The Lorentz-invariant phase-space element for the process is
Using the three-momentum -function, we can eliminate the integral over
On the right-hand side and are no longer independent variables but are determined by the conditions
Next, convert to angular variables.
where now stands for the magnitude of the three-momentum. The energy and momentum are related by
With all these changes we arrive at the result (valid in any frame)
We now specialize to the center-of-mass (CM) frame, for which
Introduce the variable (since is only constrained to be equal to after performing the integral over the energy-conserving -function). Then
Thus the factor
and we arrive at the important result
for two-body phase space in the CM frame.