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.

Precision Of Top Quark Mass Measurement Improved

A new analysis combines all of the top quark mass measurement data from the CDF and D0 experiments at the now closed Tevatron collider and the ATLAS and CMS experiments at the Large Hadron Collider (LHC).

Bottom line: the mass of the top quark is 173,340 ± 760 MeV (combined, the error is ± 270 MeV statistical and ± 710 MeV systemic error). This is a precision of one part in 228 (i.e. ± 0.044%).

The top quark mass is the only quark mass that can be directly measured.  All other quark masses must be inferred from the masses of hadrons believed to contain those quarks in a model dependent manner.

While the top quark mass is the most precisely known quark mass on a percentage basis, is known with only slight less precision than the Higgs boson mass, and is known more precisely than any of the neutrino masses (the masses of the W boson, Z boson and charged leptons, as well as the Higgs vacuum expectation value ("vev") are known more precisely).

But, as explained below, about 61% of the uncertainty in the sum of the absolute value of the Standard Model fundamental particle masses (including the Higgs vev), and about 72% of the uncertainty in the sum of the square square of the Standard Model fundamental particle masses (including the square of the Higgs vev) is due to uncertainty in the mass of the top quark.  All but 6.3% and 0.85% of the balance of these uncertainties is due to uncertainty in the mass of the Higgs boson.

Thus, even a modest improvement in the precision of the top quark mass improves the overall precision of the measurements o the Standard Model fundamental particle masses considerably.

Previous Estimates of the Top Quark Mass

Previous Direct Estimates of the Top Quark Mass

The top quark mass is the only quark mass that can be directly measured.  All other quark masses must be inferred from the masses of hadrons believed to contain those quarks in a model dependent manner.

The previous best estimate based upon direct measurements from the Particle Data Group had been 173,070 ± 888 MeV (combined, ± 520 MeV statistical, ± 720 MeV systemic).

The new result is 270 MeV higher than the previous best estimate (0.3 standard deviations from the previous best estimate, which is unsurprising since the new result uses most of the same data as the old result and merely analyzes it more rigorously and precisely) and has a 14% smaller margin of error (with almost all of the improvement coming from a much smaller reduced statistical error in a pooled data set).

The final estimate of the top quark mass from Tevatron alone (CDF and D0 combined) has been 173,200 ± 600 ± 800 MeV.

Previous Top Quark Mass Global Fits

A previous global electroweak observable based fit including early LHC data in 2012 had come up with a top quark mass of 173,520 ± 0.880 MeV which was about 450 MeV more than the previous best estimate and about 180 MeV more than the current combined analysis. These global fits are based upon Standard Model relationships between the Higgs boson mass, top quark mass and W boson mass. These fits are most sensitive to the W boson mass, then to the top quark mass, and are least sensitive to the precise value of the Higgs boson mass.


The diagonal line shows the combinations of the top quark mass and W boson mass that are expected in the Standard Model at a Higgs boson mass of about 125,000 MeV, with the thickness of the line reflecting the range of uncertainty in that measurement. The latest estimate of the Higgs boson mass shift that line imperceptibly to the right, while the latest estimate of the top quark mass shift the center point of the 1 standard deviation confidence interval ellipse to the right by about a third of a hash mark.  The greatest impact of a global fit is to favor a low end estimate of the W boson mass of about 80,362 MeV, rather than the current best estimate from the Particle Data Group of 80,385 ± 15 MeV.

The Extended Koide's Rule Fit To the Top Quark Masses (and Other Quark Masses)

An extended Koide's rule estimate of the top quark mass using only the electron and muon masses as inputs, predicted a top quark mass of 173,263.947 ± 0.006 MeV, which is about 80 MeV less than the latest direct measurement.  This is within 0.1 standard deviations from the new directly measured value.

Prior to the new combined measurement, the extended Koide's rule estimate was 0.22 standard deviations from the measured value.

The fact that the extended Koide's rule estimate has become more precise than it was when it was devised, as the experimental value has been measured more precisely, is impressive.  Indeed, the extended Koide's rule estimate was closer to the new measurement than it was to the old one.

On the other hand, new precision estimates of the bottom and charm quark masses, mentioned below, increase the number of standard deviations in the gap between the experimentally measured values of these masses and the extended Koide's rule estimates for them (from 0.58 sigma to 3.57 sigma for the bottom quark, and from 3.38 sigma to 14.4 sigma for the charm quark).  In the case of the bottom quark, the extended Koide's rule estimate is about 0.7% too high.  In the case of the charm quark, the extended Koide's rule estimate is about 6.8% too high).

To four significant digits the t-b-c triple's Koide ratio which is predicted to be 0.6667 to four significant digits is 0.6695 at the PDG values and is 0.6699 using the new combined value for the t quark mass, and the new precision values for the b quark and c quark masses.  This is still a better fit than any of the other quark triples, although it is a less good fit than it was before the new precision measurements were reported.

The accuracy of the extended Koide's rule estimate for the strange, down and up quark masses is unchanged since there are no new estimates of these masses.  The strange quark estimate is 2.9% low (0.55 standard deviations), the down quark estimate is 10.8% high (1.3 standard deviations), and the up quark estimate is 98.5% low (2.26 standard deviations).

The original Koide's rule predicts a mass of the tau lepton from the electron and muon masses that is within 0.93 standard deviations of the currently measured value.

Other Standard Model Fundamental Particle Mass Measurements.

Other Quark Mass Uncertainties

All of the other measured values of the Standard Model quark masses are model dependent estimates based on QCD and hadron masses, and definition issues arise because a quark's mass is a function of the energy scale at which it is measured.  The pole mass of a quark is roughly speaking, the mass of a quark at an energy scale equal to its own rest mass and this is the number quotes for the bottom quark and charm quark as well as for the top quark.  The pole masses of the strange quark, down quark and up quark are ill defined and instead their masses in what is known as the MS scheme at an energy scale of 2 GeV (i.e. slightly more than the mass of two protons, two neutrons or one proton and one neutron), is normally used instead.

In the case of the bottom quark, the second heaviest of the quarks, the PDG estimate of the bottom quark mass has an uncertainty of 30 MeV (4,180 ± 30 MeV), but a new and consistent precision estimate using improved QCD approaches claims an uncertainty of just 8 MeV (4,169 ± 8 MeV). Similarly, in the case of the charm quark, the third heaviest of the quarks, the PDG estimate has an uncertainty of 25 MeV (1275 ± 25 MeV), but a new and consistent precision estimate using improved QCD approaches claims an uncertainty of just 6 MeV (1,273 ± 6 MeV). The best estimates of the up and down quark masses are about 2.3-0.5+0.7 MeV (about 25% precision) and 4.8-0.3+0.5 MeV (about 8% precision) respectively, but in each case the uncertainty in absolute terms is less than 1 MeV.

The strange quark mass (95 ± 5 MeV per the Particle Data Group) is known only to a roughly 5% precision.  The QCD approaches used to reduce uncertainties in bottom quark and charm quark mass produce estimates of only about 10% precision in the case of the strange quark mass.

Charged Lepton Mass Uncertainties

The measured tau lepton mass is 1776.82 MeV with an uncertainty in the mass of the tau lepton is 0.16 MeV.  The measured muon mass is 105.6583715 Mev with an uncertainty in the muon mass is 0.0000035 MeV. The measured electron mass is 0.510998928 MeV with an uncertainty in the electron mass is 0.000000011 MeV

Standard Model Neutrino Mass Uncertainties

The two differences in masses between the neutrino mass states are known to a precision of less than 10^-5 MeV and 10^-4 MeV respectively (the smaller one is about 0.007 eV, while the larger one is about 0.046 eV).

The absolute value of the lightest neutrino mass has been directly measured to be less than 2*10^-6 MeV (i.e. 2 eV), and is constrained in a model dependent way by cosmic background radiation and other astronomy measurements to be less than 10^-7 MeV (i.e. 0.1 eV).  If the neutrinos have a "normal" rather than "inverted" mass hierarchy lightest electron neutrino mass is probably on the order of 0.001 eV.

Gauge Boson Mass and Higgs vev Uncertainties

The newly discovered Higgs boson has a global best fit measured mass of 125,900 MeV with an uncertainty about about ± 400 MeV, which is just over half of the absolute value of the uncertainty in the top quark mass.  This is 79 MeV lower than the expectation if the Higgs boson mass is exactly equal to the W boson mass plus 1/2 of the Z boson mass (about 0.2 standard deviations from the measured value).

