Are the Dressed Gluon and Ghost Propagators in the Landau Gauge presently determined in the confinement regime of QCD?
AAre the Dressed Gluon and Ghost Propagators in the Landau Gaugepresently determined in the confinement regime of QCD?
M.R. Pennington a and D.J. Wilson b a Theory Center, Thomas Jefferson National Accelerator Facility,12000 Jefferson Avenue, Newport News, Virginia 23606, USA. b Argonne National Laboratory, Argonne, Illinois 60439, USA. (Dated: September 10, 2018)The Gluon and Ghost propagators in Landau gauge QCD are investigated using the Schwinger-Dyson equation approach. Working in Euclidean spacetime, we solve for these propagators usinga selection of vertex inputs, initially for the ghost equation alone and then for both propagatorssimultaneously. The results are shown to be highly sensitive to the choices of vertices. We favorthe infrared finite ghost solution from studying the ghost equation alone where we argue for aspecific unique solution. In order to solve this simultaneously with the gluon using a dressed-one-loop truncation, we find that a non-trivial full ghost-gluon vertex is required in the vanishing gluonmomentum limit. The self-consistent solutions we obtain correspond to having a mass-like term inthe gluon propagator dressing, in agreement with similar studies supporting the long held proposalof Cornwall.
PACS numbers: 12.38.Aw, 12.38.Lg, 14.70.Dj
I. INTRODUCTION
The Gluon Schwinger-Dyson equation and its gaugefixing counterpart, the Ghost Schwinger-Dyson equation,are principal tools in investigating the theoretical prop-erties of strongly coupled QCD. The transition from theweak coupling regime at large momenta, where pertur-bation theory applies, down to strong coupling at smallmomenta gives rise to the key non-perturbative phe-nomena of confinement and dynamical chiral symmetrybreaking. The latter endows quarks, and perhaps glu-ons, with mass. These are not physical pole masses, butrather effective Euclidean masses dynamically generatedby non-perturbative interactions [1, 2]. The main reasonfor studying these propagators is that they are a nec-essary input for the Bethe-Salpeter (BSE) and Faddeevequations used for calculating physical quantities, suchas hadron masses and form-factors [3, 4].The infinite tower of Schwinger-Dyson equations(SDEs), and its necessary truncations in QCD, can bea troublesome beast. This paper is intended to stimulatediscussion about the assumptions commonly made andthe solutions to which these lead. We believe that im-portant issues remain unaddressed and these are outlinedbelow.The solutions provided by the SDEs may be comparedwith the results of complementary techniques, such asLattice QCD. A benefit of this is that the theoreticalstrengths and weaknesses of each method fall in differentareas. Both may be formulated in precisely the same way,using the same gauge and freely varying the quark con-tent. While lattice calculations are inevitably restrictedto larger quark mass, an advantage of the SDE methodis that there are no major computational obstacles to ex-tending these to the physical light quarks with masses of O (MeV).The paper is organised as follows, in Sec. 2, we intro- duce the Schwinger-Dyson equations for the gluon andghost system and proceed to give details regarding trun-cations. In Sec. 3 we solve the ghost equation for simpletruncations and show that solutions may be robustly ob-tained for a wide range of vertex inputs. In Sec. 4 we out-line the gluon equation and some of the issues we deemimportant that deserve immediate attention. We presentsimultaneous solutions for the gluon and ghost propaga-tors for a range of assumptions to indicate the level ofprecision currently available. In Sec. 5 we compare ourresults to those obtained using Lattice QCD simulations.Section 6 briefly gives our conclusions and outlook. II. SCHWINGER-DYSON EQUATIONS FORGLUONS AND GHOSTS
The full equation for the gluon propagator, in the ab-sence of quarks, is represented in Fig. 1. Without ap-proximations, this is given by, D abµν ( p ) − = D ab, (0) µν ( p ) − + Π ab c µν ( p ) + Π ab g µν ( p )+ Π ab g µν ( p ) + Π ab g µν ( p ) + Π ab g µν ( p ) , (1)where latin indices denote colour and greek letters de-note the Lorentz suffixes. The loop integrations Π( p )are labeled by the number of particles in the loop with c corresponding to ghosts and g corresponding to glu-ons. Eq. (1) is represented graphically in Fig. 1. Themomenta flowing through the internal propagators arearbitrary. However we use a symmetric routing for theone-loop diagrams with (cid:96) ± = (cid:96) ± p/
2, where (cid:96) is the loopintegration momentum and p is the external momentumthat is divided equally through each propagator. Whenworking with a large finite cutoff as is often done numer-ically, then a symmetric arrangement is often essential topreserve translation invariance. a r X i v : . [ h e p - ph ] D ec − = − + → p → p ++++ → p → p → ℓ → ւ ℓ ℓ ր ℓ →→ ℓ ℓ ℓ →տ ℓ →← ℓ − ℓ + →← ℓ − ℓ + → p → p → p → p → p → p → p → p FIG. 1. The Schwinger–Dyson Equation for the Gluon in theabsence of quarks.
We work in covariant gauges. Then neglecting quarks,the gluon equation only couples to ghosts, and these havetheir own Schwinger-Dyson equation, which is depictedin Fig. 2 and given by, D ( p ) − = D (0) ( p ) − + Π gc ( p ) , (2)where Π gc contains just one simple loop integration.Solving these equations depends non-trivially on thevertices, which in turn depend on higher n -point Green’sfunctions and therein lies the key issue. Truncation ofthis sequence of dependent Green’s functions is necessaryto be able to make predictions. However truncating hasthe potential to introduce errors and violate the physicalproperties of the theory. A. Sketching the general features of the equations
The interplay of the various terms in the equationsand identifying which terms are important will be ourprimary concern. For the most part we will adopt a trun-cation in which only one loop dressing of the gluon prop-agator will be considered. This is a favorite truncationscheme [5–11]. There are then four quantities of inter-est seen in Fig. 1: the two propagator dressings that theequations should determine, and the two unknown ver-tices that are required inputs. There are important inter-relations between these terms constrained by the struc- − = − + → p → p →← ℓ − ℓ + → p → p FIG. 2. The Schwinger–Dyson Equation for the Ghost. ture of the SDEs and the Ward-Slavnov-Taylor identities(WSTIs).The propagators are remarkably simple. The ghostand gluon dressings are just functions of p with the gluondressing multiplied by a tensor structure transverse to itsmomentum p . In the Landau gauge these are, D abµν ( p ) = δ ab G (cid:96) ( p ) p (cid:18) g µν − p µ p ν p (cid:19) (3) D ab ( p ) = − δ ab G h ( p ) p (4)where D µν is the full gluon propagator, D is the ghostpropagator. The a, b indices relate to the color carriedby each propagator. The functions G (cid:96) and G h are therespective dressing functions, for which we will be solvinglater. These functions contain all of the non-perturbativephysics of these two Green’s functions.The two propagator dressing functions are determinedfrom the various quantities that make up their equations,which are represented diagrammatically in Fig. 1 andFig. 2. The triple-gluon vertex only appears in the gluonequation. However the two propagators and the ghost-gluon vertex appear in both equations. While the dress-ing functions are not expected to change sign over thewhole Euclidean momentum region, the different contri-butions from the loop diagrams can have different signs,and indeed they do. The ghost-loop has an additional( −
