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 mk. 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 Vcd 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 Vst* 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.