**Do electroweak self-energy fields explain the masses of the first generation Standard Model fermions?**

One observation that I have now seen in quite a few papers in the last couple of weeks resurrects a century old notion about the rest mass of the electron, which, it turns out, is very close to the mass associated with the electromagnetic potential energy field that an electron generates if one assumes that rather than being point-like it has a radius which dimensional analysis naturally suggests, and make some clever choices about how to do the math as one recent preprint on the topic that I read found a way to manage.

The empirical electron mass fixes an electron radius (as expressed already more than 100 years ago by Lorentz (and Poincare)) by the formula e^2/r ≈ me*c^2: for r ≈ nuclear radius (= 2.8 · 10−15 m), the mass comes out to be ≈ 1/2 MeV.In other words, perhaps the electron is massless except for the inertia generated by dragging its potential energy field around using the field between two electrons to estimate it.

This charge radius, by the way, isn't simply a physically irrelevant curiosity. While we now think of the electron and other fundamental particles as fundamentally point-like, the charge radius physically governs the scattering cross section (i.e. likelihood of fundamental particles a given distance from each other interacting) in a matter heuristically similar to the probabilities of conventional spheres of the same size colliding with each other if quantum mechanics did not apply.

The weak force between two neutrinos is about 10^-5 weaker than the force between two electrons. Since this factor is squared in the self-energy equation, assuming that the radius of a neutrino is about the same as the radius of an electron, you get the experimental result that the electron neutrino mass which is about 10^-10 less than the electron mass.

Alternately, the 10^-10 ratio of the electron mass to the electron neutrino mass, give or take, is really 10^-5 to 10^-6 due to the ratio of the electromagnetic force strength to the weak force strength, and the balance of the ratio comes from the fact that the appropriate effective radius of a neutrino, which is about 10^-15 for an electron in the proper units, the same as an electron, for a neutrino in those same units.

One can also imagine neutrino rest masses arising entirely from the masses of their W boson virtual particle fields, which while non-zero, are very, very small because quantum tunneling over the huge virtual particle energy hurdle from the near zero neutrino mass to the W boson mass and back again dramatically suppresses the probability of generating virtual particles that contribute to its self-energy. In contrast, since the electron has no virtual particle hurdle to generate at all to cross to generate a zero rest mass photon, and the weak force self-energy is only a little bit more than that of the electron neutrino because the gap between the electron mass of 0.000511 GeV and the W boson mass of about 80 GeV is still so immense, and because it has no color charge self-energy, the self-energy inertia that an electron generates very cleanly matches its rest mass. All the complicating confounds to muddy a self-energy source for first generation fundamental fermion mass in this case are so negligible or absent that the one clean relationship between rest mass and self-energy that remains stands out true and clear (just as the Koide relationship of the charged leptons, with such a negligible neutrino contribution to muddy up the relationship, is so exact as explained in more depth below).

Of course, the trouble with this terribly elegant source for the ground state lepton masses is that it still leaves us without a clue as to why second and third generation leptons of the same type are so much heavier than those of the first generation. Also, it leaves unclear what the source of the up and down quark masses should be.

The up quark, of course, has a charge of +2/3, while the down quark has a charge of -1/3. And, empirically, our best estimates of the relative masses of the up quark and down quark suppose a down quark that is twice as heavy as the up quark, which is inversely proportionate to the relative charges of the particles - the opposite of the result we would naively expect based upon an electroweak self-energy relationship if the up and down quarks had the same classical radius.

The fact that the up and down quark rest masses are roughly four and eight times heavier respectively than the electron mass (0.511 MeV), i.e. about 2.44 MeV for the up quark and about 4.88 MeV for the down quark, given best fit experimental measurements which admittedly aren't very precise, when the up quark has an electromagnetic potential field that would be 4/9ths as strong as the electron, and the down quark has an electromagnetic potential field that would be 1/9th as strong as the electron, respectively, if they each had the same charge radius of the electron, is notable.