1) owing to the antisymmetry of the ghost field.The gluon propagator dressing determined in the mo-mentum subtraction scheme is given by, G (cid:96) ( p ) − = G (cid:96) ( µ ) − + Π c ( p , µ ) + Π g ( p , µ ) (5)where µ is the subtractive-renormalization point,Π c ( p , µ ) is the ghost-loop contribution subtractedfrom itself at µ , similarly for Π g ( p , µ ) which is thegluon loop contribution. The precise details may befound in the appendices.In the definitions we use, the gluon propagator dress-ing function G (cid:96) ( p ) is positive everywhere. However, thedressed-gluon-loop diagram gives a negative contributionin the momentum subtraction scheme, below the sub-traction point. This is important because it tells us thatthe two other terms on the right of Eq. (5), G (cid:96) ( µ ) andΠ c ( p , µ ) must add to give an overall positive contribu-tion of greater magnitude than that of Π g ( p , µ ). Nowin the perturbative region G (cid:96) ( µ ) = 1 is more than largeenough to ensure that G (cid:96) ( p ) remains positive, but aswe evolve down to infrared (IR) momenta, then the twoone-loop contributions are both large. Quite genericallyfor many sensible vertices, the ghost-loop term must thengive a large positive contribution in order to be able tosolve the equation. A dichotomy of explanations exist:the large contribution either arises from a singular ghostpropagator, or a non-trivial ghost-gluon vertex. Despitethis distinction, little change is observed in the gluondressing function since the overall contribution from theghost loop remains similar.The full ghost-gluon vertex has two structures thatmay be written as follows,Γ µ ( k, p, q ) = igf abc [ q µ α ( k, p, q ) + k µ β ( k, p, q ) ] , (6)where the momenta are defined in Fig. 3 and α is the non-perturbative function dressing the outgoing ghost mo-mentum and β is the non-perturbative function dressingthe outgoing gluon term. This can be arranged in a num-ber of ways since the momenta are related by k = p − q .In the ghost equation the term β does not contribute inLandau gauge, however in a properly projected gluon the β term can be important. It appears in the ghost-loop,as we emphasise later.The ghost-gluon scattering kernel ˜Γ µν is an importantquantity since it appears in dressing both vertices. It isdirectly related to the ghost-gluon vertex,Γ µ ( k, p, q ) = igf abc q ν ˜Γ νµ ( k, p, q ) . (7)Hence the bare version is ˜Γ µν = g µν in order to yieldthe tree-level vertex. The same function also appears inthe WSTI for the triple-gluon vertex, which is a featureof the interconnections between terms in a gauge theory.The precise relation will probably be necessary to removeall gauge-dependence from the results. We do not givefurther details here as these are fully covered in [6, 12]and references therein. B. Gauge fixing
We work in Landau gauge since it happens to be themost theoretically appealing for several reasons. An obvi-ous and oft-suggested alternative is to use a gauge definedby an axial vector n µ , where ghosts do not appear. How-ever these have other drawbacks, not the least of whichis the breaking of Lorentz symmetry, and in particularin numerical studies where 1 / ( n.p ) terms are a complica-tion. Additionally, direct comparison to Lattice QCD isless straightforward in these gauges. The radiative cor-rections to the Gluon propagator are always transverse inthe external momentum due to the Slavnov-Taylor iden-tity, and in the Landau gauge the Gluon itself is alsotransverse, which is a simplifying feature. Furthermore,there are theoretical statements regarding confinement,gauge fixing and the vanishing momentum limit of QCD.These are usually formulated in Landau gauge and somedetails are given below.Ultimately we hope that these methods will be usefulin making physical predictions where the gauge depen-dence must drop out as in perturbation theory. However,the gluon propagator dressing function is a gauge depen-dent object and one expects it will combine with all ofthe other gauge dependent functions in making gauge-independent physical observables.A deeper analysis of the gauge fixing procedure is re-quired on theoretical grounds, since the Faddeev-Popov procedure may not be sufficient non-perturbatively. Ear-lier studies by Gribov, Kugo, Ojima, and Zwanziger havereceived refinement in recent years in light of the seriousdebate over the behavior of the propagators in the van-ishing momentum limit [13–16]. Formulated in Landaugauge, the Gribov-Zwanziger gauge fixing procedure con-siders the technical issues related to gauge field Gribovcopies and restriction of the gauge field space to selectone representative configuration. The broad conclusionat present is that the dressed gluon propagator carryingmomentum p is suppressed as p vanishes, with respectto its bare counterpart. Moreover, many studies finda propagator that goes to a constant, or equivalently apropagator dressing function that is proportional to p in the small p limit.Coupled with the earlier conclusions of Kugo andOjima [17] regarding a singular ghost dressing being asignal of confinement, SDE practioners were led to searchfor a particular class of solution where the leading powersin the IR of the ghost and gluon propagators are directlylinked. More details of this class of solution will be out-lined below, although the key feature is the singular ghostdressing function. Recently, new studies combining thesetwo earlier studies, and also additional refinements usingStochastic quantization techniques, have found that a fi-nite ghost dressing solution is certainly allowed and mayeven be preferred [18–22]. C. Existing Truncation Schemes and Solutions
We now turn to the act of truncation and briefly sum-marise some of the ideas previously considered for theseequations. Most truncation schemes for solving the gluonequation are motivated by the practicalities of findingsolutions. For example, in all but one of the referencesin the literature [23] the dressed-two-loop diagrams ofFig. 1 are entirely neglected. This is not due to someconvenient power counting scheme, but rather becausetheir inclusion is both numerically challenging and com-putationally intensive. Clearly in the perturbative ultra-violet (UV) regime, these terms are subleading. Howeverin the intermediate and vanishing momentum limit, theircontribution is essentially unknown. Na¨ıvely, one mightexpect that if the gluon is heavily suppressed in the smallmomentum limit then these terms may be unimportant.In this case of an IR suppressed gluon, the most impor-tant region for these terms would be the mid-momentumregion ( i.e. p ∼ p ), then the situation is lessclear. The answer depends on the vertex ans¨atze adoptedand the full solution of the gluon dressing function. Themid-momentum region is also the most physically rele-vant so eventually these terms must be properly included.A recent one-loop perturbative study, in which an effec-tive gluon mass term is artificially introduced, has foundgood agreement [24] with results from Lattice QCD. Thisdiffers from a typical SDE study since the mass is insertedby hand and iterations are not performed. Otherwise theequations are very closely related. In order to performiterations required for self-consistent solutions (see Ap-pendix A), some term in the gluon equation is required todynamically generate the mass term. In the SDEs, thistype of contribution could come from any of the loops inthe gluon equation including the two loop graphs.Nevertheless, we proceed by neglecting these two-loopdressing terms and turn to the remaining two unknownvertices that appear in the ghost and gluon equations.The ideal situation would be that the propagators de-pend only weakly on the ans¨atze for these vertices, andit is the propagator itself and the structure of the SDEthat determines the result. In studies in QED the situa-tion is indeed close to this ideal, since the Ward-Green-Takahashi identity determines the key vertex dressings interms of inverse propagators [25–30]. QCD is inevitablymore complicated. The Ward-Slavnov-Taylor Identitiesthat relate vertices and propagators do not admit suchsimple solutions as in QED, and for the full vertex thereis a complex interplay between the gluon and ghost sec-tors. ↑ k ← p ← q µ FIG. 3. The ghost-gluon vertex indicating the momentumdefinition we adopt, the outgoing ghost momentum is q andthe gluon momentum is k . In QCD the starting point has often been bare ver-tices, or vertices dressed by simple ratios of dressing func-tions [3, 5, 8, 31–34]. In particular, one option is to usethe bare ghost-gluon vertex in place of the full vertex,Γ abcµ ( k, p, q ) → Γ (0) abcµ ( k, p, q ) = igf abc q µ , (8)which at first sight seems like an over-simplification, sincethis cannot be correct even at the perturbative one-looporder [35, 36], let alone non-perturbatively. However, itis expected that the ghost-gluon vertex should be rela-tively simple in the Landau gauge, by appealing to Tay-lor’s non-renormalization theorem [37] which is valid incarefully chosen renormalization schemes [38, 39]. Thefull vertex is also expected to reduce to its bare form forvanishing incoming ghost momenta [7, 37, 38],lim p → Γ abcµ ( k, p, q ) = Γ (0) abcµ ( k, p, q ) , (9)the momentum definition used throughout being indi-cated in Fig. 3. This