# 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 $\epsilon_0$ ?

• As far as I know, neither the color charge nor the weak isospin have units (cf. this post for color charge at least). – Kyle Kanos Nov 19 '14 at 14:52
• Kyle Kanos: "As far as I know, neither the color charge nor the weak isospin have units" -- That's consistent with what I know; and that's certainly convenient. (Perhaps even: a convenient choice.) The only consequence is: the hypothetical "corresponding Planck units" which I'm asking about should also be plain real numbers (of some non-zero value); and the "corresponding permittivities($\epsilon_W$, $\epsilon_S$)" (as far as those are sensible quantities at all) should both have "suitable dimensions". But what would be their values, e.g. in terms of unit $\frac{1}{\text{J m}}$ ? – user12262 Nov 19 '14 at 15:31
• This is out of my field and perhaps I am indeed wrong, but it seems like you're assuming a 1:1 correlation between E&M and QCD/QFD when I don't believe that there is such a correlation (specifically about the "permittivities"). The coupling constants are unitless regardless of unit system you work in, so I don't think you're going to find it in "units of 1/J/m". – Kyle Kanos Nov 19 '14 at 15:36

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.

• Cosmas Zachos: "[...] you'd reinstate $\sqrt{\hbar c}$ in a rate or cross section, etc..." -- That's indeed pretty much the whole point; and that "the rest is taken care of" by dimensionless coupling parameters "$\alpha$" and a few additional discrete numbers (cmp. en.wikipedia.org/wiki/Weak_hypercharge). So thanks for your answer, I'll accept it as it stands. (And sorry for having taken a while before responding.) p.s. [... contd.] – user12262 Feb 23 '17 at 22:12
• p.s.: Concerning "your" el.-weak formulas $$e=g~\text{Sin}~\theta_W = g'~\text{Cos}~\theta_W$$ -- that's of course alright; cmp. Weinberg QFT2, eq. (21.3.19). Still I wonder whether $$\sqrt{\alpha_a}=g~\text{Sin}~\theta_W = g'~\text{Cos}~\theta_W$$ might not be more appropriate in this place; because at least by Weinberg QFT2 eq. (21.3.30): $$v = \frac{2~m_W}{g} = 247~\text{GeV}$$ the parameter $g$ appears manifestly dimensionless, too. – user12262 Feb 23 '17 at 22:12
• p.p.s.: Cosmas Zachos: "The strong charge may likewise be inferred from experiment and is larger than the EW couplings at LHC energies, infinite at the confinement radius" -- May I suggest that this describes the strong coupling parameter $\alpha_s$ rather than "the strong charge"; cmp. en.wikipedia.org/wiki/Color_charge#Coupling_constant_and_charge etc. – user12262 Feb 23 '17 at 22:21
• I'm not sure you are not asking different questions now. α -type couplings are deprecated, once the "dimensionless" (in units of energy) numbers for e, g, g', g(strong), are adopted. As you see from that list, the strong charge is twice the weak g at the Z mass, but grows explosively at lower energies. – Cosmas Zachos Feb 23 '17 at 22:57
• Cosmas Zachos: "$\alpha$-type couplings are deprecated, once the "dim.-less" (in units of energy) numbers for $e, g, g', g(strong)$, are adopted." -- Well, I seem hesitant to adopt this convention when thinking of "elementary charge e"; while (by the eq. shown above, actually: QFT2, eq. (21.3.35)) $g, g'$ are strictly numbers, as are the $\alpha$-s. "As you see from that list, the strong charge is [... varying.]" -- Which list, please? (Is there such a list there?) – user12262 Feb 24 '17 at 0:56

One other thing to note here:

We worked out a quantitative theory of electric charge in the 17th century. For the subsequent 250 odd years, we knew nothing about fundamental charge. So, a whole infrastructure was built up to describe charge, that then had to be retrofitted into the microscopic world. If we knew about atoms first, we surely would have just chosen $e$ as the fundamental electric charge (and probably something like a fraction of $N_{A}e$ for macroscopic amounts of charge), and built up everything from there, and been in gaussian units for electricity, and not even be aware of any coupling other than the fine structure coupling.

Well, this is precisely the case for the weak and strong forces -- we ONLY know of them at the microscopic level, and we never accumulate more than a few units of weak charge, and in principle, cannot accumulate even a single unit of net strong charge. So, we can happily describe these things using only fundamental units, and never have any reason to do otherwise.

All are dimensionless constants.

With $\frac{e^2}{\hbar c} \approx \frac{1}{137.036}$

Similarly there are constants for weak and colour charge. These basically are the probability over time of a particle emitting a photon, W (or Z)-boson or gluon respectively.

The weak constant is of the same order as the electromagnetic constant. The colour constant is closer to 1.

Nobody yet knows how to calculate these numbers from scratch.

• zooby: "All are dimensionless constants." -- The Planck units for weak or strong "charge" are dimensionless constants? What are the (experimentally determined) values of these two (real) constant numbers?? What are the concrete expressions for these two numbers, involving (by definition) Planck's constant ($h$; or the "reduced form" $\hbar$)? (Btw. Planck's constant $h$ and the reduced Planck constant $\hbar$ are a dimensionful quantities.) Do these expressions also involve corresponding "permittivities $\epsilon$" of suitable dimension? – user12262 Sep 28 '15 at 5:20
• You don't need c and h to define these numbers since they have no dimension. Just like a probability has no dimension, is is just a number. e.g. 1/6 chance of rolling a 6. Or you could define everything in terms of the electron charge and call that -1 for example. Then 3 electrons will have charge -3. There is no such thing as a Planck unit for charge. Just like there is no Planck unit for probability. With the combination of constants you suggest all the units cancel out leaving a pure number like above which none knows why it has this value... yet. – zooby Sep 28 '15 at 12:55
• zooby: "You don't need c and h to define these numbers [...]" -- Comparing with $$q_P^{el.-mag.} = \sqrt{4~\pi~\epsilon_0~\hbar~c} = \frac{e}{\sqrt{\alpha_{el.-mag.}}}$$ from the definition of (el.-mag.) Planck charge, what's stopping us from expressing $$q_P^{weak} = \sqrt{4~\pi~\epsilon_{weak}~\hbar~c} = \frac{Y_W}{\sqrt{\alpha_{weak}[~0~]}}$$, where $Y_W$ is the weak hypercharge, or $$q_P^{strong} = \sqrt{4~\pi~\epsilon_{strong}~\hbar~c} = \frac{1}{\sqrt{\alpha_{strong}[~0~]}}$$? – user12262 Sep 28 '15 at 20:47