Primordial magnetic field and kinetic theory with Berry curvature

We study the generation of magnetic field in the primordial plasma of the standard model (SM) particles at temperature $T>80$~TeV much higher than the electroweak scale. It is assumed that there is an excess number of right-handed electrons over left-handed positrons in the plasma. Using the Berry-curvature modified kinetic theory to incorporate the effect of the Abelian anomaly, we show that this chiral-imbalance leads to generation of hyper-magnetic field in the plasma in both the collision dominated and the collisionless regimes. It is shown that in the collision dominated regime the chiral-vorticity effect can generate finite vorticity in the plasma together with the magnetic field. Typical strength of the generated magnetic field is $10^{27}$~Gauss at $T\sim 80$~TeV with the length scale $10^5/T$ whereas the Hubble length scale is $10^{13}/T$. Further the instability can also generate the magnetic field of order $10^{31}$~Gauss at typical length scale $10/T$. But there may not be any vorticity generation in this regime. We show that the estimated values of the magnetic field are consistent with the bounds obtained from present observations.