Chiral Lagrangians from lattice gauge theories in the strong coupling limit

Abstract

We derive nonlinear $\ensuremath{\sigma}$ models (chiral Lagrangians) over symmetric spaces $\mathrm{U}(n),$ $\mathrm{U}(2n)/\mathrm{Sp}(2n),$ and $\mathrm{U}(2n)/\mathrm{O}(2n)$ from $\mathrm{U}(N),$ $\mathrm{O}(N),$ and $\mathrm{Sp}(2N)$ lattice gauge theories coupled to n flavors of staggered fermions, in the large-$N$ and ${g}^{2}N$ limit. To this end, we employ Zirnbauer's color-flavor transformation. We prove the spatial homogeneity of the vacuum configurations of mesons by explicitly solving the large-$N$ saddle point equations, and thus establish these patterns of spontaneous chiral symmetry breaking in the above limit.