Towards a Theory of the QCD String

TL;DR

This work investigates whether confining flux tubes in four-dimensional gauge theories can be described by integrable worldsheet dynamics. It shows that, absent extra gapless modes, universal low-energy amplitudes violate the softness required for integrability; the phase of the worldsheet S-matrix is uniquely fixed by symmetry in certain dimensions, constraining viable integrable strings. The authors construct two explicit integrable extensions of the minimal string in 4D—a linear-dilaton (scalar) and a worldsheet axion—and argue for a broader family of models with additional worldsheet fields that preserve integrability. They confront these ideas with lattice Yang–Mills data, where a worldsheet axion with coupling $Q_a$ is observed and intriguingly agrees with the integrable value $Q=rac{7}{16\, extpi}^{1/2}\approx 0.373$, suggesting a potential integrable axionic QCD string in the planar limit. The results motivate further lattice and theoretical work to determine whether planar QCD strings are indeed governed by an integrable worldsheet theory and to map out the broader family of such models.

Abstract

We construct a new model of four-dimensional relativistic strings with integrable dynamics on the worldsheet. In addition to translational modes this model contains a single massless pseudoscalar worldsheet field - the worldsheet axion. The axion couples to a topological density which counts the self-intersection number of a string. The corresponding coupling is fixed by integrability to $Q=\sqrt{7\over 16π}\approx 0.37$. We argue that this model is a member of a larger family of relativistic non-critical integrable string models. This family includes and extends conventional non-critical strings described by the linear dilaton CFT. Intriguingly, recent lattice data in $SU(3)$ and $SU(5)$ gluodynamics reveals the presence of a massive pseudoscalar axion on the worldsheet of confining flux tubes. The value of the corresponding coupling, as determined from the lattice data, is equal to $Q_L\approx0.38\pm0.04$.

Figures (7)

• Figure 1: At large number of colors $N_c$ the worldsheet theory remains unitary up to a scale $\Lambda_{N_c}$, which is parametrically heavier than the mass of the lightest glueball.
• Figure 2: Two type of contributions giving rise to collinear singularities in the shift current Ward identities.
• Figure 3: The on-shell diagram giving rise to the Coleman--Thun contribution into the Ward identities.
• Figure 4: A bilinear contribution into the Ward identities.
• Figure 5: This plot shows $\Delta E=E-R/\ell_s^2$ as a function of length of the $SU(3)$ flux tube for the two particle states in red, blue, green and dark green for $0^{--}$, $0^{++}$, $2^{+-}$ and $2^{++}$ states respectively. The data is taken from Athenodorou:2010cs. The solid lines show the theoretical predictions derived from phase shift (\ref{['delta_res']}) with the best fit value corresponding to $2^{++}$ states (left) and $2^{+-}$ states (right). The dashed lines correspond to the fit with higher derivative corrections (\ref{['delta2']}). Shorter dashing indicates Goldstone momenta above $1.85\ell_s$ where the one loop contribution into the phase shift becomes equal to the tree level one. Dotted lines show theoretical predictions without the resonance.