QCD manual (development version)

Load FeynCalc and the necessary add-ons or other packages

description = "Q Qbar -> Gl Gl with ghosts, 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];

FeynCalc   10.0.0 (dev version, 2023-12-20 22:40:59 +01:00, dff3b835). For help, use the online  documentation‾  , check out the wiki‾   or visit the forum.‾\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}.}

Please check our FAQ‾   for answers to some common FeynCalc questions and have a look at the supplied examples.‾\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}.}

If you use FeynCalc in your research, please evaluate FeynCalcHowToCite[] to learn how to cite this software.\text{If you use FeynCalc in your research, please evaluate FeynCalcHowToCite[] to learn how to cite this software.}

Please keep in mind that the proper academic attribution of our work is crucial to ensure the future development of this package!\text{Please keep in mind that the proper academic attribution of our work is crucial to ensure the future development of this package!}

FeynArts   3.11 (3 Aug 2020) patched for use with FeynCalc, for documentation see the manual‾   or visit www.feynarts.de.‾\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}.}

If you use FeynArts in your research, please cite\text{If you use FeynArts in your research, please cite}

 ∙ T. Hahn, Comput. Phys. Commun., 140, 418-431, 2001, arXiv:hep-ph/0012260\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}]} -> 
            {V[5], V[5]}, InsertionLevel -> {Classes}, Model -> "SMQCD"]; 
 
Paint[diags, ColumnsXRows -> {2, 1}, Numbering -> Simple, 
    SheetHeader -> None, ImageSize -> {512, 256}];

1mxk9tunkc9hh

0t9kx7zt7rmjd

diagsGh1 = InsertFields[CreateTopologies[0, 2 -> 2], {F[3, {1}], -F[3, {1}]} -> 
            {-U[5], U[5]}, InsertionLevel -> {Classes}, Model -> "SMQCD"];
Paint[diagsGh1, ColumnsXRows -> {2, 1}, Numbering -> Simple, 
    SheetHeader -> None, ImageSize -> {512, 256}];

1fid4qu7f4838

diagsGh2 = InsertFields[CreateTopologies[0, 2 -> 2], {F[3, {1}], -F[3, {1}]} -> 
            {U[5], -U[5]}, InsertionLevel -> {Classes}, Model -> "SMQCD"];
Paint[diagsGh2, ColumnsXRows -> {2, 1}, Numbering -> Simple, 
    SheetHeader -> None, ImageSize -> {512, 256}];

