We study hypermagnetic helicity and lepton asymmetry evolution in plasma of the early Universe before the electroweak phase transition (EWPT) accounting for chirality flip processes via inverse Higgs decays and sphaleron transitions which violate the left lepton number and wash out the baryon asymmetry of the Universe (BAU). In the scenario where the right electron asymmetry supports the BAU alone through the conservation law B/3 − L eR = const at temperatures T > T RL ≃ 10 T eV the following universe cooling leads to the production of a non-zero left lepton (electrons and neutrinos) asymmetry. This is due to the Higgs decays becoming more faster when entering the equilibrium at T = T RL with the universe expansion, Γ RL ∼ T > H ∼ T 2 , resulting in the parallel evolution of both the right and the left electron asymmetries at T < T RL through the corresponding Abelian anomalies in SM in the presence of a seed hypermagnetic field. The hypermagnetic helicity evolution proceeds in a self-consistent way with the lepton asymmetry growth. The role of sphaleron transitions decreasing the left lepton number turns out to be negligible in given scenario. The hypermagnetic helicity can be a supply for the magnetic one in Higgs phase assuming a strong seed hypermagnetic field in symmetric phase. 1 4 Throughout the text we have neglected the bulk velocity evolution described by the Navier-Stokes equation since the length scale of the velocity variation λ v is much shorter than the correlation distance of the hypermagnetic field, λ v ≪ k −1 , or infrared modes of the hypermagnetic field are practically unaffected by the velocity of plasma. In addition, the bulk velocity v does not contribute to the helicity evolution dhThe sign for α Y and γ 5 is opposite to the sign chosen in [7,6] and coincides here with the definitions in [16] where ψ R = (1 + γ 5 )ψ is the right fermion field. See also in [14,15].
For the quadratic helicity χ (2) we present a generalization of the Arnol'd inequality which relates the magnetic energy to the quadratic helicity, which poses a lower bound. We then introduce the quadratic helicity density using the classical magnetic helicity density and its derivatives along magnetic field lines. For practical purposes we also compute the flow of the quadratic helicity and show that for an α 2 -dynamo setting it coincides with the flow of the square of the classical helicity. We then show how the quadratic helicity can be extended to obtain an invariant even under compressible deformations. Finally, we conclude with the numerical computation of χ (2) which show cases the practical usage of this higher order topological invariant.
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