2.1 Path integral

An effective technical framework to investigate quantum fluctuations around an external background configuration is provided by the Feynman path integral formalism. The partition function(al) encoding the dynamics of the interacting $\phi$ and $A _{\mu}$ fields reads

$$Z \equiv \int [ {\mathcal{D}} \phi ] [ {\mathcal{D}} A ] \exp\left\{ - \imath \int d ^{4} x {\mathcal{L}} \right\},$$ (7)

where

$${\mathcal{L}} = - \frac{1}{4} F _{\mu \nu}\, F ^{\mu \nu} +... ...artial ^{\mu} \phi + \frac{m ^{2} _{\mathrm{A}}}{2} \phi ^{2}$$ (8)

and both the gauge fixing and ghost terms are, momentarily, understood in the functional measure $[{\mathcal{D}} A]$. For the sake of generality, we assigned a non-vanishing mass to the pseudo scalar field $\phi$. Since the path integral (7) is gaussian in $\phi$, the scalar field can be integrated away exactly, i.e.

$$\int [ {\mathcal{D}} \phi ] \exp \left\{ - \imath \int d ^{4} x \... ... F _{\rho \sigma} + \frac{m _{\mathrm{A}} ^{2}}{2} \phi ^{2} \right] \right\} =$$

$$\qquad = \left[ \det \left( \frac{ \Box + m ^{2} _{\mathrm{A}}} {... ...{\alpha \beta \gamma \delta} F _{\alpha\beta} F_{\gamma\delta} \right\} ,\qquad$$ (9)

where $\mu$ is a mass scale coming from the definition of the measure $[ {\mathcal{D}} \phi ]$. Integrating out the $\phi$ field induces a non-local effective action for the $A$ field. We stress that this effective action, being obtained at the non-perturbative level, takes into account (as an effective action for $A$) all the effects due to the presence of the pseudoscalars at all perturbative orders. This is a crucial difference with many of the previous works on the subject. As is clear from (9), the resulting path integral is quartic in $A$ but, even if it cannot be computed in a closed form, the background field method provides a reliable approximation scheme to deal with this problem. We thus split $F _{\mu \nu}$ in the sum of a classical background $\langle F _{\mu \nu} \rangle$ and a small fluctuation $f_{\mu\nu}$:

$$F _{\mu \nu} = \langle F _{\mu \nu} \rangle + f _{\mu \nu} \,.$$ (10)

In the case of a pure electric or a pure magnetic background we have

$$\epsilon ^{\mu \nu \alpha \beta} \langle F _{\mu \nu} \rangle \langle F _{\alpha \beta} \rangle = 0 \,.$$ (11)

Thus, expanding ${\mathcal{L}}$ up to quadratic terms and dropping a total divergence, we obtain for the effective lagrangian of the fluctuations of the electromagnetic field

$${\mathcal{L}} ^{(2)} = - \frac{1}{4} \langle F _{\mu \nu} \... ...a} \frac{1}{\Box + m ^{2} _{\mathrm{A}}} f _{\gamma\delta}\,.$$ (12)

As we will see in a while, there is an important sign difference between the magnetic and the electric case.