dimension = ... : the dimension of spacetime around which dimensional regularization is performed. Default value is the dimension indicated by $\mathrm{Physics}:-\mathrm{Setup}\left(\mathrm{dimension}\right)$

expanddimension = ... : the right-hand side can be true or false (default), to expand the result in series up to order $\mathrm{O}\left(\mathrm{ϵ}\right)$ before returning it. The $\mathrm{ϵ}$ = FeynmanIntegral:-varepsilon is the dimensional parameter, and the integration is performed at dimension $d=\mathrm{Physics}:-\mathrm{Setup}\left(\mathrm{dimension}\right)-2\mathrm{ϵ}$, and be default $\mathrm{Physics}:-\mathrm{Setup}\left(\mathrm{dimension}\right)=4$.

feynmanparametersintegrated = ... : the right-hand side can be true or false (default), to return right after the Feynman parameters are integrated. This option does not work with alpha parameters.

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

loopmomentumintegrated = ... : the right-hand side can be true or false (default), to return right after the loop momentum integration is performed. This option does not work with alpha parameters.

mbrepresentation = ... : the right-hand side can be true or false (default), to return right after the Mellin Barnes representation of the Feynman integral is constructed. This option does not work with alpha parameters.

parametrizedform = ... : the right-hand side can be true or false (default), to return the parametrized form after swapping the order of integration: first the loop momentum, then the Feynman or alpha parameters.

quiet = ... : the right-hand side can be true or false (default), to display or not information related to matching keywords

shiftedform = ... : the right-hand side can be true or false (default), to return right after the shift of the loop momentum variable is performed and before performing the Wick rotation. This option does not work with alpha parameters.

wickrotated = ... : the right-hand side can be true or false (default), to return right after the performing the Wick rotation, before performing the momentum integration. This option does not work with alpha parameters.

NOTE: 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. shifted for shiftedform.

A scattering matrix $S$ relates the initial and final states, $\left|i\right.⟩$ and $\left|f\right.⟩$, of an interacting system. In an N-dimensional spacetime with coordinates $X$, $S$ can be written as:

where $i$ is the imaginary unit and $L$ is the interaction Lagrangian, written in terms of quantum fields depending on the spacetime coordinates $X$. The T symbol means time-ordered. For the terminology used in this page, see for instance chapters IV and V, "The Scattering Matrix" and "The Feynman Rules and Diagrams", in ref.[1]. This exponential can be expanded as

where

and $T\left(L\left(X_1\right)\,\mathrm{...}\,L\left(X_n\right)\right)$ is the time-ordered product of n interaction Lagrangians evaluated at different points. Note the factor $\frac{i^n}{n!}$ in this definition of $S_n$ used here.

The FeynmanIntegral(integrand, [[X1], [X2], ... [Xn]]) command is thus a computational representation for $S_n$ in coordinates representation. Note the factor $\frac{i^n}{n!}$ included in the definition used here for $S_n$. That factor is automatically displayed when the second argument, [[X1], [X2], ... [Xn]], is a list of lists of coordinate systems. For details on the algebraic structure of integrand see FeynmanDiagrams.

The FeynmanIntegral(integrand, [[p1], [p2], ... [pn]]) command, where [[p1], [p2], ... [pn]] is a list of lists of spacetime tensors pj, of the form p__n, is a computational representation for the integrals in momentum representation that enter the S-matrix elements $⟨f|S|i⟩$ with initial and final states $\left|i\right.⟩$ and $\left|f\right.⟩$, respectively with s initial particles with defined momentum $p_i$ and r final particles with defined momentum $p_f$.

The conventions for representing external and internal (loop integration) momenta are, respectively, that they are symbols of the form P__n and p__n, where n is an integer.

In addition to representing the integral, when working in momentum representation, the command FeynmanIntegral(integrand, [[p1], [p2], ... [pn]]) actually computes the integral, by first introducing Feynman parameters then performing the loop integral using dimensional regularization, using FeynmanIntegral:-Evaluate (see the Examples section).

The Evaluate command also provides a way to investigate the different steps of the computation of a Feynman integral, something of interest in general and also for the case where some of the steps cannot be performed automatically.

Both Evaluate and FeynmanIntegral can compute tensor Feynman integrals by first reducing them to scalar integrals using TensorReduce, as shown in the last example.

