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A Koide, neutrinos, and phenomenology

  1. Jan 6, 2016 #1
    Compared to the other fermions, I have always had much less interest in Koide mass formulas for neutrinos. There is less data, and neutrino mass works differently anyway (Dirac plus Majorana, whereas the other fermions are just Dirac).

    But today that has changed. First, today we have a paper that predicts Dirac and Majorana masses, using a Koide-like ansatz:

    Some New Symmetric Relations and the Prediction of Left and Right Handed Neutrino Masses using Koide's Relation
    Yong-Chang Huang, Syeda Tehreem Iqbal, Zhen Lei, Wen-Yu Wang
    (Submitted on 5 Jan 2016)
    Masses of the three generations of charged leptons are known to completely satisfy the Koide's mass relation. But the question remains if such a relation exists for neutrinos? In this paper, by considering SeeSaw mechanism as the mechanism generating tiny neutrino masses, we show how neutrinos satisfy the Koide's mass relation, on the basis of which we systematically give exact values of not only left but also right handed neutrino masses.

    And second, we have been discussing various minimal BSM theories, in which the masses of the right-handed neutrinos are constrained by astrophysics. We have discussed NMSM and vMSM, and I'd also like to mention this model which is SM + RH neutrinos + axion.

    With a completely predictive Koide-like ansatz and a definite physical framework, we can ask if the predicted numbers are compatible with the framework, yes or no. And that's progress.

    So we may be entering a time when there can be a tighter interaction between Koide ansatze, and BSM phenomenology.
  2. jcsd
  3. Jan 6, 2016 #2
    I've seen the Kiode mass formula before. It just seems like numerology at present, but its features are uncanny. Like the pi/4 angle from (1,1,1).

    Those right handed neutrinos are extremely massive: good dark matter candidates.
    Last edited: Jan 6, 2016
  4. Jan 8, 2016 #3
    Actually, in the typical seesaw model, superheavy neutrinos like that cannot be the dark matter, because they decay immediately into lighter particles. In the typical seesaw model, where those particles count is in the very early universe, when it's so hot and dense that no particle lives very long. Under those conditions, the short lifetime isn't a handicap with respect to physical relevance, and their presence in the mix can explain the subsequent matter/antimatter imbalance in the universe (this is the "leptogenesis" theory of where the imbalance comes from).

    But if you want the present-day dark matter to be made of right-handed neutrinos, you normally have to suppose that they are much much lighter than that. I am aware of precisely one model in which superheavy neutrinos are the dark matter, and it places them in an extra dimension in order to make them long-lived.

    Also, after thinking about this paper, I have decided that I find its way of generalizing the Koide formula to the neutrinos, to be unlikely.

    All the other Koide formulas involve Dirac masses. In the type I seesaw, the observed neutrino mass is a derived quantity, the smaller eigenvalue in a matrix whose elements are the Dirac mass and the Majorana mass. Those matrix elements are the more fundamental quantities. So both precedent, and logic, suggest that one should look first for Koide relations among the neutrino Dirac masses, and perhaps among the Majorana masses too.

    But in this paper, the hypothesis is that it's the seesaw eigenvalues which are connected in that way. I have found that François Goffinet pointed out the problem with this in his 2008 thesis. But it seems like Alejandro Rivero is the only one so far, to make a concrete proposal of the more "logical" kind.
    Last edited: Jan 8, 2016
  5. Jan 8, 2016 #4
    Is there any validity to the koide formula for leptons besides "this kind of works out nicely"? Also you mention other koide formulas. Are there more besides what is mentioned in this paper?
  6. Jan 9, 2016 #5


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    Actually I am a bit annoyed that all this papers keep quoting my old arXiv:hep-ph/0505220 and none of the new notes on the topic. I had hoped at least the people to click in the author names and find arXiv:1111.7232. A more recent note with a lot of references is my talk http://es.slideshare.net/alejandrorivero/koide2014talk on an online seminar (babling video of myself here: cosmovia)

    Last attempt to produce more formulae, by Koide itself, is Phys. Rev. D 92, 111301 (2015).
    Last edited: Jan 9, 2016
  7. Jan 19, 2016 #6


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    Empirically, Koide's formula is so right on the money in an area where the data are quite precise, and equally important the data have grown much more precise today than they were when it was devised (with the fit significantly improving), so I don't really have any serious doubts that Koide's formula for charged leptons is real. If there is an error in Koide's formula, my expectation is that it would be on the order of magnitude of the mass of the neutrinos relative to the mass of the charged leptons (one or two parts per million or so).

    Moreover, the extension of the formula to quarks is sufficiently close to suggest that the quark mass hierarchy derives from the same first order source as the charged leptons, but with an additional second order complication of some sort.

