This example uses a custom QED model created with FeynRules.
description = "Renormalization, QED, MSbar, 2-loop";
If[ $FrontEnd === Null,
$FeynCalcStartupMessages = False;
Print[description];
];
If[ $Notebooks === False,
$FeynCalcStartupMessages = False
];
LaunchKernels[8];
$LoadAddOns = {"FeynArts", "FeynHelpers"};
<< FeynCalc`
$FAVerbose = 0;
$ParallelizeFeynCalc = True;
FCCheckVersion[10, 2, 0];
If[ToExpression[StringSplit[$FeynHelpersVersion, "."]][[1]] < 2,
Print["You need at least FeynHelpers 2.0 to run this example."];
Abort[];
]\text{FeynCalc }\;\text{10.2.0 (dev version, 2026-05-18 15:58:48 +02:00, 1a8e687c). 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!}
\text{FeynArts }\;\text{3.12 (27 Mar 2025) patched for use with FeynCalc, for documentation see the }\underline{\text{manual}}\;\text{ or visit }\underline{\text{www}.\text{feynarts}.\text{de}.}
\text{If you use FeynArts in your research, please cite}
\text{ $\bullet $ T. Hahn, Comput. Phys. Commun., 140, 418-431, 2001, arXiv:hep-ph/0012260}
\text{FeynHelpers }\;\text{2.0.0 (2026-02-05 17:03:01 +02:00, 5db84fbb). 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 FeynHelpers in your research, please evaluate FeynHelpersHowToCite[] to learn how to cite this work.}
modelDir = FileNameJoin[{$UserBaseDirectory, "Applications", "FeynCalc", "Examples", "Models", "QED"}]\text{/home/vs/.Wolfram/Applications/FeynCalc/Examples/Models/QED}
FAPatch[PatchModelsOnly -> True, FAModelsDirectory -> modelDir];
(*Successfully patched FeynArts.*)renConstants = Zpsi | ZA | ZAmxt | Ze | Zxi\text{Zpsi}|\text{ZA}|\text{ZAmxt}|\text{Ze}|\text{Zxi}
Nicer typesetting
FCAttachTypesettingRule[mu, "\[Mu]"];
FCAttachTypesettingRule[nu, "\[Nu]"];
FCAttachTypesettingRule[rho, "\[Rho]"];
FCAttachTypesettingRule[si, "\[Sigma]"];diagLeptonSE = InsertFields[CreateTopologies[2, 1 -> 1,
ExcludeTopologies -> {Tadpoles, WFCorrections, WFCorrectionCTs}], {F[2, {1}]} -> {F[2, {1}]},
InsertionLevel -> {Particles}, Model -> FileNameJoin[{modelDir, "QED"}],
GenericModel -> FileNameJoin[{modelDir, "QED"}], ExcludeParticles -> {F[2, {2 | 3}]}];diagLeptonSECT = InsertFields[CreateCTTopologies[2, 1 -> 1,
ExcludeTopologies -> {Tadpoles, WFCorrections, WFCorrectionCTs}], {F[2, {1}]} -> {F[2, {1}]},
InsertionLevel -> {Particles}, Model -> FileNameJoin[{modelDir, "QED"}],
GenericModel -> FileNameJoin[{modelDir, "QED"}], ExcludeParticles -> {F[2, {2 | 3}]}];diagLeptonTreeSECT = InsertFields[CreateCTTopologies[1, 1 -> 1,
ExcludeTopologies -> {Tadpoles, WFCorrections, WFCorrectionCTs}], {F[2, {1}]} -> {F[2, {1}]},
InsertionLevel -> {Particles}, Model -> FileNameJoin[{modelDir, "QED"}],
GenericModel -> FileNameJoin[{modelDir, "QED"}], ExcludeParticles -> {F[2, {2 | 3}]}];Self-energy diagrams
Paint[diagLeptonSE, ColumnsXRows -> {3, 1}, SheetHeader -> None,
Numbering -> Simple, ImageSize -> 128 {3, 1}];
1-loop counter-term diagrams
Paint[diagLeptonSECT, ColumnsXRows -> {4, 1}, SheetHeader -> None,
