Conjectures Regarding Unsolved Mysteries Of The Standard Model

This post recaps some of my observations, conjectures and working hypotheses about the source of the Standard Model physical constants and the deeper workings of the model. 

It is notable that all fundamental SM particles with non-zero rest mass have weak force interactions, and that all fundamental SM particles with zero rest mass do not have tree level weak force interactions.

Neutrinos which lack electromagnetic charge have negligible mass, while charged leptons that do have electromagnetic charge much larger masses (by a factor on the order of one million).

The magnitude of the strong force QCD color charge of all three colors of all six types of quarks (ands their antiparticles) is identical; the magnitude of the strong force coupling of all eight kinds of gluons are identical to each other. Yet, quarks have very different rest masses from each other and gluons have no rest mass.

In the SM, the Higgs mechanism is part of the electroweak part of the SM and has no real interaction with the strong force and QCD interactions of the model. 

The Higgs vev is a function of the W boson mass and the weak force coupling constant. 

The Yukawas of the SM which are the coupling constants of the SM Higgs boson are particle rest masses normalized by the square of the Higgs vev.

In the SM electroweak theory, the mass of the Z boson relative to the W boson is a function of the electromagnetic and weak force coupling constants.

To a high degree of accuracy, the probability of a CKM matrix transition from the first generation to the third generation is equal to the probability of a CKM matrix transition from the first generation to the second generation, times the probability of a CKM matrix transition from the second generation to the third generation.

Taken together this presents a strong suggestive case that the fundamental particle rest masses in the SM are entirely a product of the electroweak sector of the SM (apart from possible negligible magnitude higher order loop factors), as the Higgs mechanism of the SM itself illustrates, rather than having anything meaningful to do with the QCD sector of the SM.

The principle that a particle's coupling to gravity is a universal function of its mass-energy is also well established and since no particle is treated differently than any other under this coupling relationship, it suggests that the electroweak sector of the SM is the sole meaningful source of the fundamental particle rest masses in the SM.

So, the fundamental masses of the SM flow from an SU(2) x U(1) group alone.

The reasonably close empirical fit of the LP & C relationship to the known rest masses (i.e. that the sum of the square of the fundamental particle masses is equal to the square of the Higgs vev) is suggestive of the hypothesis that the overall mass scale of the SM fundamental particle rest masses is strongly connected to a mass scale established by the weak force, since the Higgs vev is a function of the W boson mass and weak force coupling constant, both of which are exclusively part of the SU(2) weak force sector of the SM.

The large magnitude of the Higgs boson mass and top quark mass relative to the other fundamental SM particle masses can be understood as necessary and natural to fill out the LP & C totals in the heaviest particle in each category.

(The fact that the sum of the square of the fundamental boson masses exceeds half of the square of the Higgs vev, while the sum of the square of the fundamental fermion masses is less than half of the square of the Higgs vev, to high statistical significance, also suggests that any SM symmetry between fermions and bosons is not exact, as it is in SUSY theories, but instead, is broken or only approximate.)

Hadronic masses, in contrast, have a very significant contributions from QCD. Gluons and virtual sea quarks provide the predominant source of rest mass for light hadrons, and are a significant source of rest mass for heavy hadrons. This different mechanism for hadron mass in the SM (which can be calculated in principle from fundamental SM constants in the SM), and fundamental particle masses in the SM, suggests that comparisons between hadron masses may not be providing much insight.

The fundamental quark and charged lepton masses have a "normal hierarchy" (i.e. higher generation particles of a given electromagnetic charge are more massive than lower generation particles of the same type). 

The structure of the CKM matrix also shows that this normal hierarchy for quarks is not just a product of arranging quarks in order of mass. Generation assignments correspond to similar CKM matrix elements. 

All observational evidence also favors a "normal hierarchy" of the fundamental neutrino mass eigenstates, although the preference isn't terribly strong simply because the masses are so tiny and the precision of the available measurements is limited.

The CKM matrix appears to be "more fundamental" than the quark rest masses. CKM matrix elements at a given generation are similar despite the up and down quarks of a given generation having very different rest masses.

