Transition radiation of neutrino with TDM

The neutrinos with the TDM as well as the neutrino with the magnetic moment produce the transition radiation (TR) when crossing the interface between two media, see Fig. 4, with the plasma frequencies $\omega_{1}$ and $\omega_{2}$ $(\omega_{1} \gg \omega_{2})$ [7]. We calculate the probability of the TR of the (Majorana) neutrino TDM. The (contact) electromagnetic interaction of the neutrino field $\Psi$ with the classical external current is described by the Hamiltonian density:

$${\cal H} =-e T(q^2) \bar{\Psi} \gamma_{\mu} \gamma_{5} \Psi J^{ext}_{\mu}$$ (1)

It is shown in Ref. [7] that TR becomes possible if there is a plane interface at $z=0$, where the refractive index suddenly changes from $n_1 (z < 0)$ to $n_2 (z > 0)$. Using eq. (1) in the Born approximation, we can express the transition matrix element as

$$\vert S\vert _{fi}^2 = (2\pi)^3L^2t\frac{m_\nu}{E_1V}\frac{m_\nu}{E_2V} \frac{(1-n^2)^2\omega^4}{2\omega n^2 V}$$

$$\times \delta(p_{1x,y}-p_{2x,y}-k_{x,y}) \delta(E_1-E_2-\omega)$$

$$\times \left\vert\int^{L/2}_{-L/2}dz \exp[i(p_{1z}-p_{2z}-k_z)z]{\cal M}_{fi} \right\vert.$$ (2)

Here ${\cal M}_{fi}=eT(0)\overline{u}_2\widehat{\varepsilon}\gamma_5 u_1$ is the amplitude, and $t$, $L$ and $V=L^3$ denoted the time, length and volume respectively. Here $L=\beta t$, where $\beta=p/E$ is the velocity of the neutrino. In connection with the phase in the integrand of (2) the formation zone length of the medium may be defined as

$$Z(n)=(p_{1z}-p_{2z}-k_z)^{-1}=(p_{1z}-p_{2z}-n\omega\cos\theta)^{-1},$$

$$p_{2z}=\sqrt{E_2^2-m_\nu^2-n^2\omega^2\sin^2\theta},\quad E_2=E_1-\omega,$$

where $\theta$ is the angle between the photon and the direction of the incident neutrino. The details of further calculations are the same as in [7] and we present here the final results for the total energy of the transition radiation

$$\begin{eqnarray*} \frac{d^2 S}{d\theta d\omega} &=& \frac{T^2(0) \omega^6\sin\... ...) +\frac{E_\nu E_2}{pp_{2z}}-1+\frac{m_\nu^2}{pp_{2z}}\right\}, \end{eqnarray*}$$

where

$$R_i=\frac{1-n_i^2}{n_i}\frac{1}{p-p_{2z}-n_i\omega\cos\thet... ...omega\int_0^{\theta_{\rm max}} \frac{d^2 S}{d\theta d\omega}.$$

The total energy loss of the neutrino in the process of the medium - vacuum transition was computed numerically and its dependence on $m_{\nu}$ is given in Fig. 5.

The fact that massive neutrinos with the magnetic moment can produce TR is well known. It should be pointed out that the nonzero TDM of the neutrino gives rise to the TR for massive and massless neutrinos. This interesting fact requires a separate investigation.