QCD manual (development version)

Load FeynCalc and the necessary add-ons or other packages

description = "Q Gl -> Q 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}], V[5]} -> 
            {F[3, {1}], V[5]}, InsertionLevel -> {Classes}, Model -> "SMQCD"]; 
 
Paint[diags, ColumnsXRows -> {2, 2}, Numbering -> Simple, 
    SheetHeader -> None, ImageSize -> {512, 512}];

1ugxu0yezpt32

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

0imattdskb2fc

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

1pwrygnu6erz8

Obtain the amplitude

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

−gs2TCol5  Col1Glu2TCol3  Col5Glu4(φ(p2‾,mu)).(γˉ⋅εˉ∗(k2)).(γˉ⋅(k2‾+p2‾)+mu).(γˉ⋅εˉ(k1)).(φ(p1‾,mu))(−k2‾−p2‾)2−mu2−gs2TCol5  Col1Glu4TCol3  Col5Glu2(φ(p2‾,mu)).(γˉ⋅εˉ(k1)).(γˉ⋅(p2‾−k1‾)+mu).(γˉ⋅εˉ∗(k2)).(φ(p1‾,mu))(k1‾−p2‾)2−mu2+igs2TCol3  Col1Glu5fGlu2  Glu4  Glu5(2(k2‾⋅εˉ(k1))−k1‾⋅εˉ(k1))(φ(p2‾,mu)).(γˉ⋅εˉ∗(k2)).(φ(p1‾,mu))(k2‾−k1‾)2+igs2TCol3  Col1Glu5fGlu2  Glu4  Glu5(2(k1‾⋅εˉ∗(k2))−k2‾⋅εˉ∗(k2))(φ(p2‾,mu)).(γˉ⋅εˉ(k1)).(φ(p1‾,mu))(k2‾−k1‾)2+igs2TCol3  Col1Glu5fGlu2  Glu4  Glu5(εˉ(k1)⋅εˉ∗(k2))(φ(p2‾,mu)).(γˉ⋅(−k1‾−k2‾)).(φ(p1‾,mu))(k2‾−k1‾)2-\frac{g_s^2 T_{\text{Col5}\;\text{Col1}}^{\text{Glu2}} T_{\text{Col3}\;\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{k_2}-\overline{p_2}){}^2-m_u^2}-\frac{g_s^2 T_{\text{Col5}\;\text{Col1}}^{\text{Glu4}} T_{\text{Col3}\;\text{Col5}}^{\text{Glu2}} \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{p_2}-\overline{k_1}\right)+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{p_2}){}^2-m_u^2}+\frac{i g_s^2 T_{\text{Col3}\;\text{Col1}}^{\text{Glu5}} f^{\text{Glu2}\;\text{Glu4}\;\text{Glu5}} \left(2 \left(\overline{k_2}\cdot \bar{\varepsilon }\left(k_1\right)\right)-\overline{k_1}\cdot \bar{\varepsilon }\left(k_1\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_2}-\overline{k_1}){}^2}+\frac{i g_s^2 T_{\text{Col3}\;\text{Col1}}^{\text{Glu5}} f^{\text{Glu2}\;\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_2}-\overline{k_1}){}^2}+\frac{i g_s^2 T_{\text{Col3}\;\text{Col1}}^{\text{Glu5}} f^{\text{Glu2}\;\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_2}-\overline{k_1}){}^2}

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

−igs2TCol3  Col1Glu5fGlu2  Glu4  Glu5(φ(p2‾,mu)).(γˉ⋅k2‾).(φ(p1‾,mu))(k2‾−k1‾)2-\frac{i g_s^2 T_{\text{Col3}\;\text{Col1}}^{\text{Glu5}} f^{\text{Glu2}\;\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_2}-\overline{k_1}){}^2}

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

igs2TCol3  Col1Glu5fGlu2  Glu4  Glu5(φ(p2‾,mu)).(γˉ⋅k1‾).(φ(p1‾,mu))(k2‾−k1‾)2\frac{i g_s^2 T_{\text{Col3}\;\text{Col1}}^{\text{Glu5}} f^{\text{Glu2}\;\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_2}-\overline{k_1}){}^2}

