Electroweak mass difference of mesons

We consider electroweak (EW) gauge boson corrections to the masses of pseudoscalar mesons to next to leading order (NLO) in $\alpha_s$ and $1/N_C$. The pion mass shift induced by the $Z$-boson is shown to be $m_{\pi^\pm}-m_{\pi^0} = -0.00201(12)$ MeV. While being small compared to the electromagnetic mass shift, the prediction lies about a factor of $\sim 4$ above the precision of the current experimental measurement, and a factor $O(10)$ below the precision of current lattice calculations. This motivates future implementations of these EW gauge boson effects on the lattice. Finally, we consider BSM contributions to the pion mass difference.