The chiral phase transition temperature in (2+1)-flavor QCD

We present a lattice QCD based determination of the chiral phase transition temperature in QCD with two degenerate, massless quarks and a physical strange quark mass. We propose and calculate two novel estimators for the chiral transition temperature for several values of the light quark masses, corresponding to Goldstone pion masses in the range of $58~{\rm MeV}\lesssim m_\pi\lesssim 163~{\rm MeV}$. The chiral phase transition temperature is determined by extrapolating to vanishing pion mass using universal scaling analysis. Finite volume effects are controlled by extrapolating to the thermodynamic limit using spatial lattice extents in the range of $2.8$-$4.5$ times the inverse of the pion mass. Continuum extrapolations are carried out by using three different values of the lattice cut-off, corresponding to lattices with temporal extent $N_\tau=6,\ 8$ and $12$. After thermodynamic, continuum and chiral extrapolations we find the chiral phase transition temperature $T_c^0=132^{+3}_{-6}$ MeV.

Introduction.-Forphysical values of the light up, down and heavier strange quark masses strongly interacting matter undergoes a transition from a low temperature hadronic regime to a high temperature region that is best described by quark and gluon degrees of freedom.This smooth crossover between the two asymptotic regimes is not a phase transition [1].It is characterized by a pseudo-critical temperature, T pc , that has been determined in several numerical studies of Quantum Chromodynamics (QCD) [2][3][4].A recent determination of T pc extracted from the maximal fluctuations of several chiral observables gave T pc = (156.5 ± 1.5) MeV [5].
In the chiral limit of (2+1)-flavor QCD, i.e.where two (degenerate) light quark masses m l = (m u + m d )/2 approach zero but the strange quark mass m s is kept fixed to its physical value, the pseudo-critical behavior is expected to give rise to a "true" chiral phase transition [6,7].Whether this chiral phase transition is first or second order may depend crucially on the temperature dependence of the chiral anomaly [7].In the latter case critical behavior generally is expected to be controlled by the 3-d O(4) universality class, although a larger 3-d universality class [8,9] may become of relevance in case the axial anomaly gets also restored effectively at T 0 c .If the chiral phase transition is first order then a second order phase transition, belonging to the 3-d Z(2) universality class, would occur for m c l > 0. When decreasing the light to strange quark mass ratio, H = m l /m s , towards zero, this would give rise to diverging susceptibilities already for some critical mass ratio H c = m c l /m s > 0. The analysis presented here leads to a determination of the critical temperature T Hc c .However, as we do not have any evidence for H c = 0, we de-facto present a determination of the chiral phase transition temperature T 0 c .Although T 0 c appears as a fit parameter in all finite temperature scaling studies of the chiral transition in QCD [3,10,11], so far no lattice QCD calculation has carried out a systematic analysis of T 0 c by controlling thermodynamic, continuum and chiral limits.Here, we will present a first lattice QCD based determination of T 0 c in (2 + 1)-flavor QCD with controlled thermodynamic, continuum and chiral extrapolations.QCDinspired model calculations [12,13] suggest that T 0 c might be even (20 − 30) MeV lower than T pc .To mitigate this potentially large m l -dependence of T pc while approaching m l → 0, we propose two novel estimators of the pseudocritical temperature having only mild dependence on m l , leading to well-controlled chiral extrapolation.
Chiral observables.-Atlow temperatures chiral symmetry is spontaneously broken in QCD.An order parameter for the restoration of this symmetry at high temperature is the chiral condensate, which is obtained as the derivative of the partition function, Z(T, V, m u , m d , m s ), with respect to one of the quark masses, m f , The light quark chiral condensate, ψψ l = ( ψψ u + ψψ d )/2, is an order parameter for the chiral phase transition that occurs in the limit m l → 0. For non-vanishing m l this order parameter requires additive and multiplicative renormalization.We take care of this by introducing a renormalization group (RG) invariant combination of the light and strange quark chiral condensates,    [15,17,18].Also given are fG(zp), fχ(zp) and rχ(0) = fχ(0)/fχ(zp).
where the kaon decay constant, f K = 156.1(9)/√ 2 MeV, for physical values of the degenerate light and strange quark mass is used as normalization constant to define a dimensionless order parameter M .Its derivative with respect to the light quark masses defines the RG-invariant chiral susceptibility, with χ f g = ∂ m f ψψ g and χ l = 2(χ uu + χ ud ).
When approaching the chiral limit, one also needs to control the thermodynamic limit, V → ∞.In the vicinity of a 2 nd order phase transition M and χ M are given in terms of universal finite-size scaling functions f G (z, z L ) and f χ (z, z L ), which depend on scaling variables z = t/h 1/βδ and z L = l 0 /(Lh ν/βδ ).Here t = (T − T 0 c )/(t 0 T 0 c ) denotes the reduced temperature, h = H/h 0 is the symmetry breaking field and L/l 0 parametrizes the finite-size of the system, L ≡ V 1/3 .These scaling variables are expressed in terms of non-universal parameters, t 0 , h 0 , l 0 .
