二维非线性O（N）西格玛模型的Dyson不稳定性
Dyson instability for 2D nonlinear O(N) sigma models
摘要：
对于具有紧致场积分的格点模型（紧致流形上的非线性西格玛模型和具有紧致群的规范理论）并满足某种离散对称性，裸耦合g_0 ^2在零点处符号的改变只会导致平均能量的不连续性，而不是在任意大的场上积分的理论中出现的灾难性不稳定性。这表明，高阶微扰级数和非微扰贡献应该具有意想不到的特征。利用二维非线性O（N）西格玛模型的大N极限，讨论了复t Hooft耦合λ = g_0^2N的平均能量的复奇异性.与通常情况的一个显著区别是没有沿负实轴的切割。我们证明了配分函数的零点只能在复λ平面的三叶草形区域内。我们计算了态密度，并用数值方法验证了关于零点的说法。我们提出了色散表示的平均能量的导数的近似表达式的不连续性。这种不连续性是完全非微扰的，在一个色散表象中，小负耦合的贡献对于保证当λ-> 0^+时导数指数地小是必不可少的。我们讨论了它对规范理论的意义。
英文摘要：
For lattice models with compact field integration (nonlinear sigma models over compact manifolds and gauge theories with compact groups) and satisfying some discrete symmetry, the change of sign of the bare coupling g_0^2 at zero results in a mere discontinuity in the average energy rather than the catastrophic instability occurring in theories with integration over arbitrarily large fields. This indicates that the large order of perturbative series and the non-perturbative contributions should have unexpected features. Using the large-N limit of 2-dimensional nonlinear O(N) sigma model, we discuss the complex singularities of the average energy for complex 't Hooft coupling lambda= g_0^2N. A striking difference with the usual situation is the absence of cut along the negative real axis. We show that the zeros of the partition function can only be inside a clover shape region of the complex lambda plane. We calculate the density of states and use the result to verify numerically the statement about the zeros. We propose dispersive representations of the derivatives of the average energy for an approximate expression of the discontinuity. The discontinuity is purely non-perturbative and contributions at small negative coupling in one dispersive representation are essential to guarantee that the derivatives become exponentially small when lambda -> 0^+ We discuss the implications for gauge theories.
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