This example uses a custom QED model created with FeynRules.
description = "Renormalization, QED, MSbar, 1-loop";
If[ $FrontEnd === Null,
$FeynCalcStartupMessages = False;
Print[description];
];
If[ $Notebooks === False,
$FeynCalcStartupMessages = False
];
LaunchKernels[4];
$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"}];FAPatch[PatchModelsOnly -> True, FAModelsDirectory -> modelDir];
(*Successfully patched FeynArts.*)Here we define all Z-factors for renormalization constants present in the Lagrangian
renConstants = Zm | Zpsi | ZA | ZAmxt | Ze | Zxi\text{Zm}|\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[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}]}];diagPhotonSE = InsertFields[CreateTopologies[1, 1 -> 1,
ExcludeTopologies -> {Tadpoles, WFCorrections, WFCorrectionCTs}], {V[1]} -> {V[1]},
InsertionLevel -> {Particles}, Model -> FileNameJoin[{modelDir, "QED"}],
GenericModel -> FileNameJoin[{modelDir, "QED"}], ExcludeParticles -> {F[2, {2 | 3}]}];diagLeptonPhotonVTX = InsertFields[CreateTopologies[1, 2 -> 1,
ExcludeTopologies -> {Tadpoles, WFCorrections, WFCorrectionCTs}], {F[2, {1}], V[1]} -> {F[2, {1}]},
InsertionLevel -> {Particles}, Model -> FileNameJoin[{modelDir, "QED"}],
GenericModel -> FileNameJoin[{modelDir, "QED"}], ExcludeParticles -> {F[2, {2 | 3}]}];diagLeptonSECT = 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}]}];diagPhotonSECT = InsertFields[CreateCTTopologies[1, 1 -> 1,
ExcludeTopologies -> {Tadpoles, WFCorrections, WFCorrectionCTs}], {V[1]} -> {V[1]},
InsertionLevel -> {Particles}, Model -> FileNameJoin[{modelDir, "QED"}],
GenericModel -> FileNameJoin[{modelDir, "QED"}], ExcludeParticles -> {F[2, {2 | 3}]}];diagLeptonPhotonVTXCT = InsertFields[CreateCTTopologies[1, 2 -> 1,
ExcludeTopologies -> {Tadpoles, WFCorrections, WFCorrectionCTs}], {F[2, {1}], V[1]} -> {F[2, {1}]},
InsertionLevel -> {Particles}, Model -> FileNameJoin[{modelDir, "QED"}],
GenericModel -> FileNameJoin[{modelDir, "QED"}], ExcludeParticles -> {F[2, {2 | 3}]}];Self-energy and vertex diagrams
Paint[diagLeptonSE, ColumnsXRows -> {2, 1}, SheetHeader -> None,
Numbering -> Simple, ImageSize -> 128 {2, 1}];
Paint[diagPhotonSE, ColumnsXRows -> {4, 1}, SheetHeader -> None,
Numbering -> Simple, ImageSize -> 128 {4, 1}];
Paint[diagLeptonPhotonVTX, ColumnsXRows -> {4, 1}, SheetHeader -> None,
Numbering -> Simple, ImageSize -> 128 {4, 1}];
Counter-term diagrams
Paint[diagLeptonSECT, ColumnsXRows -> {2, 1}, SheetHeader -> None,
Numbering -> Simple, ImageSize -> 128 {2, 1}];
Paint[diagPhotonSECT, ColumnsXRows -> {4, 1}, SheetHeader -> None,
Numbering -> Simple, ImageSize -> 128 {4, 1}];
Paint[diagLeptonPhotonVTXCT, ColumnsXRows -> {4, 1}, SheetHeader -> None,
Numbering -> Simple, ImageSize -> 128 {4, 1}];
The only required masters are 1-loop tadpoles
tadpoleMaster = Get[FileNameJoin[{$FeynCalcDirectory, "Examples", "MasterIntegrals","Tadpoles", "tad1LxFx1x1xxEp999x.m"}]];tadpoleMaster1 = tadpoleMaster /. m1 -> ml /. tad1LxFx1x1xxEp999x -> "tad1Lv1";
