Symmetries, Sum Rules and Constraints on Effective Field Theories

TL;DR

This paper derives universal sum rules for scattering amplitudes in theories with symmetries, linking IR parameters to UV cross sections, and applies these to effective field theories and models like the Higgs sector.

Contribution

It systematically determines all independent sum rules and positivity constraints for symmetric theories, clarifying the effects of gauging and providing practical bounds on EFT parameters.

Findings

Sum rules relate IR amplitude coefficients to UV cross sections.

Constraints on Higgs couplings from custodial symmetry.

Impact of gauging on Goldstone boson sum rules.

Abstract

Using unitarity, analyticity and crossing symmetry, we derive universal sum rules for scattering amplitudes in theories invariant under an arbitrary symmetry group. The sum rules relate the coefficients of the energy expansion of the scattering amplitudes in the IR to total cross sections integrated all the way up to the UV. Exploiting the group structure of the symmetry, we systematically determine all the independent sum rules and positivity conditions on the expansion coefficients. For effective field theories the amplitudes in the IR are calculable and hence the sum rules set constraints on the parameters of the effective Lagrangian. We clarify the impact of gauging on the sum rules for Goldstone bosons in spontaneously broken gauge theories. We discuss explicit examples that are relevant for WW-scattering, composite Higgs models, and chiral perturbation theory. Certain sum rules based on custodial symmetry and its extensions provide constraints on the Higgs boson coupling to the electroweak gauge bosons.