= "Ga -> Ga, massless QED, 2-loops";
description If[ $FrontEnd === Null,
= False;
$FeynCalcStartupMessages Print[description];
];
If[ $Notebooks === False,
= False
$FeynCalcStartupMessages ];
= {"FeynArts"};
$LoadAddOns
<< FeynCalc`= 0;
$FAVerbose
[10, 0, 0]; FCCheckVersion
\text{FeynCalc }\;\text{10.0.0 (dev version, 2023-12-20 22:40:59 +01:00, dff3b835). For help, use the }\underline{\text{online} \;\text{documentation}}\;\text{, check out the }\underline{\text{wiki}}\;\text{ or visit the }\underline{\text{forum}.}
\text{Please check our }\underline{\text{FAQ}}\;\text{ for answers to some common FeynCalc questions and have a look at the supplied }\underline{\text{examples}.}
\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{3.11 (3 Aug 2020) 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}
Nicer typesetting
= InsertFields[CreateTopologies[2, 1 -> 1, ExcludeTopologies -> {Tadpoles}], {V[1]} -> {V[1]},
diags -> {Particles}, ExcludeParticles -> {V[2 | 3], S[_], U[_], F[1 | 3 | 4]}]; InsertionLevel
[DiagramExtract[diags, {1, 4, 7}], ColumnsXRows -> {3, 1}, SheetHeader -> False,
Paint-> None, ImageSize -> {768, 256}]; Numbering
= FCFAConvert[CreateFeynAmp[DiagramExtract[diags, {1, 4, 7}], Truncated -> True, GaugeRules -> {},
ampRaw -> 1], IncomingMomenta -> {p}, OutgoingMomenta -> {p},LoopMomenta -> {q1, q2},
PreFactor -> True, ChangeDimension -> D, List -> True, SMP -> True,
UndoChiralSplittings -> True] // SMPToSymbol; DropSumOver
[];
FCClearScalarProducts[p, p] = pp; ScalarProduct
AbsoluteTiming[ampSimp = DiracSimplify[ampRaw /. me -> 0];]
\{0.571108,\text{Null}\}
{amp, topos} = FCLoopFindTopologies[ampSimp, {q1, q2}];
\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
= FCLoopFindSubtopologies[topos]; subtopos
= FCLoopFindTopologyMappings[topos, PreferredTopologies -> subtopos]; mappings
\text{FCLoopFindTopologyMappings: }\;\text{Found }1\text{ mapping relations }
\text{FCLoopFindTopologyMappings: }\;\text{Final number of independent topologies: }1
AbsoluteTiming[ampReduced = FCLoopTensorReduce[amp, topos];]
\{0.430326,\text{Null}\}
AbsoluteTiming[ampPreFinal = FCLoopApplyTopologyMappings[ampReduced, mappings];]
\{0.344109,\text{Null}\}
AbsoluteTiming[ampFinal = ampPreFinal // DiracSimplify;]
\{0.007818,\text{Null}\}
(*FCReloadAddOns[{"FeynHelpers"}];
FIREPrepareStartFile[mappings[[2]],FCGetNotebookDirectory[]]
FIRECreateStartFile[FCGetNotebookDirectory[],mappings[[2]]]
FIRECreateConfigFile[mappings[[2]],FCGetNotebookDirectory[]]
FIRECreateIntegralFile[Cases2[ampPreFinal,GLI],mappings[[2]],FCGetNotebookDirectory[]]
FIRERunReduction[FCGetNotebookDirectory[],mappings[[2]]]
tables=FIREImportResults[mappings[[2]],FCGetNotebookDirectory[]]//Flatten;
Put[tables,FileNameJoin[{FCGetNotebookDirectory[],"ReductionTable-Ga-Ga.m"}]];*)
= Get[FileNameJoin[{FCGetNotebookDirectory[], "ReductionTable-Ga-Ga.m"}]]; reductionTable
= Collect2[Total[ampFinal /. reductionTable], GLI] resPreFinal
