Universal Confining Strings: From Compact QED to the Hadron Spectrum

Abstract

We investigate the description of quark confinement in terms of confining strings or flux tubes. We show that compact QED with a topological $\theta$-term, in the dyon condensation phase, is described by a {\it massive} two-form field $B_{\mu \nu }$ that gives rise to a string theory with an IR Brazovskii-Lifshitz fixed point at strong coupling, reached along a double-scaling renormalization trajectory on which the cutoff can be removed. This corresponds to a UV-complete quantum string in (3+1) dimensions, representing the dual of asymptotic freedom in the UV. Contrary to critical strings, which correspond to trivial Gaussian fixed points, this string is stabilized by a finite thickness, determined by the mass of the $B_{\mu \nu }$ field, instead of living in a higher-dimensional space. Interestingly, it contains a world-sheet excitation, in addition to the Nambu-Goto phonons, whose nature depends on the sign of the string stiffness: for the negative stiffness favoured by lattice data it appears as a broad, overdamped mode rather than a sharp resonance, consistent with the broad phase shift seen in lattice $SU(N)$ scattering data. On the other hand, we determine the confining potential and show that it reproduces a generalized Arvis potential $V(L)=aL\sqrt{1-c/L^2}$ with running parameters $a(L),c(L)$. With this, we compute the mass difference ratios for the heaviest quarkonium and find 2.5\% agreement with experiment already at the infrared fixed point, from a two-parameter fit of both the reduced quark mass and a boundary-condition parameter, to this fixed functional form. We also compute the intercept of Regge trajectories and find that the thickness of Brazovskii-Lifshitz strings tends to increase it from the Nambu-Goto value ${\alpha }_0=1/12$. Overall, our findings strongly support Polyakov's longstanding conjecture on universality of confining gauge theories in the IR.

28 pages, 5 figures, clarifying remarks added, slightly expanded, added referenece