Global charge conservation in the symmetric phase of the early Universe

TL;DR

The paper investigates how Abelian and non-Abelian anomalies in the Standard Model at high temperatures drive violations of $B$ and $L$ (with $B-L$ conserved) in the presence of helical hypermagnetic fields, within anomalous magnetohydrodynamics. It derives the FRW anomaly equations, constructs a coupled evolution system for all fermion asymmetries, the Higgs asymmetry, hypermagnetic helicity, and the weak CS number, and identifies a conserved global Matter-Gauge charge $\\eta_{\rm MG}=\\eta_{ B+L} +\\eta_{\\vec{A}_{Y}\\cdot\\vec{B}_{Y}}+\\eta_{\\rm CS,w}$. Numerical solutions in the range $100\,\mathrm{GeV} \le T \le 10\,\mathrm{TeV}$ with an initial hypermagnetic field show that hypermagnetic helicity decays predominantly into $N_{\rm CS,w}$, with only about $10^{-3}$ of the content converting into $B+L$, while $\\eta_{\rm MG}$ remains conserved. The work highlights the explicit interplay between gauge topology and matter asymmetries and quantifies how gauge-field topology channels energy and asymmetry among sectors under near-equilibrium sphaleron constraints.

Abstract

In the Standard Model at high temperatures, anomalous effects contribute to the violation of baryon number ($B$) and lepton number ($L$), separately, while $B-L$ remains conserved. There are also corresponding changes in the helicity of the hypermagnetic field ($h_B$) and the Chern-Simons numbers of the non-Abelian gauge fields ($N_{\rm CS,w}$ and $N_{\rm CS,s}$). In this study, we investigate a baryogenesis process in the symmetric phase of the early Universe by taking into account the Abelian and non-Abelian anomalous effects as well as the perturbative chirality-flip processes of all fermions. We calculate the time evolution of all relevant physical quantities, including the asymmetries of all fermions and the Higgs, as well as $h_B$ and $N_{\rm CS,w}$. We present a method to compute the latter, for which it is crucial to consider the minute departure from equilibrium of the sphaleron processes. We then verify explicitly the conservation of a global charge, involving the total matter-antimatter asymmetry $B+L$, the hypermagnetic helicity $h_B$ and $N_{\rm CS,w}$, the existence of which had been inferred earlier. In particular we show that, in the scenario that we study, an initial $h_B$ decays mostly to $N_{\rm CS,w}$, with only $10^{-3}$ conversion ratio into $B+L$ asymmetry.

Figures (2)

• Figure 1: Time plots of: (a) electron, muon, and tau lepton asymmetry, $\eta_{L_e}(x)=\eta_{L_{\mu}}(x)=\eta_{L_{\tau}}(x)$, (b) baryon and lepton asymmetries $\eta_{ B}(x)=\eta_{L}(x)$, (c) baryon plus lepton asymmetry $\eta_{ B+L}(x)$, (d) hypermagnetic helicity denoted by $\eta_{\vec{A}_Y\cdot\vec{B}_Y}(x)$, (e) Chern-Simons number $\eta_{\rm CS,w}(x)$ , (f) total conserved charge $\eta_{\rm MG}(x)$, for various values of the amplitude of helical components. The initial conditions are: $k=10^{-7}$, $\eta_{\rm f}^{(0)}=0$. The large-dashed (red) line is obtained for $B_{a}(x_0)=10^{21} \hbox{G}$ and $B_{d}(x_0)=0$, the dashed (green) line for $B_{a}(x_0)=10^{21} \hbox{G}$ and $B_{d}(x_0)=10^{20} \hbox{G}$, the dashed-doted (purple) line for $B_{a}(x_0)=B_{d}(x_0)=10^{21} \hbox{G}$, the dotted (blue) line for $B_{a}(x_0)=10^{20} \hbox{G}$ and $B_{d}(x_0)=10^{21} \hbox{G}$, and the thin-doted (cyan) line for $B_{a}(x_0)=0$ and $B_{d}(x_0)=10^{21} \hbox{G}$.
• Figure 2: This figure displays the time evolution of the $\eta_{\rm ws}(x)$ under the same initial conditions as in Figure \ref{['fig1']}.