The Saga of Landau-Gauge Propagators: Gathering New Ammo Attilio Cucchieri and Tereza Mendes InstitutodeFísicadeSãoCarlos,UniversidadedeSãoPaulo 1 CaixaPostal369,13560-970SãoCarlos,SP,Brazil 1 0 Abstract. Compelling evidence has recently emerged from lattice simulations in favor of the massive solution of the 2 Schwinger-DysonequationsofLandau-gaugeQCD.ThemainobjectionstotheselatticeresultsarebasedonpossibleGribov- n copyeffects.WerecentlyinstalledatIFSC-USPanewGPUclusterdedicatedtothestudyofGreen’sfunctions.Wepresent a hereourpointofviewontheSagaandthestatusofourproject.Wealsoshowdataforthe2Dcaseona25602lattice. J Keywords: Colorconfinement,Green’sfunctions,Landaugauge,Gribovcopies 5 PACS: 12.38.Aw12.38.Gc12.38.Lg 2 ] THE SAGA pact)definitionandthe(non-compact)stereographicpro- t a jection [10]. Their main conclusions are: “We further- l Intuitively,the explanationof color confinementshould more demonstrate that the massive branch observedfor - p beencodedintheinfrared(IR)behaviorofQCDGreen’s a2q2 < 1 does depend on the lattice definition of the e functions. The Landau-gauge Gribov-Zwanziger (GZ) gluon fields, and that it is thus not unambiguously de- h confinement scenario and the scaling solution obtained fined....Onemight still hope that this ambiguity will go [ bysolvingSchwinger-Dysonequations(SDEs)demand away at non-zero b in the scaling limit. While this is 1 —foranyspace-timedimensionsD≥2—anullgluon trueatlargemomenta,wedemonstrate...thattheambigu- v propagatoratzeromomentumandanIR-enhancedghost ity is still presentin the low-momentumregion,at least 9 propagator [1]. At present, there is wide agreement [2] for commonly used values of the lattice coupling such 7 that numerical simulations in minimal Landau gauge as b =2.3 or b =2.5 in SU(2)....The scaling proper- 7 show(intheinfinite-volumelimit):1)anIR-finitegluon ties such as exponent and coupling, on the other hand, 4 . propagatorD(p) in D =3,4 and a null D(0) in 2D, 2) appeartoberobustundervariationsofthediscretization 1 violationofreflectionpositivityforthegluonpropagator of the gauge fields...This emphasizes the importance of 0 inD=2,3,4and3)anessentiallyfreeghostpropagator understandingany discretization ambiguityof the asso- 1 1 G(p) in D =3,4 but IR-enhancedin 2D. Thus, the 3D ciated gluon mass, before concluding that this mass is : and4DresultssupportthemassivesolutionoftheSDEs now firmly established.” However, nowhere in Ref. [6] v [3,4],whilethe2Dcasehasascalingbehavior.Then,a are data at b =2.5 shown. On the other hand, data for i X naturalquestioniswhythe2Dcaseisdifferentfromthe a lattice volume 324 at b =2.5 in the SU(2) case are r 3D and 4D ones. At the moment, a possible answer to presentedinRef.[10]forthetwopropagators,usingthe a thisquestionhasonlybeenpresentedin[5]. standarddiscretizationandthestereographicprojection. Recently, three works [6, 7, 8] have allegedly shown Theconclusionof[10]isthat“...therearehardlyanydif- evidenceof the scaling IR behavioralso in 3D and 4D. ferencesbetweenthepropagatorsobtainedineachcase”. Here,wewillcommentonthesethreeworks. Thus,referringto thelastsentencereportedabovefrom Ref. [6], there are no discretization ambiguities in the evaluation of these propagators and the existence of a The b =0Case gluonmassisnowfirmlyestablished. We have already criticized Ref. [6] in our work [9]. Since that criticism has not been answered, we The Absolute Landau Gauge will repeat it here. The authors of Ref. [6] study the Landau-gaugegluonandghostpropagatorsinthestrong- Ref.