Symmetry Breaking and False Vacuum Decay after Hybrid Inflation

TL;DR

This work analyzes symmetry breaking at the end of hybrid inflation by tracking Higgs fluctuations through a quantum-to-classical transition into a Gaussian random field. It then employs real-time lattice simulations to follow the fully non-linear evolution, showing that infrared modes become quasi-classical and form lumps that rapidly evolve into bubbles, transferring energy to higher-momentum modes and achieving thermalization in the true vacuum. Ultraviolet divergences are renormalized and matched to a renormalized classical theory, enabling reliable simulations of the nonlinear phase. The results illuminate tachyonic preheating, bubble formation, and energy redistribution with potential implications for electroweak baryogenesis and gravitational-wave production.

Abstract

We discuss the onset of symmetry breaking from the false vacuum in generic scenarios in which the mass squared of the symmetry breaking (Higgs) field depends linearly with time, as it occurs, via the evolution of the inflaton, in models of hybrid inflation. We show that the Higgs fluctuations evolve from quantum to classical during the initial stages. This justifies the subsequent use of real-time lattice simulations to describe the fully non-perturbative and non-linear process of symmetry breaking. The early distribution of the Higgs field is that of a smooth classical gaussian random field, and consists of lumps whose shape and distribution is well understood analytically. The lumps grow with time and develop into ``bubbles'' which eventually collide among themselves, thus populating the high momentum modes, in their way towards thermalization at the true vacuum. With the help of some approximations we are able to provide a quasi-analytic understanding of this process.

Figures (14)

• Figure 1: We compare the phase $|F_k|$ with the occupation number for different times, in the whole range of interest in momenta $k$. Clearly, for large times $\tau\gg1$, the two coincide, as discussed in the text. Note that, after $\tau \simeq 2$, all long wavelength modes are essentially classical, $|F_k|\gg1$.
• Figure 2: The time for which a given mode $k$ can be treated as classical $|F_k(\tau_{\rm cl})| \equiv 1$ is above the line in this figure. It is clear that long wavelength modes with $0 < k \mathrel{\mathop {\hbox{$\sim$$<$}}} 1$ become classical very early, at $\tau_{\rm cl} \simeq 2$, while there remains, at any given time, a high energy spectrum of quantum modes, for $k \gg 1$.
• Figure 3: The power spectrum of the Higgs quantum fluctuations, $P(k,\tau)/k^2 \equiv k\, |f_k(\tau)|^2$, at different times in the evolution. The dotted vertical lines indicate the value of the cut-off, at $k=\sqrt{\tau}$, where the classical spectrum is truncated. Also shown is the excellent approximation (\ref{['Papp']}) in the region of long wave modes.
• Figure 4: Top: Comparison between $A_{\hbox{\tiny ren}}(\tau,\mu\!=\!\sqrt{\tau_i})$, Eqs. (\ref{['areneq']}) and (\ref{['match']}), and $A_{\hbox{\tiny clas}}(\tau,\tau_i)$, the approximation obtained by truncating the spectrum at $\mu=\sqrt{\tau_i}$. Bottom: Left: Relative error induced by truncating the spectrum at $\mu=\sqrt{\tau_i}$, as a function of $\tau_i$. Right: Difference between two choices of the initial time $\tau_i$ for the fixed-time renormalization scheme, Eqs. (\ref{['areneq']}) and (\ref{['match']}).
• Figure 5: The radial profile of the Higgs peak for $\lambda=0.11/4$ and $V=0.003$, at time $\tau=2.54$, corresponding to $m t = 14$, obtained with our lattice simulation (with error bars, from averaging over several realizations), and compared with the analytical result (\ref{['peak']}). We have also included the rms Higgs value (\ref{['phirms']}) at that time. Note that we are still in the linear regime, where (\ref{['peak']}) gives a very good approximation. The higher tail corresponds to an averaging out of several lower peaks.