The accepted value of the Higgs vev is 246,227.9579 MeV with an uncertainty on the order of 0.001 MeV. This is measured via measurements of the lifetime of the mean lifetime of the muon (the Higgs vev is the inverse of the square root of two times the Fermi coupling constant). Both the Higgs vev and Fermi coupling constant are functions of the W boson mass and the weak force coupling constant, but the combined impact of these two factors is known much more precisely than the exact value of either of them.

As noted above, the uncertainty in the measured mass of the W boson is about 15 MeV (from a base number 80,385 MeV). The uncertainty in the Z boson mass is 2.1 MeV (from a base number of 91,187.6 MeV).

Why Is The Accurate Determination Of The Top Quark Mass So Important?

The Absolute Value of the Top Quark Mass Uncertainty Is The Largest In The Standard Model

In evaluating relationships between Standard Model fundamental particle masses, the uncertainty in the top quark mass is one of the dominant sources of uncertainty, because the top quark mass is the heaviest fundamental particle in the Standard Model and the absolute value of the uncertainty in this value is greater than that for any other Standard Model particle.

For example, the accuracy of the strange quark mass measurement is about 100 times less precise on a percentage basis than the top quark mass measurement, but the absolute value of the uncertainty in the top quark mass is still 760 MeV, compared to just 5 MeV for the strange quark mass.

The absolute value of the uncertainty in the top quark mass (760 MeV, which is about 61% of the total) is greater than the sum of the absolute values of the uncertainty in all of the other Standard Model fundamental particles combined (478.261 MeV, which is about 39% of the total), of which 400 MeV is the uncertainty in the Higgs boson mass and 78.261 MeV (which is about 6.3% of the total).

Thus, two fundamental particle mass measurements, one of which is just a couple of years old, account for 93.7% of all of the absolute value of the uncertainty in Standard Model particle mass measurements.

The Top Mass Squared Uncertainty Is Even More Dominant

The dominance of any imprecision in the top quark mass to overall model fits is further amplified in cases where the quantities compared are the square of the masses rather than the masses themselves (e.g. comparing the sum of squares of the Standard Model particle masses to the almost precisely identical square of the vacuum expectation value of the Higgs field).

About 72% of this imprecision is due to the top quark mass and about 99.15% of the imprecision is due to the top quark mass and Higgs boson masses combined.

The difference between the high end and low end of the square of the current combined estimate of various fundamental particle masses (i.e. their uncertainty) is as follows (in MeV^2, using PDG error bars for cases other than the top quark):

t quark mass squared: 526,953,600

All other Standard Model fundamental particles mass squared combined: 207,683,572 (about 28% of the total).

Higgs boson mass squared: 201,440,000

All other masses square except the t quark and Higgs boson masses: 6,243,572 (about 0.85% of the total).

W boson mass squared: 4,823,100
Z boson mass squared: 765,976
b quark mass squared: 501,600
c quark mass squared: 127,500
tau lepton mass squared: 22,497
s quark mass squared: 1,900
Higgs vev squared: 984.912
d quark mass squared: 7.84
u quark mass squared: 5.76
muon mass squared: 0.0015
electron mass squared: 0.0000000225
neutrino masses squared (combined): < 0.0000000003

A Final Prediction Re Top Quark Mass

1.  Assume that the W boson mass has its global fit value of 80,362 MeV, rather than its best fit measured value of 80,385 MeV (a 1.53 standard deviation shift).
2. Assume that the Higgs boson mass is exactly equal to the W boson mass plus one half of the Z boson mass (i.e. that it is 125,955.8 MeV) (a 0.14 standard deviation shift).
3. Assume that the Tau lepton mass has its Koide's rule predicted value of 1776.97 MeV, rather than the 1776.82 MeV value that it is measured at today (a 0.93 standard deviation shift).
4. Assume that the bottom quark has the precision value of 4,169 MeV (a 0.37 standard deviation shift)
5. Assume that the charm quark has the precision value of 1,273 MeV (a 0.08 standard deviation shift)
6. Assume that the sum of the squares of the masses of the fundamental particles in the Standard Model equals the sum of the squares of the Higgs vev.
7. Assume that the electron, muon, up quark, down quark, and strange quark have their PDG values and that the neutrinos have masses of less than 2 eV each.
8. Assume that there are no fundamental particles beyond the Standard Model that contribute to the Higgs vev.

What is the best fit value for the top quark mass?

Answer: 173,112.5 ± 2.5 MeV (a 0.29 standard deviation shift of 227.5 MeV from the newly announced value).

N.B.  The value of the top quark mass necessary to make the sum of the squares of the fermion masses equal to the sum of the square of the boson masses would be about 174,974 MeV under the same set of assumptions, about 174,646 MeV with a Higgs boson mass at the 125,500 MeV low end of the current 68% confidence interval for the Higgs boson mass, and about 175,222 MeV with a Higgs boson mass at the 126,300 MeV high end of the 68% confidence interval for the Higgs boson mass.  These are about 2.07, 1.65, and 2.38 standard deviations, respectively, from the measured value of the mass of the top quark, and thus is not grossly inconsistent with the evidence, despite being a less good fit the the Higgs vev contribution hypothesis (which I also find more compelling theoretically).  But, in order to achieve this, the sum of the squares of all of the fundamental Standard Model particle masses must be about 0.7% or more in excess of the square of the Higgs vev.

While both relationships are possible within experimental error, they cannot be true simultaneously despite being quite similar, at least for the pole masses.  It is possible to imagine some running mass scale where they might coincide, however, and if there is some energy scale at which this happens, this might be viewed as the energy scale at which fermion-boson symmetry (much like that of supersymmetry, but without the extra particles) breaks down.

The running of the charged lepton masses is almost 2% up to the top quark mass and 3.6% over fourteen orders of magnitude.  In contrast, the Higgs boson self-coupling runs to zero at the GUT scale, and the W and Z boson masses at high energies appear to be functions of the running of the electromagnetic and weak force coupling constants.  The electromagnetic force coupling constant gets stronger (from about 1/137 to 1/125 up to the electroweak scale) while the weak force coupling constant gets weaker.  This appears to be more dramatic than the running of the fermion masses, so the equalization of masses between bosons and fermions shouldn't require too high an energy scale.

Footnote:  The measured Higgs boson mass is very nearly the mass that minimizes the second loop corrections necessary to convert the mass of a gauge boson from an MS scheme to a pole mass scheme.

FLAG 19, LC & P, And The Extended Koide Formula

Abstract

As the quark masses and Higgs boson mass have been measured more precisely over the last few years at the LHC as summarize by the Particle Data Group through its 2018 data, and in lattice determinations using the quark mass measurements in the FLAG 19 report, it is worth re-examining two phenomenological relationships between them: the LC & P hypothesis and the extended Koide's formula for quarks.

The LC & P Hypothesis

The global LC & P relationship between the fundamental particle masses and the Higgs vev proposed in 2013 that the sum of the Yukawas of the fundamental particles of the Standard Model is exactly 1 still holds to within 1.3 sigma of the currently measured masses of the top quark and Higgs boson.

This hypothesis, if true, strongly constrains the parameter space of beyond the Standard Model massive particles that derive their mass from the Higgs mechanism to a mass range that has been thoroughly tested in a manner that rules out non-Standard Model particles of most kinds.

But, it is increasingly clear that the contributions of the fundamental fermions and the fundamental bosons are not equal, although this equality is still not entirely ruled out. The sum of the Yukawas of the fundamental fermions using the best available data is 0.4940 and the sum of the Yukawas of the fundamental bosons using the best available data is 0.5022. It takes a top quark value about 2.6 sigma from the measured value and a Higgs boson value about 3.3 sigma from the measured value to bring both of them to 0.5 exactly. 

The approximate symmetry between fermions and bosons in this relationship echos supersymmetry and may help explain why supersymmetric models can serve as approximations of Standard Model phenomena. This also suggests a new unsolved question in physics (that really isn't truly a problem any more than the hierarchy problem or naturalness problem are), which is:

"Why are fundamental boson masses slightly heavier relative to fundamental fermion masses than we might naively expect?"

