Deconfining temperatures in SO(N) and SU(N) 5 1 0 gauge theories 2 n a J 0 3 t] RichardLau∗†andMichaelTeper a l RudolfPeierlsCentreforTheoreticalPhysics,UniversityofOxford - p E-mail: [email protected] e E-mail: [email protected] h [ 1 We presentourcurrentresultsforthedeconfiningtemperaturesinSO(N)gaugetheoriesin2+1 v dimensions. SO(2N)theoriesmayhelpustounderstandQCD atfinitechemicalpotentialsince 8 2 thereisalarge-NorbifoldequivalencebetweenSO(2N)QCD-liketheoriesandSU(N)QCD,and 8 SO(2N)theoriesdonothavethesignproblempresentinQCD.Weshowthatthedeconfiningtem- 7 0 peraturesinthesetwotheoriesmatchatthelarge-Nlimit. WealsopresentresultsforSO(2N+1) . 1 gaugetheoriesandcompareresultsfor SO(6)with SU(4)gaugetheories, whichhavethe same 0 Liealgebrasbutdifferentcentres. 5 1 : v i X r a The32ndInternationalSymposiumonLatticeFieldTheory, 23-28June,2014 ColumbiaUniversityNewYork,NY Speaker. ∗ †FundedbytheScienceandTechnologyFacilitiesCouncil. (cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/ DeconfiningtemperaturesinSO(N)andSU(N)gaugetheories RichardLau 1. Introduction SO(N)gaugetheoriesdonothaveafermionsignproblem[1],areorbifoldequivalenttoSU(N) QCD [2], and share a common large-N limit with SU(N)gauge theories [3]. Some SO(N) gauge groups also share Lie algebra equivalences with SU(N) gauge groups such as SO(4) SU(2) ∼ × SU(2)orSO(6) SU(4). AllthisindicatesthatwecouldinvestigateSU(N)QCDatfinitechemical ∼ potential through considering theequivalent SO(N)gaugetheories. Thereisalarge-N orbifold equivalence betweenSO(2N)QCD-liketheoriesandSU(N)QCD [1]. This equivalence holds if we take the large-N limit while relating the couplings g in the two theories by g2 = g2 . Using this result, along with knowing that the leading SU(N ¥ ) SO(2N ¥ ) correction betw(cid:12)eenfi→niteN an(cid:12)dthel→arge-N limitisO(1/N)forSO(2N)andO(1/N2)forSU(N), (cid:12) (cid:12) wecanconstruct apossible pathconnecting SO(N)andSU(N)gaugetheories. SU(N ¥ ) ✛large-Nequivalen✲ce SO(2N ¥ ) → → ✻ ✻ O 1 corrections O(1)corrections (1.1) (cid:16)N2(cid:17) N ❄ ❄ SU(N) SO(2N) We showed at Lattice 2013 that we obtain the same large-N limits for the string tension and mass spectrum from SO(2N)and SU(N)gauge theories inD=2+1[4]. Inthis contribution, we calculate SO(N) deconfining temperatures in 2+1 dimensions and we will show that they match SU(N)valuesbetweenLiealgebraequivalences andatthelarge-N limit. Asbefore, weconsider D=2+1valuesbecause theSO(N)D=3+1bulk transition occurs at very small lattice spacings so that the volumes needed are currently too large to simulate [5]. However, in D = 2+1, the bulk transition occurs at larger lattice spacings and we can obtain continuum extrapolations at reasonable volumes [6]. We use the standard plaquette action for an SO(N)gaugetheory. S=b (cid:229) 1 1tr(U ) b = 2N (1.2) (cid:18) −N p (cid:19) ag2 p 2. Deconfinement We expect SO(N) gauge theories to deconfine at some temperature T =T , just like SU(N) c gauge theories. We can look for the deconfinement temperature by using an ‘order parameter’ O such as the ‘temporal’ Polyakov loop l [7]. The expectation value of the Polyakov loop l is P p h i not invariant under a transformation with a non-trivial element of the centre. Hence, for gauge theories with non-trivial centres, such as Z SO(2N), the expectation value is zero. This corre- 2 sponds toconfinement, whileanon-zero expectation value corresponds todeconfinement. Hence, deconfinement corresponds heretoaspontaneous breakdown oftheZ symmetry. 