Put another way, the classical radius of the up quark is 1/9th of the classical radius of the electron, and the classical radius of the down quark is 1/72nd of the classical radius of an electron. These inferred smaller classical charge radii for quarks (which can't be observed in isolation) is consistent with the fact that the classical charge radius of the electron is about 2.82*10

^{-15}m, the measured classical charge radius of the proton (with an identical and opposite charge) is about 0.88*10

^{-15}m (muonic hydrogen inexplicably, however, has a measured classical charge radius of 0.84*10

^{-15 }m). Specifically, the classical charge radius of an up quark is about 0.31*10

^{-15}m and the classical charge radius of a down quark is about 0.04*10

^{-15}m.

Apparently, something different is going on with quarks than with leptons. Perhaps, unlike first generation leptons, even first generation quarks they have some rest mass from some source beyond the rest mass arising from their electroweak interactions.

There is no real indication the QCD has anything at all to do with fundamental fermion mass in quarks, except for the fact that it needs different flavored quarks to have some kind of different properties from each other so that it can have six different distinguishable flavors, but what if? (Query for future investigation: what would 1 light or massless flavor, 3 color QCD ignoring electroweak effects look like? - with an eye towards quantum gravity applications).

But, is it really just a coincidence that all of these classical charge radii are on the same order of magnitude as the distance of a bit more than 1*10

^{-15}m at which the strong force that governs quarks, but not leptons, is strongest, when this distance has no special significance to the electromagnetic force that seemingly is the dominant contributor to the rest mass of the electron?

What if the charge radius of a "bare quark" was the same as the leptons, giving an up quark 0.23 MeV of rest mass from electromagnetic self-energy, and the down quark 0.06 MeV of mass from electromagnetic self-energy (disregarding the comparatively negligible self-energy contribution of weak force self-energy interactions via Z bosons in up quarks, down quarks and electrons which should be approximately identical to the electron neutrino mass of about 0.000000001 MeV). This would be a residual of about 2.21 MeV in an up quark and about 4.82 MeV in a down quark.

Is the simple self-energy model too naive? Perhaps the electron gets some of its mass from flavor changing W boson mediated mixings with muons and tau leptons, and only the remainder from its own self-energy from electromagnetic field potential has has a different than canonical true classical radius? And, similarly, perhaps the residual rest masses of the up and down quarks have a source in their W boson mediated mixings with other quarks.

If we could merely find some plausible reason for these scaling factors, when the classical charge radii pertinent to determining fundamental particle rest masses from the self-energies of first generation fermions, which is the same for leptons, but much smaller and different from each other, for first generation quarks, we would be well on our way to a deeper understanding of the source of the mass constants in the Standard Model.

At any rate, looking through a glass darkly on can imagine that with just two or three more leaps of insight, one could develop a fundamental calibration point from the electroweak potential self-energies for each of the first generation fermions of the Standard Model.

Of course, it also doesn't help that the light quark masses are ill defined, an issue that I've been ignoring so far. Extrapolating perturbative QCD formulas for light quarks down to the up and down quark scale would suggest "dressed" quark masses that make the mass difference between the proton and the neutron, and the low mass of the pion, impossibly small. One can (and does) use a subtraction scheme like MS, rather than the more natural "pole masses" of the up, down and strange quarks (or for that matter, for all hadronized quarks in some applications). But, it isn't at all obvious that this is the right way to think about these fundamental quantities for theoretical purposes, as opposed to for the practical work of doing QCD with confined quarks in hadrons.

Aesthetically, there is a lot to like about a self-energy driven approach to setting the masses of the first generation fermions.

1. This approach makes it possible to determine these masses from first principles removing four parameters from the Standard Model using ideas that are a century old and were seriously considered by the founders of modern physics even though they couldn't figure out how to make it work at the time.

2. It means that the entire debate over Dirac mass v. Majorana mass for neutrinos, that has in the Dirac mass case cast doubt on the universality of the Higgs mechanism itself which relies on Higgs field fueled parity oscillation to extend to these fermions even though it extends to all of the others. In this approach, the same fundamental physical process naturally gives rise to the rest mass of each of the fundamental fermions.

3. It provides a natural explanation flowing from the nature of the particles themselves for why neutrinos are so light, and a sensible motivation for why the Standard Model's natural tendency towards massless fundamental particles that it had in earlier incarnations is fundamentally unworkable in a set of consistent and universal physical laws that incorporates Einstein's observation that mass and energy are equivalent.

