Extrapolation for meson screening masses from imaginary to real chemical potential

We first extend our formulation for the calculation of $\pi$- and $\sigma$-meson screening masses to the case of finite chemical potential $\mu$. We then consider the $\mathrm{imaginary}\text{−}\mu$ approach, which is an extrapolation method from an imaginary chemical potential ($\mu =i{\mu }_{\mathrm{I}}$) to a real one ($\mu ={\mu }_{\mathrm{R}}$). The feasibility of the method is discussed based on the entanglement Polyakov-loop extended Nambu–Jona-Lasinio (EPNJL) model in 2-flavor system. As an example, we investigate how reliable the $\mathrm{imaginary}\text{−}\mu$ approach is for $\pi$- and $\sigma$-meson screening masses, comparing “screening masses at ${\mu }_{\mathrm{R}}$ in the method” with “those calculated directly at ${\mu }_{\mathrm{R}}$.” We finally propose a new extrapolation method and confirm its efficiency.

Figure 1

Singularities of ${\chi }_{\xi \xi }(0,{\tilde{q}}^2)$ in the $\mathrm{complex}\text{−}\tilde{q}$ plane. Cuts are denoted by the wavy lines and poles are denoted by points. The threshold masses correspond to the endpoints of cuts. The original contour $C$ in Eq. (20) is deformed into $C_1$, $C_2$ (cut contributions) and $C_3$ (pole contribution). For the definition of threshold masses, see Eq. (23).

Figure 2

$T$ dependence of the chiral condensate $\sigma$ and the absolute value $|\mathrm{\Phi }|$ of Polyakov loop for $\theta =0$. The horizontal axis is scaled by $T_{\mathrm{c}}^{\chi }$. The $T_0$ uncertainty is denoted by bands for $\sigma$ and $\mathrm{\Phi }$. The chiral condensate is normalized by its value ${\sigma }_0$ at $T=0$. LQCD data are taken from Refs. [46, 47]. The 10% errors come from those of $T_{\mathrm{c}}^{\mathrm{d},\mathrm{LQCD}}$ and $T_{\mathrm{c}}^{\chi ,\mathrm{LQCD}}$.

Figure 3

Deconfinement- and chiral-transition lines in the $\mathrm{imaginary}\text{−}\mu$ region. The thin solid lines with bands stand for the deconfinement- and chiral-transition lines, where the bands are originated in the ambiguity of $T_0$. The location of CEP in the deconfinement transition is limited in two bars because of the uncertainty of $T_0$. At the CEP, the deconfinement transition is the second order. The first-order RW transition line is denoted by the thick solid line.

Figure 4

$\theta$ dependence of $M_{\pi }^{\mathrm{scr}}$ and $M_{\sigma }^{\mathrm{scr}}$ for two cases (A) and (B). The $M_{\pi }^{\mathrm{scr}}$ and $M_{\sigma }^{\mathrm{scr}}$ are shown by the thin solid lines with bands. The dotted line (dot-dash line) denotes $\sigma$-meson ($\pi$-meson) screening mass at vacuum.

Figure 5

Polynomial fitting for $\theta$ dependence of $M_{\pi }^{\mathrm{scr}}$ and $M_{\sigma }^{\mathrm{scr}}$ in case (A) and case (B).

Figure 6

Comparison between the extrapolating and the direct results on ${\mu }_{\mathrm{R}}/T$ dependence of $M_{\pi }^{\mathrm{scr}}$. We draw direct-result lines only when the inequality $M_{\pi }^{\mathrm{scr}}<M_{\mathrm{th}}$ in Eq. (25) is satisfied.

Figure 7

Comparison between the extrapolating and the direct results on ${\mu }_{\mathrm{R}}/T$ dependence of $M_{\sigma }^{\mathrm{scr}}$ in case (B), i.e., $T=1.02T_{\mathrm{c}}^{\chi }$.

Figure 8

A schematic figure of PTL extrapolation and standard-extrapolation. The arrows stand for the standard extrapolation and the PTL extrapolation. Transition line for chiral symmetry restoration is denoted by the dotted line.

Figure 9

${\mu }_{\mathrm{R}}/T_{\mathrm{PTL}}^{\chi }$ dependence of $\sigma$-meson screening mass for $T_{\mathrm{PTL}}^{\chi }(0)=1.02T_{\mathrm{c}}^{\chi }(0)$.

Figure 10

${\mu }_{\mathrm{R}}/T_{\mathrm{PTL}}^{\chi }$ dependence of $\pi$-meson screening mass for $T_{\mathrm{PTL}}^{\chi }(0)=1.02T_{\mathrm{c}}^{\chi }(0)$.