When computing the scattering matrix $S$ for a particle process (momentum representation, see FeynmanDiagrams) the result, at one or more loops, contains Feynman integrals. Generally speaking, these integrals can be evaluated using dimensional regularization with the Evaluate command, whose output is corresponds to computing the integral in d-dimensions. In this context, ExpandDimension takes the dimension as $d=4-2\mathrm{ϵ}$ and expands around $\mathrm{ϵ}=0$ keeping terms up to order $\mathrm{O}\left(\mathrm{ϵ}\right)$

The expansion of sums is done by first analyzing the structure of the GAMMA poles of the summand and, depending on the case, splitting the sum into two parts, separating the part that is divergent when $d=4$. Then $d$ is taken equal to $4-2\mathrm{ϵ}$ and the expansion around $\mathrm{ϵ}=0$ is done using Series.

The Series command works as the series command but - say $z$ is the series expansion variable - it always returns up to order, so a series structure up to $\mathrm{O}\left(z^{\mathrm{order}}\right)$; for the difference with series, see the Examples section.

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

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

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

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

$L≔\mathrm{\lambda }{\mathrm{\phi }\left(X\right)}^3$

$L≔\mathrm{\lambda }{\mathrm{\phi }\left(X\right)}^3$

$\mathrm{FeynmanDiagrams}\left(L\,\mathrm{incomingparticles}=\left[\mathrm{\phi }\right]\,\mathrm{outgoingparticles}=\left[\mathrm{\phi }\right]\,\mathrm{numberofloops}=1\,\mathrm{diagrams}\right)$

$\int \frac{9{\mathrm{\lambda }}^2\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)}{8{\mathrm{\pi }}^3\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}\left({\left(\mathrm{P\_\_1}+\mathrm{p\_\_2}\right)}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }\right)\left({\mathrm{p\_\_2}}^2-{\mathrm{m\_\_\φ}}^2+\mathrm{i}\mathbf{\epsilon }\right)}\ⅆ{\mathrm{p\_\_2}}^4$

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

$\frac{\frac{9\mathrm{i}}{8}{\mathrm{\pi }}^{-1-\mathrm{ϵ}}{\mathrm{\lambda }}^2\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)\left(\sum _{n=0}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(\mathrm{ϵ}+n\right){\mathrm{m\_\_\φ}}^{-2\mathrm{ϵ}-2n}\mathrm{\Gamma }\left(n+1\right){\mathrm{P\_\_1}}^{2n}}{\mathrm{\Gamma }\left(2n+2\right)}\right)}{\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}}$

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

$\frac{\frac{9\mathrm{i}}{8}{\mathrm{\lambda }}^2\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)}{\mathrm{\pi }\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}}{\mathrm{ϵ}}^{\mathrm{-1}}+\frac{-\frac{9\mathrm{i}}{8}{\mathrm{\lambda }}^2\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)\left(2\ln \left(\mathrm{m\_\_\φ}\right)+\mathrm{\gamma }-\left(\sum _{n=1}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(n\right)\mathrm{\Gamma }\left(n+1\right){\mathrm{P\_\_1}}^{2n}}{\mathrm{\Gamma }\left(2n+2\right){\mathrm{m\_\_\φ}}^{2n}}\right)+\ln \left(\mathrm{\pi }\right)\right)}{\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}\mathrm{\pi }}+O\left(\mathrm{ϵ}\right)$

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

$\frac{\frac{9\mathrm{i}}{8}{\mathrm{\lambda }}^2\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)}{\mathrm{\pi }\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}}{\mathrm{ϵ}}^{\mathrm{-1}}+\frac{-\frac{9\mathrm{i}}{8}{\mathrm{\lambda }}^2\mathrm{\delta }\left(-\mathrm{P\_\_2}+\mathrm{P\_\_1}\right)\left(2\ln \left(\mathrm{m\_\_\φ}\right)+\mathrm{\gamma }-\left(\sum _{n=1}^{\mathrm{\infty }}\frac{\mathrm{\Gamma }\left(n\right)\mathrm{\Gamma }\left(n+1\right){\mathrm{P\_\_1}}^{2n}}{\mathrm{\Gamma }\left(2n+2\right){\mathrm{m\_\_\φ}}^{2n}}\right)+\ln \left(\mathrm{\pi }\right)\right)}{\sqrt{\mathrm{E\_\_1}\mathrm{E\_\_2}}\mathrm{\pi }}+O\left(\mathrm{ϵ}\right)$

