Going beyond the propagators of Landau gauge Yang-Mills theory 3 1 0 Markus Q. Huber∗, Lorenz von Smekal 2 InstitutfürKernphysik,TechnischeUniversitätDarmstadt,Schlossgartenstr. 2,64289 n Darmstadt,Germany a E-mail: [email protected], J [email protected] 4 1 ] We present results for the propagators and the ghost-gluon vertex of Landau gauge Yang-Mills h theoryobtainedfromDyson-Schwingerequations. Solvingthesethreequantitiessimultaneously t - p constitutesanewstepintruncatingtheseequations. Wealsointroduceanewmodelforthethree- e gluon vertex that is motivated by lattice results. It features a zero crossing which is confirmed h [ a posteriori by a Dyson-Schwinger calculation. Within our setup we can reproduce lattice data 1 very well. We establish that also for the ghost-gluon vertex a difference between decoupling v andscalingsolutionsispresent. Forthescalingsolutionwediscussthepossibilityofmodifying 0 8 the infrared exponents via an angle dependence of the ghost-gluon vertex. However, no such 0 dependence is found in our calculations. Finally, we calculate the Schwinger function for the 3 . gluonpropagator. 1 0 3 1 : v i X r a XthQuarkConfinementandtheHadronSpectrum, October8-12,2012 TUMCampusGarching,Munich,Germany ∗Speaker. (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ GoingbeyondthepropagatorsofLandaugaugeYang-Millstheory MarkusQ.Huber 1. Two-pointfunctions Extracting non-perturbative information from a quantum field theory is in most cases a hard task. One possibility is to calculate its Green functions which are related by several sets of func- tionalequations. AconcreteexampleofsuchasetaretheequationsofmotionoftheGreenfunc- tions, called Dyson-Schwinger equations (DSEs) [1, 2]. An inherent problem is that there are infinitely many equations. The usual way to make quantitative calculations is to neglect some Green functions and/or model others. The remaining ones are then solved self-consistently. The natural way to test the reliability of such a truncated system is to compare it with a larger trunca- tion. However,inpracticethisisadifficulttaskanditcantakesometimeuntilsuchacalculations becomesfeasiblebothfromtheconceptualandfromthecomputationalpointofview. Anexample for this is provided below. Nevertheless, it has to be stressed that raising the number of included Greenfunctionscanallowasystematicimprovement. Inpractice, however, itismorecommonto estimatetruncationartifactsfromcomparisonswithothermethods. Sometimes we are in an even better situation and can estimate the influence of truncations. ThemostprominentexampleissurelythehighenergyregimeofQCD.Sincethestronginteraction is asymptotically free, a perturbative loop expansion provides a reliable approximation which can be improved systematically by including higher powers of the coupling. On the other end of the energy spectrum, in the infrared (IR), it can also be possible to determine a clear hierarchy of leading and suppressed contributions, if a so-called scaling solution is realized [3, 4, 5]. Then the qualitative behavior of all Green functions can be determined in form of IR power laws. Landau gaugeYang-Millstheorymayactuallypossesssuchasolution[6,7,8,9]. Knowledgeofadditional informationlikethisisofcourseausefulguidelineinconstructingtruncations. InthepresentworkweconsiderLandaugaugeYang-Millstheory. Itisillustrativeinthelight ofthediscussionabovetorecallthedevelopmentoftruncationsinthiscase. Inaverybasicapprox- imation the ghosts and all four-point functions were neglected [10]. The only DSE to be solved is the gluon propagator DSE. The resulting gluon propagator is IR divergent, which is nowadays known to be wrong. The next step is the inclusion of ghosts, so that a coupled system of two DSEs has to be solved. This step was taken 18 years later [6, 7] and resulted in a drastic change, as the gluon dressing became IR suppressed. Based on analytic and numeric results from various methods,wecanbeconfidentthatatruncationatthislevelyieldsqualitativelyreliableresults. Within this class of truncations two types of solutions can be obtained. However, this is not relatedtothetruncationitselfbuttotheboundaryconditionsimposedontheDSEsandadifference isonlyvisibleat small momenta[8]. The decouplingtypeconsistsofafamilyof