Abstract
We study different estimators of the radius of convergence of the Taylor series of the pressure in finite density QCD. We adopt the approach in which the radius of convergence is estimated first in a finite volume, and the infinite-volume limit is taken later. This requires an estimator for the radius of convergence that is reliable in a finite volume. Based on general arguments about the analytic structure of the partition function in a finite volume, we demonstrate that the ratio estimator cannot work in this approach, and propose three new estimators, capable of extracting reliably the radius of convergence, which coincides with the distance from the origin of the closest Lee-Yang zero. We also provide an estimator for the phase of the closest Lee-Yang zero, necessary to assess whether the leading singularity is a true critical point. We demonstrate the usage of these estimators on a toy model, namely 4 flavors of unimproved staggered fermions on a small $6^3 \times 4$ lattice, where both the radius of convergence and the Taylor coefficients to any order can be obtained by a direct determination of the Lee-Yang zeros. Interestingly, while the relative statistical error of the Taylor expansion coefficients steadily grows with order, that of our estimators stabilizes, allowing for an accurate estimate of the radius of convergence. In particular, we show that despite the more than 100\% error bars on high-order Taylor coefficients, the given ensemble contains enough information about the radius of convergence.
Highlights
One of the most important unsolved problems in QCD is the determination of its phase diagram at finite baryonic density
An open question is whether the analytic crossover of the chiral transition at zero chemical potential turns into a genuine phase transition sufficiently deep in the μ − T plane, and, if so, where the critical end point lies
Nonperturbative studies of these questions on the lattice are notoriously hampered by the sign problem, which prevents the use of standard Monte Carlo techniques to probe QCD directly at finite density
Summary
One of the most important unsolved problems in QCD is the determination of its phase diagram at finite baryonic density. The radius of convergence coincides with the distance from the origin of the closest Lee-Yang zero [30], i.e., the zero of the partition function closest to the origin in the complex μ plane. It is clear that while the Lee-Yang zeros determine both the critical points (if any) of the theory and the radius of convergence of the Taylor series of the pressure, nothing guarantees that the same zeros are involved in the two cases. In the case of QCD with staggered fermions, the fermionic determinant appearing in the functional-integral representation of the partition function is positive definite for purely imaginary μ, and so no zeros will appear on the imaginary μaxis, or equivalently on the unit circle in the complex fugacity plane
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