Introduction

$\quad$ The electromagnetic properties of a spin-$1/2$ charged particle are described by four independent dipole moments while the neutrino properties by three moments: magnetic ($\mu$), electric ($d$) and toroid ($\tau$) dipole moments [1,2]. In the Standard Model (SM) they are induced by radiative corrections and have the following theoretical predictions for the electron neutrino [3,4,5]

$$\mu_{\nu_e}\! = \frac{3eG_F m_{\nu_e}}{8\sqrt{2}\pi^2}= 3\times10^{-19}\left( \frac{m_{\nu_e}}{1\;{\rm eV}} \right)\mu_B,$$

$$d_{\nu_e} \! = 0,\quad E(0)\propto\Delta m,$$

$$\tau_{\nu_e}\! \approx e\frac{\sqrt{2}G_F}{\pi^2}%%\left(\frac{h}{c}\right)^2 = e\cdot6.5\times10^{-34}\;{\rm cm}^2 = 8.5\times10^{-13} \mu_B\lambdabar_e,$$ (1)

where $G_F$, $E$, $\mu_B$ and $\lambdabar_e$ are the Fermi constant, the electric form factor of the neutrino, the Bohr magneton and the Compton wavelength of the electron, respectively. At the same time, the experimental bounds on these moments, which can be extracted in diverse ways [6], are really poor [7].

The magnetic and electric dipole moments of neutrinos are well known, but the third electromagnetic characteristic of a neutrino, the toroid dipole moment (TDM), is still under discussion in the literature, see for example [4,5,8] and the references therein. We know that the TDM is the electromagnetic characteristic which the Dirac and Majorana neutrinos posses in both the massive and massless limits. In the non-relativistic limit the interaction energy ${\cal H}=-\mbox{\boldmath$\tau$}\cdot{\bf J} =-\tau\varphi^{\dag }\mbox{\boldmath$\sigma$} \varphi\left({\rm curl}\,{\bf B} -\dot{{\bf E}}\right)$ represents a T-invariant electromagnetic interaction of the particle induced by its TDM which does not conserve P- and C-parity individually. It is useful to remark also that in the massless limit, the electromagnetic properties of Dirac neutrinos are represented by the TDM and the neutrino charge radius which coincide numerically [9]. According to [4,5], the spatial size of the toroid dipole moment (TDM) is formed by the mass of the weak intermidiate boson $M_W$ and does not depend on the inert mass of the particle under consideration. So in the physical hierarchy the TDM is closer to the electric charge than to the magnetic moment.

The permitted forms of coupling for the electromagnetic current $J^{\rm EM}_\mu$ are

$$J^{\rm EM}_\mu(q)_{\rm Dirac} = \left[ \overline{u}_f({\bf p}')\Gamma_\mu(q) u_i({\bf p})\right],$$

$$J^{\rm EM}_\mu(q)_{\rm Majorana} = J^{\rm EM}_\mu(q)_{\rm Dirac} +\left[ \overline{v}_i({\bf p})\Gamma_\mu(q) v_f({\bf p}')\right],$$ (2)

where the matrix elements are taken between the Dirac or Majorana neutrino states with different masses. A Lorentz-covariant structure of the dressed vertex operator $\Gamma_\mu(q)$ in the toroid parametrization [1,8] is given by

$$\Gamma_\mu(q) = F(q^2)\gamma_\mu +M(q^2)\sigma_{\mu\nu}q^\... ...ilon_{\mu \nu \lambda \rho} P_{\nu}q_{\lambda}\gamma_{\rho},$$ (3)

where $F$, $M$, $E$ and ${\cal T}$ are the normal, anomalous magnetic, electric and toroid dipole form factors respectively, $P_{\nu}= p_{\nu}+p'_{\nu}$ and $\epsilon_{\mu \nu \lambda \rho}$ is the antisymmetric tensor. In the anapole parametrization [10] the vertex operator reads

$$\Gamma_\mu(q) = F(q^2)\gamma_\mu +M(q^2)\sigma_{\mu\nu}q^\... ...^\nu\gamma_5 +A(q^2)[q^2\gamma_\mu-\widehat{q}q_\mu]\gamma_5$$ (4)

where $A(q^2)$ is the anapole form factor. Using the following identity

$$\overline{u}_f({\bf p}')\Bigl\{ \Delta m\sigma_{\mu\nu}q^\nu + \left(q^2\gamma_\mu-\widehat{q}q_\mu\right)$$

$$- i\varepsilon_{\mu\nu\lambda\rho}P^\nu q^\lambda \gamma^\rho\gamma_5 \Bigr\}\gamma_5 u_i({\bf p}) = 0,$$ (5)

we see that the TDM and anapole coincide in the static limit when the initial and final masses of neutrinos are equal to each other [1,8].

It is easy to check, using CPT invariance of $\Gamma_\mu$ and C-, P- and T-properties of each contribution in eqs. (3, 4), that for the Majorana current only the toroid dipole form factor survives [2] and the value of the toroid dipole moment of the Dirac neutrino is just half of the Majorana one. For the above reasons we have not specified the nature of the neutrino and as TDM is a more simple (multipolar) characteristic than anapole, which has the composite structure as it follows from (5), we shall subsequently only use the term TDM. In addition, in the forthcoming calculations the numerical value of TDM from eq. (1) will be used [4,5].

If the toroid dipole moment is observable, what physical consequences does it lead to? Among the several possibilities are the Vavilov-Cherenkov and transition radiations (TR) of particles induced by their dipole moments. This problem for the Dirac neutrino with non-zero magnetic moment was considered in [11,12]. In 1985, Ginzburg and Tsytovich [13], using a classical approach, showed that the macroscopic toroid dipole moment moving in a medium generates Vavilov-Cherenkov and TR radiations as well. Here we present the first quantitative discussion of the transition radiation of a neutrino having non-zero TDM in the framework of quantum theory along the lines of [12].