<PERSON>-Higgs Mechnism
Consider an abelian gauge field coupled with a complex field: $$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+(D\varphi)^\dagger D\varphi+\mu^2 \varphi^\dagger\varphi-\lambda(\varphi^\dagger\varphi)^2.$$ In polar coordinates $\varphi=\rho e^{i\theta}$, we have $$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\rho_0^2(\partial_\mu\theta-eA_\mu)^2+\cdots$$ where I have ignored the fluctuation of $\rho$ by setting $\rho=\rho_0$.
(I). The standard way to deal with it is to redefine $$B_\mu\equiv A_\mu-\frac{1}{e}\partial_\mu\theta$$ thus we have $$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+e^2\rho_0^2 B^2$$ thus the massless gauge field eats a Nambu-Goldstone boson and becomes a massive gauge field which has 3 degrees of freedom.
(II). Instead of doing that, let us expand $(\partial_\mu\theta-eA_\mu)^2$ directly and impose Lorentz gauge at the same time $\partial_\mu A^\mu=0$, we have $$\mathcal{L}=-\frac{1}{4}F^2+\rho_0^2e^2 A^2+\rho_0^2 (\partial_\mu \theta)^2,$$ where the cross term vanishes because of the <PERSON> gauge. And this <PERSON> gauge is satisfied by the equation of motion of the gauge field $A_\mu$ naturally. 474
It seems we have both massive gauge field and a Goldstone boson.
My questions are:
1. How to reconcile the two approaches above?
2. If we want to examine the dynamics of the bosons (for example, apply a boson current and see how this current propagates), will this current be a supercurrent in the presence of a dynamic gauge field? This boson current will be a supercurrent(like a current in a superconductor) in the absence of a dynamic gauge field. When we couple this boson system to a dynamic gauge field, the dynamics of phase $\theta$ of the boson field is hidden somehow in the approach (I) and it is not clear whether this boson current will be a supercurrent or not. But from approach (II), it seems this boson current will still be a supercurrent. 298