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. In this context, ToAbstractRepresentation rewrites 1-loop Feynman integrals in an abstract form, convenient for performing the integral's tensor reduction, that in the context of the FeynmanIntegral package can be performed using the TensorReduce command.

The FromAbstractRepresentation command reverses the operation performed by ToAbstractRepresentation.

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

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

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

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

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

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

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

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

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

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

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

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

$\mathrm{TensorReduce}\left(\,\mathrm{step}=2\right)$

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

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

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

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

$\mathrm{TensorReduce}\left(\,\mathrm{step}=4\right)$

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

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

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

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

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

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

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

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

[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.