# What is new with Koide sum rules?

1. Mar 13, 2018

### arivero

Hmm, I should avoid to type from the phone. Well, anyway, the point was that perhaps GUT scale is not relevant for Koide. It is amusing that the main argument that we have (had?) for GUT is another numerical coincidence, the one of the coupling constants, but there was nothing about coincidence of yukawas... at most, variations on the theme of Jarslkog and Georgi https://en.wikipedia.org/wiki/Georgi–Jarlskog_mass_relation.

Another problem for quarks is that the pole mass is not directly measurable. Worse, Koide formula seems to work better with MSbar masses. Taking as input 4.18 and 1.28 GeV, Koide formula predicts 168.9 GeV for the top quark, while taking the pole masses 4.78 and 1.67 the prediction goes off to 203.2 GeV. (we nail it with intermediate mixes, eg input 4.18 and 1.37 predicts 173.3). Note that we now suspect that the MSbar mass of the top has a very noticeable EW contribution; Jegerlehner says that it actually counterweights the QCD contribution.

Last edited: Mar 13, 2018
2. May 21, 2018

### ohwilleke

A new Koide paper:

3. May 25, 2018

### ohwilleke

Koide considers the possibility that his charged lepton rule could be a function of SUSY physics. https://arxiv.org/abs/1805.09533

4. Jun 29, 2018

### mitchell porter

While Strings 2018 convened in Okinawa, Koide gave a talk at Osaka University (PDF) reviewing very succinctly the nature of his relation, the contribution of Sumino, and the very latest theoretical ideas.

5. Jun 29, 2018

### ohwilleke

Thanks. The presentation is a riot! Such humor and humility.

6. Jul 16, 2018

### ohwilleke

I didn't know that Twitter links were categorically forbidden, even top flight newspapers use them now and a lot of worthwhile discussion among experts in the field also occurs by Twitter before it ends up being published if it is published at all. Surely there must be some appropriate way to note where other people are discussing an idea. The link isn't being used as a source of authority in this case, it is being used as a link to a discussion elsewhere, in much the same way that someone might link to another Physics Forum thread or a link to leaked information about an imminent announcement.

A skeptical lot. I don't think they give sufficient credit to the fact that Koide's rule was proposed in 1981 when it was a poor fit to the tau mass which has consistently improved for 37 years of increased precision in measurement (even from 2012 to 2018), or to the fact that the number of significant digits of match is high and consistent to MOE with data when it wasn't built to match existing data.

But, credit to them for getting to a lot of the key related articles quickly (Descarte's circle and quark mass relations) and hitting on some key points quickly.

-1 for the guy saying that 0.999999... is not equal to 1.

Is there merit to the analytic expression they reference? How accurate is it? How old is it?

Also, the other bit of numerology with the analytical expressions of the lepton masses in terms of the fine structure constant and pi was interesting.

If I knew Twitter links were forbidden across the board, I would have included a more direct sourcing by clicking through to the references therein and the references within the referenced material. It is a bit irksome not to know that in advance and have to recreate a reference. I would also urge the Mods to reconsider a category ban on Twitter links as a matter of moderation policy, and to make it more clear if it is to be a policy. Mostly I was simply trying to save myself the tedium of trying to type it a formula accurately using LaTeX.

The interesting series of formulas are for the ratio of the muon mass to the electron mass, of the tau mass to the muon mass, and of the tau mass to the electron mass which are compared using 1998 CODATA and PDG sources.

There are three expressions shared by the three formulas:

A = 1-4pi(alpha^2)
B = 1 + (alpha/2)
C = 1 + 2pi*(alpha/2) = 1+ pi*alpha

The muon mass/electron mass formula is (1/(2*pi*alpha2))2/3*(C/B)

It purports to have a difference of 1 in the 7th significant digit from the PDG value.

The tau mass/muon mass formula is (1/2*alpha)2/3*(B/A)

It purports to match a 5 significant digit PDG value.

The tau mass/electron mass formula is (1/4pi*alpha3)2/3*(C/A)

It purports to have a difference of 1 in the 5th significant digit from the PDG value.

For what it is worth, I haven't confirmed the calculations or the referenced CODATA and PDG constants.

PDG for the tau mass is 1776.82 +/ 0.12 MeV

Koide's prediction for the tau mass is 1776.968921 +/- 0.000158 MeV

This formula predicts a tau mass of 1776.896635 MeV, which is about 0.07 MeV less than the Koide prediction, although there might be some rounding error issues and I don't have a MOE for the formula number. I used the five significant digit estimate of the tau mass to electron mass ratio in the illustration, so a difference in the sixth significant digit could be simply rounding error.

