Thursday, March 14, 2013

An Ansatz Re The Basis Of Koide's Formula

Why should an e-u-d Koide triplet be real?  Does a massless up quark make sense? 

This post is really a response to Mitchell's post in response to my Koide triple post of yesterday that he links in that comment to that post.  I wanted to simply make a comment at his blog, but would have had to broken my discussion into multiple comments there due to blogger limitations and decided to just do a response post instead. 

This lengthy post lays out at considerable length and with a fair amount of detail and some examples my personal conjectures about the kind of theoretical ideas that I think could be at work in given rise to the observed phenomenological relationships of the Standard Model.

I aspire to have done this in a way that illustrates that I genuinely do have some sense of the kind of things that could be going on at a deeper level in the Standard Model that would make sense.  I also aspire to have done so in a manner in which someone with more skill than I could conceivably translate into a pretty respectable Withing The Standard Model (WSM) extension of it that builds on its well established existing approaches to resolve a number of unsolved questions for theoretical physics that lurk there.

As a WSM rather than a BSM this analysis proposes ways to secure more theoretical coherence and elegance in the SM, but does not predict much, if anything, in the way of "new physics."  The problems it solves in the SM are "why" questions, not "what"  questions. 

If these mechanisms were better understood along these lines, however, it would take the steam out of the sails of many BSM models that are mostly motivated by "why" questions and only provide new physics at something close to GUT scales that can't be practically tested or applied, anyway.  Acceptance of this kind of WSM theory might also reprioritize the kinds of theoretical and experimental work in physics that would seem most fruitful to pursue first given limited resources.

The gist of the concerns Mitchell raised and the details of the underlying theoretical ideas are below the jump.

The gut of his post is that he considers this triple approach to be:
rather unlikely (and it depends on the up mass being zero, an old idea which features in Rivero's Koide waterfall but which is out of fashion, presumably because it is empirically disfavored).
He also makes a distinction, which I do not for reasons that I discuss below between:
what I call "family triples" like the original electron/muon/tauon, and "sequential triples" like top/bottom/charm; I suppose this is a "generational triple". 
The outlines of my imagined mechanism

I haven't spelled out in writing the somewhat vague ansantz that I think may heuristically explain the deeper theoretical basis for Koide formulas that are probably at work.  Here it goes.

Fermion mixing matrixes and fermion mass matrixes fully determine each other 

In my view, the mass matrixes of the Standard Model and the CKM and PMNS mixing matrixes of the Standard Model are deeply intertwined as are a few other principles like the generally "democratic" nature of Higgs, W and Z boson decays subject to mass-energy conservation limitations.  There is a fair amount of scattered and sporadic, but published and legitimate theoretical work pointing to this notion, although theorists have had trouble formulating a model that properly characterizes how this work.

|V_{ud}| & |V_{us}| & |V_{ub}| \\
|V_{cd}| & |V_{cs}| & |V_{cb}| \\
|V_{td}| & |V_{ts}| & |V_{tb}|
\end{bmatrix} =
0.97428 \pm 0.00015 & 0.2253 \pm 0.0007 & 0.00347^{+0.00016}_{-0.00012} \\
0.2252 \pm 0.0007 & 0.97345^{+0.00015}_{-0.00016} & 0.0410^{+0.0011}_{-0.0007} \\
0.00862^{+0.00026}_{-0.00020} & 0.0403^{+0.0011}_{-0.0007} & 0.999152^{+0.000030}_{-0.000045}

The unparameterized CKM matrix from K. Nakamura et al. (2010). "Review of Particles Physics: The CKM Quark-Mixing Matrix". Journal of Physics G 37 (75021): 150. via Wikipedia.

Fundamental fermion masses derive from W boson interactions that balance the mass of a fundamental particle with the mass of particles it can become via W boson couplings.

My suspicion is that Koide triples (the sequential triples in quarks are the clearest example) are driven mostly by W boson interactions that dynamically generate an equalibrium balance in fundamental masses between a fundamental particle, the particle that most often decays to that fundamental particle and the particle to which that particle itself most often decays whose precise value is a function of a single weakly interacting particle mass scale and the mixing matrix parameters.  The W boson is the only particle that can transform of quark or lepton into a different kind of quark or lepton, so its interactions are the only forum in which the masses of different flavors of fermions can be balanced against each other dynamically.

