Flux Tube Spectra from Approximate Integrability at Low Energies

TL;DR

The paper introduces a method to compute flux-tube spectra by first perturbatively obtaining the worldsheet S-matrix and then using excited-state Thermodynamic Bethe Ansatz to extract finite-volume energies, achieving superior convergence over traditional perturbation theory. Applying this to lattice data in D=4 and D=3 reveals that the ground-state spectrum is largely GGRT-like with universal corrections, while excited states expose a worldsheet axion resonance in D=4 and resonance structures in k-strings in D=3. The approach yields quantitative phase shifts and resonance parameters that reconcile many lattice observations without free-parameter tuning beyond the string tension, and it motivates further high-precision simulations to map out worldsheet dynamics and confinement physics. Overall, the work provides a robust, nonperturbative framework linking effective string theory, integrability, and lattice QCD flux-tube spectra, with implications for open strings and possible meson spectra.

Abstract

We provide a detailed introduction to a method we recently proposed for calculating the spectrum of excitations of effective strings such as QCD flux tubes. The method relies on the approximate integrability of the low energy effective theory describing the flux tube excitations and is is based on the Thermodynamic Bethe Ansatz (TBA). The approximate integrability is a consequence of the Lorentz symmetry of QCD. For excited states the convergence of the TBA technique is significantly better than that of the traditional perturbative approach. We apply the new technique to the lattice spectra for fundamental flux tubes in gluodynamics in D=3+1 and D=2+1, and to k-strings in gluodynamics in D=2+1. We identify a massive pseudoscalar resonance on the world sheet of the confining strings in SU(3) gluodynamics in D=3+1, and massive scalar resonances on the world sheet of k=2,3 strings in SU(6) gluodynamics in D=2+1.

Figures (19)

• Figure 1: $\Delta E=E-R/\ell_s^2$ for the ground state of the flux tube. The value of $\ell_s$ was determined from the lattice data. The dotted line shows the prediction of a derivative expansion. The dashed line shows the prediction of the GGRT theory.
• Figure 2: $\Delta E=E-R/\ell_s^2$ for excited states of the flux tube with one and two units of KK momentum in orange and red, respectively . The dotted lines show the prediction of a derivative expansion. The dashed lines show the prediction of the GGRT theory.
• Figure 3: $\Delta E=E-R/\ell_s^2$ for an excited state with one left- and one right-mover, each with one unit of KK-momentum. The dotted lines show the prediction of a derivative expansion. The dashed lines show the prediction of the GGRT theory. The green color represents a state that is a symmetric tensor with respect to SO(2), the blue color represents the states the scalar with respect to SO(2) and the red data points represent the anti-symmetric tensor with respect to SO(2). All states are predicted to be degenerate in the GGRT theory. In the derivative expansion, the scalar and anti-symmetric tensor are still predicted to be degenerate as indicated by the blue dotted line. The degeneracy with the symmetric state is lifted, which is predicted to have higher energies as shown by the green dotted line.
• Figure 4: Propagator for a virtual quantum in the presence of the real left-moving particles indicated in the figure by crosses.
• Figure 5: $\Delta E=E-R/\ell_s^2$ for excited states of the GGRT theory with one and two units of KK momentum in orange and red, respectively. The dotted lines show the prediction of a derivative expansion, the longer dashes show the prediction of the GGRT theory, and the shorter darker dashes represent our diagrammatic approximation. The diagrammatic approximation and the exact result are virtually indistinguishable.