Spin and Parity Analysis at All Production Angles
Murray Peshkin · Physical Review · 1964
Bohr's symmetry method is applied to an unstable spin-$j$ state $X$, which is produced in a reaction $A+B\ensuremath{\rightarrow}C+X$ and then decays according to $X\ensuremath{\rightarrow}D+E$. Particles $A$, $B$, $C$, $D$ are assumed to be spinless, and $E$ is either a spinless particle or a gamma ray. Parity is conserved in production, but not necessarily in decay. The angular distribution of $E$, in the rest system of $X$, is $I(\ensuremath{\theta})=\frac{1}{2}\ensuremath{\Sigma}{a}_{L}{P}_{L}(cos\ensuremath{\theta})$, where $L<~2j$ and the polar angle $\ensuremath{\theta}$ is measured from the normal to the production plane. The coefficients ${a}_{L}$ depend upon the production angle $\ensuremath{\delta}$ and upon the dynamics of the production. It is proved here that the sign of the maximum-complexity coefficient ${a}_{2j}$ depends only upon the parity of $X$, and that the magnitude of ${a}_{2j}$ is not zero but lies between bounds which depend upon $j$ and the parity alone. The implied test for $j$ and the parity has the following advantages: (1) The spin $j$ is equal to half the largest $L$ in $I(\ensuremath{\theta})$. Addition of a small amount of a higher ${P}_{L}$, which always improves the fit, is forbidden by the lower bound of ${a}_{2j}$. (2) The bounds of ${a}_{2j}$ are independent of $\ensuremath{\delta}$. Any (perhaps biased) average over $\ensuremath{\delta}$ may be performed before expanding $I(\ensuremath{\theta})$ in the ${P}_{L}$. (3) All the data are condensed into a single test quantity ${a}_{2j}$, whose statistical error is reliably known.