The estimation of interference of weak-electromagnetic contributions of Majorana neutrino in $\nu + e \rightarrow \nu + e$ cross-section

The toroid interaction is a part of the radiative corrections to the scattering of neutrinos on electrons, quarks and nuclei both in the vacuum and in a medium. Therefore, it gives a way to gain the experimental information about the magnitude of neutrino toroid moments in the low-energy scattering processes. A very elegant expression for the $\nu + e \rightarrow \nu + e$ scattering cross-section in the Born approximation with allowance for all three dipoles of the neutrino was given in [16]. Based on these expressions, P. Salati [6] found the laboratory and astrophysical constraints on the values of the electromagnetic characteristics of neutrinos of different types. Here we estimate once again the TDM contribution of the interference term due to the electromagnetic interaction of the neutrino and the electron through the TDM and weak interaction due to direct exchange of $\nu$ and $e$ by the neutral vector boson $Z$, see Fig. 2.

The calculations of the contributions of different diagrams, see Fig. 2 and its interference terms lead to:

$$\sigma^\gamma_{\rm tor}=\frac{8\alpha^2}{\pi^3}G^2_{F}m_{e}... ...}f_2(z), \; \sigma_{SM}=\frac{G^{2}_{F}}{\pi}m_{e}E_{\nu},$$

where $z=E_\nu/m_e$ and

$$\begin{eqnarray*} f_1(z) \!\! &=&\!\! \left[ \frac{(1+2z)^2}{2z^2} \ln \left(1... ...^2}-\frac{1+2z}{z^2} \ln \left(1+\frac{2z^2}{(1+2z)} \right), \end{eqnarray*}$$

It is convenient to consider the following ratios:

$$\frac{\sigma^\gamma_{\rm tor}}{\sigma_{SM}}= \frac{8\alpha... ..._{\rm tor}}{\sigma_{SM}}= \frac{\alpha}{\pi}\frac{f_2(z)}{z}.$$

(We use $\tau_{\nu_{e}}\approx e \sqrt{2}G_{F}/\pi^2$ throughout the paper.) These ratios are given in Fig. 3. We see that the direct interference contribution to the total cross section $\nu + e \rightarrow \nu + e$ is negligible.