simple vertex appears to work for the infinite IR ghost dressing function [8, 23]. However inthe finite solutions a problem arises. In practical terms,the large contribution from the infinite ghost must be re-placed by a large contribution from the vertices in orderfor the gluon dressing function to remain positive, andfor self-consistent solutions to be obtained. There aretwo forms given in the literature that accomplish this.Firstly, a vertex that is transverse to the gluon momen-tum in the IR is suggested as a possible solution [7, 11].This is motivated by solutions in which the ghost is dom-inant in the IR region. Then the ghost loop controls cor-rections to the gluon propagator, see Fig. 1, which mustthen be transverse on its own in this regime. This canbe achieved by making the ghost-gluon vertex transverseto the gluon momentum in the IR. A form that achievesthis is [7, 11],Γ (1) µ = igf abc (cid:18) q µ − k µ k.qk F IR ( k, p, q ) (cid:19) . (10)where F IR is a smoothed step function defined inEq. (B15) that switches on this behavior in the IR, butensures it does not affect the UV. F IR = 1 in the IRand must vanish in the UV in order to reproduce theperturbative results. To achieve this, the function F IR contains a free parameter that controls the momentumof the switchover point. This has to be fixed and somesensitivity to its value will be present in the solutions.This extra term has no effect in the ghost equation, wherethe transverse gluon is contracted with this vertex, andso it automatically drops out. It can however appear inthe gluon equation as we demonstrate below. The super-script label (1) in Eq. (10) will be used to refer to thisvertex later.The second form considered in the literature involvesinserting massless poles into the vertex that give an IRenhancement [40–42]. This is in an alternative trunca-tion where the Feynman diagrams are collected into in-dividually transverse groups using pinch-technique andbackground field method rearrangements [10, 43]. Thereis not a direct comparison of such a vertex insertion inthis formulation with the standard SDEs we use. We canhowever make similar considerations and investigate theeffects of massless poles in the vertex.Several different choices are made for the triple-gluonvertex in the literature, which is perhaps a little supris-ing since sensible WSTI solutions are available. How-ever these depend on an unknown contribution from theghost-gluon vertex. When solved self-consistently, theSDE solutions should be matched on to the resummedleading-logarithm from perturbation theory. However us-ing symmetric triple-gluon vertices, it has been foundthat the anomalous dimension of this logarithm is notexactly reproduced [5].Proposed solutions to this issue include solving anequation relating the vertex dressing and the anomalousdimension in the perturbative region in order to choose avertex dressing that gives the correct outcome. Typicallythese vertices sacrifice Bose-symmetry between the inter-nally contracted and external legs of the vertex. Whena dressed symmetric vertex is considered, this diagramthen contributes in the IR region of the gluon equation.This in turn requires an interplay between the ghost andgluon loops in the IR for the gluon equation and ar-guments about an individually transverse ghost-diagrammay not necessarily be valid [7, 11]. Infinite Ghost DressingFinite GhostDressing Gh (cid:73) p (cid:77) Typical GluonDressing Gl (cid:73) p (cid:77) p FIG. 4. A sketch of typical dressing functions. The momen-tum scale is arbitrary, but can be thought of as ∼ There are two broad classes of solution in existenceand these have been strongly debated [10, 11, 44]. Thegluon, shown in Fig. 4, is qualitatively similar in mostrecent studies. However the ghost in the IR can varywildly: from a finite value to infinity. We consider theghost equation in detail in the next section. There arealso solutions from Lattice QCD that are produced inthe absence of quarks. At present these point to a fi-nite ghost propagator dressing function [45]. We givean example of some solutions in Fig. 4 including a typi-cal gluon dressing with the logarithm from perturbationtheory at large p and a suppression at small p . Theghost is less certain and a wide range of solutions maybe found. The most important region is around 1 GeV,where QCD becomes confining. How well the functionscan be determined in this region will be a key concern.The vanishing momentum region is physically less rele-vant, although the interplay between the ghost and gluonthere does determine whether or not self-consistent solu-tions can be obtained. Importantly, the method we useto find such self-consistent solutions is outlined in Ap-pendix A. III. THE GHOST EQUATION
Non-linear integral equations may admit multiple solu-tions and it is apparent that the ghost propagator equa-tion falls into this class. Early attempts to include ghostsin the solution of the gluon equation used the well knowninfrared (IR) power law assumption for the vanishing p limit of the propagators [46, 47],lim p → G (cid:96) ( p ) = a g ( p ) κ g , (11)lim p → G h ( p ) = a c ( p ) κ c , (12)the relation that is found is κ g = 2 κ and κ c = − κ .This behavior, where the power κ is linked, is derivedfrom matching powers on both sides of the ghost equa-tion using a bare ghost-gluon vertex. In solving the ghostequation the coefficients a g and a c are left unconstrained.However the gluon equation fixes a relation between thetwo. Typically, vanishing IR gluon propagators are thenconsidered and hence a singular ghost arises for this spe-cific solution.More recent studies [11, 38, 48–50] have found a secondsolution where the IR powers are not linked; a vanishinggluon admits a finite ghost solution also. There the pow-ers are typically κ g = 1 and κ c = 0. The solution selectedby physical QCD is still an open question however and atpresent there are ideas about the singular ghost solutionbeing a critical endpoint of a family of solutions includingthe finite ones [51], and also suggestions of an additionalgauge parameter [44, 52].It is possible that solutions in the deep IR are notsolutions for the whole momentum region and this willbecome important when we consider the gluon equationitself. However, both of these classes of solutions can befound using numerical methods in Euclidean space usingstandard techniques. There is one key distinguishing fea-ture between the two however. The infinite IR ghost so-lution requires the ghost to be subtracted (renormalised)at zero momentum. Its value for physically relevant mo-menta is then specified. How the ghost evolves is fixed.The finite solution may be found by subtraction at anyvalue of momentum.We presently do not state a strong preference for ei-ther solution. However we note that only the finite solu-tion is allowed by a perturbative renormalization schemewithout fine-tuning. The infinite solution never ariseswithout such tuning, one must search for it by specifyingthe infinite value at zero momentum and sacrificing suchfreedom in the UV. We will elaborate on this below. A. Solutions of the ghost equation alone
Using a fixed gluon input we may solve the Ghost SDEalone and investigate its sensitivity to a range of inputvertices and gluons. This is useful because this type ofSDE, or gap equation, is very simple to solve, and itteaches us what to expect when solving the more com-plicated, coupled equations self-consistently.In doing this we utilise the bare vertex of Eq. (8), andthe following model gluon [53], G (cid:96) ( p ) = p m + p (cid:16) π N c g π Log (cid:16) p + ρm µ (cid:17)(cid:17) (13) (cid:45) (cid:45) p FIG. 5. (Color Online) Examples of ghost solutions on a log-log plot subtracting at zero momentum. Only the specifiedsubtraction value G h (0) is varied between the solutions whichcan be read off. The G h ( p = 0) value is fixed and the ghostequation solved with the depicted gluon until the ghost inputsand outputs are self-consistent. The gluon dressing G (cid:96) ( p ), isthe dashed curve and the solid curves are the different ghostdressings G h ( p ). The units of p are arbitrary since we havenot fixed the coupling to the physical value, but may be con-sidered to be O (1 GeV ). which reproduces the correct UV perturbatively re-summed logarithm, see Eq. (B16), and provides an IRmass term necessary to produce a gluon propagatordressing function that behaves as p in that limit. Weutilize the parameters µ = 10 , m = 0 . g ( µ ) / π =0 .
