Thursday, November 10, 2011

Musings On Higgs Boson Coupling Constants

The Higgs mechanism for creating mass in the Standard Model supposes that each fundamental particle has a coupling constant with the Higgs boson that is a function of its mass. In other words, mass is to the Higgs boson what weak isospin is to the weak force, what electric charge is to electromagnetism, what color charge is to the strong force.

This is really quite odd.

All fundamental particles have one of five values for weak isospin: zero, +/- 1/2, or +/- 1. Every fundamental particle has one of seven values for electromagnetic charge; zero, +/- 1/3, +/- 2/3, +/- 1. Any given quark has one of six possible color charges, there are eight differen possible color charges for gluons, all of which are some kind of linear combination of the six color charges available for quarks, and all other fundamental particles have no color charge.

The notion that Higgs boson coupling constants could come in values so wildly at odds with pattern for all other force carrying bosons. The fundamental particles of the Standard Model come in seventeen different values for mass, and no two of those masses are neat integer or simple fraction variants of the other.

There is obviously some method to the madness that produces the seventeen particle masses (twelve fermion masses, two weak force masses, the zero photon and gluon masses, and a Higgs boson mass) in the Standard Model. The two mass values associated with the weak force bosons could be predicted with considerable accuracy before their discovery from the other constants of the Standard Model. Higher generation particles are always heavier than lower generation particles of the same type. All neutrinos are lighter than all other fundamental particles. Charged leptons appear to be lighter than either of the quarks of the same generation (although the uncertainty in the empirically measured mass of the strange quark, which has a central value almost identical to the muon but an uncertainty of about +/- 25%, makes it somewhat unclear if it is heavier or lighter than the muon). Charged particles of one generation have more mass than all charged particles of all lower generations. Particles and their antiparticles have the same mass. Quarks and gluons have the same mass, regardless of color charge. Some of the ratios of one kind of particle to another approximate the dimensionless coupling constants of the fundamental forces. One can devise formulas that approximately predict the masses of different fundamental particles relative to each other. In the case of Koide's formula relating the masses of the charged leptons to each other, the relationship appears to be exact. There seems to be a relationship between the values of the CKM matrix and the PMNS matrix to the masses of the fundamental fermions.

All unitary three by three matrixes in which every entry must be positive or zero, including the CKM matrix and the PMNS matrix (which must meet this condition because they code the probabilities of a complete set of possibilities) have four degrees of freedom - i.e. it takes four numbers to determine the value of all nine cells. One could do this by simply determing the value of any four arbitrary entries in the nine by nine matrix, but a parameterization of the matrixes in terms of four different angles is the more conventional way to go about characterizing these matrixes.

A hypothesis called quark-lepton complementarity, first proposed in 1990, states that if you parameterize the CKM matrix and PMNS matrix in a particular way, that for each angle in the CKM matrix parameterization, there is a corresponding angle in the PMNS matrix parameterization, and that the sum of these two angles is in each case equal 45 degrees.

If this hypothesis is correct, then the CKM matrix and PMNS matrix combined actually have only four degrees of freedom and can be fully characterized by a single unit vector in the +,+,+,+ quadrant of four dimensional space (and appropriate quardrant limitation because probabilities are always positive). It also implies that the CKM matrix and PMNs matrix, when looked at in a particular way in this four dimensional probability space are orthogontal, or half-orthogonal (i.e. at 45 degree angles) to each other.

The same line of thinking involved in the notion of quark-lepton complementarity, also implies that a suitable form of matrix multiplication of the CKM matrix, the PMNS matrix, and another trivial matrix (the correlation matrix) to transform the product into the proper form, can produce the mass matrix of all of the fundamental fermions as a function of an arbitrarily chosen mass of any one particle, in some versions, with some modest complicating analysis (also here by the same author).

It is a bit hard to test this hypothesis empirically because there is a great deal of uncertainty in the empirically determined masses of the five lightest quarks, because they can only be measured as part of composite particles, and because the values of the entries in the PMNS matrix are not known to any great accuracy. For example, this method has been used to predict one of the values of the PMNS matrix angles.

But, suppose for a moment that this hypothesis is true, and that knowing these values (and suitable coupling constants), that one can infer the rest masses of th W and Z bosons (as was actually done in fact). This implies that:

* The seventeen Higgs boson couplings and eight degrees of freedom in the CKM and PMNS matrixes are actually fully determined by just a single four dimensional, all positive valued unit vector (one might call it the "weak force transition vector"), and an arbitrarily chosen single reference mass.

* In this situation, the mass matrix of the fundamental fermions and weak force bosons are predominantly a function of the three of the four Standard Model Higgs bosons that are "eaten" by the W+ boson, W- boson, and Z boson respectively. The remaining scalar Higgs boson simply sets the overall mass scale of the fundamental particles generally, without having any impact on the relative masses of particular fundamental particles to each other.

* Twenty-four of these constants, determined by five independent constants (and perhaps also the weak and electromagnetic force coupling constants, but probably independent of the strong force coupling constant), seem to be independent of the Higgs boson mass (although, as I have suggested in a prior post, it isn't too hard to imagine formulas by which one could derive a Higgs boson mass from the other masses, just as one can in the case of the W and Z bosons).

* If one uses this system to express Higgs boson couplings of fundamental particles functionally, it would be possible to attribute the arbitrarily chosen single reference mass to a single universal coupling constant of the Higgs boson with all massive fundamental particles. If one was truly clever and lucky one might even be able to express this universal coupling constant in terms of something else.

* In this schema, the constants of the Standard Model (in addition to the various equations of the three fundamental forces that it describes and the Lie groups of the particles that it contains) are four coupling constants (strong, weak, electromagnetic, Higgs boson fundamental particle mass scale) which could perhaps themselves be expressed as a four component vector, the four component weak force transition vector, the speed of light, and possibly some constants involved in the running of the strong, weak, and electromagnetic force coupling constants.

* It also seems possible that it might be possible to dispense with the Higgs boson entirely and derive fundamental particle masses entirely from weak force interactions involve the four electroweak bosons, through some mechanism other than the Higgs boson. If we were lucky, could such a mechanism were devised, it might also "coincidentally" resolve other pathologies of the Standard Model, such as TeV scale mathematically pathologies and the hierarchy problem.

I have some doubts about the viability of establishing the fundamental fermion masses from the CKM and PMNS matrixes, although it is a beautiful thing, and even more doubt about the grand unification models that inspired the concept (e.g. here), but I am inclined to think that the likelihood that quark-lepton complementarily will be confirmed experimentally as the precision with which we can determine the values in the PMNS matrix improves.

My guess is that calculations made on this assumption, using the combination of CKM and PMNS angle experimental values to reduce the uncertainty in the theoretical values of each, will consistently be more accurate than calculations made using the raw values of each matrix entry independently.

We already have enough data to quite precisely describe a four component weak force transition vector, and pooling the data for corresponding components would slightly improve that precision (the CKM matrix entries are mostly much more precisely known than the PMNS matrix entries, but for a couple of the parameters, some enhancement of the CKM matrix values might be possible). So far, that data is consistent with the hypothesis.

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