4. A calibration of the first generation fermions from self-energies would allow the absolute scale of the Standard Model particle masses to be set in a bottom up, rather than a top down, manner as the Standard Model does today (thereby solving the hierarchy problem), This would give us a firm footing from which we could renormalize the Standard Model up to a GUT or Planck scale (although a quantum gravity adjustment to each of those those beta functions would probably be necessary), rather than the current system where more often than not we assume that the laws of the universe are set at a high energy scale chosen somewhat arbitrarily without empirical guidance, to give rise to a low energy effective field. This would put our speculations about a higher energy regime in the early universe on a more firm foundation.

Of course, this step alone doesn't get us to the promised land of explaining the rest of the physical constants for fundamental particle mass in the Standard Model, but it is progress at least.

Expressing a similar sentiment with another approach see here.

**Is Sterile Neutrino Dark Matter Theoretically Impossible?**

The article also makes the observation that in nature, everything that is possible is also mandatory, and that rest mass is possible, and therefore mandatory, for all charged particles. But, it fails to make the even more exact observation:

All weakly interacting particles have positive rest mass, while all particles that do not participate in the weak interaction have zero rest mass.As an aside, it is worth noting that there is a hierarchy of forces based upon their particle interactions.

Gravity interacts with all of the Standard Model particles and the hypothetical graviton also interacts with other gravitons on the same basis. All Standard Model particles with rest mass interact via the weak force, while no particles that lack rest mass (including the hypothetical graviton) interact via the weak force. The Standard Model Higgs field interacts with all particles with rest mass except neutrinos (since they can't engage in mass generating, Higgs field driven, parity oscillation), including the Z boson that lacks electric charge. All particle that interact via the Higgs field also interact via the weak force and gravity. All charged fundamental particles also interact via the weak force and the Higgs field and gravity. Quarks interact via the strong force, the electromagnetic force, the Higgs field, the weak force and gravity.

There is only one exception to this hierarchical pattern in the Standard Model as extended by a hypothetical graviton. Gluons interact via the strong force and via gravity, but they do not interact via the electromagnetic force, the Higgs field, or the weak force. They lack electric charge, weak force charge and rest mass, although they can acquire mass dynamically in infrared confinement situations. (The creation of a Higgs boson via "gluon fusion" is an indirect process. The gluons create intermediate particles that unlike gluons do interact with the Higgs boson, and it is the intermediate particles that the gluons produce, rather than the gluons themselves that produce the Higgs bosons.)

One takeaway conclusion that flows from that observation is that truly collisionless dark matter particles such as sterile neutrinos (as opposed to WIMPS that interact via the weak force), which have no strong force interactions, no couplings to photons or electric charge, and no weak force interactions, must also have a zero rest mass, because all particles that do not interact via the weak force lack rest mass.

Due to a lack of rest mass, sterile dark matter particles of any kind would always move at the speed of light.

Dark matter particles, by hypothesis, do interact via gravity, so they would interact with gravity with a strength equal to proportional to their own wavelength times Planck's constant, since this would have to be the only source of their own mass-energy in the absence of a rest mass.

In other words, if one applies the logic that weak force interactions are necessary for a particle to have rest mass as is true in every other case, and join that to the standard assumptions that define dark matter particles, then a dark matter particle must have all of the properties, with the possible exception of total angular momentum J aka spin, as a graviton. Indeed, they would look exactly like hypothetical Neo-Newtonian, spin-0 gravitons, in the non-self-interacting case, and exactly like the hypothetical spin-2 graviton in the self-interacting case. But, this reasoning is not dependent upon these dark matter particles being scalars. It applies with equal force to any particle, boson or femion alike, that lacks strong, electromagnetic and weak force interactions.

Of course, whatever its spin, any massless sterile neutrino would be "hot dark matter", due to its speed of light velocity. But, this is inconsistent with one of the axiomatic observational requirement for dark matter, because hot dark matter would blur out all small and not so small scale structure in the universe if it was a common as the Planck satellite observations and galactic rotation curve observations suggest.

Therefore, if the only way that fundamental particles can obtain a rest mass is through electroweak interactions, it follows that dark matter that does not consist of WIMPS cannot exist.

