QCD manual (development version)

Load FeynCalc and the necessary add-ons or other packages

description = "Gl -> Gl, QCD, only UV divergences, 1-loop";
If[ $FrontEnd === Null, 
    $FeynCalcStartupMessages = False; 
    Print[description]; 
  ];
If[ $Notebooks === False, 
    $FeynCalcStartupMessages = False 
  ];
$LoadAddOns = {"FeynArts"};
<< FeynCalc`
$FAVerbose = 0; 
 
FCCheckVersion[9, 3, 1];

\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}

Configure some options

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

$KeepLogDivergentScalelessIntegrals = True;

Generate Feynman diagrams

Nicer typesetting

MakeBoxes[mu, TraditionalForm] := "\[Mu]";
MakeBoxes[nu, TraditionalForm] := "\[Nu]";
diags = InsertFields[CreateTopologies[1, 1 -> 1, ExcludeTopologies -> {Tadpoles}], 
            {V[5]} -> {V[5]}, InsertionLevel -> {Particles}, Model -> "SMQCD", 
            ExcludeParticles -> {S[_], V[2 | 3], F[4], F[3, {2 | 3}]}]; 
 
Paint[diags, ColumnsXRows -> {2, 2}, Numbering -> Simple, 
    SheetHeader -> None, ImageSize -> {512, 512}];

0d71mzxqsstdl

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}, 
    LorentzIndexNames -> {mu, nu}, UndoChiralSplittings -> True, 
    ChangeDimension -> D, List -> True, SMP -> True, DropSumOver -> True, 
    Contract -> True, FinalSubstitutions -> {SMP["m_u"] -> SMP["m_q"]}]

