Extracting the q ¯ q condensate for light quarks beyond the chiral limit in models of QCD
aa r X i v : . [ h e p - ph ] A p r Extracting the qq condensate for light quarksbeyond the chiral limit in models of QCD R. Williams , C. S. Fischer , , and M. R. Pennington Institute for Particle Physics Phenomenology, Durham University, Durham DH1 3LE, UK and Institut f¨ur Kernphysik, Darmstadt University of Technology,Schlossgartenstraße 9, 64289 Darmstadt, Germany
It has recently been suggested [1] that a reliable and unambiguous definition of the non-perturbative massive quark condensate could be provided by considering a non positive-definiteclass of solutions to the Schwinger Dyson Equation for the quark propagator. In this paper we showthat this definition is incomplete without considering a third class of solutions. Indeed, studyingthese three classes reveals a degeneracy of possible condensate definitions leading to a whole rangeof values. However, we show that the physical condensate may in fact be extracted by simple fittingto the Operator Product Expansion, a procedure which is stabilised by considering the three classesof solution together. We find that for current quark masses in the range from zero to 25 MeV or so(defined at a scale of 2 GeV in the MS scheme), the dynamically generated condensate increasesfrom the chiral limit in a wide range of phenomenologically successful models of the confining QCDinteraction. Lastly, the role of a fourth class of noded solutions is briefly discussed. PACS numbers: 12.38.-t, 11.30.Rd, 12.38.Aw, 12.38.Lg
I. INTRODUCTION
A remarkable aspect of strong coupling field theory isthe possibility that masses can be largely, or even en-tirely, created by interactions. It is by this mechanismthat a fermion mass gap is generated for the light flavoursin QCD, leading to dynamical chiral symmetry break-ing (DCSB). Indeed, the mass gap generated far exceedsthe scale of the current quark masses present in the La-grangian. The strong coupling induces long range qq cor-relations that polarise the vacuum. It is the scale of thesecondensates that determines the mass generated, and thisgap persists even in the chiral limit. As long known thisunderpins much of QCD phenomenology. In this paperwe investigate this mass generation using the Schwinger-Dyson equations. The aim is to extract the behaviour ofthe qq condensate beyond the limit of zero quark mass.The interest in the value of such a condensate arisesin the context of QCD sum-rules. There the OperatorProduct Expansion (OPE) is used to approximate theshort distance behaviour of QCD. In studying currentslike that of q i γ µ ( γ ) q j , with q i = s and q j = u, d , thevacuum expectation values of uu , dd and ss operatorsnaturally arise [2, 3, 4]. In the chiral limit, the value ofthe qq condensate for the u and d quarks is well deter-mined to be − (235 ±
15 MeV) by experiment — in par-ticular from the low energy behaviour of ππ scattering [5].However, in the OPE it is the value of the condensatesaway from the chiral limit that actually enters. Sincethe current masses of the u and d quarks are only a fewMeV, the resulting condensate is expected to be close toits value in the chiral limit, but how close? For the first20 years of QCD sum-rules their accuracy was never suf-ficient for it to matter whether this difference was a fewpercent, 10% or even 20% effect. This equally applied tothe estimate by Shifman, Vainshtein and Zakharov [6, 7] that the ss condensate was (0 . ± .
3) of the uu and dd values. It is the greater precision brought about by thestudies of Refs. [4, 8, 9, 10], for instance, that motivatethe need to learn about how the qq condensate dependson the current quark mass. Indeed, Dominguez, Ram-lakan and Schilcher [8] compute the ss condensate to bejust (0 . ± .
1) times that for uu and dd . In the lightof a better understanding [11, 12, 13, 14] of strong cou-pling QCD how robust is this? First results were alreadypresented in Ref. [15].For light quarks, u , d and s , studying the Schwinger-Dyson equation for the fermion propagator in the contin-uum is essential, until computation with large lattice vol-umes become feasible. Since the continuum Schwinger-Dyson equations can be solved for any value of the quarkmass, they also provide a natural way to bridge the gapbetween lattice data at larger masses and the chiral limitof phenomenological importance. Our primary focus is,of course, on QCD, but we shall draw on the NJL modelwhere necessary. II. SCHWINGER-DYSON EQUATIONS
Solving the Schwinger-Dyson Equations for QCD is anentirely non-trivial process, see [11, 12, 13, 14] for re-views. They comprise an infinite tower of coupled inte-gral equations, that can be solved analytically only for (cid:0)1 = (cid:0)1 +
