The Lorentz-invariant phase-space element for the process is

(7.119) |

Using the three-momentum -function, we can eliminate the integral over

(7.120) |

On the right-hand side and are no longer independent variables but are determined by the conditions

(7.121) |

Next, convert to angular variables.

(7.122) |

where now stands for the magnitude of the three-momentum. The energy and momentum are related by

(7.123) |

so that

(7.124) |

With all these changes we arrive at the result (valid in any frame)

(7.125) |

We now specialize to the center-of-mass (CM) frame, for which

(7.126) | |||

(7.127) |

and

(7.128) |

Introduce the variable (since is only constrained to be equal to after performing the integral over the energy-conserving -function). Then

(7.129) |

Thus the factor

(7.130) |

becomes

(7.131) |

and we arrive at the important result

(7.132) |

for two-body phase space in the CM frame.

2004-03-18