Numbering -> Simple, ImageSize -> 128 {4, 1}];
Tree-level counter-term diagrams
Paint[diagLeptonTreeSECT, ColumnsXRows -> {4, 1}, SheetHeader -> None,
Numbering -> Simple, ImageSize -> 128 {4, 1}];
The only required masters are 1- and 2-loop tadpoles
tadpoleMaster = Get[FileNameJoin[{$FeynCalcDirectory, "Examples", "MasterIntegrals","Tadpoles", "tad1LxFx1x1xxEp999x.m"}]];tadpoleMaster1 = tadpoleMaster /. m1 -> ml /. tad1LxFx1x1xxEp999x -> "tad1Lv1";
tadpoleMaster2 = tadpoleMaster /. m1 -> mxt /. tad1LxFx1x1xxEp999x -> "tad1Lv2";tadpoleMaster1\left\{G^{\text{tad1Lv1}}(1)\to -e^{\gamma \;\text{ep}} \left(\text{ml}^2\right)^{1-\text{ep}} \Gamma (\text{ep}-1),\left\{\text{FCTopology}\left(\text{tad1Lv1},\left\{\frac{1}{(\text{k1}^2-\text{ml}^2+i \eta )}\right\},\{\text{k1}\},\{\},\{\},\{\}\right)\right\}\right\}
tadpoleMaster2\left\{G^{\text{tad1Lv2}}(1)\to -e^{\gamma \;\text{ep}} \left(\text{mxt}^2\right)^{1-\text{ep}} \Gamma (\text{ep}-1),\left\{\text{FCTopology}\left(\text{tad1Lv2},\left\{\frac{1}{(\text{k1}^2-\text{mxt}^2+i \eta )}\right\},\{\text{k1}\},\{\},\{\},\{\}\right)\right\}\right\}
tadpoleMaster3 = Get[FileNameJoin[{$FeynCalcDirectory, "Examples", "MasterIntegrals","Tadpoles",
"tad2LxFx111x111xxEp1x.m"}]] /. m1 -> ml /. tad2LxFx111x111xxEp1x -> "tad2Lv1";tadpoleMaster4 = Get[FileNameJoin[{$FeynCalcDirectory, "Examples", "MasterIntegrals","Tadpoles",
"tad2LxFx111x111xxEp1x.m"}]] /. m1 -> mxt /. tad2LxFx111x111xxEp1x -> "tad2Lv2";tadpoleMaster5 = Get[FileNameJoin[{$FeynCalcDirectory, "Examples", "MasterIntegrals","Tadpoles",
"tad2LxAm2m1o4x111x122xxEp1x.m"}]] /. {m1 -> ml, m2 -> mxt} /. tad2LxAm2m1o4x111x122xxEp1x -> "tad2Lv3";tadpoleMaster6 = Get[FileNameJoin[{$FeynCalcDirectory, "Examples", "MasterIntegrals","Tadpoles",
"tad2LxAm2m1o4x111x112xxEp1x.m"}]] /. {m1 -> ml, m2 -> mxt} /. tad2LxAm2m1o4x111x112xxEp1x -> "tad2Lv4";{leptonSE$RawAmp, leptonSECT$RawAmp, diagLeptonTreeSECT$RawAmp} =
FCFAConvert[CreateFeynAmp[#, Truncated -> True,
GaugeRules -> {}, PreFactor -> 1],
IncomingMomenta -> {p}, OutgoingMomenta -> {p},
LorentzIndexNames -> {mu, nu}, DropSumOver -> True,
LoopMomenta -> {k1, k2}, UndoChiralSplittings -> True,
ChangeDimension -> D, SMP -> True,
FinalSubstitutions -> {SMP["m_e"] -> 0, SMP["e"] -> 4 Pi Sqrt[a4]}] & /@ {
diagLeptonSE, diagLeptonSECT, diagLeptonTreeSECT};The 2-loop lepton self-energy has superficial degree of divergence equal to 1
FCClearScalarProducts[];
divDegree = 1;
aux1 = FCLoopGetFeynAmpDenominators[Join[leptonSE$RawAmp[[1 ;; 1]], Nf leptonSE$RawAmp[[2 ;; 2]], leptonSE$RawAmp[[3 ;; 3]]],
{k1, k2}, denHead, Momentum -> {p}, "Massless" -> True];