\left\{\frac{g^{\mu \nu } g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{\$AL\$12977}} f^{\text{Glu2}\;\text{Glu3}\;\text{\$AL\$12977}}}{2 q^2}+\frac{\left(\xi _g-1\right) q^{\mu } q^{\nu } g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{\$AL\$12977}} f^{\text{Glu2}\;\text{Glu3}\;\text{\$AL\$12977}}}{2 \left(q^2\right)^2}-\frac{D g^{\mu \nu } g_s^2 \left(f^{\text{Glu1}\;\text{Glu3}\;\text{\$AL\$12978}} f^{\text{Glu2}\;\text{Glu3}\;\text{\$AL\$12978}}+f^{\text{Glu1}\;\text{Glu3}\;\text{\$AL\$12979}} f^{\text{Glu2}\;\text{Glu3}\;\text{\$AL\$12979}}\right)}{2 q^2}-\frac{\left(\xi _g-1\right) g^{\mu \nu } q^2 g_s^2 \left(f^{\text{Glu1}\;\text{Glu3}\;\text{\$AL\$12978}} f^{\text{Glu2}\;\text{Glu3}\;\text{\$AL\$12978}}+f^{\text{Glu1}\;\text{Glu3}\;\text{\$AL\$12979}} f^{\text{Glu2}\;\text{Glu3}\;\text{\$AL\$12979}}\right)}{2 \left(q^2\right)^2}+\frac{g^{\mu \nu } g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{\$AL\$12981}} f^{\text{Glu2}\;\text{Glu3}\;\text{\$AL\$12981}}}{2 q^2}+\frac{\left(\xi _g-1\right) q^{\mu } q^{\nu } g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{\$AL\$12981}} f^{\text{Glu2}\;\text{Glu3}\;\text{\$AL\$12981}}}{2 \left(q^2\right)^2},\frac{\text{tr}\left(\left(m_q-\gamma \cdot q\right).\left(-i \gamma ^{\nu } g_s T_{\text{Col3}\;\text{Col4}}^{\text{Glu2}}\right).\left(\gamma \cdot (p-q)+m_q\right).\left(-i \gamma ^{\mu } g_s T_{\text{Col4}\;\text{Col3}}^{\text{Glu1}}\right)\right)}{\left(q^2-m_q^2\right).\left((q-p)^2-m_q^2\right)},-\frac{(q-p)^{\mu } q^{\nu } g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{q^2.(q-p)^2},-\frac{\left(1-\xi _g\right){}^2 p^{\mu } p^{\nu } (p\cdot q)^2 g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 \left(q^2\right)^2.(q-p)^4}+\frac{\left(1-\xi _g\right){}^2 p^{\mu } q^{\nu } (p\cdot q)^2 g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{\left(q^2\right)^2.(q-p)^4}+\frac{D p^{\mu } p^{\nu } g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^2}-\frac{3 p^{\mu } p^{\nu } g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{q^2.(q-p)^2}-\frac{D q^{\mu } p^{\nu } g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{q^2.(q-p)^2}+\frac{3 q^{\mu } p^{\nu } g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^2}-\frac{D p^{\mu } q^{\nu } g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{q^2.(q-p)^2}+\frac{3 p^{\mu } q^{\nu } g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^2}+\frac{2 D q^{\mu } q^{\nu } g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{q^2.(q-p)^2}-\frac{3 q^{\mu } q^{\nu } g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{q^2.(q-p)^2}+\frac{\left(1-\xi _g\right) p^{\mu } p^{\nu } p^2 g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{q^2.(q-p)^4}-\frac{\left(1-\xi _g\right) p^{\mu } q^{\nu } p^2 g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{q^2.(q-p)^4}-\frac{\left(1-\xi _g\right) q^{\mu } q^{\nu } p^2 g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^4}-\frac{\left(1-\xi _g\right) p^{\mu } p^{\nu } (p\cdot q) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^4}+\frac{\left(1-\xi _g\right) q^{\mu } p^{\nu } (p\cdot q) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 \left(q^2\right)^2.(q-p)^2}-\frac{\left(1-\xi _g\right) q^{\mu } p^{\nu } (p\cdot q) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^4}+\frac{\left(1-\xi _g\right) p^{\mu } q^{\nu } (p\cdot q) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{\left(q^2\right)^2.(q-p)^2}+\frac{\left(1-\xi _g\right) p^{\mu } q^{\nu } (p\cdot q) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{q^2.(q-p)^4}+\frac{\left(1-\xi _g\right) q^{\mu } q^{\nu } (p\cdot q) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{q^2.(q-p)^4}+\frac{\left(1-\xi _g\right){}^2 q^{\mu } p^{\nu } p^2 (p\cdot q) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 \left(q^2\right)^2.(q-p)^4}-\frac{\left(1-\xi _g\right){}^2 q^{\mu } q^{\nu } p^2 (p\cdot q) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{\left(q^2\right)^2.(q-p)^4}-\frac{g^{\mu \nu } \left(p\cdot q-2 p^2\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{q^2.(q-p)^2}+\frac{\left(1-\xi _g\right) p^{\mu } p^{\nu } \left(p\cdot q-2 p^2\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^4}-\frac{\left(1-\xi _g\right) p^{\mu } q^{\nu } \left(p\cdot q-2 p^2\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^4}+\frac{g^{\mu \nu } \left(p^2+p\cdot q\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^2}+\frac{\left(1-\xi _g\right) p^{\mu } p^{\nu } \left(p^2+p\cdot q\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^4}-\frac{\left(1-\xi _g\right) q^{\mu } q^{\nu } \left(p^2+p\cdot q\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 \left(q^2\right)^2.