Strangeness content of the pion in the U(3) Nambu-Jona-Lasinio model

TL;DR

This paper investigates the strangeness content of the pion within the U(3) Nambu-Jona-Lasinio model, considering flavor-dependent interactions and quark mixing effects to estimate the strange quark contribution to pion properties.

Contribution

It introduces a self-consistent approach with flavor-dependent couplings in the NJL model to analyze strangeness content and meson mixing effects, providing new estimates for these contributions.

Findings

Strangeness significantly affects the pion's properties.

Mixing interactions induce $ ext{π}^0$-$ ext{η}$ and $ ext{η}$-$ ext{η'}$ mixings.

$ ext{η}$-$ ext{π}^0$ mixing indicates strangeness in the pion.

Abstract

The Nambu-Jona-Lasinio model is considered with flavor-dependent coupling constants $G_{ij} \left[ ( \bar{\psi} \lambda_i \psi ) ( \bar{\psi} \lambda_j \psi )+ ( \bar{\psi} i\gamma_5 \lambda_i \psi ) ( \bar{\psi} i \gamma_5\lambda_j \psi )\right]$ for $i,j=0,1..N_f^2-1$, and $N_f=3$. A self consistent calculation of quark effective masses and coupling constants is performed making the strange quark effective mass to vary considerably. Quantum mechanical mixings between up, down and strange constituent quarks yields a strangeness content of the light u and d quarks constituent and of the pion. Different types of estimates for the strangeness contribution for the pion mass are provided. Mixing type interactions, $G_{i\neq j}$, induce the light mesons mixings and estimates for the $\pi^0-\eta$ and $\eta-\eta'$ mixings are provided. The $\eta-\pi^0$ mixing is argued to be an indication of the strangeness content of the pion.