2.2 The signature of the mass generation: ordinary versus tachyonic

In this subsection we turn to the analysis of the contribution

$$\epsilon ^{\mu \nu \alpha \beta} \langle F _{\mu \nu} \rang... ...^{\rho \sigma \gamma \delta} \langle F _{\rho \sigma} \rangle$$ (13)

of equation (12), which will be responsible for ordinary or tachyonic mass generation according to whether $\langle F _{\mu \nu} \rangle$ represents an external magnetic or electric field. We notice that our results are not inconsistent with those that can be deduced from [8, equation (31)], from which it is clear that the sign of the contribution from the interaction lagrangian changes in the case of purely electric or purely magnetic background.

Let us now start from the case of a constant magnetic field $B$. Then, without loosing generality, we can rotate the reference frame to align an axis, say $x ^{1}$, with $B$. Accordingly, $\langle F _{\mu \nu} \rangle = B \delta _{[ \mu \vert 2} \delta _{ \nu ] 3}$, so that

$$\epsilon ^{\mu \nu \alpha \beta} \langle F _{\mu \nu} \rang... ...\epsilon ^{2 3 \alpha \beta} \epsilon ^{2 3 \gamma \delta}\,.$$ (14)

It is clear thus that $\alpha$, $\beta$, $\gamma$ and $\delta$ in (14) can take only the values $0$, $1$. Denoting by ${}^{(2)} \eta ^{\alpha \beta}$ the $2 \times 2$ Minkowski tensor, we must then have that

$$\epsilon ^{\alpha \beta 2 3} \epsilon ^{\gamma \delta 2 3} = A \left( {}^{(2)} \eta ^{\alpha \gamma} {}^{(2)} \eta ^{\beta \delta} - {}^{(2)} \eta ^{\alpha \delta} {}^{(2)} \eta ^{\beta \gamma} \right)$$

$$= A \left( P _{(10)} ^{\alpha \gamma} P _{(10)} ^{\beta \delta} - P _{(10)} ^{\alpha \delta} P _{(10)} ^{\beta \gamma} \right),$$ (15)

where $P _{(10)} ^{\alpha \gamma}$ is the projector onto the $(0-1)$-plane, i.e. $P _{(10)} ^{\alpha \gamma} = {}^{(2)} \eta ^{\alpha \gamma}$. Contracting in relation (15) the couples of indices $({}^{\alpha \gamma})$ and $({}^{\beta \delta})$ we obtain

$$\epsilon ^{\alpha \beta 2 3} \epsilon _{\alpha \beta} {}^{2 3} = A (2 \times 2 - 2) = 2 A \,;$$

since in lowering the indices $\alpha \beta$ there is the time involved, $\epsilon ^{\alpha \beta 2 3} \epsilon _{\alpha \beta} {}^{2 3} = - 2$, so that $A = -1$.

Analogously in the electric case we can take $F _{\mu \nu} = E \delta _{[\mu \vert 0} \delta _{1 \vert \nu]}$. Then

$$\epsilon ^{\alpha \beta \mu \nu} \langle F _{\mu \nu} \rang... ...\epsilon ^{\alpha \beta 0 1} \epsilon ^{\gamma \delta 0 1}\,,$$ (16)

with the indices $\alpha$, $\beta$, $\gamma$, $\delta$ taking only the spatial values $2$, $3$. Now, following the same procedure and observing that no minus signs are involved in lowering spatial indices, we find

$$\epsilon ^{\alpha \beta 0 1} \epsilon ^{\gamma \delta 0 1} = \left( {}^{(2)} \delta ^{\alpha \gamma} {}^{(2)} \delta ^{\beta \delta} - {}^{(2)} \delta ^{\alpha \delta} {}^{(2)} \delta ^{\beta \gamma} \right)$$

$$= P ^{\alpha \gamma} _{(23)} P ^{\beta \delta} _{(23)} - P ^{\alpha \delta} _{(23)} P ^{\beta \gamma} _{(23)} \,;$$

again we introduced the projector notation $P ^{\mu \nu} _{(23)} \equiv {}^{(2)} \delta ^{\mu \nu}$, where ${}^{(2)} \delta ^{\mu \nu}$ is the $2 \times 2$ Kronecker delta.

