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vsht committed Aug 8, 2024
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Expand Up @@ -559,6 +559,7 @@ \chapter{Loop integrals}
\subfile{pages/FCLoopPropagatorPowersExpand.tex}
\subfile{pages/FCLoopPropagatorsToTopology.tex}
\subfile{pages/FCLoopRemovePropagator.tex}
\subfile{pages/FCLoopReplaceQuadraticEikonalPropagators.tex}
\subfile{pages/FCLoopSamePropagatorHeadsQ.tex}
\subfile{pages/FCLoopRemoveNegativePropagatorPowers.tex}
\subfile{pages/FCLoopSelectTopology.tex}
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119 changes: 114 additions & 5 deletions pages/FCLoopFindMomentumShifts.tex
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51 changes: 28 additions & 23 deletions pages/FCLoopFindTopologyMappings.tex
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Expand Up @@ -89,9 +89,9 @@ \subsection{Examples}
\begin{dmath*}\breakingcomma
\left(
\begin{array}{ccc}
\;\text{FCTopology}\left(\text{fctopology3},\left\{\frac{1}{(\text{p3}^2+i \eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{((\text{p2}+\text{p3})^2+i \eta )},\frac{1}{((\text{p1}+\text{p3})^2+i \eta )},\frac{1}{((\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p2}+\text{p3}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p3}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}+\text{p3}-Q)^2+i \eta )}\right\},\{\text{p1},\text{p2},\text{p3}\},\{Q\},\{\},\{\}\right) & \{\text{p1}\to -\text{p1}-\text{p3}+Q,\text{p2}\to -\text{p2}-\text{p3}+Q,\text{p3}\to \;\text{p3}\} & G^{\text{fctopology3}}(\text{n1$\_$},\text{n7$\_$},\text{n8$\_$},\text{n5$\_$},\text{n6$\_$},\text{n4$\_$},\text{n2$\_$},\text{n3$\_$},\text{n9$\_$}):\to G^{\text{fctopology1}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5},\text{n6},\text{n7},\text{n8},\text{n9}) \\
\;\text{FCTopology}\left(\text{fctopology3},\left\{\frac{1}{(\text{p3}^2+i \eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{((\text{p2}+\text{p3})^2+i \eta )},\frac{1}{((\text{p1}+\text{p3})^2+i \eta )},\frac{1}{((\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p2}+\text{p3}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p3}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}+\text{p3}-Q)^2+i \eta )}\right\},\{\text{p1},\text{p2},\text{p3}\},\{Q\},\{\},\{\}\right) & \{\text{p1}\to -\text{p1}-\text{p3}+Q,\text{p2}\to -\text{p2}-\text{p3}+Q\} & G^{\text{fctopology3}}(\text{n1$\_$},\text{n7$\_$},\text{n8$\_$},\text{n5$\_$},\text{n6$\_$},\text{n4$\_$},\text{n2$\_$},\text{n3$\_$},\text{n9$\_$}):\to G^{\text{fctopology1}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5},\text{n6},\text{n7},\text{n8},\text{n9}) \\
\;\text{FCTopology}\left(\text{fctopology4},\left\{\frac{1}{(\text{p3}^2+i \eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{((\text{p2}+\text{p3})^2+i \eta )},\frac{1}{((\text{p1}+\text{p3})^2+i \eta )},\frac{1}{((\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p1}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p3}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}+\text{p3}-Q)^2+i \eta )}\right\},\{\text{p1},\text{p2},\text{p3}\},\{Q\},\{\},\{\}\right) & \{\text{p1}\to Q-\text{p2},\text{p2}\to Q-\text{p1},\text{p3}\to -\text{p3}\} & G^{\text{fctopology4}}(\text{n1$\_$},\text{n6$\_$},\text{n5$\_$},\text{n8$\_$},\text{n7$\_$},\text{n3$\_$},\text{n2$\_$},\text{n4$\_$},\text{n9$\_$}):\to G^{\text{fctopology1}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5},\text{n6},\text{n7},\text{n8},\text{n9}) \\