FIG. 1: Schwinger-Dyson equation for the quark propagator specific kinematical situations, for instance in the far in-frared and in the ultraviolet [16, 17, 18, 19]. In general,the solutions have to be found numerically and then onlyafter some form of truncation has been applied. To thisend we require some suitable ansatz for the three-pointfunctions in order to allow us to solve self-consistentlyfor propagators. In the following sections we will usea number of truncation schemes that have been testedelsewhere [20, 21, 22], within which we may solve forthe fermion propagator. Our chief aim is to calculatethe mass function of the quark propagator for a range ofcurrent masses. Our starting point is the renormalizedSchwinger-Dyson equation for the quark propagator asdepicted in Fig. 1: S − F ( p ) = Z h S (0) ( p ) i − − C F ˜ Z Z ˜ Z g (2 π ) Z d k × γ µ S F ( k )Γ ν ( k, p ) D µν ( p − k ) . (1)In the Landau gauge we can choose [23] ˜ Z = 1. Theinverse propagator S − F ( p ) is specified by two scalar func-tions A and M : S − F ( p ) = A ( p ) (cid:0) p + M ( p ) (cid:1) . (2)While A is also a function of the renormalisation point µ and so strictly A ( p , µ ), the quark mass function M ( p )is renormalisation group invariant. Projecting out thesetwo functions from Eq. (1), we have two coupled equa-tions to solve.The operator product for the mass function has anexpansion at large momenta given symbolically by: M ( p ) ≃ m ( p ) + const p h qq ( p ) i + · · · , (3)where the first term corresponds to the explicit mass inthe Lagrangian, and the second to the lowest dimensionvacuum condensate. For now we just show the momen-tum dependence given by the canonical dimensions, andleave for later the implications of the anomalous dimen-sions of QCD. Having computed the mass function usingthe Schwinger-Dyson equations, the essential problem ishow to separate these two terms in Eq. (3) with any ac-curacy if m is non-zero. We note that the mass functionfor a physically meaningful solution is expected to bepositive definite.In the chiral limit, there exist three solutions for themass function M ( p ). These correspond to the Wignermode (the only solution accessible to perturbation the-ory), and two non-perturbative solutions of equal magni-tude generated by the dynamical breaking of chiral sym-metry. These we denote by: M ( p ) = ( M W ( p ) = 0 M ± ( p ) = ± M ( p ) . (4) Such multiple solutions have been found in the contextof QED by Hawes et al. [24]. One can ask whetheranalogous solutions exist in QCD as we move away fromthe chiral limit and what relevance they hold. Indeed, ina recent paper [1] it was suggested that one could make anunambiguous definition of the massive quark condensateby taking a particular combination of these solutions.The existence of these is restricted to the domain: D = { m : 0 ≤ m ≤ m cr } , (5)where only the positive-definite solution exists beyond m cr . Chang et al. [1] found that inside the critical do-main, the solutions for both M + ( p ) and M − ( p ) exhibitthe same running current-quark mass in the ultraviolet.In terms of Eq. (3), this means both solutions have thesame m term. Noting that the M − ( p ) solution had acondensate of opposite sign, they proposed a definitionof the massive quark condensate given by: σ ( m ( µ )) = lim Λ →∞ Z ( µ, Λ) N c tr D Z Λ k d k (2 π ) ×
12 [ S + − S − ] . (6)where µ denotes the renormalisation point. Since bothpropagators S ± share the same asymptotic behaviour M ( p ) → m ( p ) for large momenta, potential diver-gences in the integral cancel and the expression Eq. (6)is well defined. The resulting condensate σ ( m ( µ )) will beequal to the one for the physical M + solution provided S ± have condensates of equal magnitude away from thechiral limit. However, we will show that this assumptionis not correct. Morever, we find here that there is in factan analogous solution S W to the Wigner mode of the chi-ral limit. (This has also been noted in the revised versionof [1].) This solution has the same ultraviolet behaviourof a running current quark mass, i.e. all three, M ± and M W , have a common m ( p ) in their OPE, see Eq. (3).Consequently, the combination S ( β ) = (2 − β ) S + − βS − + 2( β − S W (7)has its asymptotics controlled by the second term of theOPE, Eq. (2), for any β . Thus, we can extend the def-inition of Chang et al. [1] to a family of condensatesparametrised by β : σ ( m ( µ ) , β ) = lim Λ →∞ Z ( µ, Λ) N c tr D Z Λ k d k (2 π ) S ( β ) , (8)where all S − , S + and S W are dependent upon the mo-mentum k , the renormalisation point µ and the quarkmass m ( µ ). The choice β = 1 corresponds to the defini-tion of Eq. (6), with β = 0 and β = 2 corresponding totwo other natural choices. In fact β could take any valuefrom −∞ to + ∞ and so we can have a whole range ofvalues for the massive quark condensate: all agreeing inthe chiral limit. Consequently, the definition proposedby Chang et al. is far from unique, and does not providea value for the condensate that corresponds to the phys-ical M + solution of interest. Eq.(6) merely defines thevalue for the difference of the condensates for M + and M − . However, we will show that when combined withthe OPE the 3 solutions will pick out a precise physical definition of this condensate. Before investigating this inthe context of models of the QCD interaction, we shalldraw analogy with the NJL model, within which a nat-ural definition of the massive quark condensate alreadyexists. III. THE NAMBU-JONA-LASINIO MODEL
