Confining Strings and Glueballs in ${\mathbb{Z}}_N$ Gauge Theories

Effective string theory has shown its universal power in the prediction of the spectrum of low-lying excited states of confining strings. Here we study confining flux tubes in ${\mathbb{Z}}_N$ gauge theories. For the $N=2$ theory, which corresponds to the 3d Ising gauge model, we compute the spectrum of low-lying excitations of confining strings and show that it agrees with the universal Nambu–Goto predictions except for an additional massive scalar resonance. This resonance, however, turns out to be a bulk glueball mixing with the flux tube excitations rather than a genuine string worldsheet state. In general ${\mathbb{Z}}_N$ gauge theories (dual to clock spin models), we observe a continuous phase transition for $N\ge 4$, while for $N>5$ it is governed by the $O(2)$ universality class. The critical behavior of the string tension and mass gap is verified to be described by a dangerously irrelevant operator. At large $N$ the glueball spectrum is expected to approach the spectrum of U(1) gauge theory, which is confirmed by our lattice data.