Flavor changing W boson couplings dynamically strike an even balance between a degenerate case of equal masses for all fermions that can couple to a W boson and another fermion, and the extreme case where one fermion has a positive value and all of the others are massless.

Koide's two-third value, midway between the two extreme values of the Koide ratio, represents a perfect balance of these interactions, in a context wherre there are only three contributing interaction inputs large enough to matter, between the degenerate case where all three particles in a decay chain have equal masses, and the trival case where one particle in the decay chain is massive and the other two are massless.  This measure would have to be generalized in a meaningful way to complete sets of four particles in which particles don't always follow the dominant W boson coupling chain.

In the case of quarks, the primary decay chain in a sequential Koide triple is so dominant that contributions from other possible couplings have negligible impacts on heavier quark masses.  The Koide formula approximation works because the omitted terms of the exact W boson mass generation process are trivial compared to the relevant fundamental fermion mass.

So, for example, a top quark with a mass of about 173.4 GeV decays to a bottom quark (99.8% of t quark decays) and can't decay directly to a charm or up quark or a lepton or neutrino, those transitions require multiple interactions that can give rise only to indirect effects).  A bottom quark with a mass of about 4.19 GeV, meanwhile, can't decay directly to a strange or a down quark, and is much more likely to decay to a charm quark  with a mass of about 1,350 MeV (about 38 times out of 40 W- boson couplings when it has sufficient mass-energy to become either a top quark or an up quark, and about 39 out of 40 W- boson couplings when it has insufficient mass-energy to become a top quark) than it is to decay to an up quark with a mass of about 2 MeV (1-2 times out of 40 W- boson couplings depending upon whether it has sufficient mass-energy to become a top quark). 

Thus, a sequential Koide's triple approximation is omitting just 0.2% of bottom quark creation interactions and just 5% of bottom quark decay interactions in cases that are not mass-energy limited, which is a good leading order approximation of all of the W boson couplings of the bottom quark.

So, the dominant direct weak force interactions of bottom quarks with other fundamental fermions are with top and charm quarks, and a Koide's triple toy model is implicitly assuming that 100% of bottom quark creation events are from top quark which decay 100% to bottom quarks, and that 100% of bottom quark decays are to charm quarks which are created 100% from bottom quark decays.

My hypothesis is that the mass of the bottom quark is predominantly a matter of CKM matrix implemented balancing between the top quark mass and the charm quark mass via W boson interactions, with next to leading order terms modifying that balancing act to reflect the possibility that bottom quarks will decay directly to up quarks rather than charm quarks. 

W bosons mix the masses of fermion flavors that it couples by dynamically adjusting the transition probabilities between those flavors of fermions in a manner reflected in the CKM and PMNS matrix, which adjusts the proportionate influence of each possible coupling on the involved particle's masses.

This balancing act plays out via the medium of CKM matrix probabilities.  The CKM matrix probabilies for bottom quarks balance the pressure exterted on the bottom quark facing a transition with three possible options (top, charm, up) in a way that produces a particular blend of masses that reflects the mass asymmetry and mass-energy limitations of the three different generations because the weak force makes a heavier version of a particle inherently unstable relative to a lower mass-energy alternative version of the same particle.  All of the "sequential" quark triples can be explained this way.

Koide triples involve cases where the contributions of at least one of the fermions that couples with a "target" fermion in the triple (the one that is alone in its "up" or "down" quark character, for example) can be disregarded without having a discernable impact on the calculated value.

Because sequential triples omit an NLO term in what is really a set of four and not three interacting quarks, the pure Koide's formula for them is probably a bit off, but because the CKM matrix elements for the omitted terms are so small relative to the terms accounted for considering only the three leading order contributions to the weak force balancing that gives rise to the CKM matrix elements, the NLO terms probably don't tweak the outcomes in measurable amounts.  It probably wouldn't be too hard to generalize the sequential triples to the actually relevant four quark mass relationships that emerge from weak force balancing.

