Physics 564 - Introductory Particle Physics

3-body phase space

A good description of 3-body kinematics can be found in the PDG review. Integrals over 3-body phase space can have limits of integration that are difficult to manipulate analytically. However, these integrals are amenable to Monte Carlo integration methods. For a spin-0 particle decaying to a three-particle final state, the independent variables can be transformed to the invariant masses of two 2-particle sub-systems, m₁₂ and m₂₃. If all final state particles were massless, then the allowed limits of integration would be 0<m²₂₃<m²₁₂. In this case, the integral over the density of states corresponds to the area of a triangle with corners at (0,0), (0,M²) and (M²,0) in the m²₁₂-m²₂₃ plane. The integral over massive final states that are kinematically allowed can be computed numerically by computing the ratio of the area that is kinematically allowed to the area of the full triangle which can be calculated analytically. This program evaluates the area in the m²₁₂-m²₂₃ plane that is kinematically allowed for the decay D⁺-->K^-π⁺π⁺:

  ~/phys564/ $ root
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  *   Version   4.04/02        4 May 2005   *
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  *          http://root.cern.ch            *
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FreeType Engine v2.1.3 used to render TrueType fonts.
Compiled for linux with thread support.

CINT/ROOT C/C++ Interpreter version 5.15.169, Mar 14 2005
Type ? for help. Commands must be C++ statements.
Enclose multiple statements between { }.
root [0] .x PhaseSpace.C
Maximum possible area = 1.74714e+06 MeV^2
Area = 917424 +- 2758.99 MeV^2
root [1]

The unit triangle is uniformly sampled using random numbers x and y where 0<y<x, which are transoformed to the m₁₂²-m₂₃² plane. From m₁₂², the allowed kinematic limits of m₂₃² are calculated and if the randomly generated value is allowed, the point is counted. In this case, the numerical estimate for the area of the allowed kinematic region is 0.917 +/- 0.003 GeV².