  We compute the $\beta$-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.