tadpoleMaster2 = tadpoleMaster /. m1 -> mxt /. tad1LxFx1x1xxEp999x -> "tad1Lv2";tadpoleMaster1\left\{G^{\text{tad1Lv1}}(1)\to -e^{\gamma \;\text{ep}} \left(\text{ml}^2\right)^{1-\text{ep}} \Gamma (\text{ep}-1),\left\{\text{FCTopology}\left(\text{tad1Lv1},\left\{\frac{1}{(\text{k1}^2-\text{ml}^2+i \eta )}\right\},\{\text{k1}\},\{\},\{\},\{\}\right)\right\}\right\}
tadpoleMaster2\left\{G^{\text{tad1Lv2}}(1)\to -e^{\gamma \;\text{ep}} \left(\text{mxt}^2\right)^{1-\text{ep}} \Gamma (\text{ep}-1),\left\{\text{FCTopology}\left(\text{tad1Lv2},\left\{\frac{1}{(\text{k1}^2-\text{mxt}^2+i \eta )}\right\},\{\text{k1}\},\{\},\{\},\{\}\right)\right\}\right\}
{leptonSE$RawAmp, photonSE$RawAmp, leptonSECT$RawAmp, photonSECT$RawAmp} =
FCFAConvert[CreateFeynAmp[#, Truncated -> True,
GaugeRules -> {}, PreFactor -> 1],
IncomingMomenta -> {p}, OutgoingMomenta -> {p},
LorentzIndexNames -> {mu, nu}, DropSumOver -> True,
LoopMomenta -> {k}, UndoChiralSplittings -> True,
ChangeDimension -> D, SMP -> True,
FinalSubstitutions -> {SMP["m_e"] -> ml, SMP["e"] -> 4 Pi Sqrt[a4]}] & /@ {
diagLeptonSE, diagPhotonSE, diagLeptonSECT, diagPhotonSECT};{leptonPhotonVTX$RawAmp, leptonPhotonVTXCT$RawAmp} =
FCFAConvert[CreateFeynAmp[#, Truncated -> True,
GaugeRules -> {}, PreFactor -> 1],
IncomingMomenta -> {p1, p2}, OutgoingMomenta -> {q},
LorentzIndexNames -> {mu, nu}, DropSumOver -> True,
LoopMomenta -> {k}, UndoChiralSplittings -> True,
ChangeDimension -> D, SMP -> True,
FinalSubstitutions -> {SMP["m_e"] -> ml, SMP["e"] -> 4 Pi Sqrt[a4]}] & /@ {
diagLeptonPhotonVTX, diagLeptonPhotonVTXCT
};Infrared rearrangement works both for massive and massless leptons. However, in both cases we get different renormalization constants for the “photon mass”. This is why both calculations are needed if we want to reuse these results at higher loops orders.
The 1-loop lepton self-energy has superficial degree of divergence equal to 1
FCClearScalarProducts[];
divDegree = 1;
aux1 = FCLoopGetFeynAmpDenominators[leptonSE$RawAmp, {k}, denHead, Momentum -> {p}, "Massless" -> True];
aux2 = FCLoopAddAuxiliaryMass[aux1[[2]], {k}, -mxt^2, 0, Head -> denHead]\left\{\text{denHead}\left(\frac{1}{((k-p)^2+i \eta )}\right)\to \frac{1}{((k-p)^2-\text{mxt}^2+i \eta )}\right\}
leptonSE$StrName = StringReplace[ToString[Hold[leptonSE$Amp]], {"Hold[" -> "", "]" -> ""}]\text{leptonSE\$Amp}
AbsoluteTiming[leptonSE$Amp = (aux1[[1]] /. aux2) // Contract[#, FCParallelize -> True] & //
SUNSimplify[#, FCParallelize -> True] & // DiracSimplify[#, FCParallelize -> True] &;]\{0.143688,\text{Null}\}
flagCheck is a safety flag to ensure that higher order terms in p (higher than the divergence degree) do not contribute to the poles
AbsoluteTiming[leptonSE$Amp1 = Collect2[leptonSE$Amp, p, IsolateNames -> KK];]
AbsoluteTiming[leptonSE$Amp2 = FourSeries[leptonSE$Amp1, {p, 0, divDegree}, FCParallelize -> True];]
AbsoluteTiming[leptonSE$Amp3 = Collect2[FRH[leptonSE$Amp2], FeynAmpDenominator, FCParallelize -> True];]\{0.038089,\text{Null}\}
\{0.076349,\text{Null}\}
\{0.023505,\text{Null}\}