Yet, as I've previously discussed at this blog, there are very deep problems with WIMP dark matter. We know from W and Z boson decays that there are no undiscovered weakly interacting particles with masses of less than 45 GeV (soon to be increased to 62 GeV once we have enough Higgs boson decay data).

We also know that direct dark matter detection experiments have ruled out dark matter particles whose weak force couplings are a strong as a neutrino of that mass to considerably higher masses than 62 GeV. The LUX experiment excludes cold dark matter particles with a weak force cross section of interaction as strong as a neutrino at masses in excess of 1,000 GeV (aka 1 TeV). Yet, the flaws of CDM theories relative to WDM theories are particularly accentuated in the case of dark matter with masses of 1 TeV or more.

Also, those exclusions are probably understated, because they assume a dark matter density in the vicinity of the solar system that is roughly half of what a recent precision observation of RAVE stars in the Milky Way galaxy used to calibrate the parameters of a cold dark matter halo for the Milky Way, and other recent precision measurements of the Milky Way, require.

But, this provides a good theoretical reason to reject the possibility of dark matter at all, in favor of gravity modification theories to explain dark matter phenomena, unless:

(1) a fifth force Yukawa self-interactions confined exclusively to the dark sector on the same order of magnitude in strength as the electromagnetic force, or

(2) the possibility that warm dark matter critics of cold dark matter theorists are wrong and resolution of almost all of the apparent flaws of CDM theories in N-body simulations with the inclusion of both baryons and dark matter particles is correct (something that sounds less plausible when one learns about some of the empirically and physically unjustified but little discussed tweaks to those models that are necessary to get them to reproduce the galactic scale of the universe even with baryon interactions), or

(3) the one other solution that comes to mind is some sort of composite dark matter particle (perhaps with a mass at the warm dark matter scale) that substitutes the binding energy holding its particles together for fundamental particle mass, in much the same way that light hadrons get 98% of their mass from the binding energy contained in massless gluons. This two would require a self-interacting fifth force restricted to the dark sector, but would have a very different character than the forces proposed in self-interacting dark matter theories. It would have to be a very stable confining force like the strong force, rather than an intermediate distance force like the kinds usually proposed in self-interacting dark matter theories. Indeed, such a confining force could even operate on already known Standard Model particles, like neutrinos, by confining a large share of them to produce composite dark matter, and thereby creating just one new particle (a short range force carrying boson), rather than two. Basically, this applies to the possibility that there is no viable fundamental particle that is a dark matter candidate, the same strategy that Technicolor theory applied to the possibility that there was no fundamental particle to serve the role of the Higgs boson in the Standard Model by proposing a composite particle alternative bound by a previously unknown fifth force.

Furthermore, keep in mind that any SUSY superpartner particle must have the same force couplings as its superpartners. All of the hypothetical SUSY superpartners except the gravitino (even neutralinos) would have mass and interact via the weak force at least as intensely as neutrinos do, and most interact via other Standard Model forces as well. And, while gravitinos are expected by many supergravity theorists to have a mass of 1 TeV or more, possibly allowing them to escape detection by current direct dark matter detection edperiments, it is well known that its is problematic to fit heavy gravitino dark matter candidate into the available cosmology and dark matter phenomena data And, of course, the mechanism by which a hypothetical gravitino could acquire mass at all, unlike its massless superpartner the graviton, is far from obvious.

The stark lack of good SUSY dark matter candidates, now that empirical evidence is starting to impose more strict qualifications on potential dark matter candidates, is particularly notable because, to hear SUSY advocates tell the story (and a large share of the theoretical physics community fits that description), SUSY is popular because it is only of the only beyond the Standard Model theories that can deviate from the Standard Model in a manner that does not directly contradict experimental data, if the parameters of a SUSY theory are chosen carefully.

In contrast, no such deep objections exist to subtle modifications of general relativity that lead to the kind of Yukawa term for self-interactions of gravitons with each other, that can reproduce all or almost all known dark matter phenomena at all scales from dwarf galaxies to galactic clusters and the Bullet Cluster, as demonstrated by people like J.W. Moffat and Alexandre Deur. And, Moffat has made some progress in showing that his gravity modifications can also produce a viable cosmology.