2 Using this order parameter, wecan look for signs of the deconfining phase transition such as changes inthehistogram peaksoftheorderparameteroverafullconfiguration run. Wedisplayan 2 DeconfiningtemperaturesinSO(N)andSU(N)gaugetheories RichardLau 0.03 0.02 0.01 Β=17.5 0. -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.02 0.01 Β=17.8 0. -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.02 0.01 Β=18.1 0. -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Figure1: Histogramsof l inanSO(6)2023volumeatseveralvaluesofb aroundb . P c h i example ofsuch histograms inFigure1foranSO(6) 2023volume. Asweincrease b ,wecansee that aprimary peak around zero disappears while twosecondary peaks at non-zero values appear. Theprimarypeakrepresentstheconfinedphasewhilethesecondarypeaksrepresentthedeconfined phase. Thetransitionbetweenthetwostatesindicatesthattheb rangeweconsideredisaroundb . c Furthermore, the coexistence of the two phases that we can see in the middle histogram indicates thattheSO(6)deconfining phasetransition isfirstorder. Wecanconstruct a‘susceptibility’ c forthePolyakovloop. lP | | c l 2 l 2 (2.1) |lP|∼D(cid:12)P(cid:12) E−(cid:10)(cid:12)P(cid:12)(cid:11) (cid:12) (cid:12) (cid:12) (cid:12) Wecancalculatethesusceptibilityfordifferentb intheregionofb . Thepeakinthissusceptibility c when plotted against b corresponds to b . The peak structure can also indicate the order of the c phasetransitionwhenwevarythefinitespatialvolumeV. AsV increases,thepeakheightincreases as the peak converges towards the non-analyticity associated with the continuum phase transition, butthecharacteristic widthchanges depending onthetransition’s order. Forfirstordertransitions, the characteristic width decreases at the same rate as the peak height increases so that the peak converges to a delta function. We can see this in Figure 2 for an SO(8) phase transition. For second order transitions, the characteristic width decreases at a slower rate than the peak height increases so that the peak converges to a divergence. We can see this in Figure 3 for an SO(4) phasetransition. Toidentify accurately b from the susceptibility peak, we use reweighting [8]. The principle c behind reweighting isthat wecan consider the generation of lattice configurations assampling an underlying density of states, which is independent of b . If we could reconstruct the density of states, then we could calculate observables at an arbitrary value of b . Reweighting allows us to calculate b veryaccurately aswecanseeinFigure4. c 3 DeconfiningtemperaturesinSO(N)andSU(N)gaugetheories RichardLau 60 æ æ æ 4825 æææ æææ 50 æ 4025 æ 40 æ 3225 ææ æ æ ææ æ 2825 æææ æ æ Χ 2300 ææææææææææææææ æ æ ææ ææ æææ æ 100 æææ æææ æææ ææææ æææ ææææ æææ ææææ æææ ææææææææ æ æ æææ æææ ææ ææææ æææ æææ 48.5 49.0 49.5 50.0 50.5 51.0 Β Figure2: SusceptibilityplotforSO(8)L =5volumes. t 50 æ 6022 æ æ 5622 æææ 40 æ 4822 ææ æ æ æ æ 4022 æ ææ Χ 30 ææ 33622222 æææææææææææææææææææ 12000 æææææææææ æææ 222æææææææææ840222222 æææææææææ æææææææææ æææææææææ æææææææææ æææææææææ æææææææææ ææææææææææææææææææææææææææææææææææææææ æææææææææ æææææææææ æææææææææ æææææææææ æææææææææ æææææææææ ææææææææ 6.40 6.45 6.50 6.55 6.60 Β Figure3: SusceptibilityplotforSO(4)L =2volumes. t 3. SO(N)measurements Forafixed‘temporal’ lengthL ,wecalculate b (V)fordifferent spatialvolumesV. Byusing t c knownresultsfromfinitesizescaling,wecanextrapolateb (V)valuesfordifferentspatialvolumes c V to theinfinite spatial volume limitV ¥ . Forfirstorder transitions, b (V)varies linearly with c → 1/V, as we can see in Figure 5. For second order transitions, b (V) varies with 1/V in a way c determined bythecriticalexponents ofthephasetransition. Oncewehavecalculatedb (V ¥ )forfixedL ,wecancalculatethecontinuumstringtension c t → atthisvalue,usingmethodssimilartothoseforSU(N)gaugetheories[9]. Wecanthenexpressthe deconfining temperature T =1/(aL ) in string tension units T /√s . This allows us to calculate c t c thecontinuumlimitforfixedSO(N)byapplyingacontinuumextrapolationina2s . Thecontinuum extrapolation is only valid in the weak coupling region. Hence, we identified the bulk transition regionforeachSO(N)gaugetheory,whichcorrespondtob regionswithananomalouslylowscalar 4 DeconfiningtemperaturesinSO(N)andSU(N)gaugetheories RichardLau æ 9 æ æ æ æ æ 8 æ æ æ æ æ 7 æ 6 Χ æ l ¤ p 5 æ æ 4 æ 3 æ 2 æ 17.5 17.6 17.7 17.8 17.9 18.0 Β Figure4: ThesusceptibilityforanSO(6)2023volumewithreweightedresults. 