The rest of the calculation follows the standard multiloop template
FCClearScalarProducts[];
SPD[p] = pp;{leptonSE$Amp4, leptonSE$Topos} = FCLoopFindTopologies[leptonSE$Amp3, {k}, 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[leptonSE$Amp5 = FCLoopTensorReduce[leptonSE$Amp4, leptonSE$Topos, FCParallelize -> True];]\{0.387859,\text{Null}\}
{leptonSE$Amp6, leptonSE$Topos2} = FCLoopRewriteOverdeterminedTopologies[leptonSE$Amp5, leptonSE$Topos];\text{FCLoopRewriteOverdeterminedTopologies: }\;\text{Found }1\text{ overdetermined topologies.}
\text{FCLoopRewriteOverdeterminedTopologies: }\;\text{Generated }9\text{ new topologies through partial fractioning.}
\text{FCLoopRewriteOverdeterminedTopologies: }\;\text{Final number of topologies: }2
leptonSE$SubTopos = FCLoopFindSubtopologies[leptonSE$Topos2, Flatten -> True, Remove -> True]\{\}
{leptonSE$TopoMappings, leptonSE$FinalTopos} = FCLoopFindTopologyMappings[leptonSE$Topos2, PreferredTopologies -> leptonSE$SubTopos];\text{FCLoopFindTopologyMappings: }\;\text{Found }0\text{ mapping relations }
\text{FCLoopFindTopologyMappings: }\;\text{Final number of independent topologies: }2
leptonSE$AmpGLI = FCLoopApplyTopologyMappings[leptonSE$Amp6, {leptonSE$TopoMappings, leptonSE$FinalTopos}, FCParallelize -> True];leptonSE$GLIs = Cases2[leptonSE$AmpGLI, GLI];leptonSE$dir = FileNameJoin[{$TemporaryDirectory, "Reduction-" <> leptonSE$StrName <> "-1L"}];
Quiet[CreateDirectory[leptonSE$dir]];KiraCreateJobFile[leptonSE$FinalTopos, leptonSE$GLIs, leptonSE$dir]\{\text{/tmp/Reduction-leptonSE\$Amp-1L/fcPFRTopology1/job.yaml},\text{/tmp/Reduction-leptonSE\$Amp-1L/fcPFRTopology2/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: }3
\text{KiraCreateIntegralFile: Number of loop integrals: }5
\{\text{/tmp/Reduction-leptonSE\$Amp-1L/fcPFRTopology1/KiraLoopIntegrals},\text{/tmp/Reduction-leptonSE\$Amp-1L/fcPFRTopology2/KiraLoopIntegrals}\}
\left( \begin{array}{cc} \;\text{/tmp/Reduction-leptonSE\$Amp-1L/fcPFRTopology1/config/integralfamilies.yaml} & \;\text{/tmp/Reduction-leptonSE\$Amp-1L/fcPFRTopology1/config/kinematics.yaml} \\ \;\text{/tmp/Reduction-leptonSE\$Amp-1L/fcPFRTopology2/config/integralfamilies.yaml} & \;\text{/tmp/Reduction-leptonSE\$Amp-1L/fcPFRTopology2/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},\text{True}\}
leptonSE$ReductionTables = KiraImportResults[leptonSE$FinalTopos, leptonSE$dir] // Flatten;leptonSE$resPreFinal = Collect2[Total[leptonSE$AmpGLI /. Dispatch[leptonSE$ReductionTables]] // FeynAmpDenominatorExplicit, GLI,
GaugeXi, flagCheck, D, DiracGamma, FCParallelize -> True];leptonSE$masters = Cases2[leptonSE$resPreFinal, GLI];leptonSE$MIMappings = FCLoopFindIntegralMappings[leptonSE$masters, Join[tadpoleMaster1[[2]], tadpoleMaster2[[2]],
leptonSE$FinalTopos], PreferredIntegrals -> {tadpoleMaster1[[1]][[1]], tadpoleMaster2[[1]][[1]]}]\left( \begin{array}{cc} G^{\text{fcPFRTopology1}}(1)\to G^{\text{tad1Lv1}}(1) & G^{\text{fcPFRTopology2}}(1)\to G^{\text{tad1Lv2}}(1) \\ G^{\text{tad1Lv1}}(1) & G^{\text{tad1Lv2}}(1) \\ \end{array} \right)
