Critical point of Nf¼3 QCD from lattice simulations in the canonical ensemble

A canonical ensemble algorithm is employed to study the phase diagram of Nf ¼ 3 QCD using lattice simulations. We lock in the desired quark number sector using an exact Fourier transform of the fermion determinant. We scan the phase space below Tc and look for an S-shape structure in the chemical potential, which signals the coexistence phase of a first order phase transition in finite volume. Applying Maxwell construction, we determine the boundaries of the coexistence phase at three temperatures and extrapolate them to locate the critical point. Using an improved gauge action and improved Wilson fermions on lattices with a spatial extent of 1.8 fm and quark masses close to that of the strange, we find the critical point at TE ¼ 0:925ð5ÞTc and baryon chemical potentialE ¼ 2:60ð8ÞTc. QCD is expected to have a rich phase diagram at finite temperature and finite density. Current lattice calculations have shown that the transition from the hadronic phase to QGP phase is a rapid crossover (1,2). For large baryon chemical potential and very low temperature, a number of models suggest that the transition is a first order. If this is the case, when the chemical potential is lowered and temperature raised, this first order phase transition is ex- pected to end as a second order phase transition point—the critical point. However, lattice QCD simulations with chemical potential are difficult due to the notorious ''sign problem''. The majority of current simulations are focus- ing on small chemical potential regionq=T � 1 where the ''sign problem'' appears to be under control. Up to now, all the Nf ¼ 3 or 2 þ 1 simulations are based on the grand canonical ensemble (T, � B as parameters) with staggered fermions. The results from the multiparameter reweighting (3), Taylor expansion with small � (4,5) and the curvature of the critical surface (6) are not settled and need to be cross-checked. Even the existence of the critical point is in question (6). We employ an algorithm, which is not re- stricted to small chemical potential because of the mitiga- tion of the sign problem under the current parameter settings, to study this problem. In this letter, we adopt an exact Monte Carlo algorithm (7-9) based on the canonical partition function (10-15) which is designed to alleviate the determinant fluctuation problems. As it turns out, the sign fluctuations are not serious on the lattices used in the present study, as we shall see later. In the canonical ensemble simulations in finite volume, the coexistence phase of a first order phase tran- sition has a characteristic S-shape as a function of density due to the surface tension. This finite-volume property has been exploited successfully to identify the phase bounda- ries via the Maxwell construction in studies of phase transition with the staggered fermions (14,15) and clover fermions (16) for the Nf ¼ 4casewhich is known to have a first order phase transition at � ¼ 0. In these benchmark studies the boundaries were identified at three temperatures below Tc, and they were extrapolated in density and tem- perature to show that the intersecting point indeed coin- cides with the independently identified first order transition point at T c and � ¼ 0 (16). In view of the success of the N f ¼ 4 study, we extend this method to the more realistic Nf ¼ 3 case (17,18). Although the real world contains two light quarks and one heavier strange quark, the three degenerate flavor case has a similar phase structure. Our primary goal in this study is to determine whether a first order phase transition exists for Nf ¼ 3 and where the critical point is located. With the aid of recently developed matrix reduction technique (19-21), we scan the chemical potential as a function of baryon number for four temperatures below Tc which is determined at zero chemical potential, and we observe clear signals for a first order phase transition for temperatures below 0:93Tc. The phase boundaries of the coexistence phase are determined and then extrapolated in temperature and density to locate the critical point at TE ¼ 0:927ð5ÞTc andE ¼ 2:60ð8ÞTc. Our results are based on simulations on 6 3 � 4 lattices with clover fer- mion action with quark masses which correspond to the pion mass from 750 MeV for the lowest temperature to 775 MeV for the highest temperature. The canonical partition function in lattice QCD can be derived from the fugacity expansion of the grand canonical partition function, ZðV; T; � Þ¼ X k ZCðV; T; kÞek=T ; (1)