UV asymptotically free QED as a broken YM theory in the unitary gauge

We compute the β-function of a YM theory, broken to U(1), by evaluating the coupling constant renormalization in the broken phase. We perform the calculation in the unitary gauge where only physical particles appear and the theory looks like a version of QED containing massive charged spin-1 particles. We consider an on-shell scattering process and after verifying that the non-renormalizable divergences which appear in the Greenʹs functions cancel in the expression of the amplitude, we show that the coupling constant renormalization is entirely due to the photon self-energy, as in QED. However we get the expected asymptotic freedom and the physical charge decreases logarithmically as a function of the symmetry breaking scale.