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

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.