Dynamical Model for Baryon Resonances

We propose a nonrelativistic three-quark ($Q$) model of baryons and their resonances with the assumptions of (i) parastatistics which, in constrast to Fermi statistics, allows totally symmetric wave functions, and (ii) operation of $Q-Q$ forces of the factorable type in $s$ and $p$ waves which facilitate an exact solution of the three-body problem. It is found that the $s$-wave force gives rise to a strong attraction in a spatially symmetric ($S$) state of $L=0\mathrm{}$, but repulsion in a mixed-symmetric ($M$) state. This provides a natural dynamical realization of the 56 representation of $\mathrm{SU}\left(6\right)\mathrm{}$ for the familiar octet and decuplet of baryons. The same $s$-wave force also predicts a set of negative-parity resonances but these are of much too high energy to be of any physical consequence. The $p$-wave force leads to a set of negative-parity $L=1\mathrm{}$ resonances via strong attraction in $M$-type spatial states of $L=1\mathrm{}$, thus making up the representation (70,3) of $\mathrm{SU}\left(6\right)\mathrm{}\times O\left(3\right)\mathrm{}$. A $p$-wave spin-orbit $Q-Q$ force splits these states in a manner which fits in rather well with Dalitz's recent analysis of experimental data for the negative-parity baryonic resonances. Finally, the $p$-wave force generates a strong attraction in a spatially antisymemtric state ($A$) of even parity and $L=1\mathrm{}$, giving rise to the representation (20,3) of $\mathrm{SU}\left(6\right)\mathrm{}\times O\left(3\right)\mathrm{}$, of which the lowest states are an octet and a singlet of $J^P={\frac{1}{2}}^{+}$, the central mass of each lying lower than the corresponding lowest mass multiplet of negative parity. This provides a natural explanation of the so-called "Roper" resonance (at 1450 MeV), and in addition, strongly predicts an even-parity singlet of a mass lower than the $Y_0^{}{}_{}{}^{*}\mathrm{}\left(1405\right)\mathrm{}$. The distinction between Fermi statistics and paraststistics is discussed in the context of the above results, and it is argued that while Fermi statistics could in principle generate negative-parity $L=1\mathrm{}$ resonances (via $M$ functions), it could not possibly account for the 56 of baryons, since with $Q-Q$ forces alone,the $A$ function of $L=0\mathrm{}$ that should go with it has a strongly repulsive kernel.