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Physics[FeynmanIntegral][ToAbstractRepresentation] - rewrite in abstract form a 1 loop Feynman integral expressed in standard form; the abstract form is used when performing the integral's tensor reduction

Physics[FeynmanIntegral][FromAbstractRepresentation] - rewrite back into standard form a given 1 loop Feynman integral expressed in abstract form

 Calling Sequence FromAbstractRepresentation(expression) ToAbstractRepresentation(expression)

Parameters

 expression - any expression, equation, set, list or matrix of them, typically involving Feynman tensor integrals

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. 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.

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 of a massive $\mathrm{\phi }$ field, the integral containing two propagators and one external momentum ${{P}_{1}}^{\mathrm{\mu }}$ to which corresponds the mass ${m}_{1}$.

 > $\mathrm{%FeynmanIntegral}\left(\frac{\mathrm{p__1}\left[\mathrm{~mu}\right]}{\left({\mathrm{p__1}}^{2}-{\mathrm{m__φ}}^{2}+i\mathrm{\epsilon }\right)\left({\left(\mathrm{p__1}-\mathrm{P__1}\right)}^{2}-{\mathrm{m__1}}^{2}+i\mathrm{\epsilon }\right)},\mathrm{p__1}\right)$
 $\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\int }\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\frac{{\mathrm{p__1}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}}{\left({\mathrm{p__1}}^{{2}}{-}{\mathrm{m__φ}}^{{2}}{+}{i}{}{\mathbf{\epsilon }}\right){}\left({\left(\mathrm{p__1}{-}\mathrm{P__1}\right)}^{{2}}{-}{\mathrm{m__1}}^{{2}}{+}{i}{}{\mathbf{\epsilon }}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{p__1}}^{{4}}$ (2)

Before reducing this tensor integral to a linear combination of scalar integrals, it is convenient to represent the integral in abstract form, implemented as follows:

 > $\mathrm{ToAbstractRepresentation}\left(\right)$
 ${{𝕋}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({2}{,}{0}{,}{\mathrm{m__φ}}^{{2}}{,}{-}\mathrm{P__1}{,}{\mathrm{m__1}}^{{2}}{,}\mathrm{p__1}{,}{0}\right)$ (3)

In this output we see the integral has 2 propagators, the first one has 0 external momentum (i.e. none) and mass ${{m}_{\mathrm{\phi }}}^{2}$. The second propagator has external momentum $-{P}_{1}$ to which corresponds the mass ${m}_{1}$. Finally the loop momentum integration variable is ${p}_{1}$ and the last operand, in this example equal to 0 means there are no contracted powers of ${p}_{1}$, the loop integration variable. To retrieve the non-abstract form from the abstract one you can use

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

The reduction of this tensor integral to a linear combination of scalar Feynman integrals uses this rewriting internally and results in

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

The TensorReduce command can optionally return intermediate steps of the reduction process, from 1 to 7. Steps 2, 3 and 4 return a result using this abstract representation. For example,

