Polyakov loop analysis with Dirac-mode expansion 3 1 0 2 n a J 4 TakumiIritani∗,ShinyaGongyo,andHideoSuganuma 1 DepartmentofPhysics,KyotoUniversity,Kitashirakawaoiwake,Sakyo,Kyoto606-8502,Japan ] E-mail:[email protected] t a l - p Inorderto investigatethe directrelationbetweenconfinementandchiralsymmetrybreakingin e QCD, we investigatethe Polyakovloopin termsof the Dirac eigenmodesin both confinedand h [ deconfinedphases. UsingtheDirac-modeexpansionmethodin SU(3)lattice QCD,we analyze the contribution of low-lying and higher Dirac-modes to the Polyakov loop, respectively. In 1 v theconfinedphasebelowT , afterremovinglow-lyingDirac-modes,thechiralcondensatehq¯qi c 5 is largely reduced, however, the Polyakov loop remains almost zero and Z -center symmetry 5 3 8 is unbroken. These results indicate that the system is still in the confined phase without low- 2 lying Dirac-modes. By higher Dirac-modes cut, the Polyakov loop also remains almost zero . 1 belowT . WealsoanalyzethePolyakovloopinthedeconfinedphaseaboveT. Wefindthatthe 0 c c 3 PolyakovloopandZ -symmetrybehaviorareinsensitivetolow-lyingandhigherDirac-modesin 3 1 bothconfinedanddeconfinedphases. : 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/ PolyakovloopanalysiswithDirac-modeexpansion TakumiIritani 1. Introduction Quantum chromodynamics (QCD) is the fundamental theory of the strong interaction, how- ever,itsnon-perturbative properties suchasconfinementandchiralsymmetrybreaking arenotyet wellunderstood. Inparticular, toclarify thecorrespondence betweenconfinement andchiralsym- metrybreakingisoneofdifficultandinteresting subjects[1,2,3,4,5,6,7]. Asanevidenceofthe close relation between them, lattice QCDcalculation shows that simultaneous deconfinement and chiralphasetransition atfinitetemperature [8]. As shown in the Banks-Casher relation [9], the chiral condensate hq¯qi is proportional to the Diraczero-mode densityas hq¯qi=−lim lim p hr (0)i, (1.1) m→0V→¥ where r (l ) is the Dirac spectral density. Thus, the low-lying Dirac eigenmodes directly relate to chiralsymmetrybreaking, however,theirrelation toconfinementisstillunclear. Therefore, itisinteresting toanalyze confinement interms oftherelevant degreesoffreedom for chiral symmetry breaking, i.e., the low-lying Dirac eigenmodes. For example, based on the Gattringer’s formula [1], the Polyakov loop can be investigated by the sum of Dirac spectra with twisted boundary condition on lattice [2, 3, 4]. In our previous studies [5, 6], we formulated the Dirac-modeexpansionmethodinlatticeQCD,andinvestigated theDirac-modedependence ofthe Wilson loopandthe interquark potential. Itisalsoreported thathadrons still existasbound states without“chiralsymmetrybreaking” byremovinglow-lyingDirac-modes[10,11]. Inthispaper,weinvestigatetheDirac-modedependenceofthePolyakovloopinbothconfined and deconfined phases at finite temperature, using Dirac-mode expansion method in lattice QCD [5, 6, 7]. In Sec. 2, we review the Dirac-mode expansion method, and formulate the Dirac-mode projectedPolyakovloop. InSec.3,weperformthelatticeQCDcalculations forthePolyakovloop withDirac-modeprojection. Section4isdevoted forthesummary. 2. Dirac-mode expansionmethod inlatticeQCD Here, weintroduce the Dirac-mode expansion technique inlattice QCD[5, 6, 7], and formu- lationoftheDirac-modeprojected Polyakovloop. 