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I'd like to know how to solve the dirac equation with some small gauge potential $\epsilon \gamma^\mu{A}_\mu(x)$ by applying perturbation theory. The equations reads as $$(\gamma^\mu\partial_\mu-m+\epsilon\gamma^\mu A_\mu(x))\psi(x) = 0.$$
The solution up to first order is
$$ \psi(x) = \psi_0(x) +\tau\int\frac{d^4p}{(2\pi)^4}\int d^4x'\frac{e^{-ip(x-x')}}{\gamma_\mu{p}^\mu-m} \gamma^\mu A_\mu(x')\psi_0(x')+\mathcal{O}(\tau^2).$$
Is there any possibility to solve this integral for an constant $$A(x)= Ee_x$$?
The solution up to first order is
$$ \psi(x) = \psi_0(x) +\tau\int\frac{d^4p}{(2\pi)^4}\int d^4x'\frac{e^{-ip(x-x')}}{\gamma_\mu{p}^\mu-m} \gamma^\mu A_\mu(x')\psi_0(x')+\mathcal{O}(\tau^2).$$
Is there any possibility to solve this integral for an constant $$A(x)= Ee_x$$?
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