3 Body Decay Kinematics (Width Calc): Examples?

[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 doesn't 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

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.

 

[AndyW]

Thank you for your question. The calculation of the width of a 1->3 decay process with all three masses included can be quite complex and there are no simple tricks or shortcuts that can be used. It requires a detailed understanding of the underlying theory and the appropriate mathematical formalisms.

One approach that can be used is to use the Feynman diagrams and Feynman rules to calculate the amplitude for the decay process. This involves writing down all possible diagrams that contribute to the process and then using the appropriate rules to calculate the amplitude. The amplitude can then be squared and integrated over the phase space to obtain the decay width.

Another approach is to use the relativistic Breit-Wigner formula, which takes into account the masses of all particles involved in the decay process. This formula takes into account the resonant nature of the decay and can be used to calculate the width of the process.

In both cases, the integrals over the phase space can become quite complicated and may require the use of numerical methods or computer programs such as Mathematica to obtain solutions. It is also important to ensure that the kinematic boundaries are correctly chosen, as you have mentioned in your post.

I recommend consulting textbooks or research papers on the specific decay process you are interested in, as they may provide worked out examples or useful techniques for calculating the width. I also suggest collaborating with colleagues or seeking help from experts in the field who may have experience with similar calculations.

I hope this helps and good luck with your calculations!