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Fine tuning and numerical coincidences July 1, 2008

Posted by dorigo in Blogroll, cosmology, games, internet, physics, science.
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The issue of fine tuning is a headache for today’s theorists in particle physics. I reported here some time ago the brilliant and simple explanation of the problem with the Higgs boson mass given by Michelangelo Mangano. In a nutshell, the Higgs boson mass is expected to be light in the Standard Model, and yet it is very surprising that it be so, given that there are a dozen very large contributions to its value, each of which could make the Higgs hugely massive: but they altogether magically cancel. They are “fine-tuned” to nullify one another like gin and Martini around the olive in a perfectly crafted drink.

A similar coincidence -and actually an even more striking one- happens with dark energy in Cosmology. Dark energy has a density which is orders and orders of magnitude smaller than what one would expect from simple arguments, calling for an explanation which is still unavailable today. Of course, the fact that neither for the Higgs boson nor for dark energy there is as of today a solid experimental evidence is no deterrent: these entities are quite hard to part with, if we insist that we have understood at least in rough terms what exists in the Universe and what is the cause of electroweak symmetry breaking in particle physics. Yet, we should not forget that there might not be a problem after all.

I came across a brilliant discussion of fine tuning in this paper today by sheer chance -or rather, by that random process I entertain myself with every once in a while, called “checking the arXiV”. For me, that simply means looking at recent hep-ph and hep-ex papers, browsing through every third page, and getting caught by the title of some other article quoted in the bibliography, then iterating the process until I remind myself I have to run for some errand.

So, take the two numbers 987654321 and 123456789: could you imagine a more random choice for two 9-digit integers ? Well, what then, if I argued with you that it is by no means a random choice but an astute one, by showing that their ratio is 8.000000073, which deviates from a perfect integer only by nine parts in a billion!

Another more mundane and better known example is the 2000 US elections: the final ballots in Florida revealed that the Republican party got 2,913,321 votes, while the Democratic votes where only 2,913,144: a difference of sixty parts in a million.

Numerical “coincidences” such as the first one above have always had a tremendous impact on the standard crackpot: a person enamoured with a discipline but missing at least in part the institutional background required to be regarded as an authoritative source. A crackpot physicist, if shown a similarly odd coincidence (imagine if those numbers represented two apparently uncorrelated measurements of different physical quantities) would certainly start to build a theory around it with the means he has at his or her disposal. This would be enough for him or her to be tagged as a true crackpot. But there is nothing wrong with trying to understand a numerical coincidence! The only difference is that acknowledged scientists only get interested when those coincidences are really, really, really odd.

Yes, the feeling of being fooled by Nature (the bitch, not the magazine) is what lies underneath. You study electroweak theory, figure that the Higgs boson cannot be much heavier than 100 GeV, and find out that to be so there has to be a highly unlikely numerical coincidence in effect: this is enough for serious physicists to build new theories. And sometimes it works!

Johann Jakob BalmerThe guy in the picture on the right, Johann Jakob Balmer, got his name in all textbooks because of discovering the ratio (in the Latin sense) of the measured hydrogen emission lines. He was no crackpot, but in earnest all he did to become textbook famous was finding out that the wavelength of Hydrogen lines in the visible part of its emission spectrum could be obtained with a simple formula involving an integer number n -none other than the principal quantum number of the Hydrogen atoms.

So, is it a vacuous occupation to try and find out the underlying reason -the ratio- of the Koidé mass formula or other coincidences ? I think it only partly depends on the tools one uses; much more on the likelihood that these observed oddities are really random or not. And, since a meaningful cut-off in the probability is impossible to determine, we should not laugh at the less compelling attempts.

As far as the numerical coincidence I quoted above is concerned, you might have guessed it: it is no coincidence! Greg Landsberg explains in a footnote to the paper I quoted above that one could in fact demonstrate, with some skill in algebra, that

“It turns out that in the base-N numerical system the ratio of the number composed of digits N through 1 in the decreasing order to the number obtained from the same digits, placed in the increasing order, is equal to N-2 with the precision asymptotically approaching 10^{-N}. Playing with a hexadecimal calculator could easily reveal this via the observation that the ratio of FEDCBA987654321 to 123456789ABCDEF is equal to 14.000000000000000183, i.e. 14 with the precision of 1.3\times 10^{-17}.”

Aptly, he concludes the note as follows:

“Whether the 10^{-38} precision needed to fine-tune the SM [Standard Model] could be a result of a similarly hidden principle is yet to be found out.”

Ah, the beauty of Math! It is so reassuring to know the absolute truth on something… Alas, too bad for Godel’s incompleteness theorem. On the opposite side, whether one can demonstrate that the Florida elections were fixed, it remains to be shown.