标题： 重新审视非物理π介子质量和高温下的$O(N)$ $σ$ 模型 显示英文标题

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

摘要： 对 $\pi\pi$散射相移的格点数据进行 Roy 方程分析，在 $m_\pi=391$兆电子伏能量下发现最低的 $f_0$粒子在这种条件下成为束缚态。此外，阈值以下由交叉对称性生成的一对复数极点 [X.-H. 曹等，Phys. Rev. D 108, 034009 (2023)]。我们使用 $N/D$方法在 $1/N$展开的领先阶次上部分恢复了 $O(N)$ $\sigma$ 模型振幅的交叉对称性，并定性地重现了 Roy 方程分析揭示的极点结构和随着 pion 质量变化的极点轨迹。 讨论了随温度变化的$\sigma$杆轨迹，发现其性质与随$m_\pi$变化时的情况类似。 随着温度升高，复数$\sigma$极点首先从第二黎曼面移动到实轴，变为两个虚态极点，然后其中一个虚态极点移动到第一面变成束缚态极点，并最终在高温下趋于π介子极点位置，这是从手征对称性恢复预期的结果。 我们的结果进一步证明了从实验和格点数据提取的最低$f_0$态在手征对称自发破缺中扮演$\sigma$介子的角色。 最后，我们还简要讨论了当$m_\pi$和温度变得较大时有效势能的问题。 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 [X.-H. Cao et al., Phys. Rev. D 108, 034009 (2023)]. 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.