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What is new with Koide sum rules?

  1. Oct 21, 2014 #126
    Except if I missed something in checking the seven pages, I didn't see the document arxiv:1201.2067v1 [physics.gen-ph], 5 January 2012: "The Koide lepton mass formula and geometry of circles". Do you know if that path has been followed in between? Do you know if someone is working on the S6 group symmetry in relation with that topic? Thanks.
  2. Oct 25, 2014 #127


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    Nobody as far as I know (which amount to this online circle and some email exchanges here and there). I liked the paper; somebody had commented the similitude of the formula with Descartes's but Kocic did a nice, precise work.
  3. Oct 31, 2014 #128


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    One of the fairly robust predictions of efforts to generalize Koide's rule to quarks by hypothesizing that there are Koide triples of quarks is that the up quark mass must be very nearly zero to be consistent with the down quark and strange quark masses. This assumption also naturally solves the strong CP problem without resort to axions. (An assumption to that up quark and lightest neutrino mass eigenvalue are both negligible also provides an easy starting point from which to construct extended Koide triples in the Standard Model fermion mass matrix without having to worry about massive interrelatedness of the equations.)

    Lattice QCD and experimental evidence tend to favor a non-zero value of the up quark mass of around 2 MeV close to half of the down quark mass, but this calculation is model and QCD methodology dependent. A new preprint argues that it is premature to rule out a negligible up quark mass if one takes another approach that is theoretically legitimate and better cordons off big uncertainties in the calculation. http://arxiv.org/pdf/1410.8505.pdf

    If the analysis is right, a more straight forward extension of Koide's rule to quarks than would otherwise be possible can be a decedent fit to the data.
  4. Oct 31, 2014 #129


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    Nice! Let me to remind the "predictions" of Koide waterfall for the masses of up, strange and down. If we assume that we want the up quark exactly zero, then setting the yukawa of the top equal to one solves all the waterfall, and predicts strange=121.95 MeV and down=8.75MeV.

    t:174.10 GeV--> b:3.64 GeV---> c:1.698 GeV --> s:121.95 MeV ---> u:0 ---> d:8.75 MeV

    Of course, then the correction [itex]\Delta m_u \propto{ m_s m_d \over \Lambda}[/itex] depends of an extra parameter, but values of about 300 MeV for the denominator look reasonable for the scale the preprint is speaking about.

    If instead of fixing the up quark to zero we want to predict it, either from the top and bottom masses, or from the electron and muon plus "orthogonality", the result is similar and it has been discussed before in the thread. The mass of the strange quark is then lower, about 95 MeV, and up quark is still small, on the order of KeVs, so the results are similar and the agreement with measured spectrum is even better. But the prediction with inputs [itex]y_t=1, y_u=0, \Lambda \approx (m_u+m_s+m_c)/6 [/itex] is, to me, more impressive.
    Last edited: Oct 31, 2014
  5. Mar 18, 2015 #130
    Koide and Nishiura have their latest yukawaon model. These are models in which the original Koide relation (for electron, muon, tauon) is assumed to derive from a multiplet of scalars whose VEVs set the yukawas for the charged leptons, and then the yukawas for the other fermions are derived similarly; but only the charged lepton masses are treated as a "Koide triple" obeying a precise relation. The other masses are approached in the fashion usual for flavor physics, as only exhibiting rough relations, e.g. order-of-magnitude hierarchies. So for generalizations of Koide, such as are discussed in this thread, in which new Koide triples are also deemed to be real and needing explanation, these yukawaon papers might seem to be misguided. I mention this one only because, in its appendices, it seems to be claiming a new way to explain the original Koide relation; and that would be of general interest. But I haven't understood it yet.

    Koide's relation also made an appearance in a recent, more conventional paper on flavor physics, in which the author tries to cast doubt on its significance by contriving a new relation (his equation 62) which he says works just as well. But it looks like he judges its success against running masses considered at the same scale, whereas the Koide formula is at its most accurate for the pole masses. Sumino's papers remain the only ones known to me, that try to explain this feature.
  6. Mar 20, 2015 #131


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    I am sorry they are not using the hints from the quark sector. In my view, either the Yukawaon or other similar scalars should give mass first to an unbroken Pati-Salam or GUT group, with a mass for each of three generations that we can call

    M3 = e up bottom
    M2= tau charm down
    M1= mu top strange

    If the masses of the this unified system agree with Koide equation, K(M1,M2,M3)=0, we have really nine Koide equations:
    K(up,charm,top)=0, K(up,charm,strange)=0, K(up,down,top)=0, K(up,down,strange)=0
    , K(bottom,charm,strange)=0, K(bottom,down,top)=0, K(bottom,down,strange)=0

    Then something breaks the unification group in a way that not all the Koide equations are broken. The ones in bold somehow survive. The model should explain how and why.
  7. Mar 21, 2015 #132


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    With more detail, what I think it is happening in the quark sector.

