Higher-order Wilson coefficients for c $\to$ u transitions in the Standard Model

TL;DR

This paper calculates the evolution of Wilson coefficients for charm to up quark transitions in the Standard Model, including the effects of integrating out the bottom quark at an intermediate scale, extending existing results from other flavor transitions.

Contribution

It provides a consistent two-step running of Wilson coefficients for $c o u \, ext{l}^+\text{l}^-$ transitions, incorporating intermediate scale matching and extending previous calculations.

Findings

Results of Wilson coefficient evolution including intermediate scale effects.

Extended anomalous dimensions and matching coefficients for charm transitions.

Framework applicable to other flavor-changing processes.

Abstract

The standard theoretical framework to deal with weak decays of heavy mesons is the so-called weak effective Hamiltonian. It involves the short-distance Wilson coefficients, which depend on the renormalisation scale $\mu$. For specific calculations one has to evolve the Wilson coefficients down from the electroweak scale $\mu=M_W$ to the typical mass scale of the decay under consideration. This is done by solving a renormalisation group equation for the effective operator basis. In this paper the results of a consistent two-step running of the $c \to u \,\ell^+\ell^-$ Wilson coefficients for dimension-6 operators are presented. This running involves the intermediate scale $\mu=m_b$ (with $M_W > m_b > m_c$) where the bottom quark is integrated out. The matching coefficients and anomalous dimensions are taken to the required order by generalizing and extending results from $b \to s$ or $s \to d$ transitions available in the literature.