Multiplicity of superconducting solutions and the cutoff-free strong-coupling regime in the Maskawa-Nakajima vector-gluon model

In 1974-75 Maskawa and Nakajima analyzed the self-consistent (Schwinger-Dyson) equation for the fermion propagator in a vector-gluon model - fermions interacting through a massive neutral vector gluon - within the lowest-order (ladder) approximation, initiating the Schwinger-Dyson approach to dynamical chiral symmetry breaking. They proved uniqueness and absence of chiral symmetry breaking at weak coupling (with cutoff, $g^2/4\pi<\pi/4$), inferred without proof the existence of many superconducting ($\beta\neq 0$) solutions at strong coupling, and showed that for $g^2/4\pi>8\pi$ without cutoff the normal-state solution would necessarily carry an unphysical singularity, implying instability of the chirally symmetric vacuum. The problem considered here is to establish rigorously the multiplicity of superconducting solutions of the Maskawa-Nakajima equation as a function of coupling, bare mass, gluon mass, and cutoff, including the physical selection among the cutoff-free continuum of solutions, and to determine the fate and the precise instability mechanism of the normal state in the cutoff-free regime $g^2/4\pi>8\pi$.

Background

By 1974 it was known from the Nambu-Jona-Lasinio (NJL) model - a contact four-fermion interaction regularized by a hard ultraviolet cutoff - that a fermion can acquire a mass dynamically: the mass function obeys a nonlinear self-consistency (gap) equation whose nonzero solutions appear only above a critical coupling, in close analogy with the Cooper instability of superconductivity. This phenomenon is dynamical chiral symmetry breaking (DCSB): a theory of massless fermions with an exact global chiral symmetry (independent rotations of left- and right-handed fermion components, exact when the fermion is massless) develops a nonzero fermion mass function, accompanied by a massless Nambu-Goldstone boson (the collective ungapped mode, playing the role of the pion in quantum chromodynamics (QCD)). In this literature a superconducting solution means a nonzero solution of the gap equation; the normal state means the identically vanishing mass function. Maskawa and Nakajima asked whether the same mechanism could be analyzed in a renormalizable gauge theory, a vector-gluon model: fermions interacting through exchange of a massive neutral vector boson (the gluon, of mass $\mu\neq 0$). The exact instrument is the Schwinger-Dyson (SD) equation, the self-consistency equation for the fermion two-point Green function: writing the inverse fermion propagator as a vector (wave-function) part $\alpha$ multiplying $\gamma\cdot p$ plus a scalar (mass) part $\beta$, the SD equation couples $\alpha$ and $\beta$ through a nonlinear integral equation. At the lowest order - a bare vector vertex and a one-gluon-exchange fermion-antifermion scattering kernel (the ladder or rainbow approximation), in a Landau-like gauge (the transverse gauge in which the vector-boson propagator has no longitudinal part), with or without an ultraviolet cutoff $\Lambda$ - their 1974 paper reduced this to the nonlinear integral equation now called the Maskawa-Nakajima (MN) equation, together with a simplified model in which the vector part is set to unity ($\alpha\equiv 1$). Their results define the present problem. With a cutoff, the equation has a unique solution for $g^2/4\pi<\pi/4$, and at zero bare mass $m_0=0$ that solution has $\beta\equiv 0$: no DCSB and no Nambu-Goldstone boson, at any value of the cutoff. Without cutoff, solutions exist only at $m_0=0$, and for $g^2/4\pi<(16/33)^2\pi$ there is a continuum of solutions - a degeneracy not attributable to freedom in fixing the mass scale, since the gluon mass $\mu$ is nonzero. At strong coupling, superconducting solutions appear (established in the $\alpha\equiv 1$ model), and the existence of many superconducting solutions was inferred from eigenvalue thresholds of the kernel, not proven. Finally, for $g^2/4\pi>8\pi$ without cutoff, the normal-state solution, even if it existed, would necessarily have an unphysical singularity, implying that the normal state is unstable at sufficiently strong coupling. The 1975 sequel organized the weak-coupling uniqueness statements as theorems but left the strong-coupling multiplicity as an inference; all conclusions were obtained within the lowest-order approximation, and the validity of a bare vertex and a one-gluon-exchange kernel at the very strong couplings in question is itself uncertain, because exact gauge invariance ties the vertex to the propagator through the Ward-Takahashi identity (the identity relating the fermion-vector-boson vertex to the derivative of the fermion propagator), which the ladder truncation violates. A later operator-theoretic analysis of the massless version of the equation with cutoffs (Watanabe 2014, ref. 13) establishes the existence of a nonzero solution for its dimensionless coupling $\lambda>2$ and uniqueness of the zero solution for $0<\lambda<1$, without treating the massive model or the cutoff-free strong-coupling regime.

Problem Statement

For the Maskawa-Nakajima vector-gluon model - fermions interacting through exchange of a massive neutral vector gluon of mass $\mu\neq 0$, treated through the self-consistent (Schwinger-Dyson) equation for the fermion propagator - settle the strong-coupling solution structure that the 1974-75 analysis left as inference within the lowest-order (ladder) approximation. The two components jointly answer one question: does spontaneous chiral symmetry breaking genuinely occur in this model at strong coupling, and with what solution structure? Resolving the problem requires the multiplicity characterization for both the $\alpha\equiv 1$ reduced equation and the full coupled $(\alpha,\beta)$ system, the criterion selecting the physically realized cutoff-free solution, and the existence, singularity, and instability mechanism of the normal state.

(1) Multiplicity of superconducting solutions. Prove or disprove, rigorously for the $\alpha\equiv 1$ reduced equation and for the full coupled $(\alpha,\beta)$ system, that at strong coupling (above the critical region of order $g^2/4\pi\approx\pi/4$, with ultraviolet cutoff $\Lambda$) there exist many distinct superconducting ($\beta\neq 0$) solutions, the exact count being part of what must be determined. Characterize the multiplicity as a function of $(g, m_0, \mu, \Lambda)$ - how it grows with $g$, whether it is finite or becomes a continuum - and how branches appear and disappear as $g$ is varied. Clarify the relation between the continuum of cutoff-free solutions (which exists for $g^2/4\pi<(16/33)^2\pi$ at $m_0=0$) and the cutoff-regulated solutions: by what limiting procedure or additional physical criterion is the physically realized solution selected, and what fixes the fermion mass scale once the gluon mass is fixed?

(2) The cutoff-free super-strong-coupling regime. Determine the behavior of the equation without cutoff for $g^2/4\pi>8\pi$: whether the normal-state ($\beta\equiv 0$) solution exists, the precise nature of the unphysical singularity it was shown to carry, and the actual mechanism of the implied instability.

References

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