无质量交换胶子模型中Maskawa-Nakajima方程的算符理论处理
An operator-theoretical treatment of the Maskawa-Nakajima equation in the massless abelian gluon model
摘要：
Maskawa-Nakajima方程在基本粒子物理学中引起了相当大的兴趣。本文从算符理论的观点出发，研究了无质量交换胶子模型中的Maskawa-Nakajima方程。首先证明了当参数$λ$满足$λ>2$时，Maskawa-Nakajima方程存在非零解.此外，我们还证明了该解是无穷可微的且是严格递减的。由此我们得出结论：无质量阿贝尔胶子模型在$λ>2$时自发产生非零夸克质量，并表现出自发的手征对称性破缺。接下来我们证明了当$0<λ<1$时，Maskawa-Nakajima方程存在唯一的解$0$，由此我们得出结论，每个夸克保持无质量，并且当$0<λ<1$时，模型实现了手征对称性。
英文摘要：
The Maskawa-Nakajima equation has attracted considerable interest in elementary particle physics. From the viewpoint of operator theory, we study the Maskawa-Nakajima equation in the massless abelian gluon model. We first show that there is a nonzero solution to the Maskawa-Nakajima equation when the parameter $λ$ satisfies $λ>2$. Moreover, we show that the solution is infinitely differentiable and strictly decreasing. We thus conclude that the massless abelian gluon model generates the nonzero quark mass spontaneously and exhibits the spontaneous chiral symmetry breaking when $λ>2$. We next show that there is a unique solution $0$ to the Maskawa-Nakajima equation when $0<λ<1$, from which we conclude that each quark remains massless and that the model realizes the chiral symmetry when $0<λ<1$.
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