(q-p)^2}-\frac{\left(1-\xi _g\right) q^{\mu } q^{\nu } \left(p^2+p\cdot q\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^4}-\frac{\left(1-\xi _g\right) g^{\mu \nu } p^2 \left(p^2+p\cdot q\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^4}+\frac{\left(1-\xi _g\right){}^2 q^{\mu } q^{\nu } p^2 \left(p^2+p\cdot q\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 \left(q^2\right)^2.(q-p)^4}-\frac{\left(1-\xi _g\right){}^2 p^{\mu } q^{\nu } (p\cdot q) \left(p^2+p\cdot q\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 \left(q^2\right)^2.(q-p)^4}-\frac{\left(1-\xi _g\right) p^{\mu } p^{\nu } q^2 g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 \left(q^2\right)^2.(q-p)^2}-\frac{\left(1-\xi _g\right) p^{\mu } p^{\nu } q^2 g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^4}+\frac{\left(1-\xi _g\right) q^{\mu } p^{\nu } q^2 g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^4}+\frac{\left(1-\xi _g\right) p^{\mu } q^{\nu } q^2 g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 \left(q^2\right)^2.(q-p)^2}-\frac{\left(1-\xi _g\right) q^{\mu } q^{\nu } q^2 g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 \left(q^2\right)^2.(q-p)^2}-\frac{\left(1-\xi _g\right) q^{\mu } q^{\nu } q^2 g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^4}-\frac{\left(1-\xi _g\right){}^2 q^{\mu } p^{\nu } p^2 q^2 g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 \left(q^2\right)^2.(q-p)^4}+\frac{\left(1-\xi _g\right){}^2 q^{\mu } q^{\nu } p^2 q^2 g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{\left(q^2\right)^2.(q-p)^4}+\frac{\left(1-\xi _g\right){}^2 p^{\mu } p^{\nu } (p\cdot q) q^2 g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 \left(q^2\right)^2.(q-p)^4}-\frac{\left(1-\xi _g\right){}^2 p^{\mu } q^{\nu } (p\cdot q) q^2 g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{\left(q^2\right)^2.(q-p)^4}+\frac{\left(1-\xi _g\right) g^{\mu \nu } \left(p^2+p\cdot q\right) q^2 g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^4}+\frac{g^{\mu \nu } \left(q^2-2 (p\cdot q)\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^2}-\frac{\left(1-\xi _g\right) q^{\mu } p^{\nu } \left(q^2-2 (p\cdot q)\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 \left(q^2\right)^2.(q-p)^2}-\frac{\left(1-\xi _g\right) q^{\mu } p^{\nu } \left(q^2-2 (p\cdot q)\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^4}-\frac{\left(1-\xi _g\right) p^{\mu } q^{\nu } \left(q^2-2 (p\cdot q)\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 \left(q^2\right)^2.(q-p)^2}+\frac{\left(1-\xi _g\right) q^{\mu } q^{\nu } \left(q^2-2 (p\cdot q)\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 \left(q^2\right)^2.(q-p)^2}+\frac{\left(1-\xi _g\right) q^{\mu } q^{\nu } \left(q^2-2 (p\cdot q)\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^4}+\frac{\left(1-\xi _g\right){}^2 q^{\mu } p^{\nu } p^2 \left(q^2-2 (p\cdot q)\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 \left(q^2\right)^2.(q-p)^4}-\frac{\left(1-\xi _g\right){}^2 q^{\mu } q^{\nu } p^2 \left(q^2-2 (p\cdot q)\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 \left(q^2\right)^2.(q-p)^4}+\frac{\left(1-\xi _g\right) g^{\mu \nu } (p\cdot q) \left(q^2-2 (p\cdot q)\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{\left(q^2\right)^2.(q-p)^2}-\frac{\left(1-\xi _g\right){}^2 p^{\mu } p^{\nu } (p\cdot q) \left(q^2-2 (p\cdot q)\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 \left(q^2\right)^2.(q-p)^4}+\frac{\left(1-\xi _g\right){}^2 p^{\mu } q^{\nu } (p\cdot q) \left(q^2-2 (p\cdot q)\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 \left(q^2\right)^2.(q-p)^4}-\frac{\left(1-\xi _g\right) g^{\mu \nu } q^2 \left(q^2-2 (p\cdot q)\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 \left(q^2\right)^2.(q-p)^2}+\frac{g^{\mu \nu } \left(p\cdot q+q^2\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^2}-\frac{\left(1-\xi _g\right) p^{\mu } p^{\nu } \left(p\cdot q+q^2\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^4}-\frac{\left(1-\xi _g\right) p^{\mu } q^{\nu } \left(p\cdot q+q^2\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 \left(q^2\right)^2.(q-p)^2}+\frac{\left(1-\xi _g\right) q^{\mu } q^{\nu } \left(p\cdot q+q^2\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 \left(q^2\right)^2.(q-p)^2}+\frac{\left(1-\xi _g\right) q^{\mu } q^{\nu } \left(p\cdot q+q^2\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^4}+\frac{\left(1-\xi _g\right) g^{\mu \nu } p^2 \left(p\cdot q+q^2\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^4}-\frac{\left(1-\xi _g\right){}^2 q^{\mu } q^{\nu } p^2 \left(p\cdot q+q^2\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 \left(q^2\right)^2.(q-p)^4}+\frac{\left(1-\xi _g\right){}^2 p^{\mu } q^{\nu } (p\cdot q) \left(p\cdot q+q^2\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 \left(q^2\right)^2.(q-p)^4}-\frac{\left(1-\xi _g\right) g^{\mu \nu } q^2 \left(p\cdot q+q^2\right) g_s^2 f^{\text{Glu1}\;\text{Glu3}\;\text{Glu4}} f^{\text{Glu2}\;\text{Glu3}\;\text{Glu4}}}{2 q^2.(q-p)^4}\right\}

