# A general framework to diagonalize vector--scalar and axial-vector--pseudoscalar transitions in the effective meson Lagrangian

## TL;DR

This paper introduces a novel mathematical framework for efficiently diagonalizing meson mixing matrices in effective Lagrangians, revealing unexpected connections with the Hadamard product and improving upon standard methods, especially with explicit chiral symmetry breaking.

## Contribution

It presents a new diagonalization method leveraging the Hadamard product, significantly enhancing efficiency over traditional techniques in meson Lagrangian models.

## Key findings

- The new framework simplifies the diagonalization process.
- It demonstrates improved efficiency in models with explicit chiral symmetry breaking.
- Application to an $SU(3)_L\times SU(3)_R$ symmetry model illustrates its effectiveness.

## Abstract

A new mathematical framework for the diagonalization of the nondiagonal vector--scalar and axial-vector--pseudoscalar mixing in the effective meson Lagrangian is described. This procedure has unexpected connections with the Hadamard product of $n\times n$ matrices describing the couplings, masses, and fields involved. The approach is shown to be much more efficient as compared with the standard methods employed previously. The difference is especially noticeable if the chiral symmetry is broken explicitly. The paper ends with an illustrative application to the chiral model with broken $SU(3)_L\times SU(3)_R$ symmetry.