TopologyIdentification manual (development version)

Load FeynCalc and the necessary add-ons or other packages

description = "B -> EtaC, QCD, topology minimization, 2-loops";
If[ $FrontEnd === Null, 
    $FeynCalcStartupMessages = False; 
    Print[description]; 
  ];
If[ $Notebooks === False, 
    $FeynCalcStartupMessages = False 
  ];
<< FeynCalc` 
 
FCCheckVersion[10, 0, 0];

$$\text{FeynCalc }\;\text{10.0.0 (dev version, 2024-08-07 16:59:34 +02:00, 2f62a22c). For help, use the }\underline{\text{online} \;\text{documentation},}\;\text{ visit the }\underline{\text{forum}}\;\text{ and have a look at the supplied }\underline{\text{examples}.}\;\text{ The PDF-version of the manual can be downloaded }\underline{\text{here}.}$$

$$\text{If you use FeynCalc in your research, please evaluate FeynCalcHowToCite[] to learn how to cite this software.}$$

$$\text{Please keep in mind that the proper academic attribution of our work is crucial to ensure the future development of this package!}$$

Load the topologies

SetDirectory[FCGetNotebookDirectory[]];

rawTopologies0 = Get["RawTopologies-B-Etac.m"];

Run the naive topology identification

$$\text{fcVariables}=\{\text{gkin},\text{meta},\text{u0b}\};$$

(DataType[#, FCVariable] = True) & /@ fcVariables;

rawTopologies = loopHead /@ (rawTopologies0);

$$\text{kinematics}=\{\text{Hold}[\text{SPD}][n]\text{-$>$}0,\text{Hold}[\text{SPD}][\text{nb}]\text{-$>$}0,\text{Hold}[\text{SPD}][n,\text{nb}]\text{-$>$}2\};$$

aux1 = FCLoopFindTopologies[rawTopologies, {k1, k2}, FCLoopIsolate -> loopHead, 
    FCLoopBasisOverdeterminedQ -> True, FinalSubstitutions -> kinematics, 
    Names -> "preTopoDia", Head -> Identity, FCLoopGetKinematicInvariants -> False, FCLoopScalelessQ -> False];

$$\text{FCLoopFindTopologies: Number of the initial candidate topologies: }248$$

$$\text{FCLoopFindTopologies: Number of the identified unique topologies: }184$$

$$\text{FCLoopFindTopologies: Number of the preferred topologies among the unique topologies: }0$$

$$\text{FCLoopFindTopologies: Number of the identified subtopologies: }64$$

$$\text{FCLoopFindTopologies: Follwing identified topologies are scaleless and will be set to zero: }\{\text{preTopoDia31},\text{preTopoDia32},\text{preTopoDia33},\text{preTopoDia34},\text{preTopoDia60},\text{preTopoDia61},\text{preTopoDia77},\text{preTopoDia86},\text{preTopoDia88},\text{preTopoDia98},\text{preTopoDia110},\text{preTopoDia111},\text{preTopoDia118},\text{preTopoDia119},\text{preTopoDia120},\text{preTopoDia123},\text{preTopoDia139},\text{preTopoDia144},\text{preTopoDia152},\text{preTopoDia155},\text{preTopoDia156},\text{preTopoDia157},\text{preTopoDia158},\text{preTopoDia159},\text{preTopoDia160},\text{preTopoDia161},\text{preTopoDia162},\text{preTopoDia163},\text{preTopoDia164},\text{preTopoDia165},\text{preTopoDia166},\text{preTopoDia170},\text{preTopoDia174},\text{preTopoDia175},\text{preTopoDia176},\text{preTopoDia177},\text{preTopoDia178},\text{preTopoDia179},\text{preTopoDia180},\text{preTopoDia181},\text{preTopoDia182},\text{preTopoDia183},\text{preTopoDia184}\}$$

This particular set of topologies contains mixed quadratic-eikonal that will cause issues with the topology minimization if we leave them as is.