The Extended Koide Formula

Rivero's extended Koide's formula from 2011 for quarks can be adjusted in a manner that I suggested in 2013 based upon the ansatz that the relative values of the charged fermion masses arises dynamically from the W boson interactions between these fermions and the fermions that they can be transformed into via W boson interactions. 

How does the adjusted extended Koide's formula for quarks compare to the experimental measurements?

Quark Type - Adjusted Extended Koide Mass - FLAG 19 Mass- PDG Mass (all in MeV)

top 172,743 v. 173,000 +/- 400 (-0.64 sigma) v. 173,000 +/- 400 (-0.64 sigma)

bottom 4192.98 v. 4,198 +/- 12 (-0.49 sigma) v. 4,180 +40-/-30 (+ 0.32 sigma)

charm 1293.21 v. 1,282 +/- 17 (+0.66 sigma) v. 1,275 +25/-35  (+ 0.73 sigma)

strange 92.274758 v. 93.12 +/- 0.69 (-1.22 sigma) v. 95+9/-3 (- 0.91 sigma)

down 5.32 v. 4.88 +/- 0.2 (+2.2 sigma) v. 4.7 + 0.5/-0.4 (+1.24 sigma)

up  0.0356 v. 2.5 +/- 0.17 (-14.5 sigma) v. 2.2 +0.5/-0.4 (- 5.4 sigma)

These values are consistent at or near the two sigma level, depending upon the measurements used for comparison purposes, with all of the quark masses except the up quark mass which is grossly at odds with the experimentally measured value.

So, while Rivero's extended Koide's formula is a good first order approximation for quarks based solely on the electron and muon masses, and my adjustments are a good second order approximation for quarks that incorporates CKM matrix data, at least one more layer of adjustment is needed to actually capture physical reality.

This also suggests a new unsolved question in physics (that really isn't truly a problem any more than the hierarchy problem or naturalness problem are), which is:

"Why does the up quark have more than a negligible mass?"

Note that even though these are merely phenomenological relationships based on some hunches about a possible underlying physical mechanism (and Koide and others have also proposed different underlying physical mechanisms), even if they are not accurate for the reasons proposed, they do give us a deeper understanding of the manner in which they quantities are related that is more than merely random.

Reductions in the uncertainty in the measurements of the top quark mass and Higgs boson mass that are likely to occur in the foreseeable future will greatly increase the extent to which these hypotheses are tested empirically in a convincing manner.

A detailed analysis of these issues appears below the fold.

Developments In My Own Koide's Rule Quark Mass Research

The masses of the charged leptons (the electron, muon and tau) observe a relationship that is exact to the extent of experimental limitations, which in the case of charged leptons is quite precise, which was proposed long before these masses were known with current experimental precision. The original Koide's rule for charged leptons applies to the familiar pole masses of those particles which always appear unconfined unlike the five non-top quark flavors of quarks. Indeed, more precise measurements of these masses have produced a better fit to this relationship, rather than a less precise fit.

The relationship, known as Koide's rule (which was proposed in 1982), is that the sum of the charged lepton masses divided by the sum of the square roots of the charged lepton masses, squared, equals exactly two-thirds.

A similar relationship has been observed with several "Koide triples" of quarks, modified by the provisio that the strange quark mass and the square root of the strange quark masses in one of the triples be assigned a negative sign rather than a positive sign, when applying Koide's rule. Brannen (2006) has proposed a similar relationship, with a negative sign before the square root of the electron neutrino mass but not the electron neutrino mass itself (cited in the article linked above).

Applying the extended Koide's rule to quarks, along with the fact that the sum of the masses of one of the quark triples and its "phase" appear to be three times that of the charged lepton triple (a result in harmony with the notion that there are three color varieties of each quark that count as separate particles in, for example, observed W and Z boson decay percentages), makes it possible to come up with a set of predicted quark masses that are a decent approximation of the experimentally observed quark masses with just two source data points - the electron mass and the muon mass.

But, these fits, while reproducing roughly the various quark masses, doesn't quite match the experimentally observed results.

Some of this may be a matter of different mass measurement conventions.

For example, consider the top quark-bottom quark-charm quark triple, one of the best fitting of the quark triples.  A fit based upon the charged lepton masses by the procedure as described above produces a prediction for the top quark mass that is quite close to its experimentally measured "pole mass", but is quite a bit in excess of its "MS mass".  According to Abazov (2011), an "MS mass" of 160.0 GeV for the top quark is equivalent to a "pole mass" of 167.5 GeV for the top quark.  The current global average measurement for the directly measured pole mass of the top quark, however, is 173.07 GeV +/- 0.89 GeV.  Thus, if the percentage difference between the two ways of reporting the top quark mass is roughly the same in this narrow mass window, the "MS mass" of the top quark is really about 165.32 +/- 0.85 GeV.  See also here (converting a 172.7 GeV pole mass for the top quark to a 163.0 MS mass for the top quark).

Application of Koide's rule to predict a top quark mass using the new precision estimates of the MS mass of the charm quark (1.273 +/- 0.006 GeV) and the bottom quark (4.169 +/- 0.008 GeV), in turn predicts a top quark mass of 168.25 GeV, and puts a nearly exact fit to Koide's rule (to less than one part per thousand) possible within 2 standard deviations of the experimentally measured MS mass value of these three quark masses.

Experimentally measured masses of the bottom quark-charm quark-strange quark triple likewise produce a reasonably close approximation of a two-thirds Koide ratio with the sign adjustment.  The result in absolute terms isn't far off for the charm quark-strange quark-up quark ratio either, although the predicted value for the up quark is on the order of 0.05 MeV when the experimentally measured value is about 2.3 GeV (more than four standard deviations from the PDG observed value), and a value very much out of line with experimentally determined ratios of the up quark mass to the strange and down quark masses, respectively.

Some of this discrepancy could be a function of the fact that light quark masses are customarily quoted at values associated with 2 GeV energy scales, rather than true rest mass values or pole mass values.  At a minimum, the use of conventional 2 GeV scale masses for the light quarks may introduce an inconsistency into the units of the inputs to an extended Koide's formula.

It is also hard to know how much or little faith should be given to current observational estimates of the light quark (up, down and strange quark) masses.  These are three of the least accurately measured parameters in the Standard Model so far, and small differences in absolute mass measured in MeV for these quarks translates into big differences in percentage precision, since the base values are low.  Naive Koide's rule fits favor a very light up quark which have the added attraction of naturally suppressing the strong force's CP violation rate without requiring that new particles like the axion be introduced to solve the strong CP problem.  And, a low value of the up quark makes about possible three Koide triples including one crossing quark-lepton lines, fit quite well, when the convention up quark mass estimate doesn't achieve that result.

I have not been terribly impressed with the theoretical frameworks that I have seen in the literature that have been used to determine these masses.

For example, all of the models use 2 or 2+1 flavor QCD models rather than full 5 flavor QCD models to make their determinations, while the extended Koide's rule suggests that the light quark masses have functional relationships to the omitted heavy quark masses (see, e.g. here).  Most of the models also assume equal up and down quark masses (and then use a mass ratio of the two quark flavors determined by other means to estimate individual light quark masses), and ignore electromagnetic effects.  The consensus does seem to rule out a zero up quark mass, but not necessarily a fairly low up quark mass.  Recent results in different papers by different investigators produce significantly different results despite having essentially the same experimental inputs.  For example, the extremes of the one sigma range for the up quark mass of the seven studies since 2006 relied upon by PDG range from 1.9 MeV to 3.1 MeV, and the significant uncertainty in the strange quark mass infects some of the estimates of the up and down quark masses (although not necessarily linearly).  And, there doesn't seem to be a clear narrowing of the predicted value over time.  The estimates now aren't all that different from what they were very early on in the formulation of the Standard Model.  A discussion of the theoretical considerations can be found in a PDG review paper.  This paper also notes at page 14:34, for example, that the light quarks are about 35% heavier at a 1 GeV energy scale than at a customary 2 GeV energy scale. So, energy scale conventions are quite important. A "pole mass" for the light quarks, to the extent that this is a concept that even makes sense given that light quarks are always hadronized into much heavier hadrons, would be heavier still.* In general, quarks get lighter at higher energy scales and heavier at lower energy scales.  The review paper also provides the exact formula for converting from MS masses to pole masses out to the three loop term which is a function of the first, second and third powers of both the strong force coupling constant and the energy scale of the interaction, as well as the value of the QCD Lagrangian quark mass parameter m_k. But, this isn't very precise. For example, in the conversion for the bottom quark, the third loop term is only about 1/3rd the size of the first loop term and about 1/2 the size of the second loop term. Many more loops would have to be calculated for a really precise conversion by that method.