Because of the complexity of QCD, it is often pru-dent to examine simpler systems exhibiting similar char-acteristics first. One such example is the Nambu-Jona-Lasinio (NJL) model. Though originally formulated todescribe nucleon interactions in the pre-QCD era, themodel can be reinterpreted by regarding the nucleons asquarks [25, 26]. It can then be used to study boundstates and so determine basic phenomenological quanti-ties of meson interactions. The NJL model shares thesame symmetry structure as QCD, and is dominated byDCSB effects at low energies too. The Lagrangian forthe NJL model with just two flavours of quarks with de-generate mass m = m u = m d is: L NJL = ψ ( x ) ( i D/ − m ) ψ ( x ) + L int . (9)The interactions are given by a four-fermion contactterm: L int = G π h(cid:0) ψ ( x ) ψ ( x ) (cid:1) + (cid:0) ψ ( x ) iγ τ a ψ ( x ) (cid:1) i , (10)with the two terms corresponding to the scalar and pseu-doscalar channels respectively. From this can be derivedthe so-called gap equation: m = m + i G π N c N f Z d p (2 π ) tr D S ( p ) , (11)where the trace is over spinor indices in D -dimensions.At this point one has a choice of how to regulate theintegrals. Since we are dealing with a non-renormalisableeffective theory, our results will depend upon the cut-offused. One may introduce a non-covariant cut-off in theEuclidean 3-momentum, or employ a variety of covariantregularisation schemes such as a four-momentum cut-off,proper time or Pauli-Villars. The four-momentum cut-off is most closely related to the scheme we will employ inthe later sections, so we choose this method. By insertingthe form of the propagator into Eq. (11) we arrive at m = m + m G π N c i Z Λ d k (2 π ) k − m , (12) m q [MeV] - < qq > / [ M e V ] G π =8.0 G π =9.0 G π =10.0 G π =11.0 G π =12.0 FIG. 2: The non-perturbative condensate as a function of thecurrent quark mass for a selection of NJL couplings G π . where we have introduced a cut-off Λ on the 4-momentum. Rotating to Euclidean space we obtain m = m + G π N c m π (cid:20) Λ − m log (cid:18) m (cid:19)(cid:21) . (13)The coupling G π and cut-off, Λ, are fixed by fitting toexperimental data. To do this we must calculate theorder parameter associated with the breaking of chiralsymmetry. Thus the chiral condensate is given by: h qq i = − i N c Z d k (2 π ) tr D S F ( k ; m ) , (14)and for non-zero current masses we define: h qq i = − i N c Z d k (2 π ) tr D [ S F ( k ; m ) − S F ( k ; m )] . (15)Working with explicitly massless quarks and employinga covariant cut-off in the Euclidean 4-momentum, thepion-decay constant is given by [27]: f π = N c m (cid:20) log (cid:18) m (cid:19) − Λ m + Λ (cid:21) . (16)By solving Eq. (14) and Eq. (16) simultaneously and de-manding that f π = 93 MeV and − h qq i / = 235 MeV,we obtain Λ = 0 .
908 GeV and m = 265 MeV. These canbe substituted into the mass gap equation of Eq. (13) andthus we can solve for the coupling, finding G π = 10 . m q [MeV] | < qq > / [ M e V ] M + M - M w FIG. 3: The quark condensate within the NJL model for thethree solutions M ± ,W as functions of quark mass. the behaviour of the condensate depends upon the cho-sen coupling. Indeed, it either increases to a maximumthen decreases, or is monotonically decreasing for largercouplings as the quark mass increases. Moreover, onefinds that the condensate can become vanishingly smallfor sufficiently large masses ( ∼
600 MeV, not shown inFig. 2).Now, within the NJL model we also have solutions cor-responding to M + , M − and M W away from the chirallimit within some domain. The extent of the domain de-pends on the parameters of the model. With the favouredchoice, m cr ≃
15 MeV in Eq. (5). One may then useEq. (15) to calculate the massive quark condensate forthese solutions individually, the result of which is shownin Fig. 3. Furthermore, using the definition, Eq. (8), givesa family of condensates, σ ( m ( µ ) , β ) , for which the resultsare shown in Fig. 4 for β = { , , } . What is clearlyevident is that the condensates for the three mass func-tions M ± , M W are not equal. Drawing analogy with theSchwinger-Dyson solutions for QCD, one should thereforenot expect the condensate of M ± to be equal in magni-tude as well as opposite in sign, as assumed by Chang etal. [1]. We show this to be the case in the next section. IV. PHENOMENOLOGICAL MODEL OF QCDINTERACTION
We now turn to QCD. Rather than solving for theghost and gluon system, one may employ some suitableansatz for the coupling which has sufficient integratedstrength in the infrared so as to achieve dynamical massgeneration. There have been many suggestions in theliterature [20, 28] which have been extensively studied. Following the lead of Maris et al. [20, 28], we will employan ansatz for g D µν ( p − k ) which has been shown to beconsistent with studies of bound state mesons. We willconsider other modellings in a later section. Since thissimple model assumes a rainbow vertex truncation, thesolutions are not multiplicatively renormalisable and sodepend on the chosen renormalisation point. For com-parison with earlier studies we take this to be µ = 19GeV. We will scale results by one loop running to 2 GeVin the modified momentum subtraction scheme relevantto the Maris-Tandy model. Thus we use: g π Z ˜ Z D µν ( q ) → α (cid:0) q (cid:1) D (0) µν ( q ) (17)where the coupling is described by: α (cid:0) q (cid:1) = πω Dq exp( − q /ω )+ 2 πγ m log (cid:18) τ + (cid:16) q / Λ QCD (cid:17) (cid:19) × (cid:2) − exp (cid:0) − q / (cid:2) m t (cid:3)(cid:1)(cid:3) , (18)with m t = 0 . , τ = e − ,γ m = 12 / (33 − N f ) , Λ QCD = 0 .
