Positivity Bounds in Scalar Effective Field Theories at One-loop Level

TL;DR

This paper investigates positivity bounds in scalar effective field theories at the one-loop level, revealing subtleties in their implications for theory parameters and experimental tests, especially regarding loop-generated operators.

Contribution

It provides a detailed analysis of positivity bounds at one-loop level, highlighting conditions under which these bounds apply and their impact on scalar EFT parameter space.

Findings

One-loop generated dimension-8 operators are not strictly bounded by positivity.

Positivity bounds require considering all contributions at the same loop order.

Implications for experimental tests of positivity bounds are significant.

Abstract

Parameters in an effective field theory can be subject to certain positivity bounds if one requires a UV completion that obeys the fundamental principles of quantum field theory. These bounds are relatively straightforward at the tree level, but would become more obscure when loop effects are important. Using scalar theories as examples, we carefully exam the positivity bounds in a case where the leading contribution to a forward elastic amplitude arises at the one-loop level, and point out certain subtleties in terms of the implications of positivity bounds on the theory parameter space. In particular, the one-loop generated dimension-8 operator coefficients (that would be positive if generated at the tree level), as well as their $\beta$-functions are generally not subject to positivity bounds as they might correspond to interference terms of the cross sections under the optical theorem, which could have either sign. A strict positivity bound can only be implied when all contributions at the same loop order are considered, including the ones from dim-4 and dim-6 operator coefficients, which have important effects at the one-loop level. Our results may have important implications on the robustness of experimental tests of positivity bounds.