TensorBasis - Maple Help

Physics[FeynmanIntegral][TensorBasis] - compute a basis of tensor structures from a given list of external momentum and another one with free spacetime indices

 Calling Sequence TensorBasis(list_of_external_momenta, list_of_spacetime_indices) TensorBasis(list_of_external_momenta, list_of_spacetime_indices, symmetrize = ..)

Parameters

 list_of_external_momenta - a list of external momenta, which by convention in the FeynmanIntegral package are written as P__n where n is an integer list_of_spacetime_indices - a list of spacetime indices, that could be covariant or contravariant (preceded by ) symmetrize = .. - (optional) the right-hand side can be true (default) or false, to symmetrize the products of external momenta that appear in the returned basis

Description

 • 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.

Examples

 > $\mathrm{with}\left(\mathrm{Physics}\right):$
 > $\mathrm{with}\left(\mathrm{FeynmanIntegral}\right)$
 $\left[{\mathrm{Evaluate}}{,}{\mathrm{ExpandDimension}}{,}{\mathrm{FromAbstractRepresentation}}{,}{\mathrm{Parametrize}}{,}{\mathrm{Series}}{,}{\mathrm{SumLookup}}{,}{\mathrm{TensorBasis}}{,}{\mathrm{TensorReduce}}{,}{\mathrm{ToAbstractRepresentation}}{,}{\mathrm{\epsilon }}{,}{\mathrm{ϵ}}\right]$ (1)

To remain closer to textbook notation, display the imaginary unit with a lowercase $i$

 > $\mathrm{interface}\left(\mathrm{imaginaryunit}=i\right):$

The simplest case is that of a single external momentum and only one spacetime index

 > $\mathrm{TensorBasis}\left(\left[\mathit{P__1}\right],\left[\mathrm{μ}\right]\right)$
 $\left[{\mathrm{P__1}}_{{\mathrm{\mu }}}\right]$ (2)

This basis allows for expressing the following tensor Feynman integral as a linear combination of the elements of the basis

 > $\mathrm{%FeynmanIntegral}\left(\frac{{\mathit{p__1}}_{\mathrm{~mu}}}{\left({\mathit{p__1}}^{2}-{\mathit{m__phi}}^{2}+i\mathrm{ε}\right)\left({\left(\mathit{p__1}-\mathit{P__1}\right)}^{2}-{\mathit{m__1}}^{2}+i\mathrm{ε}\right)},\mathit{p__1}\right)$
 ${\mathrm{%FeynmanIntegral}}{}\left(\frac{{\mathrm{p__1}}_{{\mathrm{~mu}}}}{\left({\mathrm{p__1}}^{{2}}{-}{\mathrm{m__φ}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\left(\mathrm{p__1}{-}\mathrm{P__1}\right)}^{{2}}{-}{\mathrm{m__1}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right)}{,}\mathrm{p__1}\right)$ (3)
 > $\mathrm{TensorReduce}\left(,\mathrm{step}=1\right)$
 $\mathrm{* Partial match of \text{'}}\mathrm{step}\mathrm{\text{'} against keyword \text{'}}\mathrm{outputstep}\text{'}$
 ${\mathrm{%FeynmanIntegral}}{}\left(\frac{{\mathrm{p__1}}_{{\mathrm{~mu}}}}{\left({\mathrm{p__1}}^{{2}}{-}{\mathrm{m__φ}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\left(\mathrm{p__1}{-}\mathrm{P__1}\right)}^{{2}}{-}{\mathrm{m__1}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right)}{,}\mathrm{p__1}\right){=}{{C}}_{{1}}{}{\mathrm{P__1}}_{{\mathrm{~mu}}}$ (4)