aux2 = FCLoopAddAuxiliaryMass[aux1[[2]], {k1, k2}, -mxt^2, 0, Head -> denHead]\left\{\text{denHead}\left(\frac{1}{(\text{k1}^2+i \eta )}\right)\to \frac{1}{(\text{k1}^2-\text{mxt}^2+i \eta )},\text{denHead}\left(\frac{1}{(\text{k2}^2+i \eta )}\right)\to \frac{1}{(\text{k2}^2-\text{mxt}^2+i \eta )},\text{denHead}\left(\frac{1}{((\text{k1}+\text{k2})^2+i \eta )}\right)\to \frac{1}{((\text{k1}+\text{k2})^2-\text{mxt}^2+i \eta )},\text{denHead}\left(\frac{1}{((\text{k1}-p)^2+i \eta )}\right)\to \frac{1}{((\text{k1}-p)^2-\text{mxt}^2+i \eta )},\text{denHead}\left(\frac{1}{((\text{k1}-\text{k2}-p)^2+i \eta )}\right)\to \frac{1}{((\text{k1}-\text{k2}-p)^2-\text{mxt}^2+i \eta )},\text{denHead}\left(\frac{1}{((\text{k2}-p)^2+i \eta )}\right)\to \frac{1}{((\text{k2}-p)^2-\text{mxt}^2+i \eta )},\text{denHead}\left(\frac{1}{((\text{k1}+p)^2+i \eta )}\right)\to \frac{1}{((\text{k1}+p)^2-\text{mxt}^2+i \eta )},\text{denHead}\left(\frac{1}{((-\text{k1}+\text{k2}+p)^2+i \eta )}\right)\to \frac{1}{((-\text{k1}+\text{k2}+p)^2-\text{mxt}^2+i \eta )}\right\}
AbsoluteTiming[leptonSE$Amp = (aux1[[1]] /. aux2) // Contract[#, FCParallelize -> True] & //
SUNSimplify[#, FCI -> True, FCParallelize -> True] & // DiracSimplify[#, FCI -> True, FCParallelize -> True] &;]\{0.759848,\text{Null}\}
isoSymbols = FCMakeSymbols[KK, Range[1, $KernelCount], List]\{\text{KK1},\text{KK2},\text{KK3},\text{KK4},\text{KK5},\text{KK6},\text{KK7},\text{KK8}\}
AbsoluteTiming[leptonSE$Amp1 = Collect2[leptonSE$Amp, p, IsolateNames -> isoSymbols, FCParallelize -> True];]\{0.396808,\text{Null}\}
AbsoluteTiming[leptonSE$Amp2 = FourSeries[leptonSE$Amp1, {p, 0, 1}, FCParallelize -> True];]\{0.579686,\text{Null}\}
AbsoluteTiming[leptonSE$Amp3 = Collect2[FRH2[leptonSE$Amp2, isoSymbols], FeynAmpDenominator, FCParallelize -> True];]\{0.229368,\text{Null}\}
The rest of the calculation follows the standard multiloop template
FCClearScalarProducts[]
SPD[p] = pp;AbsoluteTiming[{leptonSE$Amp4, leptonSE$Topos} = FCLoopFindTopologies[leptonSE$Amp3, {k1, k2}, FCI -> True, FCParallelize -> True,
FCLoopBasisOverdeterminedQ -> True, FinalSubstitutions -> {Hold[SPD][p] -> pp}];]\text{FCLoopFindTopologies: Number of the initial candidate topologies: }2
\text{FCLoopFindTopologies: Number of the identified unique topologies: }2
\text{FCLoopFindTopologies: Number of the preferred topologies among the unique topologies: }0
\text{FCLoopFindTopologies: Number of the identified subtopologies: }0
\text{FCLoopFindTopologyMappings: }\;\text{Final number of found topologies: }2
\{0.802641,\text{Null}\}
AbsoluteTiming[leptonSE$Amp5 = FCLoopTensorReduce[leptonSE$Amp4, leptonSE$Topos, FCParallelize -> True];]\{1.27803,\text{Null}\}
AbsoluteTiming[leptonSE$Amp6 = DiracSimplify[leptonSE$Amp5, FCParallelize -> True];]\{0.312581,\text{Null}\}
AbsoluteTiming[{leptonSE$Amp7, leptonSE$Topos2} = FCLoopRewriteOverdeterminedTopologies[leptonSE$Amp6, leptonSE$Topos, FCParallelize -> True];]\text{FCLoopRewriteIncompleteTopologies: }\;\text{No overdetermined topologies detected.}
\{0.022106,\text{Null}\}
AbsoluteTiming[{leptonSE$Amp8, leptonSE$Topos3} = FCLoopRewriteIncompleteTopologies[leptonSE$Amp7, leptonSE$Topos2, FCParallelize -> True];]\text{FCLoopRewriteIncompleteTopologies: }\;\text{No incomplete topologies detected.}
