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In the natural unit system, the pion-decay constant $f_{\pi}$ is $92.4\:\rm MeV$. But I think that a decay constant should have a dimension of $[T]^{-1}$, where $[T]$ is the dimension of time. Then, in the conventional unit system, $f_{\pi}$ should be given by $$f_{\pi} = \frac{92.4 \:\rm MeV}{\hbar},$$ where $\hbar$ is the Planck's constant, so that the unit of $f_{\pi}$ becomes $$\frac{\:\rm MeV}{\:\rm MeV \: s} = \rm s^{-1}.$$

Am I correct about dividing by the $\hbar$ here?

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Yep. That's exactly the way it works. In natural units $\hbar=1$ can be inserted without changing anything, and to go back to regular units you multiply by whatever factors of $\hbar$ (and $c$) are needed to get the correct dimensionality.

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Pardon the deconstruction, but @Emilio 's answer only addresses the straightforward dimensional analysis reclamation part of your question. Its deeply wrong-minded premise, however,

But I think that a decay constant should have a dimension of [T]−1, where [T] is the dimension of time.

merits redress. As you can easily check in your textbooks, a decay constant is virtually never proportional to a decay rate, Γ (which has the inverse time dimensions you are looking at here). It features in amplitudes, normally squared to produce rates.

In the decay of a charged pion to leptons, say, muon and its neutrino, $$ \Gamma_\pi \propto ~G_F^2 ~ f_\pi^2 ~~ m_\mu^2 m_\pi ~~. $$ So it hardly makes sense to associate this constant, merely measured from pion decay, to an inverse time. It is an auxiliary parameter firmly residing within QFT and never exiting it.

In case you were interested in the "point" of the constant itself, it is but

the square root of the coefficient in front of the kinetic term for the pion in the low-energy effective action.

In turn, it is, broadly, the scale at which a (current) quark-antiquark pair "morphs" into this pseudoscalar meson and connects it to the vacuum—so it can devolve to a virtual W which then decays to the leptons. I have never met anybody who could visualize this process in terms of times.

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  • $\begingroup$ My apologies Dr. Zachos, I am unable to understand the point here. So the pion-decay constant has the dimension of energy in both the conventional unit system (where $\hbar$ and $c$ are not set unity) and natural system (where $\hbar=c=1$)? $\endgroup$ – omehoque Jan 29 at 15:25
  • $\begingroup$ Nobody knows what units it has in anything but the natural system. That is the point. It is an intermediate "unnatural" parameter in QFT, and as such, it never connects directly to engineering. The only natural quantities you convert to "conventional" units are those that deal with observables, like lifetimes, cross sections, energies, momenta. Nobody works in QFT in anything but natural units, because they would have lost their minds, by now. This is the point in adopting natural units the very first day of your HEP course. $\endgroup$ – Cosmas Zachos Jan 29 at 15:53

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