In both coordinates and momentum representation, to each element in the list of lists of the second argument in FeynmanIntegral(integrand, [[..], [..], ... [..]]) corresponds a 4-dimensional integral, and the number of elements in that list of lists indicates the number of vertices of the Feynman diagrams corresponding to the Feynman integral.

To obtain the S-matrix elements $⟨f|S|i⟩$ in momentum representation for each $S_n$ entering the expansion of S in coordinates representation see sec. 20.1 of ref.[1].

The inert form of this command, %FeynmanIntegral, is used by the FeynmanDiagrams command to represent each $S_j$ entering $S=1+S_1+\mathrm{...}+S_n$ in coordinates representation, as well as the integrals in momentum representation entering $⟨f|S|i⟩$. For details on the algebraic structure of integrand in both representations see FeynmanDiagrams.

$\mathrm{with}\left(\mathrm{Physics}\right)\:$

$\mathrm{Setup}\left(\mathrm{mathematicalnotation}\=\mathrm{true}\,\mathrm{coordinates}\=\left[X\,Y\,Z\right]\,\mathrm{quantumoperators}\=\mathrm{\φ}\right)$

$\mathrm{Systems\; of\; spacetime\; coordinates\; are:}\left\{X=\left(\mathrm{x1}\,\mathrm{x2}\,\mathrm{x3}\,\mathrm{x4}\right)\,Y=\left(\mathrm{y1}\,\mathrm{y2}\,\mathrm{y3}\,\mathrm{y4}\right)\,Z=\left(\mathrm{z1}\,\mathrm{z2}\,\mathrm{z3}\,\mathrm{z4}\right)\right\}$

$\mathrm{\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_}$

$\left[\mathrm{coordinatesystems}=\left\{X\,Y\,Z\right\}\,\mathrm{mathematicalnotation}=\mathrm{true}\,\mathrm{quantumoperators}=\left\{\mathrm{\phi }\right\}\right]$

$L≔\mathrm{\λ}{\mathrm{\φ}\left(X\right)}^4$

$L≔\mathrm{\lambda }{\mathrm{\phi }\left(X\right)}^4$

$S_2\=\mathrm{FeynmanDiagrams}\left(L\,\mathrm{numberofvertices}\=2\,\mathrm{diagrams}\right)$

$S_2\=\mathrm{\%FeynmanIntegral}\left(96{\mathrm{\λ}}^2\mathrm{\_GF}\left(\mathrm{\_NP}\left(\mathrm{\φ}\left(X\right)\,\mathrm{\φ}\left(Y\right)\right)\,\left[\left[\mathrm{\φ}\left(X\right)\,\mathrm{\φ}\left(Y\right)\right]\,\left[\mathrm{\φ}\left(X\right)\,\mathrm{\φ}\left(Y\right)\right]\,\left[\mathrm{\φ}\left(X\right)\,\mathrm{\φ}\left(Y\right)\right]\right]\right)\+16{\mathrm{\λ}}^2\mathrm{\_GF}\left(\mathrm{\_NP}\left(\mathrm{\φ}\left(X\right)\,\mathrm{\φ}\left(X\right)\,\mathrm{\φ}\left(X\right)\,\mathrm{\φ}\left(Y\right)\,\mathrm{\φ}\left(Y\right)\,\mathrm{\φ}\left(Y\right)\right)\,\left[\left[\mathrm{\φ}\left(X\right)\,\mathrm{\φ}\left(Y\right)\right]\right]\right)\+72{\mathrm{\λ}}^2\mathrm{\_GF}\left(\mathrm{\_NP}\left(\mathrm{\φ}\left(X\right)\,\mathrm{\φ}\left(X\right)\,\mathrm{\φ}\left(Y\right)\,\mathrm{\φ}\left(Y\right)\right)\,\left[\left[\mathrm{\φ}\left(X\right)\,\mathrm{\φ}\left(Y\right)\right]\,\left[\mathrm{\φ}\left(X\right)\,\mathrm{\φ}\left(Y\right)\right]\right]\right)\,\left[\left[X\right]\,\left[Y\right]\right]\right)$

