# 3 body relativistic phase space

1. Sep 16, 2009

### Final

Hi,
Do you know if there is an explicit formula for the integrated 3 body relativistic phase space of 3 particle with the same mass? I.e. M->3m
Or an approximate one?
Thank you!

2. Sep 17, 2009

### Bob S

3. Sep 17, 2009

### Final

There is only the differential cross section... I need the total cross section i.e. the integral of your formula. In the case of 3 massive body I think I can't integrate it. I did only the numerical integration...

4. Sep 17, 2009

### Bob S

In the case of a K+ --> pi+ pi+ pi-, I think the total probability of the decay is the probability of decaying between point 1 and point 2, times the branching ratio for this decay mode (~21.5%).

5. Sep 21, 2009

### Final

I didn't understand what is point 1 and point 2...
Anyhow I found the formula for the integrated 3 body phase space:
$$\Phi= \frac{1}{\pi^3 2^7 M^2} \int_{s_2}^{s_3}{\frac{d s}{s} \sqrt{(s-s_1)(s-s_2)(s_3-s)(s_4-s)}},$$
Where M is the mass of the initial state and $$s_1=(m_1-m_2)^2, \quad s_2=(m_1+m_2)^2, \quad s_3=(M-m_3)^2, \quad s_1=(M+m_3)^2$$.
I think you can do this integral only numerical...

6. Sep 24, 2009

### sri

Last edited by a moderator: May 4, 2017
7. Sep 25, 2009

### Final

thanks a lot! ...

8. Feb 2, 2012

### bustywild

Help! Four body relativistic phase space

Hello everybody!

I urgently need the value of the integrated four body phase space, whereby the four outgoing particles all have equal mass m.
So I need the analogon of the formula for Phi which "Final" posted, but this time for four outgoing particles...

Thanks a lot,

Basti