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The Maskawa-Nakajima equation has attracted considerable interest inelementary particle physics. From the viewpoint of operator theory, we studythe Maskawa-Nakajima equation in the massless abelian gluon model. We firstshow that there is a nonzero solution to the Maskawa-Nakajima equation when theparameter $\lambda$ satisfies $\lambda>2$. Moreover, we show that the solutionis infinitely differentiable and strictly decreasing. We thus conclude that themassless abelian gluon model generates the nonzero quark mass spontaneously andexhibits the spontaneous chiral symmetry breaking when $\lambda>2$. We nextshow 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 thatthe model realizes the chiral symmetry when $0<\lambda<1$.

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