We report progress in our exploration of the finite-temperature phase structure of two-flavour lattice

QCD with twisted-mass Wilson fermions and a tree-level Symanzik-improved gauge action

for a temporal lattice size Nt = 8. Extending our investigations to a wider region of parameter

space we gain a global view of the rich phase structure. We identify the finite temperature transition/

crossover for a non-vanishing twisted-mass parameter in the neighbourhood of the zerotemperature

critical line at sufficiently high b . Our findings are consistent with Creutz’s conjecture

of a conical shape of the finite temperature transition surface. Comparing with NLO lattice

cPT we achieve an improved understanding of this shape

Ilgenfritz, Ernst-Michael

Jansen, Karl

Lombardo, Maria Paola

Müller-Preussker, Michael

Petschlies, Marcus

Philipsen, Owe

Zeidlewicz, Lars

English

Hochschulschriftenserver - Universität Frankfurt am Main

PoS(LATTICE 2008)206The ﬁnite-temperature phase structureof lattice QCD with twisted-mass Wilson fermionsThe tmfT Collaboration:Ernst-Michael Ilgenfritz∗†Institut für Physik, Humboldt-Universität zu Berlin, Newtonstr. 15, D-12489 Berlin, GermanyInstitut für Physik, Karl-Franzens-Universität Graz, Universitätsplatz 5, A-8010 Graz, AustriaE-mail: ilgenfri@physik.hu-berlin.deKarl JansenDESY Zeuthen, Platanenallee 6, D-15738 Zeuthen, GermanyE-mail: Karl.Jansen@desy.deMaria Paola LombardoLaboratori Nazionali di Frascati, INFN, via E. Fermi 40, I-100044 Frascati, Roma, ItalyE-mail: Mariapaola.Lombardo@lnf.infn.itMichael Müller-Preussker and Marcus PetschliesInstitut für Physik, Humboldt-Universität zu Berlin, Newtonstr. 15, D-12489 Berlin, GermanyE-mail: mmp@physik.hu-berlin.de,Marcus.Petschlies@physik.hu-berlin.deOwe Philipsen and Lars ZeidlewiczInstitut für Theorische Physik, Westfälische Wilhems-Universität Münster,Wilhelm-Klemm-Str. 9, D-48149 Münster, Germany,E-mail: o.philipsen@uni-muenster.de,zeidlewicz@uni-muenster.deWe report progress in our exploration of the ﬁnite-temperature phase structure of two-ﬂavour lat-tice QCD with twisted-mass Wilson fermions and a tree-level Symanzik-improved gauge actionfor a temporal lattice size Nt = 8. Extending our investigations to a wider region of parameterspace we gain a global view of the rich phase structure. We identify the ﬁnite temperature tran-sition/crossover for a non-vanishing twisted-mass parameter in the neighbourhood of the zero-temperature critical line at sufﬁciently high b. Our ﬁndings are consistent with Creutz’s conjec-ture of a conical shape of the ﬁnite temperature transition surface. Comparing with NLO latticecPT we achieve an improved understanding of this shape.The XXVI International Symposium on Lattice Field TheoryJuly 14 - 19, 2008Williamsburg, VA, USA∗Speaker.†Supported by DFG via Forschergruppe Gitter-Hadronen-Phänomenologie FOR 465.c ￿ Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/PoS(LATTICE 2008)206Finite-temperature phase structure of QCD with twisted-mass Wilson fermions Ernst-Michael Ilgenfritz1. IntroductionThe goal of the tmfT collaboration [1] is to explore the applicability of the 2-ﬂavour Wilson-twisted-mass (Wtm) fermion formulation set-up as described in Ref. [2] for investigations of lat-tice thermodynamics. We refer to [2] for the deﬁnition of the Wtm action and its parametersk and m. The staggered-fermion formulation is computationally inexpensive [3], however con-ceptually controversial [4]. On the other hand, the non-perturbatively improved Wilson fermionformulation requires the calculation of the improvement coefﬁcients and furthermore needs oper-ator improvement. Thus, the Wtm formulation, nowadays combined with a tree-level Symanzik-improved gauge action [5], appears as an appealing alternative for ﬁnite temperature lattice sim-ulations, whose potential should be explored. It offers automatic O(a)− improvement by tuningthe bare, untwisted quark