\{0.043718,\text{Null}\}
AbsoluteTiming[leptonSE$SubTopos = FCLoopFindSubtopologies[leptonSE$Topos3, Flatten -> True, Remove -> True, FCParallelize -> True];]\{0.087432,\text{Null}\}
{leptonSE$TopoMappings,
leptonSE$FinalTopos} = FCLoopFindTopologyMappings[leptonSE$Topos3, PreferredTopologies -> leptonSE$SubTopos, FCParallelize -> True];\text{FCLoopFindTopologyMappings: }\;\text{Found }1\text{ mapping relations }
\text{FCLoopFindTopologyMappings: }\;\text{Final number of independent topologies: }1
AbsoluteTiming[leptonSE$AmpGLI = FCLoopApplyTopologyMappings[leptonSE$Amp8, {leptonSE$TopoMappings,
leptonSE$FinalTopos}, FCParallelize -> True];]\{0.266187,\text{Null}\}
leptonSE$GLIs = Cases2[leptonSE$AmpGLI, GLI];leptonSE$dir = FileNameJoin[{$TemporaryDirectory, "Reduction-leptonSE-2L-massless"}];
Quiet[CreateDirectory[leptonSE$dir]];KiraCreateJobFile[leptonSE$FinalTopos, leptonSE$GLIs, leptonSE$dir]\{\text{/tmp/Reduction-leptonSE-2L-massless/fctopology1/job.yaml}\}
KiraCreateIntegralFile[leptonSE$GLIs, leptonSE$FinalTopos, leptonSE$dir]
KiraCreateConfigFiles[leptonSE$FinalTopos, leptonSE$GLIs, leptonSE$dir,
KiraMassDimensions -> {pp -> 2, ml -> 1, mxt -> 1}]\text{KiraCreateIntegralFile: Number of loop integrals: }75
\{\text{/tmp/Reduction-leptonSE-2L-massless/fctopology1/KiraLoopIntegrals}\}
\left( \begin{array}{cc} \;\text{/tmp/Reduction-leptonSE-2L-massless/fctopology1/config/integralfamilies.yaml} & \;\text{/tmp/Reduction-leptonSE-2L-massless/fctopology1/config/kinematics.yaml} \\ \end{array} \right)
KiraRunReduction[leptonSE$dir, leptonSE$FinalTopos,
KiraBinaryPath -> FileNameJoin[{$HomeDirectory, ".local", "bin", "kira"}],
KiraFermatPath -> FileNameJoin[{$HomeDirectory, "bin", "ferl64", "fer64"}]]\{\text{True}\}
leptonSE$ReductionTables = KiraImportResults[leptonSE$FinalTopos, leptonSE$dir] // Flatten;AbsoluteTiming[leptonSE$resPreFinal1 = (leptonSE$AmpGLI /. Dispatch[leptonSE$ReductionTables]);]\{0.005153,\text{Null}\}
AbsoluteTiming[leptonSE$resPreFinal2 = Map[Collect2[#, GLI, DiracGamma, FCParallelize -> True] &, leptonSE$resPreFinal1];]\{0.158533,\text{Null}\}
leptonSE$masters = Cases2[leptonSE$resPreFinal1, GLI];leptonSE$MIMappings = FCLoopFindIntegralMappings[leptonSE$masters, Join[tadpoleMaster1[[2]], tadpoleMaster2[[2]], {tadpoleMaster3[[2]]}, {tadpoleMaster4[[2]]}
, {tadpoleMaster5[[2]]}, {tadpoleMaster6[[2]]}, leptonSE$FinalTopos], PreferredIntegrals -> {tadpoleMaster2[[1]][[1]] tadpoleMaster2[[1]][[1]],
tadpoleMaster1[[1]][[1]] tadpoleMaster1[[1]][[1]], tadpoleMaster1[[1]][[1]] tadpoleMaster2[[1]][[1]],
tadpoleMaster3[[1]][[1]],
tadpoleMaster4[[1]][[1]],
tadpoleMaster5[[1]][[1]],
tadpoleMaster6[[1]][[1]]}]\left( \begin{array}{cc} G^{\text{fctopology1}}(1,1,0)\to G^{\text{tad1Lv2}}(1)^2 & G^{\text{fctopology1}}(1,1,1)\to G^{\text{tad2Lv2}}(1,1,1) \\ G^{\text{tad1Lv2}}(1)^2 & G^{\text{tad2Lv2}}(1,1,1) \\ \end{array} \right)
isoSymbols1 = FCMakeSymbols[LL, Range[1, $KernelCount], List];
