Bounds on scattering of neutral Goldstones

TL;DR

This work develops and applies the numerical S-matrix bootstrap to 2→2 scattering of massless neutral Goldstones in four dimensions, establishing universal bounds on the first two non-universal EFT Wilson coefficients $\bar{g}_3$ and $\bar{g}_4$ and constructing extremal amplitudes saturating these bounds. It reveals a rich structure including three distinct regimes (QCD-like, string-like with Regge trajectories, and spin-0 exchange) and non-perturbative Regge trajectories via complex-spin continuation, while connecting large-$\bar{g}_3$ behavior to tree-level UV completions. The paper also introduces methodological advances for massless bootstrap—improved forward-limit handling and robust unitarity projections—and extends the analysis to model-dependent bounds with EFT-cutoff constraints, providing practical templates and illustrating UV completions along the boundary. These results sharpen non-perturbative constraints on Goldstone dynamics and offer a framework for incorporating experimental or lattice data into EFT bootstrap approaches for pions and related systems.

Abstract

We study the space of $2\to 2$ scattering amplitudes of neutral Goldstone bosons in four space-time dimensions. We establish universal bounds on the first two non-universal Wilson coefficients of the low energy Effective Field Theory (EFT) for such particles. We reconstruct the analytic, crossing-symmetric, and unitary amplitudes saturating our bounds, and we study their physical content. We uncover non-perturbative Regge trajectories by continuing our numerical amplitudes to complex spins. We then explore the consequence of additional constraints arising when we impose the knowledge about the EFT up to the cut-off scale. In the process, we improve on some aspects of the numerical $S$-matrix bootstrap technology for massless particles.

Figures (14)

• Figure 1: The allowed region for the parameters $\bar{g}_3$ and $\bar{g}_4$ defined in \ref{['eq:observables']} lies above the orange line.
• Figure 2: The numerical values of the branching ratio of the cross section $\bar{g}_2^\ell$ defined in \ref{['eq:coefficients_g2l']} of the amplitudes on the lower boundary in figure \ref{['fig:bound']} as a function of $\bar{g}_3$. The colors represent different values of the angular momentum $\ell=0,2,4,6$.
• Figure 3: Trajectory of the lowest lying scalar zero as function of $\bar{g}_3$.
• Figure 4: On the left, we plot the phase shifts for both spins $\ell=0,4$ in red, and the absolute value $|\mathcal{S}_\ell|$ of the corresponding partial waves in blue. Red dashed is the one-loop EFT approximation expected to be reliable up to the scale $\bar{s}\sim1$. On the right, we plot the absolute value $|\mathcal{S}_\ell|$ in the complex $\bar{s}$ plane. There we observe the presence of zeros that we interpret as resonances.
• Figure 5: Chew-Frautschi diagram for the amplitude that minimize $\bar{g}_4$. On the top panel, the leading (in red) and sub-leading (in blue) trajectories are presented up to spin 10. The spin 0 (green) resonance is not part of a trajectory. In this plot, we show various $N_{\text{max}}$ to express the convergence of the position of the zero and we use darker color for larger $N_{\text{max}}$. On the bottom left panel, we zoom on the trajectories at low spin. The leading (red) is well converged in this region, this is not the case for the sub-leading (blue) and we expect it to move up . On the bottom right panel, we present the position of the resonance in the complex plane for both the leading (circle) and sub-leading (triangle) trajectory. In the circled region, we do not trust the position of the resonance.