[7]considerstheabsoluteLandaugauge,i.e.con- couplinglimitofpureSU(2)latticegaugetheory.These figurationsbelongingtothefundamentalmodularregion propagatorsare evaluatedusingdifferentdiscretizations L .Thisapproach,however,cannotyieldanIR-enhanced of the gluon field and, in particular, the standard (com- ghost propagator in 3D or in 4D. Actually, restricting theconfigurationspacetotheregionL makestheghost propagatorevenlesssingular[11].Thiscanbeseen,in- deed, also in Figures 5 and 12 of Ref. [7]. The author triestoexplaintheseresults,whichclearlygoagainstthe scalingsolution,bysayingthat“Thereasonforthisbe- havioroftheghostpropagator...maybeconnectedtothe volumeevolutionofthefirstGribovregionandthefun- damental modular region....The combined effect of the precise shape of the low-eigenvaluespectrum and a di- vergingnormalization of the eigenstates could be suffi- cienttoprovideamoreinfrareddivergentghostpropaga- torintheinfinite-volumeandcontinuumlimitsinabso- luteLandaugaugethaninminimalLandaugauge.”Thus, simulations in the absolute Landau gauge should agree withthescalingsolutionintheinfinite-volumelimit(for D = 3,4) due to a hypothetical diverging contribution oftheeigenstatesoftheFaddeev-Popovoperator.(Note that a possible way of quantifying this sentence would FIGURE 1. Plot of D(0)/V, together with its upper and be to prove that the lower bound of the ghost propaga- lowerbounds[13],asafunctionoftheinverselatticeside1/N. tor, introduced in Ref. [12], blows up sufficiently fast Inthethreecaseswegetabehavior1/Newithe=2.67(2). in the infinite-volume limit.) Moreover, that this effect should be important in the absolute Landau gauge and not in the minimal Landau gauge remains a mystery to predictionbyevaluatingthegluonpropagatorforlattice us, considering that any configuration belonging to the volumes up to 25602 at b =10 (i.e. with a lattice size absoluteLandaugaugeisalsoaconfigurationofthemin- L≈460fm). In Fig. 1 we plot the volume dependence imalLandaugauge.Theauthoralsoadds“Afinalproof of D(0)/V, together with the upper and lower bounds is, of course, only that in the absolute Landau gauge at introduced in Ref. [13]. The three sets of data clearly sufficientlylargevolumetheghostpropagatorwouldbe extrapolate to zero faster than 1/V, implying D(0)=0 more singular than in the minimal Landau gauge. The in2D. volume dependence of the propagator in both gauges found here is as expected if this is the case....Hence, at the currentpoint it seems more appropriateto compare The BGauges the lattice results in absolute Landaugauge,rather than inminimalLandaugauge,withthosefromfunctionalre- After considering the absolute Landau gauge in Ref. sultswhichexhibitascalingbehaviorinthefarinfrared.” [7], a new set of gauges— called B gauges— was in- Thesestatements—whicharesomewhatsibylline,since troducedby the same author [8]. In this case one looks the data do not show an IR-enhanced ghost propagator alongeachorbitforatransverseconfigurationthatyields — may be the reason why Ref. [7] is (wrongly) cited agivenvalueBfortheghostdressingfunctionD (p)= G asanumericalverificationofthescalingbehaviorforD p2G(p) at the smallest non-zero momentum p . This min >2. definitiondoesnotsolvetheGribovambiguity[8].More- over, in order to find an IR-enhanced ghost propagator oneneedstofavorconfigurationsclosertothe firstGri- The2DCase bovhorizon¶ W .Thisistheoppositeofwhatisdonein theabsoluteLandaugauge,whereonefavorsconfigura- Let us note that if the massive behavior observed in tions well inside the first Gribov region W [14]. Thus, 3D and in 4D could be related to discretization effects, if the B gaugesshould producethe scaling solution, “it as suggested by Ref. [6], or to Gribov-copy effects, as couldwellbethat...theabsoluteLandaugaugeisnotcon- reportedin[7],thentheseeffectsshouldalsobepresent nected to a scaling behavior” [8], in disagreementwith forD =2andoneshouldnotfindinthiscasea scaling Ref.