S-matrix bootstrap in 3+1 dimensions: regularization and dual convex problem

TL;DR

This paper develops a dual convex optimization approach to the S-matrix bootstrap in 3+1 dimensions, providing a practical method to bound scattering amplitudes and explore physical theories, especially for scalar fields related to pion physics.

Contribution

It introduces a dual convex minimization framework for the 3+1 dimensional S-matrix bootstrap, enabling efficient bounds and analysis of scattering amplitudes with fewer variables.

Findings

The dual approach yields strict upper bounds on scattering amplitudes.

The method simplifies numerical computations by using a small number of dual partial waves.

Application to scalar fields relevant to pion physics demonstrates practical utility.

Abstract

The S-matrix bootstrap maps out the space of S-matrices allowed by analyticity, crossing, unitarity, and other constraints. For the $2\rightarrow 2$ scattering matrix $S_{2\rightarrow 2}$ such space is an infinite dimensional convex space whose boundary can be determined by maximizing linear functionals. On the boundary interesting theories can be found, many times at vertices of the space. Here we consider $3+1$ dimensional theories and focus on the equivalent dual convex minimization problem that provides strict upper bounds for the regularized primal problem and has interesting practical and physical advantages over the primal problem. Its variables are dual partial waves $k_\ell(s)$ that are free variables, namely they do not have to obey any crossing, unitarity or other constraints. Nevertheless they are directly related to the partial waves $f_\ell(s)$, for which all crossing, unitarity and symmetry properties result from the minimization. Numerically, it requires only a few dual partial waves, much as one wants to possibly match experimental results. We consider the case of scalar fields which is related to pion physics.