Faddeev description of baryons in two-dimensional QCD with $N_c=3$. I. Chiral spectrum, isospin, and strangeness

Abstract

We develop a light-front Faddeev description of baryons in two-dimensional QCD at $N_c=3$, in which an exact coupled-channel representation identifies the quark--diquark picture as a single-channel truncation, and solve it variationally with exact matrix elements. In the chiral limit we find an exactly massless lightest baryon, in agreement with the classic valence solutions, DLCQ, and bosonization; the excitation tower reproduces Webber's spectrum state by state and sets the valence gap $1.371\,M^{\ast}_{\rm mes}$. The channel content distinguishes these states: the massless baryon is single-channel, the gapped states are genuinely multi-channel, and all baryon Regge trajectories are of the meson class. The calculation is anchored at both ends of the coupling range: the meson-to-baryon mass ratio agrees with strong-coupling bosonization to $0.3$--$4.3\%$, while the valence masses reproduce the full discretized light cone quantization (DLCQ) masses to $0.04\%$ (meson) and $0.15\%$ (baryon) at weak coupling. Extending to two degenerate flavors, we find the massless baryon to be the $\Delta$-like quartet, while the $N$-like doublet at $0.977\,M^{\ast}_{\rm mes}$ is an exact $50\!:\!50$ superposition of good- and bad-diquark channels. Adding a massive strange quark, we obtain, to our knowledge for the first time at the valence three-quark level, the hyperon spectroscopy of QCD$_2$: $\Sigma$-like states fall below $\Lambda$-like ones, inverting the four-dimensional ordering, the chiral thresholds count two kaon units per strange quark, and the Gell-Mann--Okubo relation is protected in mass squared. With all three quark masses non-degenerate, the proton-like state falls below the neutron-like one, $\Sigma^0$--$\Lambda$ mixing appears, and the Coleman--Glashow relation holds at the percent level.