    A variety of hypotheses to explain this have been advanced and none has secured a consensus.

    I am personally partial to the (unorthodox) notion that Koide's formula arises from a balancing of the masses of the various particles that a fundamental fermion can transform into via W boson interactions, which would imply that the Yukawa couplings of the Higgs boson in a deeper theory arise dynamically rather than being fixed constants of Nature, but I don't claim any strong authority for that position other than that it seems amenable to producing a decent fit for the data. In this analysis, the reason that Koide's formula is so perfect for charged leptons, and so imperfect for some of the quarks, is that any given charged lepton can only transform into the other two charged leptons with 100% probability between them, while any given quark can change into three other possible quarks. An extension of Koide's formula for quarks fits very well, for example, to the top-bottom-charm triple where the probability of decays to the next quark down the chain are very high (close to 100%) on both transformations, and the quality degrades the more there are meaningful probabilities of a decay chain other than the triple in question, particularly because the correction between the extended Koide prediction and the experimentally measured value typically is on the same order of magnitude as the probability of the triple not being the decay chain times the mass of the omitted possibility.

    Our knowledge of neutrino mass differences is sufficiently precise that we know for a fact that these do not make a charged lepton Koide triple, although Brannen has made a proposal that involves a change of sign in the relation for neutrinos. Testing any proposal experimentally will take a while, however, because while we know the mass differences quite precisely, the precision with which we can determine even the heaviest neutrino mass is only the order of 100% MOE (cosmology provides the tightest limitations and also increasingly favors the normal hierarchy statistically relative to the inverted one). It doesn't help that our understanding of neutrino oscillation is as basically a black box process. We can come up with a formula that is a good fit, but don't really have a consensus story about a mechanism of the oscillation process that fits neatly into the rest of the SM.

    FWIW, I am deeply skeptical of the proposition that neutrinos are Majorana particles and of the SeeSaw mechanism with vastly heavier right hand neutrinos as an explanation for their mass. It may be a decade or two before neutrinoless double beta decay experiments are sensitive enough to resolve the Majorana particle question, but there are basically no experimental hints whatsoever of that to date, and given the importance of neutrinos having distinct particles and antiparticles to balance the lepton number of the SM which was the basis of their prediction, I don't think it makes any sense for a neutrino to be its own antiparticle. I also see no compelling reason for right handed neutrinos with masses different than the LH neutrinos to exist.

    Heuristically, it makes a lot of sense to associate the electron mass with its electromagnetic field strength, and the neutrino mass scale with the weak force field strength. But, this heuristic argument doesn't explain why different generations of either charged leptons or neutrinos which have the same field strengths have different masses.
  8. May 20, 2016 #7
    The Koide relation has been extended to the quarks in a certain way, and I would like to see it extended analogously to the neutrinos, just to see if it is consistent with all the known constraints; but I have trouble understanding exactly how it works for the quarks. So I am hoping we can figure that out, and then deduce the implications for the neutrinos.

    This extension may be found in two Phys Rev D papers by Zenczykowski, arXiv:1210.4125 and arXiv:1301.4143. As explained in the first of these papers, what is being generalized is a trigonometric reformulation of Koide's relation due to Carl Brannen (see Z's equation 5), in which an angle of 2/9 radians appears as a parameter. The proposition is, that you can get Koide-like relations for the up-type and down-type quarks, using "angles" of 2/27 and 4/27 radians, respectively. For the record, I have to point out that all this had already appeared in an unpublished preprint by Marni Sheppeard, "On Neutral Particle Gravity with Nonassociative Braids".

    However, the second paper by Zenczykowski does break new ground, by arguing that the relations for quarks are further improved, if one considers not masses, but "pseudo-masses" as defined by François Goffinet (PhD thesis, page 72). How are pseudo-masses obtained? One takes the mixing matrix (CKM for quarks, PMNS for leptons), and factors it into a product of two unitary matrices. You then write the masses of a fermion family (e.g. the up quarks) as a 3-vector, multiply that 3-vector by the relevant matrix, and you now have a 3-vector of pseudo-masses.

    For the charged leptons, the unitary matrix employed is simply the identity, and so the formula works for the original masses. But for the quarks, Zenczykowski would have us use pseudo-masses. Following Goffinet, he associates this with working in the weak basis. Whether this association makes sense is something I would like to know; another is just how precise are these formulas for the quark masses, compared to the original Koide relation, which is impressively precise. But what I would also like to do, in the spirit of this thread, is to see what this ansatz predicts for neutrino masses, e.g. if one uses angles of 0 or 8/27 radians for Brannen's parameter.
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