Mass Matrix Musings And The Higgs Mechanism

A very completely realized paper discussing the phenomenology that might relate the CKM matrix, PMNS matrix and masses of the fundamental fermions from December 2008 in introduced and concludes with these observation (note that the authors' native language is French and not English, emphasis added and some obvious misspellings corrected):

If the Higgs particle turns out to be exactly as expected, then the Standard Model is closed from a mathematical point of view. In that case, it is conceivable that any new physics will be far beyond the reach of future colliders. On the contrary, if the data do not reveal any scalar particle, it is likely that the LHC will spot some unexpected or unexplained events since the unitarity of the theory is broken around 1 TeV. From a mathematical point of view, the first possibility is probably more appealing but the latter is certainly the most interesting one for us, now. Indeed, in spite of its simplicity, the Brout-Englert-Higgs mechanism also raises some questions.

In the Standard Model, all forces are explained in terms of boson exchange. These bosons are associated with a gauge symmetry. However, the scalar sector of the theory as it stands cannot be considered as a fifth force. The Higgs boson is moreover the only particle that knows the difference between the fermion families as well as between the generations. The coupling is indeed proportional to the mass instead of some conserved charge. This particular status is quite intriguing. Furthermore, these masses seem to be completely arbitrary and display a huge hierarchy, not to mention the astonishingly small mass of the neutrinos.

The mass generation mechanism is also intimately connected with the SU(2)L gauge symmetry. This connection is at the root of some of the most interesting properties of the Standard Model, namely flavour mixing and CP violation.

The Standard Model requires thus some new flavour physics, in particular to explain the fermion mass spectrum and the number of families and generations. These numbers must be somehow connected with the mass generation mechanism. Most of the proposed extensions of the Standard Model fail to meet these criteria. Some problems of the Standard Model have indeed been solved by grand unification theories, supersymmetry, technicolour, horizontal symmetries and others. However, it has always been at the expense of a complexification of the theory, for instance a zoo of new particles or a rather large group of symmetry. . . .

Fermion mixings arise naturally in the theory of electroweak interactions and result from a mismatch between mass- and weak-eigenstates. Within the assumption of a Higgs mechanism, mixings and masses have the same origin and are bound together in the Yukawa couplings. This mechanism however fails to give an explanation of the observed fermion spectrum. Therefore, the Standard Model is probably not the end of the story but only a low energy effective theory.

In this work, we do not aim at finding a new mechanism that could explain this spectrum, but we rather assume that fermion masses and mixings are calculable in a yet-to-be-found more fundamental theory. Our goal is to glean as much information as possible from the observed fermion masses and mixings in order to identify some hidden structures that could significantly lower the number of free parameters and help us to get some clues about what could be this fundamental theory.

To achieve this goal, we follow two distinct paths:

• The analysis of the various parametrizations of the flavour mixing matrix points us to a specific decomposition. We note that the parameters of this decomposition can be independently and accurately computed if we impose some simple textures to the Yukawa couplings. We propose then a straightforward combination of these interesting textures in order to recover the observed quark flavour mixing.

• We study the properties of a successful mass relation for the charged leptons [i.e. Koide's formula]. We propose some generalizations of this relation which also apply to the neutrinos and the quarks. One of them successfully combines the masses and mixings in a kind of weak eigenstate mass. Another one describes the lepton masses through a well-defined geometric picture. . . .

[T]hese two paths lead to similar conclusions and allow us to
speculate about some interesting properties new flavour physics should
display. . . .

If the mass matrices are actually built by squaring a more fundamental matrix, the mixing angles seem a priori computable from simple ratios. We have proposed a simple texture for this “square root” matrix which leads to a quark flavour mixing similar to the observed one. . . .

All the observations we have done plead for a deeper modification of the Standard Model than just adding new symmetries or particles to the Lagrangian. In regard to all these results, our guess is that the masses do not result from a coupling to an elementary Higgs field.

If we speculate on the mechanism responsible for the electroweak symmetry breaking, we would say that preon models could fulfill most of the properties presented here. On the one hand, a dynamical symmetry breaking could in principle lead to some relations between the pole masses. On the other hand, preons constitute a suitable framework where square roots of masses may appear. The practical way to implement such a dynamical model is beyond the scope of this work but constitutes its natural outcome.