234 GeV . Note that we work in the N f = 0 limit first since inSect. VI we will investigate the mass dependence of thecondensate using a model derived from quenched latticedata [22]. The precise value of Λ QCD is irrelevent forour current study, and we choose the parameter set ω = m q [MeV] | < qq > / [ M e V ] σ(β=1)σ(β=0)σ(β=2) FIG. 4: The condensate defined by σ ( β ) within the NJLmodel for values β = { , , } as a function of quark mass. -4 -3 -2 -1 p (GeV ) -4 -3 -2 -1 M ( p ) G e V
100 MeV70 MeV50 MeV30 MeV10 MeV3 MeVchiral chiral 3 MeV30 MeV100 MeV
FIG. 5: Euclidean mass functions for different current masses,specified at µ = 19 GeV as labelled. The plot illustrates howon a log-log plot the behaviour dramatically changes betweena current mass of 0 and 3 MeV. These results are essentiallythe same as found by Maris and Roberts [28]. . D = 0 .
933 GeV in the range considered byRef. [29].Solutions are obtained by solving the coupled systemof fermion equations for A and M of Eq. (2), which wemay write symbolically as: A ( p , µ ) = Z ( µ, Λ) − Σ D ( p, Λ) , (19) M ( p ) A ( p , µ ) = Z ( µ, Λ) Z m m R ( µ ) + Σ S ( p, Λ) . The Σ S and Σ D correspond to the scalar and spinor pro-jections of the integral in Eq. (1). For massive quarks weobtain the solution M + by eliminating the renormalisa-tion factors Z , Z m via: Z ( µ, Λ) = 1 + Σ D ( µ, Λ) , (20) Z m ( µ, Λ) = 1 Z ( µ, Λ) − Σ S ( µ, Λ) Z ( µ, Λ) m R ( µ ) . The resulting momentum dependence for different valuesof m R are shown in Fig. 5. Our purpose is to define thevalue of the qq condensate for each of these.At very large momenta the tail of the mass functionis described by the operator product expansion Eq. (3).Crucially, for a given m R we obtain the M − and M W solutions by inserting the same Z and Z m found for the M + solution. This ensures that differences in the dynam-ics of the three systems do not influence the ultravioletrunning of the current-quark mass in the context of thesubtractive renormalisation scheme used by Maris andTandy. The iteration process is performed using New-ton’s method. For the solution M W this is mandatory,since it corresponds to a local maximum of the effectiveaction and is therefore not accessible using the conven-tional fixed point iteration scheme. A representative ex-ample of the solutions is shown in Fig. 6. -4 -3 -2 -1 p [GeV ] -0.6-0.4-0.200.20.40.6 M ( p ) [ G e V ] M + M - M w FIG. 6: Momentum dependence of the three solutions M ± ( p )and M W ( p ) for a quark mass m ( µ )=16 MeV, µ = 19 GeV. The value of the critical mass is model-dependent, andis summarised in Table I. Chang et al. [1] imbue thiscritical mass with some significance for the dynamics ofQCD. However, criticality does not feature in the physical solution M + , which exists for all values of m q . It onlyoccurs in the M − and M W solutions, which appear ina strongly model-dependent region. Consequently, wefind little evidence of criticality being important to themass generation in QCD. We will comment again on thiswhen we consider more sophisticated vertex structures ina later section.The definition S ( β ) of Eq. (7) gives for each β a massfunction, for which the first term in the OPE, Eq. (3)vanishes, and so is controlled entirely by the condensateterm. However, as with the analogous NJL model, wehave an infinite set of ambiguous definitions of the quarkcondensate, one for each value of β , each of which agrees N f ω m cr ( µ = 19 GeV) 38 34 16 [MeV]0 m cr ( µ = 2 GeV) 49 44 21 [MeV]4 m cr ( µ = 19 GeV) 35 31 16 [MeV]4 m cr ( µ = 2 GeV) 49 44 23 [MeV]TABLE I: How the critical mass that defines the domain of so-lutions Eq. (5) depends on the number of quark flavours, N f ,on the gluon range parameter ω , in the Maris-Tandy model.This critical mass is listed at two different renormalizationscales, 19 GeV of Ref. [1] and 2 GeV for ease of comparisonwith other works in a momentum subtraction scheme. m R ( µ = 1 GeV) [MeV] - < qq > / [ M e V ] σ(β=1)σ(β=0)σ(β=2) FIG. 7: Renormalisation point independent quark condensateas a function of m q as defined by Eq. (8) for three values of β , showing how they are quite different despite the solutionshaving the same running current-mass and being equal in thechiral limit. with the chiral condensate in the limit m q →
0. Thisambiguity seen in Fig. 7 arises because, although eachsolution exhibits the same leading logarithmic behaviourin the ultraviolet limit, the condensates for each are notequal in magnitude, cf.