    First something divides the GUT model into three families that meet a trivial Koide equation

    [itex](u,b')[/itex] of mass zero
    [itex](t',s)[/itex] of mass [itex](1- \sqrt 3 /2) m_0[/itex]
    [itex](c,d')[/itex] of mass [itex](1+\sqrt 3 /2) m_0[/itex]

    Then GUT itself is broken sequentially:

    1) the [itex](u,b')[/itex] generation keeps in Koide relationship with the other two generations, but it moves to fill the two possible solutions, so we are left with

    [itex]u[/itex] of mass zero
    [itex](t',s)[/itex] of mass [itex](1-\sqrt 3/2) m_0[/itex]
    [itex](c,d')[/itex] of mass [itex](1+\sqrt 3/2) m_0[/itex]
    [itex]b[/itex] of mass [itex]4 m_0[/itex]

    2) the [itex](t',s)[/itex] and [itex](c,d')[/itex] break, again keeping Koide, so that
    2.1) [itex](t,s)[/itex] are two solutions of the triplet "cbx" [itex]((1+\sqrt 3/2) m_0, 4 m_0, x)[/itex]
    2.2) [itex](c,d)[/itex] are the two solutions of the triplet "usx" [itex](0, (1-\sqrt 3/2) m_0, x)[/itex]

    And then we get the spectrum:

    [itex]u[/itex] of mass zero
    [itex]d[/itex] as solution of the triplet [itex](0, (1-\sqrt 3/2) m_0, m_d)[/itex]
    [itex]s[/itex] of mass [itex](1-\sqrt 3/2) m_0[/itex]
    [itex]c[/itex] of mass [itex](1+\sqrt 3/2) m_0[/itex]
    [itex]b[/itex] of mass [itex]4 m_0[/itex]
    [itex]t[/itex] as solution of the triplet [itex]((1+\sqrt 3/2) m_0, 4 m_0, m_t)[/itex]

    If we assume as input [itex]y_t = 1[/itex], then this spectrum is the still unrealistic "waterfall"
    t:174.10 GeV--> b:3.64 GeV---> c:1.698 GeV --> s:121.95 MeV ---> u:0 ---> d:8.75 MeV

    Then, or at the same time, some second order effects move slightly the u quark out of its zero mass, choosing a particular branch of solutions of 2.1 and 2.2, and also moving the top quark out of its [itex]y_t[/itex]=1 value. The global effect, keeping the four koide equations, would be the realistic waterfall of this thread.

    Finally, u quark gets a mass contribution from other mechanism, proportional to [itex]m_s m_d / m_{\Lambda}[/itex]

    From the point of view of "discrete S4 symmetry", it could be useful to notice that the step 1 uses two intersecting triplets, (u,c,s) (c,s,b) while the step 2 uses the two not intersecting (u,d,s) and (c,b,t) of the four I have boldfaced in the previous post.
    Last edited: Mar 21, 2015
  8. Apr 21, 2015 #133


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    I think that we have never mention this one; Brannen speculated a bit with mesons but I have not found now the explicit mention to the mesonic tuples, namely

    [itex]3(B^0+D^0+\pi^0)/(2(\sqrt{B^0 }+\sqrt{D^0}-\sqrt {\pi^0})^2) = 1.005244 \pm 6 \; 10^{-6} [/itex]
    [itex]3(D^0+\pi^0+0)/(2(\sqrt{D^0 }+\sqrt{\pi^0}+\sqrt {0})^2) = 0.998829 \pm 5 \; 10^{-6})[/itex]

    compared to lepton formula [itex] e \mu \nu \to 0.999987 \pm 13 \; 10^{-6} [/itex]
    it is not bad, but while the lepton measurement is still compatible with exactness, the mesons are only in target for the 0.5%.
  9. Nov 18, 2015 #134
    Jester at Resonaances informs us that leptoquarks are the BSM flavor of the month. Coincidentally, I recently noticed that there is a "leptoquark" approach to explaining the appearance of the constituent quark mass scale in Carl Brannen's rewrite of the Koide formula (in which the sqrt-masses are eigenvalues of a particular circulant matrix).

    It comes from combining two things. First, a simple formula for the constituent mass that I found in Martin Schumacher:

    ##m_q = g_{\sigma q q} f_\pi##

    where ##g_{\sigma q q}## is the sigma-meson-mediated coupling between a bare quark and a pion condensate, and ##f_\pi## is the VEV of the pion condensate, better known as the pion decay constant. ##g_{\sigma q q}## is ##2\pi/\sqrt{3}##, and ##f_\pi## is about 90 MeV, leading to ##m_q## ~ 325 MeV.

    Second, the interaction term which produces the mass matrix of the charged leptons in Sumino's model (this approach originates with Koide himself):

    ##\bar{\psi}_{Li} \Phi_{ik} \Phi_{kj} \phi e_{Rj}##

    where ##\psi_L## and ##e_R## are left- and right-handed fermions, ##\phi## is the SM Higgs, and ##\Phi## is a 3x3-component scalar whose VEVs squared determine the yukawas.

    The idea, then, is to substitute Schumacher's coupling for ##\phi##, and to suppose that the VEVs of ##\Phi## are circulant:

    ##(\bar{\psi}_{Li} \Phi_{ik} \bar{q}) \sigma (q \Phi_{kj} e_{Rj})##

    i.e. this ##\Phi## is a leptoquark scalar.

    Incidentally, to do something analogous for the quarks, one would apparently want a diquark scalar; and in the MSSM, if you allow R-parity violation, the squarks can have leptoquark and diquark couplings.
    Last edited: Nov 18, 2015
  10. Dec 21, 2015 #135
    Another month, another anomaly that might be colored scalars - "squarks", even a diquark scalar, according to footnote 2 of that paper, though oddly the other paper cited in that footnote doesn't use the term. I think the diquark counterpart of the interaction term above would be ##(\bar{q}DN)\pi(\bar{N}Dq)##, where ##D## is the diquark scalar and ##N## is a nucleon field. (I know it would be odd to have a quark field and a nucleon field in the same effective field theory, though I have seen it done.)