solutions which have an IR finite ghost dressing and an IR finite gluon propagator [8, 11, 12, 13, 14]. The scaling solution, on the other hand, features an IR divergent ghost dressing and an IR vanishing gluon propagator[6,7,8,9]. Itisstillunderdebateifbothsolutionsexist,especiallysinceall"standard" lattice calculations recover the decoupling type of solutions. However, lattice results do depend on the method used for sampling Gribov copies, see, e.g., [15, 16]. Recently a way was found to sampleGribovcopiessuchthattheresultsaremodifiedasexpectedfromfunctionalequations[17]. Inthefollowingwewillpresentresultsfromanewtruncationthatgoesbeyondthepropagator level by including the ghost gluon-vertex dynamically and employs a model for the three-gluon vertexinagreementwithlatticeresults[9]. 2 GoingbeyondthepropagatorsofLandaugaugeYang-Millstheory MarkusQ.Huber 2. Three-gluonvertex Forthethree-gluonvertexweemploythefollowingmodelmotivatedbylatticecalculations: (cid:18)x+y+z(cid:19)α (cid:18)x+y+z(cid:19)β DA3(p,q,−p−q)=G Z +h G(x+y+z)3(f3g(x)f3g(y)f3g(z))4, IR 2 2 (2.1) withthedampingfactors Λ2 f3g(x):= 3g . (2.2) Λ2 +x 3g We choose h <0 so that the vertex becomes negative for low momenta. This is observed with IR Monte Carlo simulations in two and three dimensions [18, 19], but not in four [19], probably becausethelatticedatadoesnotreachfarenoughintotheIR.WeperformedaDSEcalculationfor thevertexbasedontheresultsobtainedforthepropagatorsandtheghost-gluonvertexandtaking into account only the IR leading diagram. We indeed found a zero crossing, see Fig. 1, which is compatiblewiththeavailablelatticedata. Thepositionofthecrossingmaystillchangeinamore advancedtruncation,butitisveryunlikelythatitwillvanish. Thisiscorroboratedbyresultsintwo dimensions,wheremoredetailedcalculationswereperformedandtheinfluenceofotherdiagrams wasfoundtobenegligibleintheIR[20]. In Eq. (2.1) the IR leading part is damped out for larger momenta by the damping functions f(x). Theothertermismotivatedbyperturbationtheory: Bychoosingα andβ appropriately,we reproducetheanomalousdimensionofthethree-gluonvertex,whichisγ =1+3δ =17/44with 3g δ =−9/44theghostanomalousdimension. Thesecondconditiontofixα andβ isIRfiniteness. Weobtainforthescalingsolutionα =−2−6δ andβ =−1−3δ andfordecouplingα =3+1/δ andβ =0. Inordertoproducethecorrectanomalousdimensionsforthepropagators,itisnecessary tomakeanadjustmenttothegluonloopinthegluonDSE,wheretherenormalizationconstantZ 1 ofthethree-gluonvertexappears. Wereplaceitbyamomentum-dependentrenormalizationgroup improvementtermthatequalstheUVpartofthethree-gluonvertexmodel[7]. Suchamodification is unavoidable in order to obtain the correct anomalous dimensions, because they result from the resummationofinfinitelymanydiagramswhichcannotbetakenintoaccountdirectlyinthepresent approach. In Fig. 1 results for the gluon dressing function for varying values of Λ are shown. The 3g influenceofthezerocrossingisclearlyvisible: Ifitspositionmovestohighermomenta,thebump in the gluon dressing function rises. In general the model shows good agreement with the lattice data, which is, however, for two colors while our calculation is for three colors. From Fig. 1 we concludethatthezerocrossing,ifchosenatavalueinagreementwithlatticedata,hasnoinfluence on the propagators. The reason is that in the gluon loop the vertex is multiplied with two gluon propagators which are IR suppressed. Besides the IR part also the Bose symmetry of the vertex has an effect on the mid-momentum regime of the propagators, see ref. [9]. In the following we willusethethree-gluonvertexmodeltomimiceffectsofthetwo-loopdiagramsinthegluonDSE bychoosingthezerocrossingsuchthatthepropagatorsagreeasgoodaspossiblewithlatticedata. Themodelwiththecorrespondingchoiceofparametersiscalledoptimizedeffectivevertex. 