## Physics 564 - Introduction to Elementary Particle Physics

### 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/phys564/Tracer.m to your own directory.

Mathematica can be invoked on the amdahl cluster as follows:

```jones105@amdahl:~\$ math

Mathematica 5.0 for Linux
-- 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/phys564/muondecay.m to your own directory. If you print it to the screen it will look like this:
```jones105@amdahl:~/phys564\$ 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:~/phys564\$
```
and if you invoke it thus, you will see the result at the end of lots of output:
```jones105@amdahl:~/phys564\$ math < muondecay.m

Mathematica 5.0 for Linux
-- 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:~/phys564\$
```
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!