Deur's analysis, meanwhile, suggests that his gravitational analysis may do little but provide an alternative means of describing dark matter without a cosmological constant, which might not need to be as big or might even be possible to eliminate due to non-linear variations in gravitional field strengths around asymmetric matter distributions. This is because the non-linear graviton self-interaction effects that he has described to explain dark matter phenomena have little or no impact in spherically symmetric systems like those in Big Bang cosmologies and the analysis of black holes where his analysis does not part ways with conventional GR research into cosmology and black holes.

Of course, an extremely modest tweak to the equations of general relativity, the cosmological constant term, is already sufficient to perfectly reproduce all observed dark energy phenomena, even if a modification of gravity that explains dark matter phenomena does not do so. Dark energy is a solved problem in fundamental physics. It is solved via terms in the GR equations that Einstein himself was aware of, not exotic particles or fields. This may not be the only way that this problem can be solved, but since it is a solved problem, finding alternative solutions to it is not an urgent problem in fundamental physics in my opinion.

**Why Are There Koide Triples And What Drives The Electroweak Boson Masses?**

My other takeaway conclusion from that observation that all weakly interacting particles have rest mass, while all particles that don't interact weakly do not, is that we may be wrong about the true source of Standard Model fundamental particle masses.

It could be that rather than largely being a function of Higgs Yukawa couplings, that this really arises dynamically mostly from W boson interactions with a slight additional Z boson contribution.

Since the W boson is the only mechanism by which a quark or charged lepton can change its flavor, in accordance with an explicitly generation considering CKM matrix, it makes all sorts of sense that the W boson ought to play a fundamental role in both electroweak mass generation of fundamental Standard Model particles in general, and in particular, in the mass differences between the masses of fundamental particles that are in all other respects identical.

One can fruitfully seek a W boson driven mass generation mechanism as at the heart of why Koide's rule works for leptons perfectly (since Koide's rule captures all of the inputs to the W boson interactions of the charged leptons weighted somehow for the masses of the respective generations, but ignoring the interacts with neutrinos since their tiny masses result in those interactions having almost zero weight), and why it works approximately for Koide quark triples, with the best fits being those where the triple captures the largest share of the potential W boson transitions that the middle quark in the quark triple could experience, with deviations from correctness on the same order as the W boson flavor changing transition probability that is omitted from the triple for its middle member, times the mass of the quark whose transition is omitted.

If one had a correct generalized form of Koide's rule available, one needs only two calibration points per Koide cascade to complete the mass matrix from first principles, something that would easily become possible if all four of the first generation fundamental fermion masses in the Standard Model could be fixed from first principles based upon the potential electroweak self-energy of these particles and some suitable theoretical justification for a scale up factor for the up and down quarks respectively.

Of course, if one sees the W boson as the driving force behind mass generation of Standard Model fermions, rather than the Higgs field, then one is free to see Higgs field interactions, not as an orchestra conductor for the Standard Model, but as a dependent variable that arises from the underlying W boson driven fundamental fermion and boson rest masses, and the Higgs boson mass itself, as a residual particle creating a degree of freedom that allows the Standard Model to make the sum of the square of the fermion masses exactly equal to the sum of the square of the boson masses, when each of those masses is renormalized to take its value at the Higgs field energy scale.

Likewise, in this analysis, one can alternately explain the heaviness of the top quark on the grounds that the top quark mass is the degree of freedom necessary to balance this out, while setting the Higgs boson mass to sum of the masses of the four electroweak bosons (W+, W-, Z and the photon), divided by the square root of four (the number of electroweak bosons) when the masses of all four of the bosons in that equation are renormalized to their values at the Higgs field energy scale. Given the propensity of bosons to mix, this seems eminently sensible.

This approach wouldn't remove the Standard Model Higgs boson or the Higgs field from the Standard Model. But, it would reframe these aspects of the Standard Model in a way that would give us a new perspective on why they matter that would put the familiar hard working W boson at center stage.

Moreover, if one can resort to self-energy from electromagenetic and weak force fields combined in the case of the electron, and from the weak force field along in the case of the electron neutrino, as a means of establishing calibration points in the fermion mass matrix, and perhaps if one can use renormalization of the Higgs boson mass from a metastable zero value at the Planck or GUT scale down to the electroweak scale, one can pick up a third fundamental calibration point for the mass matrix, and with these calibration points and a generalized Koide triple mass generation mechanism that seeks to harmonize the masses of different states that a particle can transform into via the W boson relative to each other, then one is close to deriving the entire mass matrix of the Standard Model from first principles.

