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

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