description = "Gl Gl -> Q Qbar, 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}
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], {V[5], V[5]} ->
{F[3, {1}], -F[3, {1}]}, InsertionLevel -> {Classes},
Model -> "SMQCD"];
Paint[diags, ColumnsXRows -> {2, 1}, Numbering -> Simple,
SheetHeader -> None, ImageSize -> {512, 256}];
amp[0] = FCFAConvert[CreateFeynAmp[diags], IncomingMomenta -> {p1, p2},
OutgoingMomenta -> {k1, k2}, UndoChiralSplittings -> True, ChangeDimension -> 4,
TransversePolarizationVectors -> {p1, p2}, List -> False, SMP -> True,
Contract -> True, DropSumOver -> True]\frac{g_s^2 T_{\text{Col3}\;\text{Col5}}^{\text{Glu2}} T_{\text{Col5}\;\text{Col4}}^{\text{Glu1}} \left(\varphi (\overline{k_1},m_u)\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }\left(p_2\right)\right).\left(\bar{\gamma }\cdot \left(\overline{k_1}-\overline{p_2}\right)+m_u\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }\left(p_1\right)\right).\left(\varphi (-\overline{k_2},m_u)\right)}{(\overline{p_2}-\overline{k_1}){}^2-m_u^2}+\frac{g_s^2 T_{\text{Col3}\;\text{Col5}}^{\text{Glu1}} T_{\text{Col5}\;\text{Col4}}^{\text{Glu2}} \left(\varphi (\overline{k_1},m_u)\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }\left(p_1\right)\right).\left(\bar{\gamma }\cdot \left(\overline{p_2}-\overline{k_2}\right)+m_u\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }\left(p_2\right)\right).\left(\varphi (-\overline{k_2},m_u)\right)}{(\overline{k_2}-\overline{p_2}){}^2-m_u^2}-\frac{i g_s^2 T_{\text{Col3}\;\text{Col4}}^{\text{Glu5}} f^{\text{Glu1}\;\text{Glu2}\;\text{Glu5}} \left(\overline{k_1}\cdot \bar{\varepsilon }\left(p_2\right)+\overline{k_2}\cdot \bar{\varepsilon }\left(p_2\right)+\overline{p_1}\cdot \bar{\varepsilon }\left(p_2\right)\right) \left(\varphi (\overline{k_1},m_u)\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }\left(p_1\right)\right).\left(\varphi (-\overline{k_2},m_u)\right)}{(\overline{k_1}+\overline{k_2}){}^2}-\frac{i g_s^2 T_{\text{Col3}\;\text{Col4}}^{\text{Glu5}} f^{\text{Glu1}\;\text{Glu2}\;\text{Glu5}} \left(-\left(\overline{k_1}\cdot \bar{\varepsilon }\left(p_1\right)\right)-\overline{k_2}\cdot \bar{\varepsilon }\left(p_1\right)-\overline{p_2}\cdot \bar{\varepsilon }\left(p_1\right)\right) \left(\varphi (\overline{k_1},m_u)\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }\left(p_2\right)\right).\left(\varphi (-\overline{k_2},m_u)\right)}{(\overline{k_1}+\overline{k_2}){}^2}-\frac{i g_s^2 T_{\text{Col3}\;\text{Col4}}^{\text{Glu5}} f^{\text{Glu1}\;\text{Glu2}\;\text{Glu5}} \left(\bar{\varepsilon }\left(p_1\right)\cdot \bar{\varepsilon }\left(p_2\right)\right) \left(\varphi (\overline{k_1},m_u)\right).\left(\bar{\gamma }\cdot \left(\overline{p_2}-\overline{p_1}\right)\right).\left(\varphi (-\overline{k_2},m_u)\right)}{(\overline{k_1}+\overline{k_2}){}^2}
FCClearScalarProducts[];
SetMandelstam[s, t, u, p1, p2, -k1, -k2, 0, 0, SMP["m_u"], SMP["m_u"]];ampSquared[0] = 1/((SUNN^2 - 1)^2) (amp[0] (ComplexConjugate[amp[0]])) //
FeynAmpDenominatorExplicit // SUNSimplify[#, Explicit -> True,
SUNNToCACF -> False] & // FermionSpinSum //
DiracSimplify // DoPolarizationSums[#, p1, p2,
ExtraFactor -> 1/2] & // DoPolarizationSums[#, p2, p1, ExtraFactor -> 1/2] & //
TrickMandelstam[#, {s, t, u, 2 SMP["m_u"]^2}] & // Simplify\frac{g_s^4 \left(-m_u^4 \left(3 t^2+14 t u+3 u^2\right)+m_u^2 \left(t^3+7 t^2 u+7 t u^2+u^3\right)+6 m_u^8-t u \left(t^2+u^2\right)\right) \left(2 N^2 m_u^2 (t+u)-2 N^2 m_u^4-N^2 \left(t^2+u^2\right)+s^2\right)}{2 N \left(N^2-1\right) s^2 \left(u-m_u^2\right){}^2 \left(t-m_u^2\right){}^2}
ampSquaredMassless[0] = ampSquared[0] // ReplaceAll[#, {SMP["m_u"] -> 0}] & //
TrickMandelstam[#, {s, t, u, 0}] &\frac{g_s^4 \left(t^2+u^2\right) \left(-N^2 t^2-N^2 u^2+s^2\right)}{2 N \left(1-N^2\right) s^2 t u}
ampSquaredMasslessSUNN3[0] = ampSquaredMassless[0] /. SUNN -> 3-\frac{g_s^4 \left(t^2+u^2\right) \left(s^2-9 t^2-9 u^2\right)}{48 s^2 t u}
knownResults = {
(1/6) SMP["g_s"]^4 (t^2 + u^2)/(t u) - (3/8) SMP["g_s"]^4 (t^2 + u^2)/(s^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: }43.321\text{ s.}