The Extended Koide's Formula Two Years Later

Almost exactly two years ago, I blogged about a paper by A. Rivero that used a very small number of inputs which have been measured with precision and a couple of key generalizations of Koide's formula for charged leptons to predict the quark masses.

Koide's formula provides that the square of the sum of the square roots of the three respective charged lepton masses, divided by the sum of the charged lepton masses, was equal to exactly two-thirds, a formula that has held true for decades, despite every more precise measurements of the charged lepton masses, which are now known to about seven significant digits of accuracy.

A generalization of the formula for quarks proposed that:

(1) the sum of the masses of the three charged leptons was equal to precisely three times the sum of the strange quark, charm quark and bottom quark (a natural multiple in light of the fact that each quark comes in three different colors while each charged lepton comes in only one, a fact reflected in W and Z boson decays); and

(2) each three quarks which are sequential in mass (u-s-c, s-c-b, c-b-t) form a Koide triple that obey the rule for quarks.

These assumptions produced the following results, to which I add the current Particle Data Group values for the state of the art experimentally measured values for the quark masses (in the same units of the original blog post), and a conversion of the difference between the extended Koide formula calculation and the experimentally measured values into standard deviations from the measured value to two significant digits (ignoring the negligible margin of the error due to uncertainty in the electron and muon masses in Koide's formula calculations which are for all practical purposes exact).

Inputs
me = 0.510998910 MeV ± 0.000000013 (i.e. one part per 39,307,608)
mμ = 105.6583668 MeV ± 0.0000038 (i.e. one part per 2,780,483)

Outputs
m = 1776.96894(7) MeV (Tau) - PDG 1776.82 +/- 0.16 (i.e. one part per 11,105) (0.93 SD)
mt = 173.263947(6) GeV (top) - PDG 173.070 +/- 0.888 (i.e. one part per 194) (0.22 SD)
mb = 4197.57589(15) MeV (bottom) - PDG 4180 +/- 30 (i.e. one part per 139) (0.58 SD)
mc = 1359.56428(5) MeV (charm) - PDG 1275 +/- 25 (i.e. one part per 51) (3.38 SD)
ms = 92.274758(3) MeV (strange) - PDG 95 +/- 5 (i.e. one part per 19) (0.55 SD)
md = 5.32 MeV (down) - PDG 4.8 +/- 0.4 (i.e. one part per 12) (1.3 SD)
mu = 0.0356 MeV (up) - PDG 2.3 +/- 0.6 (i.e. one part per 4) (2.26 SD)

Koide ratios of PDG mean values of selected triples is as follows:
t-b-c   0.6695
b-c-s  0.4578
c-s-u  0.622
s-c-d  0.60563
s-u-d  0.564

Best fits of any given quark triple to Koide's ratio given the range of experimental error for each input is considerable better.

Successes

The original Koide's formula for charged leptons remains true for decades without modification to within the margin of experimental error despite the fact that the least accurately known of the masses, the tau mass, is known to a one part per 11,105 precision and was known far less accurately when Koide's formula was proposed.

The extended formula, with just two experimentally measured inputs, post-dicted the mass of the tau to within one standard deviation, and the three down type quarks to within 1.3 standard deviations with an average standard deviation difference from the current mean experimental value of 0.81.  The formula also predicted the top quark mass very accurately.  The predicted ratio of the strange quark mass to the down quark mass is also within one standard deviation of the experimentally value.  All of these post-dictions would be treated as experimental confirmations of the theory if it were part of the Standard Model.

The accuracy of the Koide's formula prediction on the basis above of the tau, top, strange and down quark masses, moreover, has grown greater rather than less accurate as the precision of the experimental measurements of these quantities has improved.

The triple with the best fit of the mean value PDG data to the extended Koide's formula is the t-b-c triple which has the virtue of being the most precisely measured on a percentage basis and of being a decay path that is extremely dominant because almost every top becomes a bottom and a very high percentage of bottom's in turn become charms rather than ups or tops.

The runner up, the charm-strange-up decay chain is also a quite consistent one.  Charm to strange to up is far more common than charm to strange to top, or charm to strange to charm.

Tensions Between Koide's Formula and The Measured Results

There is more tension between the post-diction in the masses of the charm quark and up quark.

The Charm Quark Mass

The charm quark measurement of absolute charm quark mass is off by the greatest amount in terms of standard deviations of error (and even more if the precision of new charm quark measurements are accepted).  The deviation from the experimental value in absolute terms is about 6.6%.

Some of this deviation may flow from the way that the 3-1 mass ratio of charged leptons to the s-c-b triple is implemented which may be an imperfect and merely coincidental relationship.  A global fit to the experimental data can be almost perfectly consistent with the extended Koide's formula from the PDG data can be made by using a 1 SD low top mass, a 1 SD high bottom quark mass, and a 1 SD high charm quark mass.

