Next-to-Leading Order QCD Corrections to $ΔF =2$ effective Hamiltonians

TL;DR

This work tackles the renormalization of $\Delta F = 2$ four-fermion operators by computing the full next-to-leading order $QCD$ anomalous-dimension matrix in naive dimensional regularization and exploring renormalization-scheme dependencies, including RI and MSbar-like schemes. The authors derive the renormalization-group evolution of the Wilson coefficients through LO and NLO, present explicit ADM blocks for the operator basis, and provide a diagram-by-diagram method and scheme-conversion rules to connect RI, LRI, and MSbar results. They verify consistency with known results for subsets of operators, ensure scheme-independence of critical combinations, and outline practical prescriptions for applying the ADM to phenomenology. The results enable precise RG evolution of Wilson coefficients from high scales to hadronic scales, improving theoretical predictions for $K^0$-$\bar{K}^0$ and $B^0$-$\bar{B}^0$ mixing, $\Delta\Gamma_{B_s}$, and $O(1/m_b^3)$ corrections in inclusive decays.

Abstract

The most general QCD next-to-leading anomalous-dimension matrix of all possible four-fermion, dimension-six $ΔF=2$ operators is computed. The results of this calculation can be used in many phenomenological applications, among which the most important are those related to theoretical predictions of $K^0-\bar{K}^0 $ and $B^0-\bar{B}^0$ mixing in several extension of the Standard Model (Supersymmetry, left-right symmetric, multi-Higgs models, etc.), to estimates of the $B^0_s$--$\bar B^0_s$ width difference, $ΔΓ_{B_s}$, and to the calculation of the $O(1/m_b^3)$ corrections for inclusive b-hadron decay rates.

Figures (3)

• Figure 1: One-loop Feynman diagrams. We show the external quark momenta chosen to obtain the results in the FRI scheme. In the LRI scheme, all the external momenta are equal to $p$.
• Figure 2: One- and two-loop diagrams for the quark propagator.
• Figure 3: Two-loop Feynman diagrams.