QCD manual (development version)

Load FeynCalc and the necessary add-ons or other packages

description = "Q -> Q, 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

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}];

Obtain the amplitude

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, List -> False, SMP -> True, DropSumOver -> True, 
    Contract -> True, FinalSubstitutions -> {SMP["m_u"] -> SMP["m_q"]}]

$$-\frac{g_s^2 T_{\text{Col3}\;\text{Col1}}^{\text{Glu3}} T_{\text{Col2}\;\text{Col3}}^{\text{Glu3}} \gamma ^{\text{Lor2}}.\left(m_q+\gamma \cdot q\right).\gamma ^{\text{Lor2}}}{\left(q^2-m_q^2\right).(q-p)^2}-\frac{\left(1-\xi _g\right) g_s^2 T_{\text{Col3}\;\text{Col1}}^{\text{Glu3}} T_{\text{Col2}\;\text{Col3}}^{\text{Glu3}} (\gamma \cdot (p-q)).\left(m_q+\gamma \cdot q\right).(\gamma \cdot (q-p))}{\left(q^2-m_q^2\right).(q-p)^4}$$

Calculate the amplitude

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

$$\frac{1}{2 p^2}i \pi ^2 C_F g_s^2 \delta _{\text{Col1}\;\text{Col2}} \;\text{B}_0\left(p^2,0,m_q^2\right) \left(-D \left(p^2-m_q^2\right) \gamma \cdot p-2 D p^2 m_q+2 D p^2 \gamma \cdot p+\xi _g m_q^2 \gamma \cdot p-2 \xi _g m_q (\gamma \cdot p).(\gamma \cdot p)+p^2 \xi _g \gamma \cdot p+m_q^2 (-(\gamma \cdot p))+2 m_q (\gamma \cdot p).(\gamma \cdot p)+2 \left(p^2-m_q^2\right) \gamma \cdot p-5 p^2 \gamma \cdot p\right)-\frac{i \pi ^2 C_F \left(1-\xi _g\right) g_s^2 \;\text{B}_0(0,0,0) \delta _{\text{Col1}\;\text{Col2}} \left(m_q^2 (-(\gamma \cdot p))+2 m_q (\gamma \cdot p).(\gamma \cdot p)-2 p^2 m_q+p^2 \gamma \cdot p\right)}{2 p^2}+\frac{i \pi ^2 C_F \left(1-\xi _g\right) g_s^2 \delta _{\text{Col1}\;\text{Col2}} \left(-m_q^2 \left(p^2-m_q^2\right) \gamma \cdot p-4 p^2 m_q (\gamma \cdot p).(\gamma \cdot p)+2 m_q \left(p^2-m_q^2\right) (\gamma \cdot p).(\gamma \cdot p)+p^2 \left(p^2-m_q^2\right) \gamma \cdot p+2 p^2 m_q^3+2 p^4 m_q\right) \;\text{C}_0\left(0,p^2,p^2,0,0,m_q^2\right)}{2 p^2}+\frac{i \pi ^2 (2-D) C_F g_s^2 \delta _{\text{Col1}\;\text{Col2}} \gamma \cdot p \;\text{A}_0\left(m_q^2\right)}{2 p^2}$$

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

$$-\frac{C_F g_s^2 \delta _{\text{Col1}\;\text{Col2}} \left(\gamma \cdot p-4 m_q\right)}{16 \pi ^2 \varepsilon }$$

Check the final results

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).

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)^2*CF*GaugeXi[g]*(1/Epsilon)) SDF[Col1, Col2];
FCCompareResults[sigma[0], knownResult, 
  Text -> {"\tCompare to Muto, Foundations of QCD, Eq 10.41:", 
    "CORRECT.", "WRONG!"}, Interrupt -> {Hold[Quit[1]], Automatic}]
Print["\tCPU Time used: ", Round[N[TimeUsed[], 4], 0.001], " s."];

$$\text{$\backslash $tCompare to Muto, Foundations of QCD, Eq 10.41:} \;\text{CORRECT.}$$

$$\text{True}$$

$$\text{$\backslash $tCPU Time used: }23.468\text{ s.}$$