In order to improve above approximation, we proposed another method to calculate cascade decays of SUSY particles; connecting the production amplitude of and the decay amplitudes of and with taking into account helicity of each particle. Since cutting points are at and whose width is very narrow, then we can expect a good agreement with an exact method.
The method will be explained in a simple example;
assuming a particle 4 being an anti-fermion. An exact expression of this cross section is
where is helicity state of i'th particle, an amplitude of a 3-body process and a 3-body phase space. A 2-body scattering amplitude () and a 2-body decay amplitude () can be written as;
where v is a wave function of a particle 4. A four-momentum of a particle 4, , is assumed to be on-shell (), where is a mass of a particle 4. By using these expression, a 3-body cross section can be expressed as;
where is a four-momentum squared of an (internal) particle 4 (), () a 2-body scattering (decay) phase space, and the total width of particle 4. If a narrow-width approximation is applied, a numerator of a propagator will be replaced by on-shell wave functions and integration of a denominator can be performed independently as;
where () is a partial width (a branching ratio) of a decay and a total cross section of a 2-body process . A factor works as a flux factor for a decay phase-space integration. In this method, the off-diagonal part of a spin summation (for example ) is included as well as the diagonal ones. These terms will disappear in total cross section after a phase-space integration, however will remain in differential distributions.
Obtained results are listed in Table-1 and Figs.9-10. Both of a total cross sections and differential distributions show a very good agreement with an exact calculations.