title: Muon production from the decay of a |
virtual photon |
= "Ga^* -> Mu Amu, QED, total decay rate, tree";
description If[ $FrontEnd === Null,
= False;
$FeynCalcStartupMessages Print[description];
];
If[ $Notebooks === False,
= False
$FeynCalcStartupMessages ];
= {"FeynArts"};
$LoadAddOns
<< FeynCalc`= 0;
$FAVerbose
[9, 3, 1]; FCCheckVersion
\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[p, TraditionalForm] := "\!\(\*SubscriptBox[\(p\), \(1\)]\)";
MakeBoxes[k1, TraditionalForm] := "\!\(\*SubscriptBox[\(k\), \(1\)]\)";
MakeBoxes[k2, TraditionalForm] := "\!\(\*SubscriptBox[\(k\), \(2\)]\)";
= InsertFields[CreateTopologies[0, 1 -> 2], {V[1]} ->
diags {F[2, {2}], -F[2, {2}]}, InsertionLevel -> {Classes},
-> QEDOnly];
Restrictions
[diags, ColumnsXRows -> {1, 1}, Numbering -> Simple,
Paint-> None, ImageSize -> {256, 256}]; SheetHeader
[0] = FCFAConvert[CreateFeynAmp[diags], IncomingMomenta -> {p},
amp-> {k1, k2}, UndoChiralSplittings -> True, ChangeDimension -> 4,
OutgoingMomenta List -> False, SMP -> True, Contract -> True]
\text{e} \left(-\left(\varphi (\overline{k_1},m_{\mu })\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }\left(p_1\right)\right).\left(\varphi (-\overline{k_2},m_{\mu })\right)\right)
[];
FCClearScalarProducts[k1] = SMP["m_mu"]^2;
SP[k2] = SMP["m_mu"]^2;
SP[k1, k2] = (QQ - SP[k1] - SP[k2])/2; SP
[0] = (amp[0] (ComplexConjugate[amp[0]])) //
ampSquared// FermionSpinSum // DiracSimplify //
FeynAmpDenominatorExplicit [#, p, 0, VirtualBoson -> True] & // Simplify DoPolarizationSums
4 \;\text{e}^2 \left(2 m_{\mu }^2+\text{QQ}\right)
[0] = ampSquared[0] // ReplaceAll[#, {
ampSquaredMassless["m_mu"] -> 0}] & SMP
4 \;\text{e}^2 \;\text{QQ}
The differential decay rate d Gamma/ d Omega is given by
= ExpandScalarProduct[1/(64 Pi^2) 1/Sqrt[(SP[k1 + k2])]] prefac
\frac{1}{64 \pi ^2 \sqrt{\text{QQ}}}
= prefac ampSquaredMassless[0] /.
diffDecayRate ["e"]^2 -> (4 Pi SMP["alpha_fs"]) SMP
\frac{\alpha \sqrt{\text{QQ}}}{4 \pi }
The total decay-rate
= 4 Pi diffDecayRate decayRateTotal
\alpha \sqrt{\text{QQ}}
= {
knownResults ["alpha_fs"] Sqrt[QQ]
SMP};
[{decayRateTotal}, knownResults,
FCCompareResultsText -> {"\tCompare to Field, Applications of Perturbative QCD, Eq. 2.1.29",
"CORRECT.", "WRONG!"}, Interrupt -> {Hold[Quit[1]], Automatic}];
Print["\tCPU Time used: ", Round[N[TimeUsed[], 4], 0.001], " s."];
\text{$\backslash $tCompare to Field, Applications of Perturbative QCD, Eq. 2.1.29} \;\text{CORRECT.}
\text{$\backslash $tCPU Time used: }17.633\text{ s.}