Basics
Phase spacing
- One- and two-particle phase space
- 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.
- MC Integration: Sampling
- Smarter sampling methods
- Selection according to a distribution: Unweighting
- Unweighting: Hit-or-miss
- Problems