We can now write the equation of motion for the fluctuations, which from the action (12) turns out to be

$$\partial _{\mu} f ^{\mu \beta} = 8 g ^{2} \epsilon ^{\mu \n... ...\rho \sigma} \rangle \partial _{\alpha} f _{\gamma \delta}\,.$$ (17)

The above equation can then be expressed, for both the electric and magnetic case, in Fourier space and in terms of the vector potential of the fluctuations, which we will call $a _{\mu}$. Since $f _{\mu \nu} = \partial _{\mu} a _{\nu} - \partial _{\nu} a _{\mu}$, we have

$$k _{\mu} \left( k ^{\mu} a ^{\nu} - k ^{\nu} a ^{\mu} \righ... ...\mu \nu} - \bar{k} ^{\mu} \bar{k} ^{\nu} \right) a _{\mu} \,.$$ (18)

Here and in what follows we define

$$\bar{g} ^{\mu \nu} = P ^{\mu \nu} _{( \cdot \cdot )}$$ (19)

$$\bar{k} ^{\mu} = P ^{\mu \alpha} _{( \cdot \cdot )} k _{\alpha} = \bar{g} ^{\mu \alpha} k _{\alpha}$$ (20)

and

$$P ^{\mu \alpha} _{( \cdot \cdot )} = P ^{\mu \alpha} _{(10)} \,, \qquad \kappa = + 32 g ^{2} B ^{2}$$

in the magnetic case, or

$$P ^{\mu \alpha} _{( \cdot \cdot )} = P ^{\mu \alpha} _{(23)} \,, \qquad \kappa = - 32 g ^{2} E ^{2}$$

in the electric case. Equation (18) can be written also as

$${{\mathcal{D}} ^{-1}} ^{\mu \nu} ( k ) a _{\mu} = 0 \,,$$

where we have defined

$${{\mathcal{D}} ^{-1}} ^{\mu \nu} ( k ) = \left( k ^{2} g ^{... ... \bar{g} ^{\mu \nu} - \bar{k} ^{\mu} \bar{k} ^{\nu} \right) .$$ (21)

The operator in (21) can be diagonalized⁵ as shown in appendix A. Four linearly independent physical states satisfy the eigenvector equation of ${{\mathcal{D}} ^{-1}} ^{\mu \nu} (k)$; we will call them $k _{\mu}$, $\tilde{k} _{\mu}$, $\tilde{k} ^{\perp} _{\mu}$, $E ^{\mu}$. Their definitions and corresponding eigenvalues are enlisted in table 1.

Table 1: Eigenvalues and eigenvectors of the propagator.

Eigenvalues	Eigenvectors
$0 \phantom{_{\mu}}$	$k _{\mu}$
${\displaystyle{}k ^{2} - \frac{\kappa \bar{k} ^{2}}{k ^{2} - m _{\mathrm{A_{\phantom{\mbox{A}}}}} ^{2}}}$	$\tilde{k} _{\mu} = \cases{ \epsilon ^{\mu \alpha 0 1} k _{\alpha} & in the... ... \cr \cr \epsilon ^{2 3 \mu \alpha} k _{\alpha} & in the magnetic case }$
$k ^{2} _{\phantom{\mu}}$	$\tilde{k} ^{\perp} _{\mu} = \epsilon _{\mu \nu \alpha \beta} \bar{k} ^{\nu} \tilde{k} ^{\alpha} k ^{\beta}$
$k ^{2} _{\phantom{\rho}}$	$E ^{\mu} = \epsilon ^{\mu \nu \rho \sigma} k _{\nu} \tilde{k} _{\rho} \tilde{k} ^{\perp} _{\sigma}$

There are, thus, two non-zero eigenvalues and one of them, $k ^{2}$, has degeneracy two; moreover the ``gauge'' eigenvector is associated to the zero eigenvalue. This last results changes if we study the problem in the covariant $\alpha$-gauge, when the eigenvector $k ^{\mu}$ is then associated to a non-zero (but still background independent) eigenvalue, $1/\alpha$ (the corresponding inverse propagator will be called ${\mathcal{D} ^{-1}} ^{\mu \nu} ( k ; \alpha )$ as defined in what follows).

As promised above, we can now compare the obtained results with those derived in previous works on the subject. Again, we observe that a key point in our derivation is the (non-perturbative) path-integral procedure used to obtain equation (9): in this way all the effects due to the quantum fluctuations of the pseudoscalar field, at all orders, are taken into account. In particular, if we concentrate on the purely magnetic case, it is then clear that this constitutes a generalization of the results obtained in [5], where the secular equation is obtained considering plane wave solutions to the classical equations of motion associated to the Maxwell $+$ Klein-Gordon action for the coupled electromagnetic and pseudoscalar fields. An analogous result for the electric case in presence of massless pseudoscalars can, for instance, be found in [17]: here the dispersion relations are obtained under physically very sensible restrictions but are, anyway, of perturbative character. Also the more detailed analysis of [18] uses a different kind of approximation with respect to the one employed in our calculation since the starting lagrangian in [18, equation (1)] is different from (12), our quadratic approximation to the full, non-perturbative, effective result in equation (9). We can thus trace back the differences between our results for the eigenvalues of the propagator and the one already derived in the literature on the subject, to the fact that we have taken into account the effects due to the pseduscalar fields in a substantially non-perturbative way.