QCD manual (development version)

Load FeynCalc and the necessary add-ons or other packages

This example uses a custom QCD model created with FeynRules. Please evaluate the file FeynCalc/Examples/FeynRules/QCD/GenerateModelQCD.m before running it for the first time.

description = "Gl - 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;
FAPatch[PatchModelsOnly -> True];

(*Successfully patched FeynArts.*)

Generate Feynman diagrams

Nicer typesetting

MakeBoxes[mu, TraditionalForm] := "\[Mu]";
MakeBoxes[nu, TraditionalForm] := "\[Nu]";
MakeBoxes[rho, TraditionalForm] := "\[Rho]";
template = insertFields[createTopologies[1, 1 -> 2, 
            ExcludeTopologies -> {Tadpoles, WFCorrections, 
            WFCorrectionCTs, SelfEnergies}], {V[5]} -> 
            {V[5], V[5]}, InsertionLevel -> {Particles}, 
        Model -> FileNameJoin[{"QCD", "QCD"}], 
        GenericModel -> FileNameJoin[{"QCD", "QCD"}], 
        ExcludeParticles -> {F[3 | 4, {2 | 3}], F[4, {1}]}];
diags = template /. createTopologies -> CreateTopologies /. 
    insertFields -> InsertFields;
diagsCT = template /. createTopologies -> CreateCTTopologies /. 
    insertFields -> InsertFields;
Paint[diags, ColumnsXRows -> {3, 1}, SheetHeader -> None, 
   Numbering -> Simple, ImageSize -> {512, 256}];

1upk049xa8r7h

0cty7gtqz5g4m

16fxlye0ek1v9

Paint[diagsCT, ColumnsXRows -> {3, 1}, SheetHeader -> None, 
   Numbering -> Simple, ImageSize -> {512, 256}];

1pfdvzv05q15f

Obtain the amplitudes

The 1/(2Pi)^D prefactor is implicit. We keep the full gauge dependence. To simplify comparisons to the literature, we make all momenta incoming.

Quark contribution. Notice that we multiply the amplitude by Nf to account for the number of quark flavours in the loop.

amp1[0] = FCFAConvert[CreateFeynAmp[DiagramExtract[diags, {1, 2}], 
    Truncated -> True, GaugeRules -> {}, PreFactor -> 1], 
    IncomingMomenta -> {p1}, OutgoingMomenta -> {p2, p3}, 
    LorentzIndexNames -> {mu, nu, rho}, DropSumOver -> True, 
    SUNIndexNames -> {a, b, c}, LoopMomenta -> {l}, UndoChiralSplittings -> True, 
    ChangeDimension -> D, List -> False, SMP -> True, Prefactor -> Nf, 
    FinalSubstitutions -> {SMP["m_u"] -> SMP["m_q"], p2 -> -p2, p3 -> -p3}]

iNf  tr((γ⋅(l+p2+p3)+mq).(−iγρgsTCol5  Col6c).(γ⋅(l+p2)+mq).(−iγνgsTCol6  Col4b).(γ⋅l+mq).(−iγμgsTCol4  Col5a))(l2−mq2).((l+p2)2−mq2).((l+p2+p3)2−mq2)+iNf  tr((γ⋅(l+p2+p3)+mq).(iγρgsTCol6  Col5c).(γ⋅(l+p2)+mq).(iγνgsTCol4  Col6b).(γ⋅l+mq).(iγμgsTCol5  Col4a))(l2−mq2).((l+p2)2−mq2).((l+p2+p3)2−mq2)\frac{i N_f \;\text{tr}\left(\left(\gamma \cdot (l+\text{p2}+\text{p3})+m_q\right).\left(-i \gamma ^{\rho } g_s T_{\text{Col5}\;\text{Col6}}^c\right).\left(\gamma \cdot (l+\text{p2})+m_q\right).\left(-i \gamma ^{\nu } g_s T_{\text{Col6}\;\text{Col4}}^b\right).\left(\gamma \cdot l+m_q\right).\left(-i \gamma ^{\mu } g_s T_{\text{Col4}\;\text{Col5}}^a\right)\right)}{\left(l^2-m_q^2\right).\left((l+\text{p2})^2-m_q^2\right).\left((l+\text{p2}+\text{p3})^2-m_q^2\right)}+\frac{i N_f \;\text{tr}\left(\left(\gamma \cdot (l+\text{p2}+\text{p3})+m_q\right).\left(i \gamma ^{\rho } g_s T_{\text{Col6}\;\text{Col5}}^c\right).\left(\gamma \cdot (l+\text{p2})+m_q\right).\left(i \gamma ^{\nu } g_s T_{\text{Col4}\;\text{Col6}}^b\right).\left(\gamma \cdot l+m_q\right).\left(i \gamma ^{\mu } g_s T_{\text{Col5}\;\text{Col4}}^a\right)\right)}{\left(l^2-m_q^2\right).\left((l+\text{p2})^2-m_q^2\right).\left((l+\text{p2}+\text{p3})^2-m_q^2\right)}

