Nonperturbative Bounds on Scattering of Massive Scalar Particles in $d \geq 2$

TL;DR

This paper derives strong nonperturbative bounds on two-to-two scattering amplitudes of massive scalar particles in dimensions 2 to 4, using a primal numerical method that enforces full unitarity, with applications to effective field theories.

Contribution

It introduces a novel primal numerical approach to impose full non-linear unitarity constraints, resulting in tighter bounds on scattering amplitudes in various dimensions.

Findings

Bounds are significantly stronger than linearized or positivity bounds.

Perturbative $\, ext{phi}^4$ theory saturates the derived bounds.

Applicable to constraining effective field theories in multiple dimensions.

Abstract

We study two-to-two scattering amplitudes of a scalar particle of mass $m$. For simplicity, we assume the presence of $\mathbb{Z}_2$ symmetry and that the particle is $\mathbb{Z}_2$ odd. We consider two classes of amplitudes: the fully nonperturbative ones and effective field theory (EFT) ones with a cut-off scale $M$. Using the primal numerical method which allows us to impose full non-linear unitarity, we construct novel bounds on various observables in $2 \leq d \leq 4$ space-time dimensions for both classes of amplitudes. We show that our bounds are much stronger than the ones obtained by using linearized unitarity or positivity only. We discuss applications of our bounds to constraining EFTs. Finally, we compare our bounds to the amplitude in $\phi^4$ theory computed perturbatively at weak coupling, and find that they saturate the bounds.