Continuum limit of a qubit-regularized SU(3) lattice gauge theory with glueballs

We show that a simple qubit-regularized $\mathrm{SU}(3)$ lattice gauge theory (LGT) on a plaquette chain admits a continuum limit with massive glueball excitations, providing a minimal toy model of strong interactions without quarks. By mapping the plaquette-chain Hamiltonian to the three-state quantum clock model in a magnetic field, we demonstrate that the theory can be tuned to a continuum limit governed at short distances by the $\mathbb{Z}_3$ parafermion conformal field theory (CFT), which serves as the ultraviolet (UV) fixed point. A small relevant magnetic perturbation then drives the system to a massive continuum quantum field theory in the infrared (IR). The resulting relativistic massive particles can be interpreted as quasi one-dimensional analogues of glueballs. In the continuum theory we compute the ratio of the lowest glueball masses with opposite charge conjugation to be $m^{-}/m^{+} = \,1.459(2)$ and find $\sqrt{\sigma}/m^{+}\,= 0.2648(2)$, where $\sigma$ is the string tension between a static quark and antiquark.