The QCD confining string and the world-sheet axion

Recent findings from an extensive investigation of the flux tube spectrum and its implications for the axion on the worldsheet of the confining string are presented. Specifically, the latest results for both closed and open flux tubes in 4D $SU(N_c)$ gauge groups are discussed. For the closed flux tube, employing the thermodynamic Bethe ansatz (TBA) method with certain approximations, it is confirmed that a significant portion of the low-lying energy states, characterized by different sets of quantum numbers, can be effectively described by the low-energy effective theory of a long string consisting of two Goldstone bosons, known as phonons, along with a massive pseudoscalar referred to as the worldsheet axion. Regarding the open flux tube, recent findings indicate that a substantial portion of the excited states can be adequately approximated by the Nambu-Goto (NG) spectrum. However, deviations from the NG spectrum are observed in certain states, suggesting the presence of massive excitations. Notably, the mass of one of these excitations aligns with that extracted for the closed flux tube.