    But I also learned something else from this "squark" paper (start of part 3, second point), something that's a problem for the whole concept of colored yukawaons. They need a nonzero vev since by hypothesis, that's where the yukawas come from; but if a colored particle has a nonzero vev, that will break SU(3) symmetry...
  11. Jan 9, 2017 #136
    A holographic model of nucleon mass, promising from the perspective of #134, can be found in Gorsky et al, 2013.
  12. Jan 11, 2017 #137


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    The distinction between the bold and not bold sequences is obvious.

    Each of the bold sequences, when arranged in order of mass, involve the most likely W boson transition from the heaviest to the next heaviest, and the most likely W boson transition from the next heaviest to the lightest. They are the "route of least resistance" a.k.a. most probable, decay channel of the heaviest fermion in the triple.

    The italic triples are impossible through W boson decay without intermediate steps.

    A decay of top->up->down isn't impossible, although it is highly improbable.

    A decay of top->bottom->down or of top->down->bottom is impossible without intermediate steps. The decays of bottom-top-down or down-top-bottom require an extremely energy boosted starting point (unlike all of the other decays) and are also highly improbable.

    Implications For Quark Triples

    This is why I am inclined to think that the relative masses of the fermions arises from a balancing of all available weak force interactions into and out of the fermion in question to other fermions, weighted by their relative probability - basically a function similar in concept to the addition of three vectors for each quark in the quark case, balanced out to an equilibrium state that simultaneous fits values for all six quark masses at once.

    This intuition is also supported by the fact that the more dominant a share of the overall probability of decays a triple has relative to all possible decays from the heaviest fermion in the triple, the closer it comes to K(triple)=0, while the lower the share of the overall probability that the triple has, the more it deviated from K(triple)=0.

    At first order the adjustment to the mass of the middle member of the triple (when that quark is an up-type quark) is approximately the CKM matrix derived probability of a transition from that up-type quark to the down-type quark that is missing from the triple times the mass of the omitted down-type quark (and visa versa when the middle quark in the triple is a down-type quark).

    This is a bit surprising, because the relationship of the mass of the middle mass quark of the triple to the other two quarks in triple (which is approximately a Koide triple relationship) is decidedly non-linear. The correct adjustment for the missing opposite type quark is probably actually non-linear, but I just haven't come up with a clever enough idea to figure out what that is yet.

    Definitional Issues For Quarks

    Of course, the other dicey piece of making a Koide-like formula work for quarks flows from the definitions of masses that are used.

    The charged lepton formula (which is exactly correct to the limits of experimental accuracy), and the top-bottom-charm triple (which is the best fit of the quark triples) exclusively involve pole masses.

    The bottom-charm-strange triple, the charm-strange-up triple, and the strange-up-down triple (which are less good fits to the Koide triple rule) all involve a mix of pole masses for the heavy quarks (i.e. evaluated at different energy scales) and MS masses for the light quarks evaluated at a constant energy scale - since light quark pole masses are ill defined.

    So, if MS mass at 1 GeV is not the correct generalization of pole masses for light quarks to capture a Koide-like relationship for quarks, then some of the discrepancy between the light quark masses and the masses predicted by a Koide waterfall method could be (in whole or in part) due to using the wrong definitions for the light quark masses.

    But, you clearly can't just extrapolate the formula for the running for quark masses at higher energy scales to light quark masses either. This gives you light quark masses in which the mass from the quark content of a pion or kaon would far exceed the mass of the particle itself (as demonstrated in a 1994 paper by Koide).

    In contrast, QCD calculations using MS masses for the light quarks that add gluon field mass contributions as for other quarks get you in the right, much lighter, ballpark of what a Koide waterfall calculation would suggest.

    Anyway, until you generalize the concept of pole mass for light quarks in a more appropriate way, it is not just experimentally difficult, but theoretically impossible to confirm or reject a generalized Koide's formula for quarks involving pole masses.

    The Lepton Case

    In the charged lepton case, you get a perfect to the limits of experimental measurement fit, because "vectors" involving neutrinos basically make zero contribution since their masses are so tiny, so only the three charged leptons make any contributions to each others masses and have to be balanced out.

    I suspect that, in principle, non-zero neutrino masses probably cause Koide's formula to be not quite perfect, since neutrinos do have W boson interactions with charged leptons, but only at something on the order of the ratio of the charged lepton masses to the corresponding neutrino masses (e.g. 1,776,960,000,000 meV for a tau v. about 50-60 meV for the heaviest neutrino mass; 105,000,000,000 meV for a muon v. about 8 meV for the next heaviest neutrino mass, and 511,000,000 meV v. less than 1 meV for the lightest neutrino mass). But, we only know the tau mass to an accuracy of one part per 14,807, so this slight tweak is impossible to measure until our experimental measurements of the tau mass are more precise by a factor on the order of (at least) about 100,000 times what they are now.


    One set of number that would be really useful to have at hand, but which I don't have in any easy reference, is a full set of the Standard Model mass parameters evaluated not at M(Z) but at the Higgs vev of about 246.2 GeV.

    If anyone could calculate or find out these values for me, it would be greatly appreciated.

    I could probably do it. I can relatively easily track down literature with the relevant beta functions. But, doing the actual calculations is not something that I'm in a good position to do at the moment, and even if I could, it would probably take me forever as I don't have the right kind of software to do it by any means other than with a calculator or Xcel worksheet.
    Last edited: Jan 11, 2017
  13. Jan 12, 2017 #138


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    Well, the problem here is that we have a bit of circularity if we assume that the CKM matrix (and then the "most likely" paths) is fixed after or at the same time that the masses. Could it be possible to postulate another "W" with another (dual, orthogonal, reciprocal?) CKM matrix so that the same criteria should select the italic triples and reject the ones in bold?
  14. Jan 12, 2017 #139


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    The italic triples are impossible because they have the same charge and the only mechanism for flavor changing in the Standard Model requires that you alternate quark electric charges at each step. by one full unit of electric charge. This isn't circular reasoning, it just a fundamental feature of how the W boson works in the Standard Model. And, it is, in general, possible to order any combination of masses from heaviest to lightest without loss of generality.

