Revisiting Yukawa's Bilocal Field Theory for Composite Scalar Bosons

Yukawa's old bilocal field theory, with modernization in the treatment of "relative time," can describe a relativistic bound state of chiral fermions. This connects to bosonized effective chiral Lagrangians and the Nambu-Jona-Lasinio (NJL) model, providing a description of the internal dynamics of the bound states. It features a static internal wave-function, $\phi(\vec{r})$, in the center-of-mass frame that satisfies a Schr\"odinger-Klein-Gordon equation with eigenvalues $m^2$. We analyze the "coloron" model (single perturbative massive gluon exchange) which yields a UV completion of the NJL model. This has a BCS-like enhancement of its interaction, $\propto N_c$, the number of colors. It is classically critical, with $g_{critical}$ remarkably close to the NJL quantum critical coupling. Negative eigenvalues for $m^2$ lead to spontaneous symmetry breaking, and the Yukawa coupling of the bound state to constituent fermions is emergent.