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We merge together recent developments in the S-matrix bootstrap program todevelop a dual setup in 2 space-time dimensions incorporating scatteringamplitudes of massive particles and matrix elements of local operators. Inparticular, the stress energy tensor allows us to input UV constraints on IRobservables in terms of the central charge $c_{UV}$ of the UV Conformal FieldTheory. We consider two applications: (1) We establish a rigorous lower boundon $c_{UV}$ 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, bydetermining the allowed range of cubic coupling and one-particle form-factorfor fixed $c_{UV} = 1/2$ and magnetic field.

Injecting the UV into the Bootstrap: Ising Field Theory