mass only. This makes high-statistics simulations for thermodynamicalproblems affordable with pseudoscalar masses as low as mp & 300MeV [6].As a necessary preparatory step we characterise the phase structure of the model by locatingthe transition/crossover lines and surfaces in the b −k −m−coupling space. In contrast to ourreport at Lattice 2007 [7], we now ﬁnd clear evidence for the Aoki phase at b ≤ 3, lower thansearched for previously, and a ﬁrst order phase transition surface in an intermediate b region be-ginning with b ≈ 3.4, consistent with earlier ﬁndings. For larger couplings 3.6 ≤ b ≤ 3.8, weobserve a thermal transition at kT(T  = 0) near kc(T = 0), which emanates from the ﬁrst order likestructure at smaller b-values. With growing b, it splits up to become a succession of two tran-sitions conﬁnement→deconﬁnement→conﬁnement in the k-direction. The conﬁnement aspect ofthese transitions is exhibited by the Polyakov loop and its susceptibility, the chiral aspect by thepion norm (the integrated pseudoscalar correlator) and the scalar and pseudoscalar condensates.Our ultimate goal is to investigate the ﬁnite temperature transition (or crossover) at maximaltwist and physically relevant pion masses. The most obvious procedure of directly scanning thetransition at maximal twist has proven the hardest to tackle, since its precise location is yet un-known. Still, our efforts so far gave us more detailed insights than reported in [7] into the phasestructure at non-vanishing twist parameter m for a larger range of b-values. Our ﬁndings are qual-itatively consistent with predictions from cPT and continuum symmetry arguments [8].2. Preview of the phase structureThere are three sources for our expectations concerning the phase structure of 2-ﬂavour WtmLQCD at ﬁnite temperature: (i) The phase structure at zero temperature was predicted usingcPT [9, 10, 11] and subsequently veriﬁed numerically [12]; the parameter space at T = 0 wasshown to contain a phase of broken parity-ﬂavour symmetry (Aoki-scenario) in the b −k−planeat strong coupling [13] (for Wilson gauge action) and a ﬁrst order phase transition surface in theb −m−plane (the ﬁrst order scenario of Sharpe and Singleton) adjacent to the former and extendinginto the weak coupling region. Both scenarios crucially depend on the choice of fermion and gaugeaction [14]. (ii) The general properties of the phase structure for (non-) perturbatively improvedWilson fermions, as established by e.g. the CP-PACS [15] and DIK [16] collaborations, should befound inthe b −k−planeatweaker coupling, too. (iii) Based onthe argument that inthecontinuumtheory the physical transition temperature cannot depend on the twist angle, Creutz [8] conjecturedthe ﬁnite temperature phase transition/crossover surface to be closed and form a cone (cf. the right2PoS(LATTICE 2008)206Finite-temperature phase structure of QCD with twisted-mass Wilson fermions Ernst-Michael Ilgenfritzpanel of Fig. 1) around the critical line kc(b) for a range of b-values somewhere between the Aokiphase and the deconﬁnement transition in the quenched limit (bdec ≈ 4.5 at k = 0). The locationof the tip of this cone and the detailed shape of its surface are subject to lattice artifacts, as may bethe nature of the transition for different twist angles.00.050.10.150.20.251.5 2 2.5 3 3.5 4 4.5κβAoki phaseend of Aoki phase1st ordercone signal2nd thermal transitionκc(β) (ETMC)κc(β), (tmfT)κt(β), near κcκt(β)prel. βt(κ = 0.17)0 ∞Aoki phase∞conﬁnementdeconﬁnementκ thermal transition/crossover surface(possibly curved)from ﬁrst doublerκc(β,T = 0)βκc(β,T = 0)µquenched limitconﬁnement1st orderFigure 1: Left: Map of our simulations performed at m < 0.007. Right: Emerging phase structure; regionsidentiﬁed by tmfT measurements are marked by magenta symbols.3. Simulation resultsThe simulations were mainly performed on lattices of spatial size Ns = 16 and Nt = 8. Thealgorithm in use is