= "Qt -> Qb W, EW, 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[k1, TraditionalForm] := "\!\(\*SubscriptBox[\(k\), \(1\)]\)";
MakeBoxes[k2, TraditionalForm] := "\!\(\*SubscriptBox[\(k\), \(2\)]\)";
Enable CKM mixing
= True; $CKM
= InsertFields[CreateTopologies[0, 1 -> 2],
diags {F[3, {3}]} -> {F[4, {3}], -V[3]}, InsertionLevel -> {Particles}];
[diags, ColumnsXRows -> {2, 1}, Numbering -> Simple,
Paint-> None, ImageSize -> {512, 256}]; SheetHeader
[0] = FCFAConvert[CreateFeynAmp[diags], IncomingMomenta -> {p},
amp-> {k1, k2}, ChangeDimension -> 4, List -> False, SMP -> True,
OutgoingMomenta -> True, DropSumOver -> True, TransversePolarizationVectors -> {k2},
Contract -> {SMP["e"] -> Sqrt[8/Sqrt[2] SMP["G_F"] SMP["m_W"]^2 SMP["sin_W"]^2]}] FinalSubstitutions
\frac{2^{3/4} V_{\text{tb}}^* \delta _{\text{Col1}\;\text{Col2}} \sqrt{G_F m_W^2 \left(\left.\sin (\theta _W\right)\right){}^2} \left(\varphi (\overline{k_1},m_b)\right).\left(\bar{\gamma }\cdot \bar{\varepsilon }^*\left(k_2\right)\right).\bar{\gamma }^7.\left(\varphi (\overline{p},m_t)\right)}{\left.\sin (\theta _W\right)}
[]
FCClearScalarProducts[p] = SMP["m_t"]^2;
SP[k1] = SMP["m_b"]^2;
SP[k2] = SMP["m_W"]^2;
SP[k1, k2] = Simplify[(SP[p] - SP[k1] - SP[k2])/2];
SP[p, k1] = Simplify[ExpandScalarProduct[SP[k1 + k2, k1]]];
SP[p, k2] = Simplify[ExpandScalarProduct[SP[k1 + k2, k2]]]; SP
We average over the polarizations of the top quark, hence the additional factor 1/2
[0] = (amp[0] (ComplexConjugate[amp[0]])) // SUNSimplify //
ampSquared[#, ExtraFactor -> 1/2] & // DiracSimplify //
FermionSpinSum[#, k2] & // Simplify DoPolarizationSums
\sqrt{2} C_A G_F V_{\text{tb}}^* V_{\text{tb}} \left(m_b^2 \left(m_W^2-2 m_t^2\right)+m_b^4+m_t^2 m_W^2+m_t^4-2 m_W^4\right)
[m1_, m2_, M_] := 1/(16 Pi M) Sqrt[1 - (m1 + m2)^2/M^2]*
phaseSpacePrefactorSqrt[1 - (m1 - m2)^2/M^2];
= phaseSpacePrefactor[SMP["m_b"], SMP["m_W"], SMP["m_t"]]*
totalDecayRate [0] // Simplify // ReplaceAll[#, Sqrt[x_] Sqrt[y_] :>
ampSquaredSqrt[ExpandAll[x y]]] &
\frac{C_A G_F V_{\text{tb}}^* V_{\text{tb}} \sqrt{-\frac{2 m_b^2 m_W^2}{m_t^4}+\frac{m_b^4}{m_t^4}-\frac{2 m_b^2}{m_t^2}+\frac{m_W^4}{m_t^4}-\frac{2 m_W^2}{m_t^2}+1} \left(m_b^2 \left(m_W^2-2 m_t^2\right)+m_b^4+m_t^2 m_W^2+m_t^4-2 m_W^4\right)}{8 \sqrt{2} \pi m_t}
= {
knownResults ["m_t"]^3 (CA*SMP["G_F"]*Sqrt[((SMP["m_b"] - SMP["m_t"] -
SMP["m_W"])*(SMP["m_b"] + SMP["m_t"] - SMP["m_W"])*(SMP["m_b"] -
SMP["m_t"] + SMP["m_W"])*(SMP["m_b"] + SMP["m_t"] + SMP["m_W"]))/
SMP["m_t"]^4]*((1 - SMP["m_b"]^2/SMP["m_t"]^2)^2 + SMP["m_W"]^2/
SMP["m_t"]^2 (1 + SMP["m_b"]^2/SMP["m_t"]^2) - 2 SMP["m_W"]^4/SMP["m_t"]^4
SMP*SMP["V_tb", -I]*SMP["V_tb", I])/(8*Sqrt[2]*Pi)
)};
[{totalDecayRate},
FCCompareResults,
knownResultsText -> {"\tCompare to Grozin, Using REDUCE in High Energy Physics, Chapter 5.2:",
"CORRECT.", "WRONG!"}, Interrupt -> {Hold[Quit[1]], Automatic}];
Print["\tCPU Time used: ", Round[N[TimeUsed[], 3], 0.001], " s."];
\text{$\backslash $tCompare to Grozin, Using REDUCE in High Energy Physics, Chapter 5.2:} \;\text{CORRECT.}
\text{$\backslash $tCPU Time used: }16.965\text{ s.}