$\mathrm{\%eval}\left(S_2\,\mathrm{legs}\=2\right)\=\mathrm{FeynmanDiagrams}\left(L\,\mathrm{numberofvertices}\=2\,\mathrm{numberofexternallegs}\=2\right)$

$\mathrm{\%eval}\left(S_2\,\mathrm{legs}\=2\right)\=\mathrm{\%FeynmanIntegral}\left(96{\mathrm{\λ}}^2\mathrm{\_GF}\left(\mathrm{\_NP}\left(\mathrm{\φ}\left(X\right)\,\mathrm{\φ}\left(Y\right)\right)\,\left[\left[\mathrm{\φ}\left(X\right)\,\mathrm{\φ}\left(Y\right)\right]\,\left[\mathrm{\φ}\left(X\right)\,\mathrm{\φ}\left(Y\right)\right]\,\left[\mathrm{\φ}\left(X\right)\,\mathrm{\φ}\left(Y\right)\right]\right]\right)\,\left[\left[X\right]\,\left[Y\right]\right]\right)$

$\mathrm{\%eval}\left(\mathrm{\%Bracket}\left(f\,S_2\,i\right)\,\mathrm{loops}\=2\right)\=\mathrm{FeynmanDiagrams}\left(L\,\mathrm{incomingparticles}\=\left[\mathrm{\φ}\right]\,\mathrm{outgoingparticles}\=\left[\mathrm{\φ}\right]\,\mathrm{numberofloops}\=2\right)$

$\mathrm{\%eval}\left(\mathrm{\%Bracket}\left(f\,S_2\,i\right)\,\mathrm{loops}\=2\right)\=\mathrm{\%FeynmanIntegral}\left(\mathrm{\%FeynmanIntegral}\left(\frac{\frac{3}{8}\mathrm{I}{\mathrm{\λ}}^2\mathrm{Dirac}\left(-\mathrm{P\_\_2}\+\mathrm{P\_\_1}\right)}{{\mathrm{\π}}^7\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}\left({\left(\mathrm{P\_\_1}\+\mathrm{p\_\_2}\+\mathrm{p\_\_3}\right)}^2-{\mathrm{m\_\_\φ}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)\left({\mathrm{p\_\_2}}^2-{\mathrm{m\_\_\φ}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)\left({\mathrm{p\_\_3}}^2-{\mathrm{m\_\_\φ}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)}\,\left[\left[\mathrm{p\_\_2}\right]\right]\right)\,\left[\left[\mathrm{p\_\_3}\right]\right]\right)$

$\mathrm{\%eval}\left(\mathrm{\%Bracket}\left(f\,S_2\,i\right)\,\mathrm{loops}\=3\right)\=\mathrm{FeynmanDiagrams}\left(L\,\mathrm{incomingparticles}\=\left[\mathrm{\φ}\right]\,\mathrm{outgoingparticles}\=\left[\mathrm{\φ}\right]\,\mathrm{numberofloops}\=3\right)$

