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

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.

💡 Research Summary

The paper presents a complete calculation of the two‑loop renormalization‑group equations (RGEs) for the baryon‑number‑violating (BNV) dimension‑six operators in the Standard Model Effective Field Theory (SMEFT). Baryon number (B) is an exact symmetry of the renormalizable Standard Model, but many extensions such as Grand Unified Theories (GUTs) or R‑parity‑violating supersymmetry predict ΔB = ±1 processes, most notably proton decay. Because the new physics scale is typically far above the electroweak scale, the effects of BNV operators must be evolved over many orders of magnitude, requiring resummation of large logarithms. While one‑loop SMEFT RGEs and tree‑level matching onto the Low‑Energy EFT (LEFT) are known, next‑to‑leading‑log (NLL) accuracy demands two‑loop SMEFT RGEs, one‑loop SMEFT‑LEFT matching, and one‑loop matching of UV models onto SMEFT. The authors fill the missing piece by deriving the full two‑loop anomalous dimensions for all four gauge‑invariant BNV operators:

• Q_{duqℓ} = ε_{αβγ} ε_{ij}(d^T_α C u_β)(q^T_γ C ℓ_j),
• Q_{qque} = ε_{αβγ} ε_{ij}(q^T_{αi} C q_{βj})(u^T_γ C e),
• Q_{duue} = ε_{αβγ}(d^T_α C u_β)(u^T_γ C e),
• Q_{qqqℓ} = ε_{αβγ} ε_{il} ε_{jk}(q^T_{αi} C q_{βj})(q^T_{γk} C ℓ_l).

The calculation is performed in naive dimensional regularization (NDR) with an anticommuting γ₅, thereby avoiding the notorious γ₅‑scheme ambiguities. The authors introduce a systematic set of evanescent operators (both Chisholm‑type and Fierz‑type) that vanish in four dimensions but are required to close the operator basis in D = 4 − 2ε. A parameter a_ev is used to keep track of scheme dependence; the final two‑loop anomalous dimensions are shown to be independent of a_ev, confirming that physical RGEs are scheme‑independent.

All gauge interactions (SU(3)_c, SU(2)_L, U(1)_Y) are included, together with the full Yukawa sector (Y_u, Y_d, Y_e) and mixed gauge‑Yukawa contributions. The Higgs quartic coupling λ does not enter at two loops because its only contributions are scaleless tadpoles, which vanish in dimensional regularization; λ‑dependence would first appear at three loops. The authors decompose the Wilson‑coefficient tensors into irreducible representations of the flavor permutation group (symmetric S, antisymmetric A, mixed M, and vanishing N) to make the flavor structure transparent. The resulting β‑functions are expressed separately for each irreducible component, which simplifies the evolution of flavor‑non‑universal BNV effects.

To demonstrate the practical relevance of their results, the authors perform a one‑loop matching of a concrete UV completion: the scalar leptoquark S₁ (a color‑triplet, SU(2)_L singlet with hypercharge 1/3). At tree level S₁ does not generate BNV operators, but at one loop it induces all four SMEFT BNV operators with calculable Wilson coefficients. The matching is carried out at a generic scale μ₀, and the authors explicitly verify that the μ₀‑dependence of the matching coefficients cancels against the logarithmic terms generated by the two‑loop RG evolution. Likewise, the dependence on the evanescent‑scheme parameter a_ev cancels when the one‑loop matching is combined with the two‑loop anomalous dimensions. This provides a non‑trivial check of the consistency of the whole framework.

Finally, the paper places the new results in the broader context of EFT evolution. The two‑loop SMEFT RGEs derived here, together with the already known two‑loop LEFT RGEs and the one‑loop SMEFT‑LEFT matching, enable a fully consistent NLL analysis of nucleon decay. Once the non‑perturbative matrix elements of the BNV operators are known at next‑to‑leading order (e.g., from lattice QCD), one can evolve Wilson coefficients from the high BNV scale (potentially 10¹⁶ GeV) down to the hadronic scale (≈ 1 GeV) with controlled logarithmic accuracy. This paves the way for precise predictions of proton and neutron lifetimes in a wide class of BSM scenarios, and for robust interpretation of upcoming experimental searches such as Hyper‑Kamiokande and DUNE.