# What is the effective (quantum) lagrangian of a fermion field for fixed electromagnetic field?

... or, put it another way, what are the loop corrections to the dirac equation in the presence of a fixed (external) electromagnetic field?.

## Background

Let $\mathcal L=\frac{1}{2}(\partial\phi^2+m^2\phi^2)+\frac{g}{3!}\phi^3$ be the lagrangian for $\phi^3$ theory (real-scalar field $\phi)$.

The action of the theory is $$S=\frac{1}{2}\int \frac{\mathrm dk}{(2\pi)^4}\bigg[\phi(k)(k^2+m^2)\phi(k)\bigg]+\frac{g}{3!}\int\frac{\mathrm dk_1}{(2\pi)^4}\frac{\mathrm dk_2}{(2\pi)^4}\frac{\mathrm dk_3}{(2\pi)^4} \phi(k_1)\phi(k_2)\phi(k_3)(2\pi)^4\delta(k_1+k_2+k_3)$$

The amplitude for any process can be calculated, order by order in $g$, by summing all the tree and loop diagrams, using the interaction as vertices.

On the other hand, we can define the (quantum) effective action for the theory, $$\Gamma=\frac{1}{2}\int\frac{\mathrm dk}{(2\pi)^4}\bigg[\phi(k)(k^2+m^2-\Pi(k^2)\bigg]\phi(k)+\sum_{n=3}^\infty \frac{1}{n!}\int\frac{\mathrm dk_1}{(2\pi)^4}\cdots\frac{\mathrm dk_n}{(2\pi)^4} V_n(k_1,\cdots,k_n)\phi(k_1)\cdots \phi(k_n) (2\pi)^4\delta(k_1+\cdots+k_n)$$ where $V_n$ is the $n$-point vertex function (i.e., amplitude for a process with $n$ external lines that are one/two/three particle-irreducible, using the exact propagator for the lines, and the exact three-point vertex function for the interactions).

The point of the effective quantum action is that the tree-level amplitudes of $\Gamma$ are equivalent to the tree+loop amplitudes of $S$.

## My question(s)

I'm trying to find a reference of how to calculate the quantum effective action for fermions, for fixed external electromagnetic field. In other words, I'm trying to do the same as for the scalar field $\phi$, but using a fermion field $\psi$, whose interactions are mediated by a fixed electromagnetic field $A^\mu$.

I haven't been able to find a reference for what I'm trying to do, but I guess that the effective action can be written as $$\Gamma\overset{?}{=}\int \frac{\mathrm dk}{(2\pi)^4}\bar\psi(k)\bigg[\not k+m-\Sigma(\not k)\bigg]\psi+\\+\int\frac{\mathrm dk_1}{(2\pi)^4}\frac{\mathrm dk_2}{(2\pi)^4} \bar\psi(k_1)V_2(k_1,k_2)\psi(k_2)+\mathcal O(\psi^4)$$ where $V_2=V_2(A^\mu,A^\mu_{,\nu})$. My ansatz for this vertex-function is $$V_2(k_1,k_2)=(2\pi)^4\delta(k_1+k_2) \bigg[A^\mu\mathcal M_\mu+\partial_{[\mu} A_{\nu]}\, \mathcal M^{\mu\nu}+\mathcal O(A^2)\bigg]$$ where

though I'm not quite sure what $\mathcal M^{\mu\nu}$ should be.

• Question 1) Is there any nice reference for what I'm trying to calculate?
• Question 2) Is my guess for $\Gamma$ right? if so,
• Question 2) Is my guess for $V_2$ right? if so,
• Question 4) Should the external lines $k_1,k_2,k_3(=k_2-k_1)$ be taken on-shell, or off-shell?

## Update

This question didn't get much attention here, so I asked the same on overflow. I got a comment there, where they explained that the $\mathcal O(\psi^4)$ and higher order terms shouldn't be present, because there are no self-interactions for fermions, and the only possible interactions are through the electromagnetic field (and so if $A^\mu$ is fixed, there cannot be any interactions between fermions). While this sounds convincing, I still don't know what are the steps to calculate $V_2(k_1,k_2)$. Is my ansatz right? should the momenta be on-shell? any comment will be highly appreciated.

• Checking: You really mean three particle-irreducible? – Qmechanic Dec 27 '15 at 18:27
• @Qmechanic supposedly, yes: for $n>3$ we must use one/two and three particle-irreducible diagrams. Source: Srednicki's book, page 139, first paragraph. – AccidentalFourierTransform Dec 27 '15 at 18:46
• I'd like to see an answer to this too; I'm not sure I know enough to attempt to work it out for myself. But what I can say is that the only diagrams you can draw are fermion lines with an arbitrary number of field insertions, and only the basic vertex is 1PI. Also your vertex function won't have a momentum conserving delta function. – Javier Mar 7 '16 at 21:03