标题： 重新审视在非物理介子质量及高温下的$O(N)$ $σ$ 模型 显示英文标题

标题： Revisiting $O(N)$ $σ$ model at unphysical pion masses and high temperatures

摘要： Roy方程分析了在$\pi\pi$能量下$m_\pi=391$MeV 的格点数据散射相移，结果表明在该条件下最低的$f_0$介子成为一个束缚态。此外，通过交叉对称性产生了一对低于阈值的复极点（PRD 108, 034009）。我们使用$N/D$方法在$O(N)$ $\sigma$ 模型振幅的$1/N$展开的领先阶次下部分恢复交叉对称性，并定性地再现了由Roy方程分析揭示的极点结构和随着π介子质量变化的极点轨迹。 $\sigma$极点轨迹随着温度的变化也被讨论，并发现其性质与随着$m_\pi$变化时类似。 随着温度的增加，复数$\sigma$极点首先从第二黎曼面移动到实轴，成为两个虚态极点，然后一个虚态极点移动到第一面，转变为束缚态极点，最后在高温下趋于介子极点位置，这与手征对称性恢复的预期一致。 我们的结果提供了进一步的证据，表明从实验和格点数据中提取的最低$f_0$状态在手征对称性自发破缺中起到了$\sigma$介子的作用。 最后，我们还简要讨论了当$m_\pi$和温度变大时有效势的问题。 显示英文摘要

摘要： Roy-equation analyses on lattice data of $\pi\pi$ scattering phase shifts at $m_\pi=391$MeV reveals that the lowest $f_0$ meson becomes a bound state under this condition. In addition, there is a pair of complex poles below threshold generated by crossing symmetry (PRD 108, 034009). We use the $N/D$ method to partially recover crossing symmetry of the $O(N)$ $\sigma$ model amplitude at leading order of $1/N$ expansion, and qualitatively reproduce the pole structure and pole trajectories with varying pion masses as revealed by Roy-equation analyses. The $\sigma$ pole trajectory with varying temperature is also discussed and found to be similar to its properties when varying $m_\pi$. As the temperature increases, the complex $\sigma$ poles firstly move from the second Riemann sheet to the real axis becoming two virtual state poles, and then one virtual state pole moves to the first sheet turning into a bound state pole and finally tends to the pion pole position at high temperature which is as expected from the chiral symmetry restoration. Our results provide further evidences that the lowest $f_0$ state extracted from experiments and lattice data plays the role of $\sigma$ meson in the spontaneous breaking of chiral symmetry. Finally, we also briefly discuss the problems of the effective potential in the situation when $m_\pi$ and temperature get large.