An operator-theoretical treatment of the Maskawa-Nakajima equation in the massless abelian gluon model

Abstract

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 $\lambda$ satisfies $\lambda >2$. 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 $\lambda >2$. We next show that there is a unique solution $0$ to the Maskawa-Nakajima equation when $0<\lambda <1$, from which we conclude that each quark remains massless and that the model realizes the chiral symmetry when $0<\lambda <1$.

10 pages

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• The BCS Functional for General Pair Interactions
• The solution to the BCS gap equation and the second-order phase transition in superconductivity
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• An operator-theoretical study of the specific heat and the critical magnetic field in the BCS-Bogoliubov model of superconductivity
• Another operator-theoretical proof for the second-order phase transition in the BCS-Bogoliubov model of superconductivity
• An operator-theoretical study on the BCS-Bogoliubov model of superconductivity near absolute zero temperature
• Lipschitz continuity and monotone decreasingness of the solution to the BCS gap equation for superconductivity