Creation of large scale strong magnetic fields in matter with parity violation

We show that the Standard Model electroweak interaction of ultrarelativistic electrons with nucleons (eN interaction) in a neutron star (NS) permeated by a seed large-scale helical magnetic field provides its growth up to during a time comparable with the ages of young magnetars . The magnetic field instability originates from the parity violation in the eN interaction entering the generalized Dirac equation for right and left massless electrons in an external uniform magnetic field. We calculate the averaged electric current given by the solution of the modified Dirac equation containing an extra current for left and right electrons (positrons), which turns out to be directed along the magnetic field. Such current includes both a changing chiral imbalance of electrons and the eN potential given by a constant neutron density in NS. Then we derive the system of the kinetic equations for the chiral imbalance and the magnetic helicity which accounts for the eN interaction. By solving this system, we show that a sizable chiral imbalance arising in a neutron protostar due to the Urca-process diminishes very rapidly because of a huge chirality flip rate. Thus the eN term prevails the chiral effect providing a huge growth of the magnetic helicity and the helical magnetic field