Physics[FeynmanIntegral] - Package of commands for the evaluation of Feynman integrals
Description
FeynmanIntegral is both a command and a package of commands for the computation of Feynman integrals, i.e. the (loop) integrals that appear in quantum field theory when performing perturbative calculations with the S-matrix in momentum representation. Feynman integrals are often divergent and must be regularized to extract physically meaningful quantities.
In this context, the FeynmanIntegral command computes a Feynman integral using dimensional regularization, rewriting the integrand using tensor reduction, Feynman parameters, and expanding in the dimensional parameter ϵ.
As a package, FeynmanIntegral includes commands for performing the relevant steps of that computation; i.e.:
parametrizing Feynman integrals using Feynman or α parameters
exact computing integrals over loop momenta using dimensional regularization, optionally expanding the result in ϵ, the dimensional parameter.
expressing tensor integrals in a basis of scalar integrals.
The FeynmanIntegral package contains the following commands:
epsilon
Evaluate
ExpandDimension
FeynmanIntegral
FromAbstractRepresentation
Parametrize
Series
TensorBasis
TensorReduce
ToAbstractRepresentation
varepsilon
You can load the FeynmanIntegral package using the with command, or invoke FeynmanIntegral commands using the long form, e.g. as in FeynmanIntegral:-Parametrize.
Brief description of the commands of the FeynmanIntegral package
Evaluate evaluates the Feynman integrals of a given expression, typically the output of the FeynmanDiagrams command, by parametrizing each of those integrals, optionally returning the intermediate steps of the computation or expanding the dimension around d=4−2ϵ keeping terms up to Oϵ.
ExpandDimension expands the d-dimensional result returned by Evaluate around d=4−2ϵ keeping terms up to Oϵ.
FromAbstractRepresentation returns the standard form of Feynman integrals passed in abstract form. This command is the reverse of ToAbstractRepresentation.
Parametrize replaces the propagators within a Feynman integral by integrals on Feynman or alpha parameters.
Series expands in series, like the series command, but strictly returning results up to order On regardless of the existence of terms with negative powers.
TensorBasis given a list of external momenta and a list of spacetime indices, constructs a complete basis of tensorial structures using the external momenta and the metric gμ,ν.
TensorReduce expresses integrals with loop momenta in the numerator in terms of scalar integrals (with no loop momenta in the numerator).
ToAbstractRepresentation represents Feynman integrals in an abstract form suitable for performing the tensor reduction of tensor Feynman integrals. The abstract form is suitable for performing the tensor reduction of Feynman integrals implemented in TensorReduce.
FeynmanIntegral:-epsilon is the same as FeynmanDiagrams:-epsilon and is used to express the prescription used to integrate in the complex p0 plane.
FeynmanIntegral:-varepsilon represents the dimensional parameter used in the dimension regularization approach.
See Also
FeynmanDiagrams, FeynmanIntegral command, PhysicalConstants, Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, StandardModel, UsingPackages
References
[1] Smirnov, V.A., Feynman Integral Calculus. Springer, 2006.
[2] Weinberg, S., The Quantum Theory Of Fields. Cambridge University Press, 2005.
[3] Bogoliubov, N.N., and Shirkov, D.V. Quantum Fields. Benjamin Cummings, 1982.
Compatibility
The Physics[FeynmanIntegral][Overview] package was introduced in Maple 2021.
For more information on Maple 2021 changes, see Updates in Maple 2021.
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