    The structure of the CKM matrix also does seem more fundamental than the quark masses. Indeed, in the exercise that follows in the rest of this post, one can sketch out a toy model of how one could cut the number of non-neutrino parameters of the Standard Model from 19 to 6 with a slight hypothetically possible tweak to the extended Koide rule model arivero has suggested to make it more accurate, a discovery of a relationship of the aggregate fundamental fermion masses and fundamental boson masses to the Higgs vev suggested by C & LP, a possible special relationship of the two electroweak constants to each other at the Higg vev energy scale, and a tweak to the definition of the Cabibbo angle to reflect the discovery of a third generation of fundamental fermions after it was initially defined.

    To be clear, I'm not actually arguing that this toy model is the key to the relationship between the fundamental constants of the Standard Model that could greatly reduce their number. Instead, I'm illustrating what new "within the Standard Model" physics that could do that ought to look like, as a motivational exercise to suggest that we aren't as far from making really major progress in greatly reducing the universe of Standard Model physical constants than it might seem. We aren't that far from the promised land, and we are approaching the point where an Einstein-like genius could, in just a few years, reveal a lot of the connections that had been opaque or purely conjectural until we had accurate enough measurements of the fundamental constants to make provable statements about their relationships.

    Conjectures Re Fundamental SM Masses

    The fermion masses could quite conceivably emerge dynamically with just a single parameter to set the overall fundamental particle mass scale for both fundamental fermions and fundamental bosons.

    On the fundamental boson side, the Weinberg angle is the inverse tangent of the bare electromagnetic force gauge coupling constant g' divided by the bare weak force gauge coupling constant g. The magnitude of the fundamental electric charge "e", in turn, is the bare weak force gauge coupling constant g times the sine of the weak force mixing angle (and thus can be determined solely from g and g'). The mass of the W and Z bosons can be computed from g, g' and the Higgs vacuum expectation value v (246.22 GeV). The measured value of the Higgs boson mass is strongly consistent with the square of the Higgs boson mass is equal to v2/2-(MW)2-(MZ)2. So, all of the fundamental boson masses can be determined from g, g' and the Higgs vev, removing three parameters from the Standard Model.

    The sum of the square of the fermion masses, likewise, is very nearly equal to v2/2. As you, arivero, have demonstrated, a couple of extensions of Koide's rule can get you very close to all of the nine fundamental charged fermion masses from the electron mass and the muon mass, if only a quirks in the extension of Koide's rule for quarks can be ironed out properly (most likely by making the appropriate adjustment for the down type quark missing from the triple with a middle mass up type quark, or visa versa). Indeed, with the Higgs vev to set an overall mass scale, the only other parameter you need use that approach to get all nine fundamental fermion masses is the ratio of electron mass to the muon mass.

    So, it seems attainable to get all nine of the charged fundamental fermion masses, and all three of the fundamental massive boson masses, from two of the three SM coupling constants, the Higgs vev, the ratio of the electron mass to the muon mass, and the CKM matrix. This would reduce the number of experimentally measured parameters in the Standard Model by 10 out of 26.

    Conjectures Re CKM Matrix

    The CKM matrix can be expressed quite accurately in a parameterization of just one real parameter (the Cabibbo angle) and one complex parameter associated with CP violation, because in the Wolfenstein parameterization Aλ2 is equal to (2λ)4 at the 0.1 sigma level of precision, and there is no place in the Wolfenstein parameterization of the CKM matrix where this substitution cannot be made. This reduces the number of CKM matrix parameters from 4 to 3.

    Combined with the mass conjectures above, that would reduce the number of Standard Model parameters from 26 to 15 (of which 7 are for neutrinos).

    The key point about the structure of the CKM matrix as expressed in the Wolfenstein parameterization that makes it seem more fundamental is that, up to adjustments for CP violation, this parameterization suggests that the probability of a first to second generation transition (or second to first generation transition) is λ, that the probability of a second to third generation (or third to second generation) transition is (2λ)4 , and that the probability of a first to third generation (or third to first generation) transition is (2λ)5 (i.e. the product of the probability of making first one of the single generation step transitions and then the second).

    The probability of transitioning to a quark of the same generation is the residual probability after the probability of the other two options is subtracted out.

    The CKM matrix, so parameterized, suggests an almost atomic energy shell-like sequence of transition probabilities between generations that one can imagine popping out easily from some more fundamental theory that is really more straight forward.

    Crazy Talk

    Making One Electroweak Coupling Constant Derived

    g + g', the sum of the two dimensionless electroweak coupling constants, at the W boson mass, are just a wee bit over 1. But, both of these constants run with energy scale, and it is very tempting to imagine the possibility that at some energy scale, such as the Higgs vev, that g+g' are exactly equal to 1.

    If this were the case, we would replace one of the two dimensionless electroweak coupling constants in the set of Standard Model parameters with the Higgs vev, reducing the number of experimentally measured parameters of the Standard Model apart from the neutrino sector, from 8 to 7.

