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

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.

arXiv comment13 pages, 1 figure

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