To handle this situation we employ the routine FCLoopReplaceQuadraticEikonalPropagators, telling it the loop momenta, kinematic constraints and the rules for completing the square for the pure loop parts of the propagators

topoPre = FCLoopReplaceQuadraticEikonalPropagators[aux1[[2]], LoopMomenta -> {k1, k2}, 
    InitialSubstitutions -> {ExpandScalarProduct[SPD[k1 - k2]] -> SPD[k1 - k2], 
      ExpandScalarProduct[SPD[k1 + k2]] -> SPD[k1 + k2]}, IntermediateSubstitutions -> kinematics];

Handle overdetermined propagator bases

Single out topologies that have an overdetermined sets of propagators

overdeterminedToposPre = Select[topoPre, FCLoopBasisOverdeterminedQ];

overdeterminedToposPre // Length

$$105$$

Generate partial fractioning rules to be applied to the original sets of denominators

AbsoluteTiming[pfrRules = FCLoopCreatePartialFractioningRules[aux1[[1]], topoPre];]

$$\{76.2728,\text{Null}\}$$

Some examples of such rules are

pfrRules[[1]][[1 ;; 2]]

$$\left\{G^{\text{preTopoDia49}}(1,1,0,1,1,1,1)\to \frac{G^{\text{preTopoDia49PFR23}}(1,2,1,1,1)}{\text{meta} \;\text{u0b}}-\frac{G^{\text{preTopoDia49PFR35}}(1,1,2,1,1)}{\text{meta} (\text{u0b}-1)}+\frac{G^{\text{preTopoDia49PFR36}}(1,1,2,1,1)}{\text{meta} (\text{u0b}-1) \;\text{u0b}},G^{\text{preTopoDia126}}(1,1,1,1,1,1)\to \frac{G^{\text{preTopoDia126PFR2}}(1,1,2,1,1)}{\text{meta} \;\text{u0b}}-\frac{G^{\text{preTopoDia126PFR5}}(1,1,1,2,1)}{\text{meta} (\text{u0b}-1)}+\frac{G^{\text{preTopoDia126PFR6}}(1,1,1,2,1)}{\text{meta} (\text{u0b}-1) \;\text{u0b}}\right\}$$

These rules can be converted to FORM and used for simplifying the amplitude.

New topologies after partial fractioning

pfrToposPre = Union[First /@ pfrRules[[2]]];
aux2 = {aux1[[1]] /. Dispatch[pfrRules[[1]]], topoPre};

Some denominators from original topologies that do not require partial fractioning . Notice that the corresponding topologies themselves still might contain overdetermined sets of propagators

remainderDens = SelectNotFree[aux2[[1]], First /@ overdeterminedToposPre] // Union

$$\left\{G^{\text{preTopoDia100}}(1,1,0,0,1,1,1),G^{\text{preTopoDia100}}(1,1,0,0,1,2,1),G^{\text{preTopoDia105}}(1,1,1,0,0,2,1),G^{\text{preTopoDia116}}(1,1,1,1,1,1,0),G^{\text{preTopoDia126}}(1,1,1,0,0,1),G^{\text{preTopoDia126}}(1,2,1,1,1,0),G^{\text{preTopoDia130}}(1,1,0,0,2,1),G^{\text{preTopoDia142}}(1,1,0,0,2,1),G^{\text{preTopoDia143}}(1,1,1,2,0,1),G^{\text{preTopoDia147}}(1,1,0,1,1,1),G^{\text{preTopoDia147}}(1,1,0,2,1,1),G^{\text{preTopoDia149}}(1,0,1,0,2,1),G^{\text{preTopoDia2}}(1,1,1,0,1,1,0,1),G^{\text{preTopoDia22}}(1,1,0,0,0,0,0,1),G^{\text{preTopoDia23}}(1,1,1,1,1,0,1,0),G^{\text{preTopoDia25}}(1,1,0,1,1,1,0,1),G^{\text{preTopoDia28}}(1,1,0,1,1,1,0,1),G^{\text{preTopoDia39}}(1,0,1,0,1,1,1,1),G^{\text{preTopoDia39}}(1,1,1,0,1,1,0,1),G^{\text{preTopoDia4}}(1,1,0,2,0,1,1,0),G^{\text{preTopoDia49}}(1,1,0,0,1,0,1),G^{\text{preTopoDia76}}(1,1,0,0,1,2,1),G^{\text{preTopoDia79}}(1,1,1,1,1,0,1),G^{\text{preTopoDia93}}(1,1,1,0,2,0,1),G^{\text{preTopoDia95}}(1,1,0,1,1,1,0),G^{\text{preTopoDia95}}(1,1,1,0,1,1,0),G^{\text{preTopoDia96}}(1,1,1,0,0,1,1)\right\}$$