Koide's original charged lepton triple, because all of the neutrino mass contributions that are omitted are so tiny, is the "cleanest" of the Koide triples, i.e. it comes closest to the target value of 2/3rds of the Koide ratio without meaningful distortions from omitted fourth W boson coupling terms.

The charged lepton "family" triple involve a similar dynamic equalbirum balancing as series of weak force decays, which while indirect, is largely unmuddied by quark or neutrino physics.  This balancing allow for electrons to become muons or taus, muons to become electrons or taus, and taus to become muons or electrons, in connection with the W boson interactions of leptons, in a basically democratic manner that is biased by mass-energy conservation that gives heavy rapidly decaying particles more of a say than light ones that often don't have the option to "decay up", and by varying decay rates at different masses driven by the weak force.

The e-u-d Koide triple arises from energy limited beta decays, which give rise to a similar, although multiple step (and hence not leading order) four fermion interaction mediate by interactions with a single W boson that beta decay probabilities provide a means for the W boson to dynamically adjust.  The Koide triple works because  a negligable contribution from the electron-antineutrino mass term can be omitted.

The e, u, d "generational" triple emerges from the dynamic equalibrium arising from the predominant weak force interaction seen in nature - beta decays in which mass-energy conservation forbids second or third generation particle formation and thus prevents those higher order effects from playing a role.  There is probably a next to leading order electron neutrino term in this triple, just as there was probably an omitted term in the sequential quark triples, but since the electron neutrino's mass and interactions are so feeble, its contribution is probably safely neglected.

All of these triples, however, basically arise from dynamically balanced transition biases between different particles in weak force decays that are summarized in the CKM and PMNS matrixes with the balances struck that manifest in mass and the balances struck that manifest in transition probabilities being intimately intertwined.

The three to one quark-lepton Koide triple mass ratio arises from parallel processes whose respeective quark and lepton sectors are calibrated against each other mostly via energy limited beta decays. 

The 3-1 mass ratio of the sum of the b-c-s masses to charged lepton triple masses, while not a true coincidence, probably arises from two parallel examples of the same process evolving from the quark-charged lepton balanced root of both trees in the e-u-d triple, rather than mostly through direct weak force balancing acts mediated by W bosons directly between the heavy hadronizing quarks and the charged lepton sector.  It may also be driven by the 3-1 asymmetry of weak force boson decays between quarks and leptons (because quarks have three colors while leptons have just one).

The theoretical motivation for a massless up quark from the solution it provides to the strong CP problem and implications of Koide's formula are stronger than those used as justifications in inconclusive experimental measurements of light quarks confined in hadrons and in notable lattice computations, none of which are rigorously justified.

The near zero up quark mass is attractrive because it makes multiple lines of Koide formula analysis all fit together neatly while keeping the Koide formula piece clean, and because is solves the strong CP problem, while having only a highly inaccurate, not very rigorous or carefully considered basis in either experimental measurements (which are highly model dependent) or lattice calculations (which engage in a fair amount of hand waving). 

Basically, the experimental data don't clearly exclude a massless up quark, and the assumptions that drive the lattice results are pretty hand waving and ad hoc without really compelling justifications necessitating them once one weighs them relative to the somewhat compelling justifications for making other assumptions and one sees the larger mass matrix context.

A massless electron neutrino or up quark would provide an attractive starting point for Koide's formula like mass ladder calculations that are mostly from first principles.

A true massless up quark also has theoretical attractiveness because it provides one of the necessary anchors from which a Koide waterfall explaining all of the quark and charged lepton masses, and probably all of the fundamental fermion masses can be derived. One would also need a fundamental fermion mass scale constant such as the electron mass, and probably also the Weinberg angle, a weak force coupling constant, and a W boson mass to fill out the mass matrix and mixing matrixes.

It is also possible that the Koide formula massless up quark prediction arises mostly from using triples when there are actually leading order contributions from four fermion masses, not three.  The cases where Koide's formula fails to match experimental measurements are those where an omitted fermion mass term is likely to have the greatest effect.