While the universal scaling functions control the behavior of M and χ M close to a critical point at (z, z L ) = (0, 0), they also receive contributions from correctionsto-scaling and regular terms [14,15], which we represent by a function f sub (T, H, L).With this we may write As far as needed for the analysis we will specify contributions arising from f sub (T, H, L) later.
Close to the thermodynamic limit f χ (z, z L ) has a pronounced peak, which often is used to define a pseudocritical temperature, T p .In the scaling regime this peak is located at some z = z p (z L ), which defines T p , with z 0 = h 1/βδ 0 /t 0 .While the first term describes the universal quark mass dependence of T p , corrections may arise from corrections-to-scaling and regular terms, shifting the peak-location of the chiral susceptibilities.
When approaching the chiral limit, depending on the magnitude of z p /z 0 ≡ z p (0)/z 0 , T p (H, L) may change significantly with H.In the potentially large temperature interval between T 0 c and T p (H, L) regular contributions, arising from f sub (T, H, L), may also be large, and during the H → 0 extrapolation several non-universal parameters may be needed to account for contributions from f sub (T, H, L).It is, thus, advantageous to determine T 0 c using observables defined close to z 0. T p (H, L) defined through such observables for small H > 0 will have milder H-dependence, and the determination of T 0 c = T p (H → 0, L → ∞) will be well-controlled.We will consider here two estimators for T 0 c , defined at or close to z = 0. We determine temperatures T δ and T 60 by demanding, Eq. 6 has already been introduced in [16] as a tool to analyze the chiral transition in QCD, and it is understood that T 60 is determined at a temperature on the left of the peak χ max M , i.e.T 60 < T p .These relations define pseudo-critical temperatures, T X , which are close to T 0 c already for non-zero H and L −1 .They converge to the chiral phase transition temperature T 0 c , in the thermodynamic and chiral limits.For non-zero L −1 , Eqs. 6 and 7 involve scaling variables z X (z L ) which approach or are close to zero in the limit L −1 → 0, i.e. z δ ≡ z δ (0) = 0 and z 60 ≡ z 60 (0) 0. Some values for z 60 , for several universality classes, are given in Table I and the relevant scaling functions, obtained in the thermodynamic limit, z L = 0, are shown in Fig. 1.
Ignoring possible contributions from corrections-toscaling, and keeping in f sub only the leading Tindependent, infinite volume regular contribution propor- tional to H, we then find for the pseudo-critical temperatures, The universal functions, z X (z L ) may directly be determined from the ratio of scaling functions, The finite-size scaling functions f G (z, z L ), f χ (z, z L ) have been determined for the 3-d, O(4) universality class in Ref. [19].
We will present here results on T δ and T 60 obtained in lattice QCD calculations [20].We calculated the chiral order parameter M and the chiral susceptibility χ M (Eqs. 2 and 3) in (2 + 1)-flavor QCD with degenerate up and down quark masses (m u = m d ).For our lattice QCD calculations, performed with the Highly Improved Staggered Quark (HISQ) action [21] in the fermion sector and the Symanzik improved gluon action, the strange quark mass has been tuned to its physical value [22] and the light quark mass has been varied in a range m l ∈ [m s /160 : m s /20] corresponding to Goldstone pion masses in the range 58 MeV < ∼ m π < ∼ 163 MeV.At each temperature we performed calculations on lattices of size N 3 σ N τ for three different values of the lattice cut-off, aT = 1/N τ , with N τ = 6, 8 and 12.The spatial lattice extent, N σ = L/a, has been varied in the range 4 ≤ N σ /N τ ≤ 8.For each N τ we analyzed the volume dependence of M and χ M in order to perform controlled infinite volume extrapolations.
Results -In Fig. 2 (left) we show results for χ M on lattices with temporal extent N τ = 8 for 5 different values of the quark mass ratio, H = m l /m s , and the largest lattice available for each H.The increase of the peak height, χ max M , with decreasing H is apparent.This rise is consistent with the expected behavior, χ max M ∼ H 1/δ−1 + const., with δ 4.8; however a precise determination of δ is not yet possible with the current data.
In Fig. 2 (right) we show the volume dependence of χ M for H = 1/80 on lattices with temporal extent N τ = 8 and for N σ /N τ = 4, 5 and 7. Similar results have also been obtained for N τ = 6 and 12.We note that χ max M decreases slightly with increasing volume, contrary to what one would expect to find at or close to a 1 st or 2 nd order phase transition.Our current results, thus, are consistent with a continuous phase transition at H c = 0.
