# 3 Body Decay Kinematics (Width Calc): Examples?

1. May 24, 2012

### Hepth

Does anyone know of any worked out examples for the calculation of the width of a 1->3 decay process, where ALL THREE masses are included?

I can find a LOT of examples where they let the masses go to zero, (like leptonic/semi leptonic decays, decays to pions, etc). This makes the integrals over the dalitz phase space easy.

Basically I would love an example, even if its a textbook of the full calculation of a width of a particle P decaying to particles p1, p2 ,p3, where the masses are taken into account with NO approximation.

Even something simple, like if the amplitude squared was "1". I can't find an easy way to do the integrals, and I think for this case theyll be elliptic functions, and in my case actually dilogarithms, but I'm not sure. Mathematica doesnt seem to be much help either. Most of the integrals I try just seem to be spit back out at me, even with the correct \$assumptions set.

$$d \Gamma = \frac{1}{(2 \pi)^3} \frac{M}{32} |A|^2 ds dt$$

where $M^2 s = (p_1 + p_2)^2$ and $M^2 t = (p_2 +p_3)^2$.

(I chose to use unitless variables.) I get my kinematic boundaries from : http://www.slac.stanford.edu/xorg/BFLB/draft_sections/pbf-DalitzAnalyses.pdf [Broken]

Which I believe is correct.

So basically the first integration over t for unity would be trivial, and you plug in the following limits:
$$t_{min}^{max} = \frac{1}{2 s} (-s^2+(\rho_1-\rho_2) (\rho_1+\rho_2) \left(-1+\rho_3^2\right)+s)\biggl[1+\rho_1^2+\rho_2^2+\rho_3^2\pm\sqrt{\frac{\left(s-\rho_1^2\right)^2-2 \left(s+\rho_1^2\right) \rho_2^2+\rho_2^4}{s}} \sqrt{\frac{(-1+s)^2-2 (1+s) \rho_3^2+\rho_3^4}{s}}\biggr]$$

Then you're integrating over this which becomes quite complicated.

Are there tricks that I should know, or relevant formalisms used for these calculations? Any help would be appreciated.
Thank you.

Last edited by a moderator: May 6, 2017