It is possible to "obtai[n] the PMNS matrix [which governs neutrino oscillation] without having to ever talk about mass diagonalization and mismatches between flavor and mass basis." Gustavo F. S. Alves, Enrico Bertuzzo, Gabriel M. Salla, "An on-shell perspective on neutrino oscillations and non-standard interactions" arXiv (March 30, 2021). While this is largely a matter of semantics, with equivalent observational outcomes, much like the different quantum mechanics interpretations, an "on-shell perspective" can be more helpful in conceptualizing neutrino mass and neutrino oscillation in the context of an overall understanding of the neutrino oscillation process. The on-shell perspective conceptualizes neutrino oscillation in the context of a virtual W boson mediated process, essentially identical to the virtual W boson mechanism for flavor changing in the CKM matrix governed quark side of the SM.

The Standard Model forces that do not experience CP violation (i.e. the electromagnetic force and the strong force) are mediated by zero rest mass carrier bosons that in special relativity do not experience the passage of time in their own reference frame. This is also true of the hypothetical massless graviton. So, it makes sense that only the weak force, which is mediated by a massive carrier boson would exhibit CP violation. This is an argument against the "strong CP problem" being a problem.

The Higgs boson and Z boson are also massive, but unlike the W boson, they lack electromagnetic charge, lack color charge, and have even parity, so CP reversal of a Higgs boson or Z boson is not something that can be observed. So, it makes sense that the sole cause of CP violation in the Standard Model as manifested through the CKM matrix which is basically a property of the W boson, is the W boson. In an on-shell interpretation of neutrino oscillation, this process is also W boson mediated, so the PMNS matrix is also basically a property of the W boson.

Koide's Rule And Its Extensions

The original Koide tuple has a couple of things going for it that are plausibly the source of the quality of the match: (1) charged lepton universality, and (2) the negligible masses of the neutrinos relative to the charge leptons. In contrast, quark universality is not present in the SM and the masses of the up-type quarks are of the same order of magnitude as the down type quarks.

Extended Koide sum rules produce a reasonable first order approximation of all of the SM quark and charged lepton rest masses (at pole mass values) in all three generations from the electron and muon rest masses. It makes an essentially perfect prediction of the tau lepton mass that has withstood the test of time. It isn't a "correct" rule for predicting the quark masses, but the predicted masses and the actual masses have the same order of magnitude. The Extended Koide sum rule uses quark triples on the decay chain t-b-c-s-u-d. Notably, at first order, the predicted up quark mass is very nearly zero.

The error in the first order approximations of the quark masses from the Extended Koide sum rules can be significantly reduced with an adjustment.

For a given up type quark (the "target quark") there are three down type quarks it could transform into via W boson interactions. Two of those down type quarks are included in the Koide triple. Adjust the first order Extended Koide sum rule estimate by multiplying the transition probability of that up type quark to the third down type quark (the one not included in the Extended Koide quark truple used to establish its mass) which is the square of the magnitude of the relevant CKM matrix element, times the Extended Koide sum rule mass estimate for the third down type quark. (The analogous rule is use for the masses of down type quarks).

This adjustment is motivated by the fact that discrepancy between the Extended Koide Rule first order estimate, and the measured rest mass is greatest in the cases where the CKM matrix element between a target quark and a third quark not included in its Extended Koide rule first order estimate calculation is largest.

Notably, the mass of the up quark comes almost entirely from this third-quark adjustment (i.e from the unaccounted for up quark to bottom quark transition).

The intuition of this adjustment (which is not done in a perfectly mathematically elegant and rigorous way and would need to be iterated or simultaneously solved to really be done right) is that the relative magnitudes of the fundamental quark masses are the product of dynamical balancing between them via W boson interactions, and the overall mass scale of the SM fundamental particles is also a function of the weak force.

Some of the geometric interpretations of the original Koide's rule are consistent with this kind of conceptualization.

In this analysis, the W boson interaction is the driving force behind the fundamental particle masses and the Higgs boson Yukawas are really just the tail wagged by the W boson dog, or a corollary that flows from the W boson interactions, instead of the usual conceptualization that the Higgs boson interactions are driving the bus with the source of those couplings due to some unknown deeper physics of an unknown source.

