When computing the scattering matrix $S$ for a particle process (momentum representation, see FeynmanDiagrams) the result, at one or more loops, contains Feynman integrals. Depending on the fields entering the interaction Lagrangian, the numerator of the integrand of such an integral may involve the loop momentum integration variable (one or a product of them) with free spacetime indices. That is the case of a tensor Feynman integral. Generally speaking, tensor integrals are computed by first reducing them to scalar integrals, for which there exist different approaches. In this context, TensorReduce reduces to scalar integrals the tensor integrals found in expression, using the Passarino-Veltman reduction approach.

There are two key observations in the Passarino-Veltman scheme. One is that a 1-loop tensor Feynman integral can always be expressed as a linear combination of tensorial expressions constructed with the external momenta $P_n^{\mathrm{\mu }}$ and the metric $g_{\mathrm{\mu },\mathrm{\nu }}$, That is, as a finite sum of the form $\sum _{n=1}^N{C}_n{t}_n^{\left(\mathrm{\mu },\mathrm{\nu },\mathrm{...}\right)}$. The $t_n^{\mathrm{\mu },\mathrm{\nu },\mathrm{...}}$ have the free indices of (possibly a product of) the loop momentum in the numerator of the integrand. These tensorial structures are constructed by taking products of the external momenta $P_j^{\mathrm{\mu }}$ entering the Feynman integral between themselves and the metric $g_{\mathrm{\mu },\mathrm{\nu }}$. The Maple command that constructs such a tensor basis of $N$ structures $t_n^{\mathrm{\mu },\mathrm{\nu },\mathrm{...}}$ is FeynmanIntegral:-TensorBasis.

The other idea behind this algorithm is that the scalar product of the loop momentum $p_{\mathrm{\mu }}$ found in the numerator of the integrand with tensor structures constructed with the external momentum $P_{\mathrm{\mu }}$ can be expressed as a linear combination of inverse propagators.

Combining both ideas, the coefficients $C_n$ entering $\sum _{n=1}^N{C}_n{t}_n^{\left(\mathrm{\mu },\mathrm{\nu },\mathrm{...}\right)}$ are computed by first, constructing an equation with the Feynman integral on the left-hand side and that expansion with the $C_n$ as unknown coefficients on the right-hand side. Then contracting this equation with the tensorial structures that, on the left-hand side, introduce inverse propagators, a system of equations is constructed and solved for the C[n].

Two relevant computational tools for performing the reduction and other manipulations of Feynman integrals are the commands ToAbstractRepresentation and FromAbstractRepresentation. These commands map, back and forth, a given Feynman integral into and from an abstract representation that contains all the information regarding the propagators present, external momenta and corresponding masses, the loop momentum integration, and the free indices.

The Passarino-Veltman algorithm has several steps, each of which can be of interest on its own, or simply to follow the computation step-by-step. For this purpose, you can use the optional argument outputstep = n, where n is any positive integer up to 7. The result of each step is as follows:

outputstep = 1:  an equation, with the Feynman tensor integral in the left-hand side and the linear combination of tensor structures constructed with the external momentum $P_{\mathrm{\mu }}$ and the metric on the right-hand side.

outputstep equal to 2, 3 or 4: a Vector of equations, each of which is constructed by contracting the output of step 1 with tensor structures that, when contracted with the loop momentum in the numerator of the integrand, results in inverse propagators. These scalar products are first represented (step 2) then the contractions are expressed in a more convenient form on the right-hand sides (step 3), finally the actual reduction - the core of the process - that happens when performing the contraction on the left-hand sides that results in inverse propagators (step 4). In these three steps, the integrals are represented in abstract form. These results can be manipulated further using ToAbstractRepresentation and FromAbstractRepresentation.

outputstep = 5: an equation with the output of the step 1 and the a set of equations containing the solution for the coefficients $C_n$

outputstep = 6: an equation with the left-hand side equal to the given Feynman tensor integral and the right hand side already as a linear combination of scalar Feynman integrals written in the abstract representation used in the FeynmanIntegral package (see the commands ToAbstractRepresentation, FromAbstractRepresentation).

outputstep = 7: the same as omitting outputstep = ... entirely, the result is the full reduction of the tensor integral to a linear combination of scalar Feynman integrals.