2.1 Dirac-modeexpansioninlatticeQCD Usingthelink-variableUm ∈SU(Nc),theDiracoperator D/ =gm Dm isgivenby D/x,y ≡ 21a (cid:229) 4 gm (cid:2)Um (x)d x+mˆ,y−U−m (x)d x−mˆ,y(cid:3), (2.1) m =1 with a lattice spacing a, andU−m (x)≡Um†(x−mˆ). Here, g -matrix is defined to be hermitian, i.e., gm† =gm . Thus, D/ becomes an antihermitian operator, and Dirac eigenvalues are pure imaginary. Weintroduce thenormalized Diraceigenstate |ni,whichsatisfies D/|ni=il |ni, (2.2) n 2 PolyakovloopanalysiswithDirac-modeexpansion TakumiIritani withl ∈R,andaneigenfunction y (x)isexpressed as n n y (x)≡hx|ni, (2.3) n whichsatisfiesD/y =il y . n n n We introduce the operator formalism in lattice QCD [5, 6, 7], which is constructed from the link-variable operatorUˆm . Thelink-variable operatorisdefinedbythematrixelement as hx|Uˆm |yi=Um (x)d x+mˆ,y, (2.4) usingtheoriginal link-variableUm (x). WedefinetheDirac-modematrixelement hn|Uˆm |mias hn|Uˆm |mi = (cid:229) hn|xihx|Uˆm |x+mˆihx+mˆ|mi=(cid:229) y n†(x)Um (x)y m(x+mˆ), (2.5) x x withtheDiraceigenfunction y (x). n Considering thecompleteness relation (cid:229) |nihn|=1,anyoperator Oˆ canbeexpressed as n Oˆ =(cid:229) (cid:229) |nihn|Oˆ|mihm|, (2.6) n m using the Dirac-mode basis. Note that this procedure is just the insertion of unity, and it is math- ematically correct. This expansion (2.6) is the mathematical basis of the Dirac-mode expansion method [5,6,7]. Next, weconsider the Dirac-mode projection by introducing projection operator as Pˆ≡ (cid:229) |nihn|, (2.7) n∈A forarbitrary set A ofeigenmodes. Forexample,IRandUVmode-cutoperators are givenby Pˆ ≡ (cid:229) |nihn|, Pˆ ≡ (cid:229) |nihn|, (2.8) IR UV |ln|≥L IR |ln|≤L UV withtheIR/UVcutoffscaleL andL . IR UV Usingtheprojection operator Pˆ,theDirac-modeprojected link-variable operatorisgiven by UˆmP≡PˆUˆm Pˆ= (cid:229) (cid:229) |nihn|Uˆm |mihm|. (2.9) n∈A m∈A WecaninvestigatetheDirac-modedependenceofvariouskindsofquantities,e.g.,theWilsonloop [5,6],usingtheprojected link-variableUˆmP instead oftheoriginallink-variable operatorUˆm . 2.2 PolyakovloopoperatorandDirac-modeprojection Next, weformulate the Dirac-mode projected Polyakov loop. Here, weconsider the periodic SU(3) lattice with the space-time volume V =L3×N and the lattice spacing a. In the operator t formalism oflatticeQCD,thePolyakovloopoperatorisgivenby Lˆ ≡ 1 (cid:213)Nt Uˆ = 1 UˆNt (2.10) P 3V 4 3V 4 i=1 3 PolyakovloopanalysiswithDirac-modeexpansion TakumiIritani withthetemporallink-variable operatorUˆ . Bythefunctional trace“Tr”,thePolyakov loopoper- 4 atorcoincides withthestandard definitionas TrLˆ = 1 Tr{(cid:213)Nt Uˆ }= 1 tr(cid:229) h~x,t|(cid:213)Nt Uˆ |~x,ti P 4 4 3V 3V i=1 ~x,t i=1 = 1 tr(cid:229) h~x,t|Uˆ |~x,t+aih~x,t+a|Uˆ |~x,t+2ai···h~x,t+(N −1)a|Uˆ |~x,ti 4 4 t 4 3V ~x,t 1 (cid:229) = tr U (~x,t)U (~x,t+a)···U (~x,t+(N −1)a)=hL i, (2.11) 4 4 4 t P 3V ~x,t where“tr”denotes thetraceoverSU(3)colorindex. WedefinetheDirac-modeprojected PolyakovloophLproj.ias P LpProj. ≡ 31VTr{(cid:213)Nt Uˆ4P}= 31VTr(cid:8)PˆUˆ4PˆUˆ4Pˆ···PˆUˆ4Pˆ(cid:9) i=1 = 1 tr (cid:229) hn |Uˆ |n ihn |Uˆ |n i···hn |Uˆ |n i. (2.12) 3V 1 4 2 2 4 3 Nt 4 1 n1,n2,...,nNt∈A Inparticular, theIRandtheUVDirac-modeprojected Polyakovlooparedenoted as hL i ≡ 1 tr (cid:229) hn |Uˆ |n i···hn |Uˆ |n i, (2.13) P IR 3V 1 4 2 Nt 4 1 |lni|≥L IR hL i ≡ 1 tr (cid:229) hn |Uˆ |n i···hn |Uˆ |n i, (2.14) P UV 3V 1 4 2 Nt 4 1 |lni|≤L UV withtheIR/UVeigenvalue cutoffL andL . IR UV 3. Lattice QCDcalculationforDirac-modeprojected Polyakovloop Inthissection,wecalculatetheDirac-modeprojectedPolyakovloopusingSU(3)latticeQCD at the quenched level. We evaluate the full Dirac eigenmodes using LAPACK [12]. For actual calculation, weusetheeigenmode basisoftheKogut-Susskind (KS)operatorof DKx,yS≡ 21a (cid:229) 4 h m (x)(cid:2)Um (x)d x+mˆ,y−U−m (x)d x−mˆ,y(cid:3), (3.1) m =1 withh 1(x)≡1andh m (x)≡(−1)x1+···+xm−1 (m ≥2)inordertoreducethecomputational cost. The use of the KS-Dirac operator gives the same result as the original Dirac operator in Eq. (2.1) for thePolyakovloop. 