opening the way for the reduction process

 > $=\mathrm{TensorReduce}\left(\right)$
 ${\mathrm{%FeynmanIntegral}}{}\left(\frac{{\mathrm{p__1}}_{{\mathrm{~mu}}}}{\left({\mathrm{p__1}}^{{2}}{-}{\mathrm{m__φ}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\left(\mathrm{p__1}{-}\mathrm{P__1}\right)}^{{2}}{-}{\mathrm{m__1}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right)}{,}\mathrm{p__1}\right){=}{-}\frac{{1}}{{2}}{}\frac{{\mathrm{P__1}}_{{\mathrm{~mu}}}{}\left(\left({\mathrm{m__1}}^{{2}}{-}{\mathrm{m__φ}}^{{2}}{-}{\mathrm{%.}}{}\left(\mathrm{P__1}{,}\mathrm{P__1}\right)\right){}{\mathrm{%FeynmanIntegral}}{}\left(\frac{{1}}{\left({\mathrm{p__1}}^{{2}}{-}{\mathrm{m__φ}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\left(\mathrm{p__1}{-}\mathrm{P__1}\right)}^{{2}}{-}{\mathrm{m__1}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right)}{,}\mathrm{p__1}\right){+}{\mathrm{%FeynmanIntegral}}{}\left(\frac{{1}}{{\mathrm{p__1}}^{{2}}{-}{\mathrm{m__φ}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}}{,}\mathrm{p__1}\right){-}{\mathrm{%FeynmanIntegral}}{}\left(\frac{{1}}{{\left(\mathrm{p__1}{-}\mathrm{P__1}\right)}^{{2}}{-}{\mathrm{m__1}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}}{,}\mathrm{p__1}\right)\right)}{{\mathrm{%.}}{}\left(\mathrm{P__1}{,}\mathrm{P__1}\right)}$ (5)

and ultimately leading to its symbolic computation by evaluating the scalar FeynmanIntegrals above

 > $=\mathrm{Evaluate}\left(\right)$
 ${\mathrm{%FeynmanIntegral}}{}\left(\frac{{\mathrm{p__1}}_{{\mathrm{~mu}}}}{\left({\mathrm{p__1}}^{{2}}{-}{\mathrm{m__φ}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\left(\mathrm{p__1}{-}\mathrm{P__1}\right)}^{{2}}{-}{\mathrm{m__1}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right)}{,}\mathrm{p__1}\right){=}{-}\frac{{1}}{{2}}{}\frac{{\mathrm{P__1}}_{{\mathrm{~mu}}}{}\left({-}{i}{}\left({\mathrm{m__1}}^{{2}}{-}{\mathrm{m__φ}}^{{2}}{-}{\mathrm{%.}}{}\left(\mathrm{P__1}{,}\mathrm{P__1}\right)\right){}{{\mathrm{π}}}^{{2}{-}{\mathrm{ϵ}}}{}{\mathrm{%sum}}{}\left({\mathrm{%sum}}{}\left({-}\frac{{\mathrm{Γ}}{}\left({n}{+}\mathrm{n__1}{+}{1}\right){}{\mathrm{m__φ}}^{{-}{2}{}{\mathrm{ϵ}}{-}{2}{}{n}{-}{2}{}\mathrm{n__1}}{}{\mathrm{P__1}}^{{2}{}\mathrm{n__1}}{}{\left({-}{\mathrm{m__1}}^{{2}}{+}{\mathrm{m__φ}}^{{2}}\right)}^{{n}}{}{\mathrm{Γ}}{}\left({\mathrm{ϵ}}{+}{n}{+}\mathrm{n__1}\right)}{{\mathrm{Γ}}{}\left({2}{}\mathrm{n__1}{+}{n}{+}{2}\right){}{\mathrm{Γ}}{}\left({1}{+}{n}\right)}{,}\mathrm{n__1}{=}{0}{..}{\mathrm{∞}}\right){,}{n}{=}{0}{..}{\mathrm{∞}}\right){-}{i}{}{{\mathrm{π}}}^{{2}{-}{\mathrm{ϵ}}}{}{\mathrm{m__φ}}^{{2}{-}{2}{}{\mathrm{ϵ}}}{}{\mathrm{Γ}}{}\left({-}{1}{+}{\mathrm{ϵ}}\right){+}{i}{}{{\mathrm{π}}}^{{2}{-}{\mathrm{ϵ}}}{}{\mathrm{m__1}}^{{2}{-}{2}{}{\mathrm{ϵ}}}{}{\mathrm{Γ}}{}\left({-}{1}{+}{\mathrm{ϵ}}\right)\right)}{{\mathrm{%.}}{}\left(\mathrm{P__1}{,}\mathrm{P__1}\right)}$ (6)

