Misalignment dynamics of Scalar Condensates with Yukawa coupling: Particle and Entropy Production

TL;DR

This work develops a fully renormalized, energy-conserving real-time framework for the non-equilibrium dynamics of a homogeneous scalar condensate Yukawa coupled to $N_f$ fermions, highlighting the inadequacy of using a static $V_{eff}$ in dynamical misalignment. By performing a large $N_f$ analysis and introducing an adiabatic particle basis, the authors connect the derivative expansion of the effective action to dynamical particle production, showing energy transfer from the condensate to fermionic fluctuations and the emergence of an asymptotic stationary, highly entangled state with $n_k(\infty)\propto 1/k^6$ and extensive entanglement entropy. The framework reveals a radiatively induced instability in $V_{eff}$ and shows how wave-function renormalization arises naturally from dynamics, while renormalization group running of the Yukawa coupling and the renormalized fields ensure a consistent, RG-invariant description. The results imply possible cosmological implications for reheating and isocurvature perturbations and hint at dynamical spontaneous symmetry breaking in asymptotic stationary states, motivating further study in cosmological spacetimes and gauge theories.

Abstract

Misalignment dynamics, the non-equilibrium evolution of a scalar (or pseudoscalar) condensate in a potential landscape, broadly describes a solution to the strong CP problem, a mechanism for cold dark matter production and (pre) reheating post inflation. Often, radiative corrections are included phenomenologically by replacing the potential by the effective potential which is a \emph{static quantity}, its usefulness is restricted to (near) equilibrium situations. We study the misalignment dynamics of a scalar condensate Yukawa coupled to $N_f$ fermions in a manifestly energy conserving, fully renormalized Hamiltonian framework. A large $N_f$ limit allows us to focus on the fermion degrees of freedom, which yield a negative contribution to the effective potential, a radiatively induced instability and ultraviolet divergent field renormalization. We introduce an adiabatic basis and an adiabatic expansion that embodies the derivative expansion in the effective action, the zeroth order is identified with the effective potential, higher orders account for the derivative expansion including field renormalization and describe profuse particle production. Energy conserving dynamics leads to the conjecture of emergent asymptotic highly excited stationary states with a distribution function $n_k(\infty)\propto 1/k^6$ and an extensive entropy which is identified with an entanglement entropy. Subtle aspects of renormalization associated with the initial value problem are analyzed and resolved. Possible new manifestations of asymptotic spontaneous symmetry breaking (SSB) as a consequence of the dynamics even in absence of tree level (SSB), and cosmological inferences are discussed.

Figures (5)

• Figure 1: The upper diagrams are the scalar and fermion self energies, from which the wave function renormalizations $\propto y^2_0\,\ln(\Lambda); (y^2_0/N_f)\,\ln(\Lambda)$ respectively are obtained and renormalize the Yukawa coupling. The bottom diagrams show vertex renormalization of the Yukawa coupling, which is $\propto (y^2_0/N_f)\,\ln(\Lambda)$. The fermion wave function renormalization and vertex correction are suppressed by a factor $N_f$ with respect to scalar wave-function renormalization. The loops yield the same type $\ln(\Lambda)$ divergence with the ultraviolet cutoff $\Lambda$ as can be seen by power counting. Solid lines correspond to fermions, dashed lines to bosons.
• Figure 2: Fermion contribution to the one loop effective potential, there are $N_f$ fermions in the loop. The first diagram is a scalar self-energy contribution, the second a quartic self-interaction. Dashed lines correspond to the scalar mean field $\varphi \sqrt{N_f}$ the solid line with arrow is the Fermion propagator. All external lines are at zero four momentum.
• Figure 3: $\frac{V_{eff}[X] }{N_f \overline{m}^{\,4}}$ vs. $X=m_f/\overline{m}$ for $\alpha =0,0.02$
• Figure 4: $I(t) = \int^{{\Lambda}/{m_F(0)}}_0 \frac{x^4\,\cos(2\,\sqrt{x^2+1}\,m_F(0)t)}{(x^2+1)^{5/2}}\,dx$ vs. $m_F(0)\,t$ for $\Lambda/m_F(0) = 4000$.
• Figure 5: $I(\lambda;\mu t)/I(\lambda;0)$ vs $\mu t$ for $\lambda=4000$.