One of the most important open problems in the study of quantum chromodynamics (QCD) at finite temperature and density is the determination of the phase diagram of the theory in the temperature (T)-baryochemical potential (μB) plane. It is now established by first principle lattice QCD calculations that the transition at μB=0 is a smooth crossover [1,2] for physical quark masses. Due to the lack of a real phase transition, the crossover temperature is of course ambiguous, since different definitions can lead to different values for it. Observables related to chiral symmetry (i.e., the chiral condensate and its susceptibility) yield a transition temperature around 155–160 MeV [3–6].

Extending our knowledge to the μB>0 part of the phase diagram turns out to be very challenging due to the notorious sign problem. Since this makes direct simulation at finite μB impossible, the state of the art for finite density QCD on fine lattices is to use one of two extrapolation methods. The first method is the direct calculation of Taylor coefficients [7–17] using simulations at μB=0, while the second is to use simulations at imaginary chemical potentials (μB2<0), where the sign problem is absent, and later perform an extrapolation of different quantities to a real chemical potential (μB2>0) [18–31]. It is often conjectured that in the (T,μB) plane the crossover line, departing from (Tc,μB=0), eventually turns into a first-order transition line. The point (TCEP,μCEP) separating the crossover and the first-order transitions is known as the critical endpoint (CEP), where the transition is expected to be of second order. Though there have been attempts in extracting information about the location of the supposed CEP from lattice simulations [15,26,32–37], these attempts face great difficulties as extrapolation-type methods have the property of giving reliable results mostly in the immediate vicinity of μB=0.

In this Letter, we address the problem of calculating the Taylor coefficients of the crossover temperature around μB=0, parameterized as follows: Tc(μB)Tc(μB=0)=1-κ2[μBTc(μB)]2-κ4[μBTc(μB)]4…(1)along the phenomenologically relevant strangeness neutrality line. In this work we improve the uncertainty on κ4 available in the literature [16] by a factor of 6, giving a state-of-the-art determination of the crossover line in the (T,μB) plane. In particular, as we will show, at chemical potentials μB>200  MeV, the error on the Tc extrapolation is dominated by the subleading coefficients, e.g., κ4. The coefficients κ2 and κ4 can be calculated with either one of the standard extrapolation methods. A direct evaluation of the μB derivatives from μB=0 ensembles was used in Refs. [38,39]. The current state of the art using the μB=0 simulation method is Ref. [16], which includes the first continuum extrapolated results for κ4. Here we will employ an analytical continuation from imaginary μB instead and use lattices as fine as Nt=16. This is motivated by the fact that the signal-to-noise ratio of higher μB derivatives is suppressed with powers of the lattice volume; therefore the calculation of higher order derivatives requires very high statistics. Determinations of κ2 using the imaginary μB method with continuum extrapolation include Refs. [24,25]. Finally, in Ref. [30] the two methods were compared with a careful check of the systematics, and a very good agreement was found for the coefficient κ2. The transition line was also studied in chiral effective models, see, e.g., the recent Ref. [40].

We also study the strength of the crossover by extrapolating the width of the transition and the value of the chiral susceptibility at the transition to real μB in the continuum limit. While one always has to be careful not to overinterpret results from extrapolations, we currently do not see any sign of criticality up to μB≈300  MeV, as the crossover transition does not get narrower or stronger in this region.

For the lattice simulations, we use 4-stout improved staggered fermions with an aspect ratio of LT=4 and temporal lattice sizes of Nt=10, 12, 16. The details of the simulation setup can be found in the Supplemental Material [41]. The use of rooted staggered fermions may come with additional systematic effects that we did not consider in this Letter. Ideally, this work should be repeated with a chiral discretization.