described in Ref. [17]. An overview of the regions covered by our simulationis presented in the left panel of Fig. 1. It contains all transition and crossover points that we foundin our simulations and is supplemented by data describing the T = 0 critical line kc(b ≥ 3.75)provided by the ETM collaboration.After the unsuccessful search reported in [7], we have checked the existence of the Aoki phaseat the three values b ∈ {1.8, 3.0, 3.4}. The necessary and sufﬁcient condition for spontaneoussymmetry breaking readslimh→0limNs→¥  ¯ yig5t3y Ns,h  = 0. (3.1)Figure2showsthe Fisherplots [13]oftheorder parameter   ¯ yig5t3y asafunction ofh/  ¯ yig5t3y ,00.010.020.030.040.050 0.01 0.02 0.03 0.04 0.05 0.06 0.07 ¯ψıγ5τ3ψ 2hh ¯ ψıγ5τ3ψiκ = 0.23κ = 0.2400.00070.00140.00210.00280.00350.00420.00490 0.02 0.04 0.06 0.08 0.1 0.12 0.14 ¯ψıγ5τ3ψ 2hh ¯ ψıγ5τ3ψiκ = 0.203κ = 0.2035κ = 0.204Figure2: Fisher plotsfortheorderparameter  ¯ yig5t3y  describingits limit forh=2km →0; left: b =1.8,right: b = 3.0 .with h = 2km, for b = 1.8 and 3.0. The condensate has a ﬁnite limit for h → 0 (a positive in-tercept at h = 0) for suitable k values. For the smallest value b = 1.8, the left panel of Fig. 23PoS(LATTICE 2008)206Finite-temperature phase structure of QCD with twisted-mass Wilson fermions Ernst-Michael Ilgenfritzgives convincing evidence for the onset of spontaneous symmetry breaking somewhere in the range0.24 > k > 0.23. The right panel strongly indicates that at b = 3.0 the symmetry-broken phase –if it exists at all – is realized only in the interval 0.203 < k < 0.204. A volume extrapolation wouldbe welcome but is unavailable at this b−value. This marks the lower boundary klowerc (b) of theAoki phase. We have not systematically searched for its upper boundary kupperc (b).0.0060.0080.010.0120.178 0.18 0.182 0.184 0.186 ¯ψıγ5τ3ψ κβ = 3.45β = 3.40.00050.0010.00150.0020.178 0.18 0.182 0.184 0.186 Re(L) κβ = 3.45β = 3.4Figure 3: Signals of metastability at b = 3.4 and 3.45. The two-state signal left: in the pseudoscalarcondensate, right: in the real part of the Polyakov loop.At b = 3.4 we reinforce our preliminary results published in Ref. [7] that, instead of an Aokiphase, there are strong metastabilities when simulating at ﬁnite twist m ≈ 0.0068. This conﬁrmsthe ﬁrst-order Sharpe-Singleton scenario, and represents a remnant of the phase structure at T = 0(on symmetric lattices). This is shown for b = 3.4 and 3.45 in Fig. 3 for the order parameter (left)and the Polyakov loop (right). The behaviour of the latter at this transition cannot be interpreted asthermal. Also the average plaquette (as checked at b = 3.4 and k = 0.1827) is running in separate,metastable histories starting from hotter/colder neighbouring states.In the range 3.4 ≤ b ≤ 3.65 between strong and weak coupling, simulating at m = 0.0068, wesee a ﬁnite temperature transition, located at kT ≫ kc(b), i.e. far above the chiral line for T = 0.The transition is inherited from the ﬁrst doubler structure existing at higher k. The correspondingkT(b,m) evolves rapidly with b, moving closer towards the critical line kc(b) with increasing b(cf. Fig. 1). The transition is possibly of ﬁrst order as indicated by histograms of the Polyakov loopat b =3.6 in the interval 0.18<k <0.24: they show a relatively ﬂat but clear double peak structureat k = 0.22. On 323×8 lattices simulated at the same b −k −m−point one observes long-livingmetastability in the Polyakov loop. Also at higher b = 3.75 this thermal transition continues toexist, still separated from the unfolding thermal transition surface around kc(b). Finally, it appearsto join the latter in the neighbourhood of b = 4.0. Before this joining happens, Fig. 4 shows thecharacteristic behaviour of the Polyakov loop and its susceptibility measured at b = 3.75 withm = 0.005 over an interval 0.16 < k ≤ 0.21 that is covering both transition regions. The two peaksare clearly visible, but the lower susceptibility peak is not yet well resolved on the coarse k scaleof Fig. 4.It is our goal to work at maximal twist, thus ultimately we need to resolve the lower peak inFig. 