isoSymbols2 = FCMakeSymbols[LM, Range[1, $KernelCount], List];AbsoluteTiming[leptonSE$resPreFinal2 = Collect2[leptonSE$resPreFinal1, D, GLI, IsolateNames -> isoSymbols1, FCParallelize -> True] // FCReplaceD[#, D -> 4 - 2 ep] & // ReplaceAll[#, leptonSE$MIMappings[[1]]] & //
ReplaceAll[#, {tadpoleMaster1[[1]], tadpoleMaster2[[1]], tadpoleMaster3[[1]], tadpoleMaster4[[1]], tadpoleMaster5[[1]],tadpoleMaster6[[1]]}] & // Collect2[#, ep, IsolateNames -> isoSymbols2, FCParallelize -> True] &;]\{2.38576,\text{Null}\}
AbsoluteTiming[leptonSE$resPreFinal3 = leptonSE$resPreFinal2 // Series[#, {ep, 0, -1}] & // Normal // Series[(I*(4*Pi)^(-2 + ep))^2 #, {ep, 0, -1}] & // Normal;]\{0.82969,\text{Null}\}
AbsoluteTiming[leptonSE$resPreFinal4 = Collect2[FRH2[FRH2[leptonSE$resPreFinal3, isoSymbols2], isoSymbols1], DiracGamma, pp, mxt, ep, FCParallelize -> True];]\{0.123304,\text{Null}\}
isoSymbols3 = FCMakeSymbols[LH, Range[1, $KernelCount], List];AbsoluteTiming[leptonSE$resPreFinal5 = Series[Total[Collect2[leptonSE$resPreFinal4, mxt, IsolateNames -> isoSymbols3, FCParallelize -> True]], {mxt, 0, 0}] // Normal;]\{0.950428,\text{Null}\}
AbsoluteTiming[leptonSE$resPreFinal6 = Collect2[FRH2[leptonSE$resPreFinal5, isoSymbols3] // ReplaceAll[#, Log[m_Symbol^2] :> 2 Log[m]] &, DiracGamma, pp, mxt, ep, FCParallelize -> True];]\{0.084178,\text{Null}\}
leptonSE$resFinal = Collect2[Collect2[leptonSE$resPreFinal6, ep, CA, CF, ml, Nf, SUNFDelta, a4, DiracGamma, GaugeXi, Factoring -> FullSimplify],
ep, ml, mxt]\frac{i \;\text{a4}^2 \xi _{V(1)}^2 \gamma \cdot p}{2 \;\text{ep}^2}-\frac{i \;\text{a4}^2 \gamma \cdot p \left(54 N_f \xi _{V(1)}^2+32 N_f \xi _{V(1)}+54 N_f+15 \xi _{V(1)}^2+25 \xi _{V(1)}-60 \log (4 \pi ) \xi _{V(1)}^2+45\right)}{60 \;\text{ep}}-\frac{2 i \;\text{a4}^2 \log (\text{mxt}) \xi _{V(1)}^2 \gamma \cdot p}{\text{ep}}
FCClearScalarProducts[];
divDegree = 1;
aux1 = FCLoopGetFeynAmpDenominators[leptonSECT$RawAmp, {k1}, denHead, Momentum -> {p}, "Massless" -> True];
aux2 = FCLoopAddAuxiliaryMass[aux1[[2]], {k1}, -mxt^2, 0, Head -> denHead]\left\{\text{denHead}\left(\frac{1}{(\text{k1}^2+i \eta )}\right)\to \frac{1}{(\text{k1}^2-\text{mxt}^2+i \eta )},\text{denHead}\left(\frac{1}{((\text{k1}-p)^2+i \eta )}\right)\to \frac{1}{((\text{k1}-p)^2-\text{mxt}^2+i \eta )},\text{denHead}\left(\frac{1}{((\text{k1}+p)^2+i \eta )}\right)\to \frac{1}{((\text{k1}+p)^2-\text{mxt}^2+i \eta )}\right\}
leptonSECT$StrName = StringReplace[ToString[Hold[leptonSECT$Amp]], {"Hold[" -> "", "]" -> ""}]\text{leptonSECT\$Amp}
AbsoluteTiming[leptonSECT$Amp = (aux1[[1]] /. aux2) // Contract[#, FCParallelize -> True] & //
SUNSimplify[#, FCParallelize -> True] & // DiracSimplify[#, FCParallelize -> True] &;]\{0.295724,\text{Null}\}
AbsoluteTiming[leptonSECT$Amp1 = Collect2[leptonSECT$Amp, p, IsolateNames -> KK];]
AbsoluteTiming[leptonSECT$Amp2 = FourSeries[leptonSECT$Amp1, {p, 0, divDegree}, FCParallelize -> True];]
