψ j ( y ) }| i , (2.16)which in momentum space has the form( /p − m ) b S ijF ( p ) = iδ ij Z − + b K ij ( p ) , (2.17)where b K ij ( p ) := F (cid:2) h | T {K i ( x ) ψ j ( y ) }| i (cid:3) . Since K i ( x ) := − gγ µ A aµ ( x )[ t a ψ ( x )] i transforms as aDirac spinor, b K ij ( p ) has an analogous spectral representation to b S ijF ( p ) b K ij ( p ) = i Z ∞ ds π he ρ ij ( s ) + /p e ρ ij ( s ) i p − s + iǫ + h e P ij ( ∂ ) + /p e P ij ( ∂ ) i δ ( p ) . (2.18)Inserting Eqs. (2.15) and (2.18) into Eq. (2.17), and separately equating the terms involvingderivatives of δ ( p ) which have support solely at p = 0, and the terms with support outside of p = 0, one obtains the equalities (cid:0) /p − m (cid:1) h P ij ( ∂ ) + /pP ij ( ∂ ) i δ ( p ) = h e P ij ( ∂ ) + /p e P ij ( ∂ ) i δ ( p ) , (2.19) (cid:0) /p − m (cid:1) i Z ∞ ds π h ρ ij ( s ) + /pρ ij ( s ) i p − s + iǫ = iδ ij Z − + i Z ∞ ds π he ρ ij ( s ) + /p e ρ ij ( s ) i p − s + iǫ . (2.20)In order to determine the relations imposed by Eq. (2.19), let e P ij ( ∂ ) = P r ˜ a ijr ( ∂ ) r and e P ij ( ∂ ) = P s ˜ b ijs ( ∂ ) s be the polynomial terms of the propagator b K ij ( p ). By equating theterms proportional to /p and the Dirac spinor identity, one obtains the following constraints onthe coefficients of P ij and P ij a ijn = m n n ( n + 1)! n ! a ij + n − X k =0 k ( k + 1)! k ! (cid:16) m ˜ a ijk + 4( k + 1)( k + 2)˜ b ijk +1 (cid:17) m k +1) , n ≥ b ijn = m n − n ( n + 1)! n ! a ij + n − X k =0 k ( k + 1)! k ! (cid:16) m ˜ a ijk + 4( k + 1)( k + 2)˜ b ijk +1 (cid:17) m k +1) − m ˜ b ijn , n ≥ a ijn and b ijn are completely determined by a ij and thecoefficients of the singular terms in b K ij ( p ). In particular, these relations imply that if the quarkpropagator contains a δ ( p ) term (i.e. a ij = 0), or singular terms are present in the propaga-tor b K ij ( p ), this is sufficient to ensure that the quark propagator must contain terms involvingderivatives of δ ( p ). In contrast, the coefficients of terms involving derivatives of δ ( p ) in the gluonpropagator are not affected by the presence or absense of δ ( p ) terms [15]. s with Eq. (2.19) one can perform the same matching procedure for Eq. (2.20), and in doingso one obtains the following equalities i Z ∞ ds π ρ ij ( s ) + i Z ∞ ds π h sρ ij ( s ) − mρ ij ( s ) i p − s + iǫ = iδ ij Z − + i Z ∞ ds π e ρ ij ( s ) p − s + iǫ , (2.23) i Z ∞ ds π h ρ ij ( s ) − mρ ij ( s ) i p − s + iǫ = i Z ∞ ds π e ρ ij ( s ) p − s + iǫ . (2.24)Using the fact that ρ ij ( s ) satisfies the integral condition in Eq. (2.14), Eqs. (2.23) and (2.24)imply the spectral density constraints sρ ij ( s ) − mρ ij ( s ) = e ρ ij ( s ) , (2.25) ρ ij ( s ) − mρ ij ( s ) = e ρ ij ( s ) , (2.26)which can be rewritten in the form (cid:0) s − m (cid:1) ρ ij ( s ) = m e ρ ij ( s ) + s e ρ ij ( s ) , (2.27) (cid:0) s − m (cid:1) ρ ij ( s ) = e ρ ij ( s ) + m e ρ ij ( s ) . (2.28)As with the spectral densities of the gluon propagator, these distributional equations can beexplicitly solved [19], and have the following general solutions ρ ij ( s ) = A ij δ ( s − m ) + κ ij ( s ) , (2.29) ρ ij ( s ) = A ij δ ( s − m ) + κ ij ( s ) , (2.30)where the components κ ij ( s ) and κ ij ( s ) are particular solutions which satisfy the relations (cid:0) s − m (cid:1) κ ij ( s ) = m e ρ ij ( s ) + s e ρ ij ( s ) and (cid:0) s − m (cid:1) κ ij ( s ) = e ρ ij ( s ) + m e ρ ij ( s ) respectively. There-fore, κ ij ( s ) and κ ij ( s ) are completely determined by the spectral densities of b K ij ( p ).In order to fix the coefficients A ij and A ij , one must use the integral constraints on the variousspectral densities. In addition to Eq. (2.14), it turns out that e ρ ij ( s ) satisfies the sum rule Z ∞ ds e ρ ij ( s ) = 0 . (2.31)This sum rule is derived from the equal-time restricted anti-commutator correlator relation h | (cid:8) K i ( x ) , ψ j ( y ) (cid:9) | i x = y = 0 , (2.32)which itself follows from Eq. (2.12) and the fact that the gluon field A aµ has a vanishing vacuumexpectation value. Combining Eqs. (2.14) and (2.31) together with Eq. (2.26), finally gives ρ ij ( s ) = (cid:20) πm δ ij Z − − Z d ˜ s κ ij (˜ s ) (cid:21) δ ( s − m ) + κ ij ( s ) , (2.33) ρ ij ( s ) = (cid:20) πδ ij Z − − Z d ˜ s κ ij (˜ s ) (cid:21) δ ( s − m ) + κ ij ( s ) . (2.34)These equalities explicitly demonstrate that the quark spectral densities both contain a discretemass component. However, in contrast to the case of the gluon propagator [15], the coefficients infront of these components are not completely constrained, and depend explicitly on the integralsof κ ij ( s ) and κ ij ( s ). It is therefore not as clear-cut as to whether these mass components areactually present or absent in specific gauges. Non-perturbative constraints on the ghost propagator
As in the case of the quark propagator in Sec. 2, before deriving the general structural formof the ghost propagator in QCD, one must first determine the properties of an arbitrary ghostcorrelator and propagator.
Ghost C a and anti-ghost C a fields are anti-commuting scalar fields. From the general analysisin Sec. 2.1 it follows that the momentum space ghost correlator can be written b G ab ( p ) = F (cid:2) h | C a ( x ) C b ( y ) | i (cid:3) = P abC ( ∂ ) δ ( p ) + Z ∞ ds θ ( p ) δ ( p − s ) ρ abC ( s ) , (3.1)where P abC = P n g abn (cid:2) − ( x − y ) (cid:3) n is a polynomial of finite order. Taking the inverse Fouriertransform of this expression then leads to the following general representation of the positionspace correlator: h | C a ( x ) C b ( y ) | i = − Z ∞ ds π ρ abC ( s ) iD ( − ) ( x − y ; s ) + 1(2 π ) P abC (cid:2) − ( x − y ) (cid:3) . (3.2)The corresponding propagator for a general ghost field is defined by h | T { C a ( x ) C b ( y ) }| i := θ ( x − y ) h | C a ( x ) C b ( y ) | i − θ ( y − x ) h | C b ( y ) C a ( x ) | i , (3.3)where the minus sign arises because the fields are anti-commuting. Unlike the fermion propaga-tor, CPT symmetry cannot be used to directly relate the ghost h | C a ( x ) C b ( y ) | i and anti-ghost h | C b ( y ) C a ( x ) | i correlators with one another. The reason for this stems from the fact thatghost and anti-ghost fields transform as Lorentz scalars but are defined to be anti-commuting,which causes a violation of the CPT theorem [19]. The CPT operator Θ therefore does nottransform the ghost and anti-ghost fields into one another, and thus the corresponding cor-relators must be treated independently. Nevertheless, since the anti-ghost correlator has thesame distributional properties as the ghost correlator, the spectral representation has the samegeneral structure F (cid:2) h | C a ( y ) C b ( x ) | i (cid:3) = P abC ( ∂ ) δ ( p ) + Z ∞ ds θ ( p ) δ ( p − s ) ρ abC ( s ) , (3.4)where P abC is some finite order polynomial, and ρ abC ( s ) is the anti-ghost