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

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 $i,j=0,\mathrm{1..}{N}_f^2-1$, 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, $G_{i\ne j}$, induce the light mesons mixings and estimates for the ${\pi }^0-\eta$ and $\eta -{\eta }^{\prime }$ mixings are provided. The $\eta -{\pi }^0$ mixing is argued to be an indication of the strangeness content of the pion.

Comments: 30 pages, 8 figures, new comments, discussions and estimates for the strangeness content of the pion, including an estimate based on the pi-eta mixing. Few misprints/mistakes corrected