QCD manual (development version)

Load FeynCalc and the necessary add-ons or other packages

description = "Qi Qjbar -> Qi Qjbar, QCD, matrix element squared, tree";
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}

Generate Feynman diagrams

Nicer typesetting

MakeBoxes[p1, TraditionalForm] := "\!\(\*SubscriptBox[\(p\), \(1\)]\)";
MakeBoxes[p2, TraditionalForm] := "\!\(\*SubscriptBox[\(p\), \(2\)]\)";
MakeBoxes[k1, TraditionalForm] := "\!\(\*SubscriptBox[\(k\), \(1\)]\)";
MakeBoxes[k2, TraditionalForm] := "\!\(\*SubscriptBox[\(k\), \(2\)]\)";
diags = InsertFields[CreateTopologies[0, 2 -> 2],  {F[3, {1}], -F[3, {2}]} -> 
        {F[3, {1}], -F[3, {2}]}, InsertionLevel -> {Classes}, Model -> "SMQCD", 
        ExcludeParticles -> {S[_], V[1 | 2]}]; 
 
Paint[diags, ColumnsXRows -> {2, 1}, Numbering -> Simple, 
    SheetHeader -> None, ImageSize -> {512, 256}];

1m1vt5nkwa5d1

Obtain the amplitude

amp[0] = FCFAConvert[CreateFeynAmp[diags], IncomingMomenta -> {p1, p2}, 
    OutgoingMomenta -> {k1, k2}, UndoChiralSplittings -> True, ChangeDimension -> 4, 
    List -> False, SMP -> True, Contract -> True, DropSumOver -> True]

\frac{g_s^2 T_{\text{Col3}\;\text{Col1}}^{\text{Glu5}} T_{\text{Col2}\;\text{Col4}}^{\text{Glu5}} \left(\varphi (-\overline{p_2},m_c)\right).\bar{\gamma }^{\text{Lor2}}.\left(\varphi (-\overline{k_2},m_c)\right) \left(\varphi (\overline{k_1},m_u)\right).\bar{\gamma }^{\text{Lor2}}.\left(\varphi (\overline{p_1},m_u)\right)}{(\overline{k_2}-\overline{p_2}){}^2}

Fix the kinematics

FCClearScalarProducts[];
SetMandelstam[s, t, u, p1, p2, -k1, -k2, SMP["m_u"], SMP["m_c"], 
    SMP["m_u"], SMP["m_c"]];

Square the amplitude

ampSquared[0] = 1/(SUNN^2) (amp[0] (ComplexConjugate[amp[0]])) // 
        FeynAmpDenominatorExplicit // SUNSimplify[#, Explicit -> True, 
            SUNNToCACF -> False] & // FermionSpinSum[#, ExtraFactor -> 1/2^2] & // 
        DiracSimplify // TrickMandelstam[#, {s, t, u, 2 SMP["m_u"]^2 + 2 SMP["m_c"]^2}] & //Simplify

\frac{\left(N^2-1\right) g_s^4 \left(-4 m_c^2 \left(u-m_u^2\right)+2 m_c^4+2 m_u^4-4 u m_u^2+t^2+2 t u+2 u^2\right)}{2 N^2 t^2}

ampSquaredMassless[0] = ampSquared[0] // ReplaceAll[#, {SMP["m_u" | "m_c"] -> 0}] & // 
    TrickMandelstam[#, {s, t, u, 0}] &

-\frac{\left(1-N^2\right) g_s^4 \left(s^2+u^2\right)}{2 N^2 t^2}

ampSquaredMasslessSUNN3[0] = ampSquaredMassless[0] /. SUNN -> 3

\frac{4 g_s^4 \left(s^2+u^2\right)}{9 t^2}

Check the final results

knownResults = {
    ((4/9) SMP["g_s"]^4 (s^2 + u^2)/t^2) 
   };
FCCompareResults[{ampSquaredMasslessSUNN3[0]}, {knownResults}, 
  Text -> {"\tCompare to Ellis, Stirling and Weber, QCD and Collider Physics, Table 7.1:", 
    "CORRECT.", "WRONG!"}, Interrupt -> {Hold[Quit[1]], Automatic}, Factoring -> 
   Function[x, Simplify[TrickMandelstam[x, {s, t, u, 0}]]]]
Print["\tCPU Time used: ", Round[N[TimeUsed[], 3], 0.001], " s."];

\text{$\backslash $tCompare to Ellis, Stirling and Weber, QCD and Collider Physics, Table 7.1:} \;\text{CORRECT.}

\text{True}

\text{$\backslash $tCPU Time used: }17.112\text{ s.}