[7]. behavior. In this respect, still in Ref. [7], the author Themainresultofthisapproachisthattheghostprop- makesthe followingprediction:“A consequenceof this agator is strongly affected by the choice of configura- scenario is that it should be expected that also in two tion on each orbit, in such a way that its values are en- dimensions, for sufficiently large volumes and number closed in a “corridor”. In particular, in 3D and 4D the of Gribov copies, an infrared finite gluon propagator is upperboundofthiscorridor“isstronglyincreasingwith obtainedintheminimalLandaugauge.”Wecheckedthis volume”.Atthesametime,thegluonpropagatorseems to be B-independent and we should have D(0)> 0 in merically the explanation presented in [5]? A clear an- theinfinite-volumelimit.Thus,theonlyscalingsolution swertothesequestionsprobablyrequiresnewideasand that can be obtainedwith the B gaugesseems to be the betterdata(especiallyintheghostsector).Unprovenhy- onecorrespondingtoacriticalexponentk =1/2,which pothesesandhappycoincidencesshouldonthecontrary wasneverthepreferredvalueinscaling-solutionworks. betreatedwithgreatcaution. Moreover,iftheinfraredexponentis1/2thenin4Done We recently installed at IFSC–USP a new machine shouldhaveD (p)∼1/p.Since1/p ≈L, theupper with 18 CPUs Intel quadcore Xeon 2.40GHz (with In- G min boundofthecorridorshouldgrowatleastasfastasthe finiBand network and a total of 216 GB of memory) lattice size L, in order to support the scaling solution. and 8 NVIDIA Tesla S1070 boards (500 Series), each Onecanverifythatthisisnotthecasewiththe4Ddata with 960 cores and 16 GB of memory. The peak per- presentedinFig.3of[8](thecurveshouldbehyperbolic formance of the 8 Tesla boards is estimated in about asafunctionof1/L). 2.8 Tflops in double precision and 33 Tflops in single Let us note that one of the motivations for the intro- precision.Thismachinewill be usedmainlyforstudies ductionoftheBgaugesisthepossiblerelationwiththe ofGreen’sfunctionsindifferentgauges(Landau,Feyn- one-parameterfamilyofsolutionsobtainedbyfunctional manandCoulomb)forvariousSU(N )gaugegroups.In c methods[3, 15]. In this respectone shouldstress, how- particular, the GPUs will be used for the inversion of ever, that the B gauges are related to different Gribov theFaddeev-Popovmatrixusingconjugategradient.This copies on each orbit. On the other hand, the configura- computerwillallowustoperformanextensivestudyof tionspaceisnotencodedintheSDEsandthisinforma- theghostsector.Webelievethatthisnewammowillhelp tionhastobeputinbyhand.Thiscanbedoneinsimple usclarifytheissuesaddressedabove. cases [16], if all Gribov copies are known, but nobody knowshowtodoitinarealisticcase.Thus,thisrelation seemsatthemomentquiteaccidental.Evenmorefanci- ACKNOWLEDGMENTS fulseemstousthehypotheticalconnectionbetweenthe Kugo-Ojima(KO)approach[1]anda(possible)scaling WethankFAPESP(grant#2009/52213-2)andCNPqfor solution obtained using B gauges. Indeed, this connec- support. tion requires “subtle cancellation” [8], since one has to relate an average over all Gribov copies to results ob- tainedbyselectingspecificcopiesinsidethefirstGribov REFERENCES region W . In our opinion, the lack of BRST invariance whenthefunctionalspaceisrestrictedtoW [17]obscures 1. 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