I suspect that some combination of a fully weak force interaction based model that converts CKM matrix entries into fundamental masses relative to the overall fundamental particle mass scale by a formula somewhat more complete four input rather than three input formula, and a clarification of terms regarding which kinds of masses we are talking about, as well as an insistence on greater rigor and assumption clarity (including a clearly alternative hypothesis model to test) from the folks doing the lattice work and experimental work in quark mass determination could probably produce a consensus answer.

In other words, on the theoretical calculation side of a mass matrix formula for each quark involving a target quark and all three quarks that interact with it, the omission of the bottom quark in the calculation of the u-d-s triple (which accounts for only one in 83,050 W boson couplings of up quarks in the non-energy limited case, but involves a particle with a mass of about two million times the standard up quark mass estimate  for an average impact of 0.5 MeV per coupling if the probability and mass simply multiply relative to a 2 MeV measured mass) may be much more material to the correct calculation than, for example, the omission of the up quark in the calculation of the t-b-c triple (with the same 0.5 MeV per coupling impact on a measured 4,190 MeV mass), or the omission of the down quark in the b-c-s triple (a roughly one in 20 W boson coupling event involving a mass estimated at about 5 MeV modifying a mass with a measured value of about 1,350,000 MeV).  Similarly, the c-u-d triple may not work because it omits the 173,400 MeV mass top quark interaction from the calculation, even though those interactions are present in only one in 13,458 non-energy limited W boson couplings - the average impact per coupling, if that is a meaningful or correct number is ).

Only when the omitted quark is a light up or a light down will the impact of the omission be small enough to ignore - allowing you to use the incomplete Koide sequential triple estimate instead.

Neutrino masses are probably generated via the W boson interactions of neutrinos in a manner very similar to those of all of the other fundamental fermion masses.  But, the large mixing angles of the PMNS matrix and the small base neutrino masses, make ignoring any of the four terms that apply to each fermion mass determination more problematic, and may make indirect or circular NNLO or more remote terms important in a way that they are for the other fermions.  Koide triples fail for these masses (as demonstrated by Brannen) because ignoring any of the four leading order terms has a material impact.

Neutrino masses are surely interaction driven as well, but the large mixing angles of the PMNS matrix, the low masses involved, and other factors may make the NLO or NNLO terms in that analysis more relevant and less dominated by simple relationships of a couple of the terms than is the case in the sequential quarks and the generational triple.  A complete four particle input formula and possible indirect term feedback between different sets of four particle inputs that almost completely balance out and aren't suppressed the way that they are because of low transition probabilities driven by high mass differences for charged leptons and quarks, may make the neutrino mass math under a rigorously generalized W boson exchange model of fundamental particle masses quite a bit tricker and unsuitable for resolution via the Koide triple shortcut.

All particles that interact via W boson couplings have non-zero mass, even if it is close to zero.  The benefits of massless up quarks and electron neutrinos can mostly be realized even with negligible non-zero masses because of the way that Koide's formula works in practice.

There is really no very solid theoretical reason to single out electron neutrinos or up quarks for special treatment in anything other than a toy model that assigns zero masses to them.  The strong CP problem and calculation benefits of very low masses for these particles isn't greatly impaired if the masses are simply very low and not zero.

The elegant logic of the hypothesis set forth here is that every weakly interacting particle should have fundamental particle electro-weak rest mass, and that every particle that does not interact weakly should not have fundamental parrticle electro-weak rest mass (although it could acquire mass, for example via strong force interactions if it has color charge, as hypothetical glueballs and confined low energy gluons are assumed to in many lattice QCD models).  This is what we see in the Standard Model: photons and gluons which are the only particles that do not have weak force interactions (and also hypothetical gravitons) lack rest mass.

Among the notable consequences of the weak force interaction origin of fundamental particle mass is that sterile neutrinos cannot have fundamental particle rest mass unless it derives from some non-BSM interaction (such as the U(1) interaction proposed with the dark sector in a recent gravitoweak unification proposal).

The Higgs boson plays a much more minor role in generating fundamental particle masses than the Standard Model assigns to it.  It's mass, spin, and electromagnetic charge are derived in a quasi-composite manner from the aggregation of the properties of  one W+ boson, one W- boson, one Z boson and one photon according to rules analogous to those applicable to hadrons.

