Example
Date: 2020-10-07 18:10:00

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### 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

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 we need to ﬁnd one quarter of the trace of

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

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