$\mathrm{\%eval}\left(\mathrm{\%Bracket}\left(f\,S_2\,i\right)\,\mathrm{loops}\=3\right)\=\mathrm{\%FeynmanIntegral}\left(\mathrm{\%FeynmanIntegral}\left(\mathrm{\%FeynmanIntegral}\left(\frac{9}{32}\frac{{\mathrm{\λ}}^3\mathrm{Dirac}\left(-\mathrm{P\_\_2}\+\mathrm{P\_\_1}\right)}{{\mathrm{\π}}^{11}\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}\left({\left(\mathrm{p\_\_3}\+\mathrm{p\_\_4}\+\mathrm{p\_\_5}\right)}^2-{\mathrm{m\_\_\φ}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)\left({\left(-\mathrm{P\_\_1}\+\mathrm{P\_\_2}-\mathrm{p\_\_3}-\mathrm{p\_\_4}-\mathrm{p\_\_5}\right)}^2-{\mathrm{m\_\_\φ}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)\left({\mathrm{p\_\_3}}^2-{\mathrm{m\_\_\φ}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)\left({\mathrm{p\_\_4}}^2-{\mathrm{m\_\_\φ}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)\left({\mathrm{p\_\_5}}^2-{\mathrm{m\_\_\φ}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)}\,\left[\left[\mathrm{p\_\_3}\right]\right]\right)\,\left[\left[\mathrm{p\_\_4}\right]\right]\right)\,\left[\left[\mathrm{p\_\_5}\right]\right]\right)\+2\mathrm{\%FeynmanIntegral}\left(\mathrm{\%FeynmanIntegral}\left(\mathrm{\%FeynmanIntegral}\left(\frac{9}{32}\frac{{\mathrm{\λ}}^3\mathrm{Dirac}\left(-\mathrm{P\_\_2}\+\mathrm{P\_\_1}\right)}{{\mathrm{\π}}^{11}\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}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)\left({\left(-\mathrm{P\_\_1}\+\mathrm{P\_\_2}-\mathrm{p\_\_3}-\mathrm{p\_\_4}-\mathrm{p\_\_5}\right)}^2-{\mathrm{m\_\_\φ}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)\left({\mathrm{p\_\_3}}^2-{\mathrm{m\_\_\φ}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)\left({\mathrm{p\_\_4}}^2-{\mathrm{m\_\_\φ}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)\left({\mathrm{p\_\_5}}^2-{\mathrm{m\_\_\φ}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)}\,\left[\left[\mathrm{p\_\_3}\right]\right]\right)\,\left[\left[\mathrm{p\_\_4}\right]\right]\right)\,\left[\left[\mathrm{p\_\_5}\right]\right]\right)\+\mathrm{\%FeynmanIntegral}\left(\mathrm{\%FeynmanIntegral}\left(\mathrm{\%FeynmanIntegral}\left(\frac{9}{32}\frac{{\mathrm{\λ}}^3\mathrm{Dirac}\left(-\mathrm{P\_\_2}\+\mathrm{P\_\_1}\right)}{{\mathrm{\π}}^{11}\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}\left({\left(-\mathrm{P\_\_1}-\mathrm{p\_\_2}\+\mathrm{P\_\_2}-\mathrm{p\_\_4}-\mathrm{p\_\_5}\right)}^2-{\mathrm{m\_\_\φ}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)\left({\mathrm{p\_\_2}}^2-{\mathrm{m\_\_\φ}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)\left({\left(-\mathrm{P\_\_2}\+\mathrm{p\_\_4}\+\mathrm{p\_\_5}\right)}^2-{\mathrm{m\_\_\φ}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)\left({\mathrm{p\_\_4}}^2-{\mathrm{m\_\_\φ}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)\left({\mathrm{p\_\_5}}^2-{\mathrm{m\_\_\φ}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)}\,\left[\left[\mathrm{p\_\_2}\right]\right]\right)\,\left[\left[\mathrm{p\_\_4}\right]\right]\right)\,\left[\left[\mathrm{p\_\_5}\right]\right]\right)$

$\mathrm{with}\left(\mathrm{FeynmanIntegral}\right)$

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

$\mathrm{\%FeynmanIntegral}\left(\frac{1}{\left(-m^2\+{\mathit{p\_\_1}}^2\+I\mathrm{\ε}\right)\left({\mathit{p\_\_1}}^2\+I\mathrm{\ε}\right)}\,\mathit{p\_\_1}\right)$

$\mathrm{\%FeynmanIntegral}\left(\frac{1}{\left(-m^2\+{\mathrm{p\_\_1}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)\left({\mathrm{p\_\_1}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)}\,\mathrm{p\_\_1}\right)$

$\mathrm{Evaluate}\left(\right)$

$\frac{\mathrm{I}{\mathrm{\pi }}^{2-\mathrm{ϵ}}\mathrm{\Gamma }\left(\mathrm{ϵ}\right){\left(m^2\right)}^{-\mathrm{ϵ}}\mathrm{\Gamma }\left(-\mathrm{ϵ}+1\right)}{\mathrm{\Gamma }\left(2-\mathrm{ϵ}\right)}$

$\frac{{\mathrm{Pi}}^{2-\mathrm{\ϵ}}I\mathrm{GAMMA}\left(\mathrm{\ϵ}\right)\mathrm{GAMMA}\left(1-\mathrm{\ϵ}\right)\cdot 1\cdot 1}{\mathrm{GAMMA}\left(2-\mathrm{\ϵ}\right){\left(m^2\right)}^{\mathrm{\ϵ}}}$