Also, while we hypothesize that each of the neutrinos have mass (because they interact via the weak force), one could imagine that the electron neutrino actually violates this rule and really has a zero mass with provides an anchor for the other mass matrix masses, even though the other neutrinos have mass (although having mused over this possibility for hours on multiple occasions, I'm ultimately pretty convinced that this will not turn out to be the case, mostly because W boson interacts of electron neutrinos with charge bosons, while suppressed by the quantum tunneling issue, have more than a zero probability, and because neutrino oscillation may add a layer of mass mediation between neutrino flavors in addition to W boson exchanges).

**Do The Mixing Matrixes Use All Of Their Possible Degrees Of Freedom?**

Meanwhile, the probabilities that come into play in W boson interactions arise from the CKM matrix. And, it so happens, as pointed out in a previous conjecture at this blog, that it appears to be possible to parameterize the entire CKM matrix apart from CP violation, with a single parameter (basically the Weinberg angle), even though, in principle, it could require as many as three parameters to do so.

And, it also looks reasonably plausible that it may be possible to parameterize CP violation in both the CKM matrix and the PMNS matrix with a single shared CP violation parameter.

If a similar non-maximal parameterization can be achieved in the PMNS matrix, or better yet, if the remaining CKM and PMNS matrix parameters can somehow be derived from the coupling constants, or have singular parameters that have some functional relationship to each other, leaving just two mixing matrix parameters that might in turn somehow be possible to derive from the Standard Model coupling constants in a degree of freedom reducing way.

If so, the plethora of Standard Model constants starts looking much more tame and logical, with seemingly "unnatural" relationships of constants in it having perfectly sensible relationships once the fundamental values that drive the true value of these constants is better understood.

I am also particularly struck by how close the CKM matrix decreed probability of a first to third generation transition is to the product of the probability of a first to second generation transition times a second to third generation transition.

It also would only take one more leap of logic to find a well motivated functional relationship between the Cabibbo angle (which basically relates to the probability of a down quark decaying to something other than an up quark) and the electroweak mixing angle (which relates among other things to the ratio of the W and Z boson masses), which are very similar, but not identical to a high degree of statistical significance, allowing us to remove yet another parameter from the Standard Model. The electroweak mixing angle also governs the relative strengths of the strong force and weak force coupling constants. So, this innovation, together with the other ones discussed above, if the leap of insight could be secured, would allow us to derive all of the Standard Model masses and mixing angles from its three dimensionless coupling constants alone.

For another recent effort to find method in the CKM and PMNS matrixes, consider this paper reviewing the progress of the most recent data towards his 2007 prediction.

**Dynamical Gluon Mass, C, J, CP violation and the Strong CP problem**

The Boya and Rivera article also barely touches on another fascinating issue which is that in the non-perturbative infrared QCD regime explored with lattice QCD for quark behavior within confined hadrons where they are asymptotically free, gluons dynamically acquire significant mass, with most of a hadron's mass being localized "in the glue" despite the fact that gluons by definition and to make the equations of QCD work have a zero mass at "rest", just like a photon.

I've hypothesized that the main reason that the fundamental QCD CP violation phase is zero is that the strong force, since it is mediated by a massless boson, the gluon which is always traveling at the speed of light, like the photon, does not experience time and hence cannot have a process which serves as an arrow of time. In contrast, the W boson, because it is massive, can experience this arrow of time, and does have a CP violating phase.

But, it is worth observing that the only time CP violation is actually observed is in electrically neutral pseudoscalar mesons, which contain confined gluons that are in the infrared regime where they dynamically acquire mass.

Thus, while we shouldn't expect the massless gluons that we observe in the ultraviolet perturbative QCD regime to experience CP violation since those gluons don't experience time, that doesn't mean that we shouldn't expect be surprised if the dynamically massive gluons of confined quarks in the infrared regime that can experience time should provide an arrow of time.

It could be that most of the observed CP violating phase is actually coming from CP violation in these dynamically massive gluons, rather than from an electroweak source.

How could this be?

