Effective Strings in QED$_3$

Effective string theory describes the physics of long confining strings in theories, like Yang-Mills theory, where the mass gap $M_{gap}^2$ is of the same order as the string tension $T$. In $2+1$ dimensions, there is a class of confining theories, including massive QED$_3$ as first analyzed by Polyakov, for which $M_{gap}^2\ll T$. These theories are weakly coupled at low energies of order $M_{gap}$, and may be analyzed perturbatively. In this paper, we analyze the physics of strings in such theories, focusing on QED$_3$, at energies of order $M_{gap}$ (but still well below $\sqrt{T}$). We argue that the width of the string in these theories should be of order $1/M_{gap}$ independently of its length, as long as the string is not exponentially long. We also compute at leading order in perturbation theory the ground state energy of a confining string on a circle, and the scattering of Nambu-Goldstone bosons on the string worldsheet.