ExpandDimension - Maple Help

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Physics[FeynmanIntegral][ExpandDimension] - expand a d-dimensional result for a Feynman integral keeping terms up to order 0 in the dimensional parameter FeynmanIntegral:-varepsilon

Physics[FeynmanIntegral][Series] - expand in series returning a result always up to the order specified

 Calling Sequence ExpandDimension(expression) Series(expression, z, order)

Parameters

 expression - any expression, equation, set, list or matrix of them, typically involving FeynmanIntegral:-varepsilon dimensional parameter z - the expansion variable order - (optional) a nonnegative integer specifying the order, so that the result is always up to $\mathrm{O}\left({z}^{\mathrm{order}}\right)$

Description

 • 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.

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 $i$

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

An example departing from an interaction Lagrangian

 > $L≔\mathrm{λ}{\mathrm{φ}\left(X\right)}^{3}$
 ${L}{≔}{\mathrm{\lambda }}{}{{\mathrm{\phi }}{}\left({X}\right)}^{{3}}$ (2)

A process with one incoming and one outgoing particle a 1-loop

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

 ${\mathrm{%FeynmanIntegral}}{}\left(\frac{{9}}{{8}}{}\frac{{{\mathrm{λ}}}^{{2}}{}{\mathrm{Dirac}}{}\left({-}\mathrm{P__2}{+}\mathrm{P__1}\right)}{{{\mathrm{π}}}^{{3}}{}\sqrt{\mathrm{E__1}{}\mathrm{E__2}}{}\left({\left(\mathrm{P__1}{+}\mathrm{p__2}\right)}^{{2}}{-}{\mathrm{m__φ}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right){}\left({\mathrm{p__2}}^{{2}}{-}{\mathrm{m__φ}}^{{2}}{+}{i}{}{\mathrm{Physics}}{:-}{\mathrm{FeynmanDiagrams}}{:-}{\mathrm{ε}}\right)}{,}\left[\left[\mathrm{p__2}\right]\right]\right)$ (3)

To evaluate the integral, using dimensional regularization, computing the integral over the loop momentum in dimension $d$, you can use  Evaluate

 > $\mathrm{Evaluate}\left(\right)$
 $\frac{\frac{{9}{}{i}}{{8}}{}{{\mathrm{\pi }}}^{{-}{1}{-}{\mathrm{ϵ}}}{}{{\mathrm{\lambda }}}^{{2}}{}{\mathrm{\delta }}{}\left({-}\mathrm{P__2}{+}\mathrm{P__1}\right){}\left({\mathrm{%sum}}{}\left(\frac{{\mathrm{Γ}}{}\left({\mathrm{ϵ}}{+}{n}\right){}{\mathrm{m__φ}}^{{-}{2}{}{\mathrm{ϵ}}{-}{2}{}{n}}{}{\mathrm{P__1}}^{{2}{}{n}}{}{\mathrm{Γ}}{}\left({n}{+}{1}\right)}{{\mathrm{Γ}}{}\left({2}{}{n}{+}{2}\right)}{,}{n}{=}{0}{..}{\mathrm{∞}}\right)\right)}{\sqrt{\mathrm{E__1}{}\mathrm{E__2}}}$ (4)

This result contains the dimensional parameter $\mathrm{ϵ}$. To expand the dimension of this result around $d=4-2\mathrm{ϵ}$ keeping terms up to order 0 in $\mathrm{ϵ}$ you can use

