5 Discussion and conclusions

In this paper we have discussed effects of mass generation in an external magnetic field and of tachyonic mass generation in an external electric field in the case in which there is a pseudo scalar field with a pseudo scalar coupling $g \phi F _{\mu \nu} \tilde{F} ^{\mu \nu}$. The effects due to the presence of the psedo scalars are fully considered: indeed in the derivation of an effective theory for the fluctuations of the electromagnetic field, the contributions from quantum fluctuations of the pseudo scalars are taken into account using a non-perturbative approach.

In the purely magnetic case the mass production for the electromagnetic fluctuations can be interpreted as in the cited works already present in the literature: the differences that we find in the eigenvalues of the propagator can be traced back to the non-perturbative character of our approach, as opposed to the perturbative analysis performed elsewhere.

A more careful discussion is instead required for the purely electric background, in connection with what we have called tachyonic mass generation. Indeed the appearance of an imaginary part in the free energy suggests the presence of an instability for homogeneous electric fields beyond some threshold due to pseudo scalar coupling. This means that the vacuum state must be redefined, to obtain a correct ground state for the theory. The analysis of what becomes the true ground state, a genuinely non perturbative effect, is beyond the scope of this manuscript. Nevertheless we would like, at least, to suggest a possible and simple, although incomplete, answer to this question. Indeed an analysis of Cornwall [22] indicates that in the case of a $3$-dimensional, euclidean, tachyonic mass term (which in our discussion is generated by the time dependent scalar field of equation (2) but can have also another origin) the final result is the formation of an inhomogeneous state. Then, properly generalizing to our set-up the results described in [8], where inhomogeneous electromagnetic fields are shown to decay in pseudoscalars, it is not unreasonable to understand under what we have called tachyonic mass generation a real ``tachyonic instability of the vacuum''.

From the result of equation (31) it is possible to see that the free energy is well-defined in the limit of vanishing background fields ($\kappa \to 0$). Thus our effect is a genuinely non-perturbative one and does not relate to a particular choice of the regularization scheme. Moreover it is worth pointing out again that this ``tachyonic instability of the vacuum'' for fluctuations around a constant external electric field, is characterized by a threshold effect, i.e. the tachyonic mass generation is switched on for electric fields high enough, so that $\kappa + m _{\mathrm{A}} ^{2} < 0$. In the case of the neutral pion, we can obtain the value of the effective pion-photon coupling, defined by equation (2), from the observed value of the neutral pion lifetime [20] and from the value of the decay rate given the coupling (2) [21]. This gives us the values

$$g = 2.53 \cdot 10 ^{-5} \,\rm {MeV} ^{-1} \,,$$

$$m _{\pi} = 134.97 \,\rm {MeV} \,,$$

$$\Longrightarrow E _{\mathrm{crit.}} = \left( 1 \,\rm {GeV} \right) ^{2} \,.$$ (33)

This is a very high electric field, not available in normal laboratory conditions. Furthermore, if it were available, it would reveal the composite structure of the pion and the effective $g \phi F _{\mu \nu} \tilde{F} ^{\mu \nu}$ coupling, used here, would not be applicable any more.

A different question would be then the study of this effect in the case of hypothetical axion particles. In this case, the threshold for the tachyonic mass generation to be set up becomes lower as lower values for the mass of the axion are considered.

Apart from the purely electric case, which is more subtle and, maybe, more exciting because of the exotic tachyonic mass term, it could be interesting a more detailed analysis of the magnetic case in connection with the set-up of PVLAS a presently running experiment at the Legnaro I.N.F.N. laboratories, near Venice, Italy.

Finally a totally different role for these effects, could be in the context of QCD. There, it is known that an external chromo-magnetic fields presents tachyonic instability. If we were to add a particle with coupling to $\epsilon ^{\mu \nu \alpha \beta} F ^{a} _{\mu \nu} F ^{a} _{\alpha \beta}$ (this particle could represent a pseudo scalar bound state of quark and anti-quark pairs), we know that the effect of the external chromo-magnetic field together with the pseudo scalar coupling is of generating mass. The interplay of these two effects could then be an interesting subject for further research.