Our master integrals are calculated using the standard multiloop normalization. To convert it back to the textbook normalization we need to multiply by I*(4 Pi)^(ep-2)
leptonSE$resFinal = Collect2[leptonSE$resPreFinal, D, GLI, IsolateNames -> KK] // FCReplaceD[#, D -> 4 - 2 ep] & //
ReplaceAll[#, leptonSE$MIMappings[[1]]] & // ReplaceAll[#, {tadpoleMaster1[[1]], tadpoleMaster2[[1]]}] & //
Collect2[#, ep, IsolateNames -> KK2] & // Series[(I*(4*Pi)^(-2 + ep)) #, {ep, 0, -1}] & // Normal // FRH //Collect2[#, DiracGamma] &\frac{i \;\text{a4} \xi _{V(1)} \gamma \cdot p}{\text{ep}}-\frac{i \;\text{a4} \;\text{ml} \left(\xi _{V(1)}+3\right)}{\text{ep}}
leptonSE$RenConstants = (leptonSE$resFinal + Total[leptonSECT$RawAmp]) // ReplaceRepeated[#, {
(h : renConstants) :> 1 + alpha a4 rc[ToExpression["del" <> ToString[h]], 1]}] & //
Series[#, {alpha, 0, 1}] & // Normal //
ReplaceAll[#, alpha -> 1] & // Collect2[#, DiracGamma] & //
FCMatchSolve[#, {ep, CF, DiracGamma, ml, mxt, SUNDelta, SUNFDelta, GaugeXi, a4}] &\text{FCMatchSolve: Solving for: }\{\text{rc}(\text{delZm},1),\text{rc}(\text{delZpsi},1)\}
\text{FCMatchSolve: A solution exists.}
\left\{\text{rc}(\text{delZm},1)\to -\frac{3}{\text{ep}},\text{rc}(\text{delZpsi},1)\to -\frac{\xi _{V(1)}}{\text{ep}}\right\}
The 1-loop photon self-energy has superficial degree of divergence equal to 2. We also add the number of flavors by hand by multiplying the corresponding diagrams with Nf.
FCClearScalarProducts[];
photonSE$RawAmp2 = {Nf photonSE$RawAmp[[1]]};
divDegree = 2;
aux1 = FCLoopGetFeynAmpDenominators[photonSE$RawAmp2, {k}, denHead, Momentum -> {p}, "Massless" -> True];
aux2 = FCLoopAddAuxiliaryMass[aux1[[2]], {k}, -mxt^2, 0, Head -> denHead]\left\{\text{denHead}\left(\frac{1}{((k-p)^2-\text{ml}^2+i \eta )}\right)\to \frac{1}{((k-p)^2-\text{ml}^2+i \eta )}\right\}
photonSE$StrName = StringReplace[ToString[Hold[photonSE$Amp]], {"Hold[" -> "", "]" -> ""}]\text{photonSE\$Amp}
AbsoluteTiming[photonSE$Amp = (aux1[[1]] /. aux2) // Contract[#, FCParallelize -> True] & //
SUNSimplify[#, FCParallelize -> True] & // DiracSimplify[#, FCParallelize -> True] &;]\{0.1356,\text{Null}\}
flagCheck is a safety flag to ensure that higher order terms in p (higher than the divergence degree) do not contribute to the poles
AbsoluteTiming[photonSE$Amp1 = Collect2[photonSE$Amp, p, IsolateNames -> KK];]
AbsoluteTiming[photonSE$Amp2 = FourSeries[photonSE$Amp1, {p, 0, divDegree}, FCParallelize -> True];]
AbsoluteTiming[photonSE$Amp3 = Collect2[FRH[photonSE$Amp2], FeynAmpDenominator, FCParallelize -> True];]\{0.012634,\text{Null}\}
\{0.087919,\text{Null}\}
\{0.035541,\text{Null}\}
The rest of the calculation follows the standard multiloop template
FCClearScalarProducts[];
SPD[p] = pp;{photonSE$Amp4, photonSE$Topos} = FCLoopFindTopologies[photonSE$Amp3, {k}, FCParallelize -> True,
