REDUCE

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

   (   ) (           )(                             )
α2-  k′ 2  γ-⋅pf +-m-   γ-⋅e′γ-⋅eγ-⋅ki  γ-⋅eγ-⋅e′γ-⋅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 α2
2m2-( k′)
  k-2 we need to find one quarter of the trace of

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

           (                             )
            γ-⋅kiγ ⋅eγ-⋅e′  γ ⋅kfγ-⋅e′γ ⋅e-
(γ ⋅pi + m)     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, p1.pf= m^2+ki.kf, p1.ki= m*k,p1.kf=  
     m*kp, pf.e= -kf.e, pf.ep= ki.ep, pf.ki= m*kp, pf.kf=  
     m*k, ki.e= 0, ki.kf= m*(k-kp), kf.ep= 0, e.e= -1,  
     ep.ep=-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)/  
     (2*kf.p1))$  
 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