Similarly, the u-s-c triple can fit values within 1 SD (high) of the charm quark mass and about 0.5 SD (low) of the strange quark mass, although it again produces a negligible up quark mass.

The Up Quark Mass

The absolute up quark mass measurement is off by fewer standard deviations (almost within the two standard deviation theory confirmation range) and just 2.3 MeV - less of an error in terms of absolute number of eVs of error than the accuracy to which any of the second or third generation quarks are known.

But, the magnitude of the difference between the Koide's formula predicted value and the measured value is sixty-five fold.  Likewise, the ratio to the down quark mass to the up quark mass (0.38-0.58 according to the PDG with margins of error of 5%-20% in the various measurements that contribute to the global average) is off by 747 standard deviations (the similarity to the model number of the largest commercial aircraft in regular service is just a coincidence).

The average of the up and down quark mass is about 4 standard deviations below the PDG summary of the experimental data, due to the low up quark mass prediction.  If the up quark mass were the experimentally measured value, the average of the up and down quark masses would be 0.44 SD from the mean).

The experimental measurement of the up quark mass is the least accurately measured of the experimentally measured masses of the Standard Model other than the neutrino masses, and while the absolute neutrino mass and mass hierarchy of the neutrino masses aren't known with certainty, the differences in mass between the three neutrino mass eigenvalues is known much more precisely than the differences in mass between quark flavors.

It is tempting to think that the negligible but non-zero up quark mass is correct.  The techniques used to estimate the up quark mass are rather crude relative to the quantity measured.  And, this would provide the added benefit of solving the strong CP problem because CP violations in strong force QCD interactions are naturally suppressed by a negligible up quark mass with resorting to fine tuning of the chiral quark mass phase in the QCD Lagrangian or requiring the introduction of new particles such as axions.

Where do we stand?

The very simple extended Koide's formula closely approximates all of the charged fermion masses from just two charged lepton masses which have been measured with great precision.  Even the two masses it gets wrong are tolerably close to the experimentally measured values for many purposes.

For example, the experimentally measured absolute values of the up quark and charm quark mass have only been known precisely enough to contradict the extended Koide's formula at a more than two standard deviation level for less than two or three years.  No other theory predicts the charged fermion masses so accurately with so little fine tuning.

But, Koide's formula is also wrong in these two cases.

The fact that the simple extended Koide's formula reasonably approximates the texture of the Standard Model quark mass matrix, without any quark mass inputs at all, suggests that it is at least more or less on the right track, as a first order approximation.

Is there a way to make the extended Koide's formula more accurate where it errs without sacrificing throwing off currently correct predictions or sacrificing the elegance of the concept?

Assume that the extended Koide's formula does such a good job of predicting the charged fermion masses because it is doing something right as a first order approximation, but that the omission of next to leading order corrections is throwing off the result for the charm and up quarks materially, and my slightly influence the formula's predictions for the other four masses.

Could terms that are well motivated theoretically be added to the formula to refine it in a way that would not throw off the other terms?

I think that the answer is yes.  But, I can't say that I'm confident that I've found it.

Heuristically, I think that what Koide's formula reflects in the quark sector is a process whereby charged fermion masses (or charged fermion Yukawas, if you would prefer that level of analysis) are the emergent result of a dynamic balancing of the masses of particles that produce a quark of a particular type in W boson interactions, and the masses of particles that a quark of a particular type produces in W boson interactions.

For example, top quarks almost always decay into bottom quarks which overwhelmingly tend to decay into charm quarks.  Hence, the bottom quark mass represents a balanced average (in the general sense of intermediate value that can be computed by any of a number of means) between the top quark mass and the charm quark mass.  Likewise, bottom quarks tend to decay into charm quark which in turn tend to decay into strange quarks.

In the case of the charged leptons, the decay chain is particularly uncluttered.  Taus decay into muons or electrons (with almost equal probability), muons decay into electrons, and the reverse (an electron that becomes a muon or a tau, or a muon that becomes a tau) almost never happens.

Quark flavors mix much more readily than charged lepton flavors.  For example, while about 95% of the time, charm quarks decay into strange quarks, about 4.9% of the time they decay into down quarks and about 0.1% of the time a charm quark emits a W+ boson and becomes a bottom quark (conservation of energy permitting).

The mass of a down quark relative to that of a strange quark is negligible, and a somewhat less than 4.9% downward adjustment in the charm quark mass due to second order terms in an extended Koide's formula terms would bring the predicted value much closer to the experimentally measured value.

Similarly, using a u-s-c triple to determine the up quark mass omits the roughly 1.1 in a 1000 chance that an up quark will become a bottom quark (PDG mass 4,180 MeV), and the dominant possibility that an up quark will become a down quark.  Using a u-d-s triple likewise omits the bottom quark impact.  Crudely, this probability times the bottom quark mass would suggest an upward adjustment on the order of 4.6 MeV which is much closer in order of magnitude to the PDG value.

