18.6 Example

We give here as an example of a simple calculation in high energy physics the computation of the Compton scattering cross-section as given in Bjorken and Drell Eqs. (7.72) through (7.74). We wish to compute the trace of

  2(  ′)2 (           ) (     ′                    ′    )
α--  k-     γ-⋅ pf-+-m    γ-⋅ e-γ ⋅-eγ ⋅-ki γ ⋅-eγ ⋅-eγ-⋅ kf
 2    k         2m            2k.pi     +     2k ′ ⋅ pi

(           ) (             ′            ′    )
  γ-⋅ pi +-m    γ ⋅-kiγ-⋅ eγ-⋅ e-+ γ-⋅ kfγ-⋅ e-γ-⋅ e
     2m             2k.pi           2k′ ⋅ pi

where ki and kf are the four-momenta of incoming and outgoing photons (with polarization vectors e and e and laboratory energies k and k respectively) and pi, pf are incident and final electron four-momenta.

Omitting therefore an overall factor   2
2m2(  ′)
  k2 we need to find one quarter of the trace of

            (     ′                   ′     )
(γ ⋅ pf + m)  γ ⋅-eγ-⋅ eγ-⋅ ki-+ γ-⋅ eγ-⋅ e-γ ⋅-kf ×
                  2k.pi           2k ′.pi

            (              ′           ′    )
(γ ⋅ pi + m ) γ-⋅ kiγ ⋅-eγ ⋅-e-+ γ-⋅ kf-γ ⋅ e-γ-⋅ e
                  2k.pi            2k′.pi

A straightforward REDUCE program for this, with appropriate substitutions (using P1 for pi, PF for pf, KI for ki and KF for kf) is

 on div; % this gives output in same form as Bjorken and Drell.  
 mass ki= 0, kf= 0, p1= m, pf= m; vector e,ep;  
 % if e is used as a vector, it loses its scalar identity  
 %      as the base of natural logarithms.  
 mshell ki,kf,p1,pf;  
 let p1.e= 0, p1.ep= 0, m^2+ki.kf, m*k,p1.kf=  
     m*kp, pf.e= -kf.e, pf.ep= ki.ep, m*kp, pf.kf=  
     m*k, ki.e= 0, ki.kf= m*(k-kp), kf.ep= 0, e.e= -1,  
 operator gp;  
 for all p let gp(p)= g(l,p)+m;  
 comment this is just to save us a lot of writing;  
 gp(pf)*(g(l,ep,e,ki)/(2*ki.p1) + g(l,e,ep,kf)/(2*kf.p1))  
   * gp(p1)*(g(l,ki,e,ep)/(2*ki.p1) + g(l,kf,ep,e)/  
 write ~The Compton cxn is ~,ws;

(We use P1 instead of PI in the above to avoid confusion with the reserved variable PI).

This program will print the following result

                         2    1      -1    1   -1  
The Compton cxn is 2*E.EP  + ---*K*KP   + ---*K  *KP - 1  
                              2            2