QCD Worldsheet Axion from the Bootstrap

TL;DR

The work addresses how confining QCD flux tubes can be described by a worldsheet theory of branons augmented by a light axion. Using non-perturbative S-matrix bootstrap for 2→2 branon scattering, the authors connect EFT expectations, lattice results, and extremal bootstrap amplitudes, revealing an axion-dominated UV completion in 4D for large β_3 and a consistent axion–EFT matching in lower-energy regimes. Dispersion relations and sum rules show that the axion controls key observables beyond naive EFT validity, explaining the near-coincidence of lattice and critical axion couplings and hinting at a near-integrable UV structure for the QCD string. Across dimensions, the analysis clarifies how the axion’s presence shapes the analytic, unitary S-matrix near the flux-tube spectrum and provides a quantitative framework to test UV completions against lattice data.

Abstract

The worldsheet axion plays a crucial role in the dynamics of the Yang-Mills confining flux tubes. According to the lattice measurements, its mass is of order the string tension and its coupling is close to a certain critical value. Using the S-matrix Bootstrap, we construct non-perturbative $2 \to 2$ branon scattering amplitudes which also feature a weakly coupled axion resonance with these properties. We study the extremal bootstrap amplitudes in detail and show that the axion plays a dominant role in their UV completion in two distinct regimes, in one of which it cannot be considered a parametrically light particle. We conjecture that the actual flux tube amplitudes exhibit a similar behavior.

Figures (11)

• Figure 1: Left: in green, the region in the $\gamma_3-\gamma_5$ space allowed by the bootstrap constraints for $D=3$ flux tubes. Right: in green, the region in the $\beta_3-\alpha_3$ space allowed by the bootstrap constraints for $D=4$ flux tubes; in red, the region excluded analytically using the Schwarz-Pick theorem EliasMiro:2019kyf. With the bar we denote S-matrix coefficients rescaled as $\bar{\alpha}_n=2^{n+1} (4\pi)^{n-1}\alpha_n$, and the same for $\beta$ and $\gamma$. See below for the definition of these coefficients.
• Figure 2: S-matrix in the antisymmetric channel as a function of $s$ obtained by minimizing $\alpha_3$ at fixed $\bar{\beta}_3=22.74$. Left: result of the optimization using an Ansatz without rescaling. Right: result using an Ansatz with the rescaling $s_0=|\alpha_2|/\beta_3$. All dimensionful quantities appearing in the figures are expressed in units of $\ell_s=1$.
• Figure 3: Left: minimum of $\bar{\alpha}_3$ as a function of the dimension $D$ (greed dots). We denote the allowed region of $\bar{\alpha}_3$ in green. In red, the region excluded by the analytic Schwarz-Pick inequality EliasMiro:2019kyf applied to the symmetric channel S-matrix. Right: boundary of the allowed region in the $\bar{\beta}_3 -\bar{\alpha}_3$ plane for different dimensions $D$. In red, the analytically excluded region using the Schwarz-Pick inequality applied to the two crossing symmetric components $\sigma_2$ and $\sigma_1+\sigma_2+\sigma_3$ which gives $\bar{\alpha}_3 > -\pi^2/3 +|\bar{\beta}_3|$.
• Figure 4: Left: the black dots correspond to the data for $\bar{\alpha}_3-\bar{\beta}_3$ as a function of $\bar{\beta}_3$; the coloured curves to the fits obtained using the ansatz in \ref{['ansatzalpha3']} for different values of the truncation $n$; the coloured horizontal lines are the extrapolated asymptotic values of $\bar{\alpha}_3-\bar{\beta}_3$. The dashed black line is the analytic guess $-\pi^2/3$. Right: the blue dots are the extrapolated values of $\bar{\alpha}_3^{(0)}$ (top), and $\bar{\alpha}_3^{(1)}$ (bottom) with the corresponding error bars. The blue curve is the power law Ansatz used to further extrapolate in $n$. The blue bands represent the final extrapolated values with the error bar.
• Figure 5: Left: black dots are the values of the ratio $\bar{\alpha}_4/\bar{\beta}_3^2$. We extrapolate the data using eq. \ref{['eq:extrapolation_alpha4']} for different values of $n$. Curves with different colours are obtained with different $n$. The horizontal lines represent the asymptotic value extrapolated. Right: the extrapolated value of $a_4^{(0)}$ with the error bars. The blue curve is the power law fit of the data. The blue band is the final result. The black dashed line is the EFT prediction.