FCLoopBasisOverdeterminedQ -> True, FinalSubstitutions -> {Hold[SPD][p] -> pp}, Names -> photonSEtopo];\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[photonSE$Amp5 = FCLoopTensorReduce[photonSE$Amp4, photonSE$Topos, FCParallelize -> True];]\{0.283148,\text{Null}\}
{photonSE$Amp6, photonSE$Topos2} = FCLoopRewriteOverdeterminedTopologies[photonSE$Amp5, photonSE$Topos];\text{FCLoopRewriteIncompleteTopologies: }\;\text{No overdetermined topologies detected.}
photonSE$SubTopos = FCLoopFindSubtopologies[photonSE$Topos2, Flatten -> True, Remove -> True]\{\}
{photonSE$TopoMappings, photonSE$FinalTopos} = FCLoopFindTopologyMappings[photonSE$Topos2, PreferredTopologies -> photonSE$SubTopos];\text{FCLoopFindTopologyMappings: }\;\text{Found }0\text{ mapping relations }
\text{FCLoopFindTopologyMappings: }\;\text{Final number of independent topologies: }1
photonSE$AmpGLI = FCLoopApplyTopologyMappings[photonSE$Amp6, {photonSE$TopoMappings, photonSE$FinalTopos}, FCParallelize -> True];photonSE$GLIs = Cases2[photonSE$AmpGLI, GLI];photonSE$dir = FileNameJoin[{$TemporaryDirectory, "Reduction-" <> photonSE$StrName <> "-1L"}];
Quiet[CreateDirectory[photonSE$dir]];KiraCreateJobFile[photonSE$FinalTopos, photonSE$GLIs, photonSE$dir]\{\text{/tmp/Reduction-photonSE\$Amp-1L/photonSEtopo1/job.yaml}\}
KiraCreateIntegralFile[photonSE$GLIs, photonSE$FinalTopos, photonSE$dir]
KiraCreateConfigFiles[photonSE$FinalTopos, photonSE$GLIs, photonSE$dir,
KiraMassDimensions -> {pp -> 2, ml -> 1, mxt -> 1}]\text{KiraCreateIntegralFile: Number of loop integrals: }4
\{\text{/tmp/Reduction-photonSE\$Amp-1L/photonSEtopo1/KiraLoopIntegrals}\}
\left( \begin{array}{cc} \;\text{/tmp/Reduction-photonSE\$Amp-1L/photonSEtopo1/config/integralfamilies.yaml} & \;\text{/tmp/Reduction-photonSE\$Amp-1L/photonSEtopo1/config/kinematics.yaml} \\ \end{array} \right)
KiraRunReduction[photonSE$dir, photonSE$FinalTopos,
KiraBinaryPath -> FileNameJoin[{$HomeDirectory, ".local", "bin", "kira"}],
KiraFermatPath -> FileNameJoin[{$HomeDirectory, "bin", "ferl64", "fer64"}]]\{\text{True}\}
photonSE$ReductionTables = KiraImportResults[photonSE$FinalTopos, photonSE$dir] // Flatten;photonSE$resPreFinal = Collect2[Total[photonSE$AmpGLI /. Dispatch[photonSE$ReductionTables]] // FeynAmpDenominatorExplicit, GLI,
GaugeXi, flagCheck, D, DiracGamma, FCParallelize -> True];photonSE$masters = Cases2[photonSE$resPreFinal, GLI];photonSE$MIMappings = FCLoopFindIntegralMappings[photonSE$masters, Join[tadpoleMaster1[[2]], tadpoleMaster2[[2]],
photonSE$FinalTopos], PreferredIntegrals -> {tadpoleMaster1[[1]][[1]], tadpoleMaster2[[1]][[1]]}]\left( \begin{array}{c} G^{\text{photonSEtopo1}}(1)\to G^{\text{tad1Lv1}}(1) \\ G^{\text{tad1Lv1}}(1) \\ \end{array} \right)
Our master integrals are calculated using the standard multiloop normalization. To convert it back to the textbook normalization we need to multiply by I*(4 Pi)^(ep-2)
photonSE$resFinal = Collect2[photonSE$resPreFinal, D, GLI, IsolateNames -> KK] // FCReplaceD[#, D -> 4 - 2 ep] & //
ReplaceAll[#, photonSE$MIMappings[[1]]] & // ReplaceAll[#, {tadpoleMaster1[[1]], tadpoleMaster2[[1]]}] & //
