## Tuesday, January 28, 2014

### A New Precision Estimate Of The Bottom Quark Mass

A new study estimates the mass of the bottom quark to be 4,169 +/- 8 MeV using QCD sum rules (0.2% precision).  Another recent study estimated the bottom quark mass at 4,166 +/- 43 MeV (consistent with the new result). The new bottom quark mass estimate compares to and is consistent with a Particle Data Group world average value of 4,180 +/- 30 MeV in 2013, but is about four times as precise.

Most of the the remaining uncertainty in the latest bottom quark mass estimate is attributable to uncertainty in hadron mass measurements and a very similar but slightly larger uncertainty arising from uncertainty in the strong force coupling constant which is currently about 0.1184 +/- 0.0007 (0.6% precision) at the W boson mass.

Other Recent Quark Mass Estimates

The best global fit estimate of the top quark mass (as opposed to the best direct measurement of this quantity) is 173,200 MeV (about 0.1% precision), and I suspect based upon Higgs vev considerations that it is probably closer to 173,180 MeV (if correct, about 0.01% precision).  Based upon direct measurements the world average of the top quark mass is 173,070 +/- 888 MeV (about 0.5% precision).

This is one of many recent claims of high precision quark mass measurements, including a charm quark mass of 1,273 +/- 6 MeV (0.5% precision).

The claims include a QCD sum ruled based strange quark mass determination of 94 +/- 9 MeV (10% precision).  The Particle Data group world average of the Strange quark mass is 95 +/- 5 MeV (5% precision).  Another recent lattice QCD calculation estimates the strange quark mass at 99.2 +/- 3.9 MeV.

The best estimates of the up and down quark masses are about 2.3-0.5+0.7 MeV (about 25% precision) and 4.8-0.3+0.5 MeV (about 8% precision) respectively.  The average of the up and down quark masses is 3.5 MeV -0.2+0.7 MeV (about 11% precision).  The estimated ratio of the up quark is estimated to be 38% to 58% of the down quark mass (about 20% precision).

The Charged Lepton Masses

The charged lepton masses are known with far greater precision for the most part (per the Particle Data Group):

The electron mass is 0.510998928 +/- 0.000000011 MeV (0.0000002% precision)
The muon mass is 105.6583715 +/- 0.0000035 MeV (0.000003% precision)
The tau mass is 1776.82 +/- 0.16 MeV (0.01% precision)

The Standard Model Massive Bosons

The masses of the massive bosons of the Standard Model are also known fairly precisely (per the Particle Data Group except for global fits and the Higgs boson mass):

The W boson has a best fit mass that is 80,385 +/- 15 MeV with a global fit value of about 80,376 MeV.
The Z boson has a mass of 91,187.6 MeV +/- 2.1 MeV.

The Higgs boson has a mass of about 125,600 MeV +/- 450 MeV (0.4% precision).  My strong personal conjecture is that it is in fact exactly equal to the W boson mass plus one half of the Z boson mass (i.e. about 125,979 MeV for the best fit isolated W boson mass estimate and 125,970 MeV for the global fit W boson mass estimate with precision on the order of 0.03%).

The Neutrino Masses

We know the following about the neutrino masses, but have not yet definitively determined their absolute masses or mass hierarchy.  But, the very plausible assumption of a normal hierarchy somewhat similar to that of the quarks and charged leptons allows us to make some pretty precise estimates of these masses.

The square of the difference between the first and second neutrino mass states is 7.50 +/- 0.20 * 10^-5 eV^2.  The square root of this squared difference is 8.7-0.2+0.1 meV (about 2% precision).

The square of the difference between the second and third neutrino mass states is 0.00232 - 0.00008 + 0.00012 eV^2   The square root of this squared difference is 48.2 - 0.5 +1.2 meV (about 2% precision) .  This tends to imply a difference between the first and third neutrino mass state in a normal hierarchy of 57 meV.

In a normal hierarchy, if the pattern of the other fundamental fermion masses are any guide, we expect the mass of the first neutrino mass state to be << 8 meV and that the sum of the three mass states of the neutrinos is less than 60 meV which is consistent with cosmology data to date.

Other Fundamental Constants

The electromagnetic force coupling constant alpha is 0.0072973525698(24).

The Fermi coupling constant Gf/(hc)^3 is 1.1663787(6) * 10^-5 GeV^-2 (a precision of 0.00006%).

The Higgs vacuum expectation value, weak mixing angle, and weak force coupling constant can be derived from the other constants already provided.  The relationship between the Fermi coupling constant Gf, the weak force coupling constant g, and the W boson mass Mw is Gf/sqrt(2)=g/8Mw.  The Higgs field vacuum expectation value is the square root of a quantity equal to the Fermi coupling times the square root of two.  The cosine of the weak mixing angle is equal to the mass of the W boson divided by the mass of the Z boson.