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(p__1[~mu]/((p__1^2 - m__phi^2 + i * epsilon)*((p__1 - P__1)^2 - m__1^2 + i * epsilon)), p__1);

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

(2) = TensorReduce((2));

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

(2) = Evaluate(rhs((3)));

Evaluate((2));

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

ToAbstractRepresentation((2));

${𝕋}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(2\,0\,{\mathrm{m\_\_\φ}}^2\,-\mathrm{P\_\_1}\,{\mathrm{m\_\_1}}^2\,\mathrm{p\_\_1}\,0\right)$

FromAbstractRepresentation((6));

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

TensorReduce((2), step = 1);

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{step}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{outputstep}\text{'}$

$\int \frac{{\mathrm{p\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}}{\left({\mathrm{p\_\_1}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }\right)\left({\left(\mathrm{p\_\_1}-\mathrm{P\_\_1}\right)}^2-{\mathrm{m\_\_1}}^2+\mathrm{i}\mathbf{\epsilon }\right)}\ⅆ{\mathrm{p\_\_1}}^4=C_1{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}$

TensorBasis([P__1], [~mu]);

$\left[{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right]$

TensorReduce((2), step = 2);

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{step}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{outputstep}\text{'}$

$\left[\begin{array}{c}{\mathrm{P\_\_1}}_{\mathrm{\mu }}{𝕋}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(2\,0\,{\mathrm{m\_\_\φ}}^2\,-\mathrm{P\_\_1}\,{\mathrm{m\_\_1}}^2\,\mathrm{p\_\_1}\,0\right)={\mathrm{P\_\_1}}_{\mathrm{\mu }}{C}_1{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\end{array}\right]$

TensorReduce((2), step = 3);

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{step}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{outputstep}\text{'}$

$\left[\begin{array}{c}{\mathrm{P\_\_1}}_{\mathrm{\mu }}{𝕋}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\left(2\,0\,{\mathrm{m\_\_\φ}}^2\,-\mathrm{P\_\_1}\,{\mathrm{m\_\_1}}^2\,\mathrm{p\_\_1}\,0\right)=C_1\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)\end{array}\right]$

TensorReduce((2), step = 4);

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{step}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{outputstep}\text{'}$

$\left[\begin{array}{c}-\frac{𝕋\left[\right]\left(1\,0\,{\mathrm{m\_\_\φ}}^2\,\mathrm{p\_\_1}\,0\right)}{2}+\frac{𝕋\left[\right]\left(1\,-\mathrm{P\_\_1}\,{\mathrm{m\_\_1}}^2\,\mathrm{p\_\_1}\,0\right)}{2}+\frac{\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}-{\mathrm{m\_\_1}}^2+{\mathrm{m\_\_\φ}}^2\right)𝕋\left[\right]\left(2\,0\,{\mathrm{m\_\_\φ}}^2\,-\mathrm{P\_\_1}\,{\mathrm{m\_\_1}}^2\,\mathrm{p\_\_1}\,0\right)}{2}=C_1\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)\end{array}\right]$

FromAbstractRepresentation((12));