Calculate the amplitude

The gluon tadpole

This contribution is zero in dimensional regularization, because the loop integrals have no scale (and they are not log divergent)

amp1[0] = TID[amp[0][[1]], q, ToPaVe -> True]

0

FCCompareResults[amp1[0], 0, 
   Text -> {"\tThe gluon tadpole vanishes:", 
     "CORRECT.", "WRONG!"}, Interrupt -> {Hold[Quit[1]], Automatic}];

\text{$\backslash $tThe gluon tadpole vanishes:} \;\text{CORRECT.}

The quark loop

amp2[0] = amp[0][[2]] // SUNSimplify // TID[#, q, ToPaVe -> True] &

\frac{i \pi ^2 g_s^2 \delta ^{\text{Glu1}\;\text{Glu2}} \;\text{B}_0\left(p^2,m_q^2,m_q^2\right) \left(-\left((1-D) p^4 g^{\mu \nu }\right)+2 (1-D) p^2 p^{\mu } p^{\nu }+D p^2 p^{\mu } p^{\nu }+4 p^2 m_q^2 g^{\mu \nu }-p^4 g^{\mu \nu }-4 m_q^2 p^{\mu } p^{\nu }\right)}{(1-D) p^2}-\frac{2 i \pi ^2 g_s^2 \delta ^{\text{Glu1}\;\text{Glu2}} \;\text{A}_0\left(m_q^2\right) \left(-(1-D) p^2 g^{\mu \nu }-D p^{\mu } p^{\nu }-p^2 g^{\mu \nu }+2 p^{\mu } p^{\nu }\right)}{(1-D) p^2}

The contribution of the quark loop alone is gauge invariant.

tmp = Contract[FVD[p, mu] FVD[p, nu] amp2[0]] // Factor
FCCompareResults[tmp, 0, 
   Text -> {"\tThe quark loop contribution is gauge invariant:", 
     "CORRECT.", "WRONG!"}, Interrupt -> {Hold[Quit[1]], Automatic}];

0

\text{$\backslash $tThe quark loop contribution is gauge invariant:} \;\text{CORRECT.}

The ghost loop

amp3[0] = amp[0][[3]] // SUNSimplify // TID[#, q, ToPaVe -> True] &

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

The contribution of the gluon loop alone is not gauge invariant.