Ghost contribution

amp2[0] = FCFAConvert[CreateFeynAmp[DiagramExtract[diags, {3, 4}], 
    Truncated -> True, GaugeRules -> {}, PreFactor -> 1], 
    IncomingMomenta -> {p1}, OutgoingMomenta -> {p2, p3}, 
    LorentzIndexNames -> {mu, nu, rho}, SUNIndexNames -> {a, b, c}, 
    DropSumOver -> True, LoopMomenta -> {l}, UndoChiralSplittings -> True, 
    ChangeDimension -> D, List -> False, SMP -> True, 
    FinalSubstitutions -> {SMP["m_u"] -> SMP["m_q"], p2 -> -p2, p3 -> -p3}]

igs3lν(l+p2)ρfaGlu4  Glu5fbGlu4  Glu6fcGlu5  Glu6(l+p2+p3)μl2.(l+p2)2.(l+p2+p3)2+igs3lμ(−l−p2)νfaGlu4  Glu5fbGlu4  Glu6fcGlu5  Glu6(−l−p2−p3)ρl2.(l+p2)2.(l+p2+p3)2\frac{i g_s^3 l^{\nu } (l+\text{p2})^{\rho } f^{a\text{Glu4}\;\text{Glu5}} f^{b\text{Glu4}\;\text{Glu6}} f^{c\text{Glu5}\;\text{Glu6}} (l+\text{p2}+\text{p3})^{\mu }}{l^2.(l+\text{p2})^2.(l+\text{p2}+\text{p3})^2}+\frac{i g_s^3 l^{\mu } (-l-\text{p2})^{\nu } f^{a\text{Glu4}\;\text{Glu5}} f^{b\text{Glu4}\;\text{Glu6}} f^{c\text{Glu5}\;\text{Glu6}} (-l-\text{p2}-\text{p3})^{\rho }}{l^2.(l+\text{p2})^2.(l+\text{p2}+\text{p3})^2}

Gluon contribution

amp3[0] = FCFAConvert[CreateFeynAmp[DiagramExtract[diags, {5, 6, 7, 8}], 
        Truncated -> True, GaugeRules -> {}, PreFactor -> 1], 
    IncomingMomenta -> {p1}, OutgoingMomenta -> {p2, p3}, 
    LorentzIndexNames -> {mu, nu, rho}, SUNIndexNames -> {a, b, c}, 
    DropSumOver -> True, LoopMomenta -> {l}, UndoChiralSplittings -> True, 
    ChangeDimension -> D, List -> False, SMP -> True, 
    FinalSubstitutions -> {SMP["m_u"] -> SMP["m_q"], p2 -> -p2, p3 -> -p3}];

Counter-term

amp4[0] = FCFAConvert[CreateFeynAmp[diagsCT, 
    Truncated -> True, GaugeRules -> {}, PreFactor -> 1], 
    IncomingMomenta -> {p1}, OutgoingMomenta -> {p2, p3}, 
    LorentzIndexNames -> {mu, nu, rho}, SUNIndexNames -> {a, b, c}, 
    DropSumOver -> True, LoopMomenta -> {l}, UndoChiralSplittings -> True, 
    ChangeDimension -> D, List -> False, SMP -> True, 
    FinalSubstitutions -> {SMP["m_u"] -> SMP["m_q"], p2 -> -p2, p3 -> -p3, 
        ZA -> SMP["Z_A"], Zg -> SMP["Z_g"]}]

