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

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

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

 > interface(imaginaryunit = i):

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

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

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

 > %FeynmanIntegral(p__1[~mu]/((p__1^2 - m__phi^2 + i * epsilon)*((p__1 - P__1)^2 - m__1^2 + i * epsilon)), p__1);
 $\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\int }\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\frac{{\mathrm{p__1}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}}{\left({\mathrm{p__1}}^{{2}}{-}{\mathrm{m__φ}}^{{2}}{+}{i}{}{\mathbf{\epsilon }}\right){}\left({\left(\mathrm{p__1}{-}\mathrm{P__1}\right)}^{{2}}{-}{\mathrm{m__1}}^{{2}}{+}{i}{}{\mathbf{\epsilon }}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{p__1}}^{{4}}$ (3)
 > TensorReduce((3), step = 1);
 $\mathrm{* Partial match of \text{'}}\mathrm{step}\mathrm{\text{'} against keyword \text{'}}\mathrm{outputstep}\text{'}$
 $\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\int }\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\frac{{\mathrm{p__1}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}}{\left({\mathrm{p__1}}^{{2}}{-}{\mathrm{m__φ}}^{{2}}{+}{i}{}{\mathbf{\epsilon }}\right){}\left({\left(\mathrm{p__1}{-}\mathrm{P__1}\right)}^{{2}}{-}{\mathrm{m__1}}^{{2}}{+}{i}{}{\mathbf{\epsilon }}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{p__1}}^{{4}}{=}{{C}}_{{1}}{}{\mathrm{P__1}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}$ (4)

opening the way for the reduction process

 > (3) = TensorReduce((3));
 $\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\int }\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\frac{{\mathrm{p__1}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}}{\left({\mathrm{p__1}}^{{2}}{-}{\mathrm{m__φ}}^{{2}}{+}{i}{}{\mathbf{\epsilon }}\right){}\left({\left(\mathrm{p__1}{-}\mathrm{P__1}\right)}^{{2}}{-}{\mathrm{m__1}}^{{2}}{+}{i}{}{\mathbf{\epsilon }}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{p__1}}^{{4}}{=}{-}\frac{{\mathrm{P__1}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left(\left({\mathrm{m__1}}^{{2}}{-}{\mathrm{m__φ}}^{{2}}{-}\mathrm{P__1}{·}\mathrm{P__1}\right){}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\int }\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\frac{{1}}{\left({\mathrm{p__1}}^{{2}}{-}{\mathrm{m__φ}}^{{2}}{+}{i}{}{\mathbf{\epsilon }}\right){}\left({\left(\mathrm{p__1}{-}\mathrm{P__1}\right)}^{{2}}{-}{\mathrm{m__1}}^{{2}}{+}{i}{}{\mathbf{\epsilon }}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{p__1}}^{{4}}{+}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\int }\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\frac{{1}}{{\mathrm{p__1}}^{{2}}{-}{\mathrm{m__φ}}^{{2}}{+}{i}{}{\mathbf{\epsilon }}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{p__1}}^{{4}}{-}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\int }\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\frac{{1}}{{\left(\mathrm{p__1}{-}\mathrm{P__1}\right)}^{{2}}{-}{\mathrm{m__1}}^{{2}}{+}{i}{}{\mathbf{\epsilon }}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{p__1}}^{{4}}\right)}{{2}{}\left(\mathrm{P__1}{·}\mathrm{P__1}\right)}$ (5)

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

 > (3) = Evaluate((3));
  (6)

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

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

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

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

The non-symmetrized form of this basis

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

Two more realistic examples

 > TensorBasis([P__1, P__2, P__3], [mu, nu]);
 $\left[{{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{,}{\mathrm{P__1}}_{{\mathrm{\mu }}}{}{\mathrm{P__1}}_{{\mathrm{\nu }}}{,}{\mathrm{P__1}}_{{\mathrm{\mu }}}{}{\mathrm{P__2}}_{{\mathrm{\nu }}}{+}{\mathrm{P__1}}_{{\mathrm{\nu }}}{}{\mathrm{P__2}}_{{\mathrm{\mu }}}{,}{\mathrm{P__1}}_{{\mathrm{\mu }}}{}{\mathrm{P__3}}_{{\mathrm{\nu }}}{+}{\mathrm{P__1}}_{{\mathrm{\nu }}}{}{\mathrm{P__3}}_{{\mathrm{\mu }}}{,}{\mathrm{P__2}}_{{\mathrm{\mu }}}{}{\mathrm{P__2}}_{{\mathrm{\nu }}}{,}{\mathrm{P__2}}_{{\mathrm{\mu }}}{}{\mathrm{P__3}}_{{\mathrm{\nu }}}{+}{\mathrm{P__2}}_{{\mathrm{\nu }}}{}{\mathrm{P__3}}_{{\mathrm{\mu }}}{,}{\mathrm{P__3}}_{{\mathrm{\mu }}}{}{\mathrm{P__3}}_{{\mathrm{\nu }}}\right]$ (10)
 > TensorBasis([P__1], [mu, nu, alpha]);
 $\left[{{g}}_{{\mathrm{\mu }}{,}{\mathrm{\nu }}}{}{\mathrm{P__1}}_{{\mathrm{\alpha }}}{+}{{g}}_{{\mathrm{\alpha }}{,}{\mathrm{\nu }}}{}{\mathrm{P__1}}_{{\mathrm{\mu }}}{+}{{g}}_{{\mathrm{\alpha }}{,}{\mathrm{\mu }}}{}{\mathrm{P__1}}_{{\mathrm{\nu }}}{,}{\mathrm{P__1}}_{{\mathrm{\mu }}}{}{\mathrm{P__1}}_{{\mathrm{\nu }}}{}{\mathrm{P__1}}_{{\mathrm{\alpha }}}\right]$ (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.