Collect2[#, ep, IsolateNames -> KK2] & // Series[(I*(4*Pi)^(-2 + ep)) #, {ep, 0, -1}] & // Normal // FRH //Collect2[#, DiracGamma] &-\frac{4 i \;\text{a4} N_f \left(\text{pp} g^{\mu \nu }-p^{\mu } p^{\nu }\right)}{3 \;\text{ep}}
photonSE$RenConstants = (photonSE$resFinal + Total[photonSECT$RawAmp]) // ReplaceRepeated[#, {
(h : renConstants) :> 1 + alpha a4 rc[ToExpression["del" <> ToString[h]], 1]}] & //
Series[#, {alpha, 0, 1}] & // Normal //
ReplaceAll[#, alpha -> 1] & // Collect2[#, DiracGamma, pp, Pair, mxt] & //
FCMatchSolve[#, {ep, CF, DiracGamma, ml, mxt, SUNDelta, SUNFDelta, CA, GaugeXi, a4, Pair, pp, Nf}] &\text{FCMatchSolve: Following coefficients trivially vanish: }\{\text{rc}(\text{delZAmxt},1)\to 0\}
\text{FCMatchSolve: Solving for: }\{\text{rc}(\text{delZA},1),\text{rc}(\text{delZxi},1)\}
\text{FCMatchSolve: A solution exists.}
\left\{\text{rc}(\text{delZAmxt},1)\to 0,\text{rc}(\text{delZA},1)\to -\frac{4 N_f}{3 \;\text{ep}},\text{rc}(\text{delZxi},1)\to -\frac{4 N_f}{3 \;\text{ep}}\right\}
The 1-loop lepton-photon-vertex has superficial degree of divergence equal to 0. We set q=0, so that p+p2=q yields p=-p2
FCClearScalarProducts[];
divDegree = 0;
aux1 = FCLoopGetFeynAmpDenominators[leptonPhotonVTX$RawAmp /. q -> 0 /. p2 -> -p, {k}, denHead, Momentum -> {p}, "Massless" -> True];
aux2 = FCLoopAddAuxiliaryMass[aux1[[2]], {k}, -mxt^2, 0, Head -> denHead]\left\{\text{denHead}\left(\frac{1}{((k-p)^2+i \eta )}\right)\to \frac{1}{((k-p)^2-\text{mxt}^2+i \eta )},\text{denHead}\left(\frac{1}{((k-p)^2-\text{ml}^2+i \eta )}\right)\to \frac{1}{((k-p)^2-\text{ml}^2+i \eta )}\right\}
leptonPhotonVTX$StrName = StringReplace[ToString[Hold[leptonPhotonVTX$Amp]], {"Hold[" -> "", "]" -> ""}]\text{leptonPhotonVTX\$Amp}
AbsoluteTiming[leptonPhotonVTX$Amp = (aux1[[1]] /. aux2) // Contract[#, FCParallelize -> True] & //
SUNSimplify[#, FCParallelize -> True] & // DiracSimplify[#, FCParallelize -> True] &;]\{0.234239,\text{Null}\}
AbsoluteTiming[leptonPhotonVTX$Amp1 = Collect2[leptonPhotonVTX$Amp, p, IsolateNames -> KK];]
AbsoluteTiming[leptonPhotonVTX$Amp2 = FourSeries[leptonPhotonVTX$Amp1, {p, 0, divDegree}, FCParallelize -> True];]
AbsoluteTiming[leptonPhotonVTX$Amp3 = Collect2[FRH[leptonPhotonVTX$Amp2], FeynAmpDenominator, FCParallelize -> True];]\{0.069615,\text{Null}\}
\{0.029393,\text{Null}\}
\{0.045947,\text{Null}\}
The rest of the calculation follows the standard multiloop template
FCClearScalarProducts[];
SPD[p] = pp;{leptonPhotonVTX$Amp4, leptonPhotonVTX$Topos} = FCLoopFindTopologies[leptonPhotonVTX$Amp3, {k}, FCParallelize -> True,
FCLoopBasisOverdeterminedQ -> True, FinalSubstitutions -> {Hold[SPD][p] -> pp}, Names -> leptonPhotonVTXtopo];\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[leptonPhotonVTX$Amp5 = FCLoopTensorReduce[leptonPhotonVTX$Amp4, leptonPhotonVTX$Topos, FCParallelize -> True];]\{0.340732,\text{Null}\}
{leptonPhotonVTX$Amp6, leptonPhotonVTX$Topos2} = FCLoopRewriteOverdeterminedTopologies[leptonPhotonVTX$Amp5, leptonPhotonVTX$Topos];\text{FCLoopRewriteOverdeterminedTopologies: }\;\text{Found }1\text{ overdetermined topologies.}