13iyr66ctlz4j

Obtain the amplitudes

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

−gs2TCol5  Col1Glu3TCol2  Col5Glu4(φ(−p2‾,mu)).(γˉ⋅εˉ∗(k2)).(γˉ⋅(k2‾−p2‾)+mu).(γˉ⋅εˉ∗(k1)).(φ(p1‾,mu))(p2‾−k2‾)2−mu2−gs2TCol5  Col1Glu4TCol2  Col5Glu3(φ(−p2‾,mu)).(γˉ⋅εˉ∗(k1)).(γˉ⋅(k1‾−p2‾)+mu).(γˉ⋅εˉ∗(k2)).(φ(p1‾,mu))(p2‾−k1‾)2−mu2+igs2TCol2  Col1Glu5fGlu3  Glu4  Glu5(k1‾⋅εˉ∗(k1)+2(k2‾⋅εˉ∗(k1)))(φ(−p2‾,mu)).(γˉ⋅εˉ∗(k2)).(φ(p1‾,mu))(k1‾+k2‾)2+igs2TCol2  Col1Glu5fGlu3  Glu4  Glu5(−2(k1‾⋅εˉ∗(k2))−k2‾⋅εˉ∗(k2))(φ(−p2‾,mu)).(γˉ⋅εˉ∗(k1)).(φ(p1‾,mu))(k1‾+k2‾)2+igs2TCol2  Col1Glu5fGlu3  Glu4  Glu5(εˉ∗(k1)⋅εˉ∗(k2))(φ(−p2‾,mu)).(γˉ⋅(k1‾−k2‾)).(φ(p1‾,mu))(k1‾+k2‾)2-\frac{g_s^2 T_{\text{Col5}\;\text{Col1}}^{\text{Glu3}} T_{\text{Col2}\;\text{Col5}}^{\text{Glu4}} \left(\varphi (-\overline{p_2},m_u)\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }^*\left(k_2\right)\right).\left(\bar{\gamma }\cdot \left(\overline{k_2}-\overline{p_2}\right)+m_u\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }^*\left(k_1\right)\right).\left(\varphi (\overline{p_1},m_u)\right)}{(\overline{p_2}-\overline{k_2}){}^2-m_u^2}-\frac{g_s^2 T_{\text{Col5}\;\text{Col1}}^{\text{Glu4}} T_{\text{Col2}\;\text{Col5}}^{\text{Glu3}} \left(\varphi (-\overline{p_2},m_u)\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }^*\left(k_1\right)\right).\left(\bar{\gamma }\cdot \left(\overline{k_1}-\overline{p_2}\right)+m_u\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }^*\left(k_2\right)\right).\left(\varphi (\overline{p_1},m_u)\right)}{(\overline{p_2}-\overline{k_1}){}^2-m_u^2}+\frac{i g_s^2 T_{\text{Col2}\;\text{Col1}}^{\text{Glu5}} f^{\text{Glu3}\;\text{Glu4}\;\text{Glu5}} \left(\overline{k_1}\cdot \bar{\varepsilon }^*\left(k_1\right)+2 \left(\overline{k_2}\cdot \bar{\varepsilon }^*\left(k_1\right)\right)\right) \left(\varphi (-\overline{p_2},m_u)\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }^*\left(k_2\right)\right).\left(\varphi (\overline{p_1},m_u)\right)}{(\overline{k_1}+\overline{k_2}){}^2}+\frac{i g_s^2 T_{\text{Col2}\;\text{Col1}}^{\text{Glu5}} f^{\text{Glu3}\;\text{Glu4}\;\text{Glu5}} \left(-2 \left(\overline{k_1}\cdot \bar{\varepsilon }^*\left(k_2\right)\right)-\overline{k_2}\cdot \bar{\varepsilon }^*\left(k_2\right)\right) \left(\varphi (-\overline{p_2},m_u)\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }^*\left(k_1\right)\right).\left(\varphi (\overline{p_1},m_u)\right)}{(\overline{k_1}+\overline{k_2}){}^2}+\frac{i g_s^2 T_{\text{Col2}\;\text{Col1}}^{\text{Glu5}} f^{\text{Glu3}\;\text{Glu4}\;\text{Glu5}} \left(\bar{\varepsilon }^*\left(k_1\right)\cdot \bar{\varepsilon }^*\left(k_2\right)\right) \left(\varphi (-\overline{p_2},m_u)\right).\left(\bar{\gamma }\cdot \left(\overline{k_1}-\overline{k_2}\right)\right).\left(\varphi (\overline{p_1},m_u)\right)}{(\overline{k_1}+\overline{k_2}){}^2}

ampGh1[0] = FCFAConvert[CreateFeynAmp[diagsGh1], IncomingMomenta -> {p1, p2}, 
    OutgoingMomenta -> {k1, k2}, UndoChiralSplittings -> True, ChangeDimension -> 4, 
    List -> False, SMP -> True, Contract -> True, DropSumOver -> True]

igs2TCol2  Col1Glu5fGlu3  Glu4  Glu5(φ(−p2‾,mu)).(γˉ⋅k2‾).(φ(p1‾,mu))(k1‾+k2‾)2\frac{i g_s^2 T_{\text{Col2}\;\text{Col1}}^{\text{Glu5}} f^{\text{Glu3}\;\text{Glu4}\;\text{Glu5}} \left(\varphi (-\overline{p_2},m_u)\right).\left(\bar{\gamma }\cdot \overline{k_2}\right).\left(\varphi (\overline{p_1},m_u)\right)}{(\overline{k_1}+\overline{k_2}){}^2}