 > $\mathrm{ExpandDimension}\left(\right)$
 $\frac{\frac{{9}{}{i}}{{8}}{}{{\mathrm{\lambda }}}^{{2}}{}{\mathrm{\delta }}{}\left({-}\mathrm{P__2}{+}\mathrm{P__1}\right)}{{\mathrm{\pi }}{}\sqrt{\mathrm{E__1}{}\mathrm{E__2}}}{}{{\mathrm{ϵ}}}^{{-1}}{+}\frac{{-}\frac{{9}{}{i}}{{8}}{}{{\mathrm{\lambda }}}^{{2}}{}{\mathrm{\delta }}{}\left({-}\mathrm{P__2}{+}\mathrm{P__1}\right){}\left({2}{}{\mathrm{ln}}{}\left(\mathrm{m__φ}\right){+}{\mathrm{\gamma }}{-}\left({\mathrm{%sum}}{}\left(\frac{{\mathrm{Γ}}{}\left({n}\right){}{\mathrm{P__1}}^{{2}{}{n}}{}{\mathrm{Γ}}{}\left({n}{+}{1}\right)}{{\left({\mathrm{m__φ}}^{{n}}\right)}^{{2}}{}{\mathrm{Γ}}{}\left({2}{}{n}{+}{2}\right)}{,}{n}{=}{1}{..}{\mathrm{∞}}\right)\right){+}{\mathrm{ln}}{}\left({\mathrm{\pi }}\right)\right)}{\sqrt{\mathrm{E__1}{}\mathrm{E__2}}{}{\mathrm{\pi }}}{+}{O}{}\left({\mathrm{ϵ}}\right)$ (5)

Computing the integral without expanding and expand in a second step allows for better control and follow-up of the computation. Alternatively, you can compute the two steps in one go using the expanddimension option of Evaluate:

 > $\mathrm{Evaluate}\left(,\mathrm{expanddimension}\right)$
 $\frac{\frac{{9}{}{i}}{{8}}{}{{\mathrm{\lambda }}}^{{2}}{}{\mathrm{\delta }}{}\left({-}\mathrm{P__2}{+}\mathrm{P__1}\right)}{{\mathrm{\pi }}{}\sqrt{\mathrm{E__1}{}\mathrm{E__2}}}{}{{\mathrm{ϵ}}}^{{-1}}{+}\frac{{-}\frac{{9}{}{i}}{{8}}{}{{\mathrm{\lambda }}}^{{2}}{}{\mathrm{\delta }}{}\left({-}\mathrm{P__2}{+}\mathrm{P__1}\right){}\left({2}{}{\mathrm{ln}}{}\left(\mathrm{m__φ}\right){+}{\mathrm{\gamma }}{-}\left({\mathrm{%sum}}{}\left(\frac{{\mathrm{Γ}}{}\left({n}\right){}{\mathrm{P__1}}^{{2}{}{n}}{}{\mathrm{Γ}}{}\left({n}{+}{1}\right)}{{\left({\mathrm{m__φ}}^{{n}}\right)}^{{2}}{}{\mathrm{Γ}}{}\left({2}{}{n}{+}{2}\right)}{,}{n}{=}{1}{..}{\mathrm{∞}}\right)\right){+}{\mathrm{ln}}{}\left({\mathrm{\pi }}\right)\right)}{\sqrt{\mathrm{E__1}{}\mathrm{E__2}}{}{\mathrm{\pi }}}{+}{O}{}\left({\mathrm{ϵ}}\right)$ (6)

The Series command has the same syntax as series but always return up to $\mathrm{O}\left({z}^{\mathrm{order}}\right)$. Consider for instance:

 > ${ⅇ}^{z}\mathrm{GAMMA}\left(z\right)$
 ${{ⅇ}}^{{z}}{}{\mathrm{\Gamma }}{}\left({z}\right)$ (7)

Indicating the order as equal to 3, the output by series, however, can be relative; in this example it starts at $\frac{1}{z}$ and goes up to $\mathrm{O}\left({z}^{2}\right)$:

 > $\mathrm{series}\left(,z,3\right)$
 ${{z}}^{{-1}}{+}{-}{\mathrm{\gamma }}{+}{1}{+}\left(\frac{{1}}{{12}}{}{{\mathrm{\pi }}}^{{2}}{+}\frac{{1}}{{2}}{}{{\mathrm{\gamma }}}^{{2}}{-}{\mathrm{\gamma }}{+}\frac{{1}}{{2}}\right){}{z}{+}{O}{}\left({{z}}^{{2}}\right)$ (8)

The output of Series is up to $\mathrm{O}\left({z}^{\mathrm{order}}\right)$ regardless of degree of the first term of the series

