## Physics 565 - *Introduction to Elementary Particle Physics II*

### Using Mathematica to simplify gamma matrix algebra

The package `Tracer.m` can be downloaded by following the link. The documentation,
`tracer.ps` can also be downloaded.
On the amdahl cluster, you can also copy a version of `Tracer.m` that
has some minor fixes from `/home/jones105/phys565/Tracer.m` to your
own directory.
Mathematica can be invoked on the amdahl
cluster as follows:

jones105@amdahl:~$ math
Mathematica 5.0 for Linux
Copyright 1988-2003 Wolfram Research, Inc.
-- Motif graphics initialized --
In[1]:=

however, because there is a limit of two users at a time, I recommend preparing
your Mathematica input as a script and invoking Mathematica as in the following
examples. First, copy the example `/home/jones105/phys565/muondecay.m`
to your own directory. If you print it to the screen it will look like this:
jones105@amdahl:~/phys565$ cat muondecay.m
<<Tracer.m (* Load the Tracer package *)
VectorDimension[4] (* We only need 4-dimensional gamma matrices *)
AntiCommute[on] (* In 4-d, gamma5 will anti-commute *)
Spur[l] (* Evaluate traces labeled by 'l' *)
lmuon = GammaTrace[l,p3,{mu},p1,{nu},(U - G5)] (* Muon trace tensor *)
lelectron = 2 GammaTrace[l,p4,{mu},p2,{nu},(U-G5)] (* Electron trace tensor *)
m = G^2/2 lmuon lelectron (* Spin averaged squared invariant amplitude *)
Simplify[m] (* Simplify the result *)
%/.{p1.p2->s/2,p3.p4->s/2} (* Substitute in the previous result *)
%/.{s->mass^2} (* more substitutions *)
jones105@amdahl:~/phys565$

and if you invoke it thus, you will see the result at the end of lots of
output:
jones105@amdahl:~/phys565$ math < muondecay.m
Mathematica 5.0 for Linux
Copyright 1988-2003 Wolfram Research, Inc.
-- Motif graphics initialized --
In[1]:=
Remove::rmnsm: There are no symbols matching "Tracer`Private`*".
T R A C E R
=============
A MATHEMATICA PACKAGE FOR GAMMA-ALGEBRA IN ARBITRARY DIMENSIONS
by M. Jamin and M.E. Lautenbacher
Physics Dept. T31, Technical University Munich
Version 1.1.1 from Mon Dec 30 15:36:00 MET 1991
(based on MATHEMATICA Version 1.2)
(lots of other output not shown here...)
In[8]:=
2
Out[8]= 64 G p1.p2 p3.p4
In[9]:=
2 2
Out[9]= 16 G s
In[10]:=
2 4
Out[10]= 16 G mass
In[11]:=
jones105@amdahl:~/phys565$

which has evaluated the spin averaged matrix element:
The Tracer documentation provides more explanation of how to represent
4-vectors, gamma matrices and traces in a form that it can work with.
Please send me mail if you
do not understand how to translate a particular expression into a form
that Tracer can understand.
Have fun!