The QCD crossover temperature and curvature coefficient from unbiased exponential resummation at physical quark masses in (2+1)-flavor lattice QCD

Technical Summary: Unbiased Exponential Resummation in (2+1)-Flavor Lattice QCD

Problem Statement
Determining the thermodynamics of strongly interacting matter at non-vanishing temperature and baryon chemical potential ($\mu_B$) is a central goal of Quantum Chromodynamics (QCD). A primary challenge is locating the QCD crossover line and its curvature near $\mu_B = 0$, which is essential for constraining the QCD phase diagram and interpreting heavy-ion collision experiments. Standard lattice QCD approaches are hindered by the sign problem at finite $\mu_B$. Consequently, most first-principles studies rely on Taylor expansions of thermodynamic observables around $\mu_B = 0$ or analytic continuation from imaginary $\mu_B$. While the "unbiased exponential resummation" framework has been developed to reconstruct finite-density representations of the QCD partition function, its application to pseudocritical observables—specifically the determination of the pseudocritical temperature ($T_{pc}$) and the leading curvature coefficient ($\kappa_2^B$)—remains unexplored.

Methodology
The author applies the unbiased exponential resummation method to (2+1)-flavor lattice QCD simulations with physical quark masses on a $32^3 \times 8$ lattice using the Highly Improved Staggered Quark (HISQ) action. The temperature range spans $T \in [135, 176]$ MeV.

The core of the methodology involves reconstructing the ratio of the partition function at finite $\hat{\mu}_B$ to that at zero chemical potential, $R_N$, truncated at the $N$-th order:
$$R_N = \frac{Z_N(\hat{\mu}_B)}{Z(0)} = \left\langle \exp \left[ \sum_{n=1}^N \frac{D_n}{n!} \hat{\mu}_B^n \right] \right\rangle$$
where $D_n$ are derivatives of the fermion determinant logarithm at $\hat{\mu}_B=0$. The author uses $N=4$ to reconstruct the baryon number susceptibility $\hat{\chi}_2^B$ up to $\mathcal{O}(\hat{\mu}_B^4)$, which reproduces the Taylor expansion exactly up to $\mathcal{O}(\hat{\mu}_B^2)$ while generating higher-order terms via the exponential structure.

Two distinct approaches are employed to determine $T_{pc}$ and $\kappa_2^B$:

1. Thermal Derivative Analysis: The temperature dependence of $\hat{\chi}_2^B$ is reconstructed for $\hat{\mu}_B \in [0, 0.5]$. Two thermal derivative observables, $\hat{\chi}_T = T \partial \hat{\chi}_2^B / \partial T$ and $\hat{\chi}_C = T_c \partial \hat{\chi}_2^B / \partial T$ (with $T_c = 145$ MeV), are constructed. The peak positions of these derivatives at $\hat{\mu}_B = 0$ define the pseudocritical temperatures $T_{pc}^T$ and $T_{pc}^C$.
2. Scaling Relations and $\hat{\chi}_4^B$: The fourth-order baryon number susceptibility $\hat{\chi}_4^B$ is extracted from the $\hat{\mu}_B$ dependence of the reconstructed $\hat{\chi}_2^B$. The peak of $\hat{\chi}_4^B$ provides a third estimate, $T_{pc}^\chi$.
3. Curvature Determination:
   • Scaling-based: Using scaling relations between $\hat{\chi}_4^B$ and the thermal derivatives of $\hat{\chi}_2^B$, the curvature is estimated as $\kappa_2^B = \frac{1}{6} \frac{\hat{\chi}_4^B}{\hat{\chi}_C}$. This method relies on the cancellation of non-universal scaling parameters in the scaling regime.
   • Direct Fit: The explicit $\hat{\mu}_B$ dependence of the peak positions ($T_{pc}^T$ and $T_{pc}^C$) is fitted to the form $T_{pc}^x(\hat{\mu}_B) = T_{pc}^x(0) [1 - \kappa_{2,x}^B \hat{\mu}_B^2 + \mathcal{O}(\hat{\mu}_B^4)]$ over the range $\hat{\mu}_B \in [0, 0.3]$.

Key Results

• Pseudocritical Temperatures: At $\hat{\mu}_B = 0$, the thermal derivative analysis yields:
  • $T_{pc}^T = 160.2(4)(3)$ MeV
  • $T_{pc}^C = 158.1(5)(3)$ MeV
    The peak of the fourth-order susceptibility yields $T_{pc}^\chi = 159.4(10)(12)$ MeV (fit method) and $159.4(9)(8)$ MeV (fixed method). These values are consistent with each other and with previous Taylor expansion results (e.g., HotQCD collaboration).
• Curvature Coefficient ($\kappa_2^B$):
  • Scaling Relations: The scaling-based estimates are stable across different $T_{pc}$ definitions:
    • $\kappa_2^B(T_{pc}^T) = 0.02082(14)$
    • $\kappa_2^B(T_{pc}^C) = 0.02064(15)$
    • $\kappa_2^B(T_{pc}^\chi) = 0.02075(14)$
  • Direct Fit: The direct analysis of the $\hat{\mu}_B$ dependence yields larger central values but with significantly larger uncertainties: $\kappa_{2,T}^B = 0.051(35)$ and $\kappa_{2,C}^B = 0.051(34)$.
• Consistency: Despite the larger uncertainties in the direct fit, the results remain statistically compatible with the scaling-based estimates. The scaling-based curvature values are also consistent with those obtained from Taylor expansion methods at comparable temperatures.

Significance and Claims
The paper claims to present the first application of unbiased exponential resummation to the determination of the QCD pseudocritical temperature and curvature at physical quark masses. The primary significance lies in demonstrating that this resummation framework is consistent with expected scaling behaviors and previous lattice QCD results based on Taylor expansions.

Specifically, the author establishes that:

1. Unbiased exponential resummation can successfully reconstruct finite-density susceptibilities and their thermal derivatives without explicitly computing all higher-order Taylor coefficients.
2. The observables derived from this method satisfy the scaling relations required for the curvature coefficient, providing a non-trivial cross-check of the resummed description.
3. The method serves as a complementary approach to Taylor expansions for probing the QCD crossover at small finite baryon densities.

The author remains modest regarding the precision of the direct curvature determination via the pseudocritical line, noting that the current results are "substantially less constrained" than the scaling-based estimates. They conclude that while the framework is established as a viable tool, further work is required to improve the definition of pseudocritical temperatures at finite $\hat{\mu}_B$, increase statistics, and perform continuum extrapolations.