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Calculating the branching fractions of a Higgs boson $m_h=125\,\text{GeV}$ into a muon and an anti-muon, and a $b$-quark and an anti-$b$-quark from the decay formula
$$\Gamma=\frac{\alpha_{w}}{8 \hbar} m_{h} c^{2}\left(\frac{m}{M_{W}}\right)^{2}\left[1-\left(\frac{2 m}{m_{h}}\right)^{2}\right]^{3 / 2}$$

gives a different ratio in comparison with the ratio from this table SM Higgs Branching Ratios.

So basically why the ratio Γ(h→bb¯)/Γ(h→μμ¯) using the formula is not as the ratio BR(h→bb¯)/BR(h→μμ¯) found from the table? I'm getting 4633 while the table gives about 2630

what is special about $m_h=125\,\text{GeV}$? maybe something related to the experiment?

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    $\begingroup$ Is the difference a factor of 3? You are missing a color factor of 3 for the bb final state. On top of that, the book contains QCD corrections which could be moderate to the bb state $\endgroup$ – innisfree Apr 29 at 2:36
  • $\begingroup$ I'm aware of the color factor. Thanks $\endgroup$ – hiery Apr 29 at 3:37
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    $\begingroup$ Perhaps the masses at the 125 GeV energy scales should be used? $\endgroup$ – ohwilleke Apr 29 at 4:08
  • $\begingroup$ No, it's not wrong, there is a reason behind this difference which I'm trying to find. $\endgroup$ – hiery Apr 29 at 4:09
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    $\begingroup$ @ohwilleke is probably right; going beyond tree-level, you should be using $m_b(m_h)$ which differs considerably from $m_b(m_b)$. You can find further discussion about the calculation in cds.cern.ch/record/340786/files/9712334.pdf $m_b(100\,\text{GeV}) \simeq 3\,\text{GeV}$. If these details more or less fix it, please write it up and answer your own question here. $\endgroup$ – innisfree Apr 29 at 4:20
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The discrepancy is because you need to use the running masses of the bottom quark and muon at the 125 GeV energy scale, rather than the pole masses of those particles in the formula that you cite (see generally here).

The running mass of the bottom quark at the 125 GeV energy scale is roughly 3.0 GeV (v. 4.18 GeV pole mass) and the running mass of the muon at that energy scale is about .104 GeV (v. a pole mass of .1057 GeV). See this 1997 paper, and then adjust for the differences between the current pole mass measurements and the ones used in the 1997 paper and for the closest scale they calculate (the Z boson mass energy scale) and the 125 GeV target energy scale, realizing that the running of masses in the Standard Model with energy scale is roughly logarithmic.

The 125 GeV energy scale is special because the experimentally measured value of the Higgs boson mass is currently 125.18 GeV +/ 0.16 GeV.

I get a ratio of 2543 calculating from the table at 125.0 GeV, and I get a ratio of 2488 using the running values of these masses, which is the same to within the uncertainty of the inputs used in the respective calculations (my calculations were made using Microsoft Xcel, the formula provided in the question with a factor of three adjustment for the three b quark colors, and data noted in this answer).

All currently measured masses are from the live edition of the Particle Data Group's website.

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