gs  p1ν(ZA3/2Zg−1)gμρfabc−gs  p1ρ(ZA3/2Zg−1)gμνfabc+gs(−p2μ)(ZA3/2Zg−1)gνρfabc+gs  p2ρ(ZA3/2Zg−1)gμνfabc+gs  p3μ(ZA3/2Zg−1)gνρfabc−gs  p3ν(ZA3/2Zg−1)gμρfabcg_s \;\text{p1}^{\nu } \left(Z_A^{3/2} Z_g-1\right) g^{\mu \rho } f^{abc}-g_s \;\text{p1}^{\rho } \left(Z_A^{3/2} Z_g-1\right) g^{\mu \nu } f^{abc}+g_s \left(-\text{p2}^{\mu }\right) \left(Z_A^{3/2} Z_g-1\right) g^{\nu \rho } f^{abc}+g_s \;\text{p2}^{\rho } \left(Z_A^{3/2} Z_g-1\right) g^{\mu \nu } f^{abc}+g_s \;\text{p3}^{\mu } \left(Z_A^{3/2} Z_g-1\right) g^{\nu \rho } f^{abc}-g_s \;\text{p3}^{\nu } \left(Z_A^{3/2} Z_g-1\right) g^{\mu \rho } f^{abc}

Calculate the amplitudes

Quark contribution

AbsoluteTiming[amp1[1] = TID[FCE[amp1[0]] /. {p2 + p3 -> -p1, -p2 - p3 -> p1}, l, 
        UsePaVeBasis -> True, ToPaVe -> True];]

{1.59185,Null}\{1.59185,\text{Null}\}

amp1Div[0] = amp1[1] // PaVeUVPart[#, Prefactor -> 1/(2 Pi)^D, FCLoopExtract -> False] &;
amp1Div[1] = amp1Div[0] // SUNSimplify[#, Explicit -> True] & // ReplaceAll[#, 
            SUNTrace[x__] :> SUNTrace[x, Explicit -> True]] & // 
            FCReplaceD[#, D -> 4 - 2 Epsilon] & // Series[#, {Epsilon, 0, 0}] & // 
        Normal // FCHideEpsilon // SelectNotFree2[#, SMP["Delta"]] & //FCE // 
    Collect2[#, MTD, Factoring -> Function[x, MomentumCombine[Factor[x]]]] &

−ΔNfgs3gνρ(p1+2  p2)μfabc24π2+ΔNfgs3gμρ(2  p1+p2)νfabc24π2−ΔNfgs3gμν(p1−p2)ρfabc24π2-\frac{\Delta N_f g_s^3 g^{\nu \rho } (\text{p1}+2 \;\text{p2})^{\mu } f^{abc}}{24 \pi ^2}+\frac{\Delta N_f g_s^3 g^{\mu \rho } (2 \;\text{p1}+\text{p2})^{\nu } f^{abc}}{24 \pi ^2}-\frac{\Delta N_f g_s^3 g^{\mu \nu } (\text{p1}-\text{p2})^{\rho } f^{abc}}{24 \pi ^2}

In the calculation p3 was eliminated via the 4-momentum conservation p1+p2+p3=0. Now we need to reintroduce it

amp1Div[2] = amp1Div[1] /. {2 p1 + p2 -> p1 - p3, p1 + 2 p2 -> p2 - p3}

−ΔNfgs3gμν(p1−p2)ρfabc24π2+ΔNfgs3gμρ(p1−p3)νfabc24π2−ΔNfgs3gνρ(p2−p3)μfabc24π2-\frac{\Delta N_f g_s^3 g^{\mu \nu } (\text{p1}-\text{p2})^{\rho } f^{abc}}{24 \pi ^2}+\frac{\Delta N_f g_s^3 g^{\mu \rho } (\text{p1}-\text{p3})^{\nu } f^{abc}}{24 \pi ^2}-\frac{\Delta N_f g_s^3 g^{\nu \rho } (\text{p2}-\text{p3})^{\mu } f^{abc}}{24 \pi ^2}

Ghost contribution

AbsoluteTiming[amp2[1] = TID[FCE[amp2[0]] /. {p2 + p3 -> -p1, -p2 - p3 -> p1}, l, 
        UsePaVeBasis -> True, ToPaVe -> True];]

