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];

FeynCalc   10.0.0 (dev version, 2023-12-20 22:40:59 +01:00, dff3b835). For help, use the online  documentation‾  , check out the wiki‾   or visit the forum.‾\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}.}

Please check our FAQ‾   for answers to some common FeynCalc questions and have a look at the supplied examples.‾\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}.}

If you use FeynCalc in your research, please evaluate FeynCalcHowToCite[] to learn how to cite this software.\text{If you use FeynCalc in your research, please evaluate FeynCalcHowToCite[] to learn how to cite this software.}

Please keep in mind that the proper academic attribution of our work is crucial to ensure the future development of this package!\text{Please keep in mind that the proper academic attribution of our work is crucial to ensure the future development of this package!}

FeynArts   3.11 (3 Aug 2020) patched for use with FeynCalc, for documentation see the manual‾   or visit www.feynarts.de.‾\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}.}

If you use FeynArts in your research, please cite\text{If you use FeynArts in your research, please cite}

 ∙ T. Hahn, Comput. Phys. Commun., 140, 418-431, 2001, arXiv:hep-ph/0012260\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"]}]

{gμνgs2fGlu1  Glu3  $AL$12977fGlu2  Glu3  $AL$129772q2+(ξg−1)qμqνgs2fGlu1  Glu3  $AL$12977fGlu2  Glu3  $AL$129772(q2)2−Dgμνgs2(fGlu1  Glu3  $AL$12978fGlu2  Glu3  $AL$12978+fGlu1  Glu3  $AL$12979fGlu2  Glu3  $AL$12979)2q2−(ξg−1)gμνq2gs2(fGlu1  Glu3  $AL$12978fGlu2  Glu3  $AL$12978+fGlu1  Glu3  $AL$12979fGlu2  Glu3  $AL$12979)2(q2)2+gμνgs2fGlu1  Glu3  $AL$12981fGlu2  Glu3  $AL$129812q2+(ξg−1)qμqνgs2fGlu1  Glu3  $AL$12981fGlu2  Glu3  $AL$129812(q2)2,tr((mq−γ⋅q).(−iγνgsTCol3  Col4Glu2).(γ⋅(p−q)+mq).(−iγμgsTCol4  Col3Glu1))(q2−mq2).((q−p)2−mq2),−(q−p)μqνgs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu4q2.(q−p)2,−(1−ξg)2pμpν(p⋅q)2gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42(q2)2.(q−p)4+(1−ξg)2pμqν(p⋅q)2gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu4(q2)2.(q−p)4+Dpμpνgs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)2−3pμpνgs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu4q2.(q−p)2−Dqμpνgs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu4q2.(q−p)2+3qμpνgs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)2−Dpμqνgs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu4q2.(q−p)2+3pμqνgs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)2+2Dqμqνgs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu4q2.(q−p)2−3qμqνgs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu4q2.(q−p)2+(1−ξg)pμpνp2gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu4q2.(q−p)4−(1−ξg)pμqνp2gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu4q2.(q−p)4−(1−ξg)qμqνp2gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)4−(1−ξg)pμpν(p⋅q)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)4+(1−ξg)qμpν(p⋅q)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42(q2)2.(q−p)2−(1−ξg)qμpν(p⋅q)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)4+(1−ξg)pμqν(p⋅q)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu4(q2)2.(q−p)2+(1−ξg)pμqν(p⋅q)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu4q2.(q−p)4+(1−ξg)qμqν(p⋅q)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu4q2.(q−p)4+(1−ξg)2qμpνp2(p⋅q)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42(q2)2.(q−p)4−(1−ξg)2qμqνp2(p⋅q)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu4(q2)2.(q−p)4−gμν(p⋅q−2p2)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu4q2.(q−p)2+(1−ξg)pμpν(p⋅q−2p2)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)4−(1−ξg)pμqν(p⋅q−2p2)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)4+gμν(p2+p⋅q)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)2+(1−ξg)pμpν(p2+p⋅q)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)4−(1−ξg)qμqν(p2+p⋅q)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42(q2)2.(q−p)2−(1−ξg)qμqν(p2+p⋅q)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)4−(1−ξg)gμνp2(p2+p⋅q)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)4+(1−ξg)2qμqνp2(p2+p⋅q)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42(q2)2.(q−p)4−(1−ξg)2pμqν(p⋅q)(p2+p⋅q)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42(q2)2.(q−p)4−(1−ξg)pμpνq2gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42(q2)2.(q−p)2−(1−ξg)pμpνq2gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)4+(1−ξg)qμpνq2gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)4+(1−ξg)pμqνq2gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42(q2)2.(q−p)2−(1−ξg)qμqνq2gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42(q2)2.(q−p)2−(1−ξg)qμqνq2gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)4−(1−ξg)2qμpνp2q2gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42(q2)2.(q−p)4+(1−ξg)2qμqνp2q2gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu4(q2)2.(q−p)4+(1−ξg)2pμpν(p⋅q)q2gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42(q2)2.(q−p)4−(1−ξg)2pμqν(p⋅q)q2gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu4(q2)2.(q−p)4+(1−ξg)gμν(p2+p⋅q)q2gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)4+gμν(q2−2(p⋅q))gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)2−(1−ξg)qμpν(q2−2(p⋅q))gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42(q2)2.(q−p)2−(1−ξg)qμpν(q2−2(p⋅q))gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)4−(1−ξg)pμqν(q2−2(p⋅q))gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42(q2)2.(q−p)2+(1−ξg)qμqν(q2−2(p⋅q))gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42(q2)2.(q−p)2+(1−ξg)qμqν(q2−2(p⋅q))gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)4+(1−ξg)2qμpνp2(q2−2(p⋅q))gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42(q2)2.(q−p)4−(1−ξg)2qμqνp2(q2−2(p⋅q))gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42(q2)2.(q−p)4+(1−ξg)gμν(p⋅q)(q2−2(p⋅q))gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu4(q2)2.(q−p)2−(1−ξg)2pμpν(p⋅q)(q2−2(p⋅q))gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42(q2)2.(q−p)4+(1−ξg)2pμqν(p⋅q)(q2−2(p⋅q))gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42(q2)2.(q−p)4−(1−ξg)gμνq2(q2−2(p⋅q))gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42(q2)2.(q−p)2+gμν(p⋅q+q2)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)2−(1−ξg)pμpν(p⋅q+q2)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)4−(1−ξg)pμqν(p⋅q+q2)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42(q2)2.(q−p)2+(1−ξg)qμqν(p⋅q+q2)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42(q2)2.(q−p)2+(1−ξg)qμqν(p⋅q+q2)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)4+(1−ξg)gμνp2(p⋅q+q2)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)4−(1−ξg)2qμqνp2(p⋅q+q2)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42(q2)2.(q−p)4+(1−ξg)2pμqν(p⋅q)(p⋅q+q2)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42(q2)2.(q−p)4−(1−ξg)gμνq2(p⋅q+q2)gs2fGlu1  Glu3  Glu4fGlu2  Glu3  Glu42q2.(q−p)4}\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]

