Bootstrapping Pions at Large $N$. Part II: Background Gauge Fields and the Chiral Anomaly

TL;DR

The paper extends the large $N$ hadronic bootstrap by incorporating background electromagnetic fields, formulating a covariant, mixed EFT for massless pions and external photons. It derives dispersion relations and a comprehensive set of sum rules and null constraints, then implements semidefinite-programming positivity bounds to constrain Wilson coefficients and the chiral anomaly, including explicit $N$-dependent bounds via anomaly matching. The analysis reveals novel large $N$ selection rules, Goldstone constraints, and improved Regge channels that sharpen theoretical limits, while identifying numerical obstructions that motivate future refinements (e.g., off-shell photons and richer background fields). Overall, the work demonstrates how S-matrix bootstrap methods can meaningfully constrain low-energy QCD-like EFT data in the large $N$ limit and points to concrete directions for tightening bounds and connecting to UV completions. The anomaly coefficient plays a central role, enabling inhomogeneous, $N$-dependent bounds that tie low-energy EFTs to the microscopic theory.

Abstract

We continue the program [1] of carving out the space of large $N$ confining gauge theories by modern S-matrix bootstrap methods, with the ultimate goal of cornering large $N$ QCD. In this paper, we focus on the effective field theory of massless pions coupled to background electromagnetic fields. We derive the full set of positivity constraints encoded in the system of 2 $\to$ 2 scattering amplitudes of pions and photons. This system probes a larger set of intermediate meson states, and is thus sensitive to intricate large $N$ selection rules, especially when supplemented with expectations from Regge theory. It also has access to the coefficient of the chiral anomaly. We find novel numerical bounds on several ratios of Wilson coefficients, in units of the rho mass. By matching the chiral anomaly with the microscopic theory, we also derive bounds that contain an explicit $N$ dependence.

Figures (14)

• Figure 1: Our old exclusion plot Albert:2022oes for the normalized four-derivative couplings $\tilde{g}_2$ and $\tilde{g}'_2$ in the chiral Lagrangian, now color-coded to represent the bound on the normalized anomaly coefficient $|\widetilde{B}_{\text{ch}}|$. The bound is maximized along the straight oblique segment ($\tilde{g}_2 = \frac{1}{4}\tilde{g}_2'$ up to the kink) and decreases away from it.
• Figure 2: Schematic representation of the basic disk amplitude for large $N$ four-photon scattering.
• Figure 3: Schematic representation of the basic disk amplitudes for the scattering of two pions and two photons. There are two inequivalent orderings of the external legs.
• Figure 4: Disk amplitude for the scattering of three pions and one photon.
• Figure 5: At large $N$ also three-point couplings must be planar with a quark loop running along the boundary, making the coupling proportional to a single trace.