Undressing Confining Flux Tubes with $T\bar T$

TL;DR

The paper presents a systematic framework to extract confining flux-tube worldsheet S-matrices from lattice spectra by leveraging a gravitational/$T\bar{T}$ dressing equivalence. It provides a practical recipe: compute undressed multi-particle spectra from the worldsheet using ABA/Lüscher, then dress the energies through a hydrodynamic equation to include leading polarization effects for arbitrary inelasticity and particle number. Applications to $D=4$ and $D=3$ Yang–Mills show that the method reproduces lattice data for the two-particle sector and reveals inelastic effects in multi-particle states, including the impact of the worldsheet axion and higher-derivative operators. The work also demonstrates how three-particle states can probe inelasticity and outlines a program to obtain leading multiparticle amplitudes from lattice data, setting the stage for broader connections to soft gluon dynamics in gauge theories.

Abstract

Lattice QCD simulations provide crucial information about the worldsheet dynamics of confining strings (flux tubes). An accurate extraction of the worldsheet $S$-matrix from lattice spectra requires accounting for polarization effects. Approximate integrability of the low energy worldsheet theory makes it possible to apply the Thermodynamic Bethe Ansatz to incorporate polarization effects at all orders in the number of windings and at the leading order in the derivative expansion. However, a systematic application of this technique in the presence of non-integrable effects and for multiparticle states becomes increasingly challenging. We point out that a recently understood equivalence between gravitational dressing and $T\bar{T}$ deformation provides a fully systematic and straightforward recipe to incorporate the leading polarization effects in the presence of an arbitrary inelasticity and for any number of particles. We illustrate this technique with several examples.

Figures (5)

• Figure 1: The energy gap between the lowest two particle excitations and the ground state on the worldsheet. Blue color refers to scalar, red to pseudoscalar and green to spin 2 excitations w.r.t. to the transverse $O(2)$ rotation group. The left panel shows one loop predictions of the minimal Nambu--Goto theory (in this case pseudoscalar and scalar levels are predicted to be degenerate), and the right panel includes the effect of the worldsheet axion. Dashed lines on both panels show the tree level Nambu--Goto prediction (all states are degenerate in this approximation). Lattice data is from Athenodorou:2010cs.
• Figure 2: Lattice data and fits for $\ell_s\Delta E \equiv \ell_s E - R/\ell_s$ as a function of $R/l_s$ for the $N=1,2$ states of the flux-tube. Dashed black lines show the GGRT spectrum. The left panel shows the first and second excited states with two particles (purple and blue markers) and the first excited state with four particles (green markers). The solid colored lines stand for the corresponding theoretical curves. The right panel shows the first two three particle excited states. The orange dashed line results from the dressed ABA calculation. The green and blue lines correspond to including perturbatively the effects of the inelasticity.
• Figure 3: Types of diagrams contributing to 2 to 4 particle amplitude at order $l_{s}^{8}$
• Figure 4: Input data used for determination of the contact three particle scattering amplitude $\mathcal{M}_{3\rightarrow 3}$. a) Curves obtained from the best fit values to the coefficients in (\ref{['fitEform']}) (red) and $1\sigma$ variations in best fit parameters (orange). b) Dressed phase shift as a function of momentum. The dots represent lattice data points for first and second excited two particle states (purple and blue dots respectively) and first four particle state (green dots). The black curve corresponds to the GGRT phase-shift and the blue curve to the GGRT phase shift with the $\ell_s^6\gamma p^6$ correction. The red curve corresponds to the best fit parameters for the parametrization given by \ref{['fitpsform']}.
• Figure 5: Particle production probability as a function of collision energy. The blue line gives the result from using the undresed lagrangian \ref{['3Dundlag']}. The green line is the result extracted from the data using the theoretical $p^6$ phase-shift but fitting the data for the energies. The red curve is obtained by fitting both the phase shift and energies from the data. The orange curves account for fitting uncertainties around this last curve.