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Dalitz plots & mass distributions

  • Thread starter JonReeDO
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1. Homework Statement

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. Homework 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.
 

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