ampGh2[0] = FCFAConvert[CreateFeynAmp[diagsGh2], IncomingMomenta -> {p1, p2}, 
    OutgoingMomenta -> {k1, k2}, UndoChiralSplittings -> True, ChangeDimension -> 4, 
    List -> False, SMP -> True, Contract -> True, DropSumOver -> True]

igs2TCol2  Col1Glu5fGlu3  Glu4  Glu5(φ(−p2‾,mu)).(γˉ⋅k1‾).(φ(p1‾,mu))(k1‾+k2‾)2\frac{i g_s^2 T_{\text{Col2}\;\text{Col1}}^{\text{Glu5}} f^{\text{Glu3}\;\text{Glu4}\;\text{Glu5}} \left(\varphi (-\overline{p_2},m_u)\right).\left(\bar{\gamma }\cdot \overline{k_1}\right).\left(\varphi (\overline{p_1},m_u)\right)}{(\overline{k_1}+\overline{k_2}){}^2}

Fix the kinematics

FCClearScalarProducts[];
SetMandelstam[s, t, u, p1, p2, -k1, -k2, SMP["m_u"], SMP["m_u"], 0, 0];

Square the amplitudes

ampSquaredUnphys[0] = 1/(SUNN^2) (amp[0] (ComplexConjugate[amp[0]])) // 
            FeynAmpDenominatorExplicit // SUNSimplify[#, Explicit -> True, 
            SUNNToCACF -> False] & // FermionSpinSum[#, ExtraFactor -> 1/2^2] & // 
        DiracSimplify // DoPolarizationSums[#, k1, 0] & // 
        DoPolarizationSums[#, k2, 0] & // TrickMandelstam[#, {s, t, u, 2  SMP["m_u"]^2}] & // 
    Simplify

12N3s2(u−mu2)2(t−mu2)2(N2−1)gs4(mu8(N2(3t2+37tu+3u2)+6s2)+mu4(N2(5t4+35t3u+43t2u2+35tu3+5u4)−s2(3t2+14tu+3u2))−mu2(t+u)(N2(t4+8t3u+5t2u2+8tu3+u4)−s2(t2+6tu+u2))−3N2mu6(3t3+19t2u+19tu2+3u3)+9N2mu10(t+u)−11N2mu12+tu(N2(t4+3t2u2+u4)−s2(t2+u2)))\frac{1}{2 N^3 s^2 \left(u-m_u^2\right){}^2 \left(t-m_u^2\right){}^2}\left(N^2-1\right) g_s^4 \left(m_u^8 \left(N^2 \left(3 t^2+37 t u+3 u^2\right)+6 s^2\right)+m_u^4 \left(N^2 \left(5 t^4+35 t^3 u+43 t^2 u^2+35 t u^3+5 u^4\right)-s^2 \left(3 t^2+14 t u+3 u^2\right)\right)-m_u^2 (t+u) \left(N^2 \left(t^4+8 t^3 u+5 t^2 u^2+8 t u^3+u^4\right)-s^2 \left(t^2+6 t u+u^2\right)\right)-3 N^2 m_u^6 \left(3 t^3+19 t^2 u+19 t u^2+3 u^3\right)+9 N^2 m_u^{10} (t+u)-11 N^2 m_u^{12}+t u \left(N^2 \left(t^4+3 t^2 u^2+u^4\right)-s^2 \left(t^2+u^2\right)\right)\right)

ampSquaredGh1[0] = 1/(SUNN^2) (ampGh1[0] (ComplexConjugate[ampGh1[0]])) // 
        FeynAmpDenominatorExplicit // SUNSimplify[#, Explicit -> True, 
            SUNNToCACF -> False] & // FermionSpinSum[#, ExtraFactor -> 1/2^2] & // 
        DiracSimplify // TrickMandelstam[#, {s, t, u, 2  SMP["m_u"]^2}] & // 
    Simplify