3 GoingbeyondthepropagatorsofLandaugaugeYang-Millstheory MarkusQ.Huber Z p2 DA3 p2,p2,p2 4 2.0 (cid:236) (cid:242) (cid:72) (cid:76) 3 1.5 (cid:236) (cid:72) (cid:76) (cid:242) (cid:242) (cid:242) (cid:242) 01..50 (cid:236)(cid:242)(cid:230)(cid:236)(cid:224)(cid:236)(cid:230) (cid:236)(cid:242) (cid:224) (cid:242)(cid:224) (cid:224) (cid:224) 2 (cid:236)(cid:230) (cid:230) 0.0 1 2(cid:230)(cid:236)(cid:230)(cid:230) 3 4 5 pGeV (cid:224)1 (cid:45)0.5 (cid:236) 0 pGeV (cid:45)1.0 (cid:64) (cid:68) 0 1 2 3 4 5 Figure 1: Comparison of different parameters for the three-gluon vertex model: h = −1, Λ = IR (cid:64)3g (cid:68) 0,0.86,1.08,1.31,1.53,1.7GeV. Left: Symmetricconfigurationofthevertexmodel. Thicklinesfromleft torightcorrespondtorisingvaluesoftheparameterΛ . ThethinlineistheIRleadingcontributionofthe 3g three-gluonvertexcalculatedfromitsDSEshowingazerocrossingatverylowmomenta. Atzeromomen- tumitisfinite. Individualpointsarelatticedatafromref.[19]: RedcirclescorrespondtoN=16/β =2.2, greensquarestoN=16/β =2.5,bluediamondstoN=22/β =2.2andorangetrianglestoN=22/β =2.5. Right: Thecorrespondinggluondressingfunction. CurvesfrombottomtotopcorrespondtovaluesofΛ 3g fromlowtohigh. 3. Ghost-gluonvertex The ghost-gluon vertex is an important quantity in modern truncation schemes of functional equations, because it is the only primitively divergent vertex via which the ghost interacts. The reason for the success of modern truncation schemes is that this vertex has only a very mild mo- mentum dependence. Thus approximating it by its bare expression produces reasonable results. Non-perturbative modifications of the vertex were investigated with various approaches: Monte- Carlosimulations[15,18,19],semi-perturbativeDSEcalculations[21,22],OPEanalysis[23]and based on this indirect studies via effects in the ghost DSE [24]. Here we will include the vertex dynamically, i.e., we solve the DSEs of the two propagators and the vertex simultaneously. We employthefollowingbasisforthevertex: ΓAc¯c,abc(k;p,q):= fabcΓAc¯c(k;p,q):=igfabc(cid:0)p A(k;p,q)+k B(k;p,q)(cid:1), (3.1) µ µ µ µ where the momenta k, p and q refer to the gluon, the anti-ghost and the ghost, respectively. Due to the transverse gluon propagator of Landau gauge, only the dressing A(k;p,q) is relevant. The correspondingDSEaswellasthoseforthepropagatorscanbefoundinref.[9]. Theeffectofthedynamicghost-gluonvertexisillustratedinFig.2,wherecalculationswitha bare ghost-gluon vertex are compared to one with a dynamic vertex. The red/continuous line cor- respondstothesolutionwiththeoptimizedeffectivethree-gluonvertexandadynamicghost-gluon vertex. The corresponding calculation with the same parameters for the three-gluon vertex but a bare ghost-gluon vertex is represented by the blue/dotted line. However, although there is quite some change in the mid-momentum regime of the gluon propagator, this only provides a quali- tative picture, since the model for the three-gluon vertex depends on the propagators and is thus differentinthetwocalculations. Finally,wenotethatitispossibletoapproximatelyaccommodate 4 GoingbeyondthepropagatorsofLandaugaugeYang-Millstheory MarkusQ.Huber Z p2 G p2 5 4 (cid:72) (cid:76) (cid:72) (cid:76) 4 3 3 2 1 2 0 pGeV 1 pGeV 0 1 2 3 4 5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Figure 2: Comparison of the gluon propagator dressing function Z(p2) and the ghost dressing function (cid:64) (cid:68) (cid:64) (cid:68) G(p2)fordifferentsetups. Red/continuousline:dynamicghost-gluonvertex,Λ =2.1GeV. Green/dashed 3g line: bareghost-gluonvertex,Λ =1.8GeV. Blue/dottedline: bareghost-gluonvertex,Λ =2.1GeV. 3g 3g the modifications introduced by a dynamic vertex also in the three-gluon vertex model. This can beseenfromthegreen/dashedline,forwhichtheparametersofthemodelwereadjustedsuchthat thegluonpropagatorfromthelatticeisreproducedusingabareghost-gluonvertex. 4. Results Technicaldetailsonthecalculationscanbefoundinref.[9]. TheprogramsDoFun[2,25]and CrasyDSE[26]wereusedforderivingandsolvingtheDSEs. The resulting dressing function for the propagators are shown in Fig. 3. As can be seen, it is possibletochoosetheparameterΛ suchthatweobtainaverygoodagreementwithlatticedata. 