Effect of RGUP on the nonlinear Klein-Gordon model with spontaneous symmetry breaking

TL;DR

This paper investigates how the relativistic generalized uncertainty principle (RGUP) influences the nonlinear Klein-Gordon model with spontaneous symmetry breaking, leading to new solutions and insights into quantum gravity effects on field theories.

Contribution

It introduces a generalized tanh-method to find solutions of the RGUP-modified Klein-Gordon equation, extending previous work and analyzing the energy spectrum of solitary solutions.

Findings

Derived new analytical solutions for the RGUP-modified Klein-Gordon equation.

Estimated the RGUP parameter using the Higgs boson rest energy.

Identified normalizable and bounded solitary field solutions.

Abstract

In the framework of the relativistic generalized uncertainty principle (RGUP), we study the nonlinear Klein-Gordon field with $\phi^4$ self-interaction. This generalization comes from the quantum gravity theories that predict the existence of a minimal measurable length scale. The quantum gravity effects slightly modify the momentum and change the Lagrangian density of the field. This model, specifically in the context of spontaneous symmetry breaking, is so fruitful and leads us to many interesting phenomena. Due to the complexity of the consequent equations, the model is normally studied via perturbation mechanisms. However, here, we apply a generalized $tanh$-method which supposed that the field solutions should be $sech$-functions of segments of space and time. By this extended method, we find some more general solutions with respect to our previous work \cite{Mir1}. Several solutions can be found but we select only the normalizable and bounded solitary fields. The energy spectrum of each field solution is obtained in an analytical form and discussed. The modification parameter of the corresponding RGUP is estimated by using the rest energy of the mass of the Higgs boson.