Cascade-decay invariant-mass endpoint ($(m_{st}^2{)}_{\max }=(m_C^2-m_B^2)(m_B^2-m_A^2)/m_B^2$)

For sequential on-shell two-body decays $C\to B+t$ and $B\to A+s$, with massless $s,t$, evaluate their energies in the B rest frame. Their pair mass is $2E_s{E}_t(1-\cos \theta )$, so its squared invariant mass ranges from zero to the displayed endpoint. The mass ordering is $m_C>m_B>m_A>0$. spin correlations can alter the distribution inside these limits, but cannot move the kinematic endpoints while the same on-shell decay chain applies.

For the sequential decay defining the cascade-decay invariant-mass endpoint, momentum conservation gives the displayed eventwise identity. Thus $m_{At}^2$ has minimum $m_A^2{m}_C^2/m_B^2$ and maximum $m_A^2+m_C^2-m_B^2$, attained at the opposite relative angles from the $s,t$ pair endpoints. The final A particle is massive, so its lower pair-mass bound need not vanish.