{0.432995,Null}\{0.432995,\text{Null}\}

amp2Div[0] = amp2[1] // PaVeUVPart[#, Prefactor -> 1/(2 Pi)^D, FCLoopExtract -> False] &;
amp2Div[1] = amp2Div[0] // SUNSimplify[#, Explicit -> True] & // ReplaceAll[#, 
            SUNTrace[x__] :> SUNTrace[x, Explicit -> True]] & // 
            FCReplaceD[#, D -> 4 - 2 Epsilon] & // Series[#, {Epsilon, 0, 0}] & // 
        Normal // FCHideEpsilon // SelectNotFree2[#, SMP["Delta"]] & //FCE // 
    Collect2[#, MTD, Factoring -> Function[x, MomentumCombine[Factor[x]]]] &

ΔCAgs3gνρ(p1+2  p2)μfabc384π2−ΔCAgs3gμρ(2  p1+p2)νfabc384π2+ΔCAgs3gμν(p1−p2)ρfabc384π2\frac{\Delta C_A g_s^3 g^{\nu \rho } (\text{p1}+2 \;\text{p2})^{\mu } f^{abc}}{384 \pi ^2}-\frac{\Delta C_A g_s^3 g^{\mu \rho } (2 \;\text{p1}+\text{p2})^{\nu } f^{abc}}{384 \pi ^2}+\frac{\Delta C_A g_s^3 g^{\mu \nu } (\text{p1}-\text{p2})^{\rho } f^{abc}}{384 \pi ^2}

In the calculation p3 was eliminated via the 4-momentum conservation p1+p2+p3=0. Now we need to reintroduce it

amp2Div[2] = amp2Div[1] /. {2 p1 + p2 -> p1 - p3, p1 + 2 p2 -> p2 - p3}

ΔCAgs3gμν(p1−p2)ρfabc384π2−ΔCAgs3gμρ(p1−p3)νfabc384π2+ΔCAgs3gνρ(p2−p3)μfabc384π2\frac{\Delta C_A g_s^3 g^{\mu \nu } (\text{p1}-\text{p2})^{\rho } f^{abc}}{384 \pi ^2}-\frac{\Delta C_A g_s^3 g^{\mu \rho } (\text{p1}-\text{p3})^{\nu } f^{abc}}{384 \pi ^2}+\frac{\Delta C_A g_s^3 g^{\nu \rho } (\text{p2}-\text{p3})^{\mu } f^{abc}}{384 \pi ^2}

Gluon contribution

This calculation requires about 70 seconds on a modern laptop

AbsoluteTiming[amp3[1] = TID[FCE[amp3[0]] /. {p2 + p3 -> -p1, -p2 - p3 -> p1}, l, 
        UsePaVeBasis -> True, ExpandScalarProduct -> False, ToPaVe -> True];]

{56.6716,Null}\{56.6716,\text{Null}\}

amp3Div[0] = amp3[1] // PaVeUVPart[#, Prefactor -> 1/(2 Pi)^D, FCLoopExtract -> False] &;
amp3Div[1] = amp3Div[0] // SUNSimplify // FCReplaceD[#, D -> 4 - 2 Epsilon] & // 
        Series[#, {Epsilon, 0, 0}] & // Normal // FCHideEpsilon // 
        SelectNotFree2[#, SMP["Delta"]] & // FCE // 
    Collect2[#, MTD, Factoring -> Function[x, MomentumCombine[Factor[x]]]] &

−ΔCAgs3gνρfabc(6ξG  p1μ−3ξG(p1+p2+p3)μ+12ξG  p2μ+(4  p1−7  p2+15  p3)μ)128π2+ΔCAgs3gμρfabc(13ξG  p1ν−4ξG(p1+p2)ν+7ξG  p2ν−3ξG  p3ν+(−7  p1+4  p2+15  p3)ν)128π2−ΔCA(6ξG−11)gs3gμν(p1−p2)ρfabc128π2-\frac{\Delta C_A g_s^3 g^{\nu \rho } f^{abc} \left(6 \xi _{\text{G}} \;\text{p1}^{\mu }-3 \xi _{\text{G}} (\text{p1}+\text{p2}+\text{p3})^{\mu }+12 \xi _{\text{G}} \;\text{p2}^{\mu }+(4 \;\text{p1}-7 \;\text{p2}+15 \;\text{p3})^{\mu }\right)}{128 \pi ^2}+\frac{\Delta C_A g_s^3 g^{\mu \rho } f^{abc} \left(13 \xi _{\text{G}} \;\text{p1}^{\nu }-4 \xi _{\text{G}} (\text{p1}+\text{p2})^{\nu }+7 \xi _{\text{G}} \;\text{p2}^{\nu }-3 \xi _{\text{G}} \;\text{p3}^{\nu }+(-7 \;\text{p1}+4 \;\text{p2}+15 \;\text{p3})^{\nu }\right)}{128 \pi ^2}-\frac{\Delta C_A \left(6 \xi _{\text{G}}-11\right) g_s^3 g^{\mu \nu } (\text{p1}-\text{p2})^{\rho } f^{abc}}{128 \pi ^2}

