Short Distance QCD Contribution to the Electroweak Mass Difference of Pions

It is known that the short distance QCD contribution to the mass difference of pions is quadratic on the quark masses, and irrelevant with respect to the long distance part. It is also considered in the literature that its calculation contain infinities, which should be absorbed by the quark mass renormalization. Following a prescription by Craigie, Narison and Riazuddin, of a renormalization group improved perturbation theory to deal with the electromagnetic mass shift problem in QCD, we show that the short distance QCD contribution to the electroweak pion mass difference (with $m_u=m_d\neq0$) is finite and, of course, its value is negligible compared to other contributions.