Mathematica in Quantum Field Theory Calculations:
An Example
Abstract
The following gives a simple example of the use
of Mathematica in the calculation of the one loop "fish"
diagram in
theory. One approach of many possible ones is given. This is a potential
starting point for further investigations by the interested reader.
Introduction
There are a number of Mathematica tools available for quantum
field theory calculations. Several of these are available free of
charge from MathSource,
and the third party commercial product FeynCalc
is available from its developer. However, independent of or in conjunction
with these useful tools, Mathematica provides all of the
needed functionality for doing quantum field theory calculations
from scratch and for implementing the needed transformations and
knowledge base of your specific field.
Problem Set Up
A basic example is the use of dimensional regularization to calculate
a one loop diagram in
theory. This is the diagram.
The action for the
theory is
The feynman rules for this theory give the following expression
for this diagram .
The traditional Feynman parameter trick allows us to write the
product of the propagators in the form
(An example of a function that does this automatically is given
in the Appendix.)
When the square is completed in the denominator and the momentum
integration variable is changed via the linear transformation ,
the following integral is obtained (where the dimensional
angular integration has been performed).
The Dimensional
Integration
The radial integration gives,
Check to be sure that the requirements of the integration are
met.
The dimension of the diagram is ,
so we pull out a factor of so
as not to perform any series expansions on this term in the following.
We will return it to its rightful place at the end.
Expand the result in a series around
to extract the constant and pole parts.
Feynman Parameter Integration
We now integrate over the Feynman parameter. We include the option
to instruct the integration routine about the properties of these
parameters.
The raw result that Mathematica returns from this integration
is a bit long, so we wrap it in Short
and give it the name temporary.
Perform some simplifications using the same assumptions as in
the integration above (note that FullSimplify
accepts its assumptions as an argument rather than as an option).
Rearrange terms using pattern matching and replacement rules.
An easy way to isolate the
pole term is to expand this result in series once again. We use
Simplify while
we are at it and reintroduce the
factor.
Finally, one small rearrangement to make everything look nice.
This is the textbook result. Thus we have, in
dimensions,
APPENDIX: Feynman Parameterization Function
Here is one way to automatically generate the Feynman parameterization
of a product of scalar propagators. The function takes as its first
argument a list of denominators and their exponents. The
second argument is a list of integration parameters  one for each
denominator.