3.1 Theconfinedphase First, we analyze the Polyakov loop in the confined phase below T . Here, we use 64 lattice c withb =5.6, whichcorresponds tothe lattice spacing a≃0.25 fmand T ≡1/(Na)≃0.13 GeV t [5,6,7]. 4 PolyakovloopanalysiswithDirac-modeexpansion TakumiIritani 1400 64 lattice with β = 5.6 1400 ΛIR = 0.5a-1 1400 ΛUV = 2.0a-1 1200 1200 1200 1000 1000 1000 ρλVa () [] 468000000 ρλVa () []IR 468000000 ρλVa () []UV 468000000 200 200 200 0 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 λ [a-1] λ [a-1] λ [a-1] Figure1: Thespectraldensityr (l )oftheDiracoperatoron64 latticewithb =5.6,i.e.,a=0.25fm: (a) originalspectraldensity,(b)IR-cutr (l )atL =0.5a−1,(c)UV-cutr (l )atL =2.0a−1. IR IR UV UV 0.3 0.3 0.3 β = 5.6 IR cut Λ = 0.5a-1 UV cut Λ = 2.0a-1 IR UV 0.2 0.2 0.2 0.1 R 0.1 V 0.1 L hiP 0 L iPI 0 L iPU 0 Im -0.1 m h -0.1 m h -0.1 I I -0.2 -0.2 -0.2 -0.3 -0.3 -0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Re hL i Re hL i Re hL i P P IR P UV Figure 2: The scatter plot of the Polyakov loop in the confined phase on 64 lattice with b =5.6, i.e., a=0.25 fm and T ≡1/(Na)≃0.2 GeV. (a) The original (no Dirac-mode cut) Polyakov loop. (b) The t low-lyingDirac-modecutatL =0.5a−1. (c)ThehigherDirac-modecutatL =2.0a−1. IR UV Figure 1 shows the lattice QCD results for the Dirac spectral density r (l ), and IR/UV-cut spectraldensityr (l )≡r (l )q (|l |−L ),r (l )≡r (l )q (L −|l |)withL =0.5a−1 and IR IR UV UV IR L =2.0a−1. ThetotalnumberoftheKSDirac-modeisL3×N ×3=3888,andbothmodecuts UV t correspond toremovingabout400modes. Figure 2 shows the scatter plot of the original Polyakov loop, low-lying Dirac-modes cut at L = 0.5a−1, and the higher Dirac-modes cut at L = 2.0a−1, respectively. As shown in IR UV Fig.2(a),thePolyakovloopisalmostzero,i.e.,hL i≃0,whichindicates theconfinedphase. P Then, weconsider low-lying Dirac-modes projection, which leads tothe effective restoration ofchiralsymmetrybreaking[6,9,10,11]. Inthepresence of theIRcutL ,thequarkcondensate IR isgivenby 1 (cid:229) 2m hq¯qi =− . (3.2) IR V l 2+m2 ln≥L IR n At the IR cut parameter L =0.5a−1 ≃0.4 GeV, only 2% of the quark condensate remains as IR hq¯qi /hq¯qi≃0.02 around the physical region m≃5 MeV [6]. However, as shown in Fig. 2(b), IR thePolyakovloophL i remainsalmostzeroandZ -centersymmetryisunbroken,andthesefacts P IR 3 indicatethatthesystemstillremainsintheconfinedphase,evenwithoutchiralsymmetrybreaking. In addition to the low-lying mode cut, we show the higher Dirac-modes cut in Fig. 2(c). In thiscase,thechiralcondensationisalmostunchanged, andthePolyakovloopremainsalmostzero. Therefore, thePolyakovloopisinsensitive tobothlow-lyingandhigherDiraceigenmodes. 5 PolyakovloopanalysiswithDirac-modeexpansion TakumiIritani 0.3 0.3 0.3 β = 6.0 IR cut Λ = 0.5a-1 UV cut Λ = 2.0a-1 IR UV 0.2 0.2 0.2 0.1 R 0.1 V 0.1 L hiP 0 L iPI 0 L iPU 0 Im -0.1 m h -0.1 m h -0.1 I I -0.2 -0.2 -0.2 -0.3 -0.3 -0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Re hL i Re hL i Re hL i P P IR P UV Figure 3: The scatter plot of the Polyakov loop in the deconfinedphase on 63×4 lattice with b =6.0, i.e.,a=0.10fmandT ≡1/(Na)≃0.5GeV.