Example

REDUCE

22.6 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 ⋅kf-γ ⋅e-γ ⋅e 2m 2k.pi 2k′ ⋅pi

where k_i and k_f are the four-momenta of incoming and outgoing photons (with polarization vectors e and e^′ and laboratory energies k and k^′ respectively) and p_i, p_f 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.pi + 2k′.pi ×

( 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 p_i, pf for p_f, ki for k_i and kf for k_f) 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

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