Thursday, September 26, 2013

The Not Quite Boson-Fermion Mass Symmetry Considered


The sum of the square of each of the fundamental boson masses, plus the sum of the square of each of the fundamental fermion masses, equals to the square of the Higgs vacuum expectation value to a precision of 0.012% (easily within the experimental measurement error of the source masses), four significant digits if:

* one assumes that the mass of the Higgs boson is actually one half of the W boson mass and the Z boson mass (91.1876 GeV), and
* one also uses global fit values for the W boson mass (80.376 GeV/c^2) and the top quark mass (173.2 GeV/c^2) rather than taking individual best estimates of these masses without considering global fit considerations, and
* one calculates the Higgs vacuum expectation value of 242.29 GeV that these best fit values imply (together with the canonical value of the electromagnetic force coupling constant and its beta function governing its running).

The Higgs boson mass if this relationship is correct is 125.97 GeV (well within experimental bounds and accurate to five significant digits).  The four significant digit accuracy of the Higgs vev to the sum of the fundamental particle masses squared is as accurate as the least accurately known of the inputs that have a significant impact on the total (the top quark mass).

In my view, this is quite compelling evidence that both the 2H=2W+Z and Higgs vev squared equals sum of fundamental particle rest mass squared relationships are true and fundamental rather than mere coincidences (which isn't to say that I have all of the details of the mechanism that illustrates how these fundamental relationships arise).  We are moving into an era where these relationships can be evaluated with precision.

The Not Quite Boson-Fermion Mass Symmetry

If these relationships are true, then the sum of the square of the three boson masses (W, Z, and Higgs) is about 2.1% greater than the sum of the square of the masses of the twelve fermions (six quarks, three charged leptons and three neutrinos, but dominated by the top quark mass which accounts for 99.94% of the total).

If the first two relationships that almost precisely match the best possible estimates of the Standard Model parameters are true, it cannot be true that the sum of the squares of the boson masses and the sum of the square of the fermion masses are equal.  Or at any rate, it can't be true for rest masses and the coupling constant strength at the energy level ordinarily used by physicists.

This is a frustrating thing.  If the two were equal, we would have this profound boson-fermion mass symmetry - hinting that the boson-fermion symmetries of SUSY might be present in the plain old Standard Model, although more subtly.  If the two were wildly different, we wouldn't even think to look at the relationship of the two quantities.  But, they are instead close, but not quite close enough to possibly be the same.

It might be that there is a profound boson-fermion mass symmetry, but that it only manifests under the right conditions or from the right perspective.  For example, there might be some energy scale at which the boson masses and fermion masses and coupling constants run at which 2H=2W+Z and the sum of squares relationship to Higgs vev, and the boson-fermion mass symmetry all hold simultaneously, and below which the boson-fermion mass symmetry is broken.  Perhaps above this threshold, there is even a subtle shift in the running of the electroweak coupling constants that causes the three Standard Model coupling constants to converge at a single point.

One could also imagine this approximate symmetry corresponding to one or more other approximate symmetries in physics, which perhaps together produce an exact symmetry.

For example, perhaps the magnitude of boson-fermion mass asymmetry in the Standard Model exactly corresponds to the aggregate amount of CP violation found in the Standard Model when measured appropriately, or to some appropriate function of the Weinberg angle that governs electroweak unification, or to the amount by which quark-lepton complementarity does not quite perfectly hold in the CKM and PMNS matrixes.

Or, perhaps boson-fermion mass asymmetry is the key to the puzzle of matter-antimatter asymmetry in the universe.

Or, perhaps the "missing fermion rest mass" (about 25 GeV if all from a single particle), corresponds to the sum of the square of the rest masses of fundamental particles other than those in the Standard Model which do not participate in electroweak interactions and hence aren't part of equations involving the Higgs field - perhaps something in the "dark matter" sector, or a see-saw sterile neutrino, or the some significant measure of the dynamically created mass of gluons that are not "at rest" that have a negative rather than a positive contribution to the boson side of the equation.

Or, perhaps this imbalance relates to the cosmological constant in some way (another constant that is almost, but not quite, zero).


Mitchell said...

Nice post.

andrew said...

This post extends previous analysis here.

andrew said...

I used global best fit values for the W and t masses and adjusted the conventionally accepted Higgs vev for the difference in the W boson mass (in the Standard Model Higgs vev (W, g)=f(g)*W).

The numbers match using the currently accepted PDG values of the masses (in the case of the Higgs vev, a precise consensus value based on measuring the Fermi coupling constant from muon decay, from pre-print rather than the less exact PDG value). Specifically (in GeV/c^2 units):

Higgs vev=246.2279579
c=1.275 (pole mass 1.67)
b=4.18 (pole mass 4.78)

The following masses are ignored:
tau neutrino=52 meV
muon neutrino=8 meV
electron neutrino=1 meV

The sum of squared masses for these six fundamental fermions is < 0.0001 GeV^2 and I made my calculations to an already spurious accuracy of 0.01 GeV^2. In general, the formula is insensitive to a small number of new particles so long as they have masses < O(0.07 GeV).

The pole masses are a slightly better fit using the PDG value than the MS masses.

Using all values but the t quark mass and fitting the t mass to the equation gives a t mass of 173.066 using pole masses and 173.085 using MS masses. But, rounding errors and uncertainties in the underlying values make these values insignificantly different from each other statistically. The measured t quark mass of 173.07 is to +/- 0.888 GeV.

The Higgs mass used of 125.9788 is consistent with the current measured value of 125.9+/-0.4 per PDG.

Recent precision measurements have put b=4.169 on an MS mass basis and c=1.273 on an MS mass basis, both with greater precision. The percentage differences from this greater precision in these two very tiny values is pretty much immaterial since it is so small relative to other uncertainties (dominated by the t mass uncertainty).