Koide (1994) calculates the running of the three light quark masses down to their pole masses, even though these values have little practical application, in both a five quark and three quark flavor model. In the five quark flavor model he comes up with pole masses for the up quark of 346.3 MeV, for the down quark of 352.4 MeV and for the strange quark of 489 MeV. In a three quark flavor model he comes up with pole masses of 163.1 MeV for the up quark, 169 MeV for the down quark, and 338 MeV for the strange quark. One could get somewhat lower light quark pole mass values still in a two quark flavor model. Koide updated these calculations in 1997 and concluded that the pole mass of the up quark was 0.501 GeV, the pole mass of the down quark was 0.517 GeV and the pole mass of the strange quark was 0.687 GeV (based on their measured values at other energy scales), although all sub-1 GeV values were noted with an "*" mark. A more recent update of the calculations can be found at Xing (2008) (which does not consider masses running to very low energy scales for light quarks, explaining that "The pole masses of three light quarks are not listed, simply because the perturbative QCD calculation is not reliable in that energy region.").

Of course, the problem with these purely theoretical and basically unphysical values is that at those values, the sum of the quark content for the lightest mesons, the pions (ca. 140 MeV) exceeds the masses of the two light quarks that go into them. Light quarks do not appear in nature at energy scales as low as that of their own masses outside a hadron. But, this is a good indication that quark pole masses, which are completely unphysical in at least two cases, cannot be as fundamental as folks like Lubos Motl like to claim that they are, even though they have the attractive feature of being independent of any arbitrarily imposed scale. This does leave open, however, the question of at which scale it is most appropriate to apply mass formulas like an extended Koide's rule, when the seemingly most neutral choice (pole mass) turns out to be unphysical and absurd.

To take one example, suppose that we renormalize to a theoretical W boson mass value. Koide (1994) quotes quark masses at that energy scale in a five quark flavor model of u=2.45 MeV, d=4.33, s=87 MeV, c=653 MeV, b=3063 MeV and t=185 GeV (although it is particularly silly to apply a five quark model to masses for six quarks, and the c and b values may be below "rest masses" for those quarks and hence be unphysical as well), a scale that produces no good fits to a Koide ratio of two-thirds at all. In general, the fit of Koide's rule to reality is fermion mass running energy scale dependent. A general analysis of the relationship between Koide's rule and the running of fermion masses is discussed here by Xing (2006).

Similarly, one of the investigators whose 2011 paper was used by PDG uses to determine the up quark mass, also published a paper in 2012 concluding that the ratio of the charm quark mass to the strange quark mass is 11.27 +/- 0.39. Yet, given precision estimates of the charm quark mass of 1273 MeV, this would imply a strange quark mass of 108 Mev to 118 MeV (including the 6 MeV uncertainty in the charm quark mass as well as the uncertainty in the ratio of the masses) with a mean value of 113 MeV, which is substantially above the currently accepted value of 95 +/- 5 MeV, a 2.6 sigma discrepancy (update: to clarify their point after reading the paper, the 1273 MeV charm quark mass corresponds to a mass at the charm quark mass; at 2 GeV, they see the masses run to a 1.093(13) GeV MS mass which corresponds to a 97(3.6) MeV strange quark mass, which is closer to the conventional mean and illustrates how important it is to be clear about the conventions one is using when providing mass figures).  A paper coming up with a much lighter strange quark mass (and closer to the Koide formula preferred value) ishere.

The extreme one sigma range of the fifteen studies since 2006 relied upon by the PDG estimate for the strange quark mass range from 81 MeV to 128 GeV, and produce results in which some of the studies are inconsistent with the results of other studies that are relied upon(the extreme results differ by 2.96 sigma from each other and there may be bigger sigma discrepancies between closer values with lower claimed uncertainty). So, the true theoretical uncertainty of the theoretical strange quark mass may be meaningfully understated, and any error in this determination influences the correct result from 2+1 quark flavor masses for the up and down quarks.

On the other hand, there are plausible reasons within the heuristic framework that I've constructed, for the mere leading order prediction for the up quark mass to be one of the least accurate. Even fairly next to leading order term adjustments in this mass have the potential to greatly alter the predicted mass of the up and down and strange quarks.

I've observed that the higher the probability that the heaviest quark in a extended Koide rule quark triple will decay to the other two quarks in the triple, the more closely the Koide's ratio for the triple will tend to match two-thirds. In contrast, triples that have a low probability of being decay paths for the heaviest quark in the triple have Koide ratios that differ greatly from two-thirds. Dominant decay paths with roughly 90% probabilities or more are decent Koide fits, while decay paths with 10% probabilities are less are poor Koide fits. There is little middle ground, although paths involving down quark to up quark and strange quark to up quark masses may have similar probabilities in the middle ground region.

Moreover, when a Koide triple that is a decent fit, involves an up type quark and two down type quarks (for example, b-c-s), the discrepancy between the mass of the up type quark predicted from the masses of the two down type quarks and the experimentally measured value, is on the order of the probability of the up type quark becoming the missing down type quark according to the CKM matrix times the mass of the omitted quark (for example, the product of the square of CKM matrix element V_cd and the mass of the down quark).

Similarly, when a Koide triple that is a decent fit, involves a down type quark and two up type quarks (for example c-s-u), the fit is improved by multiplying the mass of the omitted up type quark times the probability of the down type quark transitioning into the omitted up type quark (for example, the product of the square of inverse CKM matrix element V_st* and the mass of the top quark).

Hence, a pure Koide triple based evaluation of the up quark mass, for example, might be too low because it omits a term that captures the effect of a very low probability up to bottom quark transitions that has a high weight due to the high bottom quark mass.

Thus, the data strong suggest that the extended Koide's rule for quarks using consistent mass units provides a leading order approximation of the quark masses that is most accurate to the extent it most completely captures the W boson quark flavor transitions that occur per the CKM matrix, and that some kind of next to leading order  approximation involving all of the quark flavors that a quark type may transition pursuant to flavor changing W boson interactions that the quark may engage in can improve that fit.

Alternately, the right approach may be to combine extended Koide's rule fits for all decay paths from the heaviest quark in the triple in a manner weighted in some fashion by a function of the probability of a particular decay path.

These scenarios fit the larger heuristic notion that Koide's rule is ultimately a function of the fundamental charged fermion masses being emergent in a way that involves the W boson somehow dynamically balancing the masses of the particles involved in the flavor changing transitions that it makes possible.

This said, I am hardly in the promised land of finding next to leading order and possible further terms of an extended Koide's rule fit to the quark masses that are a really excellent fit to experimental observation, and I have likewise failed to find a formula that does an excellent job of predicting CKM matrix elements from the quark mass matrix.  Very crude and simple applications of my proposals to generate next to leading order terms don't produced very good fits without using formulas that are hard to justify theoretically in any plausible way. There is certainly no shame in this lack of results. Koide's himself has been pursuing possibilities involving essentially the same research program for twenty years without success either.

On the other hand, the absence of other approaches in the literature that do as good of a job of formulating these within the Standard Model relationship and post-dicting the elements of the fundamental fermion mass matrix with any accuracy makes this line of analysis appear to me to be the most fruitful one to pursue at this time.

There are several reasons that this general line of research is so tempting, of course.

First, if one could find the right relationship, the number of experimentally measured free parameters in the Standard Model could be dramatically reduced, while the precision with which we know of all but a couple of the newly dependent parameters could be increased.

Second, the nature of the relationship discovered should shed light on the deeper structure embedded in the more fundamental theory for which the Standard Model is a low energy effective theory.

And, third, the fact that there is a formula that comes reasonably close to the right results, and is wrong when it is wrong in somewhat predictable ways, already suggests that the Standard Model parameters are not merely anarchistic and that we are on the right track to finding it.

