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With \mathem{\mmaDef{FSimplify}} we introduce an algorithm for the simplification of large sums of diagrams. \mathem{\mmaDef{FSimplify}} can detect identical diagrams and performs the following steps to find all matches and sumthem:
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With \mathem{\mmaDef{FSimplify}} we introduce an algorithm for the simplification of large sums of connected diagrams. \mathem{\mmaDef{FSimplify}} can detect identical diagrams and performs the following steps to find all matches and sum~them:
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\begin{enumerate}
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\item All Diagrams with exactly the same content of objects and the same set of open indices are grouped together.
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\begin{itemize}
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\item If no undetermined fields \mathem{\mmaDef{AnyField}} are present, apply the truncation table directly and return the result. This zeros all objects whose field content is not listed in the setup's key \mathem{"Truncation"}.
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\item\textbf{Propagator truncation} (primary path): For each propagator-like object (\mathem{\mmaDef{Propagator}}, \mathem{\mmaDef{Rdot}}, \mathem{\mmaDef{R}}) with \mathem{\mmaDef{AnyField}}, enumerate its explicit field alternatives from the truncation table.
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\item\textbf{Propagator truncation}: For each propagator-like object (\mathem{\mmaDef{Propagator}}, \mathem{\mmaDef{Rdot}}, \mathem{\mmaDef{R}}) with \mathem{\mmaDef{AnyField}}, enumerate its explicit field alternatives from the truncation table.
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Incrementally replace each unresolved propagator with the sum of its alternatives, propagate the field assignments to connected objects, and fully expand the product over the sum.
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After each expansion step, apply the truncation table. This early pruning prevents combinatorial blowup by discarding invalid branches before subsequent propagators are expanded.
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\item\textbf{Vertex truncation}: For terms containing no propagator-like objects, valid field assignments are instead enumerated by imposing consistency at every closed index incrementally and validating each surviving assignment against the truncation table.
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\end{align}
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and it can be readily obtained in \FunKit by using the \mathem{\mmaDef{FMakeDSE}[...]}, providing a setup and a field that is inserted for the superindex $c$.
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To obtain an equation for the gluon two-point function, we choose $c = A^a_{\mu}$ and take a further functional derivative with respect to $A$ (note that we don't provide a setup, as a global one has been specified before):
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Note that \mathem{\mmaDef{FMakeDSE}} can be readily constructed also by the user: First, $\delta S / \delta\phi^c$ can be directly created from a given classical action, e.g. using \mathem{\mmaDef{FMakeClassicalAction}}. Next, all fields can be replaced with the rule \mathem{f[a_]->\mmaDef{FEx}[\mmaDef{FTerm}[f[a]],\mmaDef{FTerm}[\mmaDef{Propagator}[{f,\mmaDef{AnyField}},{a,b}],\mmaDef{FDOp}[\mmaDef{AnyField}[b]]]]}. Finally, one can resolve all derivative operators with \mathem{\mmaDef{FResolveDerivatives}}. This is precisely what the convenience function \mathem{\mmaDef{FMakeDSE}} internally does.
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Now, to obtain an equation for the gluon two-point function, we choose $c = A^a_{\mu}$ and take a further functional derivative with respect to $A$ (note that we don't provide a setup, as a global one has been specified before):
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