12 and ρ = 1 which give the gluon shown in Fig. 5. Di-mensionful quantities can be thought of as having unitsclose to GeV. However we do not match to a phys-ical scale at this stage. The physical scale is deter-mined by the value of the coupling at the renormaliza-tion point. This of course only has true physical mean-ing when quarks are included. This gluon, Eq. (13),is qualitatively similar to that found in recent LatticeQCD studies [45, 54] and in other Schwinger-Dyson stud-ies [11, 12, 49, 53, 55].When solving the ghost equation and subtracting atzero momentum, we find its value at the subtractionpoint does not change its UV values between many ofthe solutions. Thus, for ghosts with IR values in therange 2 < G h (0) < ∞ we find their UV differences to benegligble. Note that the precise value is dependent uponthe coupling and the gluon equation. For a fixed gluon, alarger coupling gives a larger critical value of G h (0) abovewhich all the dressings are practically indistinguishablein the perturbative region. This can be seen in Fig. 5where the largest three ghost dressings, although differ-ent in the IR, are identical in the UV. Below this criticalvalue, the ghost equation admits solutions that differ inboth the UV and the IR. They still exhibit the same per-turbative logarithm although it is hardly visible on thisscale since its coefficient is reduced when G h (0) is set tobe small. Plotted on a linear vertical axis, the effect is clearer as can be seen in Fig. 6.This is interesting and a point that has not often beenstressed. If we subtract at some perturbative value andimpose the perturbative condition G h ( µ ) = 1, and dothe same for the gluon then none of the curves in Fig. 5are reproduced. The solution that connects to the stan-dard perturbative solution with the standard momentumsubtraction condition is unique and is separate from boththe singular ghost and the zero-momentum subtractionfinite solutions (except for the single fine-tuned value sat-ified by this unique solution).Setting G h ( µ ) = 1 is not essential as we can in prin-ciple renormalise at any point and the renormalizationgroup tells us how these differently renormalised solu-tions are connected to each other. However, it is notpossible to run down from the solution that we have ob-tained subtracting at a perturbative point and enforcing G h ( µ ) = 1 to any general solution subtracted at zeromomentum. Fine-tuned examples exist but these essen-tially predetermine the IR value.Starting from the perturbative solution and runningdown into the IR, we would then stay on the finite solu-tion for the ghost dressing for all momenta. The expec-tation is that asymptotically free QCD at large momen-tum transfers is accurately described by the perturbativesolution, so this is the one we favor. In this example,a perturbative subtraction point never admits the infi-nite IR ghost solution. This effect is depicted in Fig. 6,where the solid curve is the unique solution selected byrenormalising G h ( µ ) = 1 at the same µ point as thegluon. Hence, given similar conclusions from other stud-ies [11, 38, 48–50, 56], we carry forward this solution toinvestigate the gluon equation.In contrast, subtracting at zero momentum and speci-fying different values for G h (0) maps out infinitely manyother solutions even with the same fixed gluon input.Evolving these solutions up in momentum yields differentcurves in the perturbative region. We have verified thatthis effect is also present for fixed gluon inputs that van-ish more or less rapidly in the IR, including the κ (cid:39) . B. Other ghost-gluon vertices in the ghost equation
It has been noted [38] that Taylor’s theorem, whichis often used to infer a bare ghost-gluon vertex, actuallyonly places a restriction on the sum of the two functionsdressing the components of the vertex, Eq. (10). The con-dition is in the simplest terms, that any corrections van-ish when the incoming ghost-momentum vanishes. Thismotivates the modification to Eq. (10),Γ (2) µ = igf abc (cid:18) q µ − p µ k.qk N IR F IR ( k, p, q ) (cid:19) , (14)which is clearly not transverse to the gluon momentum.However, the piece additional to the bare term now van-ishes when the incoming ghost momentum p vanishes. (cid:45) (cid:45) p FIG. 6. (Color Online) The dotted curves correspond to thezero momentum subtracted solutions and the colours matchFig. 5. The dashed curve is the gluon input, G (cid:96) ( p ), andthe solid curve is the physically relevant solution of the ghostequation, G h ( p ). The units of p are arbitrary. The non-perturbative normalization parameter N IR isalso introduced into the modeling. Unlike dressing formsthat add only to the k µ term, this form may affect thesolutions of the ghost equation, as we demonstrate belowin Fig. 7. Adopting the same method as above, we fixthe gluon using the previous model and parameters, andsolve the ghost until self-consistent with this input andits own equation. (cid:45) (cid:45) p FIG. 7. (Color Online) A comparison of two solutions nor-malised using the perturbative condition. The (red) dottedcurve is the ghost solution, Gh ( p ), with a bare vertex ortransverse vertex. The (blue) dashed curve is the ghost dress-ing obtained using Eq. (14). The (black) solid curve is thefixed gluon dressing input, G (cid:96) ( p ) from Eq. (13). The unitsof p are arbitrary internal units. We find that in principle the ghost equation is depen-dent upon the choice of vertex and fairly large correctionsare possible as found for the example shown in Fig. 7. Many other extensions are possible and these exist in theliterature. Clearly a precise form for this vertex wouldbe useful. However at present we must consider this partof the uncertainty on the result of the calculation. Cor-rections of 20% over the bare vertex appear to be al-lowed. Nevertheless we have found the ghost equation tobe solvable for a wide range of parameters and choices ofvertices.
IV. COUPLING GLUONS AND GHOSTS
After the ease of solving the ghost equation, we nowturn to the gluon, where matters are very different.It is the opinion of the present authors that no com-pletely satisfactory solution exists for the gluon propa-gator equation at present. Solutions exist, but they arehighly dependent on arbitrary vertex choices, and dis-appointingly this is an issue rarely addressed. The pre-dictive power of the equations is often lost through theintroduction of arbitrary parameters that are not derivedfrom the fundamental field theory. The simplest possi-ble model just modifies the gluon propagator, such thatit contains an IR mass term that vanishes in the UV.Modeling vertices with a number of parameters and arange of possible forms is unfortunately no more predic-tive. However, it can be useful in guiding where to lookand in understanding those quantities we may need toknow precisely and those that are less important.Although we intend to subtract and renormalise at aperturbative point, which excludes the singular ghost so-lution, it is worth saying a few words about one partic-ular singular solution, especially in the light of recentcriticism [57]. In what follows, we note that this is notdirected at all singular solutions, just this particular oneand any others that contain the same flaw we describe.The problem arises due to the combined effects of trun-cation and projection of the dressed gluon propagator inthe Landau gauge, D µν ( p ) = G (cid:96) ( p ) p (cid:18) g µν − p µ p ν p (cid:19) , (15)= A ( p ) g µν − B ( p ) p µ p ν p . (16)The structure of the gauge theory demands that A ( p ) = G (cid:96) ( p ) = B ( p ), which in a full treatment with no trun-cation would be the case. However, in an incompletetreatment where uncontrolled approximations are madegreat care must be taken, because individual diagramscontain quadratic divergences that are no longer guaran-teed to cancel.The simplest solution has been long known and isstraightforward [32, 33, 58]. In solving the gluon equa-tion we only need to determine A or B . As is knownfrom perturbation theory, the problematic quadratic di-vergences only occur in the A term, so we just projectonto D µν of Eqs. (15,16) in such a way that only the B term is retained. This is achieved using the so-called Brown-Pennington projector [59], P µν ( p ) = g µν − d p µ p ν p (17)where d is the number of dimensions in which we work.Using the A term leaves any solution exposed to unphys-ical quadratic divergences, and these are present in oneset of solutions in the literature [8, 57]. The solutionsare obtained by the insertion of an additional step inthe iterative procedure, where the quadratic divergenceis subtracted from the result of the gluon loop in thegluon equation. This is not necessarily a safe thing todo since in the non-perturbative region the integrationresults are a priori unknown. This point has been madepreviously [23] and is of fundamental importance.In the IR analysis which is common to this solution andall subsequent refinements, the gluon determined from B differs from the gluon determined from A signaling thebreaking of transversality of the propagator. This may beseen in [23] in the IR analysis, where the A and B termsare analysed separately and the difference is clear. It mayalso be seen by solving self-consistently for the singularsolution defined by the A function, and then seeing howthe solutions change when we switch from the A term andthe B term. This we show in Fig. 8 and we see a clear dif-ference between the two gluon dressings, signalling thereis an issue with transversality. Moreover the B solutionis not self-consistent. If the usual iterative procedure isfollowed then no self-consistent solutions can be foundwhere the B term satisfies the set of equations [60].We thus determine our gluon from the p µ p ν term of thepropagator which is known to yield the correct logarithmsin the UV limit. Reference [11] addresses this issue byintroducing parameters into their vertices to make A and B come together. A. Self-consistent solutions of both propagators ina one-loop only system
We proceed in the traditional manner by starting withthe simplest conceivable system, then attempt to obtainsolutions. In this system, we neglect the two two-loopdressed contributions in Fig. 1 to the gluon propagatorand also drop the quark interaction for now. The re-quired input is the triple-gluon vertex and the ingredientsdescribed above for the ghost equation.There are several sources of information regarding thetriple-gluon vertex. One property that we consider im-portant that tends to be neglected is the Bose symmetryof the vertex. Bose symmetry is, of course, present at allorders in perturbation theory, and in the solution of theWSTI for this vertex [6, 36].We proceed by considering three dressings of thetriple-gluon vertex of increasing complexity, we factor off igf abc , and write the bare vertex Lorentz structure as