    Deriving Wolfenstein CKM Parameter λ From The Electroweak Coupling Constants

    It is also tempting to think that the sine of the Cabibbo angle could have a functional relationship of some kind to the Weinberg angle, in some way that could reconcile their 2.48% discrepancy, perhaps by redefining the Cabibbo angle. For example, one could imagine redefining it as the inverse tangent of (the absolute value of CKM matrix element Vus plus the absolute value of CKM matrix element Vub) divided by the absolute value of CKM matrix element Vud which would increase the sine of the Cabibbo angle to about 0.22867, and then multiplying this time one plus the fine structure constant (which is roughly 1/137), which would bring it to 0.23034. This would be within one standard deviation of the square of the sine of the Weinberg angle at the Z boson energy scale given the precision of current experimental measurements (the precision of the Weinberg angle measurement is about six times greater than the precision of the Cabibbo angle measurement).

    The extension of the definition of the Cabibbo angle to include the addition of CKM matrix element Vub is very natural. The Cabibbo angle was originally defined before the third generation of Standard Model fermions was discovered. In a two fermion generation Standard Model that Cabibbo angle was simultaneous the probability of a transition to a non-first generation quark and the probability of a transition from a first to a second generation quark. Including the CKM matrix element for a transition to a third generation would generalize it using the latter interpretation of its meaning, rather than the former, which were both identical in the two generation case.

    The inclusion of a factor of one plus the fine structure constant is less obvious and somewhat arbitrary. But, given that we are talking about an electroweak process that always involves a W boson with has both a weak force coupling and an electromagnetic coupling, it would hardly be stunning that a formula to derive from first principles a probability of quark generation transitions from one generation to another might involve both the weak mixing angle and the electromagnetic coupling constant.

    This would make the Cabibbo angle a function of the two electroweak coupling constants, and they, in turn, could conceivably be a function of either one of those constants and the Higgs vev. You could then work out the Wolfenstein parameter λ, from the redefined Cabibbo angle.

    This would mean that the Standard Model experimentally measured parameters (outside the neutrino sector) could be reduced to just six if something along the lines of the kind of toy models I am discussing as conjectures could be worked out:

    1. The strong force coupling constant.
    2. The value of one of the electroweak coupling constant at the Higgs vev energy scale.
    3. The Higgs vev.
    4. The ratio of the muon mass to the electron mass.
    5.-6. The complex valued CP violating parameter of the CKM matrix in the Wolfenstein parameterization.

    We know 2, 3 and 4 to extreme precision. We know 1, 5 and 6 to moderate precision.

    This would still leave in the neutrino sector three mass eigenvalues and four PMNS matrix parameters. Discovery of a Koide-like relationship for the neutrino masses might reduce the number of neutrino sector parameters from 7 to 6 that could be confirmed when they could actually be measured more precisely. But, we are too far away from having sufficiently accurate measurements of the PMNS matrix parameters (particularly the Dirac CP violating phase) to be able to speculate about possible relationships between them in more than an idle way at this point.

    Really, Really Crazy Talk

    For what it's worth, I think we are also at a similar juncture with the dark matter-dark energy problem. I think we could find a modification of gravity that solves both problems in one fell stroke while eliminating the cosmological constant as well, in just a few inspired years of articles from the right scholar (my eyes are on Alexandre Deur or someone follow up on his insights).

    Between the SM's 26 constants, GR's 2 constants (Newton's constant and the cosmological constant), Plank's constant and the speed of light, we current have a total of 30 fundamental experimentally measured constants.

    I think we could cut the total down to 15 while simultaneously solving the dark matter and dark energy issues. And, given that 6 of those remaining 15 would be for the neutrino sector, some of those could probably be trimmed somehow as well with one or two more breakthroughs in the neutrino sector.

    Of course, all of those extra relationships and the reduced number of pieces of the puzzle, might in turn increase the likelihood that someone could find a yet deeper relationship that is even more reductionist and fundamental.
  15. Jan 13, 2017 #140


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    I was speculating that we could have two kinds of W particles, seeing differently the charge of some of our unconventional "families", so that one of the W grants the Koide equation for the first group of triplets, the boldface ones, and other for the second group. But it is unclear, to see the least. Also, the breaking of (bds) seems small, empirically, so perhaps the unbroken triplets are not really the ones listed above, but bds instead of uds. Or both :-(

    btw, Koide has new paper, on Sumino models. https://arxiv.org/abs/1701.01921
  16. Jan 24, 2017 #141
    Koide and Nishiura's latest (it came out today) contains new numerology.

    In their model, each fermion family e (e,μ,τ), u (u,c,t), and d (d,s,b) gets its masses as eigenvalues of a matrix [itex]Z (1 + b_f X)^{-1} Z[/itex], mutiplied by a mass scale [itex]m_{0f}[/itex], where
    [tex]Z = \frac 1 {\sqrt{m_e +m_μ + m_τ}}
    \sqrt{m_e} & 0 & 0 \\
    0 & \sqrt{m_μ} & 0 \\
    0 & 0 & \sqrt{m_τ}
    [tex]X = \frac 1 3
    1 & 1 & 1 \\
    1 & 1 & 1 \\
    1 & 1 & 1
    and [itex]b_f[/itex] is a free parameter. f indicates the family, e, u, or d, and [itex]b_f[/itex] and [itex]m_{0f}[/itex] take different values according to the family.

    For the e family, [itex]b_e[/itex] is just zero, so the matrix is [itex]Z^2[/itex], and it just gives the (e,μ,τ) masses by construction. For the d family, [itex]b_d[/itex] is a random-looking number. But for the u family, we have
    [tex]b_u = -1.011[/tex]
    [tex]\frac {m_{0u}} {m_{0e}} = 3.121[/tex]
    i.e. very close to the integer values, -1 and 3.