Eq. (3). Indeed, the solutions M − , M W have negative condensates, but we cannot di-rectly use combinations of these mass functions to forma well-defined and unique condensate that coincides withthe true condensate contained within M + ( p ). V. EXTRACTING THE CONDENSATE
At very large momenta the tail of the mass function isdescribed by the operator product expansion of Eq. (3).For QCD, let us introduce the appropriate anomalousdimension factors explicitly, so that M ( p ) asym = m (cid:2) log (cid:0) p / Λ (cid:1)(cid:3) − γ m + 2 π γ m C p (cid:20)
12 log (cid:0) p / Λ (cid:1)(cid:21) γ m − . (21)where m is related to the quantity m R ( µ ) via some renor-malisation factors. This provides an excellent represen-tation of all our solutions. If we included the expressionto all orders then the scales Λ , and Λ would both beequal to Λ QCD . However, the leading order forms inEq. (21) absorb different higher order contributions intothe two terms and so Λ and Λ are in practice different,as we will discuss below. For large masses the conden-sate piece, C , is irrelevant and so it is the leading termthat describes the mass function well. In contrast in the chiral limit, m = 0 and so the second term of the OPEdescribes the behaviour of the mass function. This thenaccurately determines the scale Λ . Indeed, its value isequal to Λ QCD . We can then easily extract the renormal-isation point independent condensate,
C ≡ − h qq i , fromthe asymptotics — see Fig. 5 for the chiral limit.In this latter case, strictly in the chiral limit, we mayalso extract the condensate via: − h qq i µ = Z ( µ, Λ) Z m ( µ, Λ) N c tr D Z Λ d k (2 π ) S ( k, µ ) , (22)where h qq i µ is the renormalisation dependent quark con-densate. At one-loop, this is related to the renormalisa-tion point independent quark condensate: h qq i µ =
12 log µ Λ QCD ! γ m h qq i . (23)which we compare with the asymptotic extraction togood agreement.However, for small quark masses, where the conden-sate is believed to play a sizeable role, we cannot ap-ply Eq. (22), since it acquires a quadratic divergence, cf. Eq. (3). Indeed, it is the elimination of this that inspiredthe original Eq. (6) and later Eq. (8), which we have seenlead to a wholly ambiguous definition of the physical con-densate. Nevertheless, one can attempt to fit both termsof the OPE in Eq. (21) to the tail of the mass function, M + for instance. While a value for the condensate canthen be extracted, this procedure is not at all reliablebecause of the difficulty in resolving the two functions in m R ( µ =2 GeV) [ MeV] | < qq > | / [ M e V ] M + M - M w FIG. 8: Condensate extracted through simultaneous fitting ofthe three solutions to the fermion mass-function in the Maris-Tandy model with N f = 0 and ω = 0 . m R ( µ =2 GeV) [ MeV] | < qq > | / [ M e V ] M + M - M W FIG. 9: Condensate for Maris-Tandy Model with N f = 4, ω = 0 . the OPE from one another and in fixing the appropriatescales, Λ and Λ .It is to this last point which we now turn. Instead ofone single solution, we now have three solutions to thesame model, each with identical running of the current-quark mass (the first term in Eq. (21)) in the ultravioletregion and differing only by their values of the conden-sate. Thus it is possible to fit Eq. (21) simultaneously tothe three mass functions M ± , M W . The scales Λ andΛ are determined separately for each value of the cur-rent mass. Remarkably, within the given model, they areexactly the same for the range of quark masses we con-sider with Λ ∼ QCD and Λ ∼ Λ QCD respectively.The condensates C ± and C W are then determined in anaccurate and stable way. For this to work the solutionshave to be found to an accuracy of 1 part in 10 . This fit-ting is performed using a modified Levenberg-Marquardtalgorithm with appropriate weights added to give betterbehaviour at large momentum in accord with perturba-tion theory. The results for the phenomenological modelemployed here are given in Fig. 8. The error bars re-flect the accuracy with which the mass functions, repre-sentable by two terms in the OPE expression, Eq. (21),are separable with the anomalous dimensions specified.In contrast to the condensate defined by Eq. (6), we findthat in the limited mass range investigated, the conden-sate increases as a function of m q . At the critical point m cr ( µ = 2 GeV)=44 MeV, we find the ratio for the con-densate to the chiral limit with N f = 0 to be (Fig. 8): h qq i m =50 MeV / h qq i m =0 = 1 .
24 (24)To estimate the errors in this determination, we can formcombinations of these condensates in the same way as de-fined in Eq. (8), favourably reproducing the same results of Fig. 7. In Fig. 9 is a similar plot with N f = 4 and ω = 0 . m cr changes comparedwith Fig. 8.We see that within errors the condensate is found toincrease with quark mass. This rise at small masses wasanticipated by Novikov et al. [30] combining a pertur-bative chiral expansion with QCD sum-rule arguments.That the chiral logs relevant at very small m q are barelyseen is due to the quenching of the gluon and the rain-bow approximation of Eq. (1). As we will show in thenext section when we model more complex interactions,including matching with the lattice, this effect remainssmall. VI. MORE SOPHISTICATED MODELS OF QCDINTERACTION
We now consider the consequences of using more so-phisticated vertex structure for the quark-gluon interac-tion in the quark Dyson-Schwinger equation, Fig. 1. Thefirst framework we study is a truncation scheme intro-duced in [21, 31]. It involves replacing the bare quark-gluon vertex of Sect. IV with the Curtis-Pennington (CP)vertex [32], thus ensuring multiplicative renormalizabil-ity for the fermion propagator. In the Yang-Mills sectorof QCD ans¨atze for ghost and gluon interactions havebeen introduced, which enable a self-consistent solutionfor the ghost and gluon propagators. The second schemewe shall investigate is an ansatz for the quark-gluon ver-tex, which has been fitted to lattice results, and was pre-viously employed in Ref [22, 33].