It is impossible to observe CP violation in parity even bosons, such as the CP even Higgs bosons, the Z boson, the photon, true scalar hadrons like the sigma meson, or a hypothetical glueball. This is because symmetry makes a difference in forward and backward decay rates indistinguishable. Likewise, it is impossible to observe CP violation, due to symmetry considerations, in a boson that is a linear combination of exactly opposite asymmetric states that we see in the neutral pion, which is an equal blend of pseudoscalar up-antidown and down-antiup mesons.

One can also hypothesize that somehow or other, the presence of a non-zero net electric charge in a hadron, or the fractional integer total angular momentum J of a hadron, dramatically suppresses CP violation, perhaps by some mechanism similar to the GIM mechanism the suppresses flavor changing neutral currents, for example.

If these rules hold, then there are only three possible hadron ground states in which CP violations can arise: (1) the neutral kaon (which is a mix of states, K long and K short, but not a symmetrical mix of states like the neutral pion), (2) the neutral D meson, and (3) the neutral B meson. We are pretty sure that we have seen all three in practice, but it is very subtle in the neutral D and B mesons relative to the neutral kaon.

Also, it is notable that in all thee of these cases, the asymmetric transitions between different configurations of the same neutral mesons that lead to CP violating decays, are strong force transitions in which the dynamically massive gluons inside these confined hadrons play a role. And, the CP violating decays are strongest in the neutral meson where the dynamically generated gluon mass is greatest (the kaon which is the deepest into the infrared) and are more subtle in the CP violating decays where the gluons are more energetic and therefore acquire less dynamical gluon mass.

A CKM matrix CP violating phase has been fit to these three special cases in the Standard Model, but it seems absurd (however correct it may be) that one must add a new parameter that has some tiny effect on every single weak force interaction in the Standard Model in order to accommodate a phenomena that is observed in nature, even with ultra-precise measurements, in just three of the more than a hundred possible ground state hadrons of the Standard Model that can be constructed from constituent quarks and gluons.

And, at the time that a CP violating phase in the CKM matrix was proposed as a solution to this CP violation in neutral meson decays, lattice QCD simulations had not yet revealed that confined low energy gluons dynamically acquire mass in a way that is greater when the energy scales are lower, or even suspected that such a thing could be true, so there was no reason to consider this potential alternative source of CP violation in the Standard Model.

Of course, the two CP violating phases are not mutually exclusive. It could be that observed CP violation is partially due to weak force decay CP violation from the CKM matrix CP violating phase (after all CP violation does make sense in a massive particle that is maximally parity discriminating itself) and is partially due to strong force CP violation that is only possible when gluons dynamically acquire mass that is somehow suppressed in electrically charged mesons and in baryons.

It might take both of these factors to have enough of an impact to be detectable in a system as heavy as a neutral kaon, neutral D meson or neutral B meson (several GeV). But, the CKM/PMNS derived CP violating phase arising because of the mass of the W and Z boson force carriers, might be sufficient in the case of neutrinos where there is much less mass to force to engage in a CP violating flavor change.

Also, one possible mechanism for flavor change in neutrinos might involve a pair of virtual W boson interactions, whose virtual charged leptons cancel each other out, and which make CP violation in charged lepton decays so negligible that they can't be observed since it takes too much energy to pull that off with charged leptons relative to featherlight neutrinos (and because as charged fermions, the CP violation mechanism I suggest phenomenologically doubly suppresses CP violation). But, to get CP violation in a hadron decay, there is only one, on shell W boson involved, so the CP violating process is less suppressed.

I don't have the technical proficiency to create the QCD model, and then do the calculations with it, to determine if a strong force CP violation phase that only operates when gluons acquire mass dynamically in electrically neutral bosons, rather than via a tiny CP violating phase in the CKM matrix that influences every single W boson interaction in principle, could be fit to the data, although my intuition is that it could and that doing so would greatly simplify electroweak theory calculations and make a single parameter fit for the CKM matrix even easier than expected.

It is also worth noting, that while I am on record predicting a particular CP violating phase in neutrino oscillation physics, that zero CP violation in neutrino oscillations via the PMNS matrix is currently fully consistent with experimental evidence at this point. So, it could even be that CP violation, contrary to the Standard Model consensus for decades, is actually a strong force process rather than a weak force process and is driven by a CP violating phase in the QCD Lagrangian, rather than the electroweak Lagrangian.

