Einstein–Cartan–Holst–Proca dynamos massive photons as dark matter and GWs

Abstract

In this paper, we extend Shaposhnikov et al.'s parity violating the Einstein–Cartan–Holst action by adding to Holst term H, a Proca photon–torsion coupling χpt = 10−24 as computed by De Sabbata, Garcia de Andrade, and Sivaram. The strength of Holst term is obtained, in the case of early universe torsion as, T ∼ 1 MeV. The gravity part is the Planck mass product with the Holst term. The photon mass in this case yields m2 γ H ∼ 10−69 GeV4. Therefore, in this case, the photon interacts very weakly with spin-torsion matter in Einstein–Cartan (EC) gravity and the strength of the Maxwell–Proca–Holst interaction is much weaker than in the pure gravity sector. For dark photons, this situation changes drastically since a dark photon mass reaches 1.5 GeV, and from this mass and T = 10−3 GeV, one obtains a Maxwell–Proca–Holst strength of 10−6 GeV4, which is now comparable with the gravity sector. Dynamo mechanism competes with the chirality and dissipation by the handness of the magnetic field with respect to the torsion trace vector to regenerate the magnetic field decay. Bombacigno and Mantovani recently found a sign of the Immirzi parameter in the Holst–Nieh–Yan extension of EC gravity. Here, we also find torsion waves for the Immirzi inverse parameter in the sense that this divergence of torsion trace is porportional to the Immirzi field.