tmp1 = Contract[FVD[p, mu] FVD[p, nu] amp3[0]] // Factor

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

The gluon loop

amp4[0] = amp[0][[4]] // SUNSimplify // TID[#, q, ToPaVe -> True] &

-\frac{i C_A \pi ^2 \;\text{D}_0\left(0,p^2,0,p^2,p^2,p^2,0,0,0,0\right) \left(1-\xi _g\right){}^2 p^4 \left((1-D) p^{\mu } p^{\nu }+D p^{\mu } p^{\nu }-g^{\mu \nu } p^2\right) \delta ^{\text{Glu1}\;\text{Glu2}} g_s^2}{8 (1-D)}-\frac{i C_A \pi ^2 \;\text{B}_0(0,0,0) \left(1-\xi _g\right) \left(7 (1-D) p^{\mu } p^{\nu }+3 D p^{\mu } p^{\nu }-(1-D) \xi _g p^{\mu } p^{\nu }-D \xi _g p^{\mu } p^{\nu }-4 (1-D) g^{\mu \nu } p^2+\xi _g g^{\mu \nu } p^2-3 g^{\mu \nu } p^2\right) \delta ^{\text{Glu1}\;\text{Glu2}} g_s^2}{4 (1-D)}+\frac{i C_A \pi ^2 \;\text{C}_0\left(0,p^2,p^2,0,0,0\right) \left(1-\xi _g\right) p^2 \left(3 (1-D) p^{\mu } p^{\nu }+D p^{\mu } p^{\nu }-\xi _g p^{\mu } p^{\nu }+p^{\mu } p^{\nu }-2 (1-D) g^{\mu \nu } p^2+\xi _g g^{\mu \nu } p^2-2 g^{\mu \nu } p^2\right) \delta ^{\text{Glu1}\;\text{Glu2}} g_s^2}{2 (1-D)}-\frac{1}{4 (1-D)}i C_A \pi ^2 \;\text{B}_0\left(p^2,0,0\right) \left(2 p^{\mu } p^{\nu } D^2-\xi _g^2 p^{\mu } p^{\nu } D+2 (1-D) p^{\mu } p^{\nu } D+6 \xi _g p^{\mu } p^{\nu } D-8 p^{\mu } p^{\nu } D-2 g^{\mu \nu } p^2 D-(1-D) \xi _g^2 p^{\mu } p^{\nu }+(1-D) p^{\mu } p^{\nu }+6 (1-D) \xi _g p^{\mu } p^{\nu }-8 \xi _g p^{\mu } p^{\nu }+8 p^{\mu } p^{\nu }+\xi _g^2 g^{\mu \nu } p^2-8 (1-D) g^{\mu \nu } p^2+2 \xi _g g^{\mu \nu } p^2\right) \delta ^{\text{Glu1}\;\text{Glu2}} g_s^2

The contribution of the gluon loop alone is not gauge invariant. Notice, however, that the sum of the ghost and gluon contributions is gauge invariant!

tmp2 = Contract[FVD[p, mu] FVD[p, nu] amp4[0]] // Factor

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

FCCompareResults[tmp1 + tmp2, 0, 
   Text -> {"\tThe sum of the ghost and gluon loop contributions is gauge invariant:", 
     "CORRECT.", "WRONG!"}, Interrupt -> {Hold[Quit[1]], Automatic}];

\text{$\backslash $tThe sum of the ghost and gluon loop contributions is gauge invariant:} \;\text{CORRECT.}

Putting everything together

When adding all the contributions together, we multiply the quark contribution by N_f to account for the 6 quark flavors that actually run in that loop. We ignore the fact that different flavors have different masses, since the divergent piece of the gluon self-energy will not depend on the quark mass.

amp[1] = Nf amp2[0] + amp3[0] + amp4[0]

