Below are some selected interesting relationships of physical constants for boson masses driven by some conjectures that would motivate these relationships theoretically.
Higgs boson mass relationships
(1) Higgs boson mass is approximately equal to the mean of the top quark mass plus W boson mass (i.e. 126.65 GeV).
One formula for the Higgs boson mass heavily weights the top quark mass relative to all other fermionic contributions. The top quark accounts for about 95%+ of the average mass of all fermions in the Standard Model and slightly more when quarks are weighted three to one relative to leptons which is their relative proportion in weak force decays (since there are three colors of each quark but only one "color" of lepton). Adjustments for the other five quarks, the six leptons and the Z boson and photon, in appropriate proportions, could tweak this value, particularly contributions from the bottom quark mass, charm quark mass, the tau (a third generation electron) and the Z boson. The collective masses of the up, down and strange quarks, the electron, the muon, the neutrinos, and the massless photon by comparison, are negligible by comparison.
The notion here is that supersymmetry may not be necessary to resolve the hierarchy problem because hidden structure in the relationships between the fermion and boson masses cause them to cancel out in a way that reflects supersymmetry-like cancellations between these bosons and fermions, which is not obvious when you formulate your Higgs boson mass formula in a way that doesn't take advantage of this hidden structure.
For example, if a fundamental particle's mass can be stated in terms of a formula from some other particle's masses, it may be possible to pull common factors out of the infinite series formulas that combined determine the Higgs boson mass and cancel them out, making it possible to arrange the terms into one or more infinite series which we know to converge to a particular value that can be expressed in some other way that are easier to demonstrate cancel each other out.
Perhaps the Higgs boson mass is simply equal to whatever is necessary to balance the scales between the right sets of fermions and bosons on each side of the scale.
In the same vein, it may not be coincidental that there are twelve basic kinds of spin-1/2 fermions (three generations each of up and down quarks and three generations each of charged and uncharged leptons respectively), and that there are twelve basic kinds of spin-1 bosons (photons, three kinds of weak force bosons, and eight kinds of gluons). Like supersymmetry, in the Standard Model itself, there is one kind of boson for every kind of fermion and one kind of fermion for every kind of boson, although which fermion is a partner to which boson isn't necessarily obvious, and this coincidence could certainly be spurious or misguided.
(2) Higgs boson mass is approximately equal to one half of the sum of two times the W boson mass and one time the Z boson mass (i.e. 125.99 GeV). Another way this could be stated is as the sum of the four electroweak boson masses (W+, W-, Z and the photon) divided by the square root of the number of electroweak bosons.
The inference would be that the Higgs boson mass could be equivalent to the mass of a linear combination of the weak force bosons which could also produce a combined spin of zero. Given the electroweak unification itself describes the four electroweak bosons themselves as linear combinations of other more fundamental bosons in the unified electroweak theory, this seems like a reasonable approach.
The notion of dividing the sum by the square root of the number of particles involved, while summing charges and considering all permissible sums of plus or minus the intrinsic spin of each particle in the combination derives from the way that linear combinations of mesons (e.g. different kinds of neutral kaons in linear combinations with each other) are handled in QCD.
The linear combination notion, if there is anything to it, is also suggestive of the possibility that there might be spin-2 and perhaps even spin-4 variants on the Higgs boson with the same mass that make up a tiny percentage of the total percentage of all Higgs bosons produced. It isn't obvious that linear combinations of bosons are subject to the same limitations on the spin of fundamental particles discussed in a post yesterday. They are at least similar to composite particles even if they may lack internal geometric structure of the kind found, for example, in a classical chemistry molecule, or a proton-neutron-electron model with electron shells atom.
Formulas (1) and (2) are consistent with each other to within one standard deviation of experimental uncertainty in the source values (particularly due to uncertainty in the mass of the top quark which is on the order of 1 GeV from all sources combined) and are also consistent with the measured value of the Higgs boson within the range of experimental uncertainty in that measurement (the current measurement has an uncertainty on the order of +/- 1 GeV).
The Z boson mass
(3) The Z boson mass is about 2% smaller than the sum of the 2 times the W boson mass plus the photon mass (i.e. zero) divided by the square root of three (for three bosons in the numerator).
The inference would be that the Z boson mass could be related in some way to the mass of a linear combination of the W+, the W- and the photon , which could also produce a combined spin of one and a neutral electromagnetic charge. Some form of binding energy or synergistic effect could account for a discrepency.