Charm Quark and Bottom (a.k.a. Beauty) Quark Mass Definitions

Executive Summary

It is possible in principle to convert the MS bar mass of a top quark, bottom quark (a.k.a. beauty quark) or charm quark to the pole mass of that quark, in practice, the current state of the art formula for making that conversion introduces a theoretical uncertainty into the pole mass value of several hundred MeV. This is only about 0.1% of the mass of the top quark, but is about 3% for the bottom quark and about 9% or more for the charm quark. The MS bar mass determinations for the bottom and charm quarks are currently ten times as precise.

But, while the pole mass of a top quark can be determined directly, since it does not hadronize, the pole masses for all other quarks cannot be determined directly because they are always confined in hadrons.

The theoretical error introduced by the conversion factor undermines the immense progress that has been made in recent years in precisely determining the MS bar masses of the heavy quarks by perturbative QCD methods in recent years.

This is a problem because some important particle physics calculations that involve perturbative renormalization physics that involve high energies work best if you use the pole mass rather than the MS bar mass. So, if we are going to calculate the pole masses of bottom quarks and charm quarks, we still need to use the non-perturbative QCD methods of lattice QCD to determine them instead.

Background

There are multiple different definitions of the mass of a quark which are particular relevant in perturbative QCD calculations (i.e. calculations involving the Standard Model equations for interactions involving quarks and gluons at high energy scales). 

Perturbative QCD is less accurate to the point of becoming invalid at low energies, because the more precisely non-perturbative QCD effects that it ignores for mathematical expediency become significant at low energies and in circumstances like conversions from MS bar mass to pole mass. The math is too hard to calculate the full non-perturbative QCD equations analytically (i.e. exactly using algebra and calculus), however, so to make non-perturbative QCD calculations we have to make a discrete approximation to these continuous equations and work out a numerical approximation of the results using methods known as "lattice QCD."

Theoretically, the "pole mass" which is the rest mass of a free quark of a particular type at an energy scale equal to its mass, is the most "natural" definition of the quark mass and is important in some kinds of calculations. The normal definition of the masses of the leptons (electrons, muons, taus, and corresponding neutrinos) is equivalent to the pole mass.

But, pole mass can't be measured directly for any kind of quark other than a top quark, as all other kinds of quarks are found only found in hadrons (e.g. protons, neutrons, pions and kaons), rather than in a free state.

For these quarks, it is easier to use another mass definition, the most popular of which is the MS bar definition.

MS bar masses at an energy scale equal to MS bar mass are of the same order of magnitude as pole masses, and in principle can be converted according to an exact formula (which involves the sum of terms in an infinite series). MS bar masses are defined in a way that is always less than the pole mass for heavy quarks.

Pole Masses v. MS bar Masses for Heavy Quarks

In the case of the top quark, the conversion formula is very precise, converges rapidly, and has an uncertainty of only about one part per thousand.

The conversion formula works less well for the bottom quark (a.k.a. beauty quark), converging slowly and providing less precision, and works even less well for the charm quark where the formula diverges almost immediately. 

The first few terms of the formula are as follows, assuming a strong force coupling constant at the Z boson mass energy scale of the global average value of 0.118:

mt pole = (mt MS bar)(1 + 0.046 + 0.010 + 0.003 + 0.001 + · · ·)
mb pole= (mb MS bar)(1 + 0.096 + 0.048 + 0.035 + 0.033 + · · ·)
mc pole= (mc MS bar)(1 + 0.16 + 0.16 + 0.22 + 0.39 + · · ·)

Basically, the conversion for MS bar mass to pole mass introduces an uncertainty of hundreds of MeVs to the mass of the bottom and charm quarks (whose masses are about 4,200 MeV and 1,275 MeV respectively).

The good news noted in the paper discussed below, is that for processes with only virtual bottom and charm quarks, the more precisely determinable MS bar mass can be used, making the calculation of PDFs (parton distribution functions) which tell you what particles come out of a smashed hadron (often including particles other than the valence quarks of the hadron, for example, causing charm quarks to be emitted from a proton or neutron which has only up or down quarks as valence quarks) more accurate when this is possible.

The bad news is that if the process has an end state that includes bottom and/or charm quarks, there is no escaping the need to use the pole mass and the accuracy of the calculations cannot be improved by using the more precisely known MS bar mass.

The Paper

This is explained in Richard D. Ball, "Charm Production: Pole Mass or Running Mass" (December 12, 2016), and the abstract of the paper is shoddy to the point of being misleading, but the body of the paper tells the story well enough.

The introduction of the paper explains that:

Inclusive processes involving massive quarks are an important ingredient of LHC physics, not least because of their role in determining PDFs. Uncertainties in heavy quark masses can lead to substantial contributions to PDF uncertainties, and thus to uncertainties in predictions for LHC crosssections for a wide range of processes. 

Perturbative coefficient functions can be renormalized to depend on either the pole mass m or the MS running mass m(µ). The perturbative relation between them is now known to four loops, and for top the choice is essentially immaterial. However for charm and beauty nonperturbative corrections are more substantial, and while the MS charm and beauty masses can be determined rather precisely (to a few tens of MeV) through nonperturbative lattice or sum rule calculations, their pole masses are subject to large uncertainties (a few hundreds of MeV). 

Global PDF fits traditionally use pole masses. This is because the experimental observables used in the fits are generally inclusive, to avoid large uncertainties from final state effects. Heavy quarks in the final state are dealt with in the perturbative calculations by putting them on-shell: hadronisation corrections are then of relative order Λ/m, and thus power suppressed. The on-shell condition naturally leads to the mass dependence from heavy quarks in the final state being expressed in terms of the pole mass. In this short note, we will re-examine the possible use of the MS running mass in PDF determinations, and consider other ways in which uncertainties due to charm mass dependence might be reduced at LHC.

The conclusion of the paper explains that:

We have shown that, while for processes with only internal charm quark lines (such as processes with no charm in the final state, or semi-inclusive processes) it is straightforward to calculate using either pole mass or running mass in the hard cross-section, for inclusive processes with charm in the final state there no advantage to using the running mass, since the kinematics produces a nontrivial dependence on the pole mass which cannot be avoided without spoiling the factorized perturbative expansion. It follows that any empirical determination of the charm quark mass from inclusive charm production data has an intrinsic limitation due to nonperturbative corrections of a few hundred MeV. It is easy to see that these considerations generalise straightforwardly to inclusive hadronic processes such as W c or Z c-anti-c  production, and indeed to inclusive beauty production, though here the effect will be less significant. 

A number of recent perturbative determinations of the MS charm mass from inclusive data claim an uncertainty as small as 50 MeV, competitive with the nonperturbative results. The reason for this small uncertainty is probably the use of the theoretical assumption that charm is produced entirely perturbatively, which greatly increases sensitivity to the charm mass. However it is clear from the poor convergence of Equation (4) that perturbation theory close to the charm threshold is unreliable: charm production is subject to large nonperturbative corrections. Relaxing the assumption by fitting a charm PDF, significantly reduces the dependence of the PDFs on the charm mass. This in turn reduces the dependence of high energy cross-sections required at LHC on the charm mass, and thus increases their precision. It will also presumably increase the uncertainty on any empirical determination of the charm mass from inclusive data to a few hundred MeV.

Bottom Quark Mass And The LP & C Relation

The pole masses of the top quark, Higgs boson, Z boson, W boson, tau lepton, muon, and electron are relatively straight forward to measure directly, and are known to decent to excellent relative precision.

In contrast, the bottom, charm, strange, down, and up quarks are always confined in a hadron. This means that their masses can't be measured directly and instead have to be reverse engineered from hadron properties according to some self-consistent scheme, one in which the pole masses of isolated particles is not even necessarily well defined.

The most common scheme for determining the masses of the five less massive quarks is the MS-bar mass a.ka. the modified minimal subtraction scheme. But this isn't the only scheme for determining their masses. Another one is the "on-shell mass" of bottom and charm quarks which can be determined more or less exactly and to almost the same precision as the MS-bar mass upon which a lot of good data has been assembled. And, it more fundamental, and hence more appropriate to use for theoretical purposes (although the "on-shell" mass of the three lightest quarks is ill-defined). As explained in the introduction of the linked paper:

In perturbative QCD (pQCD) theory, two schemes are frequently adopted for renormalizing the quark masses, e.g. the on-shell (OS) scheme and the modified minimal subtraction (MS) scheme. 

The OS mass, also known as the pole mass, offers the advantage of being grounded in a physical definition which is gauge-parameter independent and scheme independent. It ensures that the inverse heavy-quark propagator exhibits a zero at the location of the pole mass to any order in the perturbative expansion. 

