(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

We are given a reaction [tex]X_{1} + X_{2} \rightarrow Y_{1} + Y_{2} + Y_{3}[/tex]. The quantities [tex]m_{Y_{1}Y_{2}}^2[/tex] and [tex]m_{Y_{2}Y_{3}}^2[/tex] are plotted in a Dalitz plot. [tex]Y_{1}[/tex] + [tex]Y_{2}[/tex] resonate at a fixed mass [tex]m_{Y_{1}Y_{2}}[/tex].

Show how this resonance leads to a mass distribution for the "wrong pairing" of particles [tex]m_{Y_{2}Y_{3}}^2[/tex] where [tex]m_{Y_{2}Y_{3}}^2 = A + Bcos\theta[/tex]. [tex]A[/tex] and [tex]B[/tex] are constants and [tex]\theta[/tex] is the angle between the outgoing [tex]Y_{3}[/tex] and either [tex]Y_{1}[/tex] or [tex]Y_{2}[/tex]. What are [tex]A[/tex] and [tex]B[/tex]?

2. Relevant equations

I'm not sure ... I think these are relevant:

In a 3-body system, we have, by definition, [tex]m_{13}^2 = (P_{1} + P_{3})^2 = (E_{1} + E_{3})^2 - (\vec{p_{1}} + \vec{p_{3}})^2 = m_{1}^2 + m_{3}^2 + 2(E_{1}E_{3} - p_{1}p_{3}cos\theta)[/tex]. So here's the [tex]\theta[/tex] between two particles.

The Dalitz plot plots [tex]m_{Y_{1}Y_{2}}^2[/tex] against [tex]m_{Y_{2}Y_{3}}^2[/tex].

Comparing the particles going in and coming out, we can find the energy release [tex]Q[/tex], which is equal to the kinetic energies of the final state particles [tex]T_{i}[/tex].

3. The attempt at a solution

I'm trying to show that [tex]m_{Y_{2}Y_{3}}^2 = A + Bcos\theta[/tex] ... but I'm not understanding exactly where these values come from. I assume I need to somehow find [tex]m_{Y_{2}Y_{3}}^2[/tex] in terms of the various variables above -- how do I do that? I'll also have [tex]\theta[/tex] in terms of the fixed masses -- and momenta and energies?

I don't think that'll give me enough to find [tex]A[/tex] and [tex]B[/tex], except in terms of so many variables that I'm sure this approach is wrong. I'm probably quite a bit off the mark, but what am I missing here?

Thanks in advance for any help.

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Dalitz plots & mass distributions

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**