## Wednesday, November 28, 2012

### How Many Standard Model Constants Are There?

The Standard Model has lots of moving parts.  They are categorized and described below, together with discussion of how the number of independent moving parts might be reduced.

I.  Exact Standard Model Constants

Some of the moving parts, like the number of strong nuclear force colors, the quantum numbers for the four kinds of fermion charge, weak isospin, the number of generations of particles, the zero rest mass of photons and gluons, the conservation laws, the mass of particles relative to their antiparticles.

There are a number of abstract algebra concepts that can reproduce all of the particles of the Standard Model with the property exact Standard Model properties in a compact way described sometimes as SU(3) x SU(2) x U(1) in the Standard Model, which can be embedded in an even more compact abstract algebra representation such as SU(5) or SU(10), but these more compact representations have their own difficulties when one tries to translate them into a "grand unified theory", rather than a patchwork of Quantum Chromodynamics (i.e. SU(3)) to describe the strong nuclear force, and Electroweak theory (i.e. SU(2) x U(1)) to describe the electromagnetic force and weak nuclear force.

Of course, the form of the Standard Model equations, such as the Lagrangians that describe the operation of the fundamental forces, the zero value of the strong nuclear force CP violation term, and the zero masses are set forth exactly by the Standard Model.

At the one loop level, at least, the running of the Standard Model coupling constants (aka the beta function) with the energy scale of the interaction, relative to the basic coupling constant value for that force is also exact.  For example, the QCD beta function expressed to the "three loop" level depends only on the "unadjusted" strong force coupling constant and the number of QCD colors in the model.  I have sometimes, confusingly stated that the beta function constants of the Standard Model are among the constants that are moving parts in the Standard Model, but this flows in part from my failure to really clearly deliniate between "exact" Standard Model constants that could turn out to be wrong when compared to the experimental evidence, and "measured experimental constants" in the Standard Model that can't even in principle be determined any other way if the Standard Model is correct.

The beta functions, while in principle exact within the Standard Model, have not been rigorously tested at high energies and are not necessarily worked out for arbitrarily many "loops" of corrections beyond the next to next leading order (i.e. three loop) level.

II.  Measured Standard Model Constants

Other are experimentally measured and the theory does not describe them exactly, and the elements of the CKM and PMNS matrixes are the principle measured constants of the Standard Model (as well as the speed of light and Planck's constant).

But, the minimum number of measurements necessary to describe all of the experimentally measured constants is considerably less than the total number of the experimentally measured constants, because they are related to each other by a number of exact relationship.

A. The Three Measured Coupling Constants

There is a coupling constant for each of the three Standard Model forces that governs how strong the electromagnetic force, the weak nuclear force and the strong nuclear force, respectively, are in practice that must be determined experimentally.

One of the criticisms of electroweak unification in the Standard Model is that it is not possible to describe the electromagnetic coupling constant and weak force coupling constant with a single measured constant.

The Dim Prospects For A Single Measured Coupling Constant In The Near Future

Many theorists think that the three coupling constants actually converge to a single value (i.e. gauge coupling unification) at very high energy levels (the GUT scale).  The Standard Model comes close to, but does not actually reach gauge coupling unification, although this could simply be because there is something wrong with the beta functions of the Standard Model such as a failure to take into account quantum gravity effects at high energy levels.  SUSY models, generically, does have a gauge coupling unification.

If indeed a suitably modified exact Standard Model beta function did produce a gauge coupling unification, it would in principle be possible to express all three of the Standard Model coupling constants in the form of a single, measured, grand unified coupling constant at the GUT scale and the exactly stated beta function for each of the three Standard Model forces for sub-GUT energy scales.   But, since one needs to know both the strength of the GUT level coupling constant and the precise energy level at which gauge coupling unification occurs, this doesn't actually reduce the number of measured coupling constants in the Standard Model - it just reparameterizes them.

Also, any quantum gravity correction to the beta functions would very likely introduce at least one gravitational constant, so it might not be possible to reduce the number of experimentally measured constants related to coupling strength in the Standard Model by these means, and even SUSY only reduced the number of measured coupling constant parameters for the three Standard Model forces from three to two while introducing other measured constants.