Continuum studies: CP vertex
In this truncation scheme we use explicit solutions forthe Dyson-Schwinger equations for the ghost and gluonpropagators, given diagrammatically in Fig. 10. This sys-tem of equations has been solved numerically in [34]. Therelevant ans¨atze for the ghost-gluon and triple-gluon ver-tices have been discussed in the literature [14, 21, 34].The solutions for the ghost and gluon propagator D G ( p ) = − G ( p ) p , (25) D µν ( p ) = (cid:18) δ µν − p µ p ν p (cid:19) Z ( p ) p , (26)can be represented accurately by Z ( p ) = (cid:18) α ( p ) α ( µ ) (cid:19) δ R ( p ) , (27) G ( p ) = (cid:18) α ( p ) α ( µ ) (cid:19) − δ R − ( p ) , (28)with R ( p ) = c ( p / Λ Y M ) κ + d ( p / Λ Y M ) κ c ( p / Λ Y M ) κ + d ( p / Λ Y M ) κ , (29)with the scale Λ Y M = 0 .
658 GeV, the coupling α ( µ ) =0 .
97 and the parameters c = 1 .
269 and d = 2 .
105 inthe auxiliary function R ( p ). The quenched anomalousdimension γ of the gluon is related to the anomalous di-mension δ of the ghost by γ = − − δ and δ = − /
44 for N f = 0. The infrared exponent κ = (93 − √ / ≈ .
595 [16]. The running coupling α ( p ) is defined via thenonperturbative ghost-gluon vertex, α ( p ) = α ( µ ) G ( p ) Z ( p ) (30)and can be represented by α ( p ) = 11 + p / Λ Y M (cid:20) α (0) + p / Λ Y M × πβ (cid:18) p / Λ Y M ) − p / Λ Y M − (cid:19)(cid:21) . (31)The value α (0) ≈ . /N c is known from an analyticalinfrared analysis [16].In the quark-DSE Eq. (1) we use the solution ofEq. (27) for the gluon propagator together with an ansatzof the formΓ ν ( q, k ) = V Abelν ( p, q, k ) W ¬ Abel ( p, q, k ) , (32)where p and q denote the quark momenta and k the gluonmomentum. The ansatz factorises into an Abelian andan non-Abelian part which are specified and discussedin detail in [21]. Here we only need to remark that theAbelian part V abelν is identical to the CP vertex [32]. Thisconstruction carries further tensor structure in addition − = − - 12 - 12 - 16- 12 + − = − - FIG. 10: Dyson-Schwinger equations for the gluon and ghostpropagator. Filled circles denote dressed propagators andempty circles denote dressed vertex functions. to the γ µ -piece, which makes it an interesting ansatz incomparison with the simple model in Sect. IV. The cor-responding numerical solutions for the quark propagatorare discussed in detail in [21]. Here we are only inter-ested in the chiral condensate as a function of the currentquark mass. The corresponding results can be found inFig. 11. Despite the complicated tensor structure of thevertex Eq. (32) we find similar results for the condensateas previously. We were able to extract the condensatefrom all three solutions M ± and M W , again for a re-stricted region of m q < m cr . The physical condensaterises again slightly for small current quark masses andbends down for larger ones. The critical value of m cr isfound to be 20 MeV at µ = 19 GeV. This corresponds to30 MeV at µ = 2 GeV. With the parameters of [21], thecondensate in the chiral limit is 270 MeV, rather thanthe phenomenological 235 MeV we have used. However,its dependence with quark mass hardly depends on thisexact value and so at the critical point we obtain theratio h qq i m =30 MeV / h qq i m =0 = 1 . . (33)As with the phenomenological model considered inSect. IV we find a considerable increase of the chiral con-densate with the current quark mass. Lattice Model
The third model we investigate has been defined in[22]. The idea is to solve the coupled system of gluon,ghost and quark Dyson-Schwinger equations on a com-pact manifold with periodic boundary conditions, similarto lattice QCD. For the vertices in the Yang-Mills sectorthe same truncation scheme as in the last section is em-ployed. However, for the quark-gluon vertex an ansatz m R ( µ =2 GeV) [ MeV] | < qq > | / [ M e V ] M + M - M W FIG. 11: Condensate for CP-Vertex Model. N f = 0 m R ( µ = 2 GeV) [ MeV] | < qq > | / [ M e V ] M + M - M w FIG. 12: Condensate for Lattice Model. N f = 0 has been specified such that lattice