On the other hand, the MS scheme focuses solely on removing the subtraction term 1/ǫ+ln(4π)−γE from the quantum corrections to the quark two-point function. And by combining this with the bare mass, one can derive the expression for the renormalized MS mass.

In high-energy processes, the MS mass is preferred for its lack of intrinsic uncertainties. It has been found that for the high-energy processes involving the bottom quark, such as the B meson decays, when their typical scales are lower than the bottom quark mass, the using of MS mass becomes less suitable and the OS mass is usually adopted. 

Practically, the perturbative series using the OS mass is plagued by renormalon ambiguities, resulting in a perturbative series with poor convergence. Thus for precision tests of the Standard Model, accurate determination of the OS mass is important. 

It is noted that the OS mass can be related to the MS mass by using the perturbative relation between the bare quark mass (mq,0) and the renormalized mass in either the OS or MS scheme, where q denotes the heavy charm, bottom, and top quark, respectively.

The MS-bar mass of the bottom quark is 4.18 + 0.03 - 0.02 GeV. A new paper determines that this is equivalent to an on-shell mass of the bottom quark of 5.36 + 0.10 - 0.07 GeV. The new preprint that makes this conversion and its abstract are as follows:

Shun-Yue Ma, Xu-Dong Huang, Xu-Chang Zheng, Xing-Gang Wu, "Precise determination of the bottom-quark on-shell mass using its four-loop relation to the MS bar scheme running mass" arXiv:2406.18025 (June 26, 2024).

The on-shell mass of the charm quark (calculated somewhat less precisely) is 2.486 + 0.126 - 0.109 GeV.

Why care which definition of quark mass is used?

One reason is that this is relevant to a hypothesized relationship between the masses of the Standard Model fundamental particles and the Higgs vacuum expectation value (Higgs vev) known as the LP & C relation. 

This hypothesis holds that the sum of the square of the fundamental particle masses is equal to the square of the Higgs vev. This is equivalent to saying that the Higgs field Yukawas of the fundamental particles in the Standard Model add up to exactly one.

Putting best fit measurements into this formula comes up just a little short in a way that is principally due to the top quark mass being too light, when MS-bar scheme running masses are used for the other five quarks.

The current best fit measurement of the top quark mass is 172.690 ± 0.3 GeV.

But, to make the LP & C relation work with the best fit masses of all of the other fundamental particles, the preferred value is 173.615 GeV, which is about a 3.1 sigma tension.

If the on-shell mass of the bottom quark is used instead, however, this eases up these tensions somewhat. 

For example, fitting the top quark mass alone requires a top quark mass of 173.583 GeV when using the on-shell mass of the bottom quark, which is a little bit less than a 3.0 sigma tension. Using the on-shell masses of the charm quark as well would require a top quark mass of 173.570 GeV, which is a bit more than a 2.9 sigma tension.

Another way to make the numbers fit while reducing the tensions for the mass measurements of individual particles is to use top quark masses and Higgs boson masses. In the case of the Higgs boson, the current world average mass is 125.25 ± 0.17 GeV.

If both the top quark mass and Higgs boson mass are increased above their best fit measurements by equal numbers of standard deviations, which reduces the tension to about 2.6 sigma, and that could be reduced to a tension of about 2.5 sigma or less for both the top quark and the Higgs boson, using on-shell masses, which is significantly more mild than a 3.1 sigma tension in the top quark mass.

With all three adjustments, the LP & C relation could fit with a top quark mass of 173.44 GeV and the Higgs boson mass of 125.675 GeV, neither of which is a huge stretch, which means that the LP & C relation is still a viable theory, even though it is not perfectly consistent with the latest mass measurements.

Using the MS-bar mass rather than the on-shell masses of the three light quarks turns out to be immaterial in evaluating the LP & C relation, in which the uncertainties are dominated by the uncertainties in the largest absolute fundamental particle masses.

Alternatively, the LP & C relation could hold because the list of Standard Model fundamental particles is not complete, in which case it estimates the sum of the square of the masses of the missing fundamental particles, subject to the relative uncertainties in the known fundamental particle masses, in a global test of the completeness of the Standard Model.

The best fit to this gap, if concentrated in a single particle, would be a particle with a mass of about 17.5 GeV, but with a great uncertainty, mostly due to the uncertainties in the top quark and Higgs boson masses, and to a lesser extent the W boson mass uncertainty.

The trouble is, of course, that this mass range is well-explored, has produced no fundamental particles in this mass range, and would wildly throw off the observed branching fractions of the Higgs boson, Z boson, and W boson if it did exist, so it probably doesn't.

On-shell masses also make sense to use when exploring generalizations of Koide's rule to quark masses.

The Latest Charm Quark Mass Measurement

A new analysis combining experimental data, and a combined lattice QCD and QED analysis, has produced the latest measurement of the charm quark mass adjusted to the energy scale of the charm quarks own mass (i.e. it's "pole" mass) from a measurement of its mass at a 3 GeV energy scale (at which it is more than 20% less massive than at its pole mass, because quark masses decline at higher energy scales).

It determined that charm quark mass to be 1,272.3 ± 7.8 MeV.

This compares to previous global estimates from multiple sources previously reported at this blog as follows:

2018 PDG charm quark mass 1,275 +25/-35 MeV

2019 PDG charm quark mass 1,270 ± 20 MeV

FLAG19 charm quark 1,282 ± 17 MeV

The new measurement is consistent at the one sigma level with the 2018 and 2019 estimates from the Particle Data Group and with the 2019 estimate from the FLAG group, but purports to be more than twice as precise as any of those estimates.

The FLAG2019 paper provides a chart that graphically shows some of its source data for its global average estimate of the charm quark mass including the margins of errors involved. Notably, the outliers (which tend to run heavy) have the largest margins of error and tend to be older.

Theoretical Bench Marks

Rivero's extended Koide's formula from 2011 predicts a charm quark mass of 1,359.56428 MeV which has long been known to be too high, and lacks of the precision and accuracy of Koide's rule for charged leptons. But, comes pretty close for a formula generalizing the Koide's charged lepton rule with only the electron and muon masses as inputs.

My first order adjustment of Rivero's formula based upon the CKM matrix and quark masses not considered in Rivero's formula predicts a charm quark mass of 1,293.21 MeV, which is also probably too high (although it is within one standard deviation of the 2018 PDG value and the FLAG19 value, and within 1.2 sigma of the 2019 PDG value), but is significantly closer to the experimentally measured value. Probably, even if my first order adjustment of Rivero's formula is correct, it also needs at least a second order adjustment as well that I haven't worked out.

Is There An Electron, Up, Down Koide Triple?

Koide's Formula and Related Extensions

Koide's formula in its original form asserts that:

(sqrt(electron mass)+sqrt(muon mass)+sqrt(tau mass))^2/(electron mass+muon mass+tau mass)=2/3.

This is true to the highest levels of precision determined to date, which for the charged leptons is very great.

A Koide triple is any three sets of particle masses that satisfy that relationship.

The hypothesis that there are Koide triples among the quarks, which is not inconsistent with the data to current level of precision (which isn't very great) is that the following are Koide triples:

top, bottom, charm
bottom, charm, strange
charm, strange, down

A related observation is that the combined mass of the bottom, charm, strange triple is almost precisely three times the mass of the tau, muon, electron triple (a notion that corresponds to the fact that in weak force decays three times as many quarks, one for each color, are produced as leptons).

Koide's Formula, the Up Quark Mass and a Possibile Up, Down, Electron Triple.

Implications of zero mass or neutrino scale mass for up quarks.

The final conceivable triple following that patterns are charm, strange, up, and strange, down, up.  Koide's formula predicts a near zero value for the up quark mass from a c, s, u triple.  But, if that value is carried through to the down quark in the s, u, d triple, it produces a value within the measured range of the down quark mass.

Using central values of t=172.9 GeV (a hair low with the latest data) and b=4.19 GeV. Then,
Koide(t,b,c) implies c=1.356 GeV (PDG value 1.180-1.340 GeV)
Koide(b,c,s) implies s= 92 MeV (PDG value 80-130 MeV)
Koide(c,s,u) implies u= 36 KeV (PDG value 1,700 to 3,100 KeV)
Koide(s,u,d) implies d= 5.3 MeV (PDG value 4.1-5.7 MeV)

You can also form a Koide triple of an electron, up and down if you use an electron mass of about 0.511 MeV, an up quark mass of zero, and a down quark mass of 6.7 MeV. 

