Spin and Parity Analysis at All Production Angles

Bohr's symmetry method is applied to an unstable spin-$j$ state $X$, which is produced in a reaction $A+B\to C+X$ and then decays according to $X\to D+E$. 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 $I\left(\theta \right)=\frac{1}{2}\Sigma a_L{P}_L\left(cos\theta \right)$, where $L<~2j$ and the polar angle $\theta$ is measured from the normal to the production plane. The coefficients $a_L$ depend upon the production angle $\delta$ 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 $I\left(\theta \right)$. 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 $\delta$. Any (perhaps biased) average over $\delta$ may be performed before expanding $I\left(\theta \right)$ in the $P_L$. (3) All the data are condensed into a single test quantity $a_{2j}$, whose statistical error is reliably known.