    All that is from section 4.1 (page 12). The model itself is a seesaw as displayed in section 2.1 (page 5 forward). There is no explanation for the values of those numbers.
  17. Jan 24, 2017 #142


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    It looks very much as the discarded corrections for the lepton sector.

    For b=-1 the matrix is singular, so perhaps it is just a signal of how massive the top quark is.
  18. May 15, 2017 #143
    A few times in this thread (e.g. #129, #132), @arivero mentions that in a first approximation to his "waterfall" mass ansatz, the mass of the bottom quark comes out as exactly 12 times the constituent quark mass. Earlier this month, in a speculative new framework for hadron masses and systematics, I ran across the claim that the mass of the rho meson is sqrt(6) times the constituent quark mass (see footnote 6, bottom of page 6). This seems to be derivable within the framework of a so-called "gauged quark-level linear sigma model". The original linear sigma model is a famous 1960 creation of Gell-Mann and Levy, with nucleon, pion, and sigma meson fields. The quark-level linear sigma model employs quark fields in the place of the nucleon fields, and the gauged version of this adds vector mesons as well. This is, more or less, the framework I was citing in #134.

    What I want to observe here, is that this first approximation to the bottom quark mass, as 12 times the constituent quark mass, could be expressed as 2 . sqrt(6) . sqrt(6) times the constituent quark mass. Furthermore, all these quantities (in the extended LSM framework) come from couplings to QCD condensates, where the value of the coupling is an exact quantity determined nonperturbatively. Also, the double appearance of one factor, is reminiscent of how Koide's yukawaon models work. So maybe this is a hint regarding how to realize the waterfall.
    Last edited: May 15, 2017
  19. Jun 6, 2017 #144
    The aspect of the waterfall that has confounded and tantalized me the most, is that it alternates between charge +2/3 and charge -1/3, whereas the original Koide triple is all charge +1. The original Koide triple naturally fits a family symmetry like Z3 or U(3), but these alternating triples don't; I haven't seen anything like them in the literature. The closest thing I have seen, are radiative models of fermion mass in which only the top couples directly to the Higgs, and then the lighter fermions get their masses through top loops; the lighter the particle, the higher the order at which the top appears.

    I do think it is probably possible to construct a field theory in which a waterfall of Koide triples is produced, I just worry that we would only be able to do it in a very complicated and artificial way (by engineering an appropriate potential for the SM yukawa couplings, through the introduction of new charges, symmetries, and fields). That's the confounding part, that makes me doubt the worth of the enterprise. But the tantalizing part is when I remember that the principle of its construction is simple. Yes, it is at odds with the sort of mechanisms that are familiar in BSM theory, but perhaps we just haven't found the right mechanism. So I thought I would review a few of the possibilities that exist.

    In #137 and #139, @ohwilleke said that the sequence resembles a cascade of W boson decays, and speculated that the Koide relation (for a consecutive triple in the sequence) might be due to a sort of equilibrium between the three quark fields involved. This makes me wonder: the radiative models of mass generation, in which the lighter quarks get their mass from top loops, usually involve new scalars. Could you have such a model, in which some of the scalars involved are the Goldstone bosons that arise from the Higgs field, and which are "eaten" by the W and Z bosons to become their spin-0 components?

    Another underused representation of the Koide relation is the geometric representation due to Robert Foot. (It does make an appearance in @arivero's original paper introducing the waterfall.) The Koide formula holds if the 3-vector of square roots of masses forms an angle of 45 degrees with respect to the vector (1,1,1). For a set of six sqrt-masses, forming four overlapping triples, one may envisage a set of four such relationships in a six-dimensional space. This could be the basis of a potential for the yukawa couplings, or perhaps even a model with extra dimensions. The same goes for Jerzy Kocik's Cartesian formula mentioned by @Blackforest in #126.

    A further possibility is that it might be done nonperturbatively with condensates, in a Nambu-Jona-Lasinio model. This would fit some recent thoughts (#134 and #143), and I wrote about it here. Finally, there's @arivero's own scenario, described in #131-#132.
  20. Jun 6, 2017 #145
    May I first thank you for your résumé on the topic and for re-mentioning an article on which I proposed to think about a few years ago? Being myself not a professional I get the biggest difficulty to meet the criteria allowing an intervention on your forums. This doesn’t impeach me to think about some of the topics which are discussed here.

    Concerning the Koïde formula and the article related to an old Descartes theorem, I would like to add that diverse personal thoughts are pushing my intuition into a direction questioning the role of the tetrahedrons into that discussion. I don’t mean that tetrahedrons are paving our space-time (a view that would feet with the branch which is working on tetrahedral meshes [e.g.: Data structures for geometric and topological aspects of finite element in Progress In Electromagnetics Research, PIER 32, 151–169, 2001]).

    No; referring to articles studying the altitudes of that object, I just wonder about some of their fascinating properties and ask myself in which way the trace-less quadratic forms of rank 3 which are sometimes associated with these platonic objects [e.g.: N.A. Court, Notes on the orthocentric tetrahedron, Amer. Math. Monthly 41 (1934) 499–502] may eventually be an alternative and useful tool for the description of the propagation of light?

    Although my thoughts are seemingly a little bit off-topic, I would like to remark that four spheroids may deform the four faces of a tetrahedron and, because of that, be involved into a discussion roughly related to the Descartes theorem mentioned in the article at post 126.