 > $\mathrm{TensorReduce}\left(,\mathrm{step}=2\right)$
 $\mathrm{* Partial match of \text{'}}\mathrm{step}\mathrm{\text{'} against keyword \text{'}}\mathrm{outputstep}\text{'}$
 $\left[\begin{array}{c}{\mathrm{P__1}}_{{\mathrm{\mu }}}{}{{𝕋}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({2}{,}{0}{,}{\mathrm{m__φ}}^{{2}}{,}{-}\mathrm{P__1}{,}{\mathrm{m__1}}^{{2}}{,}\mathrm{p__1}{,}{0}\right){=}{\mathrm{P__1}}_{{\mathrm{\mu }}}{}{{C}}_{{1}}{}{\mathrm{P__1}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}\end{array}\right]$ (6)
 > $\mathrm{FromAbstractRepresentation}\left(\right)$
 $\left[\begin{array}{c}{\mathrm{P__1}}_{{\mathrm{\mu }}}{}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\int }\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\frac{{\mathrm{p__1}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}}{\left({\mathrm{p__1}}^{{2}}{-}{\mathrm{m__φ}}^{{2}}{+}{i}{}{\mathbf{\epsilon }}\right){}\left({\left(\mathrm{p__1}{-}\mathrm{P__1}\right)}^{{2}}{-}{\mathrm{m__1}}^{{2}}{+}{i}{}{\mathbf{\epsilon }}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{p__1}}^{{4}}{=}{\mathrm{P__1}}_{{\mathrm{\mu }}}{}{{C}}_{{1}}{}{\mathrm{P__1}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}\end{array}\right]$ (7)
 > $\mathrm{TensorReduce}\left(,\mathrm{step}=4\right)$
 $\mathrm{* Partial match of \text{'}}\mathrm{step}\mathrm{\text{'} against keyword \text{'}}\mathrm{outputstep}\text{'}$
 $\left[\begin{array}{c}{-}\frac{{𝕋}\left[\right]{}\left({1}{,}{0}{,}{\mathrm{m__φ}}^{{2}}{,}\mathrm{p__1}{,}{0}\right)}{{2}}{+}\frac{{𝕋}\left[\right]{}\left({1}{,}{-}\mathrm{P__1}{,}{\mathrm{m__1}}^{{2}}{,}\mathrm{p__1}{,}{0}\right)}{{2}}{+}\frac{\left(\mathrm{P__1}{·}\mathrm{P__1}{-}{\mathrm{m__1}}^{{2}}{+}{\mathrm{m__φ}}^{{2}}\right){}{𝕋}\left[\right]{}\left({2}{,}{0}{,}{\mathrm{m__φ}}^{{2}}{,}{-}\mathrm{P__1}{,}{\mathrm{m__1}}^{{2}}{,}\mathrm{p__1}{,}{0}\right)}{{2}}{=}{{C}}_{{1}}{}\left(\mathrm{P__1}{·}\mathrm{P__1}\right)\end{array}\right]$ (8)
 > $\mathrm{FromAbstractRepresentation}\left(\right)$
 $\left[\begin{array}{c}{-}\frac{\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\int }\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\frac{{1}}{{\mathrm{p__1}}^{{2}}{-}{\mathrm{m__φ}}^{{2}}{+}{i}{}{\mathbf{\epsilon }}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{p__1}}^{{4}}}{{2}}{+}\frac{\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\int }\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\frac{{1}}{{\left(\mathrm{p__1}{-}\mathrm{P__1}\right)}^{{2}}{-}{\mathrm{m__1}}^{{2}}{+}{i}{}{\mathbf{\epsilon }}}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{p__1}}^{{4}}}{{2}}{+}\frac{\left(\mathrm{P__1}{·}\mathrm{P__1}{-}{\mathrm{m__1}}^{{2}}{+}{\mathrm{m__φ}}^{{2}}\right){}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{\int }\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}\frac{{1}}{\left({\mathrm{p__1}}^{{2}}{-}{\mathrm{m__φ}}^{{2}}{+}{i}{}{\mathbf{\epsilon }}\right){}\left({\left(\mathrm{p__1}{-}\mathrm{P__1}\right)}^{{2}}{-}{\mathrm{m__1}}^{{2}}{+}{i}{}{\mathbf{\epsilon }}\right)}\phantom{\rule[-0.0ex]{0.3em}{0.0ex}}{ⅆ}{\mathrm{p__1}}^{{4}}}{{2}}{=}{{C}}_{{1}}{}\left(\mathrm{P__1}{·}\mathrm{P__1}\right)\end{array}\right]$ (9)

Back to the reduction process, note that, by design, TensorReduce does not evaluate the integrals so that one can follow the computational process clearly. The evaluation can be performed next by passing this result to Evaluate