Less elegantly, even in the existing Standard Model, it is possible from the exact beta functions of the Standard Model at the point at which the coupling constant strength of any two of the three coupling constants are identical, to determine both of those coupling constants.

B. The Thirteen Independent Measured Masses

There are twelve non-zero masses of the fermions, and three measured weak force boson masses (the W, the Z and the Higgs boson mass).

According to electroweak unification theory, the photon, W+, W- and Z boson are linear combinations of two more fundamental massless electroweak bosons (the neutral W and neutral B) which are mixed according to the weak mixing angle (experimentally the sine of the weak mixing angle is about 0.24), that "eat" three of the four "Goldstone bosons" predicted by electroweak unification theory, with the fourth giving rise to the Higgs boson that imparts mass to the W and the Z and all other weakly interacting fundamental particles in the Standard Model.

The weak mixing angle that governs the relative masses of the W and Z bosons (and is also one of the major factor in computing W and Z boson branching fractions), is a function of the electromagnetic and weak force coupling constants (the cosine of the weak mixing angle is equal to one of the coupling constants divided by the square root of the sum of the two coupling constants, and the sine of the weak mixing angle is equal to the other electroweak coupling constant divided by teh square root of the sum of hte two coupling constants).  Thus, for example, it is possible, in principle, to derive the weak mixing angle (aka the Weinberg angle) and the Z boson mass, from other exact and measured Standard Model constants.  So, only one measured mass is necessary to describe both the W and Z boson masses.

Electroweak unification theory also claims that the Higgs boson mass is derived from all of the other masses, but in practice, the formula is not such that direct calculation of the Higgs boson mass from it is possible.

The original version of Koide's formula, which is widely believed to  be true (although it isn't clear why), provides a way to determine all three charged lepton masses (apparently exactly) from any two charged lepton masses.

The neutrino masses are simply not known with sufficient accuracy to claim that any particular rule describes them and theoretical basis for Koide's rule is unknown.  indeed, it isn't even known with any experimental certainty if the mass hierarchy of neutrino masses is "normal" (i.e. the third generation is heavier than the second generation is heavier than the first generation), or "inverted" (i.e. the rank order of the neutrino masses is not "normal").

Thus, there are eleven independent fermion mass parameters and two independent boson mass parameters (either the W or Z boson mass and the Higgs boson mass) that are measured parameters in the Standard model.

SUSY models, generically, have far more massive fermions and bosons than the Standard Model (none of which have been experimentally observed) but also has more comprehensive ways of deriving some of these masses in many versions of these theories.

Simple Proposals To Narrow The Measured SM Masses From Thirteen To Seven.

But, it appears that it may be possible with a simple extended version of Koide's formula to exactly derive the masses of the four of the six quarks as well as the masses of the three charged leptons from these two charged lepton masses as well (the up and down quarks' measured values do not match the extended Koide's formula predictions).  There have also been proposals for versions of the Koide's formula that apply to neutrino masses that would allow a third neutrino mass to be derived from the other two neutrino masses, even though they have not yet been tested.

It also appears that the Higgs boson mass may have a much simpler functional relationship to other measured measures than previously supposed, and may be possible to derive exactly entirely from the masses of other fermion and/or boson masses via the simpler formula than the traditional one that gives rise to the hierarchy problem.

Thus, it is entirely plausible that the thirteen measured masses of the Standard Model may in fact be possible to calculate from just six fermion masses (the up, the down, the electron, the muon, the electron neutrino and the muon neutrino) and just one boson mass using already proposed theoretical formulas.  This could reduce the number of independent measured mass constants in the Standard Model from thirteen to seven.

C. The Eight Parameters of the Mixing Matrixes.

The CKM matrix describes the probability that a W boson interaction will change any particular kind of quark into a particular different kind of quark.  Any of the three up type quarks when it emits a W boson can become any of the three down type quarks and visa versa.  The PMNS matrix describes the analogous probabilities for leptons.

While the CKM and PMNS matrixes have nine elements each, since they are unitary matrixes (i.e. the probability of any given particle that emits a W boson becoming one of the three other possible particles is 100%), each can be perfectly described with four parameters that can be chosen in any number of ways.  So, there are no more than eight measured Standard Model mixing matrix constants.