And, if you use a value of zero rather than 36 KeV for the up quark, and use the 6.7 MeV value for the down quark predicted by the electron, up, down triple, the formula predicts a strange quark mass of 92 MeV. 

This strange quark mass derived from the electron, up, down triple and the assmption that the up quark has a zero mass is consistent with the experimentally measured mass value of the strange quark, is consistent with a "top quark down" calculation of the strange quark mass, and is consistent with an estimate based upon a mass for the bottom, charm, strange triple that is the charged lepton mass triple.

Even a modest mass of 36 KeV for the up quark makes a significant different in the estimated value of the down quark mass via an electron, up and down Koide triple or a strange, up and down Koide triple.  But, an up quark mass on the order of magnitude of 1 eV or less does not throw off the Koide triple by more than can be easily made up with tiny tweaks to calibration points elsewhere. 

This is important because there are a variety of theoretical reasons why an up quark with a non-zero but negligible rest mass, even if it was just 1 eV, would involve a far more modest tweak to the Standard Model than a truly zero mass up quark.

The Koide's formula's prediction does not alter the experimentally estimated combined up and down quark mass.

The 6.7 MeV estimate for the down quark mass from applying Koide's formula naively is also not far from the Particle Data Group (PDG) mid-range value for the up quark and mid-range value for the down quark mass combined, which is 7.3 MeV.  The sum of the lower extremes of the PDG estimates for the up and down quark masses is 5.8 MeV and the sum of the upper extremes of the PDG estimates for the up and down quark masses is 8.8 MeV.   Twice the PDG estimate of the mean up and down quark masses is 6.0 MeV to 9.6 MeV, a range within which the 6.7 MeV Koide's formula value fits comfortably.

Thus, the Koide formula predicted value for the sum of the up and down quark masses when the up quark is assumed to have a mass of zero is well within the PDG value.  Koide's formula simply allocates all of the combined mass to the down quark rather than assigning a mass to the up quark of 35% to 60% of the down quark mass.  It also does nothing to alter the longstanding assumption based largely on the fact that the proton is lighter than the neutron, that the down quark is heavier than the up quark.

Reconsidering the experimental estimate of the up quark mass.

Keep in mind that up quarks are always confined and can't be measured in isolation the way that top quarks  can be, and that almost all of the mass in hadrons (two quark mesons and three quark baryons) is derived from the strong force binding energy carried by gluons and not from the quarks themselves.  This is particularly true in the case of hadrons that have only up and down quarks like the proton and the neutron for which the measured hadron masses that contribute to the estimates are most precise.

Since up quarks are always confined, any estimate of the up quark mass is necessarily model dependent.  Yet, computations of quantities like the proton or neutron mass from first principles using QCD alone have a precision of only about 1%, making them far less precise than the experimentally measured masses of hadrons.

Also, the experimental uncertainty in the mass of all quarks except the up quark equals or exceeds the low end experimental value per the PDG of the up quark mass.  So, in hadrons with some quarks other than up quarks in them, the up quark number has an impact on the total mass which is generally lower than the total uncertainty in the fundamental quark mass contribution to the hadron's total mass.

The strength of the strong force, weak force, electromagnetic forces are so great at the scale of a hadron relative to the masses of the quarks involved in all but the most exotic hadrons with heavy quarks in them, that an up quark's color charge, weak isospin and electromagnetic charge all have more relevance to its behavior when confined in a hadron than its fundamental mass (except insofar is its mass influences its weak force decays).

Obviously, if the Koide's formula prediction conflicts with the experimental data then there is simply something wrong with the formula.  But, the existence of consistent predictions from an electron, up, down triple with those of series of quark triples, and with the predictions of quark masses from the masses of the lepton triple all argue for revisiting the model dependent assumptions that went into making the PDG estimate of the up quark's mass (which is in any case has extremes that vary by a factor of two anyway).

Figuring out how the up quark mass was estimated and what practical implications the up quark mass has in the Standard Model is clearly near the top of my to do list.

Implications of a zero mass for the up quark.

If the up quark mass were assumed to be zero, as a non-measured Standard Model constant, rather than an experimentally measured one, and the other quark masses were estimated based upon this model dependent assumption, how would the estimated quark masses different and what experiments, if any, that were the basis for the PDG estimate would be contradicted?

Some of the issues of how up quark mass is determined and what this implies in practice when doing QCD are discussed in this 2004 paper and another paper in 2010 and in 2011 by the same author, Michael Creutz who together with the authors of this 2002 paper are interested in the possibility that a massless up quark could explain the strong CP problem. This 2003 paper (possibly identical) also investigates the possibility of a massless up quark and makes a mass calculation for the up quark using lattice QCD.  This paper from 2001 disfavors that solution in a model limited to two quarks (the 2002-2003 analysis is a three quark flavor analysis).

This 1997 paper gets into the guts of mass renormalization for quarks.  A 2009 model dependent estimate of the up and down quark masses shows how these quantities are derived in QCD. A 2011 paper uses the up-down mass difference and applies it to neutrino scattering. This 2011 paper discusses relevant source data in the context of a BSM Higgs mass generation idea.

 (I've omit papers by Koide himself in this review).  Thinking similar to that of Koide's on mass matrixes is found in a 2013 paper and in this 2012 paper and this 2012 paper.  A BSM model from 2011 explores similar ideas.  A 1999 paper considers implications for quintessence theories.

Current light quark mass ratio estimates don't differ materially from those devised by Weinberg and discussed in this 1986 paper whose abstract stated:

We investigate the current-mass ratios of the light quarks by fitting the squares of meson masses to second order in chiral-symmetry breaking, determining corrections to Weinberg's first-order values: m_u/m_d=0.56, m_s/m_d=20.1. We find that to this order, m_s/m_d is a known function of m_u/m_d. The values of the quark-mass ratios can be constrained by limiting the size of second-order corrections to the squares of meson masses. We find that for specific values of presently unmeasured phenomenological parameters one can have a massless u quark. In that case 30% of the squares of meson masses arise from operators second order in chiral-symmetry breaking.
 

A 1979 estimate is also not that different in its early estimation of light quark masses as is this 1996 paper or a 1994 paper.  The 1994 paper's abstract stated that: "the claim that

mu = 0 leads to a coherent picture for the low energy structure of QCD is examined in detail. It is pointed out that this picture leads to violent flavour asymmetries in the matrix elements of the scalar and pseudoscalar operators, which are in conflict with the hypothesis that the light quark masses may be treated as perturbations."

This 1978 paper's abstract states:

We consider, within the framework of current algebra, the possibility that the up-quark mass vanishes (as an alternative to the axion). We argue that the contrary current-algebra value, m_u/m_d=1/1.8, is unreliable. A critical analysis leads to the conclusion that m_u=0 is not unreasonable and furthermore leads to a surprisingly good prediction for the δ-meson mass.

A massless up quark has been considered a viable option seriously considered since 1978.  It was considered an open possible solution to the strong CP problem in 1994 it found that:

We conclude that at the level of precision (order of magnitude) of nonperturbative QCD calculations available to us at present, low-energy phenomenology is completely compatible with a vanishing value of the high-energy up quark mass.

Only a nonperturbative calculation in QCD can prove or disprove the phenomenological viability of mu = 0. Therefore, in view of the recent progress in numerical methods in lattice gauge theory, we would like to encourage a detailed analysis of the possibility of a massless up quark by these methods.
 

A 2000 paper considered ways to test the massless up quark hypothesis using lattice QCD methods and does this 2002 paper which finds that lattice calculations disfavor a massless up quark but that experiments don't resolve the issues apart from a first principles analysis.  A 2001 paper notes that useful theoretical QCD predictions can be done with massless quarks entirely.  A 2007 paper quantifies the impact of quark mass on QCD predictions from massless models.

A zero or non-zero mass for the up quark might help explain why proton decay is so surprisingly rare.

A zero mass for the up quark together with the extended Koide's formula that motivates it, would imply that the masses of the six quarks and all three charged leptons can be calculated via the extended Koide's formula (including the mass relationship of the charged lepton triple to one of the quark triples and the electron, up down triple) and the assumption that the up quark has zero mass from the mass of the electron using high school algebra to accuracies greater than those available for any of the experimentally measured quark masses (even the top quark whose mass is currently known to 0.6% accuracy). 

