Hadrons in group expansion

TL;DR

This paper investigates how approximate flavor symmetries like SU(4), SU(3), and SU(2) influence the mass spectrum of ground-state baryons by analyzing their flavor representation mixtures and symmetry breaking effects.

Contribution

It systematically calculates transition matrices for flavor SU(4) representations and applies them to understand baryon mass spectra and symmetry breaking.

Findings

Baryon states are mixtures of various flavor representations.

Flavor symmetry breaking induces specific mixing rules.

Deviations from exact SU(3) decuplet are characterized by SU(3) octet.

Abstract

Various approximate symmetries exist in nature. For example, the flavor $SU(4)$ symmetry involving the $up/down/strange/charm$ quarks is severely broken, the flavor $SU(3)$ symmetry involving the $up/down/strange$ quarks is moderately broken, and the isospin $SU(2)$ symmetry involving the $up/down$ quarks is slightly broken. These broken symmetries are primarily governed by the strong interaction, making them an ideal platform for investigating the general behavior of approximate symmetries. To explore the application of the flavor $SU(4)$ group to ground-state baryons, we systematically calculate the transition matrices associated with various flavor $SU(4)$ representations as well as the matrices that describe their connections. These matrices are then employed to analyze the mass spectrum of ground-state baryons. Our results indicate that these states can be described as mixtures of various flavor representations, such as $\Sigma_c/\Xi_c^\prime/\Omega_c \sim \mathbf{20_M} \oplus \mathbf{20_S}\oplus \mathbf{\bar{4}_A}~[SU(4)]$, $\Xi_c/\Xi_c^\prime \sim \mathbf{\bar 3_A} \oplus \mathbf{6_S}~[SU(3)]$, $\Lambda^0/\Sigma^0 \sim \mathbf{1_A} \oplus \mathbf{3_S}~[SU(2)]$, where the subscripts $\mathbf{S}$, $\mathbf{A}$, and $\mathbf{M}$ denote the symmetric, antisymmetric, and mixed flavor wave functions, respectively. Our results also indicate that the flavor symmetries, as they break, necessitate the mixing of these flavor representations according to specific rules. For example, the approximate $SU(3)$ flavor decuplet, with one of its flavor components slightly differing from the other two, deviates from the exact $SU(3)$ flavor decuplet, and this deviation is characterized by the exact $SU(3)$ flavor octet.