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Spontaneous magnetization of a vacuum in the hot Universe and intergalactic magnetic fields

We review the spontaneous magnetization of the vacuum of non-Abelian gauge fields at high temperature. The standard model of particles is investigated as a particular example. By using both analytic methods of quantum field theory and gauge field theory on a lattice, we determine the Abelian (chromo)magnetic fields in the restored phase of the model at high temperatures $T \geq T_{ew}$. The fields are stable and temperature dependent, $B = B(T)$. We investigate the mechanisms of the field stabilization in detail. The screening parameters for electric and magnetic fields - the Debye, $m_D(B,T),$ and magnetic, $m_{magn.}(B,T)$, masses - are calculated. It is shown that, in the field presence, the former one is smaller than at zero field. The magnetic mass of the (chromo)magnetic fields is determined to be zero, as for usual $U(1)$ magnetic field. We also show that the vacuum magnetization stops at temperatures below the electroweak phase transition temperature, $T \leq T_{ew}$, when a scalar condensate creates. These properties make reasonable a possibility that the intergalactic magnetic fields observed recently were spontaneously generated in the hot Universe at the reheating epoch due to vacuum polarization of non-Abelian gauge fields. We present a procedure for estimating the field strengths $B(T)$ at different temperatures. In particular, the value of $B(T_{ew}) \sim 10^{14} G$, at $T_{ew}$ is estimated with taking into consideration the observed intergalactic magnetic field $B_0 \sim 10^{- 15} G$. The magnetic field scale is also estimated. Some model dependent peculiarities of the phenomena studied are briefly discussed.