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We study the generation of magnetic field in the primordial plasma of thestandard model (SM) particles at temperature $T>80$~TeV much higher than theelectroweak scale. It is assumed that there is an excess number of right-handedelectrons over left-handed positrons in the plasma. Using the Berry-curvaturemodified kinetic theory to incorporate the effect of the Abelian anomaly, weshow that this chiral-imbalance leads to generation of hyper-magnetic field inthe plasma in both the collision dominated and the collisionless regimes. It isshown that in the collision dominated regime the chiral-vorticity effect cangenerate finite vorticity in the plasma together with the magnetic field.Typical strength of the generated magnetic field is $10^{27}$~Gauss at $T\sim80$~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 oforder $10^{31}$~Gauss at typical length scale $10/T$. But there may not be anyvorticity generation in this regime. We show that the estimated values of themagnetic field are consistent with the bounds obtained from presentobservations.

Primordial magnetic field and kinetic theory with Berry curvature