Two-Loop Anomalous Dimensions in the LEFT: Dimension-Six Four-Fermion Operators in NDR

TL;DR

This work delivers the complete two-loop anomalous-dimension matrix for all dimension-six four-fermion operators in the LEFT within the NDR scheme and JMS basis, including $O(\alpha_s^2)$, $O(\alpha_s\alpha)$, and $O(\alpha^2)$ corrections. The authors unify results from multiple literature sources through explicit basis changes, flavor-symmetry arguments, and pole-coefficient tables, and address the intricacies of traces with $\gamma_5$ via evanescent operators. They categorize the full ADM into RG-invariant sectors (ΔF=2, ΔF=1, ΔF=1^{\bar f f}, ΔF=0, ΔL and ΔB/ΔL violations), providing detailed block structures for each sector and flavor version LEFT(5), LEFT(4), and LEFT(3). The results are validated across methods, implemented in DsixTools, and include new findings for baryon- and lepton-number-violating sectors and $\alpha^2$ contributions, with quantitative numerical analyses illustrating the size of two-loop effects. These results enable precise RG evolution of LEFT Wilson coefficients and facilitate higher-order phenomenology in low-energy flavor and baryon/lepton-violating processes.

Abstract

We derive the complete set of two-loop anomalous dimensions describing the mixing of four-fermion operators in the Low Energy Effective Field Theory (LEFT). The calculation is performed in Naive Dimensional Regularization with anticommuting $γ_5$ (the NDR scheme), and the results are given in the "JMS basis" of dimension-six operators. The derivation relies on known results for UV poles in two-loop diagrams in QCD, which are then used to derive the two-loop Anomalous Dimension Matrix (ADM) for the full set of four-fermion operators including $O(α_s^2)$, $O(α_sα)$ and $O(α^2)$ corrections. The method employed is an extension of a common approach to deal with traces containing $γ_5$ in NDR. Our results have been implemented in the public code DsixTools. We also discuss and provide the results in the LEFT with 5, 4 and 3 active quark flavors.

Figures (6)

• Figure 1: Flowchart outlining the full procedure to obtain the two-loop ADMs starting from the tables of pole coefficients of Appendix \ref{['app:Pole Tables']}. In the figure, "CoB" is short for "Change of Basis", and the dashed arrows represent the steps which use surface-level flavor universality for the manipulation of the matrices.
• Figure 2: Diagrams contributing to the cancellation given in Eq. (\ref{['eq:Flavor Decoupling NLO']}), with (a) contributing to the first term and (b) contributing to the second term. The square represents an insertion of $Q_i$, the black blob represents the gluon/photon self energy, and the crossed circle is an EFT counterterm corresponding to $p^{(1,0;0)}_{Q_kQ_j}$.
• Figure 3: Class of diagrams exclusive to operators in the $\Delta F = 0$ sector. Their contribution to the penguin ADM vanishes in any flavor-decoupled basis, in exactly the same way as in the diagrams in Fig. \ref{['fig:Flavor Diagrams']}.
• Figure 4: Running of $L_{eu}^{V,LR}(\mu)$ from the EW scale $\mu = M_Z$ down to the bottom mass scale $\mu = m_b$ with LL (blue) and NLL (red) resummation and the corresponding LO (blue-dashed) and NLO (red-dashed) fixed-order approximations. The Wilson coefficient is shown normalized to 1 at the matching scale $\mu = M_Z$, that is $L_{eu}^{V,LR}(\mu)/L_{eu}^{V,LR}(M_Z)$.
• Figure 5: Running from the EW scale $\mu = M_Z$ down to the bottom mass scale $\mu = m_b$ with LL (blue) and NLL (red) resummation, and the corresponding LO (blue-dashed) and NLO (red-dashed) fixed-order approximations. The Wilson coefficients are shown normalized to 1 at the matching scale $\mu = M_Z$.