This would reduce the number of experimentally measured physical constants related to fermion mass in the Standard Model + Extended Koide Model from fifteen to four (the electron and the three neutrino masses).

If the supposition that the Higgs boson mass is equal to half of the sum of the masses of the W+, W- and Z bosons (which is currently accurate to within all current bounds of experimental precision and is closer to the experimentally measured mark than any of the prediscovery mass predictions for the Higgs boson mass), then the number of experimentally measured physical constants related to mass in the Standard Model would fall from three to two, one of which (the Weinberg angle that relates to W and Z boson masses) isn't even a mass value itself.

Thus, we could be on the verge of going from having eighteen measured Standard Model mass constants to having just six, and having much more accurate theoretical values than experimental values for many of those constants.

This would also motivate strongly a Koide derived formula for neutrino masses that if devised and confirmed by experimental evidence would cut the number of experimentally measured mass constants in the Standard Model from six to not more than five (one of which is an angle rather than a mass), and possibly to as few as three if a way to derive the neutrino masses from first principles using the masses of the other fermions and Standard Model bosons (and perhaps the PMNS and/or CKM matrix elements and/or the Standard Model coupling constants) was devised.

Extensions To A Standard Model With Four Generations

Extending the Koide's formula allows one to make useful, constrained and testable predictions regarding a fourth generation of Standard Model particles.

Fourth generation Standard Model particles that would have the masses a naive extension of Koide's formula would imply are experimentally forbidden because the lepton sector is inconsistent with experimental data.  This is a conclusion that has already been reached for the large part by the fundamental physics community already based on other grounds.

Fourth Generation Koide Quarks

If one extends the formula based upon recent data on the mass of the bottom and top quarks and presumes that there is a b', t, b triple, and uses masses of 173,400 GeV for the top quark and 4,190 for the bottom quark, then the predicted b' mass would be 3,563 GeV and the predicted t' mass would be about 83.75 TeV (i.e. 83,750 GeV). 

Since they would be produced a t'-anti-t' and b'-anti-b' pairs, it would take about 167.5 TeV of energy to produce a t' and 7.1 TeV of energy to produce a b'.  Producing a t' would be far beyond the capababilities of the LHC.  But, it could conceivably produce a few b' quark events of the Koide's formula predicted mass.  These would be unmistakeable unless the extreme speeds of the decay products prevented them from decaying (as a result of special relativity effects) until they reached a point beyond the most remote LHC detectors.  This probably wouldn't happen for a b' decay which is within the design parameters of the LHC, but might happen in the case of a fluke t' decay, which is far outside of its design parameters.

The up to the minute direct exclusion range at the LHC for the b' and t' is that there can be no b' with a mass of less than 670 GeV and no t' with a mass of less than 656 GeV (per ATLAS) and the comparable exclusions from CMS are similar (well under 1 TeV).

Koide t' and b' quark decays

A simple fourth generation b' quark or t' quark, that otherwise fits the Standard Model, of that mass would decay so rapidly that it woud not hadronize (i.e. not form composite QCD particles via strong force gluon interactions).  Instead, the t' would decay almost exclusively to the b' and the b'  would decay almost exclusively to the t, with both interactions happening almost instantaneously. 

A t' decay to a b' would produce a highly energetic W+ boson that would carry much of the energy of 80 TeV of rest mass being converted into  kinetic energy for the W+ and b' produced in the decay, immediately followed by a highly energetic W- boson produced in the b' to t decay in which about 3,390 GeV of kinetic energy was created from rest mass, followed by the usual immediate t quark to b quark decay with an emission of a W+ converting about 169.2 GeV of rest mass into kinetic energy for the W+ and b quark.  There would be an exactly parallel set of reactions for the decay chain of the anti-t' particle. 

This highly energetic emission and subsequent decays of the t' quark to a b quark would produce 3 W+ bosons and 3 W- bosons at three discrete and equal energy levels would all take place in about 10^-23 seconds.  This is because the lifetime of a t quark is 0.5 * 10^-24 seconds, the lifetime of a W boson is 0.3 * 10^-24 seconds, and the lifetime of a fourth generation Standard Model t' or b' quark would be less than that of the top quark (probably much, much less).  The bottom quark and a large share of the heavy decay products of the highly energetic W boson decays (such as b', c and b quarks, and tau prime and tau charged leptons and their antiparticles which have lifetimes of 10^-12 to 10^-13 seconds, with b and c quarks hadronizing into exotic and short lived hadrons before decaying further) would in turn decay by about the time that they had traversed a distance roughly equal to the distance from the center of a gold atom to its outmost orbiting electrons within about 10^-12 seconds.  Strange quarks decay in about 10^-10 seconds and muons decay in about 10^-6 seconds. 

In the absence of special relativity, this would take place within a sphere of a diameter of less than 10^-16 meters (i.e. about 1-2% of the diameter of a nucleus of a gold atom, a number derived from the decay time of 10^-23 seconds for the first three decays times the speed of light), the strange quark decays would start to happen about a foot from the original site of the decay, and the muon decays would peak about 300 meters away.  But, since particle decay takes place in the reference frame of the particle, which is moving at speeds near the speed of light, the decays would take place over a far more extended area because time would pass more slowly for the fast moving t' decay products.  The extreme kinetic energies of the particles would cause the their decays to happen at much greater distances from the initial t' production and decay site than the ordinary LHC decays - indeed they might make it past the detectors entirely. 

Also, while a 167.5 TeV event does involve a lot of energy in a concentrated place, a single event of that size involves only about 3*10-4 joules of energy, about the amount of kinetic energy of a single grape on the verge of hitting the ground after falling from a vine at waist height, so if it made it past the detectors due to special relativistic extensions of decay times it would be virtually invisible to observers in the area around the impact site and beyond the detectors.

So, even if the LHC was able due to a fluke fluctuation that led to a collision more than ten times as energetic as its design limitations with a spectacular decay chain, it might be missed entirely or almost entirely except as a completely unprecedented amount of missing energy that might be attributed to an equipment failure rather than a real physics event because it was so far beyond the designed detection range of the scientific equipment in the facility.

Fourth generation Koide leptons

The extension for charged leptons (a muon, tau, tau prime triple), however, would imply a 43.7 GeV tau prime, which has been excluded at the 95% confidence level for masses of less than 100.8 GeV and with far greater confidence at 43.7 GeV (which would be produced at a significant and easy to measure freuquency in Z boson decays).

A simple Koide's rule formula for neutrinos using the muon neutrino mass (of 7.5 * 10^-5 eV + 0.08 eV +/- 0.09 eV) and tau neutrino mass (of 2.4 * 10^-3 eV + 0.08 eV +/- 0.09 eV) (with absolute masses derived from accurately measured mass differences between types and a 0.51 eV limit on the sum of the electron neutrino, muon neutrino and tau neutrino masses if there are only three kinds of neutrinos - less if there are more generations of neutrinos), would yield a tau prime neutrino mass of far less than 43.7 GeV.  A naive extension of Koide's formula with an electron neutrino of near zero mass would lead to a fourth generation neutrino of about 0.05 eV and would have a mass of up to 11.6 eV in the nearly degenerate case where all three neutrino species had almost precisely the same mass.  But, this would contradict the cosmological data constraint that limits to 0.51 eV for the sum of the masses all of the species of light neutrinos combined (which is about 1/1,000,000th the rest mass of an electron).  So, instead, 0.51 eV would be the realistic upper limit of a Standard Model weakly interacting fourth neutrino generation.

Given the cosmological constraint on the sum of neutrino masses, the possibility that the naive Koide's formula needs a sign modification or something like it for neutrinos (which it probably does) is irrelevant.

Yet, any simple, fourth generation, weakly interacting tau prime neutrinos of any rest mass less than 45 GeV can be excluded on the basis of Z boson decays, so this scenario is definitively excluded if Koide's formula is even remotely an accurate way of estimating the mass of a hypothetical fourth generation Standard Model neutrino.

These theoretical considerations make it highly unlikely that there is any fourth generation of Standard Model fermions at all.  The Standard Model makes fermions an entire generation at a time, and this would require a fourth generation charge fermion far in excess of the Koide formula extension predicted value.