Determine which topologies related to these denominators are overdetermined

overdeterminedTopos = FCLoopSelectTopology[remainderDens, overdeterminedToposPre];

Group the remaining denominators together with the corresponding topologies. Remove the now irrelevant propagators from the leftover topologies

toRemoveList = {#, First@SelectNotFree[overdeterminedTopos, #[[1]]], First /@ Position[#[[2]], 0]} & /@ remainderDens;
newNoPfrGLIs = (FCLoopRemovePropagator[#[[1]], #[[3]]] & /@ toRemoveList);
newNoPfrTopos = (FCLoopRemovePropagator[#[[2]], #[[3]]] & /@ toRemoveList);

List of all resulting topologies upon doing partial fractioning

pfrTopos = Union[pfrToposPre, First /@ newNoPfrTopos];

Replacement rule for renaming preTopo - topologies (with PFR - suffixes from partial fractioning) to pfrTopo topologies

pfrToposNew = Table["pfrTopo" <> ToString[i], {i, 1, Length[pfrTopos]}];
pfrRenRu = Thread[Rule[pfrTopos, pfrToposNew]];

An extra rule for mapping the remaining denominators to the corresponding topologies with removed propagators

gliRulePfr = Thread[Rule[remainderDens, newNoPfrGLIs]] /. pfrRenRu;

Final list of topologies upon doing partial fractioning

relevantPFrTopos = Union[Cases[aux2[[1]] /. Dispatch[gliRulePfr], GLI[id_, ___] :> id, Infinity]];
finalPreToposPfrRaw = SelectNotFree[Join[topoPre, pfrRules[[2]], newNoPfrTopos /. Dispatch[pfrRenRu]],relevantPFrTopos] // Union;

Identify scaleless topologies among them

scalelessPfrTopos = Select[finalPreToposPfrRaw, FCLoopScalelessQ] /. Dispatch[pfrRenRu];

Remove scaleless topologies. This gives us the final list of topologies after partial fractioning

finalPreTopos = SelectFree[finalPreToposPfrRaw /. pfrRenRu, scalelessPfrTopos] // Union;

Check that there are no overdetermined topologies left

If[Union[FCLoopBasisOverdeterminedQ /@ finalPreTopos] =!= {False}, 
    Print["ERROR! Not all overdetermined topologies were eliminated."]; 
 ]

Finally use FCLoopFindTopologyMappings to find mappings between topologies

AbsoluteTiming[mappedTopos = FCLoopFindTopologyMappings[finalPreTopos];]

$$\text{FCLoopFindMomentumShifts: }\;\text{Failed to derive the momentum shifts between topologies pfrTopo378 and pfrTopo114. This can be due to the presence of nonquadratic propagators or because shifts in external momenta are also necessary.}$$

$$\text{FCLoopFindMomentumShifts: }\;\text{Failed to derive the momentum shifts between topologies pfrTopo413 and pfrTopo114. This can be due to the presence of nonquadratic propagators or because shifts in external momenta are also necessary.}$$

$$\text{FCLoopFindMomentumShifts: }\;\text{Failed to derive the momentum shifts between topologies pfrTopo76 and pfrTopo114. This can be due to the presence of nonquadratic propagators or because shifts in external momenta are also necessary.}$$

$$\text{FCLoopFindMomentumShifts: }\;\text{Failed to derive the momentum shifts between topologies pfrTopo82 and pfrTopo114. This can be due to the presence of nonquadratic propagators or because shifts in external momenta are also necessary.}$$

$$\text{FCLoopFindMomentumShifts: }\;\text{Failed to derive the momentum shifts between topologies pfrTopo97 and pfrTopo114. This can be due to the presence of nonquadratic propagators or because shifts in external momenta are also necessary.}$$

$$\text{FCLoopFindMomentumShifts: }\;\text{Failed to derive the momentum shifts between topologies pfrTopo207 and pfrTopo154. This can be due to the presence of nonquadratic propagators or because shifts in external momenta are also necessary.}$$

$$\text{FCLoopFindMomentumShifts: }\;\text{Failed to derive the momentum shifts between topologies pfrTopo320 and pfrTopo154. This can be due to the presence of nonquadratic propagators or because shifts in external momenta are also necessary.}$$