Higgs boson interactions are, I suspect, actually a sideshow in this process and the view that arbitrary yukawan interaction strength terms of the Higgs boson with different fundamental particles is the source of their fundamental mass is probably wrong. The Higgs boson while probably technically fundamental (in the same way that the W and Z and photon are conceived of as combinations of more fundamental electroweak bosons) is conceptually likely to be best understood a composite or fused version of the four electroweak bosons in a very pure way unaffected by strong force or gravitational physics. The Higgs is probably basically a bit like a four point Feyman diagram that is a theoretical possibility that is mathematically necessary to consider when calculating high energy physics interaction results, but only every so slightly tweaks the overall outcome. The real story in mass generation, however, in my view, mostly involves W and Z boson interactions with the Standard Model Higgs boson interaction having only a bit part.

This hypothesis is motivated by the neat coincidence of the Higgs boson properties especially its newly determined mass with these proposed rules.  The rules are:

*to get the combined mass add the contributing boson masses and divided by the square root of the number of bosons involved in the combination (this is analogous to a formula used in linear combinations of hadrons, but cleaner due to the absence of color charge considerations; there is no mass rule for hadrons and there is a different mass rule for atomic nuclei and for molecules which have additive mass modified by binding energy which is trivial outside atomic nuclei)
* the combined charge is equal to the charge of the component bosons (this is the same as the rule for hadrons)
* the combined spin is equal to the sum of the component boson spins with particles and antiparticles in the combination taking opposite spin values and each other boson capable of taking either a negative or positive spin (this is the same as the rule for hadrons)
* every particle in the combined particle must have a coupling of some form to at least one other particle in the combination (this is also true in other composite particles, although not in such a general form - in hadrons and atomic nuclei the primary coupling is the strong force or is derived from it, in molecules electromagnetic forces create the coupling).
* a particle and antiparticle of the same type cannot be present in combination unless at least one other particle has couplings to both of them (e.g. both the photon and Z couple to both the W+ and W- thereby buffering them from each other) (in hadrons made of particles and antiparticles gluons provide that buffering).

Problems addressed by the implied underpinnings of Koide's formula

This approach eliminates the premise upon which the hierarchy problem which requires all of the fermion and boson contributions to the Higgs boson mass to almost perfectly cancel because this hypothesis supposes that the Higgs boson mass has a different origin.

If the Higgs boson mass is actually a simple combination of the weak force boson masses, rather than primarily a horrendously complex set of cancelling interactions with all of the fundamental fermions, then the hierarchy problem, as it is usually formulated goes away, although really, it just migrates to a more understandable location.  The question becomes not, "why do the Higgs boson terms magically cancel out to produce the observed result for Higgs boson mass", but "why does the W boson have the mass that it does?", which would probably have a ready answer and not remain a mysterious unsolved problem in physics.  The hierarchy problem is probably just a product of misconceptions about how the Higgs boson really gets its mass.

This model dramatically reduces the number of experimentally measured Standard Model constants

The masses of every particle are contrained by three other particle masses interacting via the weak force, all of which must balance simultaneously.  The weak force CKM and PMNS transition probabilities must be unitary.  The mixing matrixes are determined by the mass matrixes and the mass matrixes are determined by the mixing matrixes.  The W and Z boson masses themselves may be possible to tease out of their interaction profiles with all of the weakly interacting particles in the Standard Model so these masses must be consistent throughout.  The W and Z boson interactions in addition to mixing matrix terms, are governed by the weak isospin and electromagnetic charges of the Standard Model particles (which are discrete theoretically specified properties of the various fundamental particles that can be derived from appropriate Lie group representations) and by the weak force coupling constant.

These contraints, and a few principles relating to the three generation mass hierarchy of quarks and leptons probably completely constrains the relative masses and mixing matrix enteries of the Standard Model, or perhaps reduces these values to a one dimensional parameter space.  The renormalization group running of the electromagnetic and weak force coupling constants according to their theoretically specified beta functions up to the GUT scale provides a dimensionful scale for electroweak interactions that probably fixes a point in this one dimensional mass matrix and mixing matrix parameter space.