$\exp \left(z\right)\mathrm{\Gamma }\left(z\right)$

${\ⅇ}^z\mathrm{\Gamma }\left(z\right)$

$\mathrm{series}\left(\,z\,3\right)$

$z^{\mathrm{-1}}+-\mathrm{\gamma }+1+\left(\frac{1}{12}{\mathrm{\pi }}^2+\frac{1}{2}{\mathrm{\gamma }}^2-\mathrm{\gamma }+\frac{1}{2}\right)z+O\left(z^2\right)$

$\mathrm{Series}\left(\,z\,3\right)$

$z^{\mathrm{-1}}+-\mathrm{\gamma }+1+\left(\frac{1}{12}{\mathrm{\pi }}^2+\frac{1}{2}{\mathrm{\gamma }}^2-\mathrm{\gamma }+\frac{1}{2}\right)z+\left(-\frac{\mathrm{\zeta }\left(3\right)}{3}-\frac{{\mathrm{\pi }}^2\mathrm{\gamma }}{12}-\frac{{\mathrm{\gamma }}^3}{6}+\frac{{\mathrm{\pi }}^2}{12}+\frac{{\mathrm{\gamma }}^2}{2}-\frac{\mathrm{\gamma }}{2}+\frac{1}{6}\right)z^2+O\left(z^3\right)$

$\mathrm{series}\left(\mathrm{\Gamma }\left(\mathrm{ϵ}+n\right)\,\mathrm{ϵ}\,1\right)\mathbf{assuming}n::\mathrm{nonposint}$

$\frac{1}{{\left(\mathrm{-1}\right)}^n\mathrm{\Gamma }\left(1-n\right)}{\mathrm{ϵ}}^{\mathrm{-1}}+\frac{\mathrm{\Psi }\left(1-n\right)}{{\left(\mathrm{-1}\right)}^n\mathrm{\Gamma }\left(1-n\right)}+\frac{\frac{{\mathrm{\pi }}^2}{6{\left(\mathrm{-1}\right)}^n}-\frac{\frac{{\mathrm{\Psi }}^{\left(1\right)}\left(1-n\right)\mathrm{\Gamma }\left(1-n\right)}{2}+\frac{{\mathrm{\Psi }\left(1-n\right)}^2\mathrm{\Gamma }\left(1-n\right)}{2}}{{\left(\mathrm{-1}\right)}^n\mathrm{\Gamma }\left(1-n\right)}+\frac{{\mathrm{\Psi }\left(1-n\right)}^2}{{\left(\mathrm{-1}\right)}^n}}{\mathrm{\Gamma }\left(1-n\right)}\mathrm{ϵ}+O\left({\mathrm{ϵ}}^2\right)$

$\mathrm{Series}\left(\mathrm{\Gamma }\left(\mathrm{ϵ}+n\right)\,\mathrm{ϵ}\,1\right)\mathbf{assuming}n::\mathrm{nonposint}$

$\frac{1}{{\left(\mathrm{-1}\right)}^n\mathrm{\Gamma }\left(1-n\right)}{\mathrm{ϵ}}^{\mathrm{-1}}+\frac{\mathrm{\Psi }\left(1-n\right)}{{\left(\mathrm{-1}\right)}^n\mathrm{\Gamma }\left(1-n\right)}+O\left(\mathrm{ϵ}\right)$

[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.

The Physics[FeynmanIntegral][ExpandDimension] and Physics[FeynmanIntegral][Series] commands were introduced in Maple 2021.

For more information on Maple 2021 changes, see Updates in Maple 2021.