Matching the Weak Mixing Angle at Low Energies

The running weak mixing angle is used as a convenient tool to keep control of dominating logarithmic corrections in the Standard Model of the electroweak interactions connecting measurements at largely differing energy scales. To relate the solutions of the renormalization group equation for the weak mixing angle above and below particle thresholds one needs matching conditions. We suggest to define the matching at the W-boson threshold by comparing the full Standard Model with a low-energy effective theory where the weak neutral current, in particular for parity-violating fermion scattering, is described by 4-fermion operators and calculate the corresponding matching relation.