What to make of Dirac's 1937 Conjecture?

Dirac's conjecture on the electron radius v. size of the universe being roughly the same as the fine structure constant v. Newton's constant is also intriguing.

The conjecture called the Dirac Large Numbers Hypothesis is discussed at Wikipedia here: https://en.wikipedia.org/wiki/Dirac_large_numbers_hypothesis

An analysis that explores the same thing with a bit more clear language is here: http://www.jgiesen.de/astro/stars/diracnumber.htm

A 2017 preprint with eight citations discusses it here: https://arxiv.org/pdf/1707.07563.pdf

A 2003 paper touches on it at https://www.jstor.org/stable/41134170?seq=1#page_scan_tab_contents

I didn't know that twitter links were categorically forbidden and would purge the ads if I knew how. It seemed a convenient way to link to an academically explored idea. Also, without the link the latest insights of very notable commentator, and mathematical physicist Baez are harder to present. If the latest commentary of leading scientists on scientific issues isn't acceptable to reference, it should be. Is it permissible to cut and paste a post from a Twitter thread by someone like Baez?

Baez notes that even though this coincidence holds at the moment, that we have enough data to know that the magnitude of Newton's constant has not changed that dramatically over the history of the universe.

Neutrino Mass and Koide?

By the way - do you have links to any of the Koide-ish neutrino mass papers? The mass measurements are quite a bit more constrained that they were then (with normal hierarchy strongly favored, some sense of the CP violating phase, pretty accurate relative mass differences and a fairly tight sum of three neutrino masses cap) so it would be interesting to compare. Plugging in all of those constraints you get:

Mv1 0-7.6 meV
Mv2 8.42-16.1 meV
Mv3 56.92-66.2 meV

The CP violating phase seems to be centered around -pi.

Which is more information than it seems because most of the Mv2 an Mv3 mass ranges are perfectly correlated with the Mv1 mass range.

One ought to be able to look at the Koide-ish neutrino mass papers (which flip a +/- sign IIRC) and numerically run through the allowed range for Mv1 to see what the best fit is and use that to make a prediction for all three absolute neutrino masses.

Never mind, found it: http://brannenworks.com/MASSES.pdf It puts a negative sign in front of the square root of Mv1 in the denominator and comes up with:

m1 = 0.000383462480(38) eV
m2 = 0.00891348724(79) eV
m3 = 0.0507118044(45) eV (I think this maybe an error in the original as it doesn't seem to be consistent with the Mv3 squared - Mv2 squared value predicted, I think it should be 0.05962528 . . .).

m22 − m12 = 7.930321129(141) × 10−5 eV2 ------ PDG Value 7.53±0.18 (a 2.22 sigma difference - i.e. a modest tension)
m32 − m2 2= 2.49223685(44) × 10−3 eV2 ------ PDG Value 2.51±0.05 (less than 1 sigma different)

There is no value of Mv1 which can make the Koide formula without a sign flip work. I tried to reproduce his calculation and came up with Mv1 of 0.31 meV using current PDG numbers for the M1-M2 and M2-M3 mass gaps, which isn't far off from Brannen's estimate.

Last edited: Jul 18, 2018
7. Jul 17, 2018

### nikkkom

I looked closely at Mills and his "hydrino" paper. Mills is a fraudster. I assume a deliberate one. Elaborate one, too - you need to look rather closely to find blatant inconsistencies in his formulas, but when I found a place where he said "this quantity needs to be imaginary, so just insert 'i' multiplier here", it is a dead giveaway. No actual honest scientist would ever do that. If by the logic of your theory something has to be imaginary, it must come out imaginary from the math. Inserting multipliers where you need them is nonsense.

His mass formulas you link to are probably constructed by trying combinations of fine structure constant, pi, and various powers of them until a "match" is "found". E.g. multiplying by (1-alpha) fudges your result by ~0,9% down. Multiplying by sqrt(1-alpha) fudges your result by ~0,3% down. Divisions fudge it up, etc. This way a "formula" for any value may be constructed.

Last edited by a moderator: Jul 17, 2018
8. Jul 18, 2018

### ohwilleke

On further review this is a 1998 formula from a rather disreputable source but may very well still hold.

