Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 3 / 33B / c / Solution

Parity conservation and the negative parity of $X$ require odd orbital angular momentum $L$. Since the two $Y$ particles are identical bosons, their spin state must then be antisymmetric, so their total spin is $S=1$. Total angular momentum zero forces $L=S=1$.

Conditioning on spin projection $m_S=1$ leaves orbital projection $m_L=-1$. The angular density is therefore proportional to $|Y_{1,-1}(\theta ,\phi ){|}^2\propto {\sin }^2\theta$. Including the solid-angle measure, the requested probability is