Obtaining the Gauge Invariant Kinetic Term for a SU(n)$_U$ x SU(m)$_V$ Lagrangian

TL;DR

This paper introduces a generalized formal method to derive gauge-invariant kinetic terms for Lagrangians with $SU(n)_U imes SU(m)_V$ symmetry, exemplified by the Standard Model, aiding future studies of complex gauge symmetries.

Contribution

It presents a new formal approach to obtain gauge-invariant kinetic terms for combined $SU(n)_U imes SU(m)_V$ gauge symmetries, extending standard methods.

Findings

Successfully reproduces the Standard Model Lagrangian using the formalism.

Provides a framework for studying generalized gauge transformations.

Facilitates the analysis of complex gauge symmetry structures.

Abstract

We propose a generalized way to formally obtain the gauge invariance of the kinetic part of a field Lagrangian over which a gauge transformation ruled by an $SU(n)_{U} \otimes SU(m)_{V}$ coupling symmetry is applied. As an illustrative example, we employ such a formal construction for reproducing the standard model Lagrangian. This generalized formulation is supposed to contribute for initiating the study of gauge transformation applied to generalized $SU(n)_{U} \otimes SU(m)_{V}$ symmetries as well as for complementing an introductory study of the standard model of elementary particles.