-\frac{1}{3 (D-4)^2 (D-1) \;\text{pp}}2 i (D-2) e^4 G^{\text{fctopology1}}(0,1,1,0,1) \left(3 D^3 \;\text{pp} \xi _A g^{\text{Lor1}\;\text{Lor2}}-4 D^3 \xi _A p^{\text{Lor1}} p^{\text{Lor2}}-23 D^2 \;\text{pp} \xi _A g^{\text{Lor1}\;\text{Lor2}}+32 D^2 \xi _A p^{\text{Lor1}} p^{\text{Lor2}}+52 D \;\text{pp} \xi _A g^{\text{Lor1}\;\text{Lor2}}-76 D \xi _A p^{\text{Lor1}} p^{\text{Lor2}}-32 \;\text{pp} \xi _A g^{\text{Lor1}\;\text{Lor2}}+48 \xi _A p^{\text{Lor1}} p^{\text{Lor2}}+6 D^3 \;\text{pp} g^{\text{Lor1}\;\text{Lor2}}-6 D^3 p^{\text{Lor1}} p^{\text{Lor2}}-42 D^2 \;\text{pp} g^{\text{Lor1}\;\text{Lor2}}+42 D^2 p^{\text{Lor1}} p^{\text{Lor2}}+120 D \;\text{pp} g^{\text{Lor1}\;\text{Lor2}}-120 D p^{\text{Lor1}} p^{\text{Lor2}}-144 \;\text{pp} g^{\text{Lor1}\;\text{Lor2}}+144 p^{\text{Lor1}} p^{\text{Lor2}}\right)+\frac{1}{3 (D-4)^2 (D-1) \;\text{pp}}2 i (D-2) e^4 G^{\text{fctopology1}}(1,0,1,1,0) \left(3 D^3 \;\text{pp} \xi _A g^{\text{Lor1}\;\text{Lor2}}-4 D^3 \xi _A p^{\text{Lor1}} p^{\text{Lor2}}-23 D^2 \;\text{pp} \xi _A g^{\text{Lor1}\;\text{Lor2}}+32 D^2 \xi _A p^{\text{Lor1}} p^{\text{Lor2}}+52 D \;\text{pp} \xi _A g^{\text{Lor1}\;\text{Lor2}}-76 D \xi _A p^{\text{Lor1}} p^{\text{Lor2}}-32 \;\text{pp} \xi _A g^{\text{Lor1}\;\text{Lor2}}+48 \xi _A p^{\text{Lor1}} p^{\text{Lor2}}-6 D^3 \;\text{pp} g^{\text{Lor1}\;\text{Lor2}}+6 D^3 p^{\text{Lor1}} p^{\text{Lor2}}+42 D^2 \;\text{pp} g^{\text{Lor1}\;\text{Lor2}}-42 D^2 p^{\text{Lor1}} p^{\text{Lor2}}-120 D \;\text{pp} g^{\text{Lor1}\;\text{Lor2}}+120 D p^{\text{Lor1}} p^{\text{Lor2}}+144 \;\text{pp} g^{\text{Lor1}\;\text{Lor2}}-144 p^{\text{Lor1}} p^{\text{Lor2}}\right)+\frac{2 i (D-2) \left(D^2-7 D+16\right) e^4 G^{\text{fctopology1}}(1,1,0,1,1) \left(\text{pp} g^{\text{Lor1}\;\text{Lor2}}-p^{\text{Lor1}} p^{\text{Lor2}}\right)}{(D-4) (D-1)}
= FCLoopFindIntegralMappings[Cases2[resPreFinal, GLI], mappings[[2]]] integralMappings
\left\{\left\{G^{\text{fctopology1}}(1,0,1,1,0)\to G^{\text{fctopology1}}(0,1,1,0,1)\right\},\left\{G^{\text{fctopology1}}(0,1,1,0,1),G^{\text{fctopology1}}(1,1,0,1,1)\right\}\right\}
= Collect2[resPreFinal /. integralMappings[[1]], GLI] resFinal
\frac{2 i (D-2) \left(D^2-7 D+16\right) e^4 G^{\text{fctopology1}}(1,1,0,1,1) \left(\text{pp} g^{\text{Lor1}\;\text{Lor2}}-p^{\text{Lor1}} p^{\text{Lor2}}\right)}{(D-4) (D-1)}-\frac{8 i (D-3) (D-2) \left(D^2-4 D+8\right) e^4 G^{\text{fctopology1}}(0,1,1,0,1) \left(\text{pp} g^{\text{Lor1}\;\text{Lor2}}-p^{\text{Lor1}} p^{\text{Lor2}}\right)}{(D-4)^2 (D-1) \;\text{pp}}
= -I FCI[e^4 2 (D - 2)/((D - 1) (D - 4)) (-(D^2 - 7 D + 16) GLI["fctopology1", {1, 1, 0, 1, 1}] +
resGrozinVacuumPol 4 (D - 3) (D^2 - 4 D + 8)/(D - 4) (1/SPD[p, p]) GLI["fctopology1", {0, 1, 1, 0, 1}]) (-(FVD[p, Lor1]*FVD[p, Lor2]) +
*MTD[Lor1, Lor2])];
pp[resFinal, resGrozinVacuumPol,
FCCompareResultsText -> {"\tCompare to Grozin's Lectures on QED and QCD, hep-ph/0508242, Eq. 5.18:",
"CORRECT.", "WRONG!"}, Interrupt -> {Hold[Quit[1]], Automatic}, Factoring -> Simplify];
Print["\tCPU Time used: ", Round[N[TimeUsed[], 4], 0.001], " s."];
\text{$\backslash $tCompare to Grozin's Lectures on QED and QCD, hep-ph/0508242, Eq. 5.18:} \;\text{CORRECT.}
\text{$\backslash $tCPU Time used: }29.186\text{ s.}