$\frac{\mathrm{I}{\mathrm{\pi }}^{2-\mathrm{ϵ}}\mathrm{\Gamma }\left(\mathrm{ϵ}\right)\mathrm{\Gamma }\left(-\mathrm{ϵ}+1\right)}{\mathrm{\Gamma }\left(2-\mathrm{ϵ}\right){\left(m^2\right)}^{\mathrm{ϵ}}}$

$\mathrm{simplify}\left(-\right)$

$0$

$\mathrm{ExpandDimension}\left(\right)$

$\mathrm{I}{\mathrm{\pi }}^2{\mathrm{ϵ}}^{\mathrm{-1}}+\mathrm{-I}{\mathrm{\pi }}^2\left(\ln \left(m^2\right)+\ln \left(\mathrm{\pi }\right)+\mathrm{\gamma }-1\right)+O\left(\mathrm{ϵ}\right)$

$\mathrm{Evaluate}\left(\,\mathrm{expanddimension}\right)$

$\mathrm{I}{\mathrm{\pi }}^2{\mathrm{ϵ}}^{\mathrm{-1}}+\mathrm{-I}{\mathrm{\pi }}^2\left(\ln \left(m^2\right)+\ln \left(\mathrm{\pi }\right)+\mathrm{\gamma }-1\right)+O\left(\mathrm{ϵ}\right)$

$L≔\mathrm{\λ}{\mathrm{\φ}\left(X\right)}^3$

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

$\mathrm{FeynmanDiagrams}\left(L\,\mathrm{incomingparticles}\=\left[\mathrm{\φ}\right]\,\mathrm{outgoingparticles}\=\left[\mathrm{\φ}\right]\,\mathrm{numberofloops}\=1\,\mathrm{diagrams}\right)$

$\mathrm{\%FeynmanIntegral}\left(\frac{9}{8}\frac{{\mathrm{\λ}}^2\mathrm{Dirac}\left(-\mathrm{P\_\_2}\+\mathrm{P\_\_1}\right)}{{\mathrm{\π}}^3\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}\left({\left(\mathrm{P\_\_1}\+\mathrm{p\_\_2}\right)}^2-{\mathrm{m\_\_\φ}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)\left({\mathrm{p\_\_2}}^2-{\mathrm{m\_\_\φ}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)}\,\left[\left[\mathrm{p\_\_2}\right]\right]\right)$

$\mathrm{Parametrize}\left(\right)$

$\mathrm{\%FeynmanIntegral}\left(\frac{9}{8}\frac{{\mathrm{\λ}}^2\mathrm{Dirac}\left(-\mathrm{P\_\_2}\+\mathrm{P\_\_1}\right)\mathrm{\%int}\left(\mathrm{\%int}\left(\frac{\mathrm{Dirac}\left(-1\+\mathrm{\ξ\_\_1}\+\mathrm{\ξ\_\_2}\right)}{{\left(\mathrm{\ξ\_\_1}\left({\left(\mathrm{P\_\_1}\+\mathrm{p\_\_2}\right)}^2-{\mathrm{m\_\_\φ}}^2\right)\+\mathrm{\ξ\_\_2}\left(-{\mathrm{m\_\_\φ}}^2\+{\mathrm{p\_\_2}}^2\right)\right)}^2}\,\mathrm{\ξ\_\_2}\=0..1\right)\,\mathrm{\ξ\_\_1}\=0..1\right)}{{\mathrm{\π}}^3\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}}\,\left[\left[\mathrm{p\_\_2}\right]\right]\right)$

$\mathrm{Evaluate}\left(\,\mathrm{parametrizedform}\right)$

$\frac{9}{8}\frac{{\mathrm{\λ}}^2\mathrm{Dirac}\left(-\mathrm{P\_\_2}\+\mathrm{P\_\_1}\right)\mathrm{\%int}\left(\mathrm{\%int}\left(\mathrm{Dirac}\left(-1\+\mathrm{\ξ\_\_1}\+\mathrm{\ξ\_\_2}\right)\mathrm{\%FeynmanIntegral}\left(\frac{1}{{\left(\mathrm{\ξ\_\_1}\left({\left(\mathrm{P\_\_1}\+\mathrm{p\_\_2}\right)}^2-{\mathrm{m\_\_\φ}}^2\right)\+\mathrm{\ξ\_\_2}\left(-{\mathrm{m\_\_\φ}}^2\+{\mathrm{p\_\_2}}^2\right)\right)}^2}\,\left[\left[\mathrm{p\_\_2}\right]\right]\right)\,\mathrm{\ξ\_\_2}\=0..1\right)\,\mathrm{\ξ\_\_1}\=0..1\right)}{{\mathrm{\π}}^3\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}}$

