Injecting the UV into the Bootstrap: Ising Field Theory

We merge together recent developments in the S-matrix bootstrap program to develop a dual setup in 2 space-time dimensions incorporating scattering amplitudes of massive particles and matrix elements of local operators. In particular, the stress energy tensor allows us to input UV constraints on IR observables in terms of the central charge $c_{UV}$ of the UV Conformal Field Theory. We consider two applications: (1) We establish a rigorous lower bound on $c_{UV}$ of a class of $\mathbb{Z}_2$ symmetric scalar theories in the IR (including $\phi^4$); (2) We target Ising Field Theory by, first, minimizing $c_{UV}$ for different values of the magnetic field and, secondly, by determining the allowed range of cubic coupling and one-particle form-factor for fixed $c_{UV} = 1/2$ and magnetic field.