Inverse magnetic catalysis effect and current quark mass effect on mass spectra and Mott transitions of pions under external magnetic field

With our fitted $G(eB)$ in Fig.1, we solve the gap equation (7) and pole equation (5) to obtain the $\pi^{0}$ meson mass at finite temperature and magnetic field with IMC effect. The results with and without IMC effect are plotted in red and blue, respectively.

Figure 2 depicts the $\pi^{0}$ meson mass $m_{\pi^{0}}$ as a function of magnetic field at vanishing temperature with (red solid line) and without (blue dashed line) inverse magnetic catalysis effect. Because in both cases the magnetic field enhances the breaking of chiral symmetry in vacuum. As the pseudo-Goldstone boson, the $\pi^{0}$ meson masses are decreasing functions of magnetic field with and without IMC effect. We observe a lower value for $m_{\pi^{0}}$, when including the IMC effect. Similar conclusion is obtained in Refs meson ; su3meson4 , where inverse magnetic catalysis effect is introduced into the two-flavor and three-flavor NJL models.

In Fig.3, we plot the $\pi^{0}$ meson mass $m_{\pi^{0}}$ at finite temperature and fixed magnetic field $eB=20m_{\pi}^{2}$. With (red lines) and without (blue lines) inverse magnetic catalysis effect, the $\pi^{0}$ mass spectra have the similar structure. $m_{\pi^{0}}$ increases at low temperature region and jumps at the Mott transition $T=T^{0}_{m}$. After that, it firstly decreases and then increases with temperature. The inverse magnetic catalysis effect leads to some quantitative modifications. For instance, Mott transition temperature $T^{0}_{m}$ is shifted to a lower value, and $m_{\pi^{0}}$ becomes lower (higher) at $T<T^{0}_{m}$ ($T>T^{0}_{m}$).

Figure 4 shows the Mott transition temperature of $\pi^{0}$ meson $T^{0}_{m}$ as a function of magnetic field, with (red solid line) and without (blue dashed line) inverse magnetic catalysis effect. Without IMC effect, the $T^{0}_{m}$ decreases with the magnetic field in weak magnetic field region $(eB\leq 5m^{2}_{\pi})$, becomes flat in medium magnetic field region $(5m^{2}_{\pi}\leq eB\leq 16m^{2}_{\pi})$ and decreases in strong magnetic field region $(eB\geq 16m^{2}_{\pi})$. With IMC effect, the flat structure of Mott transition temperature disappears and $T^{0}_{m}$ monotonically decreases with magnetic field. Furthermore, at fixed magnetic field, $T^{0}_{m}$ has a lower value when including IMC effect.

With our fitted $G(eB)$ in Fig.1, we solve the gap equation (7) and pole equation (10) to obtain $\pi^{+}$ meson mass at finite temperature and magnetic field with IMC effect. The results with and without IMC effect are plotted in red and blue, respectively.

As shown in Fig.5, with and without IMC effect, $m_{\pi^{+}}$ are increasing functions of magnetic field, and no decreasing behavior is observed at vanishing temperature. With IMC effect, $m_{\pi^{+}}$ has a lower value than without IMC effect, and the deviation becomes larger at stronger magnetic field. Similar results are obtained in three-flavor NJL model including IMC effect su3meson4 .

In Fig.6, we make comparison of $m_{\pi^{+}}$ as a function of temperature at fixed magnetic field $eB=20m_{\pi}^{2}$ with (red lines) and without (blue lines) inverse magnetic catalysis effect. They show similar structure. $m_{\pi^{+}}$ decreases in the low temperature region. At the Mott transition $T^{+}_{m}$, $m_{\pi^{+}}$ shows the first jump, and two other mass jumps happen at $T^{+}_{1}$ and $T^{+}_{2}$. With $T^{+}_{m}<T<T^{+}_{1}$, $m_{\pi^{+}}$ first decreases and then increases with temperature. With $T^{+}_{1}<T<T^{+}_{2}$ and $T>T^{+}_{2}$, $m_{\pi^{+}}$ decreases with temperature. At high enough temperature, $m_{\pi^{+}}$ becomes independent of temperature and IMC effect. In the low and high temperature regions, $m_{\pi^{+}}$ with IMC effect is smaller than without IMC effect. The value of $T^{+}_{m}=124,160$ MeV, $T^{+}_{1}=136,172$ MeV and $T^{+}_{2}=157,187$ MeV are different for the cases with and without IMC effect, which indicates that IMC effect lowers down the temperatures of ${\pi^{+}}$ meson mass jumps.

