Neutrino oscillations

The vacuum TDM may play a very important role for neutrino oscillations. For instance, let us consider the evolution equation of three Majorana neutrinos $\vec{\nu} = (\nu_e,\nu_\mu,\nu_\tau)^T$ in the presence of electromagnetic interactions induced by their toroid dipole moment (further we will call this type of interactions as the toroid ones)

$$i\frac{d\vec{\nu}}{dt}= K\left[\frac{1}{2E}{\rm diag} \left(m_1^2,m_2^2,m_3^2\right)+W(t) \right]K^{\dag }\vec{\nu},$$

where $t$ is a time, $K$ is the mixing matrix connecting the flavor basis, $\nu_\ell$ $(\ell=e,\mu,\tau)$, and the mass basis, $N_i$ $(i=1,...,k)$, of Majorana neutrinos as $\nu_\ell=\sum_i K_{\ell i} N_i$, and has $3(k+1)$ mixing angles and $3(k+1)$ CP-violating phases. The matrix $W$ is, in general, a $3\times3$ matrix, whose elements

$$W_{if}(t) \propto \tau_{if}\,\, \mbox{\boldmath$\sigma$}\cdot{\rm curl}\,{\bf B}(t),$$ (1)

are functions of time $t$ different from zero in the presence of the inhomogeneous vortex magnetic field which, in a concrete experimental situation, may be realized according to Maxwell's equations as the displacement current or the current of the particle colliding with the neutrino at the space point where the interaction (1) is determined.

In this sense, this problem is an analog of the well-known Wolfenstein equation for the propagation of neutrinos through a medium [12], but the resonance conversion of neutrinos can occur even in the vacuum due to the toroid interactions of neutrinos (if $W_{ii}\neq W_{jj}$). The off-diagonal matrix elements $W_{if}$, induced by the transition toroid moments, are nontrivial factors in the Wolfenstein equation and had no previous analogs in the SM (beyond the scope of SM this role was played by the so-called flavor changing neutral currents). Since the Hamiltonian of the evolution of the three neutrino flavors contains at least one time-dependent external parameter $\mbox{\boldmath$\sigma$}\cdot{\rm curl}\,{\bf B}(t)$, we should take into account the topological phases in the evolution operator [13], which may be very important for neutrino oscillations [14]. If, in addition, a neutrino beam intersects some density fluctuations and element compositions in the background of matter, a new phenomenon, geometric resonance, occurs in neutrino oscillations [15]. The two time-dependent parameters of the Hamiltonian (varying independently of each other), which are needed for the geometric resonance, can be the external electromagnetic field (in the medium, it can be the electron current and/or intrinsic sources) and the medium itself ( $n_{\nu_\ell}\neq1$). For example, if ${\rm curl}\,{\bf B}(t)$ and the particle number density $\rho(t)$ vary cyclically when the neutrino beam propagates through the medium, i.e., ${\rm curl}\,{\bf B}(t)={\rm curl}\,{\bf B}(0)$ and $\rho(t)=\rho(0)$ for some time $t$, they form a closed contour on the plane ( $\mbox{\boldmath$\sigma$}\cdot{\rm curl}\,{\bf B}$, $\rho$), and for some neutrino momentum the geometric resonance takes place (for details, see [15]). In order to estimate the contribution of toroid interactions to the transition probability of neutrino conversion we simplify eq. (1) to two neutrino scenario. Then the probability of the $\nu_\alpha\rightarrow\nu_\beta$ conversion can be represent as

$$P(\nu_\alpha\rightarrow\nu_\beta) = \sin^22\theta\sin \int_0... ...u \mbox{\boldmath $\sigma$}\cdot {\rm curl}{\bf B}\right]dt.$$

Using the standard definition of the vacuum oscillation wavelength and defining the oscillation wavelength of the toroid interaction as

$$\ell_{\rm TDM}/\ell_V = \frac{\Delta m^2}{2E_\nu}\frac{z}{\Delta B_x} \frac{1}{\Delta a\mu_B\lambda_e}$$

for $E_\nu=1\,{\rm MeV}$, $\Delta m^2=10^{-5}\;{\rm eV}^2$, (curlB $\Rightarrow\partial B_x/\partial z$) and $\Delta a\equiv\Delta\tau/\mu_b\lambda_e=8.5\times10^{-13}$, we obtain

$$\ell_{\rm TDM} \sim 3\times10^{21} \left(\frac{\Delta z}{1\,{\rm m}} \frac{1\,{\rm Gauss}}{\Delta B}\right) \ell_V.$$ (2)

In the medium three different contributions $\Delta m^2/2E$, $\sqrt{2}G_FN_e$ and $\Delta\tau\mbox{\boldmath$\sigma$}\cdot{\rm curl}\,{\bf B}$ are present in the evolution equation. As is seen from eq. (2) and Table 2, the contribution of toroid interactions is negligibly small for cold media but becomes important for hot media like the early Universe, supernova, neutron stars, etc., where inhomogeneous vortex magnetic fields can have large values.