Massive three particle phase space

[melli1992]

If you produce three massive particles with m1=/=m2=/=m3 near threshold (beta -> 0), the cross section of the production is supressed by a factor beta^4, where beta = sqrt(1-(M_tot)^2/s) and s is COM energy. I have been trying to prove this statement, but I can't seem to manage. Could anybody help me?

 

[mfb]

Did you set up the integrals for the phase space factor and then see where an approximation $M = s(1-\epsilon)$ leads? $\beta^4 = \epsilon^2$ here.

 

[melli1992]

Did you set up the integrals for the phase space factor and then see where an approximation $M = s(1-\epsilon)$ leads? $\beta^4 = \epsilon^2$ here.

Yes I have, but my problem is that I have an integral over the square root of a polynomial of degree 3. I don't see it reducing to beta^4 that easily...

 

[mfb]

Can you post the integral you get?

 

[melli1992]

Yes:
$$\int {\rm d}s_{23} \sqrt{(s^2 + m_1^4+s_{23}^2-2ss_{23}-2m_1^2s_{23}-2sm_1^2)(s_{23}-m_2^2)}$$
We have $(m_2+m_3)^2 \leq s_{23} \leq s-m_1^2$, $s=(p_1+p_2+p_3)^2$ and $s_{23} = (p_2 + p_3)^2$. To get to this form, we have assumed that the combined state $p_{23}$ is at rest.

 

[mfb]

If p₂₃ is at rest, then particle 3 is also at rest and s₂₃ is fixed, there would be nothing to integrate over.