I don't know anything about Mills personally, and honestly don't expect that his GUT theory is right. But, I think his lepton mass formulas are interesting even though they may very well be numerology and no more. Looking at ways that physical quantities can be closely approximated often adds insight, even if the phenomenological formula has no basis in underlying theory that has been established yet.

Even if he formula is nothing more than tinkering, the number of significant digits of agreement achieved with three fairly simple looking formulas (part of which is a common factor for all three) with only one physical constant and one only one common transcendental number is still an admirable counterfeit.

It is also proof of concept that it is possible that a first principles formula that simple that did explain the quantities from a theoretical basis using only coupling constants could exist, even if it turns out that this isn't the one that is actually supported by a coherent theory. There are a great many quantities for which this is not possible even in principle.

Along the same lines, suppose that MOND is false that that we discover actual dark matter particles tomorrow. Any dark matter theory still needs to explain how it produces the very tight and simple phenomenological relationship between rotation curves and the distribution of baryonic matter in the universe that it does by some other means. The counterfeit or trial and error hypothesis can shed light on some feature of the true theory that makes it work.

Last edited: Jul 18, 2018
9. Jul 18, 2018

### ftr

suppose I give you this formula
proton electron mass ratio =3*(9/2)*(1/alpha-1) -1/3= 1836.152655 using codata for alpha
= 1836.1526734 using (1/alpha =137.036005 very close to average of codata and neutron Compton wave experiments base precision qed tests).

Can you say that this might have a physical basis or this is just a fluke. Is it possible to give probabilities for such and similar formulas.

Last edited: Jul 18, 2018
10. Jul 18, 2018

### arivero

11. Jul 18, 2018

### arivero

Are some of these relationships linked to koide formula? Can not tell. Perhaps the most promising, to me, is the mass of proton compared with the sum of electron, muon and tau. Three confined quarks vs three free leptons.

12. Jul 19, 2018

### Staff: Mentor

Using numbers from 1 to 9 plus e, pi and alpha, five different operations (+-*/^) and the option to take square roots, we have at least 10 options per operation. Even taking into account that multiple expressions can have the same result you would expect more than one additional significant figure added per operation. I count 7 in the above calculation plus one initial value. We would expect that we can get 8 significant figures just by random chance. And, surprise (?), we get 8 significant figures agreeing with measurements.

$\frac{e^8-10}{\phi} \approx 1836.153015$ - 6 significant figures (or 7 if we round) with just 3 operations.

13. Jul 19, 2018

### ftr

Ok, but relating two fundamental constants with simple numbers seems to be much more stringent, doesn't it.

14. Jul 19, 2018

### Staff: Mentor

You want the fine structure constant in?
$\displaystyle \frac{e^8-10(1+\alpha^2)}{\phi}\approx 1836.152686028$, an 8-digit approximation of 1836.15267389(17).

9 is not simpler than 8 and 10, an exponential is not very unnatural, and the golden ratio is always nice.

15. Jul 19, 2018

### ftr

16. Jul 19, 2018

### ftr

BTW does anybody know the whereabouts of Hans de Vries. Or why he drop out.

Last edited: Jul 19, 2018
17. Jul 20, 2018

### Staff: Mentor

I get 31.8 bits for 3*(9/2)*(1/alpha-1) -1/3 counting one bit for the 1 in "-1" and ld(5) for alpha. The approximation is good for 26.5 bits, worse than expected.
I get 33.8 bits for (e^8-10(1+alpha^2))/phi again counting the 1 as one bit and e and phi as ld(5). The approximation is good for 27.2 bits, similarly worse.
I get 20.7 bits for (e^8-10)/phi. The approximation is good for 22.4 bits.

The last one is the only one that beats the algorithm from @Hans de Vries you referenced. phi is too exotic? Okay, give it ld(20), then we are still at 22.7 bits for 22.4 bits, or equality.

18. Jul 20, 2018

### ftr

Although the bit calculation can be close, however, there are other considerations. For example the relation between the fundamental constants is very strong, i.e. one is the major bulk that makes the other in my equation (indicating a possible physics), in yours it affects the digits beyond the accuracy anyway, that is a very weak relation. Moreover, due to this consequence one constant is very sensitive to the accuracy you choose for the other(experimentally varying somewhat), hence the bit analysis accuracy problem. Also, if you reverse the formula, mine looks good, yours looks like ugly duckling .

Last edited: Jul 20, 2018