quantum field theory - Are there Planck units for weak or strong "charge", similar to the electromagnetic Planck charge $sqrt{4~pi~epsilon_0~hbar~c}~$?

Are there Planck units for "charge" of weak or strong interaction, similar to the Planck unit of electromagnetic charge: $\sqrt{4~\pi~\epsilon_0~\hbar~c}$ ?

Are there perhaps direct substitutes, relating to weak or strong interaction, for the electromagnetic vacuum permittivity $\epsilon_0$ ?

Answer

All couplings in QFT are measured in Lorentz-Heaviside rationalized natural units.

That is, for instance, for the electric charge,

$$\alpha = \frac{e^2}{4\pi\varepsilon_0\hbar c} \approx 1/137 .$$ In these units $\epsilon_0=1$, so the elementary electric charge is simply $$e = \sqrt{4\pi\alpha} ~\sqrt{ \hbar c} \approx 0.30282212 \sqrt{ \hbar c} \ .$$ The square root is called the Planck charge (in HEP; about a quarter of your definition above), $\sqrt{\hbar c}\approx 5.291 \times 10^{-19}$coulombs.

However, In natural units, one measures everything in units of $\hbar$ and $c$, and, e.g., GeV, a discretionary unit: the energy scale is left to itself, and is an inverse length scale, etc... Consequently, one sets $\hbar=c=1$, and the Planck charge is just 1, so e looks dimensionless in energy units---the only surviving dimension. And is about 1/3.

But you know that such charge units may be reinstated by dimensional analysis at the very end, uniquely, to produce a quantity to hand over to an engineer. In our case, if charge is really what you wish to hand over, you reinstate the above minuscule number in coulombs, (cruuuude mnemonic: recall the inverse Planck mass in GeVs).

Nevertheless, you'll probably never hand over a weak or strong charge to an engineer. In all likelihood, you'd reinstate $\sqrt{ \hbar c}$ in a rate or cross section, to make them dimensionally consistent. This is, in my book, the apotheosis of dimensional analysis. To sum up, the natural unit of charge is one.

For the electroweak interactions, you know that the above electric charge is $e= g \sin \theta_W = g' \cos \theta_W$, where g is the weak isospin coupling and g' the weak hypercharge coupling and $\theta_W$ is the weak mixing ("Weinberg") angle of about 28 degrees or so.

The strong coupling $g_s$ may likewise be inferred from experiment and is larger than the EW couplings at LHC energies, infinite at the confinement radius, and of the order of 1 in residual nuclear interactions... Think of the normalization of the Yukawa potential. I'd be shocked if you ever wished to measure it in coulombs.

To sum up, at the $M_Z$ scale, all these dimensionless SM couplings are, beyond e above: $g'\approx 0.357; ~ g\approx 0.652; ~ g_s\approx 1.221$. The last one, naturally, grows explosively with decreasing energy.