Three-body decay energy endpoints ($m_1\le E_1\le [M^2+m_1^2-(m_2+m_3{)}^2]/(2M)$)

Use natural units $c=1$ and assume open phase space, $M>m_1+m_2+m_3$. In the parent's rest frame, define $s_{23}=(p_2+p_3{)}^2$. Four-momentum conservation givesThe upper recoil-mass limit follows from $E_1\ge m_1$. For positive $m_2,m_3$, the lower recoil-mass limit follows in their pair rest frame and is attained when their relative momentum vanishes. If both recoil particles are massless, $s_{23}=2E_2{E}_3(1-\cos {\theta }_{23})$ reaches zero with nonzero parallel momenta. If exactly one is massless, $s_{23}$ approaches the nonzero daughter's squared mass only when the massless particle becomes soft. Thus that upper $E_1$ bound is a supremum for a genuine three-body event with all massless particles of positive energy. If $m_1=0$, the lower endpoint is likewise a soft limit. Multiplying the mass expressions for $E_1$ by $c^2$ restores energy units.