(a)Theoriginal(noDirac-modecut)Polyakovloop. (b)The t low-lyingDirac-modecutatL =0.5a−1. (c)ThehigherDirac-modecutatL =2.0a−1. IR UV 3.2 Thedeconfinedphaseathightemperature Next,weinvestigatethePolyakovloopinthedeconfinedphaseathightemperature. Here,we use 63×4 lattice with b =6.0, which corresponds to a=0.10 fm and T ≡1/(Na)≃0.5 GeV. t ThetotalnumberoftheKSDirac-modeisL3×N ×3=2592. t Weshow theoriginal Polyakov loop, typical low-lying modecutatL =0.5a−1,and higher IR mode cut at L =2.0a−1 in Fig. 3. These mode cuts correspond to removing about 200 eigen- UV modes. As shown in Fig. 3(a), the Polyakov loop has non-zero expectation value hL i=6 0, and P showsthecentergroupZ structureonthecomplexplane. Thesebehaviorsindicatethedeconfined 3 phase. As shown in Figs. 3(b) and (c), the Polyakov loop shows the characteristic behaviors in the deconfined phase even after removing low-lying or higher Dirac-modes. To be strict, the UV-cut Polyakov loop has a smaller absolute value than the IR-cut Polyakov loop, although the number of UV-cut modes is comparable to that of the IR-cut case. This suggests that contributions of the higher Dirac-modes are much larger than low-lying modes [2]. However, apart from the normal- ization, both IR and UV cut Polyakov loops show the characteristic Z -pattern in the deconfined 3 phase, andhencetheseDirac-modesseemtobeinsensitive tothePolyakovloopproperties. Figure4showstheDiracspectraldensitiesinconfinedanddeconfinedphaseson63×4lattice withb =5.6and6.0,respectively. Wealsocomparetheirlow-lyingspectraldensities inFig.4(c). In the deconfinement phase, the low-lying Dirac-modes are suppressed, which leads to the chiral restoration. Thechiralcondensate isalsoreducedbytheIR cutoftheDirac-modesasinEq.(3.2). On the other hand, there seems no clear correspondence between the Dirac spectral densities and thePolyakovloop. 3.3 b -dependenceoftheDirac-modeprojected Polyakovloop Finally, we investigate b -dependence of the Dirac-mode projected Polyakov loop. Here, we adopt63×4latticewithb =5.4∼6.0. Figure 5 is the absolute values of the Polyakov loop with typical Dirac-mode projections, and the original Polyakov loop data are also added for comparison. In this lattice volume, the deconfinement phase transition occurs around b =5.6∼5.7. Asshown in Fig. 5, both low-lying 6 PolyakovloopanalysiswithDirac-modeexpansion TakumiIritani 1000 63 × 4 lattice with β = 5.6 1000 63 × 4 lattice with β = 6.0 116800 6633 ×× 44 llaattttiiccee wwiitthh ββ == 65..06 800 800 140 120 Va [] 600 Va [] 600 Va [] 100 λ) λ) λ) 80 ρ ( 400 ρ ( 400 ρ ( 60 200 200 40 20 0 0 0 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 0 0.1 0.2 0.3 0.4 0.5 λ [a-1] λ [a-1] λ [a-1] Figure4: TheDiracspectraldensitiesinconfinedanddeconfinedphases,respectively(a)63×4latticewith b =5.6intheconfinedphase. (b)63×4latticewithb =6.0inthedeconfinedphase. (c)Thecomparison betweenconfinedanddeconfinedphasesonlow-lyingspectraldensities. 0.3 0.3 no cut no cut Λ = 0.5a-1 Λ = 2.0a-1 IR UV Λ = 1.0a-1 Λ = 1.7a-1 IR UV 0.2 0.2 |R |V L iPI L iPU |h 0.1 |h 0.1 0 0 5.4 5.5 5.6 5.7 5.8 5.9 6 5.4 5.5 5.6 5.7 5.8 5.9 6 β β Figure5: b -dependenceoftheabsolutevalueofthePolyakovloopon63×4lattice.