    I apologize if I have disturbed the forum with that piece of dream. Best regards.
  21. Jun 6, 2017 #146


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    Well. actually its usage is very obvious in early Koide models, as first he imposes the "couplings" [itex]z_1, z_2, z_3[/itex] to be in a plane orthogonal to (1,1,1), then he asks them to be in a circle of some radious proportional to "coupling" [itex]z_0[/itex] and finally the composites are given masses proportional to [itex](z_i + z_0)^2[/itex]. In these models the idea is that [itex]z^i[/itex]^2 is the "self coupling potential" of the preon with itself, while [itex] 2 z_a z_b[/itex] is the "interaction energy". He already uses different radius for leptons and quarks, but I think that the (1,1,1) axis is always used. By the way, a lot of years ago it was suggested to try an angle around (1,1,0) for the up-quark tuple. In this sense, changing the main axis, Foot's representation is underused, yep.

    Radiative models are always very tempting due to the relative regularity of the masses, very nicely spaced by quantities of the order of strong and em couplings. I think that this was remarked, for the s-c gap, by Gsponer when doing our colaboration, and anyway it catches the eye when one looks a log plot of the standard model:


    (btw, remember that source for this kind of plot is available in github at https://gist.github.com/arivero/e74ad3848290845de5ca )

    My point to favour a breaking instead of a radiative approach to the waterfall is that the scheme of two linked Koide tuples
    [tex](0,[1-{\sqrt 3 \over 2} , 1+{\sqrt 3 \over 2}), 4] [/tex] already covers a lot of the spectrum, as said above in https://www.physicsforums.com/threads/what-is-new-with-koide-sum-rules.551549/page-7#post-5049287 comment #132: we get (u,[s,c),b) in quarks as well as the original (e,mu,tau) tuple. But it can also be argued that the extremes, down and top, obtained by linking another two tuples left and right, become ugly:
    [tex]13 - 15 {\sqrt 3 \over 2} \leftarrow (0,[1-{\sqrt 3 \over 2} , 1+{\sqrt 3 \over 2}), 4] \rightarrow 109 + 95 {\sqrt 3 \over 2} [/tex]

    On other hand, radiative jumps could be the answer to the use of tuples with different charges, and it is not really incompatible with a "breaking" view. But the main problem, to me, is not that we have different charges, but that they need to be at the same time in different tuples; this makes the preon a very hard pill to swallow. It is true that we have also the two koide tuples (0, [pi, D), B]) which are obviously composites but here QCD makes some magic to set the pion mass as about the same size that the strange quark; honestly we could have expected the equivalent pair of tuples to be (pi,[K,D),B]

    Edit: same line, but scaling, looks a bit less ugly, but only a little bit:
    [tex] \small
    97 - 56 \sqrt 3 \leftarrow (0,[7-4 \sqrt 3 , 1), 16-8 \sqrt 3] \rightarrow 151 -28 {\sqrt 3} [/tex]
    Last edited: Jun 6, 2017
  22. Jun 7, 2017 #147


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    Just to recap the extended Koide's rule for quarks notion a bit. The idea would be that the quark masses arise dynamically from W boson interactions. Each "target" quark flavor's mass would be a blend of the three other different quark masses that the target quark could transform into via W boson interactions, weighted in some manner by the relative likelihood of each W boson transformation in the CKM matrix. Thus, no additional boson is necessary for the mass generation mechanism. A similar mechanism would apply, at least, to the charged leptons and probably to the entire lepton sector, giving rise to Koide's rule for charged leptons.

    Notably, all Standard Model fundamental particles that have mass have weak force interactions with W and Z bosons, while none of the Standard Model fundamental particles that lack mass (the gluon and the photon) have weak force interactions with W and Z bosons. This scheme is also, incidentally, an argument for the non-existence of massive right handed neutrinos, since right handed neutrinos do not interact with W and Z bosons via the weak force, so even if they exist as fundamental particles, they should not have any mass.

    In cases where almost all interactions of the target quark flavor can be attributed to just two of the three possible source quarks, Koide's formula is a very good approximation of the relative masses in a triple, e.g., in the case of the t-b-c triple where the b quark is the target and the t and c quarks are the source quarks. The likelihood of a b-u transformation via a W boson under the CKM matrix is so low that it can be disregarded in a first order approximation.

    Empirically, a linear interpolation from a three quark Koide triple estimate of a quark mass, to one that considers all three source quark masses, can be achieved by multiplying the probability o the omitted source quark mass by the probability of a CKM transformation to the omitted source quark.

    For example, the Koide waterfall applied naively give you a mass of zero or very nearly zero for an up quark derived from a u-d-s triple, which also is the dominant source of W boson transformations from a u quark to another quark flavor, energy conservation permitting. This is significantly different from the actual measured mass of the u quark on the order of 2 Mev. But, if you multiply the square of the CKM element for an up quark-bottom quark transformation via a W boson which represents the probability of such an event by the mass of the bottom quark, you get a value much closer to the experimentally measured value of the mass of the up quark.

    More generally, the Koide triples for quarks which are least accurate when compared to experiment are those for which the square of the CKM entry for the W boson transformation from the target quark to the omitted source quark is greatest.

    Of course, to get the entire quark mass matrix you need to simultaneously solve six sets of equations that set for a relationship between each of the six target quark masses relative to the three source quark masses for each at the same time, and ideally, you would do so in a manner (probably inspired by the geometric interpretation of Koide's rule) that is an exact non-linear formulation, rather than a non-linear formulation that approximates the target quark mass from two of the three source quarks and then adjusts the result with a linear approximation for the third source quark.

    If one could make this work, you could derive the relative masses of the six quarks entirely from the four CKM matrix parameters and one mass constant (either the mass of any one of the quarks or a mass scale for the quark sector as a whole).

