Tuesday, August 5, 2014

Balancing Standard Model Fermion and Boson Squared Masses

The sum of the square of the pole masses of the Standard Model fermions (the quarks, charged leptons and neutrinos) plus the sum of the square of the Standard Model bosons (W, Z and Higgs) is almost exactly equal to the square of the vacuum expectation value of the Higgs boson given current experimental data.  Equivalently, the sum of the coefficients that are multiplied by the square of the Higgs vacuum expectation value to get the square of fundamental particle masses in the Standard Model sum to one.

Using these values the contribution from the fermions is about 2-3% smaller than the contribution from the bosons.  But, in the Standard Model, particle masses run with energy scales.  In general, at higher energy scales, fermion masses gets lower, but the boson masses (or at least the Higgs boson mass) fall much more rapidly.  As I've mused before, the energy scale at which the square of the fermion masses is equal to the square of the boson masses in the Standard Model may be a natural energy scale with some sort of significance.

The Higgs boson self-coupling constant, lambda, which is directly proportionate to the Higgs boson mass, falls by about 50% from its 0.13 value at the 125-126 GeV energy scale by 10,000 GeV (i.e. 10 TeV), and falls to zero in the vicinity of the GUT scale.

The running of the W and Z boson masses should correspond to the running of the constants g and g' in the Higgs boson mass formula, which are related to the electromagnetic and weak force coupling constants, which run in opposite directions from each other at higher energy scales and converge at about 4*10^12 GeV.

In contrast, the running of the charged lepton masses is almost 2% from the Z boson mass to the top quark mass and just 3.6% over fourteen orders of magnitude.  Quarks also run more slowly than the Higgs boson masses and the Higgs vev (which also runs with energy scale).

Eye balling the numbers, it looks like this cutoff is in the rough vicinity of the top quark mass (about 173.3 GeV in the latest global average) and Higgs vacuum expectation value (about 246.22 GeV).  Certainly, this equivalence is reached somewhere around the electroweak scale and surely below 1 TeV.  An equivalence at the Higgs vev would be particularly notable and is within the range of possibility.

1 comment:

andrew said...

Another thing that looks like it might happen around the Higgs vev (of 246.22 GeV) is that the running mass of the Higgs boson which has a pole mass of 125-126 GeV might drop to the 123.11 GeV which is exactly one half of the Higgs vev. Thus, the approximate relationship might become exact when evaluated at the right energy scale.