1.1 Preliminary considerations

A dynamical mechanism providing mass to vector gauge bosons is instrumental to match theoretical models of fundamental interactions with the particle spectrum observed in high energy experiments. A blueprint of dynamical mass generation is given by the Schwinger Model, or QED$_{2}$, where fermions quantum fluctuations induce a mass term for the two-dimensional photon. Extension of this non-perturbative quantum effect to four-dimensional gauge theories has still to come because the gauge field effective action cannot be computed exactly in $4D$. In the meanwhile, the archetypal mechanism for gauge field theory mass generation is Spontaneous Symmetry Breaking, induced either by classical tachyonic mass terms [1] or by quantum radiative corrections [2]. The Coleman-Weinberg breaking of gauge symmetry avoids classical tachyonic mass terms and gives raise to a non-vanishing vacuum expectation value for massless scalar fields through radiative quantum corrections. In this paper we are going to discuss a ``complementary'' mechanism, where mass follows from the breaking of rotational invariance induced by a classical background configuration of the gauge field strength. A real, or tachyonic, mass is obtained according with the magnetic, or electric, nature of the background field. The model implementing this effect consists of a scalar field $\phi$ non-minimally coupled to a $U(1)$ gauge vector (non-abelian extensions of this model are planned for future investigations) through an interaction term of the form:

$${\mathcal{L}} _{\mathrm{I}} = \frac{g}{8}\phi \epsilon _{\mu \nu \alpha \beta} F ^{\mu \nu} F ^{\alpha \beta} \,.$$ (1)

The interaction term ${\mathcal{L}} _{\mathrm{I}}$ has a long history dating back to the celebrated ABJ anomaly and neutral pion electromagnetic decay [3]. Moreover by keeping fixed the form and varying the strength of the coupling constant, ${\mathcal{L}} _{\mathrm{I}}$ is equally well suited to describe the axion field currently appearing in many astrophysical and quantum field theoretical problems [4].

In what follows we are going to analyze the case in which the electromagnetic field is a purely electric or purely magnetic background, with special interest about the dynamics of its fluctuations. Before embarking this program and before giving a more detailed account of the main aspects of our approach, it is worth to recall some important steps already taken in the past in this direction. In the main part of this work we will stress more carefully analogies, as well as differences, with what we are proposing in this paper. In particular the fact that an external magnetic field modifies the dispersion relation of photons coupled to (pseudo)scalars was already discussed, for example, in [5]; there the authors have in mind an experimental set-up for the detection of pseudoscalars coupled to two photons based on the fact that the photon effective mass provided by the pseudoscalar coupling is responsible for an ellipticity in an initially linearly polarized beam.⁴ Concerning the situation in which a background electric field is present, recently this problem has been analyzed in [8], where the authors show under which conditions an external electric field decays to pseudoscalars and discuss some particular configurations in which their results can be applied. Postponing a deeper analysis to what follows, we think that an important point to be stressed already at this early stage, is the fact that in the above studies the discussion is perturbative whereas, in the present paper, we are going to analyze a second order effective approximation for the dynamics of the fluctuations of the electromagnetic field, only after a full, non-perturbative treatment of the pseudoscalar.

Before developing this part, we will shortly present some interesting features of the model in a naive form. We remember that, indeed, it is a special feature of the coupling (1) to generate physical masses, or tachyonic instabilities, when an appropriate classical configuration of the scalar field $\phi$ is turned on: performing an integration by parts in the action associated with (1), we end up with an interaction term of the form

$${\mathcal{L}} _{\mathrm{I}} = \frac{g}{8} \left( \partial _... ...\epsilon ^{\mu \nu \alpha \beta} F _{\mu \nu} A _{\alpha} \,.$$ (2)

We can then consider a background configuration selecting a preferred spacelike direction, e.g. $\langle \phi \rangle = \mathrm{const.} \cdot \delta^3_\mu x ^\mu$, so that (2) gives a $(2+1)$-dimensional, Chern-Simons type, mass term [9]. The resulting massive Chern-Simons model is embedded into a $(3+1)$-dimensional theory. Thus, rotational invariance in embedding space is broken [10]. It is interesting to compare this result with the case in which the background field is time dependent only, i.e. $\langle \phi \rangle \equiv \varphi (t)$. The resulting Chern-Simons model is endowed with a tachyonic mass term [11], meaning that this type of background field is unstable. This kind of instabilities could play a role in baryogenesis, as it has been argued in [11], [12]. Moreover, a time dependent axion field may also affect the growth of primordial magnetic fields [13], produce effective Lorentz and parity violating modifications of electrodynamics and affect the polarization of radiation coming from distant galaxies [14].