**The unnatural desire for naturalness**

There's a particle called the muon that's almost like the electron, except it's about 206.768 times heavier. Nobody knows why. The number 206.768 is something we measure experimentally, with no explanation so far. Theories of physics tend to involve a bunch of unexplained numbers like this. If you combine general relativity with Standard Model of particle physics, there are about 25 of these constants.

Many particle physicists prefer theories where these constants are not incredibly huge and not incredibly tiny. They call such theories

**natural**. Naturalness sounds good - just like whole wheat bread. But there's no solid evidence that this particular kind of naturalness is really a good thing. Why should the universe prefer numbers that aren't huge and aren't tiny? Nobody knows.

For example, many particle physicists get upset that the density of the vacuum is about

0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000001

Planck masses per Planck volume. They find it 'unnatural' that this number is so tiny. They think it requires 'fine-tuning', which is supposed to be bad.

I agree that it would be nice to explain this number. But it would also be nice to explain the mass of the muon. Is it really more urgent to explain a tiny number than a number like 206.768, which is neither tiny nor huge?

+Sabine Hossenfelder say no, and I tend to agree. More precisely: I see no

*a priori*reason why naturalness should be a feature of fundamental physics. If for some mysterious reason the quest for naturalness always led to good discoveries, I would support it. In science, it makes sense to do things because they tend to work, even if we're not sure why. But in fact, the quest for naturalness has not always been fruitful. Sometimes it seems to lead us into dead ends.

Besides the cosmological constant, another thing physicists worry about is the Higgs mass. Avoiding the 'unnaturalness' of this mass is a popular argument for supersymmetry... but so far that's not working so well. Sabine writes:

*Here is a different example for this idiocy. High energy physicists think it’s a problem that the mass of the Higgs is 15 orders of magnitude smaller than the Planck mass because that means you’d need two constants to cancel each other for 15 digits. That’s supposedly unlikely, but please don’t ask anyone according to which probability distribution it’s unlikely. Because they can’t answer that question. Indeed, depending on character, they’ll either walk off or talk down to you. Guess how I know.*

*Now consider for a moment that the mass of the Higgs was actually about as large as the Planck mass. To be precise, let’s say it’s 1.1370982612166126 times the Planck mass. Now you’d again have to explain how you get exactly those 16 digits. But that is, according to current lore, not a fine-tuning problem. So, erm, what was the problem again?*

Sabine explains things in such down-to-earth terms, with so few of the esoteric technicalities that usually grace discussions of naturalness, that it may be worth reading a more typical discussion of naturalness just to imbibe some of the lore.

This one is quite good, because it includes a lot of lore but doesn't try too hard to intimidate you into believing in the virtue of naturalness:

• G.F. Giudice, Naturally speaking: the naturalness criterion and physics at the LHC, available at https://arxiv.org/abs/0801.2562.

#physics

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- +Matthew Rapaport wrote: "But am I missing something in what you wrote? Most of these are masses which are not dimensionless numbers? Ratios between these are dimensionless (proton/electron mass ratio)... "

Right. I'm taking one mass as a 'standard', namely the Planck mass, and then looking at the ratios of all the other masses to this one.

This is another way of saying what +Layra Idarani said. I said it yet another way, early in my article:

"General relativity and pure quantum mechanics have no dimensionless constants, because the speed of light, the gravitational constant, and Planck's constant merely suffice to set units of mass, length and time. Thus, all the dimensionless constants come in from our wonderful, baroque theory of all the forces other than gravity: the Standard Model."

The Standard Model involves 13 masses, so we get 13 dimensionless constants by taking the ratios of these masses to the Planck mass.

Then there are 12 other dimensionless constants in the Standard Model.24w - +John Baez ok, that makes sense. Sorry I didn't read article just looked at the list. Have read about these for years but not seen them all expressed as ratios to Planck mass before that I can recall. Thanks for reply24w
- +Matt McIrvin wrote: "while I've always found naturalness arguments a little fishy myself, I can't shake the feeling that the basis of them is something sensible."

That's a nicely nuanced viewpoint. I guess I'd say that the smallness of the cosmological constant, the smallness of the Higgs mass/Planck mass ratio, and the apparent failure of the strong force to violate CP symmetry are three clues in the pile of clues we have to work with. Are they strong enough clues to push us into accepting supersymmetry and an axion? Not me, anyway.

A lot of the challenge is keeping the whole array of clues in mind simultaneously, and trying to formulate theories based on*lots*of these clues, rather than pushing very hard with a few while ignoring most, and winding up with a theory (like superstring theory) that describes a world quite different from what we actually see.24w - +John Baez - Please, write that book! Maybe you can restrict the number of combinations of parameters by discussing only those that lead to interesting worlds. I think I would learn a lot.24w
- +Steve Wenner - okay, I'll add that book to my to-write list. So far my main dreamt-of books are
*Scary Concepts in Mathematics*,*My Favorite Numbers*,*Exotic Beauty*and*What We Don't Know About Physics*.24w - I get the lack of a 100% success rate as a reason to drop naturalness as a precept for enquiry. But it is so easy to become drawn in. After all if you were presented with the Ramanujan constant, which is all but an integer in name to 17 sig figs (262537412640768743.9999999999992) rather than its deconstructed form involving three irrationals e^[pi*sqrt(163)]

you would not ask why 163 and thus not uncover the link to the 40 Euler (polynomial generator of) primes. To a numerologist 262537412640768744 surely has a quantum-like status...

wolframalpha.com - Wolfram|Alpha: Computational Knowledge Engine24w - (pi+e)/2 + exp(3 sqrt(pi)) = 206.77527390990966762810328840481

Maybe there's a simple formula which can get closer to 206.768. (The pi+e term looks really ugly).23w - Thank you for writing about this post. I too have wondered about our need for natural numbers that are not too big or too small. At first it did seem reasonable, but now I have to wonder why I ever felt that way.21w

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