$X$ Meson aka $\eta'$ and Kobayashi-Maskawa-'t Hooft Six-quark Vertex -- $U(1)_A$ Anomaly and Generalized Nambu-Jona-Lasinio Model --

TL;DR

This paper discusses the Kobayashi-Maskawa-'t Hooft six-quark vertex's role in explaining the $ ext{η'}$ meson mass, its symmetry properties, derivation, and applications in hadron physics and QCD phase transitions.

Contribution

It provides a comprehensive account of the KMT vertex's theoretical background, derivation, symmetry breaking, and recent applications in QCD-related phenomena.

Findings

The KMT vertex explains the large $ ext{η'}$ meson mass.

It explicitly breaks $U(1)_A$ symmetry in QCD.

The vertex is crucial in modeling QCD phase transitions.

Abstract

In 1970, Kobayashi and Maskawa concluded that an effective six-quark vertex with a determinantal form is necessary in the chiral effective models to account for the large mass of $X$ meson, which is now called the $\eta'$. The determinantal interaction has an $SU(3)_L \otimes SU(3)_R$ symmetry but not $U(3)_L \otimes U(3)_R$, and, hence accounts for the explicit breaking of $U(1)_A$ symmetry in quantum chromodynamics (QCD); the vertex was later derived by 't Hooft as an instanton-induced quark interaction. The vertex, which may be called the Kobayashi-Maskawa-'t Hooft (KMT) term, is widely used in quantitative analyses of hadron physics and QCD phase transitions at finite temperature and density. An account is made for the KMT term with recent extensive applications. Described are also personal experiences with Professor Maskawa and Professor Kobayashi,including an encounter with Professor Maskawa when the author first presented his work on the KMT term.