spectral density. More-over, since one defines the ghost and anti-ghost fields to be hermitian: C a ( x ) † = C a ( x ), C a ( x ) † = C a ( x ) [4], applying the hermitian operator to Eq. (3.4) and comparing this withEq. (3.2) implies the relations ρ abC ( s ) = (cid:2) ρ baC ( s ) (cid:3) † , P abC = (cid:2) P baC (cid:3) † . (3.5)Although the violation of CPT symmetry prevents the ghost and anti-ghost correlators beinglinearly related, the hermitian property of the fields implies that the ghost and anti-ghost spec-tral densities are hermitian conjugates of one another.Combining Eqs. (3.2) and (3.4) together with the definition of the propagator in Eq. (3.3), theghost propagator takes the following form h | T { C a ( x ) C b ( y ) }| i = − Z ∞ ds π ρ abC ( s ) i ∆ F ( x − y ; s ) + Z d p (2 π ) e − ip ( x − y ) P abC ( ∂ ) δ ( p ) − θ ( y − x ) Z ∞ ds π (cid:2) ρ abC ( s ) + ρ baC ( s ) (cid:3) iD (+) ( x − y ; s ) − θ ( y − x ) Z d p (2 π ) e − ip ( x − y ) (cid:2) P abC ( ∂ ) + P baC ( ∂ ) (cid:3) δ ( p ) . (3.6) ince the spectral densities ρ abC ( s ) and ρ abC ( s ) are only related via hermitian conjugation, onecannot simplify this expression further without additional constraints. Using the general spectral properties outlined in the previous section, one can now use thedynamical characteristics of BRST quantised QCD to derive explicit constraints on the structureof the QCD ghost propagator.
In BRST quantised QCD the renormalised ghost field C a satisfies the equation of motion ∂ C a = − igf abc ∂ ν ( A bν C c ) = L a , (3.7)together with the equal-time anti-commutation relations { C a ( x ) , C b ( y ) } x = y = 0 , (3.8) { ˙ C a ( x ) , C b ( y ) } x = y = δ ab e Z − δ ( x − y ) , (3.9)where e Z is the ghost renormalisation constant. Taking the vacuum expectation values ofEqs. (3.8) and (3.9), and applying Eq. (3.2), one obtains the conditions P abC = − P baC , (3.10) Z ∞ ds ρ abC ( s ) = 2 πiδ ab e Z − , (3.11) (cid:20)Z ∞ ds (cid:2) ρ abC ( s ) + ρ baC ( s ) (cid:3) D (+) ( x − y ; s ) (cid:21) x = y = 0 , (3.12) (cid:20)Z ∞ ds (cid:2) ρ abC ( s ) + ρ baC ( s ) (cid:3) ˙ D (+) ( x − y ; s ) (cid:21) x = y = 0 . (3.13)Since R ∞ ds h ρ abC ( s ) + ρ baC ( s ) i D (+) ( x − y ; s ) satisfies the Klein-Gordon equation, the solution ofthis distribution for unequal times is uniquely determined by the initial conditions in Eqs. (3.12)and (3.13) [19]. Furthermore, since this solution depends linearly on the initial conditions, bothof which are vanishing, this implies Z ∞ ds (cid:2) ρ abC ( s ) + ρ baC ( s ) (cid:3) D (+) ( x − y ; s ) = 0 . (3.14)Combining all of these constraints together with the representation in Eq. (3.6), the non-perturbative ghost propagator can then be written h | T { C a ( x ) C b ( y ) }| i = − Z ∞ ds π ρ abC ( s ) i ∆ F ( x − y ; s ) + Z d p (2 π ) e − ip ( x − y ) P abC ( ∂ ) δ ( p ) , (3.15)which in momentum space is given by b G abF ( p ) = i Z ∞ ds π ρ abC ( s ) p − s + iǫ + P abC ( ∂ ) δ ( p ) . (3.16)Since the ghost field transforms as a Lorenz scalar it is not surprising that the propagator has thesame overall structure as a scalar propagator. However, unlike with standard commuting scalarfields, the structure in Eq. (3.16) depends crucially on the equal-time anti-commutation relationsin Eqs. (3.8) and (3.9). Eq. (3.11) is equivalent to the sum rule satisfied by the gluon spectraldensity, which is proportional