(a)TheIRDirac-mode cutwithL =0.5a−1and1.0a−1. (b)TheUVDirac-modecutwithL =2.0a−1and1.7a−1. IR UV and higher Dirac-mode projected Polyakov loops show the similar b -dependence as the original data, apart from a normalization factor. This Dirac-mode insensitivity of the Polyakov loop is consistent withtheresultsintheprevious subsections. 4. Summary In this paper, we have analyzed the Polyakov loop in terms of the Dirac eigenmodes using SU(3) lattice QCD. We have carefully removed relevant degrees of freedom for chiral symmetry breaking fromthePolyakovloop. IntheconfinedphasebelowT ,thePolyakovloopisalmostzero,i.e.,hL i≃0. Byremoving c P low-lying Dirac-modes, the chiral condensate hq¯qi is largely reduced. However, we have found that the Polyakov loop remains almost zero as hL i ≃ 0 even without low-lying Dirac-modes, P IR andthisfactindicatesthatthesystemstillremainsintheconfinedphase. Wehavealsoinvestigated contributions from higher Dirac-modes to the Polyakov loop, and have found no change of the PolyakovloopwithouthigherDirac-modes. Therefore,thereseemsnospecificregionoftheDirac eigenmodes essential forthePolyakovloop. We have also investigated the Polyakov loop in the deconfined phase at high temperature. In thedeconfined phase, thePolyakovloophasnon-zero expectation value, i.e.,hL i=6 0,whichdis- P tributesaround Z elementsinthecomplexplane. Thesecharacteristic behaviorsalsoremainwith- 3 7 PolyakovloopanalysiswithDirac-modeexpansion TakumiIritani out low-lying and higher Dirac eigenmodes. Therefore, the Polyakov loop and the Z -symmetry 3 behaviordonotdependonlow-lyingandhigherDiraceigenmodesinbothconfinementanddecon- finementphases. Here, we comment on the related studies about the correspondence between the Dirac eigen- modes and confinement. In the previous studies [5, 6], we investigated Dirac-mode dependence oftheWilsonloop, andfound thattheconfining potential survives withoutlow-lying Diraceigen- modes. TheGrazgroupalsoreportedthathadrons stillremainasboundstateswithoutchiralsym- metrybreakingbyremovinglow-lyingDirac-modes[10,11],whichseemstosuggesttheexistence oftheconfiningforce. These lattice QCD studies suggest that there is no direct relation between chiral symmetry breaking and confinement through the Dirac eigenmodes. For further investigation of correspon- dence between these phenomena, it is also interesting to analyze chiral symmetry breaking from therelevanteigenmodes ofconfinement[13]. Acknowledgements ThelatticeQCDcalculationshavebeendoneonNEC-SX8andNEC-SX9atOsakaUniversity. This work is in part supported by a Grant-in-Aid for JSPSFellows [No.23-752, 24-1458] and the GrantforScientificResearch [(C)No.23540306, PriorityAreas“NewHadrons” (E01:21105006)] fromtheMinistryofEducation, Culture,ScienceandTechnology(MEXT)ofJapan. References [1] C.Gattringer,Phys.Rev.Lett.97(2006)032003[hep-lat/0605018]. [2] F.Bruckmann,C.Gattringer,andC.Hagen,Phys.Lett.B624(2007)56[hep-lat/0612020]. [3] E.BilgiciandC.Gattringer,JHEP05(2008)030[arXiv:0803.1127[hep-lat]]. [4] F.Synatschke,A.Wipf,andK.Langfeld,Phys.Rev.D77(2008)114018[arXiv:0803.0271[hep-lat]]. [5] H.Suganuma,S.Gongyo,T.Iritani,andA.Yamamoto,PoS(QCD-TNT-II) (2011) 044 [arXiv:1112.1962[hep-lat]]. [6] S.Gongyo,T.Iritani,andH.Suganuma,Phys.Rev.D86(2012)034510[arXiv:1202.4130[hep-lat]]. [7] T.Iritani,S.Gongyo,andH.Suganuma,PoS(Lattice 2012) (2012) 218[arXiv:1210.7914 [hep-lat]]. 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