$\left[\begin{array}{c}-\frac{\int \frac{1}{{\mathrm{p\_\_1}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }}\ⅆ{\mathrm{p\_\_1}}^4}{2}+\frac{\int \frac{1}{{\left(\mathrm{p\_\_1}-\mathrm{P\_\_1}\right)}^2-{\mathrm{m\_\_1}}^2+\mathrm{i}\mathbf{\epsilon }}\ⅆ{\mathrm{p\_\_1}}^4}{2}+\frac{\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}-{\mathrm{m\_\_1}}^2+{\mathrm{m\_\_\φ}}^2\right)\int \frac{1}{\left({\mathrm{p\_\_1}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }\right)\left({\left(\mathrm{p\_\_1}-\mathrm{P\_\_1}\right)}^2-{\mathrm{m\_\_1}}^2+\mathrm{i}\mathbf{\epsilon }\right)}\ⅆ{\mathrm{p\_\_1}}^4}{2}=C_1\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)\end{array}\right]$

TensorReduce((2), step = 5);

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{step}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{outputstep}\text{'}$

$\int \frac{{\mathrm{p\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}}{\left({\mathrm{p\_\_1}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }\right)\left({\left(\mathrm{p\_\_1}-\mathrm{P\_\_1}\right)}^2-{\mathrm{m\_\_1}}^2+\mathrm{i}\mathbf{\epsilon }\right)}\ⅆ{\mathrm{p\_\_1}}^4=C_1{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\mathbf{where}\left\{C_1=\frac{-𝕋\left[\right]\left(2\,0\,{\mathrm{m\_\_\φ}}^2\,-\mathrm{P\_\_1}\,{\mathrm{m\_\_1}}^2\,\mathrm{p\_\_1}\,0\right){\mathrm{m\_\_1}}^2+𝕋\left[\right]\left(2\,0\,{\mathrm{m\_\_\φ}}^2\,-\mathrm{P\_\_1}\,{\mathrm{m\_\_1}}^2\,\mathrm{p\_\_1}\,0\right){\mathrm{m\_\_\φ}}^2+\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)𝕋\left[\right]\left(2\,0\,{\mathrm{m\_\_\φ}}^2\,-\mathrm{P\_\_1}\,{\mathrm{m\_\_1}}^2\,\mathrm{p\_\_1}\,0\right)-𝕋\left[\right]\left(1\,0\,{\mathrm{m\_\_\φ}}^2\,\mathrm{p\_\_1}\,0\right)+𝕋\left[\right]\left(1\,-\mathrm{P\_\_1}\,{\mathrm{m\_\_1}}^2\,\mathrm{p\_\_1}\,0\right)}{2\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)}\right\}$

TensorReduce((2), step = 6);

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{step}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{outputstep}\text{'}$

$\int \frac{{\mathrm{p\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}}{\left({\mathrm{p\_\_1}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }\right)\left({\left(\mathrm{p\_\_1}-\mathrm{P\_\_1}\right)}^2-{\mathrm{m\_\_1}}^2+\mathrm{i}\mathbf{\epsilon }\right)}\ⅆ{\mathrm{p\_\_1}}^4=\frac{\left(-𝕋\left[\right]\left(2\,0\,{\mathrm{m\_\_\φ}}^2\,-\mathrm{P\_\_1}\,{\mathrm{m\_\_1}}^2\,\mathrm{p\_\_1}\,0\right){\mathrm{m\_\_1}}^2+𝕋\left[\right]\left(2\,0\,{\mathrm{m\_\_\φ}}^2\,-\mathrm{P\_\_1}\,{\mathrm{m\_\_1}}^2\,\mathrm{p\_\_1}\,0\right){\mathrm{m\_\_\φ}}^2+\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)𝕋\left[\right]\left(2\,0\,{\mathrm{m\_\_\φ}}^2\,-\mathrm{P\_\_1}\,{\mathrm{m\_\_1}}^2\,\mathrm{p\_\_1}\,0\right)-𝕋\left[\right]\left(1\,0\,{\mathrm{m\_\_\φ}}^2\,\mathrm{p\_\_1}\,0\right)+𝕋\left[\right]\left(1\,-\mathrm{P\_\_1}\,{\mathrm{m\_\_1}}^2\,\mathrm{p\_\_1}\,0\right)\right){\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}}{2\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)}$