The linear combination notion, if there is anything to it, is suggestive of the possibility that there might be spin-3 Z bosons of the same mass that make up a small proportion of all Z bosons (perhaps 1/8th of them). Also, it is worth noting that higher spin versions of quark hadrons always decay faster than lower spin quark hadrons, making the infrequently produced higher spin linear combinations perhaps more difficult to observe.
Other linear combinations
One could imagine trying to play the linear combination game with other combinations of fundamental electroweak bosons. But, if only bosons that have some direct interactions with some of the other bosons in the linear combination are permitted, the number of possibilities is greatly reduced, and it may be possible to devise some simple rule regarding permitted combinations that rules out other linearly combined particles that are not observed.
Thus, while it might be possible to have glueballs (composite particles bound by color charge made up entirely of gluons) since gluons have a self-interaction term, since gluons do not interact directly with the Higgs boson (because they lack mass), do not interact via the weak nuclear force (and hence don't interact with W or Z bosons), and lack electromagnetic charge (and hence don't interact with photons), one would not expect to see composite particles with gluons and electroweak bosons together.
Similarly, one wouldn't expect to see a photon which does not interact via the weak force and a Z boson which lacks electromagnetic charge, to form two boson linear combinations. The only bosons with which Z bosons interact are W and Z bosons.
Photons, similarly, because they lack electromagnetic charge themselves and only interact with charged particles, don't interact with each other. The only bosons with which photons interact are W bosons.
A W+W- combination would seemingly create an electrically scalar with a mass of about 113.7 GeV, but they might immediately annihilate rather than forming a linear combination since they are antiparticles to each other, without having some other fundamental particle in the mix to buffer them from each other, or a property like parity or color charge to distinguish them from each other. A ZZ linear combination would be an electrically neutral scalar with a mass of about 127.4 GeV which would be quite hard to distinguish experimentally from a 125-126 GeV Higgs boson - ZZ linear combinations would just look like experimental noise around Higgs boson data at current levels of experimental precision.
One could imagine a linear combination of a single photon with a W boson, or a WZ combination. Each of these would create a spin zero charged particle lighter than the Higgs boson without supersymmetry. But, particles with these properties have been searched for and ruled out in the appropriate mass ranges according to the summary of the experimental data provided by the Particle Data Group in the course of conducted supersymmetry motivated charged Higgs boson searches.
A rule which would eliminate most of the unobserved possibilities while not ruling out the possiblity that Z bosons and Higgs bosons could be linear combinations, could be as simple as one that provides that all linear combinations of bosons must have a neutral combined electromagnetic charge. This is the case, empirically, in all situations where linear combinations of mesons with integer spins are observed and described experimentally. I don't have a reason for this rule, put perhaps charge particles take too much energy to oscillate between alternative modes.
Caveats and observations
These are merely conjectures. Finding rough empirical correspondences between numbers whose values aren't known with perfect precision is much easier than someone not familiar with the exercise might guess. But, the right answer, whatever it might be, will necessarily be among the category of correspondences that are a rough match.
Also, if some heuristic, for instance, the notion that the properties of some bosons might closely resemble the properties of a linear combination of other bosons according to some straightforward rules for determining rest masses and spin and charge, is fruitful in producing multiple such relationships and has some sort of theoretical precedent, the empirical correspondences may be less random than mere brute force combinations in every possible way of input numbers in order to match the observed results. The more structured your method is, the less coincidental similarities are a valid objection to the results that you obtain.
It is also worth noting that technical adjustments to the dominant first order source behind the relationship mean that the right answer properly articulated will not always be the closest to the data when formulated only in a crude heuristic form.
For example, basically, W and Z boson decay is governed by the "democratic principle" that every possibility is equally likely with quarks of different colors counting as different possibilities. But, details of the calculations cause the exact branching fractions to not match up perfectly with this dominant first order guiding principal for calculating decay probabilities.
In a more basic example, the first order equation for determining the speed of a falling object is derived exclusively from Newton's law of gravity, which is effectively flawless at that scale (i.e. general relativity effects are truly negligable to the point where they can't be measured). But, because there are air resistance effects in real life, Newton's law of gravity would not be the closest match to what is actually observed.
The moral of the story is that one shouldn't totally abandon a heuristic notion of what is driving a physical phenomena simply because it isn't a perfect fit for the data when expressed in a simplified form and compared to a far messier reality.
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