I've heard that in the Standard Model of Particle Physics you can´t have a massive photon whatever you do but I'm having a few problems showing that. I understand that the trick to make this work is to redefine the electric charge to make sure that the electromagnetic gauge invariance of the vacuum remains unbroken, but in practice I'm not entirely sure how to do it.
My attempt: consider a doublet of KG fields which acquires two complex vacuum expectation values (vevs), ie, $<\Phi_{vac}> = (v_1 + i v_2, v_3 + i v_4)/\sqrt2$. Then consider a perturbation around the vacuum of the form $\Phi = (v_1 + i v_2 + \phi_1 + i \phi_2, v_3 + i v_4 + \phi_3 + i \phi_4)/\sqrt2$, where $\phi_i$ are real KG fields. If I insert this doublet in the Higgs potencial, at the end of the day I should get 3 Goldstone bosons (GBs) and 1 Higgs. But after calculating the quadratic terms that come from the Higgs potencial V, given by V = $\mu^2 \Phi^\dagger \Phi + \lambda (\Phi^\dagger \Phi)^2$ with $\mu^2 = - \lambda \sum_i v_i^2$ (from minimization), I end up with 2 Higgs and 2 GBs instead. So the 1st question is what am I doing wrong here.
As for verifying that the photon will remain massless, I've used the covariant derivate for the electroweak interaction which is $D^\mu = d^\mu -ig (W_1^\mu T_1 + W_2^\mu T_2 + W_3^\mu T_3) - i g' B^\mu Y$ which can be rewritten as $D^\mu = d^\mu -ig (W^{+\mu} T_+ + W^{-\mu} T_-) + ieA^\mu (T_3 + Y) - ig (c_w T_3 - s^2_w Y/c_w)Z^\mu$ since $s_w = e/g$, $c_w = -e/g'$, $W_3^\mu = -s_w A^\mu + c_w Z^\mu$ and $B^\mu = c_w A^\mu + s_w Z^\mu$. So I've calculated the mass terms from $D^\mu\Phi^\dagger D_\mu\Phi$ but this time I get four massive gauge bosons (one is the photon)...so again I'm not quite sure what am I doing (wrong)?