-\frac{i C_A \pi ^2 \;\text{D}_0\left(0,p^2,0,p^2,p^2,p^2,0,0,0,0\right) \left(1-\xi _g\right){}^2 p^4 \left((1-D) p^{\mu } p^{\nu }+D p^{\mu } p^{\nu }-g^{\mu \nu } p^2\right) \delta ^{\text{Glu1}\;\text{Glu2}} g_s^2}{8 (1-D)}+\frac{i C_A \pi ^2 \;\text{B}_0\left(p^2,0,0\right) \left(2 (1-D) p^{\mu } p^{\nu }+D p^{\mu } p^{\nu }-g^{\mu \nu } p^2\right) \delta ^{\text{Glu1}\;\text{Glu2}} g_s^2}{4 (1-D)}-\frac{i C_A \pi ^2 \;\text{B}_0(0,0,0) \left(1-\xi _g\right) \left(7 (1-D) p^{\mu } p^{\nu }+3 D p^{\mu } p^{\nu }-(1-D) \xi _g p^{\mu } p^{\nu }-D \xi _g p^{\mu } p^{\nu }-4 (1-D) g^{\mu \nu } p^2+\xi _g g^{\mu \nu } p^2-3 g^{\mu \nu } p^2\right) \delta ^{\text{Glu1}\;\text{Glu2}} g_s^2}{4 (1-D)}+\frac{i C_A \pi ^2 \;\text{C}_0\left(0,p^2,p^2,0,0,0\right) \left(1-\xi _g\right) p^2 \left(3 (1-D) p^{\mu } p^{\nu }+D p^{\mu } p^{\nu }-\xi _g p^{\mu } p^{\nu }+p^{\mu } p^{\nu }-2 (1-D) g^{\mu \nu } p^2+\xi _g g^{\mu \nu } p^2-2 g^{\mu \nu } p^2\right) \delta ^{\text{Glu1}\;\text{Glu2}} g_s^2}{2 (1-D)}-\frac{1}{4 (1-D)}i C_A \pi ^2 \;\text{B}_0\left(p^2,0,0\right) \left(2 p^{\mu } p^{\nu } D^2-\xi _g^2 p^{\mu } p^{\nu } D+2 (1-D) p^{\mu } p^{\nu } D+6 \xi _g p^{\mu } p^{\nu } D-8 p^{\mu } p^{\nu } D-2 g^{\mu \nu } p^2 D-(1-D) \xi _g^2 p^{\mu } p^{\nu }+(1-D) p^{\mu } p^{\nu }+6 (1-D) \xi _g p^{\mu } p^{\nu }-8 \xi _g p^{\mu } p^{\nu }+8 p^{\mu } p^{\nu }+\xi _g^2 g^{\mu \nu } p^2-8 (1-D) g^{\mu \nu } p^2+2 \xi _g g^{\mu \nu } p^2\right) \delta ^{\text{Glu1}\;\text{Glu2}} g_s^2+N_f \left(\frac{i \pi ^2 \;\text{B}_0\left(p^2,m_q^2,m_q^2\right) g_s^2 \left(-\left((1-D) g^{\mu \nu } p^4\right)-g^{\mu \nu } p^4+4 g^{\mu \nu } m_q^2 p^2+2 (1-D) p^{\mu } p^{\nu } p^2+D p^{\mu } p^{\nu } p^2-4 p^{\mu } p^{\nu } m_q^2\right) \delta ^{\text{Glu1}\;\text{Glu2}}}{(1-D) p^2}-\frac{2 i \pi ^2 \;\text{A}_0\left(m_q^2\right) \left(-D p^{\mu } p^{\nu }+2 p^{\mu } p^{\nu }-(1-D) g^{\mu \nu } p^2-g^{\mu \nu } p^2\right) g_s^2 \delta ^{\text{Glu1}\;\text{Glu2}}}{(1-D) p^2}\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 g_s^2 \delta ^{\text{Glu1}\;\text{Glu2}} \left(p^{\mu } p^{\nu }-p^2 g^{\mu \nu }\right) \left(3 C_A \xi _g-13 C_A+4 N_f\right)}{96 \pi ^2 \varepsilon }

The self-energy amplitude is usually defined as (p^2 g^{mu nu} - p^mu p^nu) i Pi(p^2)

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

-\frac{g_s^2 \left(3 C_A \xi _g-13 C_A+4 N_f\right)}{96 \pi ^2 \varepsilon }

Check the final results

knownResult = -(SMP["g_s"]^2/(4 Pi)^2)*(4/3*(1/2)*Nf - 
      (1/2) CA (13/3 - GaugeXi[g]))*1/Epsilon;
FCCompareResults[pi[0], knownResult, 
   Text -> {"\tCompare to Muta, Foundations of QCD, Eqs 2.5.131-2.5.132:", 
     "CORRECT.", "WRONG!"}, Interrupt -> {Hold[Quit[1]], Automatic}];
Print["\tCPU Time used: ", Round[N[TimeUsed[], 4], 0.001], " s."];

\text{$\backslash $tCompare to Muta, Foundations of QCD, Eqs 2.5.131-2.5.132:} \;\text{CORRECT.}

\text{$\backslash $tCPU Time used: }27.379\text{ s.}