Critical endline of the finite temperature phase transition for 2 + 1 flavor QCD away from the SU ( 3 )-flavor symmetric point

We investigate the critical end line of the finite temperature phase transition of QCD away from the SU(3)-flavor symmetric point at zero chemical potential. We employ the renormalization-group improved Iwasaki gauge action and non-perturbatively O(a)improved Wilson-clover fermion action. The critical end line is determined by using the intersection point of kurtosis, employing the multi-parameter, multi-ensemble reweighting method at the temporal size NT = 6 and lattice spacing as low as a ≈0.19 fm.


Introduction
The nature of the finite temperature phase transition of 2+1 flavor QCD at zero chemical potential depends on quark masses.The order of transition and universality class are summarized in the plane of light quark mass, m l and strange quark mass, m s , and it is called the Columbia plot [1].
The first order phase transition is expected in the small quark mass region [2].It is well known that the phase transition is also the first order in the heavy quark mass region and it is crossover in the medium quark mass region by many lattice QCD studies.The boundary between the first order region and crossover region is the second order phase transition of Z 2 universality class.
The nature in the lower-left corner of the Columbia plot has not been fully understood yet.The first lattice QCD calculation was done by using standard Wilson fermions at N T = 4 roughly 20 years ago.It reported the critical mass at the critical endpoint (CEP), m E , for N F = 3 is heavy , the critical quark mass m q,E = m l,E = m s,E 140 MeV or, equivalently, the critical pseudo scalar mass m PS,E = m π,E = m η s ,E 1 GeV [3].After preliminary study with standard Wilson gauge and staggered fermions which reported the bare critical mass am q,E ∼ 0.035 [4] at N T = 4, Karsch et al. reported preliminary values for the critical mass, m PS,E ∼ 290 MeV with unimproved gauge and staggered fermion actions and m PS,E ∼ 190 MeV with improved gauge and staggered fermion actions (p4-action) [5].These results were obtained by using R-algorithm [6].Afterward, the results were updated as m PS,E = 290 (20) MeV with unimproved gauge and staggered fermion actions and m PS,E = 67 (17) MeV with improved gauge and staggered fermion actions (p4-action) [7].Then, in Speaker, e-mail: nakamura@riken.jpref. [8], de Forcrand and Philipsen obtained am q,E = 0.0260(5) by using the RHMC algorithm [9,10], which is about 25% smaller than the value am q,E ≈ 0.033 quoted by works using the R-algorithm.They also performed N F = 2+1 simulations and non zero chemical potential simulations and obtained the critical line and tri-critical point, am tri s ≈ 0.7, where lattice spacing a was approximately 0.3 fm.In ref. [11] with unimproved staggered fermions, is was reported that the ration of m PS,E and the CEP temperature T E decreased from 1.680(4) to 0.954 (12) as increasing N T from 4 to 6.These results are showing very large cut off effect for the critical mass and it is important to push N T and use improved action.Further studies with improved staggered fermions have not found the first order phase transition and quoted only the bound of the critical mass, m PS,E 50 MeV, [12][13][14].Therefure, the position of the critical endline (CEL), m tri s and CEP for N F = 3 is still particularly important problem to be solved at this moment.
Recently we also have investigated the nature of the finite phase transition in small quark mass region by using non-perturbatively O(a)-improved Wilson-clover fermion fermions.We have determined CEP at N T = 4, 6, 8, 10 and upper bound of CEP in the continuum limit for N F = 3 [15,16].For N F = 2 + 1, we have studied at N T = 6 and determined CEL around the SU(3) flavor symmetric point and confirmed that the slope of CEL at the SU(3) flavor symmetric point is -2 [17].In this paper, we extend our study for CEL away form the SU(3) flavor symmetric point.

Simulations
We employ the renormalization-group improved Iwasaki gauge action [18] and non-perturbatively O(a)-improved Wilson-clover fermion action [19].CEP is determined by using the intersection point of kurtosis of chiral condensate.This method is expounded in Ref [15] and used in our recent studies [15][16][17].Expectation value, susceptibility and skewness of chiral condensate are used for confirming phase transition and determination of the transition point.Chiral condensate and its higher moments are computed from traces of inverse Wilson clover Dirac operator up to power of −4, TrD −1,−2.−3,−4, by using 10 noise vectors.We have checked that 10 noises are good enough for some parameter sets.We employ the multi-parameter, multi-ensemble reweighting method [20] to determine CEP very small statistical error.We reweight both κ l and κ s , so that we can determine many CEP without doing simulations at many parameter sets.Our simulations are performed at the temporal size N T = 6 and lattice spacing a ≈0.19 fm.The spatial size N S is 10, 12, 16, 24.We have confirmed m PS L > 4 at almost all transition points, where m PS is pseudo scalar mass and L is physical spatial extent.Simulation parameters are shown in Table 1 and Table 2. Configurations around symmetric point (κ l = κ s ) were generated in previous study [17].
We also performed O(100) zero temperature runs at β = 1.72, 1.73, 1.74 for physical scale setting which are covering almost all transition points of finite temperature simulations.

Simulation results
We show expectation value, susceptibility, skewness and kurtosis of chiral condensate at κ s = 0.128000 as example in Fig. 1.It shows that the re-weighting method works well and we can find the phase transition precisely.Kurtosis intersection plots are shown in Fig. 2. We determine CEP by assuming Z 2 universality class.

Figure 2 .
Figure 2. Kurtosis intersection as a function of β.

Table 1 .
Simulation parameters at κ s = 0.128000 (very large m s runs).
Expectation, susceptibility, skewness and kurtosis as a function of κ l at κ s = 0.128000.