4. The position and nature of the ﬁrst (i.e. lower in k) ﬁnite temperature transition/crossover,when studied at m  = 0, changes considerably between the metastability region described by Fig. 3and the scaling region. Midway between the two, at b = 3.65, one sees the Polyakov loop enteringand then leaving again a narrow deconﬁning region in k (cf. the left panel of Fig. 5). Approaching4PoS(LATTICE 2008)206Finite-temperature phase structure of QCD with twisted-mass Wilson fermions Ernst-Michael Ilgenfritz0.0050.010.0150.020.0250.030.0350.040.17 0.18 0.19 0.2 0.21 Re(L) κnear κclarger κ00.20.40.60.80.17 0.18 0.19 0.2 0.21χ(Re(L))κnear κclarger κFigure 4: Left: real part of the Polyakov loop, right: its susceptibility as functions of k for b = 3.75 andm = 0.005. The dashed line marks kc(b = 3.75,T = 0).the physically relevant scaling region, b & 3.75, the peak structure grows even clearer. The narrowpeak (supposed to contain the deconﬁned phase) moves gradually further to lower k with risingb = 3.75, 3.775 and b = 3.8 (cf. the right panel of Fig. 5).0.0020.0030.0040.0050.0060.169 0.1695 0.17 0.1705 0.171 0.1715realpartofPolyakovloopκβ = 3.65 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02 0.162  0.163  0.164  0.165  0.166  0.167  0.168  0.169real part of Polyakov loopkb=3.75b=3.775b=3.8Figure 5: The real part of the Polyakov loop as function of k; left: for b = 3.65 with m = 0.0068, right:for b = 3.75, 3.775 and 3.8 (all with m = 0.005).With a large computational effort the rise and decline of the Polyakov loop begins to be re-ﬂected also in the corresponding susceptibility. The Polyakov loop susceptibility gradually de-velops a double peak which is already clearly visible at b = 3.8 (cf. the left panel of Fig. 6).Remarkably, the lower−k peak is accompanied by a peak of the pion norm (presented in the rightpanel of Fig. 6) emphasizing the entrance into the deconﬁning and chiral-symmetry restored phase.So far, within the presently available statistics, we could not detect a signiﬁcant similar chiral signalmarking the exit.4. The cone signal and comparison with lattice cPTWeinterpret thestructured signal in thePolyakov loop near kc outlined inthe preceding sectionas a numerical indication for passing through a conﬁnement→deconﬁnement crossover approach-ing kc from small positive quark mass, followed by a deconﬁnement→conﬁnement crossover atsmall negative quark mass. This picture is qualitatively consistent with the conical structure pre-dicted by Creutz (see also Fig. 1). The thermal transition takes place at a given value of the quarkmass which is determined by the parameters k and m. At tree level the according relation reads:m2q = m2+14￿1k−1kc￿2. (4.1)5PoS(LATTICE 2008)206Finite-temperature phase structure of QCD with twisted-mass Wilson fermions Ernst-Michael Ilgenfritz 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.162  0.163  0.164  0.165  0.166  0.167  0.168  0.169susceptibility of real part of Polyakov loopkb=3.75b=3.775b=3.8 2.5 2.6 2.7 2.8 2.9 3 3.1 3.2 3.3 3.4 0.162  0.163  0.164  0.165  0.166  0.167  0.168  0.169pionnormkb=3.75b=3.775b=3.8Figure 6: Left: the susceptibility of the Polyakovloop, right: the pion norm as functionsof k for b =3.75,3.775 and 3.8 (all at m = 0.005).Using lattice chiral perturbation theory (LcPT), one obtains the following modiﬁcation of (4.1) inNLO (cf. [18]):m2q = 1Z2Pm2+1Z2S14￿1k−1kc￿2!