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);

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

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));

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

Evaluate((9), expanddimension);

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

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.