$\mathrm{Evaluate}\left(\,\mathrm{shiftedform}\right)$

$\frac{9}{8}\frac{{\mathrm{\λ}}^2\mathrm{Dirac}\left(-\mathrm{P\_\_2}\+\mathrm{P\_\_1}\right)\mathrm{\%int}\left(\mathrm{\%int}\left(\mathrm{Dirac}\left(-1\+\mathrm{\ξ\_\_1}\+\mathrm{\ξ\_\_2}\right)\mathrm{\%FeynmanIntegral}\left(\frac{1}{{\left(\left({\mathrm{\ξ\_\_1}}^2-\mathrm{\ξ\_\_1}\right){\mathrm{P\_\_1}}^2-{\mathrm{p\_\_2}}^2\+{\mathrm{m\_\_\φ}}^2\right)}^2}\,\mathrm{p\_\_2}\right)\,\mathrm{\ξ\_\_2}\=0..1\right)\,\mathrm{\ξ\_\_1}\=0..1\right)}{{\mathrm{\π}}^3\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}}$

$\mathrm{Evaluate}\left(\,\mathrm{wickrotated}\right)$

$\frac{\frac{9}{8}\mathrm{I}{\mathrm{\λ}}^2\mathrm{Dirac}\left(-\mathrm{P\_\_2}\+\mathrm{P\_\_1}\right)\mathrm{\%int}\left(\mathrm{\%int}\left(\mathrm{Dirac}\left(-1\+\mathrm{\ξ\_\_1}\+\mathrm{\ξ\_\_2}\right)\mathrm{\%FeynmanIntegral}\left(\frac{1}{{\left(\left({\mathrm{\ξ\_\_1}}^2-\mathrm{\ξ\_\_1}\right){\mathrm{P\_\_1}}^2\+{\mathrm{p\_\_E}}^2\+{\mathrm{m\_\_\φ}}^2\right)}^2}\,\mathrm{p\_\_E}\right)\,\mathrm{\ξ\_\_2}\=0..1\right)\,\mathrm{\ξ\_\_1}\=0..1\right)}{{\mathrm{\π}}^3\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}}$

$\mathrm{Evaluate}\left(\,\mathrm{loopmomentumintegrated}\right)$

$\frac{\frac{9}{8}\mathrm{I}{\mathrm{\λ}}^2\mathrm{Dirac}\left(-\mathrm{P\_\_2}\+\mathrm{P\_\_1}\right){\mathrm{\π}}^{2-\mathrm{\ϵ}}\mathrm{\Γ}\left(\mathrm{\ϵ}\right)\mathrm{\%int}\left(\mathrm{\%int}\left(\mathrm{Dirac}\left(-1\+\mathrm{\ξ\_\_1}\+\mathrm{\ξ\_\_2}\right){\left(\left({\mathrm{\ξ\_\_1}}^2-\mathrm{\ξ\_\_1}\right){\mathrm{P\_\_1}}^2\+{\mathrm{m\_\_\φ}}^2\right)}^{-\mathrm{\ϵ}}\,\mathrm{\ξ\_\_2}\=0..1\right)\,\mathrm{\ξ\_\_1}\=0..1\right)}{{\mathrm{\π}}^3\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}}$

