Anisotropic neutrino effect on magnetar spin: constraint on inner toroidal field

The ultra-strong magnetic field of magnetars modifies the neutrino cross section due to the parity violation of the weak interaction and can induce asymmetric propagation of neutrinos. Such an anisotropic neutrino radiation transfers not only the linear momentum of a neutron star but also the angular momentum, if a strong toroidal field is embedded inside the stellar interior. As such, the hidden toroidal field implied by recent observations potentially affects the rotational spin evolution of new-born magnetars. We analytically solve the transport equation for neutrinos and evaluate the degree of anisotropy that causes the magnetar to spin-up or spin-down during the early neutrino cooling phase. Supposing that after the neutrino cooling phase the dominant process causing the magnetar spin-down is the canonical magnetic dipole radiation, we compare the solution with the observed present rotational periods of anomalous X-ray pulsars 1E 1841-045 and 1E 2259+586, whose poloidal (dipole) fields are $\sim 10^{15}$ G and $10^{14}$ G, respectively. Combining with the supernova remnant age associated with these magnetars, the present evaluation implies a rough constraint of global (average) toroidal field strength at $B^\phi\lesssim 10^{15}$ G.