\;\text{FCTopology}\left(\text{fctopology5},\left\{\frac{1}{(\text{p3}^2+i \eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{((\text{p1}+\text{p3})^2+i \eta )},\frac{1}{((\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p1}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p3}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}+\text{p3}-Q)^2+i \eta )}\right\},\{\text{p1},\text{p2},\text{p3}\},\{Q\},\{\},\{\}\right) & \{\text{p1}\to \;\text{p2},\text{p2}\to \;\text{p1},\text{p3}\to \;\text{p3}\} & G^{\text{fctopology5}}(\text{n1$\_$},\text{n3$\_$},\text{n2$\_$},\text{n4$\_$},\text{n6$\_$},\text{n5$\_$},\text{n7$\_$},\text{n8$\_$},\text{n9$\_$}):\to G^{\text{fctopology2}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5},\text{n6},\text{n7},\text{n8},\text{n9}) \\
\;\text{FCTopology}\left(\text{fctopology5},\left\{\frac{1}{(\text{p3}^2+i \eta )},\frac{1}{(\text{p2}^2+i \eta )},\frac{1}{(\text{p1}^2+i \eta )},\frac{1}{((\text{p1}+\text{p3})^2+i \eta )},\frac{1}{((\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p1}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p3}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}-Q)^2+i \eta )},\frac{1}{((\text{p1}+\text{p2}+\text{p3}-Q)^2+i \eta )}\right\},\{\text{p1},\text{p2},\text{p3}\},\{Q\},\{\},\{\}\right) & \{\text{p1}\to \;\text{p2},\text{p2}\to \;\text{p1}\} & G^{\text{fctopology5}}(\text{n1$\_$},\text{n3$\_$},\text{n2$\_$},\text{n4$\_$},\text{n6$\_$},\text{n5$\_$},\text{n7$\_$},\text{n8$\_$},\text{n9$\_$}):\to G^{\text{fctopology2}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5},\text{n6},\text{n7},\text{n8},\text{n9}) \\
\end{array}
\right)
\end{dmath*}
Expand Down Expand Up @@ -320,7 +320,7 @@ \subsection{Examples}
\begin{dmath*}\breakingcomma
\left(
\begin{array}{ccc}
\;\text{FCTopology}\left(\text{topo2},\left\{\frac{1}{((\text{l1}-\text{l2})^2-m^2+i \eta )},\frac{1}{((\text{l1}-\text{q2})^2+i \eta )},\frac{1}{((\text{l2}-\text{q2})^2-m^2+i \eta )},\frac{1}{((\text{l2}+\text{q1})^2-m^2+i \eta )},\frac{1}{(\text{l2}^2+i \eta )}\right\},\{\text{l1},\text{l2}\},\{\text{q1},\text{q2}\},\left\{\text{q1}^2\to 0,\text{q2}^2\to 0,\text{q1}\cdot \;\text{q2}\to \frac{s}{2}\right\},\{\}\right) & \{\text{l1}\to -\text{l1}+\text{l2}-\text{q1},\text{l2}\to \;\text{l2},\text{q1}\to -\text{q2},\text{q2}\to -\text{q1}\} & G^{\text{topo2}}(\text{n1$\_$},\text{n2$\_$},\text{n3$\_$},\text{n4$\_$},\text{n5$\_$}):\to G^{\text{topo1}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5}) \\
\;\text{FCTopology}\left(\text{topo2},\left\{\frac{1}{((\text{l1}-\text{l2})^2-m^2+i \eta )},\frac{1}{((\text{l1}-\text{q2})^2+i \eta )},\frac{1}{((\text{l2}-\text{q2})^2-m^2+i \eta )},\frac{1}{((\text{l2}+\text{q1})^2-m^2+i \eta )},\frac{1}{(\text{l2}^2+i \eta )}\right\},\{\text{l1},\text{l2}\},\{\text{q1},\text{q2}\},\left\{\text{q1}^2\to 0,\text{q2}^2\to 0,\text{q1}\cdot \;\text{q2}\to \frac{s}{2}\right\},\{\}\right) & \{\text{l1}\to -\text{l1}+\text{l2}-\text{q1},\text{q1}\to -\text{q2},\text{q2}\to -\text{q1}\} & G^{\text{topo2}}(\text{n1$\_$},\text{n2$\_$},\text{n3$\_$},\text{n4$\_$},\text{n5$\_$}):\to G^{\text{topo1}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5}) \\
\end{array}
\right)
\end{dmath*}
Expand All @@ -333,6 +333,10 @@ \subsection{Examples}
\end{Highlighting}
\end{Shaded}

