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The standard theoretical framework to deal with weak decays of heavy mesonsis the so-called weak effective Hamiltonian. It involves the short-distanceWilson coefficients, which depend on the renormalisation scale $\mu$. Forspecific calculations one has to evolve the Wilson coefficients down from theelectroweak scale $\mu=M_W$ to the typical mass scale of the decay underconsideration. This is done by solving a renormalisation group equation for theeffective operator basis. In this paper the results of a consistent two-steprunning of the $c \to u \,\ell^+\ell^-$ Wilson coefficients for dimension-6operators 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 matchingcoefficients and anomalous dimensions are taken to the required order bygeneralizing and extending results from $b \to s$ or $s \to d$ transitionsavailable in the literature.

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