GhostGluon from A Gluon from B (cid:45) p FIG. 8. (Color Online) The singular ghost solutions and vio-lation of transverality in the infrared. The dashed gluon anddotted ghost curves are obtained first using the A term andthen we switch to the B term and we find that transversalityis broken since the curves differ. The units of p are arbitraryinternal units. Γ (0) µνρ ( k, p, q ),Γ ( A ) µνρ ( k, p, q ) = Γ (0) µνρ ( k, p, q ) G h ( p ) G h ( q ) G (cid:96) ( p ) G (cid:96) ( q ) (18)Γ ( B ) µνρ ( k, p, q ) = Γ (0) µνρ ( k, p, q ) × (cid:18) G h ( k ) G (cid:96) ( k ) + G h ( p ) G (cid:96) ( p ) + G h ( q ) G (cid:96) ( q ) (cid:19) (19)Γ ( C ) µνρ ( k, p, q ) = 12 (cid:18) G h ( q ) G (cid:96) ( p ) + G h ( q ) G (cid:96) ( k ) (cid:19) g µν ( k − p ) ρ + 12 (cid:18) G h ( k ) G (cid:96) ( q ) + G h ( k ) G (cid:96) ( p ) (cid:19) g νρ ( p − q ) µ + 12 (cid:18) G h ( p ) G (cid:96) ( k ) + G h ( p ) G (cid:96) ( q ) (cid:19) g ρµ ( q − k ) ν . (20)The first vertex, Γ ( A ) , has been used to reproduce theprecise perturbatively resummed one-loop running of thegluon propagator [61], note that it does not reproducethe one-loop behavior of the triple-gluon vertex itself,nor does it have Bose-symmetry between each leg. Thesecond vertex, Γ ( B ) is a simple symmetric vertex inspiredby WSTI solutions that involve ratios of dressing func-tions. Finally, Γ ( C ) is an approximate solution to theWSTI itself using a bare ghost-gluon scattering kernel(˜Γ µν = g µν ). Since the ghost-gluon vertex WSTI andhence the ghost-gluon scattering kernel contributions arenot precisely known, we do not go further than this atpresent. A full solution of the triple-gluon WSTI usingan approximate ghost-gluon scattering kernel is availablein the literature [6, 25, 26], however the associated ghost-gluon vertex is insufficient to obtain solutions in a finiteghost system.We select a conservative set of parameters that fromexperience allow solutions for our vertices given above.These are not intended to describe the physical worldin any way, but rather to expose the salient features ofthe truncation. For sensible comparison, we only use thesame parameters, where applicable.The solutions obtained are shown in Fig. 9. We labelthe solutions using ( i, j ) to denote the vertices we use.These are defined above and correspond to Γ ( i ) µ for theghost-gluon vertex and Γ ( j ) µνρ for the triple-gluon vertex.Similarly to the solution of the ghost equation alone, wefind a strong sensitivity to the vertices we choose, partic-uarly in the all-important region that is most relevant tophysics: roughly between 0.1 and 10 in these momentum-squared units (see Figs. (4-12)). Although we do notmatch to a specific scale, the peak of the gluon dressingfunction would be expected to have a close relation to thefundamental scale Λ QCD . The differences between thesesolutions with the range of vertices { i, j } would undoubt-edly have an effect on physical quantities, for example,hadron masses and form-factors [62]. (2,B) (2,C)(2,A) (1,C)(1,A) (2,A)(2,C)(2,B)(1,A)(1,C) Gluon DressingsGhost Dressings p FIG. 9. (Color Online) The range of solutions obtained usingthe possible vertex combinations. The label ( i, j ) refers tothe vertices used in obtaining the solutions corresponding toΓ ( i ) µ for the ghost-gluon vertex and Γ ( j ) µνρ for the triple-gluonvertex. The missing curve corresponds to the Γ (1) µ and Γ ( B ) µνρ ,self-consistent solutions were not obtained there. The param-eters are not varied between the solutions, only the vertices.The units of p are arbitrary since we have not fixed the cou-pling to a physical value. B. Infrared Loop Contributions from differentvertices
We note that previous studies have found ghost-dominance in the IR region of the gluon equation. Weshow this to be false and an artifact of choosing a sim-ple vertex. We find that as in the perturbative region,the ghost and gluon loops contribute a similar numeri-cal amount. This may negate arguments relating to aghost-gluon vertex that is transverse alone. In order to explain this, we write the renormalised(subtracted) gluon equation in the following form, G (cid:96) ( p ) − = G (cid:96) ( µ ) − + Π g ( p , µ ) + Π c ( p , µ ) , (21)the functions Π( p , µ ) are the loop integrals subtractedat the renormalization point µ . These polarization func-tions are then zero at p = µ and G (cid:96) ( µ ) = 1. It isexpected that the gluon propagator dressing should bepositive for all momenta.The ghost-loop polarization function, Π c ( p , µ ), ishighly sensitive to the ghost-gluon vertex, and the gluonloop polarization function, Π g ( p ), is most sensitive tothe choice of triple-gluon vertex. This is completely asexpected. There are additional feedback effects that hap-pen when the equations are solved together as describedin appendix A, but these effects are typically smaller.These functions diverge as p (or a little less) so we mul-tiply up by p and plot the numerical results in Fig. 10.In Fig. 10 we see the key feature that we wish to elu-cidate, and that is, for the Bose-symmetric triple-gluonvertices, the IR gluon-loop contributions are non-zero.Both ghost and gluon loops contribute at a similar nu-merical order, whilst the ghost must be larger to keep thegluon dressing function positive in the IR limit. Thereis clearly some interplay between these, as can be seenby the differences between the transverse vertex and themodified version that satisfies Taylor’s theorem. An in-terdependence of this type is to be expected, since theghost-gluon scattering kernel appears separately in theWSTI for both vertices, and the idealised full solutionwould uniquely constrain this relation.The behavior of these functions is important, partic-ularly in light of the above partial cancellation, if thegluon propagator dressing function is to be positive ev-erywhere. As the loop contribution induced by the triple-gluon dressing is negative then some canceling positivecontribution is required from the loop containing theghost-gluon vertex. This is only present for the twovertices we show here. For the bare vertex or a ver-tex dressed by simple ratios, no such contribution occursand hence self-consistent solutions are not possible. Itis this extreme sensitivity, highlighted in Fig. 10, to theans¨atze for both the ghost-gluon and triple gluon verticesthat makes us conclude that consistent gluon and ghostpropagators have not yet been determined in continuumstrong coupling QCD.An alternative to this would be for a positive contri-bution to arise from the gluon loop. This would requirefurther corrections to be added to the triple-gluon ver-tex. Some evidence for this may already exist in recentlattice studies [63]. These contributions could be due tothe WSTI-unconstrained parts of the triple-gluon vertexand hence are not present in Γ ( C ) µνρ of Eq. (20), or a morecomplete WSTI solution [6].A point of note is that the coefficient of p in the IRlimit of the gluon propagator dressing is determined bythe sum of the ghost and gluon loop functions in thatlimit. In order to determine this with any precision it is0 (2,B) (2,C) (1,C) (1,A) (2,A) G m H L G m H L p P g p P c p - - p P c p P g
10 100 1000 p - - - FIG. 10. (Color Online) Gluon polarization functions, p Π c ( p ) (positive values) and p Π g ( p ) (negative values). Left: IRregion, Right: UV region. In the UV the curves are indistinguishable on this scale. In the IR, the ghost-loop curves for multiplesolutions lie on top of each other. The bare ghost-gluon vertex, not shown here, always gives a vanishing IR contribution givenan IR vanishing gluon dressing and an IR finite ghost dressing. The functions p Π c ( p , µ ) are labelled according to the inputvertices with the contributions from Γ ( i ) µ and Γ ( j ) µνρ labelled as ( i, j ) in the plot. The units of p are arbitrary. important to get the vertices right. This coefficient is ofsome importance since it determines the effective gluon‘mass’ term. The mass term, though almost certainlygauge dependent, will inevitably affect physical quanti-ties like hadron masses and form-factors. Many practicalmodels contain mass terms for the gluon; for a recentexample see [64].Considering these two contributions from the differ-ent loops we also find the reason why the gluon dressinglooks similar in both the infinite and finite ghost solu-tions. In the singular solution the bare vertex is usedand the large contribution from the ghost loop is pro-vided by the singular propagator dressing. Converselyin the finite ghost solution a non-trivial vertex providesa similar large contribution and a qualitatively similarvanishing gluon arises. V. COMPARISON TO LATTICE QCD
Primarily we have been motivated by theoretical issuesencountered in solving the Schwinger-Dyson equationsfor the gluon and ghost propagators. However a com-plementary technique is available where these quantitieshave been calculated and that is Lattice QCD. The pos-sible issues there are quite different to those that we mayinduce here by truncation, so a comparison is a usefulindependent cross-check. Importantly the lattice com-putations, for which there are extremely precise results,are for the pure gauge sector we investigate here in thecontinuum.Many recent lattice studies exist [65–67], starting withthe early work of [2]. We compare our calculations to[45] which provides results in Landau gauge for bothdressing functions. The qualitative behavior there is aswe have here with a finite ghost dressing function anda finite gluon propagator, corresponding to a vanishinggluon propagator dressing function. In obtaining these solutions we use the preferred set ofvertices, Γ ( C ) from Eq. (20) and the ghost-gluon vertexgiven in Eq. (14). We then tune the parameters to ob-tain a reasonable representation of the lattice data andthese are given in Table I. We note several numerical dis-crepancies. Most importantly in the UV, the availableparameter space does not include the coupling strengthfound on the lattice; smaller couplings have to be used.The iterative procedure described in Appendix A breaksdown at larger couplings signalling an absence of solu-tions in this given truncation. This is visible in the plotsas a smaller gradient of the perturbatively resummed logfor the SDE solutions. Secondly, the peak of the gluondressing function in Fig. 11 is not as large as found onthe lattice. It is expected that the two-loop dressingswould make a contribution here, so that could be thesource of the difference. Other effects will of course in-clude the different coupling values and uncertainties inthe vertices. We do not comment upon any differencesdue to the method used to extract the predictions fromthe lattice. Solution µ α ( µ ) Λ N IR Dotted 650 0.1313 6.0 3.5Dot-dashed 650 0.1200 3.8 3.7Dashed 650 0.1225 7.5 2.0TABLE I. Parameters used in obtaining the tuned self-consistent lattice solutions. µ and Λ IR can be regarded asbeing in GeV units. VI. OUTLOOK
The gluon and ghost propagators are the basic Green’sfunctions that embody not only the short distance be-havior determined by the asymptotically free nature of1 (cid:45) p (cid:45) p FIG. 11. (Color Online) Self-consistent solutions tuned to lattice solutions, showing the dressing functions. The solid curvesdepict a smooth fit to the lattice data. The heavier region is where the functions are represented by lattice data and thefeint region represents the natural extrapolation. The broken curves are the tuned solutions. Left: The blue peaked curvescorrespond to the gluon dressing function G (cid:96) ( p ), this vanishes as p →
0, the red monotonic curves correspond to the ghostdressing function G h ( p ). Right: The upper (blue) curves correspond to G (cid:96) ( p ) /p , this is ∼
10 as p →
0, the lower (red)curves correspond to the ghost dressing function.
