 Three-body phase space with one massive daughter

ID: three-body-phase-space-with-one-massive-daughter

For one daughter of mass

m

and two massless daughters, set

r = m^{2} / M^{2}

, where

M

is the parent mass. In natural units, integrating over all directions and all kinematically allowed final momenta gives

Φ_{3} = \frac{1}{128 π ^{3} M ^{2}} \int_{m^{2}}^{M^{2}} \frac{( M ^{2} - s ) ( s - m ^{2} )}{s} d s = \frac{M ^{2}}{256 π ^{3}} (1 - r^{2} + 2 r ln r) .

This follows by splitting the final state into a massless spectator and a two-particle cluster of invariant squared mass

s

, and using

d Φ_{3} = d Φ_{2} (M; s, 0) d Φ_{2} (s; m, 0) d s / (2 π)

. The expression counts phase space alone; a physical decay rate must weight it with the squared decay amplitude. Near threshold the dimensionless factor vanishes as

(1 - r)^{3} /3

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