An operator-theoretical treatment of the Maskawa-Nakajima equation in   the massless abelian gluon model
  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 $0<\lambda<1$. 
Shuji Watanabe
math.FA hep-ph hep-th math-ph math.MP