$\mathrm{Evaluate}\left(\,\mathrm{mbrepresentation}\right)$

$\frac{9}{16}\frac{{\mathrm{\λ}}^2\mathrm{Dirac}\left(-\mathrm{P\_\_2}\+\mathrm{P\_\_1}\right){\mathrm{\π}}^{-2-\mathrm{\ϵ}}\mathrm{\%int}\left(\mathrm{\%int}\left({\left(-\mathrm{\ξ\_\_1}\mathrm{\ξ\_\_2}{\mathrm{P\_\_1}}^2\right)}^{\mathrm{\σ\_\_1}}{\left({\mathrm{m\_\_\φ}}^2\right)}^{-\mathrm{\ϵ}-\mathrm{\σ\_\_1}}\mathrm{\Γ}\left(-\mathrm{\σ\_\_1}\right)\mathrm{\Γ}\left(\mathrm{\ϵ}\+\mathrm{\σ\_\_1}\right)\mathrm{Dirac}\left(-1\+\mathrm{\ξ\_\_1}\+\mathrm{\ξ\_\_2}\right)\,\left[\mathrm{\ξ\_\_2}\=0..1\,\mathrm{\ξ\_\_1}\=0..1\right]\right)\,\left[\mathrm{\σ\_\_1}\=-\mathrm{\∞}\mathrm{I}..\mathrm{\∞}\mathrm{I}\right]\right)}{\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}}$

$\mathrm{Evaluate}\left(\,\mathrm{feynmanparametersintegrated}\right)$

$\frac{9}{16}\frac{{\mathrm{\λ}}^2\mathrm{Dirac}\left(-\mathrm{P\_\_2}\+\mathrm{P\_\_1}\right){\mathrm{\π}}^{-2-\mathrm{\ϵ}}\mathrm{\%int}\left(\frac{{\left(-{\mathrm{P\_\_1}}^2\right)}^{\mathrm{\σ\_\_1}}{\left({\mathrm{m\_\_\φ}}^2\right)}^{-\mathrm{\ϵ}-\mathrm{\σ\_\_1}}\mathrm{\Γ}\left(-\mathrm{\σ\_\_1}\right)\mathrm{\Γ}\left(\mathrm{\ϵ}\+\mathrm{\σ\_\_1}\right){\mathrm{\Γ}\left(\mathrm{\σ\_\_1}\+1\right)}^2}{\mathrm{\Γ}\left(2\mathrm{\σ\_\_1}\+2\right)}\,\left[\mathrm{\σ\_\_1}\=-\mathrm{\∞}\mathrm{I}..\mathrm{\∞}\mathrm{I}\right]\right)}{\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}}$

$\mathrm{Evaluate}\left(\right)$

$\frac{\frac{9\mathrm{I}}{8}{\mathrm{\pi }}^{-1-\mathrm{ϵ}}{\mathrm{\lambda }}^2\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)\left(\mathrm{\%sum}\left(\frac{{\mathrm{m\_\_\φ}}^{-2\mathrm{\ϵ}-2n}\mathrm{\Γ}\left(n\+1\right)\mathrm{\Γ}\left(\mathrm{\ϵ}\+n\right){\mathrm{P\_\_1}}^{2n}}{\mathrm{\Γ}\left(2n\+2\right)}\,n\=0..\mathrm{\∞}\right)\right)}{\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}}$

$\mathrm{ExpandDimension}\left(\right)$

$\frac{\frac{9\mathrm{I}}{8}{\mathrm{\lambda }}^2\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)}{\mathrm{\pi }\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}}{\mathrm{ϵ}}^{\mathrm{-1}}+\frac{-\frac{9\mathrm{I}}{8}{\mathrm{\lambda }}^2\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)\left(\mathrm{\gamma }+2\ln \left(\mathrm{m\_\_\φ}\right)-\left(\mathrm{\%sum}\left(\frac{\mathrm{\Γ}\left(n\+1\right)\mathrm{\Γ}\left(n\right){\mathrm{P\_\_1}}^{2n}}{{\left({\mathrm{m\_\_\φ}}^n\right)}^2\mathrm{\Γ}\left(2n\+2\right)}\,n\=1..\mathrm{\∞}\right)\right)+\ln \left(\mathrm{\pi }\right)\right)}{\mathrm{\pi }\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}}+O\left(\mathrm{ϵ}\right)$

$\mathrm{\%FeynmanIntegral}\left(\frac{{\mathit{p\_\_1}}_{\mathrm{`~mu`}}}{\left({\mathit{p\_\_1}}^2-{\mathit{m\_\_phi}}^2\+i\mathrm{\ε}\right)\left({\left(\mathit{p\_\_1}-\mathit{P\_\_1}\right)}^2-{\mathit{m\_\_1}}^2\+i\mathrm{\ε}\right)}\,\mathit{p\_\_1}\right)$

