Positivity in Perturbative Renormalization: an EFT $a$-theorem

TL;DR

This paper establishes a fundamental theorem constraining the sign of one-loop coupling running in effective field theories based on principles like unitarity and Lorentz invariance, with broad phenomenological implications.

Contribution

It introduces a new EFT $a$-theorem that constrains renormalization flow signs and impacts positivity bounds, extending the conceptual framework of the $a$-theorem to EFTs.

Findings

The theorem constrains the sign of one-loop running of couplings in EFTs.

It applies broadly to EFTs with arbitrary UV completions.

Implications for positivity bounds in chiral perturbation theory and SMEFT.

Abstract

We show that the direction of renormalization in effective field theory is constrained by fundamental principles in the infrared$\unicode{x2014}$unitarity, analyticity, and Lorentz invariance. Our theorem, in the spirit of the $a$-theorem in conformal field theory, determines the sign of the one-loop running of couplings in the forward limit, when one inserts two operators whose mass dimensions are identical and even. The theorem holds for a broad class of effective field theories with arbitrary ultraviolet completions. The constraint directly applies to linear positivity bounds derived using tree-level amplitudes in the IR, providing a criterion for whether renormalization effects can preserve the positivity bounds, or lead to their apparent violation. We discuss the phenomenological implications of our theorem in chiral perturbation theory and the Standard Model Effective Field Theory, where our theorem is particularly constraining for the running at dimension eight. We provide several examples and show various extensions and applications even at dimension six.