Dual EFT Bootstrap: QCD flux tubes

TL;DR

The paper develops a dual S-matrix bootstrap framework to bound Wilson coefficients in the effective field theory of confining flux tubes, going beyond traditional positivity by enforcing full two-to-two unitarity, analyticity, and crossing. By formulating primal and dual optimisation problems for the worldsheet S-matrix, it derives rigorous bounds on non-universal low-energy constants such as $oldsymbol{ extgamma_3}$ in D=3 and on $(oldsymbol{ extalpha_3},oldsymbol{ extbeta_3})$ in higher dimensions, with analytic results in simple limits and tight numerical bounds elsewhere. Saturating amplitudes are constructed that saturate unitarity and crossing, yielding interpretable S-matrices and phase shifts that reveal resonant structures (dilaton-like or axion-like) tied to the boundary of the allowed region. The approach provides a principled, scalable route to constrain EFTs in confining strings, aligns with lattice data and integrable limits, and opens paths to higher-dimensional generalisations, multi-particle processes, and potential connections to ASA and supersymmetric world-sheets.

Abstract

We develop a bootstrap approach to Effective Field Theories (EFTs) based on the concept of duality in optimisation theory. As a first application, we consider the fascinating set of EFTs for confining flux tubes. The outcome of our analysis are optimal bounds on the scattering amplitude of Goldstone excitations of the flux tube, which in turn translate into bounds on the Wilson coefficients of the EFT action. Finally, we comment on how our approach compares to EFT positivity bounds.

Figures (3)

• Figure 1: Contour of integration used to relate (\ref{['arcs']}) with (\ref{['lec1']})-(\ref{['lec2']}).
• Figure 2: Primal and dual bounds on the Wilson coefficients $\{\alpha_3, \beta_3\}$. The green region is allowed by primal numerics, the red region is excluded by the dual problem. The red lines are obtained solving the dual problem at fixed $\beta_3$ maximizing the dual functional for $N_*=5,10,\dots,30$; the dashed red lines are the analytic bounds obtained in EliasMiro:2019kyf. The green lines denote the boundary at some fixed $N_\text{max}$ from $N_\text{max}=20,40,\dots, 120$; the black line is the power law extrapolation of primal numerics at $N_*\to\infty$. In the inset we zoom around a point of the boundary to appreciate better the convergence rate of dual numerics compared to the primal one.
• Figure 3: Phase shifts $\delta_I=\tfrac{1}{2i}\log{S_I}$ as a function of $s\ell_s^2$ for some irrep $I$, with $I$={singlet, antisymmetric, symmetric} respectively in red, blue and green. In each plot, solid line is obtained from primal numerics with $N_\text{max}=120$, the dashed line is obtained from the dual with $N_*=30$. The gray lines are the predictions from the EFT up to two-loops. The left panel shows the phase shifts for an arbitrary $\beta_3<0$: in the singlet channel there is a sharp resonance, signaled by the phase shifts passing through $\pi/2$. The right panel shows the phase shifts for a fixed $\beta_3>0$: in this case we see an axion resonance in the antisymmetric channel. Notice that for both points the EFT prediction agrees well with the non perturbative completion up to the scale set the by the lightest resonance, which, for this values of $\beta_3$ we chose appears dynamically around the naive cutoff scale $s^*=4/\ell_s^2$.