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From chiral random matrix theory to chiral perturbation theory

机译：从手性随机矩阵理论到手性摄动理论

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摘要

We study the spectrum of the QCD Dirac operator by means of the valence quark mass dependence of the chiral condensate in partially quenched Chiral Perturbation Theory (pqChPT) in the supersymmetric formulation of Bernard and Golterman. We consider valence quark masses both in the ergodic domain (m_c E_c) and the diffusive domain (m_e E_c). These domains are separated by a mass scale E_c approx F~2/#SIGMA#_0L~2 (with F the pion decay constant, #SIGMA#~0 the chiral condensate and L the size of the box). In the ergodic domain the effective super-Lagrangian reproduces the microscopic spectral density of chiral Random Matrix Theory (chRMT). We obtain a natural explanation of Damgaard's relation between the spectral density and the finite volume partition function with two additional flavors. We argue that in the ergodic domain the natural measure for the superunitary integration in the pqChPT partition function is non-compact. We find that the tail of the two-point spectral correlation function derived from pqChPT agrees with the chRMT result in the ergodic domain. In the diffusive domain we extend the results for the slope of the Dirac spectrum first obtained by Smilga and Stern. We find that the spectral density diverges logarithmically for non-zero topological susceptibility. We study the transition between the ergodic and the diffusive domains and identify a range where chRMT and pqChPT coincide.
机译：我们在伯纳德和古尔特曼的超对称公式中，通过部分淬灭手性扰动理论（pqChPT）中的手性缩合物的价夸克质量依赖性，研究了QCD Dirac算子的谱。我们考虑遍历域（m_c E_c）和扩散域（m_e E_c）的价夸克质量。这些域之间的距离为E_c约F〜2 /＃SIGMA＃_0L〜2（其中P为介子衰减常数，＃SIGMA＃〜0为手性冷凝物，L为盒子的大小）。在遍历域中，有效的超拉格朗日模型再现了手性随机矩阵理论（chRMT）的微观光谱密度。我们对两种密度的丹加德光谱密度与有限体积分配函数之间的关系作了自然的解释。我们认为，在遍历域中，pqChPT分区函数中超单元积分的自然度量是非紧凑的。我们发现，从pqChPT导出的两点频谱相关函数的尾部与遍历域中的chRMT结果一致。在扩散域中，我们扩展了Smilga和Stern首先获得的狄拉克谱的斜率的结果。我们发现，对于非零拓扑敏感性，频谱密度在对数上发散。我们研究遍历遍历域和扩散域之间的过渡，并确定chRMT和pqChPT重合的范围。

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