Abstract
We derive a general formula for two-loop counterterms in Effective Field Theories (EFTs) using a geometric approach. This formula allows the two-loop results of our previous paper to be applied to a wide range of theories. The two-loop results hold for loop graphs in EFTs where the interaction vertices contain operators of arbitrarily high dimension, but at most two derivatives. We also extend our previous one-loop result to include operators with an arbitrary number of derivatives, as long as there is at most one derivative acting on each field. The final result for the two-loop counterterms is written in terms of geometric quantities such as the Riemann curvature tensor of the scalar manifold and its covariant derivatives. As applications of our results, we give the two-loop counterterms and renormalization group equations for the O(n) EFT to dimension six, the scalar sector of the Standard Model Effective Field Theory (SMEFT) to dimension six, and chiral perturbation theory to order p6.
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Acknowledgments
We thank Xiaochuan Lu, Chia-Hsien Shen, Peter Stoffer and Anders Thomsen for helpful discussions. We would also like to thank J. Bijnens, G. Colangelo and G. Ecker for helpful comments about their results in ref. [24]. This work is supported in part by the U.S. Depart- ment of Energy (DOE) under award numbers DE-SC0009919. LN gratefully acknowledges financial support by the Swiss National Science Foundation (Project No. PCEFP2_194272 and mobility grant PCEFP2_194272/2).
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Jenkins, E.E., Manohar, A.V., Naterop, L. et al. Two loop renormalization of scalar theories using a geometric approach. J. High Energ. Phys. 2024, 131 (2024). https://doi.org/10.1007/JHEP02(2024)131
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DOI: https://doi.org/10.1007/JHEP02(2024)131