Both of these examples suggest that the next to leading order term adjustment ought to have a value roughly on the order of the most important particle mass that the triple omits times the probability of a transition to that kind of particle in the CKM matrix.

Now, the precise formulas to use to implement these changes is hard to work out.  Conceptually, for example, in the b-c-s triple, the notion would be to replace the bottom quark mass in the extended Koide triple formula with a probability weighted average of particles that could be transformed by a W- boson emission into charm quarks, and to replace the strange quark mass in the formula with the probability weighted average of particles that a charm quark could be transformed into by a W+ emission from a charm quark.

There is a fairly straight forward way to do this using Standard CKM matrix elements from the charm decays.  But, the way to do this for decays of particles that become a charm quark is less obvious (since the probability matrix into a charm quark isn't necessarily exactly unitary like the elements coming out of it in CKM matrix form), and less easy to get the proper inputs for since the Standard CKM matrix covers only up type to down type quark transitions and one has to properly determine the inverse down type to up type quark transition probability matrix to get it right.

I'm also not necessarily comfortable that it is correct to simply disregard conservation of energy considerations in doing the analysis, but I'm not sure how to integrate conservation of energy considerations if I didn't disregard them.

The other vexing aspect of the next to leading order terms is that since the extended Koide's formula sets forth a non-linear relationship between the three terms in the Koide triple, using the naive weighted average approach that I have suggested seems to unduly dilute the impact of the most important missing mass term.  Some trial and error efforts on my part suggests that the weighted average approach that I suggest, while it seems to makes sense, is not the right way to integrate the information about the CKM matrix probabilities and omitted masses that it should.

Using the extended Koide's formula in the original form to determine all quark masses, and adding in each case an adjustment equal to something like the mass of the most important quark not included in the extended Koide triple (i.e. the omitted quark) for the Koide triple used to determine the mass of the quark you are solving for, and then multiplying that mass times the square of the CKM matrix element which represents the probability of a transition from the quark you are solving for in the Koide triple to the omitted quark, produces a result closer to the experimentally measured values than a weighted average.

Something like the average of all of the possible adjustments could be used as the NLO term (two one possible, one when there is only one to make), rather than putting weighted averages in the extended Koide's formula itself, and seems to work even better.

This would give:

mt=172.743 GeV PDG 173.070 +/- 0.888 per t-b-c avg adj down with ts (0.16%) and td (7.52*10^-5)
mb=4193 MeV PDG 4180 +/- 30 per b-c-s adjusted down with ub (0.11%)
mc=1293 MeV PDG 1275 +/- 25 per b-c-s adjusted down with cd (4.9%)
ms=92.55 MeV PDG 95 +/- 5 per b-c-s avg adj up with ts (0.16%) and down with us (4.97%)
md= 5.12 MeV PDG 4.8 +/- 0.4 per s-c-d avg adj of up with td (7.52*10^-5) and down with ud (94.9%)
mu=4.60 MeV PDG 2.3 +/- 0.6 per s-u-d avg adj up with ub (0.11%) and up with us (4.97%)

Now, I'll be the first to admit that this seems to take too much art and too little science, and that it produces an up quark value that is a bit too high.   It ought to be possible to iterate the process so that adjusted values are then used to readjust the predictions numerically (or analytically), and to make the adjustments more elegantly.

But, the adjusted values do bring all of the formula values for quark masses (and the tau lepton) except the up quark to within 0.8 standard deviations of the experimental values and to the right order of magnitude in the case of the up quark - now off by a factor of 2 rather than a factor of 64.6 - much closer to the mark on a percentage basis - without any experimental inputs other than the electron mass, muon mass and several of the four parameter CKM matrix element values!  Thus, the formula comes very close to reproducing the Standard Model values despite dispensing with 7 of the experimentally measured parameters of the Standard Model (seven more of which, assuming the Dirac neutrino scenario, pertain only to neutrinos).

Before v. After Adjustments Experimental Standard Deviations Between Theory and Experimental Value
top  0.22 v. 0.368 SD
bottom 0.58 v. 0.433 SD
charm 3.38 v.  0.72 SD
strange 0.55 v. 0.49 SD
down 1.3 v. 0.8 SD
up 2.26 v. 3.83 SD

Since this adjustment approach does seem to be bringing the predictions closer to the experimental values overall in a way that has some sort of heuristic theoretical motivation, it may be on the right track.

It also supports the underlying theoretical notion that Standard Model fermion masses represent a balancing of source masses and decay product masses of a particle in a manner that reflects the relative likelihood of various possibilities as reflected in the CKM matrix.  In other words, fermion masses seem to fit a pattern that makes sense if they arise dynamically via W boson interactions.

UPDATE:  There is a new Koide paper out of New Zealand noted in this thread.  It has a preon hypothesis.