Why Isn't There A Fourth Generation Of Fermions?

One plausible estimate for the expected mass of fourth generation fermions would be to use Koide's rule to predict them, since Koide's rule provides nearly exact predictions for the charged lepton masses and quite accurate predictions for the heavier quark masses that are beyond current experimental bounds. But, the fourth generation charged lepton and neutrino present a more serious problem, as due the fourth generation quark decay time periods.

As explained in this March 13, 2013 blog post:

Koide's Formula  

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. 

Related Extensiona of Koide's Formula

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).  . . . 

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) . . .

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.  . . . 

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.

The CKM matrix probability of a third to fourth generation quark would have to be quite low. Other  transitions would be equal to the sum of the intermediate transitions which were possible at a given number of loops.

If there were a fourth generation of active neutrinos with masses in the range from 45 GeV to 62 GeV, they would have appeared in Higgs boson decays already and would have been quite obvious. There is no way that the other Standard Model particle decays could be so close to correct if there was an active neutrino that massive. 

So, if there is a fourth generation of Standard Model fermions, it must include a fourth generation neutrino with a mass in excess of 62 GeV, despite the fact that none of the other neutrinos in the Standard Model have masses in excess of 60 meV, a 12 order of magnitude gap, when the between generation gap in all of the other fermions in the Standard Model is less than 140 in all eight cases for which we have data.