Complementary invariant masses in a sequential decay ($m_{At}^2+m_{st}^2=m_C^2+m_A^2-m_B^2$)

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.