results for the quarkpropagator have been reproduced on a similar manifold.Solving the system also in the infinite volume/continuumlimit one can then study volume effects in the pattern ofdynamical chiral symmetry breaking [22, 33].In the infinite volume/continuum limit, i.e on R , thesolutions for the ghost and gluon propagator are given byEq. (27) and Eq. (29). The ansatz for the quark-gluonvertex isΓ ν ( k, µ ) = γ ν Γ ( k ) Γ ( k , µ ) Γ ( k , µ ) (34)with the componentsΓ ( k ) = πγ m ln( k / Λ QCD + τ ) , (35)Γ ( k , µ ) = G ( k , µ ) G ( ζ , µ ) e Z ( µ ) (36) × h [ln( k / Λ g + τ )] δ Γ ( k , µ ) = Z ( µ ) a ( M ) + k / Λ QCD k / Λ QCD , (37)where δ = − /
44 is the (quenched) one-loop anomalousdimension of the ghost, γ m = 12 /
33 the correspondinganomalous dimension of the quark and τ = e − h Λ g Λ QCD a a a (GeV) (GeV)overlap 1.31 1.50 0.35 25.58 3.44 2.23TABLE II: Parameters used in the vertex model, Eqs. (34-37). as a convenient infrared cutoff for the logarithms. Thequark mass dependence of the vertex is parametrised by a ( M ) = a a M ( ζ ) / Λ QCD + a M ( ζ ) / Λ QCD , (38)where M ( ζ ) is determined during the iteration processat ζ = 2 . g , Λ Y M and Λ
QCD . Inthe continuum this results in a failure to reproduce theperturbatively determined anomalous dimensions in theOPE, Eq. (21). The extraction of the tiny condensateterm in the OPE is very sensitive to these anomalousdimensions. Combined with the effect the uncertaintiesin determining the multiple scales has on Λ and Λ inEq. (21) leads here to much larger errors than in our pre-viously modellings. Nevertheless, for the massive con-densate we again find solutions similar to the previoussections. The condensate corresponding to the M ± solu-tions are given in Fig. 12 and the critical value m cr is 22MeV, see Table III. The large errors point to the needfor further studies of matching lattice on a torus to thecontinuum, if we are to extract reliable infinite volume,continuum quantities like the quark condensate. Thatis for the future. The ratio of the condensates is here h qq i m =22 MeV / h qq i m =0 = 1 .
17. This is a little lowerthan for the previous models of Eqs. (24,33). However,uncertainties in extraction are considerably larger. Nev-ertheless the ratio is still bigger than one.
Model CP Lattice m cr ( µ = 19 GeV) 20 16 [MeV] m cr ( µ = 2 GeV) 30 22 [MeV]TABLE III: The critical mass for our quenched CP and Lat-tice model. VII. CONDENSATE BEYOND THE CRITICALMASS RANGE: NODED SOLUTIONS
We see in Figs. 8, 9 that the M − and M W solutionsbifurcate below m cr ≃ . .
0) MeV with ω = 0 . N f = 0(4) respectively. But what about the valueof the condensate for the physical solution M + beyondthe region where M − and M W exist, i.e. m R ( µ ) > m cr ?Having accurately determined the scales Λ and Λ in theOPE of Eq. (21) in the region where all 3 solutions exist,we could just continue to use the same values in fitting0 -6 -4 -2 p [GeV ] -0.5-0.4-0.3-0.2-0.100.10.20.30.40.50.6 M ( p ) [ G e V ] M + M - M w M w noded FIG. 13: Momentum dependence of the 4 solutions for thefermion mass-function in the Maris-Tandy model with m = 20MeV at µ = 19 GeV, N f =4, ω = 0 . the physical M + solution alone and find its condensate.Unfortunately, this would make it difficult to producerealistic errors as the quark mass increases.However, as soon as one allows for solutions for thefermion mass-function that are not positive definite, oneexposes a whole series of variants on the solutions M − , M W we have already considered. Thus there are noded solutions, which have also been discovered recently in thecontext of a simple Yukawa theory by Martin and Llanes-Estrada [37]. These noded solutions are only accessible ifsufficient numerical precision is used. For instance, repre-senting the dressing functions by a Chebyshev expansiondoes not provide the required accuracy with sufficientsmoothness to reproduce the OPE form of Eq. (21). Weillustrate this within the Maris-Tandy model, for instancewith N f = 4 and ω = 0 .