The case of two spacetime indices already results in a basis even when there are no external momenta

 > $\mathrm{TensorBasis}\left(\left[\right],\left[\mathrm{μ},\mathrm{ν}\right]\right)$
 $\left[{{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}\right]$ (7)

Products of the metric are introduced when the number of indices makes that necessary

 > $\mathrm{TensorBasis}\left(\left[\right],\left[\mathrm{μ},\mathrm{ν},\mathrm{α},\mathrm{β}\right]\right)$
 $\left[{{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{}{{g}}_{{\mathrm{\alpha }}{,}{\mathrm{\beta }}}{+}{{g}}_{{\mathrm{\alpha }}{,}{\mathrm{\mu }}}{}{{g}}_{{\mathrm{\beta }}{,}{\mathrm{\nu }}}{+}{{g}}_{{\mathrm{\alpha }}{,}{\mathrm{\nu }}}{}{{g}}_{{\mathrm{\beta }}{,}{\mathrm{\mu }}}\right]$ (8)

The non-symmetrized form of this basis

 > $\mathrm{TensorBasis}\left(\left[\right],\left[\mathrm{μ},\mathrm{ν},\mathrm{α},\mathrm{β}\right],\mathrm{symmetrize}=\mathrm{false}\right)$
 $\left[{{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{}{{g}}_{{\mathrm{\alpha }}{,}{\mathrm{\beta }}}\right]$ (9)

Two more realistic examples

 > $\mathrm{TensorBasis}\left(\left[\mathit{P__1},\mathit{P__2},\mathit{P__3}\right],\left[\mathrm{μ},\mathrm{ν}\right]\right)$
 $\left[{{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{\mathrm{P__1}}_{{\mathrm{\mu }}}{}{\mathrm{P__1}}_{{\mathrm{\nu }}}{,}{\mathrm{P__1}}_{{\mathrm{\mu }}}{}{\mathrm{P__2}}_{{\mathrm{\nu }}}{+}{\mathrm{P__1}}_{{\mathrm{\nu }}}{}{\mathrm{P__2}}_{{\mathrm{\mu }}}{,}{\mathrm{P__1}}_{{\mathrm{\mu }}}{}{\mathrm{P__3}}_{{\mathrm{\nu }}}{+}{\mathrm{P__1}}_{{\mathrm{\nu }}}{}{\mathrm{P__3}}_{{\mathrm{\mu }}}{,}{\mathrm{P__2}}_{{\mathrm{\mu }}}{}{\mathrm{P__2}}_{{\mathrm{\nu }}}{,}{\mathrm{P__2}}_{{\mathrm{\mu }}}{}{\mathrm{P__3}}_{{\mathrm{\nu }}}{+}{\mathrm{P__2}}_{{\mathrm{\nu }}}{}{\mathrm{P__3}}_{{\mathrm{\mu }}}{,}{\mathrm{P__3}}_{{\mathrm{\mu }}}{}{\mathrm{P__3}}_{{\mathrm{\nu }}}\right]$ (10)
 > $\mathrm{TensorBasis}\left(\left[\mathit{P__1}\right],\left[\mathrm{μ},\mathrm{ν},\mathrm{α}\right]\right)$
 $\left[{{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{}{\mathrm{P__1}}_{{\mathrm{\alpha }}}{+}{{g}}_{{\mathrm{\alpha }}{,}{\mathrm{\nu }}}{}{\mathrm{P__1}}_{{\mathrm{\mu }}}{+}{{g}}_{{\mathrm{\alpha }}{,}{\mathrm{\mu }}}{}{\mathrm{P__1}}_{{\mathrm{\nu }}}{,}{\mathrm{P__1}}_{{\mathrm{\mu }}}{}{\mathrm{P__1}}_{{\mathrm{\nu }}}{}{\mathrm{P__1}}_{{\mathrm{\alpha }}}\right]$ (11)

References

 [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.

Compatibility

 • The Physics[FeynmanIntegral][TensorBasis] command was introduced in Maple 2021.