1.2 A ``naive'' demonstrative approach

In this paper we want to explore ``the other side of the Moon'', that is what happens if the gauge field strength $F _{\mu \nu}$, and not the scalar/axion field $\phi$, acquires a non-vanishing background value. As an example, suppose $\langle F _{\mu \nu} \rangle = B \delta _{[ \mu \vert 2} \delta _{\nu ] 3}$, i.e. $\langle F _{\mu \nu} \rangle$ is a constant magnetic field along the $x ^{3}$ direction. This choice is not ad hoc as it could appear: we are trying to mimic the QCD vacuum, where a constant, color, magnetic field lowers the energy density with respect to the perturbative Fock vacuum, where no gluons are present [15]. By introducing the proper kinetic term for $\phi$ and the fluctuation field $f_{\mu\nu}$, we see that (2) turns into

$${\mathcal{L}} = - \frac{1}{4} f _{\mu \nu} f ^{\mu \nu} + ... ...g}{4} B\phi \epsilon _{2 3 \alpha \beta} f ^{\alpha \beta} \,.$$ (3)

Equation (3) shows as $\phi$ couples only to the $f ^{01}$ component of the fluctuation field strength. Accordingly, we can write the following effective lagrangian in the $(0-1)$-plane:

$${\mathcal{L}}_{01} \equiv - \frac{1}{2} f _{01} f ^{01} + ... ...} \phi \partial ^{\mu} \phi + \frac{g }{4} B \phi f ^{01} \,.$$ (4)

By freezing the $x^2$, $x^3$ dependence of the fields in (4) we get the bosonized Schwinger model [16] and solving the classical field equation for $\phi$ we obtain the non-local effective action for the dimensionally reduced, massive, electromagnetic field

$${\mathcal{L}}_{01} \equiv - \frac{1}{2} f _{01} f ^{01} - \frac{g^2 B^2}{32} f ^{01}\frac{1}{\Box}f _{01} \,,$$ (5)

where the generated mass square is proportional to $g^2 B^2$. Of course the full spectrum of excitations will contain more general types of spacetime dependence, so that we expect here an anisotropic mass generation effect.

We can repeat the same calculations with an electric background, e.g.

$$\langle F _{\mu \nu} \rangle = E \delta _{[ \mu \vert 0} \delta _{1 \vert \nu ]} \,.$$

Then, instead of (5), we obtain for the small excitations in the $(2-3)$-plane

$${\mathcal{L}} = - \frac{1}{2} f _{23} f ^{23} + \frac{g^2 E^2}{32} f ^{23} \frac{1}{\Box} f _{23} \,.$$ (6)

By comparing (6) with (5), we see that in the electric case there is a different sign in front of the non-local term, i.e. the generated mass is tachyonic. This situation for the electric background can be partially cured providing a non vanishing mass $m _{\mathrm{A}}$ to the axion field, so that the tachyonic mass generation sets in only around the critical electric field $g E \sim m _{\mathrm{A}}$.

As we will see in the next sections, the theory can be solved beyond the two dimensional truncated sectors mentioned here. Furthermore, it is possible to compute the vacuum energy resulting from small fluctuations around the chosen background. The vacuum energy develops an imaginary part for electric fields bigger than some critical value, of the order of $m _{\mathrm{A}} / g$. It can be worth now to remark a difference with respect to the instability in two dimensional QED. Both effects are triggered by an external electric field, but the standard Schwinger mechanism operates for any value of the constant electric field, provided it extends to sufficiently large distances. Even with a weak field, one can perform enough work to create a new pair. By contrast, in our case, the effect is present only for field strengths bigger than a critical one, regardless of whether it extends over a large region of space or not.

The paper is organized as follows. In subsection 2.1 we set up the ``sum over histories'' formulation for the system under study: in particular we show how an effective lagrangian for small fluctuations around a pure electric/magnetic background can be obtained, as a quadratic approximation, after integrating out exactly (and non-perturbatively) the scalar field and neglecting higher order terms. The next subsection 2.2 deals in more detail with the mass generation effect, fully exploiting the difference between the electric and the magnetic case and determining the propagator and its spectrum in momentum space. Then in section 3 the expression of the vacuum energy is introduced together with the exact expression for the propagator. The main steps in the computation of the free energy are outlined in section 4 and the emergence of an imaginary part in the pure electric case is analyzed. Discussion and conclusions are in section 5, with particular emphasis about the relation between the tachyonic mass-generation effect and a possible instability of homogeneous electric fields above some threshold. Three appendices follow, where technical details can be found about the solution of the eigenvalue equation for the momentum space propagator (appendix A), the determination of the propagator itself (appendix B) and the explicit computation of the free energy (appendix C), with particular attention at the behavior in the infrared and ultraviolet limits.