QED manual (development version)

Load FeynCalc and the necessary add-ons or other packages

description = "El Mu -> El Mu, QED, 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[2, {1}], F[2, {2}]} -> 
            {F[ 2, {1}], F[2, {2}]}, InsertionLevel -> {Classes}, 
            Restrictions -> QEDOnly]; 
 
Paint[diags, ColumnsXRows -> {1, 1}, Numbering -> Simple, 
    SheetHeader -> None, ImageSize -> {256, 256}];

08ja8c2z0syw7

Obtain the amplitude

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

−e2(φ(k1‾,me)).γˉLor2.(φ(p1‾,me))(φ(k2‾,mμ)).γˉLor2.(φ(p2‾,mμ))(k2‾−p2‾)2-\frac{\text{e}^2 \left(\varphi (\overline{k_1},m_e)\right).\bar{\gamma }^{\text{Lor2}}.\left(\varphi (\overline{p_1},m_e)\right) \left(\varphi (\overline{k_2},m_{\mu })\right).\bar{\gamma }^{\text{Lor2}}.\left(\varphi (\overline{p_2},m_{\mu })\right)}{(\overline{k_2}-\overline{p_2}){}^2}

Fix the kinematics

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

Square the amplitude

ampSquared[0] = (amp[0] (ComplexConjugate[amp[0]])) // 
        FeynAmpDenominatorExplicit // FermionSpinSum[#, ExtraFactor -> 1/2^2] & // 
    DiracSimplify // Simplify

2  e4(−2me2(−2mμ2+s−t+u)+2me4+2mμ4−2mμ2(s−t+u)+s2+u2)t2\frac{2 \;\text{e}^4 \left(-2 m_e^2 \left(-2 m_{\mu }^2+s-t+u\right)+2 m_e^4+2 m_{\mu }^4-2 m_{\mu }^2 (s-t+u)+s^2+u^2\right)}{t^2}

ampSquaredMassless1[0] = ampSquared[0] // ReplaceAll[#, {SMP["m_e"] -> 0}] & // 
    Simplify

2  e4(2mμ4−2mμ2(s−t+u)+s2+u2)t2\frac{2 \;\text{e}^4 \left(2 m_{\mu }^4-2 m_{\mu }^2 (s-t+u)+s^2+u^2\right)}{t^2}

ampSquaredMassless2[0] = ampSquared[0] // ReplaceAll[#, {
        SMP["m_e"] -> 0, SMP["m_mu"] -> 0}] & // Simplify

2  e4(s2+u2)t2\frac{2 \;\text{e}^4 \left(s^2+u^2\right)}{t^2}

Check the final results

knownResults = {
    (8 SMP["e"]^4 (SP[p1, k2] SP[p2, k1] + SP[p1, p2] SP[k1, k2] - 
            SMP["m_mu"]^2 SP[p1, k1]))/(SP[k1 - p1])^2 // ExpandScalarProduct // 
        ReplaceAll[#, SMP["m_e"] -> 0] &, 
    ((8 SMP["e"]^4/t^2) ((s/2)^2 + (u/2)^2)) 
   };
FCCompareResults[{ampSquaredMassless1[0], ampSquaredMassless2[0]}, knownResults, 
  Text -> {"\tCompare to Peskin and Schroeder, An Introduction to QFT, Eqs 5.61 and 5.71:", 
    "CORRECT.", "WRONG!"}, Interrupt -> {Hold[Quit[1]], Automatic}]
Print["\tCPU Time used: ", Round[N[TimeUsed[], 3], 0.001], " s."];

\tCompare to Peskin and Schroeder, An Introduction to QFT, Eqs 5.61 and 5.71:  CORRECT.\text{$\backslash $tCompare to Peskin and Schroeder, An Introduction to QFT, Eqs 5.61 and 5.71:} \;\text{CORRECT.}

True\text{True}

\tCPU Time used: 20.121 s.\text{$\backslash $tCPU Time used: }20.121\text{ s.}