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New physics realized above the electroweak scale can be encoded in a modelindependent way in the Wilson coefficients of higher dimensional operatorswhich are invariant under the Standard Model gauge group. In this article, westudy the matching of the $SU(3)_C \times SU(2)_L \times U(1)_Y$ gaugeinvariant dim-6 operators on the standard $B$ physics Hamiltonian relevant for$b \to s$ and $b\to c$ transitions. The matching is performed at theelectroweak scale (after spontaneous symmetry breaking) by integrating out thetop quark, $W$, $Z$ and the Higgs particle. We first carry out the matching ofthe dim-6 operators that give a contribution at tree level to the low energyHamiltonian. In a second step, we identify those gauge invariant operators thatdo not enter $b \to s$ transitions already at tree level, but can give relevantone-loop matching effects.

Matching of gauge invariant dimension 6 operators for $b\to s$ and $b\to c$ transitions