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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 $\mathrm{ϵ}$.
 • As a package, FeynmanIntegral includes commands for performing the relevant steps of that computation; i.e.:
 – parametrizing Feynman integrals using Feynman or $\mathrm{\alpha }$ parameters
 – exact computing integrals over loop momenta using dimensional regularization, optionally expanding the result in $\mathrm{ϵ}$, the dimensional parameter.
 – expressing tensor integrals in a basis of scalar integrals.
 • The FeynmanIntegral package contains the following commands:

 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 $d=4-2\mathrm{ϵ}$ keeping terms up to $\mathrm{O}\left(\mathrm{ϵ}\right)$.
 • ExpandDimension expands the d-dimensional result returned by Evaluate around $d=4-2\mathrm{ϵ}$ keeping terms up to $\mathrm{O}\left(\mathrm{ϵ}\right)$.
 • 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 $\mathrm{O}\left(n\right)$ 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 ${g}_{\mathrm{\mu },\mathrm{\nu }}$.
 • 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 ${p}^{0}$ plane.
 • FeynmanIntegral:-varepsilon represents the dimensional parameter used in the dimension regularization approach.

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.