Another great observation from the same paper is that "One can
also wonder how the top quark is so heavy while it is as point-like as the
electron in the Standard Model for electroweak and strong interactions.
It has indeed the same mass as one molecule of vitamin C (C6H8O6)[.]"

Analysis

Another way of putting the apparent connection between the mixing matrixes and masses is to say that we seem to know, to the extent of our ability to empirically test our theories, everything that there is to know about the strong force and electromagnetic force (i.e. the SU(3) and U(1) parts of the Standard Model, although this is muddled a bit by the intertwining of the weak and electromagnetic components of the unified electroweak force). Even if we discovered, for example, that CP violation occurs at some very low level in quantum chromodynamics it would be trivial to include a natural term in that equation to account for those experiments.

We don't have experiments indicating the presence of unexplained fundamental forces, missing fundamental particles, or statistically disagreements between experiment and calculations at the energy levels of the LHC to date or any prior high energy physics experiments. We can perfectly easily fit all observed dark energy phenomena simply by assigning a value to the cosmological constant in the equations of general relativity. We do need some mechanism that explains observed dark matter effects, which isn't a good fit to any of the forces or fundamental particles or hadronic matter of the Standard Model, but that is all we are experimentally motivate to look for at this point.

Everything we don't know about fundamental particle mass and the CKM/PMNS matrix parameters, and the Higgs mechanism, seems to arise from an incomplete description of the SU(2)L weak force part of the theory.

The existing Standard Model SU(2)L theory seems to be both underconstrained in its parameters and missing some subtle term or piece that solves its problems at higher energies. It seems as if we have left a rule or two, and a Lagrangian term or two (or perhaps a renormalization calculation step) out of what we are working with now. There ought to be an explicit quark-lepton complementarity rule or mechanism that gives rise to it, and there ought to be some sort of mass generation mechanism, perhaps in the nature of a see saw type mechanism that gives rise to fundamental particle mass from the weak force mixings themselves and should be capable of being described by far fewer free parameters.

The problem doesn't seem to be so much that what we know is wrong, at least as a low energy effective theory, but that our current formulation is incomplete. It doesn't make explicit deeper connections between its parameters that seem to exist, we are having trouble finding the Higgs boson it predicts which we may not need in a proper formulation, it ceases to predict unitary decays at the TeV level, we observe just three generations of particles but don't have good theoretical reasons for believing that there aren't more generations, we're not sure why it makes sense that there don't seem to be right handed neutrinos (or even if we can really say with comfort that there aren't any).

The other lingering issues that are pretty much purely theoretical. The hypothetical point-like nature of massive particles in quantum mechanics (which give rise to singularities) and the non-local effects in quantum mechanics (which is continous, local and causal) are at odds with general relativity. Moreover, while in principle the path to describing general relativity is a boson mediated force via a spin-2 graviton would seem to be capable of replicating general relativity, efforts to realize this in practice have not produced a consensus solution. But, these quantum gravity problems aren't obviously necessary to resolve to fix other less than ideal features of the Standard Model.

They may, of course, be related. There may be something unsound in a subtle way about shoehorning a toy model with massless fundamental point-like particles when the reality may involve non-point-like particles that are massive through a mechanism more natural and emergent than the Higgs field.

There is also a lingering sense that a Grand Unified Theory in which the three coupling constants of the Standard Model and its three Lie Groups can be understood as a spontaneously broken symmetry of a single Lie Group with a single coupling constant is possible, but if we've done our job properly with the Standard Model characterizations of the three fundamental forces, this is pretty much icing on the cake.

Indeed, one way to see the persistent failure of efforts to produce a GUT that doesn't include particles and forces that we don't see, while not explaining everything that we do see, is that the flaw in the way we have formulated the weak force is preventing the pieces from fitting together into a coherent whole in the way that they should.

It seems as if there ought to be a more elegant way to formulate this part of the Standard Model that might remove all of its pathologies in one fell swoop, and with more data that rule out many of the alternatives that we are so close to really grasping the connections that have so far eluded the entire global theoretical physics community, which has been stymied by group think pursuing dead end paths to solve this problem like SUSY and String Theory and Technicolor.