    This hypothesis also assumes that the CKM matrix is logically prior to the fundamental fermion masses in the SM, which the structure of the CKM matrix tends to support. It is pretty much impossible to go the other direction and derive the quark mass matrix from the CKM matrix because there isn't a big enough difference between the entries for up-like quarks and down-like quarks of the same generation. The Wolfenstein parameterization likewise favors an understanding of the CKM matrix that treats generations as distinct units rather than individual quarks.

    The Koide triple formula works so well in the charged lepton sector because any given charged lepton target particle has only two source particles (the other two charged leptons) rather than three, and because any contributions from W boson interactions between neutrinos and charged leptons is negligible since all of the neutrino masses are on the order of 1,000,00+ times smaller than any of the charged lepton masses, so the linear interpolation of additional elements which is material in the case of the quark masses in much smaller than the precision of experimental mass measurement accuracy in the case of the charged leptons.
    Last edited: Jun 7, 2017
  23. Jun 12, 2017 #148
    Today it occurred to me, what if the rho meson were somehow the mediator in these radiative cascades? That can't be literally true, but first let me present the argument. Again, it goes back to the fact that in @arivero's first approximation to the quark masses, the "unperturbed waterfall", the bottom quark mass is twelve times the constituent quark mass.

    As described in Schumacher 2014, page 4, one paradigm for explaining the constituent quark mass is very similar to the Higgs mechanism for fermion mass: there's a vev, and a coupling to that vev. It's there in equation 5. The coupling is pion-pion-sigma meson, and the vev is, I guess, the chiral condensate. I don't quite get how it is supposed to work - maybe the quark emits a sigma meson, which couples to two pions from the chiral condensate??

    Now it so happens that the pion-pion-rho meson coupling is about 6 (see e.g. Delbourgo & Scadron, equations 29 and 34, where it comes out as 2π). So we have that the bottom quark mass is something like the constituent quark mass, times the pion-pion-rho coupling, times 2. Schumacher already explained the constituent quark as something like a bare quark, coupling via sigma meson to two pions from the chiral condensate. Now suppose we have a sigma meson condensate as well. A sigma meson is often modeled as two quarks and two antiquarks, so it can be decomposed into two pions in two ways. Could the bottom quark be like a constituent quark that then couples to a sigma meson condensate, in those two ways, via rho meson - with its mass thereby picking up a further factor of 6+6, i.e. 12??

    More precisely, one should imagine a rho-like light vector meson. The real point is that the bottom quark would be the unexpected origin of the waterfall, the flavor that couples directly to the mass-giving condensates, with all the other quarks getting their masses from loop effects, some sort of radiative equilibrium, etc. The top quark, the traditional source of radiative cascade, is a little peculiar in this bottom-centric picture, because it's far more massive than the bottom. So some further idea about mechanism may be required; and in grand unified theories, it is common to regard top and bottom together as having a special status with respect to mass. In any case, the facts are (1) top, bottom, charm are a Koide triple (2) bottom mass apparently has the right magnitude, to be obtained from strong couplings to QCD condensates.

    A few more remarks. There are tantalizing similarities between QCD vector mesons like the rho, and the electroweak gauge bosons. The QCD vector mesons have at least a formal resemblance to an emergent gauge symmetry that has been higgsed. In holographic QCD, they are the Kaluza-Klein reduction of higher-dimensional flavor gauge bosons. In the sbootstrap, one might hope to see the W and Z emerge along with the leptons, in one of these ways... Meanwhile, one may try to implement the waterfall through methods like those of Cabo or Zubkov - by positing four-quark interactions whose couplings are fixed nonperturbatively, by consistency arguments.
  24. Jun 14, 2017 #149


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    Question, given the current trend of events in D and B mesons... does anyone knows if Brannen finished his inspection of Koide-like relationships in hadron spectroscopy? I think he published some attempt on excited states for mesons having similar quark composition, but I do not remember if he extended to different compositions, say (Pi, B, D) or similar tuples.

    Edit: remember that by the coincidence between piom and muon, and D (or charm) with tau, we have a pseudotuple (0, pion, D) and then also the "scb" one: [tex]{(-\sqrt M_{\pi^-} + \sqrt M_{D^-} + \sqrt M_{B^-})^2 \over M_{\pi^-} + M_{D^-} + M_{B^-}} = 1.486... [/tex] and so variations of these oscillate around [itex]\frac 32[/itex] above or below without any particular pattern, as far as I can see. Normally we do not look at these ones except as part of some self-consistency, but who knows, perhaps they have some role in decay puzzles.
    Last edited: Jun 15, 2017
  25. Jul 3, 2017 #150
    I have been very skeptical of Koide relations for mesons. In fact I still am. But I guess that, if we are looking for QCD-like mechanisms, it does make sense to consider whether any of these hadronic Koide triples e.g. have an explanation in which Foot's vector appears. I don't know if you could work towards that by considering the cyclic basis for SU(3), which uses circulants.

    Also, a search for diquark sum rules turned up this paper, which has mass formulas for heavy-light diquarks. If the sBootstrap is on the right path, then there is some sort of mapping between quarks and diquarks, and quark mass formulas may resemble diquark mass formulas. In the original bootstrap calculation for the rho meson mass (informal description), the rho meson is modeled as a bound state of pions, and the pion as a bound state of a pion and a rho meson. Given my remarks in #148, I might look for a "sbootstrap calculation" in which the bottom quark is dual to a light diquark (two light quarks), and a light quark is dual to a heavy-light diquark (a bottom quark plus a light quark)...
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