# Wick contraction in scalar QED

Markus Kahn
Homework Statement:
Expand
$$\left\langle 0\left|\mathrm{T}\left[\exp \left(i \int_{-T}^{T} \mathrm{d} y^{4} \mathcal{L}_{\mathrm{int}}(y)\right)\right]\right| 0\right\rangle,$$
where we have
$$\mathcal{L}_{int}(x)= iqA_\mu (\phi\partial^\mu\phi^*-\phi^*\partial^\mu\phi)+i^2q^2 A_\mu A^\mu |\phi|^2$$
up to second order in ##q##.
Relevant Equations:
All given above
While writing out the Dyson series due to the time ordering above I encountered the two expressions
$$T(\mathcal{L}_{int}(x))\quad \text{and}\quad T(\mathcal{L}_{int}(x)\mathcal{L}_{int}(y))$$
I was able to write out the first term in terms of contractions using Wick's theorem and then finally in terms of propagators... But when I try to do the same for the second term I encountered some problems. If you expand the second term by inserting the definition of ##\mathcal{L}_{int}(x)## one of the terms this will result in is
$$\langle A_\mu(x)A_\nu(y)\phi(x)[\partial^\mu \phi^*(x)]\phi(y)[\partial^\nu \phi^*(y)] \rangle,$$
where the ##\langle\dots \rangle## is to be understood as the sum over all possible contractions of the fields inside the brackets. The relations I'm familiar with in this context are
$$\langle A_\mu (x)A_\nu(y)\rangle = G_V^{\mu\nu}(x-y)\quad \langle \phi^*(x)\phi(y) \rangle=G_F(x-y)\quad \langle \phi(x)\phi(y) \rangle=0,$$
where ##G_V## is the photon propagator and ##G_F## the Feynman propagator of the real scalar field.

My issue is now, that I don't know how one is supposed to evaluate expressions like
$$\langle A_\mu(x)\phi(y)\rangle\quad \text{and}\quad \langle \phi(x)\partial^\mu\phi^*(y)\rangle,$$
I'm not even sure if these make sense since ##\partial^\mu \phi## is a vector while ##\phi## is obviously a scalar (same for ##A_\mu## and ##\phi##). But if you really can't evaluate the two examples, then how does one evaluate
$$\langle A_\mu(x)A_\nu(y)\phi(x)[\partial^\mu \phi^*(x)]\phi(y)[\partial^\nu \phi^*(y)] \rangle$$

Any hints are appreciated.