Two-loop anomalous dimensions for baryon-number-violating operators in SMEFT

TL;DR

This work addresses baryon-number violation within the SMEFT by delivering the two-loop renormalization-group equations for all baryon-number-violating dimension-six operators, including gauge, Yukawa, and Higgs-sector effects. It also provides the one-loop SMEFT–S1 leptoquark matching, illustrating how the four BNV SMEFT operators can be generated from a simple UV model and demonstrating cancellations of scheme and matching-scale dependences. Together with existing LEFT RGEs below the electroweak scale and SMEFT-to-LEFT matching, the results enable consistent next-to-leading-log analyses of nucleon decays, provided hadronic matrix elements are known at NLO. The paper establishes a complete NLL EFT framework for BNV processes and validates it through multiple cross-checks, including gauge-parameter independence and UV-model consistency, with full results available in electronic supplementary material.

Abstract

We compute the two-loop renormalization-group equations for the baryon-number-violating dimension-six operators in the SMEFT. This includes all three gauge interactions, the Yukawa, and Higgs self-interaction contributions. In addition, we present the one-loop matching of the $S_1$ scalar leptoquark on the SMEFT, which can generate the Wilson coefficients of all four gauge-invariant baryon-number-violating SMEFT operators. Using this example, we demonstrate the cancellation of scheme and matching-scale dependences. Together with the known two-loop renormalization-group evolution below the electroweak scale in the LEFT, as well as the one-loop matching of SMEFT onto LEFT, our results enable consistent next-to-leading-log analyses of nucleon decays, provided that the relevant matrix elements are known at next-to-leading-order accuracy.

Figures (2)

• Figure 1: Comparison of the LL and NLL dependence of the Wilson coefficients on the matching scale $\mu_M$, normalized to a matching scale of $\mu_M = M_\mathrm{OS}$, which is chosen at the unification scale, $M_\mathrm{OS} = 6.5 \times 10^{15} \,\text{GeV}$. We only include the contribution of the gauge interactions.
• Figure 2: Comparison of the LL and NLL running of the Wilson coefficients due to the gauge interactions, normalized to the tree-level expression. The uncertainties are obtained by varying the matching scale between $\mu_M = M_\mathrm{OS}/2$ and $\mu_M = 2 M_\mathrm{OS}$. The solid lines include a resummation of large logarithms, while the dashed lines are the re-expanded LL and NLL results, truncated at NLO and NNLO, respectively.