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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 Green's 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.