Physics[FeynmanIntegral][Parametrize] - parametrize a Feynman integral, as the ones returned by the FeynmanDiagrams command, appearing in the expansion of the Scattering matrix in momentum representation
Calling Sequence
Parametrize(inert_Feynman_integral, options)
Parameters
inert_Feynman_integral
-
the inert form of Feynman integral, that is a function whose name is %FeynmanIntegral, as the ones returned by FeynmanDiagrams when working in momentum representation.
Options
kindofparameters = ... : the kind of auxiliary parameters - Feynman (default) or α - used to parameterize Feynman integrals.
numberofpropagators = ... : the right-hand side a non-assigned name to which the number of propagators parameterized will be assigned.
parameters = ... : the right-hand side a non-assigned name to which the parameters introduced will be assigned.
quiet = ... : the right-hand side can be true or false (default), to display or not information related to matching keywords
returnintegrand = ... : the right-hand side can be true or false (default). if set to true, Parametrize will return only the integrand of the parameterized Feynman integral, omitting the integrals over the parameters. This option is frequently used together with the parameters option to also get the parameters introduced.
All the optional keywords that can have the value true on the right-hand side can be passed just as themselves, not as an equation, representing the value true. For example quiet is the same as quiet = true. Also, you don't need to use the exact spelling of any of these keywords - any unambiguous portion of them suffices, e.g. perform for performmomentumintegration.
Description
Parametrize receives a Feynman integral constructed using the inert function %FeynmanIntegral, as the ones returned by the FeynmanDiagrams command, and rewrites the integrand replacing the propagators by parameterized integrals, using Feynman (default) or α parameters. This is the first step performed by the FeynmanIntegral command towards the computation of the integral using dimensional regularization, by expanding in the dimensional parameter ϵ.
Only propagators involving a loop momentum (the integration variable of the %FeynmanIntegral) are included in the parametrization. The output is the parameterized form of the integral, or, if specified, only of the integrand.
The available parametrization schemes introduce either Feynman or alpha (also known as Schwinger) parameters. The Feynman parametrization of a product of L denominators A_l is [1]
where the ξi are the Feynman parameters, and the αi and λj are, respectively, the α parameters and the λ (possibly complex) exponents.
The alpha parameterized form is obtained by applying [1]
for each propagator involving a loop momentum.
Examples
For details about the integrands entering these Feynman integrals see FeynmanDiagrams.
See Also
Physics, Physics conventions, Physics examples, Physics Updates, Tensors - a complete guide, Mini-Course Computer Algebra for Physicists, Physics/Setup, Physics[Dgamma], Physics[FeynmanDiagrams], Physics[FeynmanIntegral][command], Physics[Psigma]
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][Parametrize] command was introduced in Maple 2021.
For more information on Maple 2021 changes, see Updates in Maple 2021.
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