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.

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

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

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

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

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

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

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

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

$S_2=\frac{{\mathrm{I}}^2}{2!}\int \int 96{\mathrm{\lambda }}^2:\mathrm{\phi }\left(X\right)\mathrm{\phi }\left(Y\right):{\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Y\right)\right]}^3+16{\mathrm{\lambda }}^2:{\mathrm{\phi }\left(X\right)}^3{\mathrm{\phi }\left(Y\right)}^3:\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Y\right)\right]+72{\mathrm{\lambda }}^2:{\mathrm{\phi }\left(X\right)}^2{\mathrm{\phi }\left(Y\right)}^2:{\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Y\right)\right]}^2\ⅆX^4\ⅆY^4$

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

$\genfrac{}{}{0ex}{}{S_2}{\phantom{\mathrm{legs}=2}}|\genfrac{}{}{0ex}{}{\phantom{S_2}}{\mathrm{legs}=2}=\frac{{\mathrm{I}}^2}{2!}\int \int 96{\mathrm{\lambda }}^2:\mathrm{\phi }\left(X\right)\mathrm{\phi }\left(Y\right):{\left[\mathrm{\phi }\left(X\right)\,\mathrm{\phi }\left(Y\right)\right]}^3\ⅆX^4\ⅆY^4$

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

$\genfrac{}{}{0ex}{}{⟨f\left|S_2\right|i⟩}{\phantom{\mathrm{loops}=2}}|\genfrac{}{}{0ex}{}{\phantom{⟨f\left|S_2\right|i⟩}}{\mathrm{loops}=2}=\int \int \frac{\frac{3\mathrm{I}}{8}{\mathrm{\lambda }}^2\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)}{{\mathrm{\pi }}^7\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}\left({\left(\mathrm{P\_\_1}+\mathrm{p\_\_2}+\mathrm{p\_\_3}\right)}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_2}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_3}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)}\ⅆ{\mathrm{p\_\_2}}^4\ⅆ{\mathrm{p\_\_3}}^4$

$\mathrm{\%eval}\left(\mathrm{\%Bracket}\left(f\,S\left[2\right]\,i\right)\,\mathrm{loops}=3\right)=\mathrm{FeynmanDiagrams}\left(L\,\mathrm{incomingparticles}=\left[\mathrm{\phi }\right]\,\mathrm{outgoingparticles}=\left[\mathrm{\phi }\right]\,\mathrm{numberofloops}=3\right)$

$\genfrac{}{}{0ex}{}{⟨f\left|S_2\right|i⟩}{\phantom{\mathrm{loops}=3}}|\genfrac{}{}{0ex}{}{\phantom{⟨f\left|S_2\right|i⟩}}{\mathrm{loops}=3}=\int \int \int \frac{9{\mathrm{\lambda }}^3\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)}{32{\mathrm{\pi }}^{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}\mathbf{\epsilon }\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}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_3}}^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\_\_3}}^4\ⅆ{\mathrm{p\_\_4}}^4\ⅆ{\mathrm{p\_\_5}}^4+2\int \int \int \frac{9{\mathrm{\lambda }}^3\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)}{32{\mathrm{\pi }}^{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}\mathbf{\epsilon }\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}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_3}}^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\_\_3}}^4\ⅆ{\mathrm{p\_\_4}}^4\ⅆ{\mathrm{p\_\_5}}^4+\int \int \int \frac{9{\mathrm{\lambda }}^3\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)}{32{\mathrm{\pi }}^{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}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_2}}^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({\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\_\_2}}^4\ⅆ{\mathrm{p\_\_4}}^4\ⅆ{\mathrm{p\_\_5}}^4$

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

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

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

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

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

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

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

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

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

$0$

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$\frac{9{\mathrm{\pi }}^{-2-\mathrm{ϵ}}\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right){\mathrm{\lambda }}^2{\int }_{-\mathrm{\infty }\mathrm{I}}^{\mathrm{\infty }\mathrm{I}}{\int }_0^1{\int }_0^1{\left(-\mathrm{\ξ\_\_1}\mathrm{\ξ\_\_2}{\mathrm{P\_\_1}}^2\right)}^{\mathrm{\σ\_\_1}}{\left({\mathrm{m\_\_\φ}}^2\right)}^{-\mathrm{ϵ}-\mathrm{\σ\_\_1}}\mathrm{\Gamma }\left(-\mathrm{\σ\_\_1}\right)\mathrm{\Gamma }\left(\mathrm{ϵ}+\mathrm{\σ\_\_1}\right)\mathrm{\delta }\left(-1+\mathrm{\ξ\_\_1}+\mathrm{\ξ\_\_2}\right)\ⅆ\mathrm{\ξ\_\_2}\ⅆ\mathrm{\ξ\_\_1}\ⅆ\mathrm{\σ\_\_1}}{16\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}}$

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

$\frac{9{\mathrm{\pi }}^{-2-\mathrm{ϵ}}\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right){\mathrm{\lambda }}^2{\int }_{-\mathrm{\infty }\mathrm{I}}^{\mathrm{\infty }\mathrm{I}}\frac{{\left(-{\mathrm{P\_\_1}}^2\right)}^{\mathrm{\σ\_\_1}}{\left({\mathrm{m\_\_\φ}}^2\right)}^{-\mathrm{ϵ}-\mathrm{\σ\_\_1}}\mathrm{\Gamma }\left(-\mathrm{\σ\_\_1}\right)\mathrm{\Gamma }\left(\mathrm{ϵ}+\mathrm{\σ\_\_1}\right){\mathrm{\Gamma }\left(\mathrm{\σ\_\_1}+1\right)}^2}{\mathrm{\Gamma }\left(2\mathrm{\σ\_\_1}+2\right)}\ⅆ\mathrm{\σ\_\_1}}{16\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}}$

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

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

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

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

$\mathrm{\%FeynmanIntegral}\left(\frac{\mathrm{p\_\_1}\left[\mathrm{~mu}\right]}{\left({\mathrm{p\_\_1}}^2-{\mathrm{m\_\_\φ}}^2+i\mathrm{\epsilon }\right)\left({\left(\mathrm{p\_\_1}-\mathrm{P\_\_1}\right)}^2-{\mathrm{m\_\_1}}^2+i\mathrm{\epsilon }\right)}\,\mathrm{p\_\_1}\right)$

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

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

$-\frac{\left(\left({\mathrm{m\_\_1}}^2-{\mathrm{m\_\_\φ}}^2-\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)\int \frac{1}{\left({\mathrm{p\_\_1}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }\right)\left({\left(\mathrm{p\_\_1}-\mathrm{P\_\_1}\right)}^2-{\mathrm{m\_\_1}}^2+\mathrm{I}\mathbf{\epsilon }\right)}\ⅆ{\mathrm{p\_\_1}}^4+\int \frac{1}{{\mathrm{p\_\_1}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{I}\mathbf{\epsilon }}\ⅆ{\mathrm{p\_\_1}}^4-\int \frac{1}{{\left(\mathrm{p\_\_1}-\mathrm{P\_\_1}\right)}^2-{\mathrm{m\_\_1}}^2+\mathrm{I}\mathbf{\epsilon }}\ⅆ{\mathrm{p\_\_1}}^4\right){\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}}{2\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)}$

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

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

[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.