Emergence of the polydeterminant in QCD

TL;DR

This work formalizes the polydeterminant, or mixed discriminant, as a determinant-like invariant $\epsilon(A_1,\dots,A_N)$ for $N$ distinct $N\times N$ matrices, and demonstrates its relevance to the chiral anomaly in QCD. It presents the key algebraic properties (symmetry, $N$-linearity, reduction to the determinant, and invariance under similarity) and explicit expressions for low-dimensional cases, enabling practical use in effective meson theories. The authors embed the polydeterminant into QCD-inspired Lagrangians, showing how terms like $\epsilon(A_1,A_1,A_2)$ and $\epsilon(A_1,A_2,A_2)$ induce anomalous interactions and mixing among meson multiplets, alongside a concrete example Lagrangian $\mathcal{L}=c_1\det A_1+c_2\det A_2+c_3\epsilon(A_1,A_1,A_2)+c_4\epsilon(A_1,A_2,A_2)+h.c.$; they also discuss condensate effects and implications for the $\eta$ and $\eta'$ system, and extend the construction to tensor fields. The results provide a systematic framework for generating chirally anomalous terms in QCD and potentially in beyond-Standard-Model theories, with a clear path to tensor generalizations and broader applications in meson phenomenology.

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.

Figures (1)

• Figure 1: Feynman diagrams of Eq. \ref{['lageff']}. The two upper ones arise from the usual determinant. The two lower ones arise from GPKJ interaction Lagrangians that involve the polydeterminant. Because of that, they involve $\phi_1$ and $\phi_2$ fields: the $c_3$-terms is of the type $\phi_1 \phi_1 \phi_2$ and the $c_4$-terms of the type $\phi_1 \phi_2 \phi_2$.