4, in Fig. 13. There the four so-lutions we have found are displayed.It is interesting to note that this noded solution is notlimited to the same domain that restricts M − and M W .These noded solutions do develop a singularity in M ( p )beyond m = 51 . µ = 2 GeV. However, thisis compensated for by a zero in A ( p ), Eq. (2), until m = 66 . M + solution, as far as m = 66 . and Λ of Eq. (21) as fixed for m < m cr arestill well-determined by our fit procedure for m > m cr and so deduce the condensates. Indeed, fitting the M + and M Wnoded at each value of m R ( µ ) with common scalesin the OPE equation, Eq. (21) allows the condensate forthe physical solution to be found for much larger quarkmasses, as shown in Fig. 14. Indeed, these fits confirm that Λ and Λ are independent of m R ( µ ). We can thenfit the remaining M + solutions shown in Fig. 5 to givethe physical condensate shown in Fig. 14 for acceptablevalues of ω as determined by [29].In Fig. 15 we scale the quark mass from µ = 2 GeV inthe (quark-gluon) MOM scheme by one loop running tothe M S scheme at 2 GeV using the relationship betweenΛ
MOM and Λ MS for 4 flavours deduced by Celmasterand Gonsalves [38]. In this latter scheme the strangequark mass is ∼
95 MeV as given in the PDG Tables [39].Within the range of the Maris-Tandy modelling of strongcoupling QCD, we find the ratio of the condensates at thestrange quark mass to the chiral limit is h qq i / m ( MS ) =95 MeV / h qq i / m =0 = ( 1 . ± . . (39)in a world with 4 independent flavours. Moreover, hereall the quarks have the same mass and there is no mixingbetween different hidden flavour pairs. Elsewhere we willillustrate the change that occurs in solving the quarkSchwinger-Dyson equations with 2 flavours of very smallmass m u,d and 1 flavour with variable mass. Of course,in the quenched case quark loops decouple and exactlyreplicate the results given here.What we have shown here is that there is a robustmethod of determining the value of the qq condensatebeyond the chiral limit based on the Operator ProductExpansion. Of course, as the quark mass increases thecontribution of the condensate to the behaviour of themass function, Fig. 5, becomes relatively less importantand so the errors on the extraction of the physical con-densate increases considerably. Nevertheless, the methodis reliable up to and beyond the strange quark mass. Al-ternative definitions are not. m R ( µ =2 GeV) [ MeV] | < qq > | / [ M e V ] M + M - M W M W noded FIG. 14: Current quark mass dependence of the condensatesfor Maris-Tandy model with N f = 4, ω = 0 . m MS ( µ =2 GeV) [ MeV] | < qq > | / [ M e V ] M + , ω =0.40 GeVM + , ω =0.45 GeVM + , ω =0.50 GeV FIG. 15: Condensate for Maris-Tandy Model with N f = 4, ω = 0 . , . , . MS scheme. While the existence of multiple solutions to the fermionSchwinger-Dyson equation is essential for our method,only the M + solution has any physical significance andthe others are mathematical curiosities. Those, like M − , M W and M Wnoded , only exist in restricted domains. Incontrast, the physical solution exists for all current quarkmasses, even if we cannot reliably extract the value of thecorresponding condensate from the OPE. While it is clearthat the radius of convergence of the chiral expansion of M + [40, 41] in terms of the quark mass has a scale oforder of Λ QCD or equally ( −h qq i ) / , is this scale set by m cr of Eq. (5)? Chang et al. claim it is [1]. However,the bifurcation point for the unphysical solutions differswhether they are noded and not, cf. Figs. 9 and 14, andin turn each is highly model-dependent. This makes itdifficult to claim that the value of m cr of Eq. (5) is thekey parameter of the radius of the convergence of thechiral expansion for M + . VIII. SUMMARY
Within the NJL model and the Schwinger-Dyson ap-proach to QCD, we have investigated the three inequiva- lent solutions, called M + , M − and M W , to the massgap equation that exist within the interval D ( m ) = { m : 0 ≤ m ≤ m cr } . By ensuring each were solved usingthe same renormalisation conditions, we found that eachsolution exhibited the same running of the current-quarkmass in the ultraviolet: differentiated solely by their in-frared behaviour and value of the quark condensate.Though it was not possible to define the condensateunambiguously by simply taking combinations of M + , M − and M W , the increased information available on thedomain D ( m ) by having three solutions permits a reliableextraction of the condensate through simultaneous fittingof these to the OPE. In addition, we were able to obtaina fourth (noded) solution. This is only possible if theequations are solved to high numerical accuracy. Thoughthis fourth solution violates the physical requirement ofpositivity, it has allowed us to extract the condensatebeyond m cr and into the region of the physical strangequark mass.We have investigated a number of models for the strongcoupling (infrared) behaviour of the quark-gluon inter-action of QCD and found in all cases that the con-densate, corresponding to the solution with a positive-definite mass function, increases moderately with currentquark mass in the region under consideration. This is incontrast to the QCD sum-rule calculations of Refs. 6-8.Typically we find an increase of 30% from the chiral limitto a current M S mass of 100 MeV at a scale of 2 GeV.Only at still larger quark masses does the condensatesignificantly decrease.
Acknowledgments
RW is grateful to the UK Particle Physics and As-tronomy Research Council (PPARC) for the award ofa research studentship. CSF acknowledges a HelmholtzYoung Investigator award VH-NG-332. We thank Ro-man Zwicky and Dominik Nickel for interesting discus-sions. This work was supported in part by the EURTN Contracts HPRN-CT-2002-00311, “EURIDICE”and MRTN-CT-2006-035482, “FLAVIAnet”. Two of us(CSF and MRP) wish to thank ECT* and the organisersof the Workshop on Quark Confinement in Trento, wherethis work was completed. [1] L. Chang, Y.-X. Liu, M.S. Bhagwat, C.D. Robertsand S.V. Wright, Phys. Rev.
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