to the inverse of the gluon field renormalisation constant [15]. ince e Z − similarly vanishes in Landau gauge, the ghost spectral density therefore also obeys theOehme-Zimmermann superconvergence relation [24, 25]. As in the case of the interacting quarkpropagator, the potential appearance of singular terms in the ghost propagator is relevant forunderstanding confinement. In fact, this is particularly true for the ghost propagator, since theinfrared behaviour of this object plays a central role in the Kugo-Ojima confinement criterion [3,4, 5]. In an analogous manner to Sec. 2.3.2, one can determine the further conditions that the equationof motion [Eq. (3.7)] imposes on the structure of the ghost propagator by deriving the form ofthe Schwinger-Dyson equation, and then using this to separately constrain the singular and non-singular terms in the propagator. Combining Eqs. (3.7), (3.8) and (3.9) together with the generaldefinition of a ghost propagator in Eq. (3.3), one obtains the coordinate space Schwinger-Dysonequation ∂ h | T { C a ( x ) C b ( y ) }| i = δ ab e Z − δ ( x − y ) + h | T {L a ( x ) C b ( y ) }| i , (3.17)which in momentum space is given by − p G abF ( p ) = δ ab e Z − + L ab ( p ) , (3.18)where L ab ( p ) = F (cid:2) h | T {L a ( x ) C b ( y ) }| i (cid:3) . Since L a has the same Lorentz transformation prop-erties as C a , it follows that L ab ( p ) has an analogous spectral representation to Eq. (3.6). More-over, because one has the following equal-time restricted anti-commutator correlator relations h |{L a ( x ) , C b ( y ) }| i x = y = 0 , h |{ ˙ L a ( x ) , C b ( y ) }| i x = y = 0 , (3.19)the spectral representation of L ab ( p ) can be written in the same manner as for the QCD ghostpropagator L ab ( p ) = i Z ∞ ds π e ρ abC ( s ) p − s + iǫ + e P abC ( ∂ ) δ ( p ) , (3.20)where now the corresponding spectral density e ρ abC ( s ) instead satisfies the constraint Z ∞ ds e ρ abC ( s ) = 0 . (3.21)Inserting Eqs. (3.16) and (3.20) into Eq. (3.18), and separately equating the terms involvingderivatives of δ ( p ) and those with support outside of p = 0, one obtains − p (cid:2) P abC ( ∂ ) δ ( p ) (cid:3) = e P abC ( ∂ ) δ ( p ) , (3.22) − p (cid:20) i Z ∞ ds π ρ abC ( s ) p − s + iǫ (cid:21) = δ ab e Z − + i Z ∞ ds π e ρ abC ( s ) p − s + iǫ . (3.23)It follows from Eq. (3.22) that the coefficients g abn and ˜ g abn of the polynomials P abC and e P abC respectively, satisfy the following constraint g abn +1 = − ˜ g abn n + 1)( n + 2) , n ≥ . (3.24)Eq. (3.24) implies that the coefficients of the singular terms in the ghost propagator are com-pletely fixed by the coefficients of the singular terms in L ab ( p ). Therefore, if L ab ( p ) contains These relations follow from Eqs. (3.8) and (3.9), together with the fact that QCD fields have vanishing vacuumexpectation values. ither δ ( p ) or non-measure defining terms involving derivatives of δ ( p ), then this is sufficient toguarantee that the ghost propagator must contain non-measure defining terms.In order to determine the constraints imposed by Eq. (3.23) one can make use of the fact thatthis expression can be written in the form − i Z ∞ ds π ρ abC ( s ) − i Z ∞ ds π sρ abC ( s ) p − s + iǫ = δ ab e Z − + i Z ∞ ds π e ρ abC ( s ) p − s + iǫ . (3.25)Since the ghost spectral density satisfies the sum rule in Eq. (3.11), the above equality thereforeimplies the following constraint sρ abC ( s ) = − e ρ abC ( s ) . (3.26)Similarly to the quark spectral densities, one can solve this distributional equation in terms of ρ abC ( s ), and one obtains the solution ρ abC ( s ) = A ab δ ( s ) + κ abC ( s ) , (3.27)where the particular solution κ abC ( s ) satisfies the relation sκ abC ( s ) = − e ρ abC ( s ). By applying thesum rule in Eq. (3.11), the ghost spectral density can then finally be written ρ abC ( s ) = (cid:20) πiδ ab e Z − − Z ∞ d ˜ s κ abC (˜ s ) (cid:21) δ ( s ) + κ abC ( s ) . (3.28)Eq. (3.28) demonstrates that the ghost spectral density contains a discrete massless component.Similarly to the quark spectral densities, the coefficient in front of this discrete component is notcompletely constrained since it depends on the integral of κ abC (˜ s ), which itself is determined by e ρ abC ( s ). This feature is particularly for understanding confinement because it turns out that inorder to violate the cluster decomposition property in QCD, this requires both the appearanceof non-measure-defining terms in the correlators of coloured fields, such as derivatives of δ ( p ),and also that the full space of states V QCD has no mass gap [20, 21]. This second requirementis still consistent with the possibility that the physical subspace V phys ⊂ V QCD has a massgap, as one would expect in QCD [4]. In Landau gauge e Z − vanishes, and therefore the onlything preventing the absence of a massless ghost pole is the non-vanishing of R ∞ d ˜ s κ abC (˜ s ).This feature is in contrast to the case of the gluon spectral density, where the coefficient ofthe massless component is entirely propotional to Z − , which vanishes in Landau gauge, andtherefore prevents the appearance of a massless gluon state [15]. Since R ∞ d ˜ s κ abC (˜ s ) can inprinciple be non-vanishing, this preserves the possibility that V QCD has no mass gap, and thatthe cluster decomposition property can be violated for coloured states, which is a sufficientcondition for confinement [23].
Although the quark and ghost propagators play an important role in QCD, the general ana-lytic structure of these objects remains largely unknown. In this work we demonstrate thatthe dynamical properties of the quark and ghost fields, and in particular their correspondingSchwinger-Dyson equations, impose non-perturbative constraints on these propagators. For thequark propagator it turns out that these constraints imply that both spectral densities neces-sarily contain massive components proportional to δ ( s − m ), and that the presence of singularterms in the propagator involving derivatives of δ ( p ) are permitted. In the case of the ghostpropagator the corresponding spectral density is constrained to contain a massless componentproportional to δ ( s ), and the appearance of singular terms is also similarly permitted. Thepotential presence of a non-vanishing massless component in the ghost spectral density, andsingular terms in the quark and ghost propagators, are of particular importance in the contextof confinement. Besides the purely theoretical relevance of these results, these constraints couldalso provide important input for improving existing parametrisations of the QCD propagators. cknowledgements This work was supported by the Swiss National Science Foundation under contract P2ZHP2 168622,and by the DOE under contract DE-AC02-76SF00515.
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