Neutral pion masses within a hot and magnetized medium in a lattice-improved soft-wall $\mathrm{AdS}/\mathrm{QCD}$ model

We investigate chiral phase transitions and the screening masses, pole masses, and thermal widths of neutral pion mesons with finite temperature $T$ and magnetic field $B$ in a lattice-improved $\mathrm{AdS}/\mathrm{QCD}$ model, which is constructed by fitting the lattice results of the pseudocritical temperatures $T_{\mathrm{pc}}(B)$. Specifically, we find that the chiral condensate $\sigma$ undergoes a crossover phase transition demonstrating distinct magnetic catalysis and inverse magnetic catalysis effects in very low- and high-temperature regions with fixed finite $B$, respectively. For the screening masses, we find that the longitudinal component decreases with $B$ at very low and high temperatures and increases with $B$ near $T_{\mathrm{pc}}$. The transverse component always increases with $B$ at fixed $T$. However, both the longitudinal and transverse screening masses increase with $T$ at fixed $B$. Furthermore, we find that the pole mass decreases with the increase of $B$ or $T$. It is interesting to note that the thermal width shows similar behavior to the longitudinal screening masses in the very-high-temperature region.

Figure 1

The $f$, $q$, and $h$ as functions of $z$ represented by red, blue, and cyan lines, respectively. (a)–(c) Magnetic field $B$ fixed at $1{\mathrm{GeV}}^2$, with temperature $T$ fixed at 8, 159, and 398 MeV, respectively. (d)–(f) Temperature $T$ fixed at a high value, $T\approx 398\mathrm{MeV}$; magnetic field $B$ fixed at 0.1, 1, and $4G{\mathrm{eV}}^2$, respectively. (h),(i) Temperature $T$ fixed at a low value, $T\approx 8\mathrm{MeV}$; magnetic field $B$ fixed at 0, 0.1, and $2{\mathrm{GeV}}^2$, respectively.

Figure 2

(a) Horizon boundary $z_h$ as a function of temperature $T$ and magnetic field $B$. (b) Distribution of relative error of $z_h$, $\delta =\mathrm{\Delta }/z_h=(z_h-{\tilde{z}}_h)/z_h$, with ${\tilde{z}}_h$ the perturbation result, between the results from the perturbative asymptotic approximation and the full solution across the plane of the temperature $T$ and magnetic field $B$.

Figure 3

(a) Chiral condensation $\sigma$ as a function of $T$ under different fixed $B$. The range of $B$ is approximately from 0 to 51.32 $M_{\pi }^2$, which corresponds to 0 to $1{\mathrm{GeV}}^2$ in our model. (b) $\mathrm{\Delta }\sigma =\sigma (B,T)-\sigma (0,T)$ as a function of $B/M_{\pi }^2$ under different fixed $T$, where the temperatures $T$ are chosen as 70, 88, 130, 140, 162, and 201 MeV, respectively.

Figure 4

Pseudocritical temperature scaled by its $B=0$ value as a function of $B/M_{\pi }^2$. These results compare with those from lattice simulations [4] and lattice-improved NJL [22]. Given that the values of $M_{\pi }$ in lattice simulations and the lattice-improved NJL model are 135 and 138 MeV, respectively, the results have been scaled accordingly for their respective $M_{\pi }$ in the extracted data.

Figure 5

The $m_{\mathrm{scr},\parallel }$ and $m_{\mathrm{scr},\perp }$ as a function of $T$ under different fixed $B$, respectively. The fixed magnetic field $B$ is about 0, 5.56, 21.04, 36.44, and 51.32 $M_{\pi }^2$, which corresponds to 0, 0.11, 0.41, 0.71, and $1{\mathrm{GeV}}^2$, respectively, in our model. The inset in (a) and (b) is a zoom for the range $T\in [0.05,0.14]\mathrm{GeV}$. For $B=0$, 5.56, 21.04, 36.44, and 51.32 $M_{\pi }^2$, their corresponding $T_{\mathrm{pc}}$ of the chiral phase transition are 161.0, 160.2, 156.0, 146.5, and 138.8 MeV, respectively.

Figure 6

Normalized screening mass, $m_{\mathrm{scr}}(B,T)/m_{\mathrm{scr}}(0,T)$, as a function of $B/M_{\pi }^2$ under different fixed $T$, where the temperatures $T$ are chosen as 70, 158, 212, and 398 MeV, respectively. Panels (a) and (b) show normalized screening masses in the longitudinal and transverse screening masses, respectively.

Figure 7

(a) Ratio of sound velocity $u_{\perp }/u_{\parallel }$ as a function of $T$ under different fixed $B$. The range of $B$ is approximately from 0 to 51.32 $M_{\pi }^2$, which corresponds to 0 to $1{\mathrm{GeV}}^2$ in our model. (b) Ratio of sound velocity $u_{\perp }/u_{\parallel }$ as a function of $B/M_{\pi }^2$ under different fixed $T$, where the temperatures $T$ are chosen as 50, 150, 201, 251, and 398 MeV, respectively. For $B=0$, 10.78, 21.04, 31.31, and 51.32 $M_{\pi }^2$, the corresponding $T_{\mathrm{pc}}$ of the chiral phase transition are 161.0, 158.7, 156.1, 150.2, and 138.8 MeV, respectively. Furthermore, we have extracted results from the lattice-improved NJL model [22], represented by dots in the graph. The displayed results from the lattice-improved NJL model have been scaled according to $M_{\pi }$ of the model, $M_{\pi }=138\mathrm{MeV}$.

Figure 8

Illustration of $m_{\mathrm{pole}}$ and $\mathrm{\Gamma }/2$ as a function of $T$ under different fixed $B$. The selection of $B$ is about 0, 15.91, 31.31, 41.57, and 51.32 $M_{\pi }^2$, which corresponds to 0, 0.51, 0.81, and $1{\mathrm{GeV}}^2$, respectively, in our model. The inset in (b) is a zoom for the range $T\in [0.05,0.14]\mathrm{GeV}$.

Figure 9

Normalized $m_{\mathrm{pole}}$ and normalized $\mathrm{\Gamma }/2$ as a function of $B/M_{\pi }^2$ under different fixed $T$, where the temperatures $T$ are chosen as 50, 80, 105, and 125 MeV, and 120, 140, 169, 185, 205, and 299 MeV respectively.