In the 2004 edition of the book "QCD as a Theory of Hadrons" by S. Narison, the author provides a value for the strong coupling at a scale of the mass of the Z boson,

$$ \alpha_s (M_Z) = 0.1181 \pm 0.0027 \tag{11.68}$$

This is a footnote explaining the choice of the scale:

[$\tau$ decays give] so far the most precise measurement of $\alpha_s$ at $M_Z$ as a modest accuracy at the $\tau$-mass becomes a precise value at the $Z$-mass because the errors decrease faster than the running of $\alpha_s$. Also, here, compared with some other determinations, we have relatively the best theoretical control including the perturbative corrections to order $\alpha_s^4$, the non-perturbative condensates and the resummation of the asymptotic series. (sic)

Apart from a possible typo in the first sentence, I do not understand this reasoning.

Also the PDG states on page 4 of their QCD review (PDF link) that

it has become standard practice to quote the value of $\alpha_s$ at a given scale (typically the mass of the $Z$ boson, $M_Z$) [...]


1 Answer 1


These are the measurements of the strong coupling constant .


Running of the Strong Force Strength

The plot shows the running, or evolution, of the strength of the strong interaction. This is characterized by the evolution of the strong coupling constant aS, plotted here versus energy scale in GeV. This plot is taken from the Particle Data Group.

Look at the measurement error at the tau,just below two GeV, and the meaurement error at the Z just below 100 GeV.

The errors for measured quantities include the statistical errors, the systematic errors of the actual measurements and the estimated errors due to the mathematical formulae used to get at the measured value.

In the case of $a_s$ the contribution to the error by the theoretical model is much better known at the Z, than at the tau energy, thus one uses the value measured at the Z. After the Z the errors are larger.

The plot justifies the the errors decrease faster than the running of $a_s$ only for the beginning of the curve, for the measured erros, so the statement is not really supported by the measurement error behavior observed in the plot. The green width must be the theoretical errors.

  • $\begingroup$ Great answer. The only thing I'm missing is how the $M_Z$ plays a role in tau decays. Since the tau is charged, I presume it would mostly interact with a W boson..? $\endgroup$
    – ersbygre1
    Mar 15, 2019 at 0:34
  • $\begingroup$ @Stephan it is the W that are involved in Z tau decays, The errors on a_strong are estimated measured at the Z , at the tau etc. The value at Z as more accurate is considered the standard (extrapolated theoretically to other mass/energies) $\endgroup$
    – anna v
    Mar 15, 2019 at 4:30
  • $\begingroup$ Ah, I see. So $\alpha_s$ gets measured at various energy scales and for some unknown reason, $\alpha_s(M_Z)$ has the smallest error bars. And therefore this value is cited. Correct? $\endgroup$
    – ersbygre1
    Mar 15, 2019 at 4:42
  • 1
    $\begingroup$ yes, though the reason is shown by the green curve, we know better the mathematics on Z which we use to get a_s. $\endgroup$
    – anna v
    Mar 15, 2019 at 4:45
  • $\begingroup$ I'm sorry, I still don't understand it completely. You wrote "the contribution to the error by the theoretical model is much better known at the Z, than at the tau energy". What part of the theory is better known at some energy scales than others? Since the tau decays weakly at first, could it be that some parameter of the weak interaction is better known at the Z mann than at some other mass? $\endgroup$
    – ersbygre1
    Mar 15, 2019 at 23:47

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