Relativistic three-body decay

A particle decay into three daughter particles subject to four-momentum conservation and the relativistic energy-momentum relation. The daughter energies generally vary continuously even in the parent's rest frame. Grouping two daughters into a recoil system of variable invariant mass turns the remaining kinematics into a relativistic two-body decay.

A boundary of phase space approached as a massless outgoing particle's energy tends to zero. A zero four-vector is not a physical positive-energy one-particle state. An energy bound requiring such a particle is a supremum or infimum, even when the closed kinematic interval is conventionally described using minimum and maximum. This distinction can matter when applying three-body formulas at mixed massive-massless recoil thresholds.

Three-body decay energy endpoints ( $m_{1} \leq E_{1} \leq [M^{2} + m_{1}^{2} - (m_{2} + m_{3})^{2}] / (2 M)$ )

 0  0

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 gives

E_{1} = \frac{M ^{2} + m _{1}^{2} - s _{23}}{2 M}, (m_{2} + m_{3})^{2} \leq s_{23} \leq (M - m_{1})^{2} .

(1)

The upper recoil-mass limit follows from

E_{1} \geq 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,