Figure 7 is the Mott transition temperature $T^{+}_{m}$ of $\pi^{+}$ meson as a function of magnetic field, with (red solid line) and without (blue dashed line) inverse magnetic catalysis effect. A fast increase of $T^{+}_{m}$ occurs when turning on external magnetic field, where a peak structure around $eB\simeq 1m^{2}_{\pi}$ shows up. Without IMC effect, the $T^{+}_{m}$ decreases with magnetic field and then increases, which are associated with some oscillations. With IMC effect, the $T^{+}_{m}$ decreases as magnetic field goes up, which is also accompanied with some oscillations. At fixed magnetic field, $T^{+}_{m}$ has a lower value, when including IMC effect, .

Figure 8 plots the $\pi^{0}$ meson mass $m_{\pi^{0}}$ and normalized $\pi^{0}$ meson mass $m_{\pi^{0}}(eB)/m_{\pi^{0}}(eB=0)$ as a function of magnetic field at vanishing temperature with different current quark mass $m_{0}=2,\ 5,\ 10,\ 20,\ 50$ MeV. Because the current quark mass determines the explicit breaking of chiral symmetry. As the pseudo-Goldstone boson, the mass of $\pi^{0}$ meson will increase with current quark mass when fixing magnetic field and vanishing temperature. On the other side, magnetic field plays the role of catalysis for spontaneous breaking of chiral symmetry when fixing current quark mass and vanishing temperature, and this will lead to a decreasing $m_{\pi^{0}}$. As shown in Fig.8, with fixed magnetic field, $m_{\pi^{0}}$ becomes larger with larger $m_{0}$. Moreover, the deviation between $m_{\pi^{0}}$ with different $m_{0}$ looks independent of the magnetic field. With fixed current quark mass, $m_{\pi^{0}}$ is a decreasing function of magnetic field. Similar as $m_{\pi^{0}}$, the normalized $\pi^{0}$ meson mass $m_{\pi^{0}}(eB)/m_{\pi^{0}}(eB=0)$ increases with the current quark mass when fixing magnetic field, and decreases with magnetic field when fixing $m_{0}$, which are consistent with LQCD results l4 .

In Fig.9, we depict the $\pi^{0}$ meson mass $m_{\pi^{0}}$ and normalized $\pi^{0}$ meson mass $m_{\pi^{0}}(T)/m_{\pi^{0}}(T=0)$ as a function of temperature with fixed magnetic field $eB=20m_{\pi}^{2}$ and different current quark mass $m_{0}=2,\ 5,\ 10,\ 20,\ 50$ MeV. They show similar structure. $m_{\pi^{0}}$ slightly increases with temperature, and a mass jump happens at the Mott transition $T=T^{0}_{m}$. After Mott transition, $m_{\pi^{0}}$ first decreases and then increases with temperature. However, there exist some quantitative difference. With larger current quark mass, the $m_{\pi^{0}}$ is larger in the whole temperature region, and the Mott transition temperature is lower. At high enough temperature, $m_{\pi^{0}}$ will become degenerate due to the strong thermal motion of constituent quarks. Different from meson mass $m_{\pi^{0}}$, the normalized $\pi^{0}$ meson mass $m_{\pi^{0}}(T)/m_{\pi^{0}}(T=0)$ is larger with smaller current quark mass, and the deviation between different $m_{0}$ is larger with higher temperature.