%FeynmanIntegral(p__1[~mu]*p__1[~nu]/((p__1^2 - m__1^2 + I*epsilon)*((p__1 - P__1)^2 - m__1^2 + i*epsilon)), p__1);

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

(16) = TensorReduce((16));

$\int \frac{{\mathrm{p\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{\mathrm{p\_\_1}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}}{\left(I\mathbf{\epsilon }-{\mathrm{m\_\_1}}^2+{\mathrm{p\_\_1}}^2\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=\frac{4\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)\left(\frac{g_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}{\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)}^2}{4}+\left(\frac{{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(-2+\mathrm{ϵ}\right){\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}}{2}-{\mathrm{m\_\_1}}^2{g}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}\right)\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)+{\mathrm{m\_\_1}}^2{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\right)\int \frac{1}{\left({\mathrm{p\_\_1}}^2-{\mathrm{m\_\_1}}^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+\left(-2g_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}{\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)}^2+\left(4{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(-1+\mathrm{ϵ}\right){\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}-{\mathrm{m\_\_1}}^2{g}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}\right)\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)-2{\mathrm{m\_\_1}}^2{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(-2+\mathrm{ϵ}\right)\right)\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+2\int \frac{1}{{\mathrm{p\_\_1}}^2-{\mathrm{m\_\_1}}^2+\mathrm{i}\mathbf{\epsilon }}\ⅆ{\mathrm{p\_\_1}}^4\left({\mathrm{P\_\_1}}^2+{\mathrm{m\_\_1}}^2-\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)\left(\frac{g_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)}{2}+{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\left(-2+\mathrm{ϵ}\right){\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}\right)}{\left(-12+8\mathrm{ϵ}\right){\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)}^2}$

TensorBasis([P__1], [~mu, ~nu]);

$\left[g_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\right]$

TensorReduce((16), step = 1);

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{step}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{outputstep}\text{'}$

$\int \frac{{\mathrm{p\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{\mathrm{p\_\_1}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}}{\left(I\mathbf{\epsilon }-{\mathrm{m\_\_1}}^2+{\mathrm{p\_\_1}}^2\right)\left({\left(\mathrm{p\_\_1}-\mathrm{P\_\_1}\right)}^2-{\mathrm{m\_\_1}}^2+\mathrm{i}\mathbf{\epsilon }\right)}\ⅆ{\mathrm{p\_\_1}}^4=C_2{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}+C_1{g}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}$

TensorReduce((16), step = 2);

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{step}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{outputstep}\text{'}$

$\left[\begin{array}{c}{g}_{\mathrm{\mu },\mathrm{\nu }}{𝕋}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}\left(2\,0\,{\mathrm{m\_\_1}}^2\,-\mathrm{P\_\_1}\,{\mathrm{m\_\_1}}^2\,\mathrm{p\_\_1}\,0\right)=C_1\left(4-2\mathrm{ϵ}\right)+C_2{\mathrm{P\_\_1}}_{\mathrm{\nu }}{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\\ {\mathrm{P\_\_1}}_{\mathrm{\mu }}{\mathrm{P\_\_1}}_{\mathrm{\nu }}{𝕋}_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}\left(2\,0\,{\mathrm{m\_\_1}}^2\,-\mathrm{P\_\_1}\,{\mathrm{m\_\_1}}^2\,\mathrm{p\_\_1}\,0\right)={\mathrm{P\_\_1}}_{\mathrm{\mu }}{\mathrm{P\_\_1}}_{\mathrm{\nu }}{C}_2{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}+C_1{\mathrm{P\_\_1}}_{\mathrm{\nu }}{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}\end{array}\right]$

TensorReduce((16), step = 4);

$\mathrm{*\; Partial\; match\; of\; \text{'}}\mathrm{step}\mathrm{\text{'}\; against\; keyword\; \text{'}}\mathrm{outputstep}\text{'}$