−(N2−1)gs4(mu2−u)(t−mu2)4Ns2-\frac{\left(N^2-1\right) g_s^4 \left(m_u^2-u\right) \left(t-m_u^2\right)}{4 N s^2}

ampSquaredGh2[0] = 1/(SUNN^2) (ampGh1[0] (ComplexConjugate[ampGh1[0]])) // 
        FeynAmpDenominatorExplicit // SUNSimplify[#, Explicit -> True, 
            SUNNToCACF -> False] & // FermionSpinSum[#, ExtraFactor -> 1/2^2] & // 
        DiracSimplify // TrickMandelstam[#, {s, t, u, 2  SMP["m_u"]^2}] & // 
    Simplify

−(N2−1)gs4(mu2−u)(t−mu2)4Ns2-\frac{\left(N^2-1\right) g_s^4 \left(m_u^2-u\right) \left(t-m_u^2\right)}{4 N s^2}

Subtract unphysical degrees of freedom using ghost contributions

ampSquared[0] = ampSquaredUnphys[0] - ampSquaredGh1[0] - ampSquaredGh2[0] // Together

12N3s2(mu2−u)2(t−mu2)2(N2−1)gs4(−N2t5mu2+5N2t4mu4−9N2t4umu2−10N2t3u2mu2−8N2t3mu6+32N2t3umu4−10N2t2u3mu2+34N2t2u2mu4−48N2t2umu6−9N2tu4mu2+32N2tu3mu4−48N2tu2mu6+12N2tmu10+28N2tumu8−N2u5mu2+5N2u4mu4−8N2u3mu6−12N2mu12+12N2umu10+s2t3mu2−3s2t2mu4+7s2t2umu2+7s2tu2mu2−14s2tumu4+s2u3mu2−3s2u2mu4+6s2mu8+N2t5u+2N2t3u3+N2tu5−s2t3u−s2tu3)\frac{1}{2 N^3 s^2 \left(m_u^2-u\right){}^2 \left(t-m_u^2\right){}^2}\left(N^2-1\right) g_s^4 \left(-N^2 t^5 m_u^2+5 N^2 t^4 m_u^4-9 N^2 t^4 u m_u^2-10 N^2 t^3 u^2 m_u^2-8 N^2 t^3 m_u^6+32 N^2 t^3 u m_u^4-10 N^2 t^2 u^3 m_u^2+34 N^2 t^2 u^2 m_u^4-48 N^2 t^2 u m_u^6-9 N^2 t u^4 m_u^2+32 N^2 t u^3 m_u^4-48 N^2 t u^2 m_u^6+12 N^2 t m_u^{10}+28 N^2 t u m_u^8-N^2 u^5 m_u^2+5 N^2 u^4 m_u^4-8 N^2 u^3 m_u^6-12 N^2 m_u^{12}+12 N^2 u m_u^{10}+s^2 t^3 m_u^2-3 s^2 t^2 m_u^4+7 s^2 t^2 u m_u^2+7 s^2 t u^2 m_u^2-14 s^2 t u m_u^4+s^2 u^3 m_u^2-3 s^2 u^2 m_u^4+6 s^2 m_u^8+N^2 t^5 u+2 N^2 t^3 u^3+N^2 t u^5-s^2 t^3 u-s^2 t u^3\right)

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

(1−N2)gs4(t2+u2)(−N2t2−N2u2+s2)2N3s2tu\frac{\left(1-N^2\right) g_s^4 \left(t^2+u^2\right) \left(-N^2 t^2-N^2 u^2+s^2\right)}{2 N^3 s^2 t u}

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

−4gs4(t2+u2)(s2−9t2−9u2)27s2tu-\frac{4 g_s^4 \left(t^2+u^2\right) \left(s^2-9 t^2-9 u^2\right)}{27 s^2 t u}

Check the final results

knownResults = {
    (32/27) SMP["g_s"]^4 (t^2 + u^2)/(t u) - (8/3) 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."];

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

True\text{True}

\tCPU Time used: 33.796 s.\text{$\backslash $tCPU Time used: }33.796\text{ s.}