 > $\mathrm{Evaluate}\left(\mathrm{rhs}\left(\right)\right)$
 ${-}\frac{{\mathrm{P__1}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({-}{i}{}\left({\mathrm{m__1}}^{{2}}{-}{\mathrm{m__φ}}^{{2}}{-}\mathrm{P__1}{·}\mathrm{P__1}\right){}{{\mathrm{\pi }}}^{{2}{-}{\mathrm{ϵ}}}{}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{\sum }_{\mathrm{n__1}{=}{0}}^{{\mathrm{\infty }}}{}\left({-}\frac{{\mathrm{P__1}}^{{2}{}\mathrm{n__1}}{}{\mathrm{\Gamma }}{}\left({\mathrm{ϵ}}{+}{n}{+}\mathrm{n__1}\right){}{\mathrm{m__φ}}^{{-}{2}{}\mathrm{n__1}{-}{2}{}{\mathrm{ϵ}}{-}{2}{}{n}}{}{\left({-}{\mathrm{m__1}}^{{2}}{+}{\mathrm{m__φ}}^{{2}}\right)}^{{n}}{}{\mathrm{\Gamma }}{}\left({n}{+}\mathrm{n__1}{+}{1}\right)}{{\mathrm{\Gamma }}{}\left({1}{+}{n}\right){}{\mathrm{\Gamma }}{}\left({2}{}\mathrm{n__1}{+}{n}{+}{2}\right)}\right)\right){-}{i}{}{{\mathrm{\pi }}}^{{2}{-}{\mathrm{ϵ}}}{}{\mathrm{m__φ}}^{{2}{-}{2}{}{\mathrm{ϵ}}}{}{\mathrm{\Gamma }}{}\left({-}{1}{+}{\mathrm{ϵ}}\right){+}{i}{}{{\mathrm{\pi }}}^{{2}{-}{\mathrm{ϵ}}}{}{\mathrm{m__1}}^{{2}{-}{2}{}{\mathrm{ϵ}}}{}{\mathrm{\Gamma }}{}\left({-}{1}{+}{\mathrm{ϵ}}\right)\right)}{{2}{}\left(\mathrm{P__1}{·}\mathrm{P__1}\right)}$ (10)

Note also that Evaluate automatically calls TensorReduce that in turn uses ToAbstractRepresentation and FromAbstractRepresentation to perform the reduction of tensor integrals when that is the case. So, passing the Feynman integral directly to Evaluate results in the same process all in one go

 > $\mathrm{Evaluate}\left(\right)$
 ${-}\frac{{\mathrm{P__1}}_{\phantom{{}}\phantom{{\mathrm{\mu }}}}^{\phantom{{}}{\mathrm{\mu }}}{}\left({-}{i}{}\left({\mathrm{m__1}}^{{2}}{-}{\mathrm{m__φ}}^{{2}}{-}\mathrm{P__1}{·}\mathrm{P__1}\right){}{{\mathrm{\pi }}}^{{2}{-}{\mathrm{ϵ}}}{}\left({\sum }_{{n}{=}{0}}^{{\mathrm{\infty }}}{}{\sum }_{\mathrm{n__1}{=}{0}}^{{\mathrm{\infty }}}{}\left({-}\frac{{\mathrm{P__1}}^{{2}{}\mathrm{n__1}}{}{\mathrm{\Gamma }}{}\left({\mathrm{ϵ}}{+}{n}{+}\mathrm{n__1}\right){}{\mathrm{m__φ}}^{{-}{2}{}\mathrm{n__1}{-}{2}{}{\mathrm{ϵ}}{-}{2}{}{n}}{}{\left({-}{\mathrm{m__1}}^{{2}}{+}{\mathrm{m__φ}}^{{2}}\right)}^{{n}}{}{\mathrm{\Gamma }}{}\left({n}{+}\mathrm{n__1}{+}{1}\right)}{{\mathrm{\Gamma }}{}\left({1}{+}{n}\right){}{\mathrm{\Gamma }}{}\left({2}{}\mathrm{n__1}{+}{n}{+}{2}\right)}\right)\right){-}{i}{}{{\mathrm{\pi }}}^{{2}{-}{\mathrm{ϵ}}}{}{\mathrm{m__φ}}^{{2}{-}{2}{}{\mathrm{ϵ}}}{}{\mathrm{\Gamma }}{}\left({-}{1}{+}{\mathrm{ϵ}}\right){+}{i}{}{{\mathrm{\pi }}}^{{2}{-}{\mathrm{ϵ}}}{}{\mathrm{m__1}}^{{2}{-}{2}{}{\mathrm{ϵ}}}{}{\mathrm{\Gamma }}{}\left({-}{1}{+}{\mathrm{ϵ}}\right)\right)}{{2}{}\left(\mathrm{P__1}{·}\mathrm{P__1}\right)}$ (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.