\text{FCLoopRewriteOverdeterminedTopologies: }\;\text{Generated }7\text{ new topologies through partial fractioning.}
\text{FCLoopRewriteOverdeterminedTopologies: }\;\text{Final number of topologies: }2
leptonPhotonVTX$SubTopos = FCLoopFindSubtopologies[leptonPhotonVTX$Topos2, Flatten -> True, Remove -> True]\{\}
{leptonPhotonVTX$TopoMappings, leptonPhotonVTX$FinalTopos} = FCLoopFindTopologyMappings[leptonPhotonVTX$Topos2, PreferredTopologies -> leptonPhotonVTX$SubTopos];\text{FCLoopFindTopologyMappings: }\;\text{Found }0\text{ mapping relations }
\text{FCLoopFindTopologyMappings: }\;\text{Final number of independent topologies: }2
leptonPhotonVTX$AmpGLI = FCLoopApplyTopologyMappings[leptonPhotonVTX$Amp6, {leptonPhotonVTX$TopoMappings, leptonPhotonVTX$FinalTopos}, FCParallelize -> True];leptonPhotonVTX$GLIs = Cases2[leptonPhotonVTX$AmpGLI, GLI];leptonPhotonVTX$dir = FileNameJoin[{$TemporaryDirectory, "Reduction-" <> leptonPhotonVTX$StrName <> "-1L"}];
Quiet[CreateDirectory[leptonPhotonVTX$dir]];KiraCreateJobFile[leptonPhotonVTX$FinalTopos, leptonPhotonVTX$GLIs, leptonPhotonVTX$dir]\{\text{/tmp/Reduction-leptonPhotonVTX\$Amp-1L/fcPFRTopology1/job.yaml},\text{/tmp/Reduction-leptonPhotonVTX\$Amp-1L/fcPFRTopology2/job.yaml}\}
KiraCreateIntegralFile[leptonPhotonVTX$GLIs, leptonPhotonVTX$FinalTopos, leptonPhotonVTX$dir]
KiraCreateConfigFiles[leptonPhotonVTX$FinalTopos, leptonPhotonVTX$GLIs, leptonPhotonVTX$dir,
KiraMassDimensions -> {pp -> 2, ml -> 1, mxt -> 1}]\text{KiraCreateIntegralFile: Number of loop integrals: }4
\text{KiraCreateIntegralFile: Number of loop integrals: }4
\{\text{/tmp/Reduction-leptonPhotonVTX\$Amp-1L/fcPFRTopology1/KiraLoopIntegrals},\text{/tmp/Reduction-leptonPhotonVTX\$Amp-1L/fcPFRTopology2/KiraLoopIntegrals}\}
\left( \begin{array}{cc} \;\text{/tmp/Reduction-leptonPhotonVTX\$Amp-1L/fcPFRTopology1/config/integralfamilies.yaml} & \;\text{/tmp/Reduction-leptonPhotonVTX\$Amp-1L/fcPFRTopology1/config/kinematics.yaml} \\ \;\text{/tmp/Reduction-leptonPhotonVTX\$Amp-1L/fcPFRTopology2/config/integralfamilies.yaml} & \;\text{/tmp/Reduction-leptonPhotonVTX\$Amp-1L/fcPFRTopology2/config/kinematics.yaml} \\ \end{array} \right)
KiraRunReduction[leptonPhotonVTX$dir, leptonPhotonVTX$FinalTopos,
KiraBinaryPath -> FileNameJoin[{$HomeDirectory, ".local", "bin", "kira"}],
KiraFermatPath -> FileNameJoin[{$HomeDirectory, "bin", "ferl64", "fer64"}]]\{\text{True},\text{True}\}
leptonPhotonVTX$ReductionTables = KiraImportResults[leptonPhotonVTX$FinalTopos, leptonPhotonVTX$dir] //Flatten;leptonPhotonVTX$resPreFinal = Collect2[Total[leptonPhotonVTX$AmpGLI /. Dispatch[leptonPhotonVTX$ReductionTables]] // FeynAmpDenominatorExplicit, GLI,
GaugeXi, flagCheck, D, DiracGamma, FCParallelize -> True];leptonPhotonVTX$masters = Cases2[leptonPhotonVTX$resPreFinal, GLI];leptonPhotonVTX$MIMappings = FCLoopFindIntegralMappings[leptonPhotonVTX$masters, Join[tadpoleMaster1[[2]], tadpoleMaster2[[2]],