$$\text{FCLoopFindMomentumShifts: }\;\text{Failed to derive the momentum shifts between topologies pfrTopo327 and pfrTopo154. This can be due to the presence of nonquadratic propagators or because shifts in external momenta are also necessary.}$$

$$\text{FCLoopFindMomentumShifts: }\;\text{Failed to derive the momentum shifts between topologies pfrTopo6 and pfrTopo154. This can be due to the presence of nonquadratic propagators or because shifts in external momenta are also necessary.}$$

$$\text{FCLoopFindMomentumShifts: }\;\text{Failed to derive the momentum shifts between topologies pfrTopo450 and pfrTopo19. This can be due to the presence of nonquadratic propagators or because shifts in external momenta are also necessary.}$$

$$\text{FCLoopFindMomentumShifts: }\;\text{Failed to derive the momentum shifts between topologies pfrTopo80 and pfrTopo19. This can be due to the presence of nonquadratic propagators or because shifts in external momenta are also necessary.}$$

$$\text{FCLoopFindMomentumShifts: }\;\text{Failed to derive the momentum shifts between topologies pfrTopo278 and pfrTopo203. This can be due to the presence of nonquadratic propagators or because shifts in external momenta are also necessary.}$$

$$\text{FCLoopFindMomentumShifts: }\;\text{Failed to derive the momentum shifts between topologies pfrTopo349 and pfrTopo203. This can be due to the presence of nonquadratic propagators or because shifts in external momenta are also necessary.}$$

$$\text{FCLoopFindMomentumShifts: }\;\text{Failed to derive the momentum shifts between topologies pfrTopo63 and pfrTopo23. This can be due to the presence of nonquadratic propagators or because shifts in external momenta are also necessary.}$$

$$\text{FCLoopFindMomentumShifts: }\;\text{Failed to derive the momentum shifts between topologies preTopoDia171 and pfrTopo23. This can be due to the presence of nonquadratic propagators or because shifts in external momenta are also necessary.}$$

$$\text{FCLoopFindTopologyMappings: }\;\text{Found }190\text{ mapping relations }$$

$$\text{FCLoopFindTopologyMappings: }\;\text{Final number of independent topologies: }241$$

$$\{22.4032,\text{Null}\}$$

Allowing for shifts of external momenta would give us even more mapping relations but this is not safe unless we explicitly know that the amplitude is symmetric under such shifts

AbsoluteTiming[mappedToposTest = FCLoopFindTopologyMappings[finalPreTopos, Momentum -> All];]

$$\text{FCLoopFindTopologyMappings: }\;\text{Found }205\text{ mapping relations }$$

$$\text{FCLoopFindTopologyMappings: }\;\text{Final number of independent topologies: }226$$

$$\{23.6277,\text{Null}\}$$

Introducing new names for the final topologies

finTopoNames = First /@ mappedTopos[[2]];
finTopoNamesNew = Table["finTopo" <> ToString[i], {i, 1, Length[finTopoNames]}];
finRenRu = Thread[Rule[finTopoNames, finTopoNamesNew]];

Some of the final topologies might be incomplete, so we need to account for that as well

finToposRenamed = mappedTopos[[2]] /. finRenRu;
incompleteTopos = Select[finToposRenamed, FCLoopBasisIncompleteQ];

For the basis completion we can use all available propagators

allProps = Union[Flatten[#[[2]] & /@ finToposRenamed]];
completedTopos = FCLoopBasisFindCompletion[incompleteTopos, Method -> allProps];

Generate basis completion rules

basisCompletionRules = FCLoopCreateRuleGLIToGLI[completedTopos, List /@ incompleteTopos] //Flatten;

Generating the ultimate list of topologies where all propagator sets now form a basis

ultimateTopos = finToposRenamed /. Thread[Rule[incompleteTopos, completedTopos]];
ultimateToposNewNames = Table["topology" <> ToString[i], {i, 1, Length[finToposRenamed]}];
ultimateToposRenamingRule = Thread[Rule[First /@ ultimateTopos, ultimateToposNewNames]];
ultimateToposRenamed = ultimateTopos /. ultimateToposRenamingRule;
fcTopologies = ultimateToposRenamed;

Finally, we also need rules to eliminate scalar products

ruGLI = Map[{#[[1]], FCLoopCreateRulesToGLI[#]} &, fcTopologies // FCLoopTopologyNameToSymbol];

Names of the final topologies

sortedTopologyNames = First /@ fcTopologies;