Spring Problem Set 1

Problem 3

Input gamma matrix macros:

<< "dirac/Tracer.m"

VectorDimension[4]

Spur[1]

Define electron current tensor:

electroncurrent = 
      G[le, p2 + me U]**G[le, {mu}]**G[le, p4 + me U]**G[le, {nu}]

Trace it:

ecurrent = electroncurrent /. le -> 1

Define tensor for scalar vertex:

scurrent = S[p1 + p3, {mu}] S[p1 + p3, {nu}]

Contract it with the electron tensor and the photon propogator to get the matrix element:

msq = 
      (((ge^4) ContractEpsGamma[ecurrent scurrent])/(2 (q^4))) /. {
          me -> 0}

Define the vectors in the lab (scalar is at rest):

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

Note: kinematic condition for elastic scattering is that the following equal M^2:

Simplify[pp3 . gmunu . pp3 /. (sint^2) -> 1 - (cost^2)]

However, do not insert it!!!

list = 
      Simplify[{p1 . p1 -> pp1 . gmunu . pp1, 
          p3 . p3 -> pp3 . gmunu . pp3, 
          p1 . p2 -> pp1 . gmunu . pp2, 
          p1 . p3 -> pp1 . gmunu . pp3, 
          p1 . p4 -> pp1 . gmunu . pp4, 
          p2 . p3 -> pp2 . gmunu . pp3, 
          p2 . p4 -> pp2 . gmunu . pp4, 
          p3 . p4 -> pp3 . gmunu . pp4, 
          (1/(q^4)) -> 
            (1/((((pp2 - pp4) . gmunu . (pp2 - pp4)))^2))}]

Replace the vector dot products:

msq1 = Simplify[msq /. list /. (sint^2) -> 1 - (cost^2)]

Put matrix element into cross section equation:

dxsectdOmega = 
      Simplify[PowerExpand[
          ((((e3/(8 π M e1))))^2) msq1 /. 
            ge -> (Sqrt[4 π alpha])]]