kindofparameters = ... : the kind of auxiliary parameters - Feynman (default) or $\mathrm{\alpha }$ - used to parametrize Feynman integrals.

numberofpropagators = ... : the right-hand side a non-assigned name to which the number of propagators parametrized 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 parametrized 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.

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 parametrized integrals, using Feynman (default) or $\mathrm{\alpha }$ parameters. This is the first step performed by the FeynmanIntegral command towards the computation of the integral using dimensional regularization.

Only propagators involving a loop momentum (the integration variable of the %FeynmanIntegral), of the form p__n, so the letter p followed by two underscores and where n is a positive integer, are included in the parametrization. The output is the parametrized 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 ${\mathrm{\xi }}_i$ are the Feynman parameters, and the ${\mathrm{\alpha }}_i$ and ${\mathrm{\lambda }}_j$ are, respectively, the $\mathrm{\alpha }$ parameters and the $\mathrm{\lambda }$ (possibly complex) exponents.

with(Physics):

with(FeynmanIntegral);

$\left[\mathrm{Evaluate}\,\mathrm{ExpandDimension}\,\mathrm{FromAbstractRepresentation}\,\mathrm{Parametrize}\,\mathrm{Series}\,\mathrm{SumLookup}\,\mathrm{TensorBasis}\,\mathrm{TensorReduce}\,\mathrm{ToAbstractRepresentation}\,\mathrm{\epsilon }\,\mathrm{ϵ}\right]$

interface(imaginaryunit = i):

%FeynmanIntegral(1/((-m^2 + p__1^2 + i * epsilon)*(p__1^2 + i * epsilon)), p__1);

$\int \frac{1}{\left(-m^2+{\mathrm{p\_\_1}}^2+\mathrm{i}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_1}}^2+\mathrm{i}\mathbf{\epsilon }\right)}\ⅆ{\mathrm{p\_\_1}}^4$

Parametrize((2));

Parametrize((2), integrand);

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{integrand}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{returnintegrand}\text{'}$

$\frac{\mathrm{\delta }\left(-1+\mathrm{\ξ\_\_1}+\mathrm{\ξ\_\_2}\right)}{{\left(\mathrm{\ξ\_\_1}\left(-m^2+{\mathrm{p\_\_1}}^2\right)+\mathrm{\ξ\_\_2}{\mathrm{p\_\_1}}^2\right)}^2}$

Parametrize((2), kindofparameters = alpha);

$\int {\int }_0^{\mathrm{\infty }}\left({\int }_0^{\mathrm{\infty }}-{\ⅇ}^{\mathrm{i}{\mathrm{p\_\_1}}^2\left(\mathrm{\α\_\_1}+\mathrm{\α\_\_2}\right)}{\ⅇ}^{\mathrm{-i}\mathrm{\α\_\_1}{m}^2}\ⅆ\mathrm{\α\_\_1}\right)\ⅆ\mathrm{\α\_\_2}\ⅆ{\mathrm{p\_\_1}}^4$

Parametrize((2), propagators = 'LP');

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{propagators}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{propagatorslist}\text{'}$

LP;

$\left[-m^2+{\mathrm{p\_\_1}}^2\,{\mathrm{p\_\_1}}^2\right]$

L := lambda*phi(X)^3;

$L≔\mathrm{\lambda }{\mathrm{\phi }\left(X\right)}^3$

FeynmanDiagrams(L, incomingparticles = [phi], outgoingparticles = [phi], numberofloops = 1, diagrams);

$\int \frac{9{\mathrm{\lambda }}^2\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)}{8{\mathrm{\pi }}^3\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}\left({\left(\mathrm{P\_\_1}+\mathrm{p\_\_2}\right)}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_2}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }\right)}\ⅆ{\mathrm{p\_\_2}}^4$

Parametrize((9));

Evaluate((9));

$\frac{\frac{9\mathrm{i}}{8}{\mathrm{\pi }}^{-1-\mathrm{ϵ}}{\mathrm{\lambda }}^2\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)\left(\sum _{n=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\mathrm{ϵ}+n\right){\mathrm{P\_\_1}}^{2n}\mathrm{\Gamma }\left(n+1\right){\mathrm{m\_\_\φ}}^{-2\mathrm{ϵ}-2n}}{\mathrm{\Gamma }\left(2n+2\right)}\right)}{\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}}$

Evaluate((9), expanddimension);

$\frac{\frac{9\mathrm{i}}{8}{\mathrm{\lambda }}^2\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)}{\mathrm{\pi }\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}}{\mathrm{ϵ}}^{\mathrm{-1}}+\frac{-\frac{9\mathrm{i}}{8}{\mathrm{\lambda }}^2\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)\left(2\ln \left(\mathrm{m\_\_\φ}\right)+\mathrm{\gamma }-\left(\sum _{n=1}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(n\right){\mathrm{P\_\_1}}^{2n}\mathrm{\Gamma }\left(n+1\right)}{{\mathrm{m\_\_\φ}}^{2n}\mathrm{\Gamma }\left(2n+2\right)}\right)+\ln \left(\mathrm{\pi }\right)\right)}{\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}\mathrm{\pi }}+O\left(\mathrm{ϵ}\right)$

FeynmanDiagrams(L, incomingparticles = [phi], outgoingparticles = [phi], numberofloops = 2);

$2\int \int \frac{\frac{81\mathrm{i}}{64}{\mathrm{\lambda }}^4\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)}{{\mathrm{\pi }}^7\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}\left({\left(\mathrm{P\_\_2}+\mathrm{p\_\_4}+\mathrm{p\_\_5}\right)}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }\right)\left({\left(\mathrm{P\_\_2}-\mathrm{P\_\_1}+\mathrm{p\_\_4}+\mathrm{p\_\_5}\right)}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }\right)\left({\left(\mathrm{p\_\_4}+\mathrm{p\_\_5}\right)}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_4}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_5}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }\right)}\ⅆ{\mathrm{p\_\_4}}^4\ⅆ{\mathrm{p\_\_5}}^4+\int \int \frac{\frac{81\mathrm{i}}{64}{\mathrm{\lambda }}^4\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)}{{\mathrm{\pi }}^7\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}\left({\left(-\mathrm{P\_\_1}+\mathrm{p\_\_4}+\mathrm{p\_\_5}\right)}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }\right)\left({\left(\mathrm{p\_\_4}+\mathrm{p\_\_5}\right)}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }\right)\left({\left(\mathrm{P\_\_2}-\mathrm{p\_\_4}\right)}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_4}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_5}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }\right)}\ⅆ{\mathrm{p\_\_4}}^4\ⅆ{\mathrm{p\_\_5}}^4+\int \int \frac{\frac{81\mathrm{i}}{64}{\mathrm{\lambda }}^4\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)}{{\mathrm{\pi }}^7\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}\left({\left(-\mathrm{P\_\_1}+\mathrm{P\_\_2}-\mathrm{p\_\_4}+\mathrm{p\_\_5}\right)}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }\right)\left({\left(-\mathrm{P\_\_2}+\mathrm{p\_\_4}-\mathrm{p\_\_5}\right)}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }\right)\left({\left(\mathrm{P\_\_2}-\mathrm{p\_\_4}\right)}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_4}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_5}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }\right)}\ⅆ{\mathrm{p\_\_4}}^4\ⅆ{\mathrm{p\_\_5}}^4$

subsindets((13), specfunc(%FeynmanIntegral), Parametrize);

[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.

The Physics[FeynmanIntegral][Parametrize] command was introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.