description = "Q -> Q, 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.}
We keep scaleless B0 functions, since otherwise the UV part would not come out right.
$KeepLogDivergentScalelessIntegrals = True;diags = InsertFields[CreateTopologies[1, 1 -> 1,
ExcludeTopologies -> Tadpoles], {F[3, {1}]} ->
{F[3, {1}]}, InsertionLevel -> {Particles}, Model -> "SMQCD",
ExcludeParticles -> {S[_], V[1 | 2 | 3]}];
Paint[diags, ColumnsXRows -> {1, 1}, Numbering -> Simple,
SheetHeader -> None, ImageSize -> {256, 256}];
The 1/(2Pi)^D prefactor is implicit. We keep the full gauge dependence.
amp[0] = FCFAConvert[CreateFeynAmp[diags, Truncated -> True,
PreFactor -> 1, GaugeRules -> {}], IncomingMomenta -> {p},
OutgoingMomenta -> {p}, LoopMomenta -> {q},
UndoChiralSplittings -> True,
ChangeDimension -> D, SMP -> True, DropSumOver -> True,
Contract -> True, FinalSubstitutions -> {SMP["m_u"] -> SMP["m_q"]}]\left\{-\frac{g_s^2 T_{\text{Col3}\;\text{Col1}}^{\text{Glu3}} T_{\text{Col2}\;\text{Col3}}^{\text{Glu3}} \gamma ^{\text{Lor2}}.\left(m_q+\gamma \cdot q\right).\gamma ^{\text{Lor2}}}{\left(q^2-m_q^2\right).(q-p)^2}-\frac{\left(1-\xi _g\right) g_s^2 T_{\text{Col3}\;\text{Col1}}^{\text{Glu3}} T_{\text{Col2}\;\text{Col3}}^{\text{Glu3}} (\gamma \cdot (p-q)).\left(m_q+\gamma \cdot q\right).(\gamma \cdot (q-p))}{\left(q^2-m_q^2\right).(q-p)^4}\right\}
amp[1] = amp[0] // SUNSimplify[#, FCParallelize -> True] & //
TID[#, q, ToPaVe -> True, FCParallelize -> True] & // Total\frac{1}{2 p^2}i \pi ^2 C_F g_s^2 \delta _{\text{Col1}\;\text{Col2}} \;\text{B}_0\left(p^2,0,m_q^2\right) \left(-D \left(p^2-m_q^2\right) \gamma \cdot p-2 D p^2 m_q+2 D p^2 \gamma \cdot p+\xi _g m_q^2 \gamma \cdot p-2 \xi _g m_q (\gamma \cdot p).(\gamma \cdot p)+p^2 \xi _g \gamma \cdot p+m_q^2 (-(\gamma \cdot p))+2 m_q (\gamma \cdot p).(\gamma \cdot p)+2 \left(p^2-m_q^2\right) \gamma \cdot p-5 p^2 \gamma \cdot p\right)-\frac{i \pi ^2 C_F \left(1-\xi _g\right) g_s^2 \;\text{B}_0(0,0,0) \delta _{\text{Col1}\;\text{Col2}} \left(m_q^2 (-(\gamma \cdot p))+2 m_q (\gamma \cdot p).(\gamma \cdot p)-2 p^2 m_q+p^2 \gamma \cdot p\right)}{2 p^2}+\frac{i \pi ^2 C_F \left(1-\xi _g\right) g_s^2 \delta _{\text{Col1}\;\text{Col2}} \left(-m_q^2 \left(p^2-m_q^2\right) \gamma \cdot p-4 p^2 m_q (\gamma \cdot p).(\gamma \cdot p)+2 m_q \left(p^2-m_q^2\right) (\gamma \cdot p).(\gamma \cdot p)+p^2 \left(p^2-m_q^2\right) \gamma \cdot p+2 p^2 m_q^3+2 p^4 m_q\right) \;\text{C}_0\left(0,p^2,p^2,0,0,m_q^2\right)}{2 p^2}+\frac{i \pi ^2 (2-D) C_F g_s^2 \delta _{\text{Col1}\;\text{Col2}} \gamma \cdot p \;\text{A}_0\left(m_q^2\right)}{2 p^2}
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 C_F g_s^2 \delta _{\text{Col1}\;\text{Col2}} \left(\xi _g \gamma \cdot p-\left(\xi _g+3\right) m_q\right)}{16 \pi ^2 \varepsilon }
The self-energy amplitude is usually defined as -i Sigma(p^2)
sigma[0] = I ampDiv[0]-\frac{C_F g_s^2 \delta _{\text{Col1}\;\text{Col2}} \left(\xi _g \gamma \cdot p-\left(\xi _g+3\right) m_q\right)}{16 \pi ^2 \varepsilon }
sigmaFeynmanGauge[0] = sigma[0] /. GaugeXi[g] -> 1-\frac{C_F g_s^2 \delta _{\text{Col1}\;\text{Col2}} \left(\gamma \cdot p-4 m_q\right)}{16 \pi ^2 \varepsilon }
Notice that the result in the book must be multiplied by (-1) due to the way how self-energy is defined there (c.f. Eq. 2.4.4 and Eq. 2.4.6).
```mathematica knownResult = -(-SMP[“g_s”]^2/(4 Pi)^2 CF(3 + GaugeXi[g]) (1/Epsilon)SMP[“m_q”] + GSD[p]SMP[“g_s”]^2/(4 Pi)^2CFGaugeXi[g](1/Epsilon)) SDF[Col1, Col2]; FCCompareResults[sigma[0], knownResult, Text -> {“to Muto, Foundations of QCD, Eq 10.41:”, “CORRECT.”, “WRONG!”}, Interrupt -> {Hold[Quit[1]], Automatic}] Print[“Time used:”, Round[N[TimeUsed[], 4], 0.001], ” s.”];
```mathematica
\text{$\backslash $tCompare to Muto, Foundations of QCD, Eq 10.41:} \;\text{CORRECT.}
\text{True}
\text{$\backslash $tCPU Time used: }18.781\text{ s.}