APPEARANCE OF TOROID DIPOLE MOMENT

It is well known that the electromagnetic properties of a particle with spin $j$ are described by $2(2j+1)$ form factors. Thus, for spin-$\frac{1}{2}$ particles different parametrizations of electromagnetic current contain four independent form factors. Using the multipole parametrization we may find its nonrelativistic counterparts and call them the charge, magnetic, toroid and electrical dipole form factors. After Ya. Zel'dovich introduced the anapole for the Dirac particle in 1957 [1], it is generally believed that a geometrical model of the anapole is poloidal currents on a torus. Indeed, a model like this corresponds to the toroid dipole form factor, rather than the anapole as the latter coincides with the toroid form factor only for the current diagonal on initial and final masses [2]. This circumstance is especially important in the framework of neutrino physics in connection with the problems of neutrino decay, oscillation processes and so on. A large number of articles are devoted to the problem of appearance of the anapole or the toroid dipole in the elementary particle theory. Nonetheless, up to date these characteristics are the most mysterious and ambiguous. For example, we may remark that the anapole cannot radiate while the toroid dipole can, because the former is a composition of two multipole parameters, the time derivative of the electric dipole moment and the toroid dipole moment, working in opposite phases so that their radiation amplitudes cancel each other. In our context it is important to cite the articles by V. Ginzburg and V. Tsytovich [3], who calculated the intensities of the Vavilov-Cherenkov and transition radiation of the toroid dipole moment within the classical approach, and by A. Gongora-T. and R. Stuart [4], who demonstrated the gauge invariance of the field-theoretical calculation of the anapole in the framework of the Standard Model. Based on this knowledge and Ref. [5], we continue the search for the possibilities of fixing the toroid moment contributions along the lines outlined in Refs. [6] and [7]. Our withdrawal from the common approach is caused by the fact that we take into account both massive and masless neutrinos and limits of low and high energies. Recall the permitted forms of couplings for the electromagnetic current $J^{\rm EM}_\mu$:

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

$$J^{\rm EM}_\mu(q)_{\rm Majorana} = J^{\rm EM}_\mu(q)_{\rm Dirac} +\left[ %%\overline{u}_f({\bf p}')\... ...mu(q) u_i({\bf p})+ \overline{v}_i({\bf p})\Gamma_\mu(q) v_f({\bf p}')\right],$$ (1)

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

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

where $F$, $M$, $E$ and $T$ 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. It is easy to check that for the Majorana current only the toroid dipole form factor survives [8]. Moreover, if the current considered is diagonal on the initial and final masses, the anapole and toroid parametrizations coincide. Then the interaction of the Majorana neutrino with the electromagnetic field is described by

$${\cal H}_{\rm int}^{\rm Maj} = eT(q^2)\overline{\psi}(x)\left[q^2\gamma_\mu -\widehat{q}q_\mu\right]\gamma_5\psi(x){\cal A}^\mu(x)$$

$$\Rightarrow eT(q^2)\overline{\psi}(x)\gamma_\mu\gamma_5 \psi(x) \frac{\partial F^{\mu\nu}(x)}{\partial x^\nu}.$$ (3)

In the nonrelativistic limit ${\bf q}^2 \rightarrow 0,$ where $\overline{\psi}\gamma_0\gamma_5\psi\rightarrow 0$ and $\overline{\psi}\mbox{\boldmath$\gamma$}\gamma_5\psi \rightarrow\varphi^{\dag }\mbox{\boldmath$\sigma$}\varphi,$ we find:

$${\cal H}_{\rm int}^{\rm Maj}= \tau\varphi^{\dag }\mbox{\boldmath$\sigma$} \varphi\,{\rm curl}\,{\bf B}.$$ (4)

In the static limit ( $m_{\nu_i}=m_{\nu_f}$) for Dirac neutrinos we may utilize the following nonrelativistic approximation for the interaction energy:

$${\cal H}_{\rm int}^{\rm Dir} \propto -\mu\left(\varphi^{\d... ...x{\boldmath$\sigma$}\varphi \cdot{\rm curl}\, {\bf B}\right),$$ (5)

where $\mu,d$ and $\tau$ are the magnetic, electric and toroid dipole moments respectively. The magnetic and electric dipole moments are odd under both the temporal and spatial reflections, and the toroid dipole moment is T invariant and does not conserve P- and C-parities individually, see Table 1.

The complete analysis of the current properties with non-diagonal masses are given [9].