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  • Letter
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Positivity in the renormalization of effective field theory

You-Peng Liao1, Jasper Roosmale Nepveu1,2, and Chia-Hsien Shen1,2,3,4

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Phys. Rev. D 114, L031901 – Published 6 August, 2026

DOI: https://doi.org/10.1103/d29d-zn65

Abstract

We show that the direction of renormalization in effective field theory is constrained by fundamental principles in the infrared—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 infrared, 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.

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Phys. Rev. D 114, L031901– Published 6 August, 2026
  • Received 21 May 2025
  • Accepted 4 June 2026
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