(1+Kcosw)2 . (4.2)From the zero temperature simulations ofthe ETMcollaboration weknow that kc(b =3.75)=0.1660(1), ZS ≈ 0.6 and ZP ≈ 0.3. K is an unknown O(a) coefﬁcient that is introduced by LcPT.The twist angle w is deﬁned by tanw = m/(0.5(1/k −1/kc)).In Fig. 7 the thermal transition points that we have found at b = 3.75 for m = 0.005 and0.007 are compared with the two relations above. In principle, K and mq can be determined froma ﬁt. However, given the rather large uncertainties of ZP and ZS and also the small number ofavailable data points, we are only able to check whether the formulae are capable of describing thedata. For the tree level formula this is not the case (in the plot: mq = 0.01, kc = 0.1660), due to thek ↔(2/kc−1/k)-symmetry of equation (4.1). However, choosing mq =0.028, K =0.5, ZS =0.6,ZP = 0.3 and kc = 0.1660 the NLO formula (4.2) is consistent with the data points.æ ææ æ0.165 0.166 0.167 0.168 0.1690.0020.0040.0060.0080.0100.012ΚΜFigure 7: Thermal transition points found at b = 3.75 with kc = 0.1660 compared to tree level (mq = 0.01,green) and LcPT (mq = 0.028, K = 0.5, ZS = 0.6, ZP = 0.3, blue) curves.5. Summary and outlookWe investigated the global phase structure of 2-ﬂavour Wilson-twisted-mass LQCD with atree-level Symanzik-improved gauge action. Our simulation results support the existence of an6PoS(LATTICE 2008)206Finite-temperature phase structure of QCD with twisted-mass Wilson fermions Ernst-Michael IlgenfritzAoki phase (at b . 3.0) as well as the ﬁrst-order Sharpe-Singleton-scenario at ﬁnite twist (forb ≈ 3.4). Moreover, we ﬁnd indications of a conical structure of the ﬁnite-temperature crossoversurface in a bounded b−interval, 3.65 ≤ b ≤ 3.8, with m = 0.005 and 0.007. Our next steps willbe the localization of additional ﬁnite temperature crossover points to reﬁne the LcPT−predictionof the transition line and to bound the value mT for the thermal crossover at maximal twist withina narrow interval. Depending on the ﬁndings of this investigation, a subsequent simulation atmaximal twist to detect mT numerically will follow.Acknowledgements. It is a pleasure to thank Steve Sharpe, André Sternbeck and Carsten Ur-bach for discussions. Inaddition, wethank Andréforjoining our phone conferences andCarsten forsupplying the HMC code. Most simulations have been done at the apeNEXT in Rome and DESY-Zeuthen. This work has been supported in part by the DFG Sonderforschungsbereich/TransregioSFB/TR9-03. O. P. and L. Z. acknowledge support by the DFG project PH 158/3-1. E.-M. I.was supported by DFG under contract FOR 465 / Mu932/2 (Forschergruppe Gitter-Hadronen-Phänomenologie). He is grateful to the Karl-Franzens-Universität Graz for the guest position heholds while this paper is written up.References[1] E.-M. Ilgenfritz et al., PoS(LAT2006)140, [arXiv:hep-lat/0610112].[2] A. Shindler, Phys. Rept. 461 (2008) 37, [arXiv:0707.4093[hep-lat]][3] K. Jansen, plenary talk at this conference.[4] M. Creutz, PoS(LAT2007)007, [arXiv:0708.1295[hep-lat]].[5] Ph. Boucaud et al. (ETM collaboration), [arXiv:0803.0224[hep-lat]].[6] Ph. Boucaud et al. (ETM collaboration), Phys. Lett. B650 (2007) 304,[arXiv:hep-lat/0701012].[7] E.-M. Ilgenfritz et al., PoS(LAT2007)238, [arXiv:0710.0569[hep-lat]].[8] M. Creutz, Phys. Rev. D76 (2007) 054501, [arXiv:0706.1207[hep-lat]].[9] M. Creutz, [arXiv:hep-lat/9608024].[10] S. R. Sharpe and J. M. S. Wu, Phys. Rev. D70 (2004) 094029, [arXiv:hep-lat/0407025].[11] G. Münster, JHEP 09 (2004) 035, [arXiv:hep-lat/0407006].[12] F. Farchioni et. al., Eur. Phys. J. C39 (2005) 421, [arXiv:hep-lat/0406039].[13] E.-M. Ilgenfritz et al., Phys. Rev. D 69 (2004) 074511, [arXiv:hep-lat/0309057].[14] F. Farchioni et. al., Eur. Phys. J. C42 (2005) 73, [arXiv:hep-lat/0410031].[15] N. Ukita et al., PoS(LAT2006)150, [arXiv:hep-lat/0610038].[16] V. G. Bornyakov, et al. (DIK collaboration), PoS(LAT2007)171,[arXiv:0711.1427[hep-lat]].[17] C. Urbach et al., Comput. Phys. Commun. 174 (2006) 87, [arXiv:hep-lat/0506011].[18] S. R. Sharpe, [arXiv:hep-lat/0607016].7

plenary talk at this conference.

The finite-temperature phase structure of lattice QCD with twisted-mass Wilson fermions

The finite-temperature phase structure of lattice QCD with twisted-mass Wilson fermions

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