leptonPhotonVTX$FinalTopos], PreferredIntegrals -> {tadpoleMaster1[[1]][[1]], tadpoleMaster2[[1]][[1]]}]\left( \begin{array}{cc} G^{\text{fcPFRTopology1}}(1)\to G^{\text{tad1Lv1}}(1) & G^{\text{fcPFRTopology2}}(1)\to G^{\text{tad1Lv2}}(1) \\ G^{\text{tad1Lv1}}(1) & G^{\text{tad1Lv2}}(1) \\ \end{array} \right)
Our master integrals are calculated using the standard multiloop normalization. To convert it back to the textbook normalization we need to multiply by I*(4 Pi)^(ep-2)
leptonPhotonVTX$resFinal = Collect2[leptonPhotonVTX$resPreFinal, D, GLI, IsolateNames -> KK] // FCReplaceD[#, D -> 4 - 2 ep] & //
ReplaceAll[#, leptonPhotonVTX$MIMappings[[1]]] & // ReplaceAll[#, {tadpoleMaster1[[1]], tadpoleMaster2[[1]]}] & //
Collect2[#, ep, IsolateNames -> KK2] & // Series[(I*(4*Pi)^(-2 + ep)) #, {ep, 0, -1}] & // Normal // FRH //Collect2[#, DiracGamma] &\frac{4 i \pi \;\text{a4}^{3/2} \gamma ^{\mu } \xi _{V(1)}}{\text{ep}}
leptonPhotonVTX$resFinal + Total[leptonPhotonVTXCT$RawAmp]\frac{4 i \pi \;\text{a4}^{3/2} \gamma ^{\mu } \xi _{V(1)}}{\text{ep}}+4 i \pi \sqrt{\text{a4}} \gamma ^{\mu } \left(\sqrt{\text{ZA}} \;\text{Ze} \;\text{Zpsi}-1\right)
leptonPhotonVTX$RenConstants = (leptonPhotonVTX$resFinal + Total[leptonPhotonVTXCT$RawAmp]) // ReplaceRepeated[#, {
(h : renConstants) :> 1 + alpha a4 rc[ToExpression["del" <> ToString[h]], 1]}] & //
Series[#, {alpha, 0, 1}] & // Normal // ReplaceAll[#, Join[photonSE$RenConstants, leptonSE$RenConstants]] & //
ReplaceAll[#, alpha -> 1] & // Collect2[#, DiracGamma, pp, Pair, mxt] & //
FCMatchSolve[#, {ep, CF, DiracGamma, ml, mxt, SUNDelta, SUNTF, SUNFDelta, CA, GaugeXi, a4, Pair, pp, Nf}] &\text{FCMatchSolve: Solving for: }\{\text{rc}(\text{delZe},1)\}
\text{FCMatchSolve: A solution exists.}
\left\{\text{rc}(\text{delZe},1)\to \frac{2 N_f}{3 \;\text{ep}}\right\}
Our final QED 1-loop renormalization constants
finalResults = Thread[Rule[List @@ renConstants,
(List @@ renConstants /. (h : renConstants) :> 1 + a4 rc[ToExpression["del" <> ToString[h]], 1]) // ReplaceAll[#, Join[photonSE$RenConstants, leptonSE$RenConstants,
leptonPhotonVTX$RenConstants]] &]]\left\{\text{Zm}\to 1-\frac{3 \;\text{a4}}{\text{ep}},\text{Zpsi}\to 1-\frac{\text{a4} \xi _{V(1)}}{\text{ep}},\text{ZA}\to 1-\frac{4 \;\text{a4} N_f}{3 \;\text{ep}},\text{ZAmxt}\to 1,\text{Ze}\to \frac{2 \;\text{a4} N_f}{3 \;\text{ep}}+1,\text{Zxi}\to 1-\frac{4 \;\text{a4} N_f}{3 \;\text{ep}}\right\}
knownResult = {
rc[delZAmxt, 1] -> 0,
rc[delZA, 1] -> (-4*Nf)/(3*ep),
rc[delZxi, 1] -> (-4*Nf)/(3*ep), rc[delZm, 1] -> -3/ep,
rc[delZpsi, 1] -> -(GaugeXi[V[1]]/ep),
rc[delZe, 1] -> (2*Nf)/(3*ep)};```mathematica FCCompareResults[Join[photonSERenConstants, leptonSERenConstants, leptonPhotonVTX$RenConstants] /. Rule -> Equal, knownResult /. Rule -> Equal, Text -> {“the final result:”, “CORRECT.”, “WRONG!”}, Interrupt -> {Hold[Quit[1]], Automatic}] Print[“Time used:”, Round[N[TimeUsed[], 4], 0.001], ” s.”];
```mathematica
\text{$\backslash $tCheck the final result:} \;\text{CORRECT.}
\text{True}
\text{$\backslash $tCPU Time used: }38.337\text{ s.}