I suspect that you wouldn't actually need additional constants for the neutrinos (and an up quark mass of zero would also suggest the likelihood of an electron neutrino mass of zero and only two rather than three massive neutrinos, although negligible masses for both of them are also attractive), and itis probably possible in this scheme to derive a W boson mass from first principles from an assumed zero mass for the electron neutrino and up quark or an electron mass or the GUT scale, together with the Weinberg angle, and a weak force coupling constant. 

There are some conceptual chicken and egg issues involving the deeply intertwined mixing matrix and mass matrixes that can be derived from each other in this extension of Koide's concepts. It is possible that these could be resolved simply via quantum numbers but this isn't obvious either. If not, we may have to accept a considerable number of constants in one sector or the other, or have to use assumptions that give good first order approximations and then iterate those approximations until an exact result is produced from first principles.  It might be intractable to simultaneously solve all of the relevant equations to get all of the necessary outputs, one might instead build a Koide's formula ladder of masses, use that to derive a mixing matrix, and then refine these values with interative numerical approximation that adjust calibration points until all of the masses and mixing matrixes are theoretically determined to a requisite level of precision.

Thus, you might be left with a "Within the Standard Model" theory that fully determines all of its currently experimentally measured parameters from nothing more than the weak force and electromagnetic coupling constants and Weinberg angle, and the theoretically set forms of the equations, quantum numbers, number of particle generations, and so on.  Indeed, I think that you can use the Weinberg angle somehow to theoretical determine one of the coupling constants from the other, so you may need just one coupling constant and one angle to fully determine every other Standard Model parameter in this approach.


The hypothetical theoretical rules sketched ouut above for deriving all or most mass matrix and mixing matrix entries from a scheme related mostly from globally considering weak force interactions and applying composite particle-like rules to the Higgs boson's properties is a pretty good day's work:

* It replaces twenty-one or more experimentally measured Standard Model constants (14 out of 15 masses and 7 out of 8 mixing matrix angles with theoretically determinable values).
* It solves the hierachy problem.
* It solves the strong CP problem.
* It clarifies the mechanism by which the mass matrix and mixing matrixes are dynamically generated in a way that may permit consideration of exotic states where mass-energy limitations or the lack thereof lead to new physics.
* It may provide an alternative way from the usual Standard Model for constraining proton decay with a very low up quark mass.
* It resolves the question of neutrino mass generation (they would have Dirac mass).

It also leaves some questions possibly unsolved, although some might not make as much sense in this context:

* It doesn't provide much insight into electroweak symmetry breaking relative to the Standard Model.
* It does not integrate the Standard Model forces into a single Lie group or Lie group representation.  It is still SU(3)*SU(2)*U(1) with three generations.
* It doesn't necessarily provide much insight into the Higgs vev (which may not be part of the revised structure), or the cosmological constant or vacuum energy or vacuum metastability.
* It is implicitly 3+1 dimensional, just like the Standard Model and General Relativity and those dimensions arae put into the theory by hand rather than emerging naturally.
* It doesn't provide any quantum gravity theory (I suspect that quantum gravity efforts of some sort in the high energy regime solve the non-convergence of the strong force coupling constant and electroweak coupling constant causing them to unify like a GUT).
* It doesn't provide any gravity modification or particle to provide a mechanism for dark matter effects (I supsect that the solution to this may have to arise within a quantum gravity sector of some kind.)
* It doesn't address how matter-antimatter asymmetry arose (although I have a dynamical scenario conjecture for that as well involving the Big Bang evolving both forward in time in our mostly matter universe and backward in time in a mostly antimatter universe).
* It doesn't provide much insight into CP asymmetries at first glance at any rate.

It also lays the framework to a more fundamental theory beyond what I have sketched out.  In this the "ultimate answer" it may be that the electroweak bosons are composites or linear combinations of fundamental electroweak bosons in some sense (as is the case in Standard Model electroweak unification).  It may also be the case that the fundamental electroweak bosons and the fundamental fermions can all be generated by (1) different modulations from a finite set of possible string vibration modes, (2) a finite set  at least up to Big Bang energies of possible pertubations of space-time itself, or (3) a very small number of preons (perhaps one to three of them). 

