2H=Z+2W?

The mass of the W boson, the Z boson, and the Weinberg angle that relate the two, are all known with increasing precision. The two masses are known to about one part in five thousand. The cosine of the Weinberg angle is equal to the ratio of their masses. The mass of what appears to be the Higgs boson is known far less accurately, to about one part in fifty or sixty. The Higg field vacuum expectation value is known with a precision of one part per hundred, at least, and perhaps even more precision.

As the error bars around these standard model constants, and others, becomes smaller, efforts to discern any apparent relationships between the constants become more interesting, because more precisely known relationships are less prone to be numerological flukes.

At one point, it looked like the mass of the Higgs boson would be half of the vacuum expection value of the Higgs field, but this now looks a little low.

Another possibility is that the Higgs boson mass that fits current data to within current error bars for the Higgs boson mass is that two times the Higgs boson mass equal the mass of the Z boson plus the mass of the W+ boson, plus the mass of the W- boson (i.e. the three weak force bosons). Equivalently, since the mass of the photon is zero, 2H might equal the mass of the four electroweak bosons, or for that matter (since the strong force bosons also have zero rest mass), 2H might equal the mass of all of the other bosons.

Of course, the Higgs boson mass uncertainty right now is sufficient that all manner of inconsistent formulas to arrive at some mass within the current error bars for its value are possible.

Also, since the Weinberg angle and a fortiori, the masses of the W and/or Z bosons "run" with the energy scale of the interaction, much as the three couple constants of the Standard Model forces do, small discrepencies between the measured figures for these constants and apparently simple relationships between them, could flow from an incorrect implementation of the running of the Weinberg angle when determining a numerical value to insert into the formula.

For example, the most commonly used value for the Weinberg angle (about thirty degrees a.k.a. pi/6 radians), is based on an energy scale of the Z boson mass. But, perhaps in the equation that is the title of this post, an energy scale equal to the sum of the Z boson mass and two times of the W boson mass is more appropriate and would tweak the relevant values a little bit.

Perhaps, both 2H=Z+2W and 2H=Higgs v.e.v. are both true if one makes the appropriate adjustments for the running of the Weinberg angle, which are not obvious. Perhaps the "energy level of 2H=Z+2W is half of the energy level of 2H=Higgs v.e.v.

If one wanted to really get numerological about it, one could even suggest that a pi/6 value for the Weinberg angle might be related in some deep way to the fact that it is a component of a formula with six bosons in it: two Higgs bosons and the four electroweak bosons. Probably not. But, who knows.

Even a tenfold increase in the accuracy with which we know what appears to be the Higgs boson mass greatly narrows the range of numerical combinations that can produce it, and if there some sort of simple formula like 2H=Z+2W that relates the masses of the respective bosons, with or without adjustments for the running of the Weinberg angle, then this suggests that there may be some deeper and previously unknown relationship between the Standard Model constants.

This, in turn, could suggest theories such as the Higgs boson as a linear combination or otherwise composite state of the electro-weak bosons. And, any newly discovered relationship of these constants, whatever its character, would make the Standard Model more parsimonious, would reduce the number of degrees of freedom in the model, and would point the way towards a more fundamental theory from which the Standard Model is emergent.

If the Higgs boson mass runs with energy scale, one could also imagine this relationship tweaking the theoretical expectations about the high energy behavior of the Standard Model, for example, to cause the coupling constants of the three Standard Model forces to converge at a single point when extrapolated to a "GUT scale" energy, something that they don't quite do under existing Standard Model formulas for the running of the coupling constants. Or, this might resolve divergent terms in Standard Model calculations at high energy levels.