$\mathrm{\%FeynmanIntegral}\left(\frac{{\mathrm{p\_\_1}}_{\mathrm{~mu}}}{\left(\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}i-{\mathrm{m\_\_\φ}}^2\+{\mathrm{p\_\_1}}^2\right)\left({\left(\mathrm{p\_\_1}-\mathrm{P\_\_1}\right)}^2-{\mathrm{m\_\_1}}^2\+i\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)}\,\mathrm{p\_\_1}\right)$

$\mathrm{TensorReduce}\left(\right)$

$-\frac{1}{2}\frac{{\mathrm{P\_\_1}}_{\mathrm{~mu}}\left(\left({\mathrm{m\_\_1}}^2-{\mathrm{m\_\_\φ}}^2-\mathrm{\%.}\left(\mathrm{P\_\_1}\,\mathrm{P\_\_1}\right)\right)\mathrm{\%FeynmanIntegral}\left(\frac{1}{\left({\mathrm{p\_\_1}}^2-{\mathrm{m\_\_\φ}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)\left({\left(\mathrm{p\_\_1}-\mathrm{P\_\_1}\right)}^2-{\mathrm{m\_\_1}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}\right)}\,\mathrm{p\_\_1}\right)\+\mathrm{\%FeynmanIntegral}\left(\frac{1}{{\mathrm{p\_\_1}}^2-{\mathrm{m\_\_\φ}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}}\,\mathrm{p\_\_1}\right)-\mathrm{\%FeynmanIntegral}\left(\frac{1}{{\left(\mathrm{p\_\_1}-\mathrm{P\_\_1}\right)}^2-{\mathrm{m\_\_1}}^2\+\mathrm{I}\mathrm{Physics}:-\mathrm{FeynmanDiagrams}:-\mathrm{\ε}}\,\mathrm{p\_\_1}\right)\right)}{\mathrm{\%.}\left(\mathrm{P\_\_1}\,\mathrm{P\_\_1}\right)}$

$\mathrm{Evaluate}\left(\right)$

$-\frac{{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(-\mathrm{I}\left({\mathrm{m\_\_1}}^2-{\mathrm{m\_\_\φ}}^2-\mathrm{\%.}\left(\mathrm{P\_\_1}\,\mathrm{P\_\_1}\right)\right){\mathrm{\pi }}^{2-\mathrm{ϵ}}\left(\mathrm{\%sum}\left(\mathrm{\%sum}\left(-\frac{\mathrm{\Γ}\left(n\+\mathrm{n\_\_1}\+1\right){\mathrm{m\_\_\φ}}^{-2\mathrm{\ϵ}-2n-2\mathrm{n\_\_1}}{\left(-{\mathrm{m\_\_1}}^2\+{\mathrm{m\_\_\φ}}^2\right)}^n\mathrm{\Γ}\left(\mathrm{\ϵ}\+n\+\mathrm{n\_\_1}\right){\mathrm{P\_\_1}}^{2\mathrm{n\_\_1}}}{\mathrm{\Γ}\left(2\mathrm{n\_\_1}\+n\+2\right)\mathrm{\Γ}\left(n\+1\right)}\,\mathrm{n\_\_1}\=0..\mathrm{\∞}\right)\,n\=0..\mathrm{\∞}\right)\right)-\mathrm{I}{\mathrm{\pi }}^{2-\mathrm{ϵ}}{\mathrm{m\_\_\φ}}^{-2\mathrm{ϵ}+2}\mathrm{\Gamma }\left(\mathrm{ϵ}-1\right)+\mathrm{I}{\mathrm{\pi }}^{2-\mathrm{ϵ}}{\mathrm{m\_\_1}}^{-2\mathrm{ϵ}+2}\mathrm{\Gamma }\left(\mathrm{ϵ}-1\right)\right)}{2\left(\mathrm{\%.}\left(\mathrm{P\_\_1}\,\mathrm{P\_\_1}\right)\right)}$

[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], Physics[FeynmanIntegral][Evaluate], Physics[FeynmanIntegral][epsilon] and Physics[FeynmanIntegral][varepsilon] commands were introduced in Maple 2021.

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