amp3Div[2] = (((amp3Div[1] /. {p3 -> -p1 - p2}) // ExpandScalarProduct // FCE // 
            Collect2[#, MTD, GaugeXi, Factoring -> Function[x, MomentumCombine[Factor[x]]]] &) /. 
            2 p1 + p2 -> p1 - p3 /. p1 + 2 p2 -> p2 - p3 /. 22 p1 + 11 p2 -> 11 p1 - 11 p3 /. 
        -22 p2 - 11 p1 -> -11 p2 + 11 p3) // ExpandScalarProduct // FCE // 
    Collect2[#, MTD, Factoring -> Function[x, MomentumCombine[Factor[x]]]] &

−ΔCA(6ξG−11)gs3gμν(p1−p2)ρfabc128π2+ΔCA(6ξG−11)gs3gμρ(p1−p3)νfabc128π2−ΔCA(6ξG−11)gs3gνρ(p2−p3)μfabc128π2-\frac{\Delta C_A \left(6 \xi _{\text{G}}-11\right) g_s^3 g^{\mu \nu } (\text{p1}-\text{p2})^{\rho } f^{abc}}{128 \pi ^2}+\frac{\Delta C_A \left(6 \xi _{\text{G}}-11\right) g_s^3 g^{\mu \rho } (\text{p1}-\text{p3})^{\nu } f^{abc}}{128 \pi ^2}-\frac{\Delta C_A \left(6 \xi _{\text{G}}-11\right) g_s^3 g^{\nu \rho } (\text{p2}-\text{p3})^{\mu } f^{abc}}{128 \pi ^2}

Counter-term

amp4[1] = amp4[0] // ReplaceAll[#, {SMP["Z_A"] -> 1 + alpha SMP["d_A"], 
            SMP["Z_g"] -> 1 + alpha SMP["d_g"]}] & // Series[#, {alpha, 0, 1}] & // 
        Normal // ReplaceAll[#, alpha -> 1] & // ExpandScalarProduct //FCE // 
    Collect2[#, MTD, GaugeXi, Factoring -> Function[x, MomentumCombine[Factor[x]]]] &

−12gs(3δA+2δg)gμν(p1−p2)ρfabc−12gs(3δA+2δg)gμρ(p3−p1)νfabc−12gs(3δA+2δg)gνρ(p2−p3)μfabc-\frac{1}{2} g_s \left(3 \delta _A+2 \delta _g\right) g^{\mu \nu } (\text{p1}-\text{p2})^{\rho } f^{abc}-\frac{1}{2} g_s \left(3 \delta _A+2 \delta _g\right) g^{\mu \rho } (\text{p3}-\text{p1})^{\nu } f^{abc}-\frac{1}{2} g_s \left(3 \delta _A+2 \delta _g\right) g^{\nu \rho } (\text{p2}-\text{p3})^{\mu } f^{abc}

Check the cancellation of the UV divergences in the MSbar scheme. The renormalization constants are obtained from another example calculation, “Renormalization.m”

renormalizationConstants = {
        SMP["d_A"] -> SMP["alpha_s"]/(4 Pi) SMP["Delta"] (1/2 CA (13/3 - GaugeXi["G"]) - 2/3 Nf), 
        SMP["d_g"] -> ((-11*CA*SMP["alpha_s"])/(24 Pi) SMP["Delta"] + (Nf*SMP["alpha_s"])/(12*Pi) SMP["Delta"]) 
    } /. SMP["alpha_s"] -> SMP["g_s"]^2/(4 Pi);
uvDiv[0] = ExpandScalarProduct[amp1Div[2] + amp2Div[2] + amp3Div[2] + amp4[1]] //FCE // 
   Collect2[#, MTD, Factoring -> Function[x, MomentumCombine[Factor[x]]]] &