Since the effect that I am suggesting is quite specific and well constrained by experimental CP violation observations in neutral mesons already, it wouldn't be hard at all, once a model was formulated, to determine the strong force CP violation parameter needed and its form, and to then compare the predictions of the existing Standard Model and this BSM variant with strong force CP violation and find some corner of rare but observable phenomena that could discriminate experimentally between the two possibilities in the lucky case that it is even possible to devise a viable strong force CP violation phase in a viable QCD Lagrangian that fits the current data at all.

Doing so, if one did, would also take the pressure off the increasingly not experimentally viable efforts to fit the data to a zero or negligible up quark mass - something that potential electromagnetic self-energy from the up quark arguably places a theoretical lower bound upon that is probably still too high, without having to resort to axions which are also not observed experimentally.

CORRECTED, UPDATED AND EXPANDED ON JANUARY 9, 2015.

## 4 comments:

Incidentaly, I am mentioned at the end of the paper :-)

I did notice.

Andrew,

There a lot of potentially good ideas in this post. I'd just like to expand on a few of them.

We know that there must be right-handed (i.e. sterile) neutrinos due to the fact that neutrinos have mass and are fermions. The mass of a fermion (due to the Higgs field) tells you how quickly it switches between the left and right spin direction (i.e. an electron switches between the left and right handed spin much faster than a neutrino.) When massive fermions are in the left-handed state, they can interact via the weak force, and when they are in the right-handed state, they do not interact via the weak nuclear force. When the Z boson decays into neutrinos, it makes a left-handed neutrino and a right-handed anti-neutrino. It does make a right-handed neutrino or a left-handed anti-neutrinos.

But eventually the left-handed neutrinos will flip-flop to being right handed, and vice versa for the anti-neutrino.

So, the question is not whether there are sterile neutrinos (there have to be right handed neutrinos if neutrinos have mass.) The question for warm dark matter is simply the following: when the neutrino flip-flops from left to right spin, does its rest mass change from meV to keV?

It seems strange, but it is possible in the see-saw and other Beyond the Standard Model theories.

As for the relationship between rest mass and charge, there is the monkey-wrench that the rest mass of the neutral Z boson is actually more than the rest mass of the W charged bosons.

For me, the question that I'm asking myself right now is:

Let's assume that all particles (both fermions and bosons) are symmetry operators of a finite group. The smallest possible group would be S4 (order 24.) The even classes of this group have size 1,3,8 and the odd classes of this group have size 6,6. This fits quite well with the structure of the fermions (6 leptons, 6 quarks) and bosons (1 photon,3 weak,8 gluons.)

However, in this model, all of the particles would have a rest mass of zero. The question is: how can you break the underlying symmetry of this group in such a way that the photon and gluons remain massless, while every other particle acquires mass. This is not easy to do because there is no subgroup of the group S4 that is of order 9 (i.e. 1+8.)

So, what one needs to do is to find a group such that when you break the symmetry of the group (via the Higgs or other mechanisms) fermions and weak bosons all acquire different masses (except the W+/- are the same) and the photon and gluons remain massless.

This is not an easy task.

Given there are are 14 different masses for the particles (6 leptons, 6 quarks, and W/Z masses), there are likely many different symmetry breaking mechanisms going on at the same time. The Higgs field is likely only one of the many symmetry breaking mechanisms.

You've fairly cogently stated the prevailing theoretical assumption about how a Higg field creates mass, which incorrectly predicted that the neutrinos were massless. But, that doesn't mean it is correct. Indeed, the existence of neutrino mass, the necessity that the neutrino not be its own anti-neutrino as the Majorana mass approach would suggest for the maintenance of lepton number (which is how the neutrino's existence was predicted in the first place) and the non-observation of anything suggesting that right handed neutrinos exist, all suggest that the prevailing theoretical understanding of the way that mass is generated (i.e. via Higgs field interactions of the kinds you describe) is in trouble. So, maybe that theory is wrong.

I am increasingly convinced that there are no sterile neutrinos, and that there is likewise no see-saw mechanism or Majorana mass for neutrinos.

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