Planck's constant is 6.62606959(29)*10^-34 J*s.

The speed of light (by definition) is 299,702,458 meters per second.

The gravitational constant is 6.70837(80)* 10^-39 hc/(GeV/c^)^2 (a precision of about 0.01%).  Some aspects of general relativity have been tested to 0.1% precision, and others have been tested to up to 0.001% precision, with the greatest precision demonstrated with the equivalence principal tested to one part per 10^13 precision.

The cosmological constant of general relativity is 10^-47 GeV^4 and is known only to about one significant digit.

We have almost all of the pieces of the puzzle

We have observed all of the particles predicted by the Standard Model and none that are not predicted by it.  We now have either measured values or very well motivated good estimates of all of the mass constants in the Standard Model, to considerable precision that simply not available even a couple of years ago.  Each of these constants, except the Strange quark mass for which we have 5% and 5 MeV precision, is known either to a better than 1% precision or to a less than 1 MeV precision.  The realistic uncertainties in each of the neutrino masses are less than 3 meV.

We also have reasonably precise values for all three of the Standard Model coupling constants (to at least 0.1% precision), for all four of the Standard Model CKM parameters (to at least 15% precision), and tolerably accurate estimates for three of the four Standard Model PMNS matrix parameters (to at least 11% precision).

While not specific to the Standard Model, we also have precise measurements of the speed of light in a vacuum and Planck's constant.  For what it is worth we also have serviceable measurements of the two experimentally measured constants of general relativity: the gravitational coupling constant (to 0.01% precision), and the cosmological constant (to an order of magnitude).

Even the permitted parameter space for dark matter models is quite constrained compared to just a few years ago.  A great many dark matter models have been ruled out by observational evidence and the parameter spaces of many of the competing approaches is quite confined.  We know how much of it there is, what kind of velocity distribution it must have, and more or less how it is distributed within galaxies.  We know it is pressureless or very nearly so.  We know that it is close to collisionless, but probably not perfectly.  If it is a gravity modification, rather than a particle, we know its approximate form and have made an estimate of the key experimentally measured constant in such a theory to about 10% precision.

Testing Within The Standard Model Theories

The era in which it is possible to make true predictions of any of the Standard Model experimentally measured constants, untainted by preliminary measurements of them, other than arguably the CP violating phase of the PMNS matrix parameter for which only extremely inaccurate and preliminary measurements have been made to date, is gone.

But, the time has now come when it is possible to rigorously test possible phenomenological relationships between these constants, such a variations on Koide's formula, with a goal of piercing through to some deeper theory that underlies the Standard Model and makes these experimentally measured constants anything other than arbitrary.

I have yet to meet anyone who really believes that all of these Standard Model parameters are simply arbitrary.  There are many "coincidences" that flow from the particular values of these parameters.

If suspected relationships between the electroweak boson masses and the Higgs boson mass, and between the Higgs vev and the aggregate squared masses of the Standard Model are correct, this removes at least three of free parameters from the Standard Model - the Higgs boson mass, the electromagnetic coupling constant and the weak force coupling constant.  On these assumptions, the Higgs boson mass can be determined from the W and Z boson masses, the weak mixing angle can be determined from the W and Z boson masses, the weak force coupling constant can be determined from the total set of fundamental particle masses in the Standard Model, and the electromagnetic coupling constant can be determined from the weak force coupling constant and the weak mixing angle.

Koide's rule for charged leptons reduces the number of free fermion mass parameters by one.

The connection between the Higgs boson mass and the value of that mass that maximizes photon production in its decays, the value of that mass that makes the vacuum metastable, and the value of that mass that runs to zero at a GUT scale all seem to coincide and possibly sets a mass scale for all of the Standard Model masses.

This still leaves thirteen mass parameters, eight mixing matrix parameters, and the strong force coupling constant, for a total of twenty-two free experimentally measured parameters.  But, eliminating four experimentally measured parameters is progress.

The sense that the mass parameters are related to each other and to the mixing matrix parameters in some formulaic way, while elusive, seems so very likely.  The extended Koide's formula, for example, comes quite close to predicting the quark masses from the two most precisely measured charged lepton masses, and a variant on its comes quite close to predicting the relative masses of the neutrinos.  It wouldn't be at all surprising if one genius tweak to that formula could make all nine of the charged fermion masses determinable from just two fermion masses.  It also doesn't seem at all implausible that the CKM matrix elements could be derived in some way from the quark masses.  There is structure there, even if no one has puzzled out its exact nature yet, perhaps in part because no answer reached could ever be conclusive without the level of precision measurements that we have now achieved.