Confining strings and the worldsheet axion from the lattice

In this paper, we present a major update on the spectrum of closed flux tubes in [Formula: see text] [Formula: see text] gauge theories. We measure the excitation spectrum of confining strings wound around a spatial dimension of a size R. We do so for the [Formula: see text] Yang–Mills theory with [Formula: see text] and for two different values of the lattice spacing. We employ the generalized eigenvalue problem in combination with an extended basis of operators; this enables us to project onto all possible irreducible representations characterized by spin [Formula: see text], transverse parity [Formula: see text], longitudinal parity [Formula: see text] as well as by longitudinal momentum [Formula: see text] and extract accurate results for approximately 35 lightest states. By applying the Thermodynamic Bethe Ansatz (TBA) technique for calculating the finite volume spectrum, we confirm that the observed states are well described by the low-energy effective theory of a long string consisting of two translational Goldstone bosons (“phonons”), along with a massive pseudoscalar (“the worldsheet axion”) coupled to phonons through a [Formula: see text]-term. Moreover, we find that the leading axion–axion and axion–phonon interactions are well approximated by the [Formula: see text] deformation of a free axion.