QCD manual (development version)

Load FeynCalc and the necessary add-ons or other packages

description = "Gh -> Gh, QCD, only UV divergences, 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, 2025-12-22 21:09:03 +01:00, fcd53f9b). 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 (2025-12-22 19:07:44 +01:00, c92fb9f5). 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.}

Configure some options

We keep scaleless B0 functions, since otherwise the UV part would not come out right.

$KeepLogDivergentScalelessIntegrals = True;

Generate Feynman diagrams

diags = InsertFields[CreateTopologies[1, 1 -> 1, ExcludeTopologies -> {Tadpoles}], 
            {U[5]} -> {U[5]}, InsertionLevel -> {Particles}, Model -> "SMQCD"]; 
 
Paint[diags, ColumnsXRows -> {1, 1}, Numbering -> Simple, 
    SheetHeader -> None, ImageSize -> {256, 256}];

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Obtain the amplitude

The 1/(2Pi)^D prefactor is implicit.

amp[0] = FCFAConvert[CreateFeynAmp[diags, Truncated -> True, GaugeRules -> {}, 
    PreFactor -> 1], IncomingMomenta -> {p}, OutgoingMomenta -> {p}, LoopMomenta -> {q}, 
    UndoChiralSplittings -> True, ChangeDimension -> D, SMP -> True, 
    DropSumOver -> True]

\left\{g_s^2 \left(-q^{\text{Lor1}}\right) p^{\text{Lor2}} f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}} \left(\frac{\left(1-\xi _g\right) (q-p)^{\text{Lor1}} (p-q)^{\text{Lor2}}}{q^2.(q-p)^4}+\frac{g^{\text{Lor1}\;\text{Lor2}}}{q^2.(q-p)^2}\right)\right\}

Calculate the amplitude

amp[1] = amp[0] // SUNSimplify[#, FCParallelize -> True] & // 
    TID[#, q, ToPaVe -> True, FCParallelize -> True] & // Total

-\frac{1}{4} i \pi ^2 p^2 C_A \left(1-\xi _g\right) g_s^2 \;\text{B}_0(0,0,0) \delta ^{\text{Glu1}\;\text{Glu2}}-\frac{1}{2} i \pi ^2 p^2 C_A g_s^2 \delta ^{\text{Glu1}\;\text{Glu2}} \;\text{B}_0\left(p^2,0,0\right)+\frac{1}{4} i \pi ^2 p^4 C_A \left(1-\xi _g\right) g_s^2 \delta ^{\text{Glu1}\;\text{Glu2}} \;\text{C}_0\left(0,p^2,p^2,0,0,0\right)

The UV divergence of the amplitude can be obtained via PaVeUVPart. Here we also need to reintroduce the implicit 1/(2Pi)^D prefactor. Hint: If you need the full result for the amplitude, use PaXEvaluate from FeynHelpers.

ampDiv[0] = PaVeUVPart[amp[1], Prefactor -> 1/(2 Pi)^D] // 
       FCReplaceD[#, D -> 4 - 2 Epsilon] & // Series[#, {Epsilon, 0, 0}] & // Normal // 
    SelectNotFree2[#, Epsilon] & // Simplify

\frac{i p^2 C_A \left(\xi _g-3\right) g_s^2 \delta ^{\text{Glu1}\;\text{Glu2}}}{64 \pi ^2 \varepsilon }

The self-energy amplitude is usually defined as (p^2 delta^ab Pi(p^2)

pi[0] = FCI[ampDiv[0]/(I SUNDelta[SUNIndex[Glu1], 
        SUNIndex[Glu2]]*SPD[p, p])] // Cancel

\frac{C_A \left(\xi _g-3\right) g_s^2}{64 \pi ^2 \varepsilon }

Check the final results

```mathematica knownResult = -SMP[“g_s”]^2/(4 Pi)^2 CA (3 - GaugeXi[g])/4*1/Epsilon; FCCompareResults[pi[0], knownResult, Text -> {“to Muta, Foundations of QCD, Eq. 2.5.136:”, “CORRECT.”, “WRONG!”}, Interrupt -> {Hold[Quit[1]], Automatic}]; Print[“Time used:”, Round[N[TimeUsed[], 4], 0.001], ” s.”];

```mathematica

\text{$\backslash $tCompare to Muta, Foundations of QCD, Eq. 2.5.136:} \;\text{CORRECT.}

\text{$\backslash $tCPU Time used: }34.508\text{ s.}