AbsoluteTiming[leptonSECT$Amp3 = Collect2[FRH[leptonSECT$Amp2], FeynAmpDenominator, FCParallelize -> True];]\{0.124104,\text{Null}\}
\{0.069278,\text{Null}\}
\{0.103986,\text{Null}\}
The rest of the calculation follows the standard multiloop template
FCClearScalarProducts[];
SPD[p] = pp;{leptonSECT$Amp4, leptonSECT$Topos} = FCLoopFindTopologies[leptonSECT$Amp3, {k1}, FCParallelize -> True,
FCLoopBasisOverdeterminedQ -> True, FinalSubstitutions -> {Hold[SPD][p] -> pp}, Names -> leptonSEtopo];\text{FCLoopFindTopologies: Number of the initial candidate topologies: }1
\text{FCLoopFindTopologies: Number of the identified unique topologies: }1
\text{FCLoopFindTopologies: Number of the preferred topologies among the unique topologies: }0
\text{FCLoopFindTopologies: Number of the identified subtopologies: }0
\text{FCLoopFindTopologyMappings: }\;\text{Final number of found topologies: }1
AbsoluteTiming[leptonSECT$Amp5 = FCLoopTensorReduce[leptonSECT$Amp4, leptonSECT$Topos, FCParallelize -> True];]\{0.486722,\text{Null}\}
AbsoluteTiming[leptonSECT$Amp6 = DiracSimplify[leptonSECT$Amp5, FCParallelize -> True];]\{0.122048,\text{Null}\}
{leptonSECT$Amp7, leptonSECT$Topos2} = FCLoopRewriteOverdeterminedTopologies[leptonSECT$Amp6, leptonSECT$Topos, FCParallelize -> True];\text{FCLoopRewriteIncompleteTopologies: }\;\text{No overdetermined topologies detected.}
{leptonSECT$Amp8, leptonSECT$Topos3} = FCLoopRewriteIncompleteTopologies[leptonSECT$Amp7, leptonSECT$Topos2, FCParallelize -> True];\text{FCLoopRewriteIncompleteTopologies: }\;\text{No incomplete topologies detected.}
AbsoluteTiming[leptonSECT$SubTopos = FCLoopFindSubtopologies[leptonSECT$Topos2, Flatten -> True, Remove -> True, FCParallelize -> True];]\{0.047655,\text{Null}\}
AbsoluteTiming[{leptonSECT$TopoMappings, leptonSECT$FinalTopos} = FCLoopFindTopologyMappings[leptonSECT$Topos2, PreferredTopologies -> leptonSECT$SubTopos, FCParallelize -> True];]\text{FCLoopFindTopologyMappings: }\;\text{Found }0\text{ mapping relations }
\text{FCLoopFindTopologyMappings: }\;\text{Final number of independent topologies: }1
\{0.055277,\text{Null}\}
AbsoluteTiming[leptonSECT$AmpGLI = FCLoopApplyTopologyMappings[leptonSECT$Amp8, {leptonSECT$TopoMappings, leptonSECT$FinalTopos},FCParallelize -> True];]\{0.118211,\text{Null}\}
leptonSECT$GLIs = Cases2[leptonSECT$AmpGLI, GLI];leptonSECT$dir = FileNameJoin[{$TemporaryDirectory, "Reduction-" <> leptonSECT$StrName <> "-1L-massless"}];
Quiet[CreateDirectory[leptonSECT$dir]];KiraCreateJobFile[leptonSECT$FinalTopos, leptonSECT$GLIs, leptonSECT$dir]\{\text{/tmp/Reduction-leptonSECT\$Amp-1L-massless/leptonSEtopo1/job.yaml}\}
KiraCreateIntegralFile[leptonSECT$GLIs, leptonSECT$FinalTopos, leptonSECT$dir]
KiraCreateConfigFiles[leptonSECT$FinalTopos, leptonSECT$GLIs, leptonSECT$dir,
KiraMassDimensions -> {pp -> 2, ml -> 1, mxt -> 1}]\text{KiraCreateIntegralFile: Number of loop integrals: }5
\{\text{/tmp/Reduction-leptonSECT\$Amp-1L-massless/leptonSEtopo1/KiraLoopIntegrals}\}