$\left[\begin{array}{c}𝕋\left[\right]\left(1\,-\mathrm{P\_\_1}\,{\mathrm{m\_\_1}}^2\,\mathrm{p\_\_1}\,0\right)+{\mathrm{m\_\_1}}^2𝕋\left[\right]\left(2\,0\,{\mathrm{m\_\_1}}^2\,-\mathrm{P\_\_1}\,{\mathrm{m\_\_1}}^2\,\mathrm{p\_\_1}\,0\right)=C_2\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)-2C_1\left(-2+\mathrm{ϵ}\right)\\ \frac{{\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)}^2𝕋\left[\right]\left(2\,0\,{\mathrm{m\_\_1}}^2\,-\mathrm{P\_\_1}\,{\mathrm{m\_\_1}}^2\,\mathrm{p\_\_1}\,0\right)}{4}+\frac{\left(-{\mathrm{m\_\_1}}^2+2\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)𝕋\left[\right]\left(1\,-\mathrm{P\_\_1}\,{\mathrm{m\_\_1}}^2\,\mathrm{p\_\_1}\,0\right)}{4}+\frac{𝕋\left[\right]\left(1\,0\,{\mathrm{m\_\_1}}^2\,\mathrm{p\_\_1}\,0\right)\left({\mathrm{P\_\_1}}^2+{\mathrm{m\_\_1}}^2-\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)}{4}=\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)\left(C_2\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)+C_1\right)\end{array}\right]$

FromAbstractRepresentation((21)[2]);

$\frac{{\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)}^2\int \frac{1}{\left({\mathrm{p\_\_1}}^2-{\mathrm{m\_\_1}}^2+\mathrm{i}\mathbf{\epsilon }\right)\left({\left(\mathrm{p\_\_1}-\mathrm{P\_\_1}\right)}^2-{\mathrm{m\_\_1}}^2+\mathrm{i}\mathbf{\epsilon }\right)}\ⅆ{\mathrm{p\_\_1}}^4}{4}+\frac{\left(-{\mathrm{m\_\_1}}^2+2\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)\int \frac{1}{{\left(\mathrm{p\_\_1}-\mathrm{P\_\_1}\right)}^2-{\mathrm{m\_\_1}}^2+\mathrm{i}\mathbf{\epsilon }}\ⅆ{\mathrm{p\_\_1}}^4}{4}+\frac{\int \frac{1}{{\mathrm{p\_\_1}}^2-{\mathrm{m\_\_1}}^2+\mathrm{i}\mathbf{\epsilon }}\ⅆ{\mathrm{p\_\_1}}^4\left({\mathrm{P\_\_1}}^2+{\mathrm{m\_\_1}}^2-\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)}{4}=\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)\left(C_2\left(\mathrm{P\_\_1}·\mathrm{P\_\_1}\right)+C_1\right)$

TensorBasis([P__1, P__2], [~mu, ~nu, ~rho], symmetrize = false);

$\left[g_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\,g_{\phantom{}\phantom{\mathrm{\mu },\mathrm{\nu }}}^{\phantom{}\mathrm{\mu },\mathrm{\nu }}{\mathrm{P\_\_2}}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\,{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\,{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{\mathrm{P\_\_2}}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\,{\mathrm{P\_\_1}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{\mathrm{P\_\_2}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{\mathrm{P\_\_2}}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\,{\mathrm{P\_\_2}}_{\phantom{}\phantom{\mathrm{\mu }}}^{\phantom{}\mathrm{\mu }}{\mathrm{P\_\_2}}_{\phantom{}\phantom{\mathrm{\nu }}}^{\phantom{}\mathrm{\nu }}{\mathrm{P\_\_2}}_{\phantom{}\phantom{\mathrm{\rho }}}^{\phantom{}\mathrm{\rho }}\right]$

nops((23));

$6$

[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.