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: mu/md=0.56, ms/md=20.1. We find that to this order, ms/md is a known function of mu/md. 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, mu/md=1/1.8, is unreliable. A critical analysis leads to the conclusion that mu=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.