# A $\phi^6$ soliton with a long-range tail

## TL;DR

This paper introduces an analytically solvable sextic potential model with non-parity symmetric solitons exhibiting unique long-range tail behavior, and analyzes their spectral properties with potential for further theoretical exploration.

## Contribution

The paper presents a new solvable $^6$ model with explicit Lambert W function solutions, revealing long-range tails and spectral characteristics, expanding understanding of non-parity symmetric solitons.

## Key findings

- Solvable model with Lambert W function solutions
- Solitons with power-law and exponential asymptotics
- Analytical spectrum of boson and fermion fields

## Abstract

We propose an analytically solvable sextic potential model with non-trivial soliton solutions connecting the trivial vacua. The model does not respect parity symmetry, and like $\phi^4$ theory has two minima. The soliton solutions and the consequent results are obtained in terms of the Lambert W function, i.e., the inverse function of $f(W) = We^W$. They have power-law asymptotics at one spatial infinity and exponential asymptotics at the other. We compare the solution with the kink of $\phi^4$ theory, which preserves the parity symmetry and has exponential asymptotics at both spatial infinities. Moreover, we study the full spectrum (bound and continuum states) of boson and fermion fields in the presence of the proposed soliton. We consider two types of coupling for the boson-soliton interaction and Yukawa coupling for the fermion-soliton interaction. Most results are derived analytically. This property renders the model a fertile ground for further study, including parity breaking related phenomena and long-range soliton-soliton interactions.