Noncyclic geometric phases and helicity transitions for neutrino oscillations in a magnetic field

We show that neutrino spin and spin-flavor transitions involve nonvanishing geometric phases. The geometric character of neutrino spin rotation is explored by studying the neutrino spin trajectory in the projective Hilbert space representation and its relation to the geometric phase. Analytical expressions are derived for noncyclic geometric phases. Several calculations are performed for different cases of rotating and nonrotating magnetic fields in the context of solar neutrinos and neutrinos produced inside neutron stars. Also the effects of adiabaticity, critical magnetic fields, and cross boundary effects in the case of neutrinos emanating out of neutron stars are examined.

Figure 1

Bloch sphere representation of neutrino spin rotation. Initially the neutrinos are produced in the left helicity state, which corresponds to a point on the pole of the sphere. Under the effect of matter and magnetic field, neutrinos undergo spin-precession ${\nu }_{eL}\to {\nu }_{eR}$ and neutrino spin-vector $\mathbf{n}$ traces out cyclic [(a) and (b)] and noncyclic curves [(c) and (d)] on the Bloch sphere depending on the relative values of ${\dot{\varphi }}_p$ and the parameters of ${\mathbf{B}}_{\mathbf{eff}}$. The circular curve describes the path of ${\mathbf{B}}_{\mathbf{eff}}$. The rotation frequency is in units of $\pi /R$, and the positive and negative signs of $\dot{\varphi }$ correspond to clockwise and anticlockwise rotation of the magnetic field about the neutrino direction, respectively. We used the following parameters: electron number density $n_e={10}^{24}{\mathrm{g}/\mathrm{cm}}^3$, neutron number density $n_n=n_e/6$, matter potential $V=\sqrt{2}{G}_F(n_e-n_n/2)$, and magnetic field strength $B={10}^6\mathrm{G}$.

Figure 2

Geometric phases associated with the curves in the Bloch sphere for neutrino spin-precession ${\nu }_{eL}\to {\nu }_{eR}$. (a) Cyclic geometric phase: Solid curve corresponds to the cyclic case ${\dot{\varphi }}_p=5\dot{\varphi }$ and the dotted curve corresponds to the resonant condition $V=-\dot{\varphi }$. (b) Noncyclic geometric phase for the cases $\dot{\varphi }=100$ (solid curve) and $\dot{\varphi }=-200$ (dotted curve).

Figure 3

(a) Density and (b) magnetic field profiles of the neutron star. Magnetic field is plotted in log scale.

Figure 4

$\log \gamma (z)$ as a function of distance (a) inside and (b) outside the NS. The adiabaticity condition $\gamma \gg 1$ is satisfied for all values of $\dot{\varphi }$ while for the outside regions adiabaticity holds only in a limited region.

Figure 5

Geometric phases neutrino propagation in NS. In (a) the flat portion of the curve corresponds to neutrino propagation inside the NS, where the geometric phase is almost zero. In (b) ${\dot{\varphi }}_{\mathrm{res}}$ corresponds to the resonant condition $V=-\dot{\varphi }$.

Figure 6

Neutrino survival probability of spin and spin-flavor precession for various values of the rotation frequency. Nonzero values of $\dot{\varphi }$ lead to suppression of transitions and the probability converges to one-half at a faster rate compared to the case when $\dot{\varphi }=0$. For the case of spin transitions in nonrotating magnetic fields the probability does not converge to 0.5 but instead approaches 1 in the limit $z\gg R$. This is because for this case $\cos {\theta }_{\mathrm{eff}}=0$ and the oscillatory term in Eq. (46) converges to 1 in the limit $z\gg R$.