QCD, but confinement dynamics at larger distances.This behavior is encoded in solutions of the appropriateSchwinger-Dyson equations. Here we first investigatedthe existence of solutions for a simplified ghost equationusing a fixed gluon input. Multiple solutions were found.However applying a condition commonly used in pertur-bative analyses, that G h ( µ ) = 1 where µ is the pointin the perturbative region at which both gluon and ghostare renormalised, we found just one solution was pre-ferred. Using this result we then investigated the neces-sary terms required in order to construct a self-consistentsolution for the coupled gluon and ghost dressing func-tions with input interaction vertices. A range of solutionsresulted that are qualitatively similar to those computedon the lattice. However, perfect quantitative agreementhas yet to be established. This is assumed to be due tothe approximations made here or on the lattice. We drewspecial attention to differences induced by vertex choicesand we have argued that the neglected two-loop dressingsare likely to give rise to significant changes. The hope isthat constraints on the vertices, particularly that of thefull ghost-gluon interaction, can be found that uniquelyspecify their structure. At present the ad hoc verticesused are motivated more by the practicalities of findingsolutions than by constraints derived from the fundamen-tal field theory. Thus we conclude that consistent gluonand ghost propagators have yet to be determined in con-tinuum strong coupling QCD.In the pure gauge sector studied here, one can define anon-perturbative running coupling following Taylor [37]from the ghost-gluon vertex renormalization, α T ( p ) = g ( µ )4 π G h ( p ) G (cid:96) ( p ) (22)where µ is the renormalization point and g ( µ ) fixes the physical scale. Experiment with 5 flavors of fermion gives α ( M ) = 0 .
118 in the modified minimal subtraction(
M S ) scheme. This can be related to momentum sub-traction with zero flavors appropriate to the calculationswe performed here, by using perturbative results fromthe two schemes [35, 68, 69]. Thus, this value of α in the M S scheme with 5 flavors translates into α ( M Z ) = 0 . IR = 2 GeV and N IR = 2 we obtain thefunction α T ( p ) shown in Fig. 12.The behavior of the gluon propagator dressings inFigs. 4, 9, 11 and 12 can be interpreted in terms of thedynamical generation of an effective gluon mass. This inturn leads to the following proposal for a correspondingdefinition of the running coupling α EC ( p ) = g ( µ )4 π p + m g ( p ) p G h ( p ) G (cid:96) ( p ) , (23)recently developed in a study of the effective chargesgiven by dressing functions with such qualitative be-havior [55]. Applying this to the solutions found using α ( M Z ) = 0 .
08, Λ IR = 2 GeV and N IR = 2 we find m g (0) = 350 MeV and the function α EC ( p ) in Fig. 12.The result is not particularly sensitive to the precise formof the function m g ( p ), however we use that given in [55]which is suppressed in the UV.Both gluon and ghost dressings do not rise as steeply asthe lattice-like solutions of Fig. 11. Nevertheless, the col-lective effect of the gluon dressing and this running cou-pling would just have sufficient strength to induce the dy-namical generation of quark mass, cf. the model of Marisand Tandy [62, 70]. The inclusion of quarks is, of course,an important aspect of making connection to physicalquantities, like the pion mass and its decay constant. A2 Α EC (cid:73) p (cid:77) Α T (cid:73) p (cid:77) Gl (cid:73) p (cid:77) Gh (cid:73) p (cid:77) p FIG. 12. (Color Online) The running coupling α T ( p ) fromEq. (22) is shown as the black solid curve. The running cou-pling α EC ( p ) from Eq. (23) is shown as the green solid curve.The gluon dressing G (cid:96) ( p ) is the dashed blue curve and thedotted red curve is the ghost dressing function G h ( p ). Theunits of p are now very close to GeV . complete study requires a simultaneous solution of thequark SDE with those of the gluon and ghost. Researchover the past 15 years indicates that we know what thequark propagator functions look like for a whole rangeof quark masses (and hence flavors) with model gluoninputs. Indeed modeling such ghost and gluon propaga-tors and their interactions, and inserting these into theBethe-Salpeter equations to compute hadronic observ-ables is now well advanced [62, 64, 70, 71]. However, acomprehensive computation with a self-consistent inves-tigation of the coupled quark, gluon and ghost equationsin continuum QCD, followed by the development of atrustworthy hadron phenomenology, is still in the future.The issues exposed by the present study show we havesome way to go before we can claim robust results fromsuch an ab initio approach. ACKNOWLEDGEMENTS
The Institute for Particle Physics Phenomenology atDurham University, UK, its staff and students are grate-fully aknowledged for providing an ideal working envi-ronment for much of this study. DJW gratefully ac-knowledges the hospitality of Jefferson Laboratory infinalising this work. This paper has in part been au-thored by Jefferson Science Associates, LLC under U. S.DOE Contract No. DE-AC05-06OR23177. This workwas also supported by the U. S. Department of En-ergy, Office of Nuclear Physics, Contract No. DE-AC02-06CH11357. We would like to thank Adnan Bashir,Ian Cl¨oet, Javier Cobos-Martinez, Craig Roberts, PeterTandy and Richard Williams for useful discussions.
Appendix A: Numerical Method
The equations are solved in Euclidean space using thewell known method set out in [72]. Using the standardspherical four-dimensional coordinate system the two in-ner angles are symmetric for the propagator integrationsso may be integrated out. We are then left with the out-ermost angle which we define as the angle between theincoming propagator momentum p and the loop coordi-nate (cid:96) .The integrals are logarithmically divergent and are reg-ularised by integrating to some momentum cutoff κ . Thisappears in the renormalization procedure. However theresults do not depend on it. The functions are subtractedat µ and this point is also used to match on to the per-turbative result. That is, above this point the one-loop-resummed perturbative result is used as an extrapolationwhen required by the loop integrations.The functions are represented by Chebychev polyno-mials. These are particularly useful for both their in-terpolation properties and for the iterative procedure.They are mapped onto a logarithmic scale typically inthe range p ∈ (cid:2) − , (cid:3) although the precise numbersare not important. The last Chebychev zero is mappedto the point µ since this point is always exactly repro-duced by the interpolation. In the IR then the functionshave to be represented by some extrapolation. In findingthese finite solutions a constant is sufficient for the ghostdressing function and the c g p term is appropriate for thegluon dressing, where c g is calculated to match smoothlyonto the lowest point represented by the Chebychev poly-nomials.In performing the iterations we use both the Newton-Raphson and natural iterative procedures. For the finiteghost solutions, then natural iterative procedures are ad-equate for finding solutions in both equations. A goodstarting point is always useful and often a requirement,particularly at large couplings. We use Eq. (13) for thegluon starting point and G h ( p ) = 1 is a sufficient start-ing point for the finite ghost dressing. Appendix B: Formulae1. The Ghost Equation
The renormalised ghost equation used is, G h ( p ) − = ˜ Z ( µ , κ ) + g N c (cid:90) d (cid:96) (2 π ) × α ( − (cid:96) − , p, (cid:96) + ) K ( p, (cid:96) + , (cid:96) − ) G h ( (cid:96) + ) G (cid:96) ( (cid:96) − ) (B1)where (cid:96) ± = (cid:96) ± p/ (cid:96) is the loop integration mo-mentum, ˜ Z is the ghost renormalization constant. Thefunction α multiplies the q µ term in the ghost-gluon ver-tex, as defined in Eq. (6). The function K arises from the3tensor contractions in the Feynman rules and is given by, K ( p, (cid:96) + , (cid:96) − ) = − p (cid:96) (cid:96) − (cid:18) p.(cid:96) + − p.(cid:96) − (cid:96) + .(cid:96) − (cid:96) − (cid:19) (B2)= − (cid:96) sin θ(cid:96) (cid:96) − where θ is the integration angle defined via (cid:96).p = | (cid:96) || p | cos θ . This equation is typically subtracted fromitself at µ which is the renormalization point. This re- moves the term ˜ Z .
