Transition amplitude in scalar QED from a point-like charge

I have a problem for you to compute. I have a classical source, $$A_\mu = \delta_\mu^0 \frac{Q}{r}$$, representing the Coulomb potential generated by a point-like charge. I want to compute the probability that a pair of scalar particles, with interaction, $$\mathcal{L}_{\it int} = ie A^\mu \phi^*\overset{\leftrightarrow}{\partial}{}_\mu \phi,$$ is produced by the source, at asymptotic time $$T\rightarrow \infty$$. The field $$A_\mu$$ is the classical source, while $$\phi$$ is quantised. I assume the source is turned on at some point, and at past infinity the field was in the vacuum. I want to know the transition amplitude for a pair particles to be produced assuming the source is there forever. First, am I right in assuming it would be, $$P(|0\rangle\rightarrow |\phi_{\vec{k}}\phi^*_{\vec{k}}\rangle) = |\mathcal{A}(|0\rangle\rightarrow |\phi_{\vec{k}}\phi^*_{\vec{k}}\rangle)|^2$$ $$\mathcal{A}(|0\rangle\rightarrow |\phi_{\vec{k}}\phi^*_{\vec{k}}\rangle)=e\int \text{d}^4x (u_{\vec{k}}(x)\overset{\leftrightarrow}{\partial}{}_\mu u^*_{\vec{k}}(x)) A^\mu\,?$$ where $$u_{\vec{k}}(x)$$ satisfies the Klein-Gordon equation, $$(\Box -m^2)u^*_{\vec{k}}(x)) =0$$. Also, is this quantity different from zero? That is, does a static classical source, produce particles in scalar QED? And what is the probability of creating $$N$$ particles?

Thank you for any imput. (Of course we are talking here by the first order computation, that's why i use the free mode functions)...

• So what you want is $\langle \mathbf{p_1,p_2}| \Omega \rangle$ to lowest order? where $p_1, p_2, \Omega$ are for the interacting theory – InertialObserver Dec 22 '18 at 3:04