\begin{dmath*}\breakingcomma
\text{FCLoopFindMomentumShifts: }\;\text{Failed to derive the momentum shifts between topologies topo2 and topo1. This can be due to the presence of nonquadratic propagators or because shifts in external momenta are also necessary.}
\end{dmath*}

\begin{dmath*}\breakingcomma
\text{FCLoopFindTopologyMappings: }\;\text{Found }0\text{ mapping relations }
\end{dmath*}
Expand Down Expand Up @@ -374,6 +378,7 @@ \subsection{Examples}

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{DataType}\OperatorTok{[}\NormalTok{gkin}\OperatorTok{,}\NormalTok{ FCVariable}\OperatorTok{]} \ExtensionTok{=} \ConstantTok{True}\NormalTok{;}
\NormalTok{DataType}\OperatorTok{[}\NormalTok{meta}\OperatorTok{,}\NormalTok{ FCVariable}\OperatorTok{]} \ExtensionTok{=} \ConstantTok{True}\NormalTok{;}
\NormalTok{DataType}\OperatorTok{[}\NormalTok{u0b}\OperatorTok{,}\NormalTok{ FCVariable}\OperatorTok{]} \ExtensionTok{=} \ConstantTok{True}\NormalTok{;}
\end{Highlighting}
Expand All @@ -387,6 +392,18 @@ \subsection{Examples}
\end{Highlighting}
\end{Shaded}

\begin{dmath*}\breakingcomma
\text{FCLoopFindMomentumShifts: }\;\text{The topologies contain following mixed quadratic-eikonal propagators that complicate the determination of the shifts: }\left\{\frac{1}{(\text{k1}^2+\text{k1}\cdot (2 \;\text{gkin} \;\text{meta} n-\text{meta} \;\text{u0b} \;\text{nb})-2 \;\text{gkin} \;\text{meta}^2 \;\text{u0b}+i \eta )},\frac{1}{(\text{k1}^2+\text{k1}\cdot (\text{meta} \;\text{u0b} \;\text{nb}-2 \;\text{gkin} \;\text{meta} n)-2 \;\text{gkin} \;\text{meta}^2 \;\text{u0b}+i \eta )},\frac{1}{(\text{k2}^2-\text{meta} \;\text{u0b} (\text{k2}\cdot \;\text{nb})+i \eta )},\frac{1}{((\text{k1}+\text{k2})^2-2 \;\text{gkin} \;\text{meta} \;\text{u0b} ((\text{k1}+\text{k2})\cdot n)+i \eta )},\frac{1}{((\text{k1}+\text{k2})^2-\text{meta} \;\text{u0b} ((\text{k1}+\text{k2})\cdot \;\text{nb})+i \eta )},\frac{1}{((\text{k1}+\text{k2})^2+(\text{k1}+\text{k2})\cdot (2 \;\text{gkin} \;\text{meta} \;\text{u0b} n-\text{meta} \;\text{u0b} \;\text{nb})-2 \;\text{gkin} \;\text{meta}^2 \;\text{u0b}^2+i \eta )}\right\}
\end{dmath*}

\begin{dmath*}\breakingcomma
\text{FCLoopFindMomentumShifts: }\;\text{You can try to trade them for purely quadratic propagators using FCLoopReplaceQuadraticEikonalPropagators.}
\end{dmath*}

\begin{dmath*}\breakingcomma
\text{FCLoopFindMomentumShifts: }\;\text{Failed to derive the momentum shifts between topologies mytopo79 and mytopo67. This can be due to the presence of nonquadratic propagators or because shifts in external momenta are also necessary.}
\end{dmath*}

\begin{dmath*}\breakingcomma
\text{FCLoopFindTopologyMappings: }\;\text{Found }0\text{ mapping relations }
\end{dmath*}
Expand All @@ -400,20 +417,18 @@ \subsection{Examples}

