Emergence of the polydeterminant in QCD

TL;DR

This paper investigates the mathematical properties and physical applications of the polydeterminant, a generalization of the determinant, in quantum chromodynamics and meson physics, introducing new Lagrangian terms and tensor extensions.

Contribution

It introduces the polydeterminant as a novel tool in high-energy physics, linking mathematical properties to physical models and extending its application to tensor functions.

Findings

Characterization of the polydeterminant's properties in quantum fields

Development of new chiral anomalous Lagrangian terms using the polydeterminant

Extension of the polydeterminant concept to tensor functions

Abstract

A generalization of the determinant appears in particle physics in effective Lagrangian interaction terms that model the chiral anomaly in Quantum Chromodynamics (PRD 97 (2018) 9, 091901 PRD 109 (2024) 7, L071502), in particular in connection to mesons. This \textit{polydeterminant function}, known in the mathematical literature as a mixed discriminant, associates $N$ distinct $N\times N$ complex matrices into a complex number and reduces to the usual determinant when all matrices are taken as equal. Here, we explore the main properties of the polydeterminant applied to (quantum) fields by using a formalism and a language close to high-energy physics approaches. We discuss its use as a tool to write down novel chiral anomalous Lagrangian terms and present an explicit illustrative model for mesons. Finally, the extension of the polydeterminant as a function of tensors is shown.