HW 9 problems/solutions

HW 9 problems/solutions - Homework 9 1 12/14/2005 Homework...

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Homework 9 1 12/14/2005 Homework IX Problems: IX.1 Calculate the matrix elements and the cross section for Compton scattering and find the d σ / d | LAB and tot (the so-called Klein-Nishina cross section; Z. Physik 52 (1927) 853). Hints: See e.g. Griffiths; or Hal- zen and Martin Sect. 6.14. Solution: Two Feynman graphs: With total amplitude 1 + 2: 2 (1) 3 4 ( () fi ip ii e u m µ γε −= 1 k + 2 12 3 4 22 21 ) m ie u ie u pk m εγ ε + +− 2 p 1 k + 1 m m + 2 2 (2) 1 4 ( fi u e u m 3 k 2 32 1 4 23 ) m ie u ie u m + −− 2 p 3 k 3 m m + 2 2 3 4 fi u eu m = 2 p 1 k + 1 m m + 1 + 2 p 3 k 3 m m + 2 u ⎡⎤ ⎢⎥ ⎣⎦ (I.1) In evaluating, we will use various tricks: 1. We work in the “transverse gauge”; i.e. 1 k 1 = 3 k 3 = 0. Also, in the laboratory these are transverse polarization directions and we have 1 p 2 = 3 p 2 = 0 as well. 2. Use: i p i p 2 ; thus = i k 0 and i = i 1, i p =− j p j p i p 2( ) ij pp +⋅ 3. Use: k 1 + p 2 = k 3 + p 4 , thus k 1 p 2 = k 3 p 4 , k 3 p 2 = k 1 p 4 , 1 p 4 = 1 k 3 , 3 p 4 = 3 k 1 . Applying (2.) and using ( γ p 2 + m ) u 2 = 0: 2 3 4 fi m = 1 2 p 1 k 1 2 m + + 3 2 p 3 k + 2 4 2 m ue u + = −⋅ 1 1 k 1 2 + 3 3 k 2 2 u (I.2) Averaging over initial electron polarizations, and summing over final electron polarizations (we will wait with averaging over photon polarizations till later) gives: 2 4 3 4 1 2 fi m = 1 1 k 1 2 + 3 3 k 4 2 uu ⎛⎞ ⎜⎟ ⎝⎠ 1 1 k 1 2 + 3 3 k 24 2 , 1 4 2 2 1 2 ss u k = 1 2 3 3 k + 3 2 1 43 4 1 1 k 1 2 + 3 3 k 2 , 1 4 2 1 Tr 2 u k e = 1 3 3 2 k + 3 1 4 2 p 3 m + 1 1 k 1 2 + 3 3 k 2 2 p m + (I.3) We recognize four terms in the trace evaluation: k 1 , 1 x t k 3 , 3 e p 2 e p 4 k 1 , 1 x t k 3 , 3
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This note was uploaded on 02/22/2009 for the course PHYSICS 557 taught by Professor Michaelrijssenbeek during the Fall '05 term at SUNY Stony Brook.

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HW 9 problems/solutions - Homework 9 1 12/14/2005 Homework...

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