\left( \begin{array}{cc} \;\text{/tmp/Reduction-leptonSECT\$Amp-1L-massless/leptonSEtopo1/config/integralfamilies.yaml} & \;\text{/tmp/Reduction-leptonSECT\$Amp-1L-massless/leptonSEtopo1/config/kinematics.yaml} \\ \end{array} \right)
KiraRunReduction[leptonSECT$dir, leptonSECT$FinalTopos,
KiraBinaryPath -> FileNameJoin[{$HomeDirectory, ".local", "bin", "kira"}],
KiraFermatPath -> FileNameJoin[{$HomeDirectory, "bin", "ferl64", "fer64"}]]\{\text{True}\}
leptonSECT$ReductionTables = KiraImportResults[leptonSECT$FinalTopos, leptonSECT$dir] // Flatten;leptonSECT$resPreFinal1 = Collect2[Total[leptonSECT$AmpGLI /. Dispatch[leptonSECT$ReductionTables]], GLI,
GaugeXi, D, DiracGamma, FCParallelize -> True];leptonSECT$masters = Cases2[leptonSECT$resPreFinal1, GLI];leptonSECT$MIMappings = FCLoopFindIntegralMappings[leptonSECT$masters, Join[tadpoleMaster1[[2]], tadpoleMaster2[[2]],
leptonSECT$FinalTopos], PreferredIntegrals -> {tadpoleMaster1[[1]][[1]], tadpoleMaster2[[1]][[1]]}]\left( \begin{array}{c} G^{\text{leptonSEtopo1}}(1)\to G^{\text{tad1Lv2}}(1) \\ G^{\text{tad1Lv2}}(1) \\ \end{array} \right)
Our master integrals are calculated using the standard multiloop normalization. To convert it back to the textbook normalization we need to multiply by I*(4 Pi)^(ep-2)
At this point we need to insert the 1-loop renormalization constants
knownResults1L = {
rc[delZA, 1] -> (-4*Nf)/(3*ep),
rc[delZAmxt, 1] -> -2 Nf/ep,
rc[delZxi, 1] -> (-4*Nf)/(3*ep),
rc[delZpsi, 1] -> -(GaugeXi[V[1]]/ep),
rc[delZe, 1] -> (2*Nf)/(3*ep)};AbsoluteTiming[leptonSECT$resPreFinal2 = Collect2[leptonSECT$resPreFinal1, D, GLI, IsolateNames -> KK] // FCReplaceD[#, D -> 4 - 2 ep] & //
ReplaceAll[#, leptonSECT$MIMappings[[1]]] & // ReplaceAll[#, {tadpoleMaster1[[1]], tadpoleMaster2[[1]]}] & //
Collect2[#, ep, IsolateNames -> KK2] & // Series[(I*(4*Pi)^(-2 + ep)) #, {ep, 0, 1}] & // Normal // FCLoopAddMissingHigherOrdersWarning[#, ep, epHelp] & // FRH //
ReplaceAll[#, {Log[mxt^2] -> 2 Log[mxt]}] &;]\{2.70048,\text{Null}\}
AbsoluteTiming[leptonSECT$resPreFinal2 = Collect2[leptonSECT$resPreFinal1, Join[{a4}, List @@ renConstants],IsolateNames -> KK] // ReplaceAll[#, Zxi -> ZA] & // ReplaceAll[#, {
(h : renConstants) :> 1 + (a4 rc[ToExpression["del" <> ToString[h]], 1] + a4^2 rc[ToExpression["del" <> ToString[h]], 2])}] & // Series[#, {a4, 0, 2}] & // Normal;]\{0.068606,\text{Null}\}
AbsoluteTiming[leptonSECT$resPreFinal3 = Collect2[leptonSECT$resPreFinal2 // FRH, {rc, D, GLI}, IsolateNames -> KK] // FCReplaceD[#, {D -> 4 - 2 ep}] & // ReplaceRepeated[#, knownResults1L] & //
ReplaceAll[#, leptonSECT$MIMappings[[1]]] & // ReplaceAll[#, {tadpoleMaster1[[1]], tadpoleMaster2[[1]]}] & // If[! FreeQ[#, GLI], Abort[], #] & // Collect2[#, ep, IsolateNames -> KK] &;]\{0.048127,\text{Null}\}
leptonSECT$resFinal = leptonSECT$resPreFinal3 // Series[(I*(4*Pi)^(-2 + ep)) #, {ep, 0, -1}] & // Normal // FRH //