By any of these means you are looking at leap directly from a pre-electroweak unification Standard Model directly to a single kind of final preon layer along the lines of a string with properties that can be varied in precisely the right number of ways which is a bit more than a hundred (72 for quarks, 8 for gluons, 12 for charged leptons, six for neutrinos, four or so for the electroweak bosons (with a composite Higgs), one the graviton (or perhaps more along the lines of graviweak theory), plus a possible gravitational sector sterile neutrino and dark matter boson).  Also at that stage it would be necessary to embed these Standard Model strings or space-time pertubations or preons in some sort of quantum gravity model that could also supply gravity and the other elements of general relativity.  Multiple approaches are available.

But, the kind of extension of the Standard Model that I have sketched out as a possible theoretical basis for a Koide's formula-like phenomonology could bring the Standard Model without going deeper than its current interactions and parricles (just reimagining them slightly at that level of resolution), would leave humanity with just one or two more layers of the Onion to peel before reaching what would amount to a "Theory of Everything" and would make the path that would need to be followed to get there well defined, soluable in principle at least, and relatively straightforward given the intermediate work already done in pursuit of BSM models in gravity and string theory, for example.

HUGE CAVEAT: This is not peer reviewed physics.  This is educated layman conjecture.

I am not saying that I have solved any of these problems or that I have a final theory or that it works.  I am saying that I have an incomplete and sketchy vision of a general outline of what a theory that would give rise to Koide's formula might look like in broad brushstrokes at a conceptual, educated layman's level. 

These are conjectures about points that are already conjectures.  Even if the real theory that makes Koide's formula work some of the time is vaguely like the one that I have outlined, there is every reason to think that one or another key detail that I have identified is just plain incorrect and does not reflect reality.  Also, even when I get something right, I may end up doing so for the wrong reasons. 

For example, I may in some sense have the conceptual balacing act that goes into a Koide's formula-like texture of the mass matrixes right, but it may really arise from a more traditional Higgs mechanism that I deemphasize the importance of in this post.

But, at least, there is some articulated sense of how a more rigorous extension of the Standard Model that is a WSM theory (as opposed to a BSM theory) could give rise to what we observe in a way that is a better fit for the feel of how quantum mechanics and the Standard Model in particular conceptualizes things and finds solutions to issues.


andrew said...

Post-script conjecture:

I wouldn't be surprised if it was possible to prove from first principles that if fundamental fermion rest masses and mixing matrixes are determined entirely by a W boson coupling mechanism, rather than the SM Higgs mechanism, that a generalized Koide ratio mass relationship for interacting particles in a necessary and unique element of any globally consistent solution that assigns values to these parameters, or that it is necessary produces a solution that is unique at any given renormalization group energy scale (providing a one dimensional parameter space of valid solutions).

Mitchell said...

If I was trying to make a theory in the image of this proposal, I would think about two potential ingredients:

Xavier Calmet's preonic theory of the weak interaction. The reason is that it places the Higgs and the W in the same family of composite bosons - see page 29 for the details - which would make something like the H,W,Z sum rule more plausible.

The other is the chiral graviweak unification discussed in this PF thread - I mean the basic idea that weak SU(2) and Ashtekar SU(2) are opposite chiral components of the same connection.

Incidentally, Calmet has also argued for a resemblance between weak and gravitational interactions.

Occasionally you get people on the forums asking how the Higgs boson is related to the graviton, since they have heard that the Higgs "creates mass". If we could combine your ansatz with chiral graviweak unification, there really would be a tight relationship between the field that creates mass and the field that responds to it.

andrew said...

A Koide triple t-b-c using existing t and b masses implies a mass of 1.356 GeV for the charm quark. This is about 7.9% more than the current 1.257 GeV experimental value for the charm quark, which is almost 4 sigma from the current value given much improved precision in this value as of 2012.

This is a real problem for quark Koide triples and even my Ansatz since the exclusion of the fairly modest probability c-d transition probably shouldn't have such a huge effect although the d is effectively zero relative to the s and b masses. However, the Koide ladder value of 92 MeV for the strange quark remains right on the money.