Dynamical mass reduction in the massive Yang-Mills spectrum in $1+1$ dimensions

The $(1+1)$-dimensional $\text{SU}(N)$ Yang-Mills Lagrangian, with bare mass $\mathcal{M}$ and gauge coupling $e$, naively describes gluons of mass $\mathcal{M}$. In fact, renormalization forces $\mathcal{M}$ to infinity. The system is in a confined phase, instead of a Higgs phase. The spectrum of this diverging-bare-mass theory contains particles of finite mass. There are an infinite number of physical particles, which are confined hadronlike bound states of fundamental colored excitations. These particles transform under irreducible representations of the global subgroup of the explicitly broken gauge symmetry. The fundamental excitations are those of the $\text{SU}(N)\times \text{SU}(N)$ principal chiral sigma model, with coupling $g_0=e/\mathcal{M}$. We find the masses of mesonlike bound states of two elementary excitations. This is done using the exact S matrix of the sigma model. We point out that the color-singlet spectrum coincides with that of the weakly coupled anisotropic $\mathrm{SU}(N)$ gauge theory in $2+1$ dimensions. We also briefly comment on how the spectrum behaves in the ’t Hooft limit, $N\to \infty$.