2. The Gluon Equation
As described in the text the gluon equation is madeup of more terms and contains two dressed one-loop in-tegrations in the formulation we have used here. We giveno details of the tadpole term since it yields results thatare proportional to the g µν term of the propagator andhence does not contribute with the method we describeabove. The gluon equation that we have used is given by, G (cid:96) ( p ) − = Z ( µ , κ ) + Π g ( p , κ ) + Π c ( p , κ ) (B3)= G (cid:96) ( µ ) − + Π g ( p , κ ) − Π g ( µ , κ ) + Π c ( p , κ ) − Π c ( µ , κ ) (B4)= G (cid:96) ( µ ) − + Π sub2 g ( p , µ ) + Π sub2 c ( p , µ ) . (B5)where in the second line we subtract the equation fromitself at the point µ and in the third line we introduce thesubtracted loop integrations Π sub j ( p , µ ) = Π j ( p , κ ) − Π j ( µ , κ ) which for each diagram are actually all thesame in a properly renormalised system when everythingabove the point µ is assumed to be described by one-loopperturbation theory, so we drop the (sub) notation below.We next define the content of the loop integrations in the Π functions. We give this only for the most complicatedcase of vertices Γ (2) and Γ ( C ) since the others may bestraightforwardly deduced from these by setting factorsto 1 and/or multiplying by the appropriate dressings ofthe vertex.First we give the ghost-loop contribution. This con-tains the two functions α ( k, p, q ) and β ( k, p, q ) from theghost-gluon vertex dressings of Eq. (6) that differ be-tween Γ (1) µ and Γ (2) µ ,Π c ( p , µ ) = ( − N c g ( µ )( d − (cid:90) d (cid:96) (2 π ) G h ( (cid:96) , µ ) G h ( (cid:96) − , µ ) p (cid:96) (cid:96) − × (cid:16) α ( − p ; (cid:96) − , (cid:96) + ) M α ( p, (cid:96) ) + β ( − p ; (cid:96) − , (cid:96) + ) M β ( p, (cid:96) ) (cid:17) , (B6)where d = 4 is the number of dimensions, g is the value of the coupling at the renormalization point, the functions α and β arise from the two terms in the ghost-gluon vertex, Eq. (6). The kinematic terms are contracted together intothe two M functions, which are derived to be, M α ( p, (cid:96) ) = − p (cid:18) p (cid:20) (cid:96) − p (cid:21) − d (cid:20) ( (cid:96).p ) − p (cid:21)(cid:19) , (B7) M β ( p, (cid:96) ) = (1 − d ) p.(cid:96) − . (B8)For the bare vertex, the β term is zero and for the transverse vertex, Γ (1) , then α ( k, p, q ) = 1. For the Taylor-respectingvertex Γ (2) then we replace p µ → k µ + q µ in Eq. (14) and read off the respective coefficients. Similarly for the gluonloop we have, Π g ( p ) = (cid:18) (cid:19) N c g ( µ )( d − (cid:90) d (cid:96) (2 π ) G (cid:96) ( (cid:96) ) G (cid:96) ( (cid:96) − )4 ( p (cid:96) (cid:96) − ) (cid:16) R Q + R Q + R Q (cid:17) (B9)where the ratios of dressing functions are chosen to be, R = 12 (cid:18) G h ( (cid:96) − ) G (cid:96) ( (cid:96) ) + G h ( (cid:96) − ) G (cid:96) ( p ) (cid:19) R = 12 (cid:18) G h ( p ) G (cid:96) ( (cid:96) ) + G h ( p ) G (cid:96) ( (cid:96) − ) (cid:19) R = 12 (cid:18) G h ( (cid:96) ) G (cid:96) ( (cid:96) − ) + G h ( (cid:96) ) G (cid:96) ( p ) (cid:19) (B10)4and the associated momentum contractions are, Q = 2( (cid:96).p − (cid:96) p ) (cid:2) (cid:96) (5 p + 2 d(cid:96).p ) + p (7 p + 2(12 − d ) (cid:96).p ) (cid:3) , (B11) Q = − d ( (cid:96).p ) + (3 d − p ( (cid:96).p ) − (cid:96) p + 24 (cid:96) (2 d(cid:96).p − p ) + (cid:96) p (8( d + 6) (cid:96).p + p ) , (B12) Q = 2( (cid:96).p − (cid:96) p ) (cid:2) (cid:96) (5 p − d(cid:96).p ) + p (7 p − − d ) (cid:96).p ) (cid:3) . (B13)In order to obtain a vertex proportional to the bareLorentz structure as in Γ ( A ) and Γ ( B ) then we just haveto make a replacement, for example for Γ ( A ) we have, R i → G h ( (cid:96) ) G h ( (cid:96) − ) G (cid:96) ( (cid:96) ) G (cid:96) ( (cid:96) − ) , (B14)for all three R i ’s, and a similar substitution is made toobtain the equation for the Γ ( B ) dressing.The smoothed step function used for interpolating be-tween IR and UV vertex functions is taken from [11].However, a range of forms may be used, F IR ( k, p, q ) = Λ ( k + Λ )( p + Λ )( q + Λ ) (B15)where Λ IR gives the change-over between the perturba-tive and non-perturbative form. The solutions are sen-sitive to the choice of this parameter. In the modifiedform of the vertex of Eq. (14) an extra parameter N IR multiplying the non-perturbative term was introduced.
3. Resummed One-loop running
In the UV we match the gluon and ghost propagatordressings on to the resummed one-loop form from per-turbation theory. This is necessary as some extrapola-tion beyond the Chebychev region in the UV is always required. The form that we find to be most useful nu-merically is given by, G (cid:96) ( p ) = G (cid:96) ( µ ) (cid:0) α ( µ ) /α ( p ) (cid:1) d g (B16) G h ( p ) = G h ( µ ) (cid:0) α ( µ ) /α ( p ) (cid:1) d c , (B17)where d g = − /
22 ( cf. the modeling in Eq. (13)) and d c = − /
44 (as appropriate for the pure gauge sector)which is applied where p > µ . Numerically, we also usethe standard relation for the running coupling,1 α ( p ) = 1 α ( µ ) + 11 N c π log p µ . (B18)where α ( µ ) is specified to fix the scale.Since only the triple-gluon vertex Γ ( A ) µνρ perfectly repro-duces the perturbatively resummed one-loop running, itis useful to interpolate between this in UV and the othervertices in the mid-momentum region and below. TheWSTI triple-gluon vertex, Γ ( C ) µνρ in particular induces asteeper gluon dressing function than would be expectedfrom perturbative studies. Some deviation is allowed dueto the effects of higher orders perturbatively and since weare still at the modeling stage then interpolating betweenthese two vertices between an IR scale Λ and µ (cid:39) M Z is currently the best that can be done. The transition isimplemented here using F IR (see Eq. (10) and Eq. (B15))and (1 − F IR ) to multiply the respective terms. [1] J. M. Cornwall, Phys. Rev. D26 , 1453 (1982)[2] J. E. Mandula and M. Ogilvie, Phys. Lett.
B185 , 127(1987)[3] C. D. Roberts and A. G. Williams, Prog. Part. Nucl.Phys. , 477 (1994), arXiv:hep-ph/9403224[4] L. Chang, C. D. Roberts, and P. C. Tandy, Chin.J.Phys. , 955 (2011), arXiv:1107.4003 [nucl-th][5] D. Atkinson and J. C. R. Bloch, Phys. Rev. D58 , 094036(1998), arXiv:hep-ph/9712459[6] L. von Smekal, A. Hauck, and R. Alkofer, Ann. Phys. , 1 (1998), arXiv:hep-ph/9707327[7] C. Lerche and L. von Smekal, Phys. Rev.
D65 , 125006(2002), arXiv:hep-ph/0202194[8] R. Alkofer, C. S. Fischer, H. Reinhardt, and L. vonSmekal, Phys. Rev.
D68 , 045003 (2003), arXiv:hep-th/0304134[9] C. S. Fischer(2003), Ph.D. thesis, (University of Tubin-gen), arXiv:hep-ph/0304233 [10] A. C. Aguilar, D. Binosi, and J. Papavassiliou, Phys. Rev.
