# A Minimal System Including Weak Sphalerons for Investigating the Evolution of Matter Asymmetries and Hypermagnetic Fields

## TL;DR

This paper develops a minimal model incorporating weak sphalerons and other processes to study the coupled evolution of hypermagnetic fields and matter asymmetries in the early universe, showing how initial conditions influence baryon asymmetry and magnetic fields.

## Contribution

It introduces a self-consistent set of evolution equations and initial conditions for hypermagnetic fields and matter asymmetries, including all relevant fast and slow processes, to analyze their interplay in the early universe.

## Key findings

- Hypermagnetic fields can generate matter asymmetries from zero initial conditions.
- Initial seed hypermagnetic fields can produce acceptable baryon asymmetry.
- Magnetic field scales are smaller than observational data, even with inverse cascade effects.

## Abstract

We study simultaneous evolution of large scale hypermagnetic fields and the asymmetries of quarks, leptons and Higgs boson in the temperature range from 10TeV to 100GeV. Above 10TeV, we identify all of the major fast interactions and use the associated conservation laws as constraints on the initial conditions at 10TeV. Below 10TeV, we identify the major processes which fall out of equilibrium or emerge as non-negligible processes and derive the relevant evolution equations. These include the Abelian anomalies which violate fermion numbers, direct and inverse Higgs decays that change the chiralities of fermions, and weak sphalerons which violate the left-handed fermion numbers. We also consider the contributions of all fermionic chemical potentials to the UY(1) Chern-Simons term which affects the evolution through the AMHD equations. Thus, we present a minimal set of self-consistent initial conditions and evolution equations which respect all constraints coming from conservation laws, fast processes and charge neutrality of the plasma. We solve the coupled evolution equations and find that initial large hypermagnetic field can produce matter asymmetries starting from zero initial value, and vice versa provided an initial seed of hypermagnetic field is present and the rate of the electron Yukawa processes is lower. We find that our model yields acceptable values for baryon asymmetry and magnetic field. However, the scale of the magnetic field obtained is much smaller than the observational data, even when the turbulence driven inverse cascade mechanism in the broken phase is taken into account.