Basics

Phase spacing

  1. One- and two-particle phase space
  2. Three and more particles

    THREE PARTICLES

    Consider a process, where a particle with momentum P and energy E decays into three particles, with momenta
    and introduce energy fractions xi defined through
    such that
    Neglecting masses of decay products, scalar products of four momenta can be expressed through the xi as
    Then the three-particle phase space can be transformed like
    where the last transformation is along the lines of what has been done for the two-particle phase space. Here, the factor 4π results from the integral over the solid angle of the direction of particle 1, yielding the reference axis for the relative angle θ, and the factor 2π is for the azimuthal integral over the direction of particle 2. The integration over the angle θ can be carried out using
    which results in
    Therefore,
    For massless particles, the x-integrals are nested, yielding


    N PARTICLES

    The reasoning above can be repeated for n massless or massive particles. An elegant way called RAMBO to calculate the n particle phase space for massless particles lends itself to an efficient sampling method. It is presented in more detail here.
    However, in the same reference a very short derivation of the n particle phase space volume is presented. In the c.m. frame of the particles, their total energy E is the only dimensionful parameter. Contrasting this with the number of dimensions in the n particle phase space
    it is clear that the phase space can be written as
    where the respective a's are dimensionless numbers. They can be constructed recursively by relating the a's for n=m and n=m-1. This can be done by first integrating over n-1 momenta. After introducing another intelligent factor of one one has
    The remaining two integrals over pn and q are nothing but a two-body phase space, with w playibng the role of a mass. Hence,
    yielding the desired recursion relation. Plugging in the well-known result for n=2 as a starting condition finally yields
    for the phase space volume of n massless particles with total c.m. energy E.

  3. MC Integration: Sampling
  4. Smarter sampling methods
  5. Selection according to a distribution: Unweighting
  6. Unweighting: Hit-or-miss
  7. Problems