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{eikRule }\ExtensionTok{=} \OperatorTok{\{}\NormalTok{SFAD}\OperatorTok{[\{\{}\NormalTok{k2}\OperatorTok{,} \SpecialCharTok{{-}}\NormalTok{meta u0b k2 . nb}\OperatorTok{\},} \OperatorTok{\{}\DecValTok{0}\OperatorTok{,} \DecValTok{1}\OperatorTok{\},} \DecValTok{1}\OperatorTok{\}]} \OtherTok{{-}\textgreater{}}\NormalTok{ SFAD}\OperatorTok{[}\NormalTok{k2 }\SpecialCharTok{{-}}\NormalTok{ meta u0b}\SpecialCharTok{/}\DecValTok{2}\NormalTok{ nb}\OperatorTok{]\}}
\NormalTok{toposNew }\ExtensionTok{=}\NormalTok{ FCLoopReplaceQuadraticEikonalPropagators}\OperatorTok{[\{}\NormalTok{topoEik1}\OperatorTok{,}\NormalTok{ topoEik2}\OperatorTok{\},}
\NormalTok{ LoopMomenta }\OtherTok{{-}\textgreater{}} \OperatorTok{\{}\NormalTok{k1}\OperatorTok{,}\NormalTok{ k2}\OperatorTok{\},}
\NormalTok{ InitialSubstitutions }\OtherTok{{-}\textgreater{}} \OperatorTok{\{}
\NormalTok{ ExpandScalarProduct}\OperatorTok{[}\NormalTok{SPD}\OperatorTok{[}\NormalTok{k1 }\SpecialCharTok{{-}}\NormalTok{ k2}\OperatorTok{]]} \OtherTok{{-}\textgreater{}}\NormalTok{ SPD}\OperatorTok{[}\NormalTok{k1 }\SpecialCharTok{{-}}\NormalTok{ k2}\OperatorTok{],}
\NormalTok{ ExpandScalarProduct}\OperatorTok{[}\NormalTok{SPD}\OperatorTok{[}\NormalTok{k1 }\SpecialCharTok{+}\NormalTok{ k2}\OperatorTok{]]} \OtherTok{{-}\textgreater{}}\NormalTok{ SPD}\OperatorTok{[}\NormalTok{k1 }\SpecialCharTok{+}\NormalTok{ k2}\OperatorTok{]\},}
\NormalTok{ IntermediateSubstitutions }\OtherTok{{-}\textgreater{}} \OperatorTok{\{}\NormalTok{SPD}\OperatorTok{[}\FunctionTok{n}\OperatorTok{]} \OtherTok{{-}\textgreater{}} \DecValTok{0}\OperatorTok{,}\NormalTok{ SPD}\OperatorTok{[}\NormalTok{nb}\OperatorTok{]} \OtherTok{{-}\textgreater{}} \DecValTok{0}\OperatorTok{,}\NormalTok{ SPD}\OperatorTok{[}\FunctionTok{n}\OperatorTok{,}\NormalTok{ nb}\OperatorTok{]} \OtherTok{{-}\textgreater{}} \DecValTok{2}\OperatorTok{\}]}\NormalTok{;}
\end{Highlighting}
\end{Shaded}

\begin{dmath*}\breakingcomma
\left\{\frac{1}{(\text{k2}^2-\text{meta} \;\text{u0b} (\text{k2}\cdot \;\text{nb})+i \eta )}\to \frac{1}{((\text{k2}-\frac{\text{meta} \;\text{u0b} \;\text{nb}}{2})^2+i \eta )}\right\}
\end{dmath*}
```mathematica eikMappings = FCLoopFindTopologyMappings{[}toposNew{]};
\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{eikMappings }\ExtensionTok{=}\NormalTok{ FCLoopFindTopologyMappings}\OperatorTok{[\{}\NormalTok{topoEik1}\OperatorTok{,}\NormalTok{ topoEik2}\OperatorTok{\},}
\NormalTok{ InitialSubstitutions }\OtherTok{{-}\textgreater{}}\NormalTok{ eikRule}\OperatorTok{]}\NormalTok{;}
\end{Highlighting}
\end{Shaded}
```mathematica
\begin{dmath*}\breakingcomma
\text{FCLoopFindTopologyMappings: }\;\text{Found }1\text{ mapping relations }
Expand All @@ -422,14 +437,4 @@ \subsection{Examples}
\begin{dmath*}\breakingcomma
\text{FCLoopFindTopologyMappings: }\;\text{Final number of independent topologies: }1
\end{dmath*}

\begin{Shaded}
\begin{Highlighting}[]
\NormalTok{eikMappings}\OperatorTok{[[}\DecValTok{1}\OperatorTok{]][[}\DecValTok{1}\OperatorTok{]][[}\DecValTok{2}\NormalTok{ ;;}\OperatorTok{]]}
\end{Highlighting}
\end{Shaded}

\begin{dmath*}\breakingcomma
\left\{\left\{\text{k1}\to -\text{k1},\text{k2}\to \frac{1}{2} (\text{meta} \;\text{nb} \;\text{u0b}-2 \;\text{k2})\right\},G^{\text{mytopo79}}(\text{n5$\_$},\text{n2$\_$},\text{n4$\_$},\text{n1$\_$},\text{n3$\_$},\text{n7$\_$},\text{n6$\_$}):\to G^{\text{mytopo67}}(\text{n1},\text{n2},\text{n3},\text{n4},\text{n5},\text{n6},\text{n7})\right\}
\end{dmath*}
\end{document}
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