Proposals To Reduce The Number Of Measured Mixing Matrix Constants

It could be, however, that some of these mixing matrix elements may have currently unknown functional relationships either to each other or to the mass matrixes.

One proposal, called quark-lepton complementarity, would derive all three or four of the PMNS matrix elements exactly from the CKM matrix elements.  Experimental evidence appears to disfavor this proposal in its naive form at the moment, but it is not definitively ruled out (since the PMNS matrix constants are not known very exactly).

Other proposals suggest other kinds of PMNS matrix structure, although experimental evidence also tends to disfavor these proposals.

Some of the proposals to reduce the number of measured mixing matrix constants do not include proposals that would eliminate the need to measure one or both of the CP violating parameters of the CKM matrix and PMNS matrixes respectively.  They would only apply to the other three parameters of each of these matrixes.

Proposals Relating Mixing Matrix Contracts and Fermion Masses

Other proposals suggest that the CKM and PMNS matrix elements may, in fact, be functionally related to the twelve fermion masses.  Thus, it might be possible to derive the twelve fermion masses either from the CKM and PMNS matrix elements and one or two "root masses" for fermions, or even from the CKM and PMNS matrix elements and a single weak force boson mass.

Many of these proposals suggest that the square root of fermion masses may be more transparently related to the mixing matrix elements than the measured fermion masses.

Of course, if a formula relates the CKM and PMNS matrix to the fermion mass matrix, the reverse is also possible.  One ought to be able to derive the CKM and PMNS matrixes from the mass matrix.

If the Koide's formula extensions discussed above hold true and the Higgs boson mass can indeed be practicably derived from other Standard Model masses, this would reduce twenty-one independent mass and mixing matrix constants in the Standard Model to just seven.

III.  Summary

There are twenty-four independent measured constants in the Standard Model that are not simply accepted as "exact" within the context of that model, plus the speed of light and Planck's constant.

There are reasonable theoretical proposals with a reasonable prospect of being confirmed in the lifetime of the readers of this blog that could reduce the number of independent measured constants in the Standard Model to as few as eight or fewer (if a comprehensive set of Koide's formula rules could be discerned for the entire mass matrix of the Standard Model).

IV. Future Research Prospects

Right now, one of the foremost issues in Standard Model physics are the ongoing efforts (1) to more accurately measure the quark masses (particularly for the quarks other than the top quark), (2) to more accurately measure the neutrino masses and PMNS matrix parameters, and (3) to experimentally confirm the accuracy of a handful of "exact" Standard Model constants such as the properties of the Higgs boson, the beta functions for the three Standard Model coupling constant, the zero value of the strong force CP violation constant, and the rules forbidding lepton number violations, proton decay, flavor changing neutral currents, and neutrinoless double beta decay (all of which are intertwined to some extent).

A very large share of the biggest gap in this knowledge, that isn't imminently about to be resolved in the next year or two at the LHC which is already well underway and more or less irrevocably so, is in the area of neutrino physics.  With regard to unfinished Standard Model physics business, the LHC will mostly be relevant to confirming the properties of the Higgs boson and refining estimates of its mass, although it may somewhat refine top quark mass estimates and to a lesser extent refining other quark mass estimates and confirming the accuracy of the Standard Model beta functions at somewhat higher energies.

In QCD experimental measurements that are indirectly related to Standard Model constants like quark masses that are known not very accurately are far more accurate than the theoretically predicted theoretical expectations for those experimental measurements which are often precise only to +/- 1%.  Particularly in the case of QCD and the beta functions of all three of the Standard Model forces, it is also critical to improve the provisions of the theoretical calculations of the Standard Model expectations so that quantities that can be measured experimentally like hadron masses, the infrared behavior of quarks and gluons, and predicted glueball composite particles, can be compared meaningfully to Standard Model predictions to determine fundamental Standard Model measured constants that can't be measured directly (e.g. due to quark confinement).

Until we have more data from neutrino physics and better calculations of the theoretically expected values in QCD for already precisely measured observables, none of the theoretical efforts to prune the twenty-four independent measurable Standard Model constants can be confirmed or ruled out definitively.