Chern-Simons number asymmetry from CP-violation during tachyonic preheating

Abstract

We consider the creation of non-zero Chern-Simons number in a model of the early Universe, where the Higgs field experiences a fast quench at the end of inflation. We perform numerical lattice simulations in the Abelian Higgs model in 1+1 dimensions and in the SU(2)-Higgs model in 3+1 dimensions with an added effective CP-violating term. We also comment on the appropriate choice of vacuum initial conditions for classical simulations.

Figures (5)

• Figure 1: Left: $\langle \phi^{*}\phi\rangle$ for different values of $t_{ro}$. At the larger roll-off times, the behavior is significantly different from the "Just the half" method, indicating that the quadratic approximation breaks down. The coupling is $\lambda/\mu^{2}=1/8$. Right: $\langle N_\textrm{cs}\rangle$ for different initial condition schemes. The thermal scheme is quantitatively different, but still shows the important features, including the "initial dip". The spinodal case at later and later $t_{ro}$ differs in the initial dip due to the coupling to the C and P violation being introduced later (at $t_{ro}$).
• Figure 2: Left: The dependence of $<N_\textrm{cs}>_\textrm{final}$ on $\kappa$ is linear (here for $m_{H}/m_{W}=1$. Right: The dependence on the mass ratio of the Higgs and W particles is very complicated.
• Figure 3: Depending on the Higgs to W mass ratio, oscillation frequencies seem to conspire to give smaller or larger final $N_\textrm{cs}$, even to the point of giving opposite signs. Here $m_{H}/m_{W}$ = 0.63 (left) and 1.06 (right); $\kappa=-0.05$.
• Figure 4: Two examples of trajectories in the SU(2)-Higgs simulation with $k=16\pi^2\kappa=0$ and $k=3/m_{H}^{2}$. Left: For this initial condition the effect of the CP-violation is a small shift in $N_\textrm{cs}$ and winding number. Right: Here, the trajectories of both observables are split to end up at different integer values.
• Figure 5: Left: Distribution of Higgs winding number and Chern-Simons number for $k=3/m_{H}^{2}$. Right: Bootstrapping the average Chern-Simons number $\langle N_\textrm{cs} \rangle_\textrm{final}$. Notice that since we used the same initial conditions for zero and non-zero $k$, the effect of CP-violation is the difference between the two Gaussians. Due to statistics, these are both shifted to the left.