Nonrelativistic Quark Model for Baryons

A nonrelativistic quark model is proposed for baryons, according to which any two quarks are assumed to interact with each other through $p$-wave forces. Such forces are shown to be capable of producing strong binding in a three-quark system in a spatially antisymmetric state of angular-momentum unity, and making the model compatible with an extension of the [56] representation of ${\mathrm{SU}}_6$. If the strength of the quark-quark force is adjusted to fit some central baryon mass ($m_0$), the model predicts a 2-quark bound state at a mass $\sim \frac{1}{2}(M+m_0)$, where $M$ is the central mass of a quark. The validity of the nonrelativistic description is shown to depend on the smallness of a certain "inverse range parameter" $\beta$ compared with the quark mass $M$, and this condition is shown to be fully compatible with the present experimental knowledge on baryon sizes, as measured by the charge radius of the proton. Further, using an ${\mathrm{SU}}_3$-invariant interaction, an "equal interval rule" for the baryon masses is shown to follow dynamically from the assumption of a mass difference between the singlet and doublet quarks, under the same condition, $\beta \ll M$, as above. It is argued that a $p$-wave quark interaction, which leads more easily to the formation of antisymmetric spatial states than of symmetric ones, gives a sort of "saturated system" at the 3-quark level. This reduces considerably the (undesirable) prospects of very strong binding of a larger number of quarks, compared to the situation with $s$-wave forces (which facilitate the formation of symmetric states in multiquark systems with stronger and stronger binding as the number of quarks is increased). By ruling out the generally stronger $s$-wave forces as the main bond between two quarks, the model leaves scope for their action in quark-antiquark systems, which should require stronger binding in order to generate the (less massive) mesons.