00

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

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

The quark loop

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

iπ2gs2δGlu1  Glu2  B0(p2,mq2,mq2)(−((1−D)p4gμν)+2(1−D)p2pμpν+Dp2pμpν+4p2mq2gμν−p4gμν−4mq2pμpν)(1−D)p2−2iπ2gs2δGlu1  Glu2  A0(mq2)(−(1−D)p2gμν−Dpμpν−p2gμν+2pμpν)(1−D)p2\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}];

00

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

The ghost loop

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

iπ2CAgs2δGlu1  Glu2  B0(p2,0,0)(2(1−D)pμpν+Dpμpν−p2gμν)4(1−D)\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

14iπ2p4CAgs2δGlu1  Glu2  B0(p2,0,0)\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] &

−iCAπ2  D0(0,p2,0,p2,p2,p2,0,0,0,0)(1−ξg)2p4((1−D)pμpν+Dpμpν−gμνp2)δGlu1  Glu2gs28(1−D)−iCAπ2  B0(0,0,0)(1−ξg)(7(1−D)pμpν+3Dpμpν−(1−D)ξgpμpν−Dξgpμpν−4(1−D)gμνp2+ξggμνp2−3gμνp2)δGlu1  Glu2gs24(1−D)+iCAπ2  C0(0,p2,p2,0,0,0)(1−ξg)p2(3(1−D)pμpν+Dpμpν−ξgpμpν+pμpν−2(1−D)gμνp2+ξggμνp2−2gμνp2)δGlu1  Glu2gs22(1−D)−14(1−D)iCAπ2  B0(p2,0,0)(2pμpνD2−ξg2pμpνD+2(1−D)pμpνD+6ξgpμpνD−8pμpνD−2gμνp2D−(1−D)ξg2pμpν+(1−D)pμpν+6(1−D)ξgpμpν−8ξgpμpν+8pμpν+ξg2gμνp2−8(1−D)gμνp2+2ξggμνp2)δGlu1  Glu2gs2-\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

−14iπ2p4CAgs2δGlu1  Glu2  B0(p2,0,0)-\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}];

\tThe sum of the ghost and gluon loop contributions is gauge invariant:  CORRECT.\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]

−iCAπ2  D0(0,p2,0,p2,p2,p2,0,0,0,0)(1−ξg)2p4((1−D)pμpν+Dpμpν−gμνp2)δGlu1  Glu2gs28(1−D)+iCAπ2  B0(p2,0,0)(2(1−D)pμpν+Dpμpν−gμνp2)δGlu1  Glu2gs24(1−D)−iCAπ2  B0(0,0,0)(1−ξg)(7(1−D)pμpν+3Dpμpν−(1−D)ξgpμpν−Dξgpμpν−4(1−D)gμνp2+ξggμνp2−3gμνp2)δGlu1  Glu2gs24(1−D)+iCAπ2  C0(0,p2,p2,0,0,0)(1−ξg)p2(3(1−D)pμpν+Dpμpν−ξgpμpν+pμpν−2(1−D)gμνp2+ξggμνp2−2gμνp2)δGlu1  Glu2gs22(1−D)−14(1−D)iCAπ2  B0(p2,0,0)(2pμpνD2−ξg2pμpνD+2(1−D)pμpνD+6ξgpμpνD−8pμpνD−2gμνp2D−(1−D)ξg2pμpν+(1−D)pμpν+6(1−D)ξgpμpν−8ξgpμpν+8pμpν+ξg2gμνp2−8(1−D)gμνp2+2ξggμνp2)δGlu1  Glu2gs2+Nf(iπ2  B0(p2,mq2,mq2)gs2(−((1−D)gμνp4)−gμνp4+4gμνmq2p2+2(1−D)pμpνp2+Dpμpνp2−4pμpνmq2)δGlu1  Glu2(1−D)p2−2iπ2  A0(mq2)(−Dpμpν+2pμpν−(1−D)gμνp2−gμνp2)gs2δGlu1  Glu2(1−D)p2)-\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

igs2δGlu1  Glu2(pμpν−p2gμν)(3CAξg−13CA+4Nf)96π2ε\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

−gs2(3CAξg−13CA+4Nf)96π2ε-\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."];

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

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