TensorBasis receives a list of external momenta, which by convention in the FeynmanIntegral package are written as P__n where n is an integer, and a list of spacetime indices, which by default are represented by greek letters (to change the kind of letter see Setup) and returns a tensor basis onto which one can expand a tensorial structure with as many indices as in list_of_spacetime_indices.

The tensor basis returned is constructed by taking the multiple-Cartesian product of the list of external momenta, and the metric $g_{\mathrm{\mu },\mathrm{\nu }}$, as many times as the number of indices in the list of spacetime indices, and discarding permutations.

The tensor basis is returned symmetrized, e.g. if a product of two tensors $P_1^{\mathrm{\mu }}{P}_2^{\mathrm{\nu }}$ appears in the basis, then the output contains $P_1^{\mathrm{\mu }}{P}_2^{\mathrm{\nu }}+P_2^{\mathrm{\mu }}{P}_1^{\mathrm{\nu }}$. To receive the tensor basis non-symmetrized pass the optional argument symmetrize = false

These tensor basis are relevant in the context of the Passarino-Veltman approach for the reduction of tensor to scalar Feynman integrals implemented in the TensorReduce command.

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):

TensorBasis([P__1], [mu]);

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

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

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

TensorReduce((3), step = 1);

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

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

(3) = TensorReduce((3));

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

(3) = Evaluate((3));

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

$\left[g_{\mathrm{\mu },\mathrm{\nu }}\right]$

TensorBasis([], [mu, nu, alpha, beta]);

$\left[g_{\mathrm{\mu },\mathrm{\nu }}{g}_{\mathrm{\alpha },\mathrm{\beta }}+g_{\mathrm{\alpha },\mathrm{\mu }}{g}_{\mathrm{\beta },\mathrm{\nu }}+g_{\mathrm{\alpha },\mathrm{\nu }}{g}_{\mathrm{\beta },\mathrm{\mu }}\right]$

TensorBasis([], [mu, nu, alpha, beta], symmetrize = false);

$\left[g_{\mathrm{\mu },\mathrm{\nu }}{g}_{\mathrm{\alpha },\mathrm{\beta }}\right]$

TensorBasis([P__1, P__2, P__3], [mu, nu]);

$\left[g_{\mathrm{\mu },\mathrm{\nu }}\,{\mathrm{P\_\_1}}_{\mathrm{\mu }}{\mathrm{P\_\_1}}_{\mathrm{\nu }}\,{\mathrm{P\_\_1}}_{\mathrm{\mu }}{\mathrm{P\_\_2}}_{\mathrm{\nu }}+{\mathrm{P\_\_1}}_{\mathrm{\nu }}{\mathrm{P\_\_2}}_{\mathrm{\mu }}\,{\mathrm{P\_\_1}}_{\mathrm{\mu }}{\mathrm{P\_\_3}}_{\mathrm{\nu }}+{\mathrm{P\_\_1}}_{\mathrm{\nu }}{\mathrm{P\_\_3}}_{\mathrm{\mu }}\,{\mathrm{P\_\_2}}_{\mathrm{\mu }}{\mathrm{P\_\_2}}_{\mathrm{\nu }}\,{\mathrm{P\_\_2}}_{\mathrm{\mu }}{\mathrm{P\_\_3}}_{\mathrm{\nu }}+{\mathrm{P\_\_2}}_{\mathrm{\nu }}{\mathrm{P\_\_3}}_{\mathrm{\mu }}\,{\mathrm{P\_\_3}}_{\mathrm{\mu }}{\mathrm{P\_\_3}}_{\mathrm{\nu }}\right]$

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

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

[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.