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<p>A generalization of the determinant appears in particle physics in effective Lagrangian interaction terms that model the chiral anomaly in quantum chromodynamics (Giacosa et al. in Phys Rev D 97(9):091901, 2018, Phys Rev D 109(7):L071502, 2024), in particular in connection to mesons. This <i>polydeterminant function</i>, known in the mathematical literature as a mixed discriminant, associates <i>N</i> distinct <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(N\times N\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <mo>×</mo> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation> 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.</p>

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Emergence of the polydeterminant in QCD

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年10月29日

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  • Francesco Giacosa,
  • Michał Zakrzewski,
  • Shahriyar Jafarzade,
  • Robert D. Pisarski

摘要

A generalization of the determinant appears in particle physics in effective Lagrangian interaction terms that model the chiral anomaly in quantum chromodynamics (Giacosa et al. in Phys Rev D 97(9):091901, 2018, Phys Rev D 109(7):L071502, 2024), in particular in connection to mesons. This polydeterminant function, known in the mathematical literature as a mixed discriminant, associates N distinct \(N\times N\) N × 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.