Vacuum expectation value (VEV) of a Gauge theory - Spontaneous Symmetry Breaking (SSB) - Higgs Mechanism

Question

I am dealing with a sort of scalar QED with a term of SSB

\begin{equation} \mathcal{L}=\left|D_{\mu} \phi\right|^{2}-\frac{1}{4}\left(F_{\mu \nu}\right)^{2}-V\left(\phi^{*} \phi\right) \end{equation}

with potential

\begin{equation} V(\phi)=-\mu^{2} \phi^{*} \phi+\frac{\lambda}{2}\left(\phi^{*} \phi\right)^{2} \end{equation}

So I have learned here that the classical potential is at tree-level a good approximation for the effective potential $V_{eff}$. Therefore we use it to calculate the vacuum expectation value of the field. The minimum of $V(\phi$) is

\begin{equation} \phi_{0}=\left(\frac{\mu^{2}}{\lambda}\right)^{1 / 2} \end{equation}

and this is a good approximation at TL for the vacuum expectation value \begin{equation} <\phi> \sim \phi_{0} \end{equation}

But now it is not clear to me why we are considering only the potential $V(\phi)$ when calculating the minimum. In fact we have terms with $\phi$ inside the covariant derivative $\left|D_{\mu} \phi\right|^{2}$ as well, meaning for example

\begin{equation} \phi\phi^*A_{\mu}A^{\mu} \end{equation}

Why these terms do not have a role in calculating the minimum of the potential? Meaning why only the self-interaction terms determine the vev?

Answer 1

The kinetic term as well as any other terms containing only derivatives that are positive definite will just raise the energy of the system. So what one states first, is that the energy is lower for constant configurations (at tree level), this a first bound. Then one can say that within the constant configurations the ones that minimize the potential terms (with no derivatives) lead to the systems minima.

For your model this is enough but however if one includes other fields they must be simultaneously minimized. What generally happens is that renormalizability of the theory imposes constraints on which sort of potentials are allowed, meaning that most fields (e.g. gauge fields and fermions) are normally set to zero so any term mixing your scalar field directly with something else will vanish anyway.