Low-energy effective field theory below the electroweak scale: matching at one loop

We compute the one-loop matching between the Standard Model Effective Field Theory and the low-energy effective field theory below the electroweak scale, where the heavy gauge bosons, the Higgs particle, and the top quark are integrated out. The complete set of matching equations is derived including effects up to dimension six in the power counting of both theories. We present the results for general flavor structures and include both the $CP$-even and $CP$-odd sectors. The matching equations express the masses, gauge couplings, as well as the coefficients of dipole, three-gluon, and four-fermion operators in the low-energy theory in terms of the parameters of the Standard Model Effective Field Theory. Using momentum insertion, we also obtain the matching for the $CP$-violating theta angles. Our results provide an ingredient for a model-independent analysis of constraints on physics beyond the Standard Model. They can be used for fixed-order calculations at one-loop accuracy and represent a first step towards a systematic next-to-leading-log analysis.