29.90 29.85 æ 29.80 æ æ æ Β 29.75 c æ 29.70 29.65 29.60 0 2 4 6 8 10 105(cid:144)V Figure5: InfinitevolumeextrapolationforSO(6)L =6volumes. t massm . Wethenappliedthecontinuumextrapolationtoweakcouplingb values. Wedisplayan 0+ exampleinFigure6fortheSO(8)continuum limit. 4. Equivalences between SO(N)andSU(N)gaugetheories Using the techniques, we can compare the deconfining temperatures between SO(N) and SU(N)gaugetheories. Weknow that SO(4) and SU(2) SU(2) share a common Lie algebra. Forthe cross product × groupSU(2) SU(2),weexpectacontributionfromeachSU(2)grouptothestringtensionsothat × weexpect s =2s . Hence,weexpectthat |SU(2) SU(2) |SU(2) × T T 1 T c c c = = (4.1) √s (cid:12) √s (cid:12) √2 √s (cid:12) (cid:12)SO(4) (cid:12)SU(2) SU(2) (cid:12)SU(2) (cid:12) (cid:12) × (cid:12) (cid:12) (cid:12) (cid:12) 5 DeconfiningtemperaturesinSO(N)andSU(N)gaugetheories RichardLau 0.84 æ æ æ Weakcoupling 0.82 æ æ Strongcoupling 0.80 æ T c 0.78 Σ 0.76 æ 0.74 0.0 0.1 0.2 0.3 0.4 0.5 a2Σ Figure6: ContinuumextrapolationforSO(8)deconfiningtemperatures. Using the known result for the SU(2) deconfining temperature T /√s =1.1238(88) [7], we can c compareourresultfortheSO(4)deconfining temperature. T c =0.7844(31) √s (cid:12) (cid:12)SO(4) (cid:12) 1 Tc (cid:12) =0.7949(58) (4.2) √2 √s (cid:12) (cid:12)SU(2) (cid:12) (cid:12) Weseethatthesevaluesarewithin1.5s ofeachother,whichisconsistent withourexpectation. Weknow that SO(6) and SU(4) share a common Lie algebra. TheSO(6) fundamental string tension isequivalent totheSU(4)k=2Astringtension [6]. Hence,weexpectthat T T c c = (4.3) √s (cid:12) √s (cid:12) f(cid:12)SO(6) 2A(cid:12)SU(4) (cid:12) (cid:12) (cid:12) (cid:12) Using the known result for the SU(4) deconfining temperature T /√s =1.1238(88) [7] together c withtheratiooftheSU(4)k=2AandfundamentalstringtensionsinD=2+1,s /s =1.355(9) 2A f [10],wecancompareourresultfortheSO(6)deconfining temperature. T c =0.8105(42) √s (cid:12) f(cid:12)SO(6) (cid:12) Tc (cid:12) =0.8163(62) (4.4) √s (cid:12) 2A(cid:12)SU(4) (cid:12) (cid:12) We see that these values are within one standard deviation of each other, which is consistent with ourexpectation. Wecanobtainalarge-N extrapolationfromourSO(2N)deconfiningtemperaturesbyapplying aO(1/N)correctionfollowinganadaptedformof’tHooft’splanardiagramargument. Wedisplay this large-N extrapolation in Figure 7. We can compare this large-N value to the large-N limit of 6 DeconfiningtemperaturesinSO(N)andSU(N)gaugetheories RichardLau 1.05 1.00 æ SOH2NL SUHNL 0.95 æ æ æ æ T 0.90 c æ Σ 0.85 æ æ 0.80 æ æ 0.75 0.70 0.0 0.1 0.2 0.3 0.4 0.5 1 N Figure7: Large-NextrapolationforSO(2N)andSU(N)deconfiningtemperatures. theSU(N)deconfining temperatures [7]. Fromthelarge-N equivalence, wewouldexpectthat T T c c = (4.5) √s (cid:12) √s (cid:12) (cid:12)SO(2N ¥ ) (cid:12)SU(N ¥ ) (cid:12) → (cid:12) → Thetwolarge-N limitsare (cid:12) (cid:12) T c =0.9076(41) √s (cid:12) (cid:12)SO(2N ¥ ) (cid:12) → Tc(cid:12) =0.9030(29) (4.6) √s (cid:12) (cid:12)SU(N ¥ ) (cid:12) → We see that these values are within one(cid:12)standard deviation of each other, which is consistent with ourexpectation. References [1] A.Cherman,M.Hanada,andD.Robles-Llana,Phys.Rev.Lett.106,091603(2011), [arXiv:1009.1623]. [2] M.UnsalandL.Yaffe,Phys.Rev.D74,105019(2006),[arXiv:hep-th/0608180]. [3] C.Lovelace,Nucl.Phys.B201(1982) [4] R.LauandM.Teper,PoS(Lattice2013)187,[arXiv:1311.1453]. [5] P.deForcrandandO.Jahn,Nucl.Phys.B651(2003)125,[arXiv:hep-lat/0211004]. [6] F.Bursa,R.Lau,andM.Teper,JHEP1305:025,2013,[arXiv:1208.4547]. [7] J.LiddleandM.Teper,[arXiv:0803.2128]. [8] A.FerrenbergandR.Swendsen,Phys.Rev.Lett.63,11951198(1989) [9] M.Teper,Phys.Rev.D.59:014512(1998),[arXiv:hep-lat/9804008]. [10] B.BringoltzandM.Teper,Phys.Lett.B663(5)(2008),[arXiv:0802.1490]. 7