The Mott transition temperature of $\pi^{0}$ meson $T^{0}_{m}$ is plotted as a function of magnetic field in Fig.10 with different current quark mass $m_{0}=2,\ 5,\ 10,\ 20,\ 50$ MeV. For a lower value of current quark mass $m_{0}=2$ MeV, the Mott transition temperature decreases with weak magnetic field $(eB\leq 2m^{2}_{\pi})$, slightly increases in medium magnetic field region $(2m^{2}_{\pi}\leq eB\leq 15m^{2}_{\pi})$ and decreases again in strong magnetic field region $(eB\geq 15m^{2}_{\pi})$. For $m_{0}=5$ MeV, we obtain a flat curve for the Mott transition temperature in medium magnetic field region $(5m^{2}_{\pi}\leq eB\leq 16m^{2}_{\pi})$, and in weak and strong magnetic field regions, the Mott transition temperature decreases. With the larger value of current quark mass $m_{0}=10,\ 20,\ 50$ MeV, the Mott transition temperatures are monotonic decreasing functions of magnetic field. In weak magnetic field region, $T^{0}_{m}$ is higher for larger $m_{0}$, but in strong magnetic field region, $T^{0}_{m}$ is lower for larger $m_{0}$.

In the end of this section, we make some comment on chiral limit with $m_{0}=0$. As discussed in our previous paper mao1 , in chiral limit, the Goldstone boson $\pi^{0}$ is massless in chiral breaking phase and its mass continuously increases with temperature in chiral restored phase. The Mott transition temperature is the same as the critical temperature of chiral restoration phase transition, and it increases with magnetic field. Moreover, no mass jump occurs for $\pi^{0}$ meson at the Mott transition with or without external magnetic field.

Figure 11 shows $\pi^{+}$ meson mass $m_{\pi^{+}}$ and normalized mass $m_{\pi^{+}}(eB)/m_{\pi^{+}}(eB=0)$ as a function of magnetic field with vanishing temperature and different current quark mass $m_{0}=2,\ 5,\ 10,\ 20,\ 50$ MeV. $m_{\pi^{+}}$ is an increasing function of magnetic field when fixing $m_{0}$. With fixed magnetic field, larger value of $m_{0}$ leads to larger $m_{\pi^{+}}$. In weak magnetic field region, the $m_{0}$ effect is more obvious than in strong magnetic field region, which is indicated by the larger difference of $m_{\pi^{+}}$ between different $m_{0}$ cases. However, with fixed magnetic field, the normalized $\pi^{+}$ meson mass $m_{\pi^{+}}(eB)/m_{\pi^{+}}(eB=0)$ decreases as $m_{0}$ goes up, which is consistent with LQCD results l4 .

$\pi^{+}$ meson mass $m_{\pi^{+}}$ is plotted as a function of temperature with fixed magnetic field $eB=20m^{2}_{\pi}$ and different current quark mass $m_{0}=2,\ 5,\ 10,\ 20,\ 50$ MeV in Fig.12. $m_{\pi^{+}}$ has several mass jumps for all considered current quark mass $m_{0}$, which happen at temperature $T_{m}^{+}$, $T_{1}^{+}$, $T_{2}^{+}$, $T_{3}^{+}$ successively. With low temperature $T<T_{m}^{+}$, $m_{\pi^{+}}$ becomes larger with larger $m_{0}$. With $T_{m}^{+}<T<T_{1}^{+}$ and $T_{1}^{+}<T<T_{2}^{+}$, $m_{\pi^{+}}$ are also larger with larger $m_{0}$, but the value of $m_{\pi^{+}}$ is very close for different $m_{0}$. With $T>T_{2}^{+}$, $m_{\pi^{+}}$ changes nonmonotonically with $m_{0}$. It is clearly shown that the temperatures with mass jump depend on current quark mass. Within the considered temperature region $0<T<0.25$ GeV, we observe three jumps in case of current quark mass $m_{0}=2,\ 5,\ 20$ MeV, four jumps in case of $m_{0}=10$ MeV. For $m_{0}=50$ MeV, the first mass jump occurs at $T=0.262$ GeV, which is beyond the scope of Fig.12.

Figure 13 plots Mott transition temperature of $\pi^{+}$ meson $T^{+}_{m}$ as a function of magnetic field with different current quark mass $m_{0}=2,\ 5,\ 10,\ 20,\ 50$ MeV. $T^{+}_{m}$ shows similar behavior when varying current quark mass. Accompanied with some oscillations, $T^{+}_{m}$ increases fast in weak magnetic field region, decreases in medium magnetic field region and slightly increases in strong magnetic field region. With larger current quark mass, Mott transition temperature $T^{+}_{m}$ becomes higher. Note that in case of $m_{0}=50$ MeV, the increasing behavior of $T^{+}_{m}$ appears in very strong magnetic field, which is beyond the scope of Fig.13.