QCD manual (development version)

Load FeynCalc and the necessary add-ons or other packages

description = "Qi Qi -> Qi Qi, 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, {1}]} -> 
        {F[3, {1}], F[3, {1}]}, InsertionLevel -> {Classes}, Model -> "SMQCD", 
        ExcludeParticles -> {S[_], V[1 | 2]}]; 
 
Paint[diags, ColumnsXRows -> {2, 1}, Numbering -> Simple, 
    SheetHeader -> None, ImageSize -> {512, 256}];

0u9ca9si01zhz

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{Col4}\;\text{Col1}}^{\text{Glu5}} T_{\text{Col3}\;\text{Col2}}^{\text{Glu5}} \left(\varphi (\overline{k_1},m_u)\right).\bar{\gamma }^{\text{Lor1}}.\left(\varphi (\overline{p_2},m_u)\right) \left(\varphi (\overline{k_2},m_u)\right).\bar{\gamma }^{\text{Lor1}}.\left(\varphi (\overline{p_1},m_u)\right)}{(\overline{k_1}-\overline{p_2}){}^2}-\frac{g_s^2 T_{\text{Col3}\;\text{Col1}}^{\text{Glu5}} T_{\text{Col4}\;\text{Col2}}^{\text{Glu5}} \left(\varphi (\overline{k_1},m_u)\right).\bar{\gamma }^{\text{Lor2}}.\left(\varphi (\overline{p_1},m_u)\right) \left(\varphi (\overline{k_2},m_u)\right).\bar{\gamma }^{\text{Lor2}}.\left(\varphi (\overline{p_2},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_u"], 
    SMP["m_u"], SMP["m_u"]];

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, 4 SMP["m_u"]^2}] & // Simplify

\frac{\left(N^2-1\right) g_s^4 \left(-4 m_u^2 \left(N \left(t^3+u^3\right)-2 s t u\right)+4 m_u^4 \left(N \left(t^2+u^2\right)-3 t u\right)+N \left(t^4+t^3 u+t^2 u^2+t u^3+u^4\right)-s^2 t u\right)}{N^3 t^2 u^2}

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

-\frac{\left(1-N^2\right) g_s^4 \left(N t^4+N t^3 u+N t^2 u^2+N t u^3+N u^4-s^2 t u\right)}{N^3 t^2 u^2}

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

\frac{8 g_s^4 \left(-s^2 t u+3 t^4+3 t^3 u+3 t^2 u^2+3 t u^3+3 u^4\right)}{27 t^2 u^2}

Check the final results

knownResults = {
    ((4/9) SMP["g_s"]^4 ((s^2 + u^2)/t^2 + (s^2 + t^2)/u^2) - (8/27) SMP["g_s"]^4 s^2/(t u)) 
   };
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: }18.94\text{ s.}