Rolling in the Higgs model and elliptic functions

Abstract

Asymptotic methods in nonlinear dynamics are used to improve perturbation theory results in the oscillations regime. However, for some problems of nonlinear dynamics, particularly in the case of Higgs (Duffing) equation and the Friedmann cosmological equations, not only small oscillations regime is of interest but also the regime of rolling (climbing), more precisely the rolling from a top (climbing to a top). In the Friedman cosmology, where the slow rolling regime is often used, the rolling from a top (not necessary slow) is of interest too. In the present work a method for approximate solution to the Higgs equation in the rolling regime is presented. It is shown that in order to improve perturbation theory in the rolling regime turns out to be effective not to use an expansion in trigonometric functions as it is done in case of small oscillations but use expansions in hyperbolic functions instead. This regime is investigated using the representation of the solution in terms of elliptic functions. An accuracy of the corresponding approximation is estimated.

Generation and decay of Higgs mode in a strongly interacting Fermi gas

“…The solutions for a classical particle moving in such potential are well known [1,45]. In the case of small amplitude oscillations around the minimum of the potential ∆ 0 = ±µ/ √ we have: where ω is the oscillator's frequency, φ is its phase, and A(t) is the amplitude, which in general may depend on time [7,12].…”

Generation and decay of Higgs mode in a strongly interacting Fermi gas

“…The aim of this section is to demonstrate that the interaction quench method induces the Higgs mode, having the well known properties. The mode is typically regarded as as oscillating system in the effective potential in the form of a Mexican hat: The solutions for a classical particle moving in such potential are well known 1 , 60 . In the case of small amplitude oscillations around the minimum of the potential we have: where is the oscillator’s frequency, is its phase, and A ( t ) is the amplitude, which in general may depend on time 13 , 26 .…”

Non-minimally coupled cosmological models with the Higgs-like potentials and negative cosmological constant

“…with the negative energy E = − ε b 4 /4, where = /(1 + 6ξ ), ε = ε/(1 + 6ξ ), has the form (see, for example, [48])…”