A Weak Argument

Perhaps, like me, you have become tired of me attacking Max Andrews' thesis.  If so, good news!  Now I want to attack something Max Andrews wrote in a blogpost.  It's quite a minor element of the post, but it blossoms out quite rapidly:

Let’s look at the weak force coupling constant, g_w = 1.43 x 10^-62.

He hasn't even got into the argument with this statement and it's taken out of context (the specific context isn't actually important) but even so there is so much to talk about.

If you take the time to look up the value of the "weak force coupling constant" you may find that α_w ≈ 10^-7.  The sharp eyed among you might see that we are talking about what appears to be apples and oranges, or g_w and α_w.  Admittedly, at first I thought that Andrews couldn't work out how to display an α and was using g instead (which would be strange).  However, if you dig deeply enough around the net you will find that

α_w = g_w² / 4π

which means that g_w ≈ 1.1 x 10^-3.  That's a big error margin when one is talking about how weak the weak force coupling constant is (10⁵⁹ times bigger than claimed).

(There is a real difference between α_w and g_w.  The former, α_w, is the "weak interaction coupling constant" or " weak nuclear force coupling constant" or "weak force coupling constant" or even "weak fine structure constant" and it represents the strength of the weak force in an interaction.  The latter, g_w, is a less frequently mentioned value.  It is known as the "effective charge of weak interaction" or "weak gauge coupling constant" and is a measure of the strength of the interaction with the weak gauge field (in gauge field theories).)

But wait … there's more.  Andrews might not be entirely wrong about this value (he is though, as we shall discover below).

Like quite a few other constants, the weak force coupling constant is not actually a constant.  It varies with distance, so it can (and should) be expressed as a function of range.  There is undoubtedly a specific range at which g_w = 1.43 x 10^-62 (one not terribly far off the radius of a proton at about 10^-15m).  The question, however, is whether Andrews - who talks about a number of constants in the article - uses the same point of reference.  Note that he refers to the strong coupling constant this way:

g_s = 15

This shares the same relationship with α_s as above, so this corresponds with α_s ≈ 13.7.  Note however, that an actual theoretical physicist, Matt Strassler, has this to say:

But at longer distances, the strong nuclear force gradually becomes (relatively!) stronger. [Again, remember what we mean by “weak” and “strong” here; the force is actually becoming weaker in absolute terms as r increases, but relative to, say, electromagnetic forces at the same distance r, it’s becoming stronger.]

§  α_strong = 0.3 (at r ~ 10^-16 meters)

That’s quite strong indeed! And by the time r reaches 10^-15 meters, the radius of a proton, α_strong is bigger than 1, and becomes impossible to define uniquely.

It's still going to fall with distance, but distance isn't the only measure against which to measure the value of the strong coupling constant.  You also need to consider the energy range, for example on the chart in this article, it can be seen that α_strong peaks at about 0.7 GeV (billions of electron volts) with a value of about 1.2 which equates to a g_s ≈ 3.9.

Andrews' suggested g_s of 15 seems quite unreasonable.  But at least this time he is only out by a factor of about four, rather than a factor of 10⁵⁹.

Which brings us back to Andrews' claim for the value of the weak force coupling constant at 1.43x10^-62.  This is, fortunately enough, a rather specific number so that when I stumbled across it again, at WikiUniversity, it leapt out at me.  But note that this value, while associated with the weak force coupling constant isn't the weak force coupling constant.  It's the Fermi coupling constant, G_F, as expressed in J.m³.  To get the weak force coupling constant, you need to go a step further ("at sufficiently small distances"):

α_w = G_F.M_p².c / ħ³ = 10^-5

This corresponds with a g_w of about 8x10^-12.

Admittedly, the value of α_w is directly proportional to the value of G_F, so Andrews could structure his argument in terms of how the extremely low value of G_F appears to be fine-tuned.  But this isn't the problem usually talked about in physics circles.  They talk about how the measured (effective or renormalised) value of G_F is so much larger than might otherwise be expected (the hierarchy problem).

Interestingly, they suggest that, once normalised, G_F lies very close to G (the gravitational constant).  Personally, this is what I'd expect at least in terms of natural units - furthermore, I'd expect both of them to have a value of precisely 1 (like the gravitational constant, the speed of light, the reduced Planck constant, all the Planck units - mass, length, time, charge, temperature - the Boltzmann constant and the Coulomb constant (which is related to the permittivity constant)).  If this is the case, then their values are not an instance of fine tuning at all, they would be just yet another expected reflection of the nature of the universe.

Even if this isn't the case, Andrews should be a lot more mindful when making bold claims about the values of fundamental constants and try to avoid being out by such huge factors.