More LC & P Papers

Lopez Castro and Pestieau have published a number of papers on interesting numerical coincidences in physics in addition to one paper that I previously blogged. Footnote 6 of that paper is also notable (but not part of the paper that I previously discussed):

(1) Higgs mass from electric charge, W boson mass, and Z boson mass.

It is interesting to note first that before the discovery of the scalar boson, the mass value mH = (124.5 ± 0.8) GeV was obtained using electroweak data available at that time (see J. Erler, arXiv:1201.0695 [hep-ph]). Second, using the measured masses of gauge bosons and the empirical relation mZ = emH/(sin θW cos θW ), with cos θW = mW /mZ and e =√4πα the electron charge, we have obtained mH = 125.4 GeV.

The experimentally measured value of mH is currently 125.6 +/- 0.4 GeV.  These values are consistent at a two sigma level.

As the comments below tend to show, much of this may be more numerology than something that indicates a deeper theoretical relationship - making them more useful as tricks to remember approximate values than as tools for penetrating fundamental physics.  But, there are still worth noting.

Here are some of them:

(2) Pure math values for electric charge and the weak mixing angle.

We propose the relations 1/e - e =3 and tan(2theta_W)=3/2, where e is the positron charge and theta_W is the weak angle. Present experimental data support these relations to a very high accuracy. We suggest that some duality relates the weak isospin and hypercharge gauge groups of the standard electroweak theory.

- "Arithmetic and the standard electroweak theory", G. Lopez Castro, J. Pestieau (Submitted on 10 Apr 1998).

This implies a theta W of about 28.155 degrees (the experimentally measured value is 28.196 degrees using current global fits for the W and Z boson masses linked below), a difference of about 0.041 degrees, with source inputs that are precise to about five significant digits.  The experimentally measured value of tan(2theta_W) is about 1.504665.

This implies an e of about 0.3027756377 which compares to an experimentally measured value of 0.30282212.  While this is close in absolute terms, since the experimental value is accurate to about 8 significant digits, this is still hundreds of standard deviations or more from the experimentally measured value.

(3) Higgs vev and electric charge from W and Z boson masses.

In the electroweak standard model we observe two remarkable empirical mass relations, m_W + m_B = v/2 and m_W - m_B = e v/2 where m^2_Z = m^2_W + m^2_B, e is the positron electric charge and v, the strength of the Higgs condensate.

- "Remarkable Mass Relations in the Electroweak Model", Jean Pestieau (Submitted on 29 May 2001)

Note that using current global fits for the W and Z boson masses, that the best estimate for the theoretical B mass discussed above is 43.0855 GeV, implying by their relation that the Higgs vacuum expectation value v=246.905 GeV and e=0.301999.  This differs from their 1/e-e=3 formula value by about 0.3% and is further from the measured value than the original formula which really isn't too impressive for a formula with five significant digit inputs.  The accepted value of the Higgs vev is 246.2279579 GeV.  This can be tweaked a fair bit, however, if one uses global electroweak best fit values for the source parameters.

(4) Effective Electroweak Mixing Angle 

A precise empirical relation between the electromagnetic coupling alpha(m_Z) and sin^2(theta) --where theta is the effective electroweak mixing angle extracted from Z leptonic decays-- is made manifest: alpha(m_Z) = sin^3(theta)*cos(theta)/(4*pi).

- "A Remarkable Relation in the Gauge Sector of Electroweakdynamics", Jean Pestieau (Submitted on 17 Jan 2003) (abstract edited to provide more readable, but equivalent, notation).

This is confirmed by the experimental data at this point.

(5) Formulas For Top Quark Mass, Higgs Boson Mass, W Boson Mass and Z Boson Mass from Electric Charge and Higgs vev.

We propose some empirical formulae relating the masses of the heaviest particles in the standard model (the W,Z,H bosons and the t quark) to the charge of the positron e and the Higgs condensate v. The relations for the masses of gauge bosons m_W = (1+e)v/4 and m_Z=sqrt{(1+e^2)/2}*(v/2) are in excellent agreement with experimental values. By requiring the electroweak standard model to be free from quadratic divergencies at the one-loop level, we find: m_t=v/sqrt{2} and m_H=v/sqrt{2e}, or the very simple ratio (m_t/m_H)^2=e.

- "The unit of electric charge and the mass hierarchy of heavy particles", G. Lopez Castro, J. Pestieau (Submitted on 13 Sep 2006) (this paper predicted, incorrectly, that the Higgs boson mass would be 317.2 GeV, a mass that had already largely been ruled out at the time).