Long-range forces : atmospheric neutrino oscillation at a magnetized detector

Among the combinations $L_e-L_\mu$, $L_e-L_\tau$ and $L_\mu-L_\tau$ any one can be gauged in anomaly free way with the standard model gauge group. The masses of these gauge bosons can be so light that it can induce long-range forces on the Earth due to the electrons in the Sun. This type of forces can be constrained significantly from neutrino oscillation. As the sign of the potential is opposite for neutrinos and antineutrinos, a magnetized iron calorimeter detector (ICAL) would be able to produce strong constraint on it. We have made conservative studies of these long-range forces with atmospheric neutrinos at ICAL considering only the muons of charge current interactions. We find stringent bounds on the couplings $ \alpha_{e\mu, e\tau} \lapp 1.65 \times 10^{-53}$ at 3$\sigma$ CL with an exposure of 1 Mton$\cdot$yr if there is no such force. For nonzero input values of the couplings we find that the potential $V_{e\mu}$ opposes and $V_{e\tau}$ helps to discriminate the mass hierarchy. However, both potentials help significantly to discriminate the octant of $\theta_{23}$. The explanation of the anomaly in recent MINOS data (the difference of $\Delta m^2_{32}$ for neutrinos and antineutrinos), using long-range force originated from the mixing of the gauge boson $Z^\prime$ of $L_\mu-L_\tau$ with the standard model gauge boson $Z$, can be tested at ICAL at more than 5$\sigma$ CL. We have also discussed how to disentangle this from the solution with CPT violation using the seasonal change of the distance between the Earth and the Sun.