−gsgμν(p1−p2)ρfabc(9ΔCAξGgs2−17ΔCAgs2+288π2δA+8ΔNfgs2+192π2δg)192π2+gsgμρ(p1−p3)νfabc(9ΔCAξGgs2−17ΔCAgs2+288π2δA+8ΔNfgs2+192π2δg)192π2−gsgνρ(p2−p3)μfabc(9ΔCAξGgs2−17ΔCAgs2+288π2δA+8ΔNfgs2+192π2δg)192π2-\frac{g_s g^{\mu \nu } (\text{p1}-\text{p2})^{\rho } f^{abc} \left(9 \Delta C_A \xi _{\text{G}} g_s^2-17 \Delta C_A g_s^2+288 \pi ^2 \delta _A+8 \Delta N_f g_s^2+192 \pi ^2 \delta _g\right)}{192 \pi ^2}+\frac{g_s g^{\mu \rho } (\text{p1}-\text{p3})^{\nu } f^{abc} \left(9 \Delta C_A \xi _{\text{G}} g_s^2-17 \Delta C_A g_s^2+288 \pi ^2 \delta _A+8 \Delta N_f g_s^2+192 \pi ^2 \delta _g\right)}{192 \pi ^2}-\frac{g_s g^{\nu \rho } (\text{p2}-\text{p3})^{\mu } f^{abc} \left(9 \Delta C_A \xi _{\text{G}} g_s^2-17 \Delta C_A g_s^2+288 \pi ^2 \delta _A+8 \Delta N_f g_s^2+192 \pi ^2 \delta _g\right)}{192 \pi ^2}

uvDiv[1] = (uvDiv[0] /. renormalizationConstants) // Simplify

00

FCCompareResults[uvDiv[1], 0, 
   Text -> {"\tThe UV divergence of the 3-gluon vertex at 1-loop is cancelled by the counter-term :", 
     "CORRECT.", "WRONG!"}, Interrupt -> {Hold[Quit[1]], Automatic}];

\tThe UV divergence of the 3-gluon vertex at 1-loop is cancelled by the counter-term :  CORRECT.\text{$\backslash $tThe UV divergence of the 3-gluon vertex at 1-loop is cancelled by the counter-term :} \;\text{CORRECT.}

Check the final results

VertexLorentzStruct[{p_, q_, k_}, {mu_, nu_, si_}, {a_, b_, c_}] := 
   -I SUNF[a, b, c] (MTD[mu, nu] FVD[p - q, si] + MTD[nu, si] FVD[q - k, mu] + MTD[si, mu] FVD[k - p, nu]);
knownResult = {
        (I SMP["g_s"]) Nf SMP["g_s"]^2/(4 Pi)^2*(-2/3) SMP["Delta"]*
            VertexLorentzStruct[{p1, p2, p3}, {mu, nu, rho}, {a, b, c}], 
     
     
        (I SMP["g_s"]) SMP["g_s"]^2/(4 Pi)^2  CA/8 (1/3) SMP["Delta"]*
            VertexLorentzStruct[{p1, p2, p3}, {mu, nu, rho}, {a, b, c}], 
     
     
        (I SMP["g_s"]) SMP["g_s"]^2/(4 Pi)^2  CA/8 SMP["Delta"]*(
            -4 - 9 GaugeXi["G"] + 15 + 3 GaugeXi["G"])*
        VertexLorentzStruct[{p1, p2, p3}, {mu, nu, rho}, {a, b, c}] 
    } // FCI;
FCCompareResults[{amp1Div[2], amp2Div[2], amp3Div[2]}, knownResult, 
    Text -> {"\tCompare to Pascual and Tarrach, QCD: Renormalization for the Practitioner, Eq III.46:", 
     "CORRECT.", "WRONG!"}, Interrupt -> {Hold[Quit[1]], Automatic}];
Print["\tCPU Time used: ", Round[N[TimeUsed[], 4], 0.001], " s."];

\tCompare to Pascual and Tarrach, QCD: Renormalization for the Practitioner, Eq III.46:  CORRECT.\text{$\backslash $tCompare to Pascual and Tarrach, QCD: Renormalization for the Practitioner, Eq III.46:} \;\text{CORRECT.}

\tCPU Time used: 92.55 s.\text{$\backslash $tCPU Time used: }92.55\text{ s.}