Collect2[#, mxt, IsolateNames -> KK] & // Series[#, {mxt, 0, 0}] & // Normal // FRH // ReplaceAll[#, Log[m_^2] :> 2 Log[m]] & // Collect2[#, ep, ml, mxt] &-\frac{i \;\text{a4}^2 \xi _{V(1)}^2 \gamma \cdot p}{\text{ep}^2}+\frac{i \;\text{a4}^2 \gamma \cdot p \left(54 N_f \xi _{V(1)}^2+32 N_f \xi _{V(1)}-6 N_f+15 \xi _{V(1)}^2+25 \xi _{V(1)}-60 \log (4 \pi ) \xi _{V(1)}^2\right)}{60 \;\text{ep}}+\frac{2 i \;\text{a4}^2 \log (\text{mxt}) \xi _{V(1)}^2 \gamma \cdot p}{\text{ep}}
diagLeptonTreeSECT$Amp = (Total[diagLeptonTreeSECT$RawAmp]) // ReplaceRepeated[#, {
(h : renConstants) :> 1 + (a4 rc[ToExpression["del" <> ToString[h]], 1] + a4^2 rc[ToExpression["del" <> ToString[h]], 2])}] & //
Series[#, {a4, 0, 2}] & // Normal // ReplaceRepeated[#, knownResults1L] &i \;\text{a4}^2 \;\text{rc}(\text{delZpsi},2) \gamma \cdot p-\frac{i \;\text{a4} \xi _{V(1)} \gamma \cdot p}{\text{ep}}
Collect2[Coefficient[SUNSimplify[leptonSE$resFinal + leptonSECT$resFinal + diagLeptonTreeSECT$Amp, SUNNToCACF -> False], a4, 2], a4, mxt, DiracGamma, Factoring -> FullSimplify]\frac{i \gamma \cdot p \left(\text{ep} \left(4 \;\text{ep} \;\text{rc}(\text{delZpsi},2)-4 N_f-3\right)-2 \xi _{V(1)}^2\right)}{4 \;\text{ep}^2}
leptonSE$RenConstants2L = Collect2[Coefficient[SUNSimplify[leptonSE$resFinal + leptonSECT$resFinal + diagLeptonTreeSECT$Amp, SUNNToCACF -> False], a4, 2], a4, mxt, DiracGamma, Factoring -> FullSimplify] //
FCMatchSolve[#, {ep, CF, DiracGamma, ml, mxt, SUNDelta, SUNTF, SUNFDelta, CA, GaugeXi, a4, Pair, pp, Nf, SUNN}] & // Collect2[#, ep] &\text{FCMatchSolve: Solving for: }\{\text{rc}(\text{delZpsi},2)\}
\text{FCMatchSolve: A solution exists.}
\left\{\text{rc}(\text{delZpsi},2)\to \frac{\xi _{V(1)}^2}{2 \;\text{ep}^2}+\frac{4 N_f+3}{4 \;\text{ep}}\right\}
Our final QED 2-loop wave-function renormalization constants
finalResults = Thread[Rule[List @@ renConstants,
(List @@ renConstants /. (h : renConstants) :> 1 + a4 rc[ToExpression["del" <> ToString[h]], 1] + a4^2 rc[ToExpression["del" <> ToString[h]], 2]) //
ReplaceAll[#, Join[SUNSimplify[knownResults1L, SUNNToCACF -> False], leptonSE$RenConstants2L]] &]] // SelectNotFree[#, Zpsi] &;finalResults // TableForm\begin{array}{l} \;\text{Zpsi}\to \;\text{a4}^2 \left(\frac{\xi _{V(1)}^2}{2 \;\text{ep}^2}+\frac{4 N_f+3}{4 \;\text{ep}}\right)-\frac{\text{a4} \xi _{V(1)}}{\text{ep}}+1 \\ \end{array}
knownResult = {rc[delZpsi, 2] -> (3 + 4*Nf)/(4*ep) + GaugeXi[V[1]]^2/(2*ep^2)}\left\{\text{rc}(\text{delZpsi},2)\to \frac{\xi _{V(1)}^2}{2 \;\text{ep}^2}+\frac{4 N_f+3}{4 \;\text{ep}}\right\}
```mathematica FCCompareResults[leptonSE$RenConstants2L, knownResult, Text -> {“to Grozing, Lectures on QED and QCD, Eq 5.52 hep-ph/0508242:”, “CORRECT.”, “WRONG!”}, Interrupt -> {Hold[Quit[1]], Automatic}] Print[“Time used:”, Round[N[TimeUsed[], 4], 0.001], ” s.”];
```mathematica
\text{$\backslash $tCompare to Grozing, Lectures on QED and QCD, Eq 5.52 hep-ph/0508242:} \;\text{CORRECT.}
\text{True}
\text{$\backslash $tCPU Time used: }48.23\text{ s.}