Perturbations against a Q-ball: Charge, energy, and additivity property

TL;DR

This paper investigates how nonlinear corrections to perturbations around Q-balls restore charge and energy conservation and demonstrate the additivity of these quantities for different modes.

Contribution

It introduces a method to account for nonlinear effects in perturbations, ensuring conservation laws and additivity without solving nonlinear equations explicitly.

Findings

Linear perturbation analysis yields non-conservation of charge and energy.

Nonlinear corrections restore conservation laws.

Additivity of charge and energy for multiple modes is confirmed.

Abstract

In the present paper, perturbations against a Q-ball solution are considered. It is shown that if we calculate the U(1) charge and the energy of the modes, which are solutions to linearized equations of motion, up to the second order in perturbations, we will get incorrect results. In particular, for the time-dependent modes we will obtain nonzero terms, which explicitly depend on time, indicating the nonconservation over time of the charge and the energy. It is shown that, as expected, this problem can be resolved by considering nonlinear equations of motion for the perturbations, providing second-order corrections to the solutions of linearized equations of motion. It turns out that contributions of these corrections to the charge and the energy can be taken into account without solving explicitly the nonlinear equations of motion for the perturbations. It is also shown that the use of such nonlinear equations not only recovers the conservation over time of the charge and the energy but also results in the additivity of the charge and the energy of different modes forming the perturbation.