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ps10soln

ps10soln - PHYS651 Problem Set 10 Solutions 1 See P.S...

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PHYS651: Problem Set 10 Solutions 1 . See P.S. section 5.1 2 . See P.S page 161-163 3 . Without realising, I answered the first part of this question in the previous solutions and I refer you to them. For the second part, we need to calculate the following diagram. e- mu- p1p p2p p2 p1 q This is 1 4 spins | M 2 | = e 4 4 q 4 Tr [( p/ 1 + m e ) γ μ ( p/ 1 + m e ) γ ν ] Tr [( p/ 2 + m μ ) γ μ ( p/ 2 + m μ ) γ ν ] Performing the trace over the dirac matrices = 8 e 4 q 2 [( p 2 · p 1 )( p 2 · p 1 ) + ( p 2 · p 1 )( p 2 · p 1 ) - m 2 μ (( p 1 · p 1 ) - m 2 e ( p 2 · p 2 ) + 2 m 2 e m 2 μ ] Now we can evaluate in the μ rest frame. Futhermore we are interested in the limit where m μ → ∞ . In this frame, we have p 2 = ( m μ , 0) p 2 = ( m μ , 0) p 1 = ( E, p ˆ z ) p 1 = ( E, p sin( θ ) , 0 , p cos( θ ))

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So I skip the algebra, but you just plug in those momenta inside the amplitude and get: 1 4 spins | M 2 | = 16 π 2 α 2 m 2 2 p 4 sin 4 ( θ/ 2) m 2 μ [1 + p 2 m 2 e cos 2 ( θ/ 2)] The cross-section for this type of process as been calculated in a previous homework: d Ω = 1 64 π 2 m 2 μ 1 4 spins | M 2 | Now we some last straighforward algebra, it is easy to see that this leads to d Ω = α 2 m 2 e 4 p 2 β 2 sin 4 ( θ/ 2) (1 - β 2 sin 2 ( θ/ 2)) 4 . We want to calculate the amplitude for a positronium to decay into 2 photons in the extreme non-relativistic limit.
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ps10soln - PHYS651 Problem Set 10 Solutions 1 See P.S...

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