A conceptualization of fundamental fermion mass generation being driven primarily by W boson interactions, rather than an explanation that is centered on Higgs boson driven parity oscillations of SM fundamental fermions, also provides a path for a source of Dirac mass for neutrinos, arising from their W boson interactions, even though neutrinos can't also have Higgs boson driven parity oscillations of charged leptons and quarks since there are no right handed neutrinos and no left handed antineutrinos. The fact that only the W boson portion, rather than both the W boson and Higgs boson portion are at work in neutrinos could also help to explain their vastly different rest mass scales.

The reason that there are no right handed neutrinos and no left handed antineutrinos is that the weak force does not interact with particles of that parity, and neutrinos have no electromagnetic or strong force interactions. The lack of parity balance, in turn, is the reason that Higgs boson parity oscillation doesn't give rise to neutrino mass, only W boson interactions and self-interactions. No see-saw mechanism is necessary to explain their smallness and they do not need to be Majorana particles to acquire mass in this way.

Why Three Generations?

The reason that there are exactly three generations of SM fermions could be basically "accidental".

There are theoretical reasons in the deep math of the SM (especially its electroweak part) which have been known since the 1970s or 1980s why any given generation of SM fermions must have four members (one up type quark, one down type quark, one charged lepton, and one neutrino).

The mechanism by which higher generation fermions decay to lower generation fermions is through W boson interactions. So, no fermion can have a mean lifetime less than the W boson. The top quark mean lifetime is only marginally longer than the mean lifetime of the W boson. If there were a hypothetical fourth generation of SM fermions, the SM mean lifetime would be less than the W boson mean lifetime. But, because this would be a contradiction, there are no fourth generation SM fermions.

Combining These Conjectures

If a correct generalization of Koide's rule for all SM fermions was determined and the LP & C relationship is correct, you would need the electron mass, the muon mass, the W boson, the SU(2) coupling constant, and the U(1) coupling constant to determine all of the rest masses of the fundamental SM particles. Indeed, it might even be possible to have just the electron mass without the muon mass, using the Higgs vev to set the overall mass scale. So, there would be one fundamental fermion mass and one fundamental boson mass in this fundamental mass generation model.

The electron mass is well approximated by a mass established by its overall self-interaction, as is the lightest neutrino mass eigenstate (which differs from the electron mass by a ratio on the order of the electromagnetic coupling constant to the weak force coupling constant). It may be true that this approach could also be applied to one or both of the first generation quark rest masses. If so, there is a path to set of fundamental SM particle rest masses with only one non-derived and experimentally determined mass dimensioned physical constant: the W boson mass. This leave four CKM matrix parameters, four PMNS matrix parameters, three SM force coupling constants in addition to the W boson mass. The other fourteen SM particle fundamental masses could be derived.

* The baryon matter-antimatter asymmetry of the Universe is best explained by a Big Bang that has a matter dominated universe extending from it in one direction of time and an antimatter dominated universe extending from it in the other direction of time. This is important because a resolution of this asymmetry removes the pressure on theorists to find additional sources of CP violation, and additional sources of baryon number and/or lepton number violation in the Standard Model.

* The fact that quarks come in three colors, and have a 3-1 ratio to leptons in W and Z boson decays, and have electromagnetic charges that are in 1/3rd multiples of the charge lepton electromagnetic charge magnitude, is probably not a coincidence and says something about the deeper structure of quarks.

* It could be that charged leptons are color charge complete, like a baryon, rather than actually lacking color charge. Similarly, could be that W bosons are color charge complete, rather than actually lacking color charge. 

* Could Z bosons and Higgs bosons be different sides of the same coin, that are color charge complete in the manner of quarkonia (or tetraquarks or hexaquarks), with Higgs bosons and Z bosons differing by spin alignments?

* A color charge-electromagnetic charge connection would then make sense, with the color charge component merely cancelling out in charged leptons and W bosons, and perhaps also in Higgs bosons and Z bosons.

* Since both color charge and electromagnetic charge have antiparticle opposites, the notion of neutrinos having either that cancel out makes less sense. Are neutrinos "empty nets"?

* Would gluons be color charge pairs bound without "the nets"?

* Is there a better way than the three color charge approach (such as a topological one), to think about color charge that addresses the fact that there are eight rather than nine kinds of gluons? I think that the answer must be yes.