Phase-Space Distributions for the4πDecay Mode of Isosinglet Bosons

Abstract

Phase-space distributions for the decay of an isosinglet resonance into four pions are developed. Such resonances have been observed in the final state of the reaction $p+\overline{p}\ensuremath{\rightarrow}2{\ensuremath{\pi}}^{+}+2{\ensuremath{\pi}}^{\ensuremath{-}}+{\ensuremath{\pi}}^{0}$, for example. The spin and parity of the resonance are of primary interest. States of four pions that observe Bose statistics and have zero isotopic spin divide into two symmetry types. In this paper, the distributions are developed phenomenologically from the simplest possible $T$ matrix for each parity and symmetry and for spins zero, one, and two. In general, eight variables characterize the four-particle decay problem. To facilitiate the experimental determination of the spin, parity, and symmetry type, partial integrations of the distributions are performed analytically. The remaining variables are the two two-particle invariant masses of the Goldhaber triangle and angles. One of the angles orients the plane of two of the pions with respect to that of the other two. The distribution in this angle alone provides a significant partial identification of the quantum numbers of the resonance. For spins one and two, two more angles which specify an external direction have been retained. In the reaction $p+\overline{p}\ensuremath{\rightarrow}2{\ensuremath{\pi}}^{+}+2{\ensuremath{\pi}}^{\ensuremath{-}}+{\ensuremath{\pi}}^{0}$, the resonance-plus-pion system has been observed to be preferentially aligned with the beam direction. If production takes place equally from the four initial spin states, and if the spin of the resonance is not zero, then conservation of angular momentum along the beam direction results in anisotropy in the decay products. When available, this anisotropy may be a sensitive test of the spin and parity of the resonance. Plots of the unpolarized distributions over the Goldhaber triangle are presented to aid further in making spin-parity assignments.