The main observables in this study are the renormalized dimensionless chiral condensate and susceptibility, respectively defined as ⟨ψ¯ψ⟩=-[⟨ψ¯ψ⟩T-⟨ψ¯ψ⟩0]mudfπ4,χ=[χT-χ0]mud2fπ4,with⟨ψ¯ψ⟩T,0=TV∂logZ∂mudχT,0=TV∂2logZ∂mud2,(2)where we assumed isospin symmetry, i.e., mu=md=mud. In the above equations, the subscripts T, 0 indicate values at finite and zero temperature, respectively. In the following, ⟨ψ¯ψ⟩ and χ are always shown after applying the correction to satisfy ns=0 with zero statistical error (see the Supplemental Material [41] for details). The peak height of the susceptibility is an indicator for the strength of the transition, while the peak position in temperature serves as a definition for the chiral crossover temperature. It was pointed out in Refs. [3,4] that different normalizations of the susceptibility, such as using 1/fπ4 or 1/T4 to define χ in Eq. (2), can shift the peak position by 11 MeV. This difference could be considered as a measure for the broadness of the chiral transition.

Our normalization choice in Eq. (2) was motivated by two observations, shown in Fig. 1 and explained below. These observations (together with the improved statistics and the more accurate tuning of μS(μB) to nS=0) allow a very precise determination of Tc as a function of imaginary chemical potential, which in turn allows a precise determination of the parameters κ2 and κ4. We explored the chiral condensate and susceptibility in a broad range of imaginary baryochemical potential. In all panels of Fig. 1, the black curves correspond to μB=0. In the left and middle panel we show the chiral condensate and susceptibility as functions of the temperature. By construction, our renormalized condensate is zero at T=0 and positive at high temperature because of the explicit vacuum subtraction and the overall negative sign in Eq. (2). In both panels, one can observe the shifting of the transition toward higher temperatures when an imaginary chemical potential is introduced. In the right panel we show the susceptibility as a function of the condensate. Here we converted the statistical error on the condensate into an additional error on the susceptibility by solving for ⟨ψ¯ψ⟩(T)=const and substituting the resulting T into χ(T) (also taking the correlation of the statistical errors into account). Our first observation on the right panel of Fig. 1 is that the form of the χ(⟨ψ¯ψ⟩) curve is simpler than that of χ(T): a low-order (e.g., third or fourth) polynomial can fit the entire transition range with an excellent fit quality. The second observation is that there is virtually no chemical potential dependence in the χ(⟨ψ¯ψ⟩) function. This way the susceptibility can be modeled as a low-order polynomial of two variables, ⟨ψ¯ψ⟩ and μ^=μB/T. Had we used a different normalization for the susceptibility, e.g., χ(T)fπ4/T4 as we did in Ref. [5], the peak height would be strongly μB dependent and the collapse of the χ(⟨ψ¯ψ⟩) curves at different (imaginary) chemical potentials would not happen.

Keeping the previous observations in mind, one can perform a precise determination of Tc, as defined by the peak of χ in Eq. (2), for various values of the imaginary chemical potential. Tc(μB2) can then be fitted for the coefficients κ2 and κ4. This requires the following steps: i) Determine the renormalized condensate ⟨ψ¯ψ⟩ and susceptibility χ in a two-dimensional parameter scan in T and ImμB using lattice simulations. Use these to obtain the susceptibility as a function of the condensate. ii) Search for the peak of χ(⟨ψ¯ψ⟩) through a low-order polynomial fit for each Nt and ImμB obtaining ⟨ψ¯ψ⟩c(Nt,ImμB). iii) Use an interpolation of ⟨ψ¯ψ⟩(T) to convert the ⟨ψ¯ψ⟩c to Tc for each ImμB/T. iv) Perform a global fit of Tc(Nt,ImμB/Tc) to determine the coefficients κ2 and κ4 for 1/Nt2=0. For this step we use various functions—all containing an independent κ6—with coefficients depending linearly on 1/Nt2.

There are ambiguities in steps i)–iv). We estimate their systematics by carrying out many versions of these steps. For all these variations and the estimation of the errors, see the Supplemental Material [41]. We finally obtain κ2=0.0153±0.0018,κ4=0.00032±0.00067.(3)We stress that the uncertainties on these two quantities are correlated. We put these results in the context of previous lattice studies in Fig. 2. The extrapolated value of Tc(μB) is shown in Fig. 3 (green band). Note that the errors on κ2 and κ4 are dominated by the statistical errors, as shown in the detailed discussion of the systematic error estimate in the Supplemental Material [41].