Fix the kinematics

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

Square the amplitude

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

−1N2t2(u−mu2)2(s−mu2)2gs4(mu8(3N2s2+37N2su+3N2u2+6t2)−3N2mu6(3s3+19s2u+19su2+3u3)+mu4(5N2s4+35N2s3u+s2(43N2u2−3t2)−7s(2t2u−5N2u3)+5N2u4−3t2u2)−mu2(s+u)(N2s4+8N2s3u−s2(t2−5N2u2)+s(8N2u3−6t2u)+N2u4−t2u2)+9N2mu10(s+u)−11N2mu12+su(N2s4−s2(t2−3N2u2)+N2u4−t2u2))-\frac{1}{N^2 t^2 \left(u-m_u^2\right){}^2 \left(s-m_u^2\right){}^2}g_s^4 \left(m_u^8 \left(3 N^2 s^2+37 N^2 s u+3 N^2 u^2+6 t^2\right)-3 N^2 m_u^6 \left(3 s^3+19 s^2 u+19 s u^2+3 u^3\right)+m_u^4 \left(5 N^2 s^4+35 N^2 s^3 u+s^2 \left(43 N^2 u^2-3 t^2\right)-7 s \left(2 t^2 u-5 N^2 u^3\right)+5 N^2 u^4-3 t^2 u^2\right)-m_u^2 (s+u) \left(N^2 s^4+8 N^2 s^3 u-s^2 \left(t^2-5 N^2 u^2\right)+s \left(8 N^2 u^3-6 t^2 u\right)+N^2 u^4-t^2 u^2\right)+9 N^2 m_u^{10} (s+u)-11 N^2 m_u^{12}+s u \left(N^2 s^4-s^2 \left(t^2-3 N^2 u^2\right)+N^2 u^4-t^2 u^2\right)\right)

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

gs4(mu2−u)(s−mu2)2t2\frac{g_s^4 \left(m_u^2-u\right) \left(s-m_u^2\right)}{2 t^2}

ampSquaredGh2[0] = 1/(SUNN (SUNN^2 - 1)) (ampGh2[0] (ComplexConjugate[ampGh2[0]])) // 
        FeynAmpDenominatorExplicit // SUNSimplify[#, Explicit -> True, 
            SUNNToCACF -> False] & // FermionSpinSum[#, ExtraFactor -> 1/2] & // 
        DiracSimplify // TrickMandelstam[#, {s, t, u, 2 SMP["m_u"]^2}] & // 
    Simplify

gs4(mu2−u)(s−mu2)2t2\frac{g_s^4 \left(m_u^2-u\right) \left(s-m_u^2\right)}{2 t^2}

Subtract unphysical degrees of freedom using ghost contributions. Notice that here the averaging over the polarizations of the initial gluons is done only after the subtraction of the unphysical contributions.

ampSquared[0] = 1/2 (ampSquaredUnphys[0] - ampSquaredGh1[0] - ampSquaredGh2[0]) // 
    Together

−12N2t2(mu2−u)2(s−mu2)2gs4(−N2s5mu2+5N2s4mu4−9N2s4umu2−10N2s3u2mu2−8N2s3mu6+32N2s3umu4−10N2s2u3mu2+34N2s2u2mu4−48N2s2umu6−9N2su4mu2+32N2su3mu4−48N2su2mu6+12N2smu10+28N2sumu8−N2u5mu2+5N2u4mu4−8N2u3mu6−12N2mu12+12N2umu10+s3t2mu2−3s2t2mu4+7s2t2umu2+7st2u2mu2−14st2umu4+t2u3mu2−3t2u2mu4+6t2mu8+N2s5u+2N2s3u3+N2su5−s3t2u−st2u3)-\frac{1}{2 N^2 t^2 \left(m_u^2-u\right){}^2 \left(s-m_u^2\right){}^2}g_s^4 \left(-N^2 s^5 m_u^2+5 N^2 s^4 m_u^4-9 N^2 s^4 u m_u^2-10 N^2 s^3 u^2 m_u^2-8 N^2 s^3 m_u^6+32 N^2 s^3 u m_u^4-10 N^2 s^2 u^3 m_u^2+34 N^2 s^2 u^2 m_u^4-48 N^2 s^2 u m_u^6-9 N^2 s u^4 m_u^2+32 N^2 s u^3 m_u^4-48 N^2 s u^2 m_u^6+12 N^2 s m_u^{10}+28 N^2 s 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^3 t^2 m_u^2-3 s^2 t^2 m_u^4+7 s^2 t^2 u m_u^2+7 s t^2 u^2 m_u^2-14 s t^2 u m_u^4+t^2 u^3 m_u^2-3 t^2 u^2 m_u^4+6 t^2 m_u^8+N^2 s^5 u+2 N^2 s^3 u^3+N^2 s u^5-s^3 t^2 u-s t^2 u^3\right)

ampSquaredMassless[0] = ampSquared[0] // ReplaceAll[#, {SMP["m_u"] -> 0}] & // 
    Simplify

−gs4(s2+u2)(N2s2+N2u2−t2)2N2st2u-\frac{g_s^4 \left(s^2+u^2\right) \left(N^2 s^2+N^2 u^2-t^2\right)}{2 N^2 s t^2 u}

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

−gs4(s2+u2)(9s2−t2+9u2)18st2u-\frac{g_s^4 \left(s^2+u^2\right) \left(9 s^2-t^2+9 u^2\right)}{18 s t^2 u}

Check the final results

knownResults = {
    -(4/9) SMP["g_s"]^4 (s^2 + u^2)/(s u) + SMP["g_s"]^4 (u^2 + s^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."];

\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: 27.861 s.\text{$\backslash $tCPU Time used: }27.861\text{ s.}