D78 , 025010 (2008), arXiv:0802.1870 [hep-ph][11] C. S. Fischer, A. Maas, and J. M. Pawlowski, AnnalsPhys. , 2408 (2009), arXiv:0810.1987 [hep-ph][12] A. C. Aguilar, D. Binosi, and J. Papavassiliou, Phys.Rev.
D84 , 085026 (2011), arXiv:1107.3968 [hep-ph][13] V. N. Gribov, Nucl. Phys.
B139 , 1 (1978)[14] D. Zwanziger, Nucl. Phys.
B364 , 127 (1991)[15] D. Zwanziger, Nucl. Phys.
B399 , 477 (1993)[16] D. Zwanziger, Nucl. Phys.
B412 , 657 (1994)[17] T. Kugo and I. Ojima, Prog. Theor. Phys. Suppl. , 1(1979)[18] D. Zwanziger, Phys. Rev. D67 , 105001 (2003),arXiv:hep-th/0206053[19] D. Dudal, J. A. Gracey, S. P. Sorella, N. Vandersickel,and H. Verschelde, Phys. Rev.
D78 , 065047 (2008),arXiv:0806.4348 [hep-th][20] K.-I. Kondo, Prog.Theor.Phys. , 1455 (2010), arXiv:0907.3249 [hep-th][21] K.-I. Kondo(2009), arXiv:0909.4866 [hep-th][22] D. Dudal, S. P. Sorella, and N. Vandersickel(2011),arXiv:1105.3371 [hep-th][23] J. C. R. Bloch, Few Body Syst. , 111 (2003), arXiv:hep-ph/0303125[24] M. Tissier and N. Wschebor, Phys.Rev. D84 , 045018(2011), arXiv:1105.2475 [hep-th][25] J. S. Ball and T.-W. Chiu, Phys. Rev.
D22 , 2542 (1980)[26] J. S. Ball and T.-W. Chiu, Phys. Rev.
D22 , 2550 (1980)[27] D. C. Curtis and M. R. Pennington, Phys. Rev.
D42 ,4165 (1990)[28] A. Bashir, A. Kizilersu, and M. R. Pennington, Phys.Rev.
D57 , 1242 (1998), arXiv:hep-ph/9707421[29] A. Bashir, Y. Concha-Sanchez, and R. Delbourgo, Phys.Rev.
D76 , 065009 (2007), arXiv:0707.2434 [hep-th][30] A. Kizilersu and M. R. Pennington, Phys. Rev.
D79 ,125020 (2009), arXiv:0904.3483 [hep-th][31] U. Bar-Gadda, Nucl. Phys.
B163 , 312 (1980)[32] N. Brown and M. R. Pennington, Phys. Rev.
D38 , 2266(1988)[33] N. Brown and M. R. Pennington, Phys. Rev.
D39 , 2723(1989)[34] R. Alkofer, C. S. Fischer, and F. J. Llanes-Estrada, Phys.Lett.
B611 , 279 (2005), arXiv:hep-th/0412330[35] W. Celmaster and R. J. Gonsalves, Phys. Rev.
D20 , 1420(1979)[36] A. I. Davydychev, P. Osland, and O. V. Tarasov, Phys.Rev.
D54 , 4087 (1996), arXiv:hep-ph/9605348[37] J. C. Taylor, Nucl. Phys.
B33 , 436 (1971)[38] P. Boucaud, J. P. Leroy, A. Le Yaouanc, A. Lokhov,J. Micheli, et al. (2005), arXiv:hep-ph/0507104 [hep-ph][39] P. Boucaud, J. P. Leroy, A. Le Yaouanc, J. Micheli,O. Pene, et al. , JHEP , 099 (2008), arXiv:0803.2161[hep-ph][40] J. S. Schwinger, Phys. Rev. , 397 (1962)[41] J. S. Schwinger, Phys. Rev. , 2425 (1962)[42] R. Jackiw and K. Johnson, Phys. Rev. D8 , 2386 (1973)[43] D. Binosi and J. Papavassiliou, Phys. Rept. , 1(2009), arXiv:0909.2536 [hep-ph][44] A. Cucchieri and T. Mendes, in The IXth Interna-tional Conference on Quark Confinement and the HadronSpectrum QCHS IX , edited by F.J. Llanes-Estradaand J.R. Pelaez, AIP Conf. Proc. , 185 (2011),arXiv:1101.4779 [hep-lat][45] I. L. Bogolubsky, E. M. Ilgenfritz, M. Muller-Preussker,and A. Sternbeck, Phys. Lett.
B676 , 69 (2009),arXiv:0901.0736 [hep-lat][46] P. Watson(2000), Ph.D. thesis, (University of Durham)[47] P. Watson and R. Alkofer, Phys. Rev. Lett. , 5239(2001), arXiv:hep-ph/0102332 [hep-ph][48] O. Pene, P. Boucaud, J. P. Leroy, A. Le Yaouanc,J. Micheli, et al. , PoS QCD-TNT09 , 035 (2009),arXiv:0911.0468 [hep-ph][49] A. C. Aguilar and J. Papavassiliou, Phys. Rev.
D81 ,034003 (2010), arXiv:0910.4142 [hep-ph] [50] J. Rodriguez-Quintero, JHEP , 105 (2011),arXiv:1005.4598 [hep-ph][51] J. Rodriguez-Quintero, in
The IXth International Con-ference on Quark Confinement and the Hadron SpectrumQCHS IX , edited by F.J. Llanes-Estrada and J.R. Pelaez,AIP Conf. Proc. , 188 (2011), arXiv:1012.0448 [hep-ph][52] A. Maas, Phys. Lett.
B689 , 107 (2010), arXiv:0907.5185[hep-lat][53] A. C. Aguilar, D. Binosi, and J. Papavassiliou, JHEP , 002 (2010), arXiv:1004.1105 [hep-ph][54] A. Cucchieri and T. Mendes, PoS
QCD-TNT09 , 026(2009), arXiv:1001.2584 [hep-lat][55] A. C. Aguilar, D. Binosi, J. Papavassiliou, andJ. Rodriguez-Quintero, Phys. Rev.
D80 , 085018 (2009),arXiv:0906.2633 [hep-ph][56] P. Watson and H. Reinhardt, Phys. Rev.
D82 , 125010(2010), arXiv:1007.2583 [hep-th][57] C. S. Fischer and L. von Smekal, AIP Conf.Proc. ,247 (2011), arXiv:1011.6482 [hep-ph][58] J. C. R. Bloch and M. R. Pennington, Mod. Phys. Lett.
A10 , 1225 (1995), arXiv:hep-ph/9501411[59] M. R. Pennington, in
The IXth International Conferenceon Quark Confinement and the Hadron Spectrum QCHSIX , edited by F.J. Llanes-Estrada and J.R. Pelaez, AIPConf. Proc. , 63 (2011), arXiv:1104.2522 [nucl-th][60] We do note however that other infinite ghost solutions ex-ist and that the functional renormalization group meth-ods also find this solution. We do not comment on thesefurther other than as stated above: we do not attemptto rule out any other infinite ghost solutions.[61] J. C. R. Bloch, Phys. Rev.
D64 , 116011 (2001),arXiv:hep-ph/0106031[62] P. Maris, C. D. Roberts, and P. C. Tandy, Phys. Lett.
B420 , 267 (1998), arXiv:nucl-th/9707003 [nucl-th][63] A. Cucchieri, A. Maas, and T. Mendes, Phys. Rev.
D77 ,094510 (2008), arXiv:0803.1798 [hep-lat][64] S.-x. Qin, L. Chang, Y.-x. Liu, C. D. Roberts, and D. J.Wilson, Phys.Rev.
C84 , 042202 (2011), arXiv:1108.0603[nucl-th][65] A. Cucchieri and T. Mendes, PoS
LAT2007 , 297 (2007),arXiv:0710.0412 [hep-lat][66] P. Bicudo and O. Oliveira, PoS
LATTICE2010 , 269(2010), arXiv:1010.1975 [hep-lat][67] A. Cucchieri and T. Mendes, PoS
LATTICE2010 , 280(2010), arXiv:1101.4537 [hep-lat][68] K. Chetyrkin and T. Seidensticker, Phys. Lett.
B495 , 74(2000), arXiv:hep-ph/0008094 [hep-ph][69] J. A. Gracey, Phys. Lett.
B700 , 79 (2011),arXiv:1104.5382 [hep-ph][70] P. Maris and P. C. Tandy, Phys. Rev.
C60 , 055214(1999), arXiv:nucl-th/9905056[71] L. Chang and C. D. Roberts, Phys. Rev. Lett.103