From effective Lagrangians, to chiral bags, to Skyrmions with the large-Nc renormalization group.

Abstract

We explicitly relate effective meson-baryon Lagrangian models, chiral bags, and Skyrmions in the following way. First, effective Lagrangians are constructed in a manner consistent with an underlying large-${\mathit{N}}_{\mathit{c}}$ QCD. An infinite set of graphs dress the bare Yukawa couplings at leading order in 1/${\mathit{N}}_{\mathit{c}}$, and are summed using semiclassical techniques. What emerges is a picture of the large-${\mathit{N}}_{\mathit{c}}$ baryon reminiscent of the chiral bag: hedgehog pions for r\ensuremath{\ge}${\mathrm{\ensuremath{\Lambda}}}^{\mathrm{\ensuremath{-}}1}$ patched onto bare nucleon degrees of freedom for r\ensuremath{\le}${\mathrm{\ensuremath{\Lambda}}}^{\mathrm{\ensuremath{-}}1}$, where the ``bag radius'' ${\mathrm{\ensuremath{\Lambda}}}^{\mathrm{\ensuremath{-}}1}$ is the UV cutoff on the graphs. Next, a novel renormalization group (RG) is derived in which the bare Yukawa couplings, baryon masses, and hyperfine baryon mass splittings run with \ensuremath{\Lambda}. Finally, this RG flow is shown to act as a filter on the renormalized Lagrangian parameters: when they are fine-tuned to obey Skyrme-model relations the continuum limit \ensuremath{\Lambda}\ensuremath{\rightarrow}\ensuremath{\infty} exists and is, in fact, a Skyrme model; otherwise there is no continuum limit.