3g We would like to stress that this is true for both dressings. The Schwinger function ∆(t) of the gluon propagator is shown in Fig. 4. It turns negative at about 2.2fm. This value is higher than observed in previous calculations [8]. Furthermore, a second zero crossing is observed. Since the Schwinger function is partially negative, positivity is violated and the gluon cannot represent an asymptoticstateofthephysicalstatespace. Results for the ghost-gluon vertex are shown in Fig. 5. The depicted momentum configura- tion corresponds to almost anti-parallel gluon and anti-ghost momenta. As expected the dressing is constant in the IR and the UV and has a small bump in the intermediate momentum regime. The ambiguity between scaling and decoupling is also present in the vertex, as the IR constant is different: For decoupling the dressing is 1, because the loop corrections are IR suppressed by the gluon propagators. For scaling, however, they yield a non-vanishing contribution which lifts the dressingabove1. Thiscanalsobeseenfromthered/continuouslineinFig.4,whereadditionally twodifferentdecouplingsolutionsareplotted. For the scaling solution the exponents of the IR power laws for all dressing functions can be calculated in terms of one variable κ [3, 4], which can be calculated analytically [29, 30]. Under theassumptionthattheIRlimitoftheghost-gluonvertexisregular,itsvalueisκ=0.595353. This value can only be modified by a non-analyticity at zero momentum [30]. One possibility for this, a dependence on the angle between two momenta, was studied in ref. [20]. Here we tried to find a deviation by not fixing the value of κ manually, as is normally done, but leaving it free. Thus 5 GoingbeyondthepropagatorsofLandaugaugeYang-Millstheory MarkusQ.Huber G p2 Z p2 6 4 (cid:230)(cid:224)(cid:224)(cid:230)(cid:224) 5(cid:72) (cid:76) (cid:72) (cid:76) (cid:230)(cid:224) (cid:224)(cid:230)(cid:224) 23 (cid:224)(cid:230)(cid:224)(cid:224)(cid:230) (cid:230)(cid:224)(cid:230)(cid:224)(cid:224)(cid:230)(cid:224)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:224)(cid:230)(cid:224)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:224)(cid:230)(cid:224)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:224)(cid:230)(cid:224)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:224)(cid:230)(cid:224)(cid:230)(cid:224)(cid:224)(cid:230)(cid:224) 34 (cid:224) (cid:224)(cid:224) (cid:224) 1 (cid:224) 2 (cid:230) (cid:224) (cid:230)(cid:224) (cid:224) (cid:230) (cid:224) 0(cid:230)(cid:224) pGeV 1 p 0 1 2 3 4 5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Figure3:ComparisonoftheresultsforthegluonpropagatordressingfunctionZ(p2)andtheghostdressing (cid:64) (cid:68) function G(p2) with lattice data [15]. The red/continuous line was obtained with a dynamic ghost-gluon vertexandtheoptimizedeffectivethree-gluonvertexwithh =−1andΛ =2.1GeV. Forreferencethe IR 3g green/dashedlineisshown, whichwasobtainedwiththevertexmodelsofrefs.[27,28]andabareghost- gluon vertex. Lattice data [15] is for β =6 and lattice sizes of L=32 (blue circles) and L=48 (orange squares). (cid:68)t A p2;p2,p2 1 1.6 (cid:200) (cid:72)(cid:76)(cid:200) 1.5 (cid:72) (cid:76) 1.4 0.01 1.3 1.2 10(cid:45)4 1.1 1.0 0.9 10(cid:45)6 tfm pGeV 0 2 4 6 8 10 0.001 0.01 0.1 1 10 (cid:64) (cid:68) (cid:64) (cid:68) Figure4:Left:TheabsolutevalueoftheSchwingerfunctionofthegluonpropagator. Intheregionbetween approximately2.2GeV and9GeV thefunctionisnegative. Right: Theghost-gluonvertexatthesymmetric point. Thered/continuouslineisfromthescalingsolution,andthegreen/dashedandblue/dottedlinesfrom twodifferentdecouplingsolutions. the ghost-gluon vertex could develop an angle dependence, which was, however, not observed. Thedifferencesbetweentheverticesasobtainedfrombothprocedureswerebelow0.1%andthus within numerical accuracy. The value of κ was found to be a little bit higher, slightly above 0.6. Thuswedidnotfindanynon-analyticitiesintheghost-gluonvertex. 5. Summaryandconclusions IngeneralDSEsareausefultoolfornon-perturbativeinvestigations. ForLandaugaugeYang- Millstheoryweshortlyrecalledthehistoryofdifferenttruncationsandexplicitlydemonstratedthat furtherimprovingthetruncationleadsonlytoquantitativeeffects. Oneimprovementpresentedhere isthedynamicalinclusionoftheghost-gluonvertex. Wefindthattheambiguityofdecouplingand 6 GoingbeyondthepropagatorsofLandaugaugeYang-Millstheory MarkusQ.Huber Figure5: Ghost-gluonvertexdressingwithalmostanti-parallelgluonandanti-ghostmomentafordecou- pling(left)andscaling(right). scaling solutions persists also at the level of three-point functions. We also introduced a model forthethree-gluonvertexthatproperlydescribeslatticedata. 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