限制、行走和共形规范理论的理想玩具模型：具有θ项的O（3）西格玛模型
An ideal toy model for confining, walking and conformal gauge theories: the O(3) sigma model with theta-term
摘要：
本文提出了一个四维非阿贝尔规范理论与大量费米子自由度耦合的玩具模型。当味数变化时，规范理论可以是限制的、行走的或共形的。模拟这一特征的玩具模型是具有θ项的二维O（3）西格玛模型。对于所有的theta，该模型是渐近自由的。对于小θ，模型被限制在红外线中，对于θ = π，模型具有非平凡的红外线固定点，因此对于θ略低于π，耦合行走。要研究玩具模型中规范理论的系统效应，这是出了名的困难，而要研究这个问题，第一步就是非微扰地确定θ参数实际上是一个相关的耦合。这是通过证明存在完全由总拓扑电荷给出的量，并且在连续极限中定义良好，并且是非零的，尽管拓扑磁化率是发散的。更确切地说，建立了拓扑电荷（累积量）的连接关联函数的差是有限的和非零的，因此在Z（θ）中只有一个发散参数，但除此之外它是有限的。这个发散常数可以通过一个适当的反项来消除，即使在θ> 0时，理论也完全有限。
英文摘要：
A toy model is proposed for four dimensional non-abelian gauge theories coupled to a large number of fermionic degrees of freedom. As the number of flavors is varied the gauge theory may be confining, walking or conformal. The toy model mimicking this feature is the two dimensional O(3) sigma model with a theta-term. For all theta the model is asymptotically free. For small theta the model is confining in the infra red, for theta = pi the model has a non-trivial infra red fixed point and consequently for theta slightly below pi the coupling walks. The first step in investigating the notoriously difficult systematic effects of the gauge theory in the toy model is to establish non-perturbatively that the theta parameter is actually a relevant coupling. This is done by showing that there exist quantities that are entirely given by the total topological charge and are well defined in the continuum limit and are non-zero, despite the fact that the topological susceptibility is divergent. More precisely it is established that the differences of connected correlation functions of the topological charge (the cumulants) are finite and non-zero and consequently there is only a single divergent parameter in Z(theta) but otherwise it is finite. This divergent constant can be removed by an appropriate counter term rendering the theory completely finite even at theta > 0.
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