 > $\mathrm{Series}\left(,z,3\right)$
 ${{z}}^{{-1}}{+}{-}{\mathrm{\gamma }}{+}{1}{+}\left(\frac{{1}}{{12}}{}{{\mathrm{\pi }}}^{{2}}{+}\frac{{1}}{{2}}{}{{\mathrm{\gamma }}}^{{2}}{-}{\mathrm{\gamma }}{+}\frac{{1}}{{2}}\right){}{z}{+}\left({-}\frac{{\mathrm{\zeta }}{}\left({3}\right)}{{3}}{-}\frac{{{\mathrm{\pi }}}^{{2}}{}{\mathrm{\gamma }}}{{12}}{-}\frac{{{\mathrm{\gamma }}}^{{3}}}{{6}}{+}\frac{{{\mathrm{\pi }}}^{{2}}}{{12}}{+}\frac{{{\mathrm{\gamma }}}^{{2}}}{{2}}{-}\frac{{\mathrm{\gamma }}}{{2}}{+}\frac{{1}}{{6}}\right){}{{z}}^{{2}}{+}{O}{}\left({{z}}^{{3}}\right)$ (9)

A different example, indicating the order equal to 1; here series default approach results in terms up to $\mathrm{O}\left({\mathrm{ϵ}}^{2}\right)$

 > $\mathrm{series}\left(\mathrm{GAMMA}\left(\mathrm{ϵ}+n\right),\mathrm{ϵ},1\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}n::\mathrm{nonposint}$
 $\frac{{1}}{{\left({-1}\right)}^{{n}}{}{\mathrm{\Gamma }}{}\left({1}{-}{n}\right)}{}{{\mathrm{ϵ}}}^{{-1}}{+}\frac{{\mathrm{\Psi }}{}\left({1}{-}{n}\right)}{{\left({-1}\right)}^{{n}}{}{\mathrm{\Gamma }}{}\left({1}{-}{n}\right)}{+}\frac{\frac{{{\mathrm{\pi }}}^{{2}}}{{6}{}{\left({-1}\right)}^{{n}}}{-}\frac{\frac{{{\mathrm{\Psi }}}^{\left({1}\right)}{}\left({1}{-}{n}\right){}{\mathrm{\Gamma }}{}\left({1}{-}{n}\right)}{{2}}{+}\frac{{{\mathrm{\Psi }}{}\left({1}{-}{n}\right)}^{{2}}{}{\mathrm{\Gamma }}{}\left({1}{-}{n}\right)}{{2}}}{{\left({-1}\right)}^{{n}}{}{\mathrm{\Gamma }}{}\left({1}{-}{n}\right)}{+}\frac{{{\mathrm{\Psi }}{}\left({1}{-}{n}\right)}^{{2}}}{{\left({-1}\right)}^{{n}}}}{{\mathrm{\Gamma }}{}\left({1}{-}{n}\right)}{}{\mathrm{ϵ}}{+}{O}{}\left({{\mathrm{ϵ}}}^{{2}}\right)$ (10)

The same computation with output up to the indicated order, $\mathrm{O}\left(\mathrm{ϵ}\right)$

 > $\mathrm{Series}\left(\mathrm{GAMMA}\left(\mathrm{ϵ}+n\right),\mathrm{ϵ},1\right)\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}assuming\phantom{\rule[-0.0ex]{0.5em}{0.0ex}}n::\mathrm{nonposint}$
 $\frac{{1}}{{\left({-1}\right)}^{{n}}{}{\mathrm{\Gamma }}{}\left({1}{-}{n}\right)}{}{{\mathrm{ϵ}}}^{{-1}}{+}\frac{{\mathrm{\Psi }}{}\left({1}{-}{n}\right)}{{\left({-1}\right)}^{{n}}{}{\mathrm{\Gamma }}{}\left({1}{-}{n}\right)}{+}{O}{}\left({\mathrm{ϵ}}\right)$ (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][ExpandDimension] and Physics[FeynmanIntegral][Series] commands were introduced in Maple 2021.