Bootstrapping Pions at Large $N$

TL;DR

The paper applies modern S-matrix bootstrap methods to massless pions in the large-N limit to bound their EFT via dispersion relations and positivity. It constructs a dual semidefinite program to derive two-sided bounds on derivative couplings, revealing a distinctive kink whose origin is explored through simple UV completions. By incorporating the rho meson, the work refines the exclusion plots, derives new rho-coupling bounds, and connects to Skyrme-type EFTs and hidden local symmetry, with comparisons to real QCD and Lovelace-Shapiro amplitudes. Overall, it demonstrates how unitarity, analyticity, and high-energy spectral data tightly constrain large-N hadronic EFTs and sketches paths to pinning down QCD-like theories within this framework.

Abstract

We revisit from a modern bootstrap perspective the longstanding problem of solving QCD in the large $N$ limit. We derive universal bounds on the effective field theory of massless pions by imposing the full set of positivity constraints that follow from $2 \to 2$ scattering. Some features of our exclusion plots have intriguing connections with hadronic phenomenology. The exclusion boundary exhibits a sharp kink, raising the tantalizing scenario that large $N$ QCD may sit at this kink. We critically examine this possibility, developing in the process a partial analytic understanding of the geometry of the bounds.

Figures (23)

• Figure 1: Exclusion plot in the space of four-derivative couplings. Healthy theories must lie inside the colored region.
• Figure 2: Upper bound on $g_{\pi\pi\rho}$ (normalized by $f_\pi$ and $m_\rho$, see \ref{['eq:normcouplingsRho']}) as a function of the gap after the rho.
• Figure 3: As $N\to \infty$ the diagrams contributing to $\pi\pi\to\pi\pi$ scattering arrange in a topological expansion in powers of $1/N$. At leading order, only planar diagrams with the topology of a disk survive. Quark loops are suppressed by $1/N$ and handles (which make the diagram non-planar) by $1/N^2$. Solid oriented lines denote quark propagators, the shaded area represents all possible (planar) dressings with gluons.
• Figure 4: Contour deformations for the two independent sets of dispersion relations. The high-energy poles correspond to the meson spectrum.
• Figure 5: The EFT amplitude for $2\to 2$ pion scattering is given by an infinite sum of four-point vertices with unknown coefficients. The sum can be arranged into a series with increasing number of derivatives in the interaction.