Since Ref. [25] we have more than doubled the statistics and introduced a more precise analysis. The overall error on κ2 has reduced slightly. The main result is the extraction of κ4. It appears to be a generic feature of deducing Taylor coefficients from polynomial fits: the increased precision on the input data leads to a sensitivity to a higher order coefficient first and only later to a reduction of the error of both coefficients. This feature is also clearly seen in the mock data analysis in the Supplemental Material [41].

In Fig. 3 we also show the comparison to the leading order Taylor expansion result (using only κ2) and the next to leading order result (using κ2 and κ4). The latter is very close to our full result (for μB<300  MeV), while the leading order result has a much smaller uncertainty. Clearly, κ2 is precise enough. At intermediate μB, the bottleneck for the precision of Tc(μB) is the error on κ4. We also fitted κ6, which turned out to be small enough to be irrelevant for μB<300  MeV.

In Ref. [25] multiple Tc definitions were considered, leading to consistent values for κ2. However, none of those definitions can match the precision of the Tc observable considered here, and precision was a prerequisite for the determination of κ4.

A natural definition of the width of the susceptibility peak is given by its second derivative at Tc as (ΔT)2=-χ(Tc)[(d2/dT2)χ]T=Tc-1. Unfortunately, evaluating this quantity is numerically difficult, so we introduce a simple width parameter σ as a proxy for ΔT via ⟨ψ¯ψ⟩(Tc±σ/2)=⟨ψ¯ψ⟩c±Δ⟨ψ¯ψ⟩/2,(4)with ⟨ψ¯ψ⟩c=0.285 and Δ⟨ψ¯ψ⟩=0.14. The choice of the range in ⟨ψ¯ψ⟩ is such that it is consistent with a linear behavior within our error bars, meaning that the ratio Δ⟨ψ¯ψ⟩/σ can be used as a proxy for (d/dT)⟨ψ¯ψ⟩|T=Tc as well. The exact range in ⟨ψ¯ψ⟩ is chosen such that σ coincides with ΔT at zero and imaginary μB. A more detailed discussion of the width parameter can be found in the Supplemental Material [41].

We conclude that the half-width of the transition—shown in the upper panel of Fig. 4—is consistent with a constant up to μB≈300  MeV within the uncertainty from the extrapolation (we note that 50% uncertainty is reached at μB≈280  MeV).

Finally, as a proxy for the strength of the crossover, we study the value of the chiral susceptibility at the crossover temperature. We get this for each ImμB and Nt as a byproduct of steps i)–ii) of the analysis for κ2 and κ4. If one then performs a continuum extrapolation of the resulting values for fixed values of ImμB, one gets the lower panel of Fig. 4. Again, we see a very mild μ^B2 dependence, consistent with a constant.

The main result of this work is a precise determination of the parameters κ2 and κ4 of the crossover line in finite density QCD. For the determination of the crossover line, we used the experimentally relevant μS(μB) tuned to keep nS=0. Based on the observation that the chiral susceptibility as a function of the condensate is a rather simple function, only weakly dependent on the imaginary chemical potential, we were able to obtain the transition temperature as a function of the imaginary chemical potential to very high accuracy. These pure lattice results can be used for further model building and are summarized in the Supplemental Material [41]. The high precision data at imaginary μB in turn allowed us to fit the μB2 and μB4 Taylor coefficients of the crossover temperature κ2 and κ4. In particular, while our determination of κ4 is still consistent with zero, the error is 6 times smaller than the one previously available in the literature and therefore represents the state of the art in the study of the phase diagram in the (T,μB) plane with current lattice techniques. As a byproduct, we also obtain the most precise value for the central temperature of the crossover at μB=0 so far, as well as the width of the transition: Tc(LT=4,μB=0)=158.0±0.6  MeVΔT(LT=4,μB=0)=15±1  MeV(5)We briefly discuss the meaning of these small errors in the Supplemental Material [41].

We also studied the strength of the phase transition as a function of μB by extrapolating our proxy for the transition width and the peak of the chiral susceptibility from imaginary chemical potentials. Even though one has to be careful with extrapolations, we see no sign of the transition getting stronger up to μB≈300  MeV.