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If I have a positron striking an electron at rest to create 2 pions(+ and -) and I want to calculate the minimum kinnetic energy that the electrons can possess to create these pions, then the created pions will be at rest correct?

This gives me two four vectors: $$[E_{e^+} + E_{e^-}, P_{e^+}C , 0 , 0]$$ the other one being $$[E_{\pi^+} +E_{\pi^-}, 0 , 0 , 0],$$ now what really confuses me is:

  1. Momentum isn't conserved? We're still in the same frame of reference ie lab frame, shouldn't momentum be conserved?

  2. Is energy conserved? If so when I equate the energies and equate the four vector length (invariant mass) I get different answers?

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  • $\begingroup$ The entire 4-momentum should be conserved, magnitude and direction. That means both energy and 3-momentum should be conserved. You're correct that if you want to know the minimum KE to create 2 pions, then after the collision the pions will have to be at rest (3-momentum is zero), so what should the 3-momentum of the electron-positron pair be to be consistent with that? $\endgroup$ – Kevin Driscoll Aug 17 '15 at 20:32
  • $\begingroup$ Also, I don't quite understand your first 4-momentum. What is $C$? Where did the momentum of the positron go? $\endgroup$ – Kevin Driscoll Aug 17 '15 at 20:36
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    $\begingroup$ If the initial electron is at rest, then the pions are not at rest after. The pions will be at rest in the centre of mass frame. $\endgroup$ – Ihle Aug 17 '15 at 20:40
  • $\begingroup$ okay, that makes more sense Ihle, $\endgroup$ – Adam Dong Aug 17 '15 at 20:44
  • $\begingroup$ Kevin, I think I was wrong in saying that the pions are at rest. Also it the C should be c ie speed of light $\endgroup$ – Adam Dong Aug 17 '15 at 20:45
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then the created pions will be at rest correct?

Well, they will be at rest in the Center of Momentum frame. But that is not the frame of reference that your problem is stated in.

Momentum is conserved, which tells you that you have written the pion four-vectors incorrectly.

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