Using results for χ M and M we constructed the ratios Hχ M /M for different lattice sizes and several values of the quark masses.This is shown in Fig. 3 (left) for the lightest quark masses used on the N τ = 12 lattices, H = 1/80.The intercepts with the horizontal line at 1/δ define T δ (H, L).For H = 1/80 and each of the three temporal lattice sizes we have results for three different volumes on which we can extrapolate T δ (H, L) to the infinite volume limit.We performed such extrapolations using (i) the O(4) ansatz given in Eq. 8 as well as (ii) an extrapolation in 1/V .The latter is appropriate if, for large L, the volume dependence predominantly arises from regular terms and the former is appropriate close to or in the continuum limit, if the singular part dominates the partition function.In the former case we use the approximation z δ (z L ) ∼ z 5.7  L , which parametrizes well the finite-size dependence of T δ in the scaling regime [19].The resulting fits are shown in Fig. 3 (middle).We note that results for fixed H tend to approach the infinite volume limit more rapidly than 1/V , which is in accordance with the behavior expected from the ratio of finite-size scaling functions.The resulting continuum limit extrapolations in 1/N 2 τ based on data for (i) all three N τ values, as well as (ii) N τ = 8 and 12 only, are shown as horizontal bars in this figure.An analogous analysis is performed for H = 1/40.Finally, we extrapolate the continuum results for T δ (H, ∞) with H = 1/40 and 1/80 to the chiral limit using Eq. 8 with z δ (0) = 0. Results obtained from these extrapolation chains, which involve either an 1/V or O(4) ansatz for the infinite volume extrapolation, and continuum limit extrapolations performed on two different data sets, lead to chiral transition temperatures T 0 are summarized in Fig. 4.
As the fits shown in Fig. 3 (middle) suggest that the O(4) scaling ansatz is appropriate for the analysis of finite volume effects already at non-zero values of the cut-off, we can attempt a combined analysis of all data available for different light quark masses and volumes at fixed N τ .This utilizes the quark mass dependence of finite-size corrections, expressed in terms of z L and, thus, intertwines continuum and chiral limit extrapolations.Using the scaling ansatz given in Eq. 8, it also allows to account for the contribution of a regular term in a single fit.Fits for fixed N τ based on this ansatz, using data for all available lattice sizes and H ≤ 1/27, are shown in Fig. 3 (right).For each N τ the fit yields results for T δ (H, L) at arbitrary H.Some bands for H = 1/40 and 1/80 are shown in the figure.As can be seen, for H = 1/80, these bands compare well with the fits shown in Fig. 3 (middle).For each N τ an arrow shows the corresponding chiral-limit result, T δ (0, ∞).We extrapolated these chiral-limit results to the continuum limit and estimated systematic errors again by including or leaving out data for N τ = 6.The resulting T 0 c , shown in Fig. 4, are in complete agreement with the corresponding numbers obtained by first taking the continuum limit and then taking the chiral limit.Within the current accuracy these two limits are interchangeable.Similarly we analyzed results for T 60 on all data sets using the same analysis strategy as for T δ .As can be seen in Fig. 4, we find for each extrapolation ansatz that the resulting values for T 0 c agree to better than 1% accuracy with the corresponding values for T δ .This corroborates that the chiral susceptibilities used for this analysis reflect basic features of the O(4) scaling functions.
Performing continuum extrapolations by either including or discarding results obtained on the coarsest (N τ = 6) lattices leads to a systematic shift of about (2-3) MeV in the estimates for T 0 c .This is reflected in the displacement of the two bands in Fig. 4, which show averages for T 0 c obtained with our different extrapolation ansätze.Averaging separately over results for T δ and T 60 obtained with both continuum extrapolation procedures and including this systematic effect we find for the chiral phase transition temperature, Conclusions.-Based on two novel estimators, we have determined the chiral phase transition temperature in QCD with two massless light quarks and a physical strange quark.Eq. 9 lists our thermodynamic-, continuum-and chiral-extrapolated result for the chiral phase transition temperature, which is about 25 MeV smaller than the pseudo-critical (crossover) temperature, T pc for physical values of the light and strange quark masses [5].
Acknowledgments.-This work was supported in part by the Deutsche Forschungsgemeinschaft (DFG) through the grant 315477589-TRR 211, the grants 05P15PBCAA and 05P18PBCA1 of the German Bundesministerium für Bildung und Forschung, grant 283286 of the European Union, the National Natural Science Foundation of China under grant numbers 11775096 and 11535012, Furthermore, this work was supported through Contract No. de-sc0012704 with the U.S. Department of Energy, through the Scientific Discovery through Advanced Computing (SciDAC) program funded by the U.S. Department of

FIG. 2 .
FIG. 2. Quark mass (left) and volume (right) dependence of the chiral susceptibility on lattices with temporal extent Nτ = 8.The left hand figure shows results for several values of the quark masses.The spatial lattices extent Nσ is increased as the light quark mass decreases: Nσ = 32 (H −1 = 20, 27), 40 (H −1 = 40), 56 (H −1 = 80, 160).The right hand figure shows results for three different spatial lattice sizes at H = 1/80.Black symbols mark the points corresponding to 60% of the peak height.

c 3 mFIG. 3 .
FIG. 3. Left:The ratio HχM /M versus temperature for Nτ = 12, m l /ms = 1/80 and different spatial volumes.Middle: Infinite volume extrapolations based on an O(4) finite-size scaling ansatz (colored bands) and fits linear in 1/V (grey bands).Horizontal bars show the continuum extrapolated results for H = 1/80.Right: Finite size scaling fits for T δ based on all data for H ≤ 1/27 and all available volumes.Arrows show chiral limit results at fixed Nτ and horizontal bars show the continuum extrapolated results for H = 0.

( 1 )
FIG. 4. Summary of fit results.For details see text.