# Multi-component scalar fields and the complete factorization of its equations of motion

## TL;DR

This paper extends the known equivalence between second-order equations of motion and first-order Bogomolnyi equations from simple scalar field theories to more complex, multi-component, and non-standard scalar field models, broadening theoretical understanding.

## Contribution

It generalizes the equivalence to a wider class of real scalar field models, including multi-component and non-standard Lagrangians, without specific functional forms.

## Key findings

- Equivalence demonstrated for single real scalar models.
- Extended to multi-component scalar field models.
- Applicable to standard and non-standard scalar field theories.

## Abstract

In the paper by Bazeia D. et al., EPL, 119 (2017) 61002, the authors demonstrate the equivalence between the second-order differential equation of motion and a family of first-order differential equations of Bogomolnyi type for the cases of single real and complex scalar field theories with non-canonical dynamics. The goal of this paper is to demonstrate that this equivalence is also valid for a more general classes of real scalar field models. We start the paper by demonstrating the equivalence in a single real scalar model. The first goal is to generalize the equivalence presented in papers by Bazeia et al. to a single real scalar field model without a specific form for its Lagrangian. The second goal is to use the setup presented in the first demonstration to show that this equivalence can be achieved also in a real multi-component scalar field model again without a specific form for its Lagrangian. The main goal of this paper is to show that this equivalence can be achieved in real scalar field scenarios that can be standard, or non-standard, with single, or multi-component, scalar fields.