Spring Problem Set 1

Problem 2

Input Tracer macros package

<< "dirac/Tracer.m"

Set vector dimension to 4

VectorDimension[4]

Contract (p1+p3).(p2+p4) to get matrix element:

mm = 
      (((ge^2) ContractEpsGamma[S[p1 + p3, {mu}] S[p2 + p4, {mu}]])/(q^2))

Define four vectors.  P2 is at rest in the lab.   E1>>m1, E3>>m3.

pp1 = {e1, 0, 0, e1}pp3 = {e3, e3 sint, 0, e3 cost}pp2 = {M, 0, 0, 0}pp4 = {M + e1 - e3, -e3 sint, 0, e1 - e3 cost}gmunu = {{1, 0, 0, 0}, {0, -1, 0, 0}, {0, 0, -1, 0}, {0, 0, 0, 
          -1}}

list = 
      Simplify[{p1 . p2 -> pp1 . gmunu . pp2, 
          p1 . p4 -> pp1 . gmunu . pp4, 
          p3 . p2 -> pp3 . gmunu . pp2, 
          p3 . p4 -> pp3 . gmunu . pp4, 
          (1/(q^2)) -> (1/((pp1 - pp3))) . gmunu . (pp1 - pp3)}]

Substitute for vector dot products:

mm1 = Simplify[Simplify[mm /. list] /. (sint^2) -> 1 - (cost^2)]

Square matrix element:

msq = Simplify[(mm1^2)]

Cross section for a massless particle1 and 3:

dxsectdOmega = 
      Simplify[PowerExpand[
          ((((e3/(8 π M e1))))^2) msq /. 
            ge -> (Sqrt[4 π alpha])]]

Rutherford scattering limit

In this limit, M-> infinity and e3->e1:

dxsectdOmega /. e3 -> e1

The (1-Cos[theta])^2 in the denominator is equal to 4 Sin[theta/2]^4.