Time for fun with fundamental constants, and in particular, potential Higgs boson mass and Higgs v.e.v. formulas. All of this, as the post title suggests, is nothing more than numerology with no real solid theoretical framework upon which is rests. But, it is interesting nonetheless.
A Higgs Boson Mass From Boson Mass Formula
It is interesting to note that within the range of Higgs boson masses not excluded by experiment is H^2=W+^2 + W-^2 + Z0^2 (i.e. a Higgs boson rest mass equal to the sum of the squares of the rest masses of the three weak force bosons). The Higgs mass under that formula would be 115.5 GeV about plus or minus 1%, just above the LEP exclusion range.
If one takes charge to come in units of plus or minus the imaginary number i, then this same equation also holds. Of course, color charge, baryon number and lepton number are zero for each term of these equations as well. This would also seem to imply some sort of special relationship between the Higgs boson the the triple W+ boson, W- boson and Z0 boson triple coupling which is indeed one of the permitted weak force couplings.
Indeed, all of this would also be true if the Higgs boson rest mass were taken to be the sum of the square of the rest masses of all four of the electroweak bosons (the W+, W-, Z0 and photon). Indeed, that relationship would suggest the square of the Higgs boson rest mass is the dot product of the electroweak boson rest mass matrix with itself, an interesting operation in and of itself, since the dot product of vectors (here we are taking the dot product of four vector boson rest masses) produces a scalar, which is notable because the Higgs boson is a scalar boson.
The dot product ansatz also has the curious suggestion that fundamental particle mass is deeply and fundamentally intertwined with the electroweak force, while continuing to set apart the gluon mediated strong force - whose massless bosons don't interact with any of the electroweak forces, which is not renormalizable while the electroweak forces are renormalizable, and whose color charge is seemingly totally independent of electroweak charges.
Higgs Boson Mass From Boson and Fermion Masses
Suppose you want to get even bigger picture and are worried that 115.5 GeV is bit too light to be a good bet. You can make H^2=(sum of squares of each of the spin 1/2 fermion masses)^1/2+(sum of squares of each of the spin-1 boson masses)^1, omitting only the top quark mass, which does not hadronize and is the only massive particle heavier than the Higgs boson, from the total. Voila! One gets a Higgs boson mass of 120.3 GeV, just perfect for a possible to detect value from the LHC. For extra cleverness, you can put the (H^2)^0 since it is a spin zero particle, which is equivalent to making the Higgs boson mass the fundamental unit of rest mass, and throw in a term for the graviton on the right hand size (or the left hand side, for that matter) of (G*2)^2, since gravitons have zero rest mass. The down side is that maybe you should be triple counting quarks as you do when you do weak force decay percentage calculations. Also, it would be wonderful to be able to relate the Higgs boson mass of 120.3 GeV (or perhaps 115.5 GeV) to the Higgs field vacuum expectation value of 246 GeV, which is not quite double the Higgs boson mass by this formula, by some trivial formula.
Higgs v.e.v. from fermion and boson masses counted correctly
There is an interesting formula for the Higgs v.e.v. from first principles that is even nicer than the 120.3 GeV Higgs boson mass formula above (the 115.5 GeV Higgs boson mass formula is prettier and more natural). In this formula the Higgs v.e.v. of 246 GeV is equal to, to within experimental measurement uncertainty, double: (the sum of squares of spin 1/2 fermion masses (except top quarks), counting each quark mass three times, one for each color)^1/2+(sum of squares of spin 1 boson masses)^1+(sum of squares of spin 2 boson masses)^2. One can justify leaving out the top quark mass, again, on all sorts of grounds, such as, for example, the fact that a top-antiquark mass pair is energetically prohibited at the Higgs v.e.v. (yes, I realize that this is somewhat circular, but the formula is also quite beautiful as formulas go, so the anomaly may be worth it if it can save the formula).
Since this Higgs v.e.v. has no dependence whatsoever on the Higgs boson mass, and three of the Higgs bosons are gobbled by the W and Z bosons that are included in the formula, maybe the Higgs boson mass doesn't even matter, and one can build a Higgsless model on that coincidence.
A Higgs Boson Mass From Hardon Interacting Particle Mass Formula
A Higgs boson mass of exactly half of the 246 GeV Higgs v.e.v. value (i.e. 123 GeV) isn't yet ruled out by LHC, but it isn't a favorite either. A few GeV less is more strongly suggested.
If we want a Higgs boson mass of above 115 GeV, the thing to do might be to take half of the Higgs v.e.v. calculated as above, and to find some legitimate reason to omit a term that accounts for about 4 GeV. For example, if you include non-top quarks, but not leptons in the calculation above, you get about 119 GeV, which is the sweet spot for Higgs boson mass if there is one.
Thus, in this bit of numerology, while all particle masses would contribute proportionate to their fermion or boson status to the Higgs v.e.v. up to the v.e.v. scale, only particles that are part of, or are emitted or absorbed directly by hadrons would contribute to the Higgs boson rest mass formula. After all, the Higgs field, like the strong force, would be a self-interacting one since Higgs bosons themselves have the mass that a Higgs field creates, so it might make sense that only self-interacting particles contribute to its mass. This would also ground the fundamental concept of mass in fundamental particles more closely to the concept of mass in hadrons that account for the vast majority of the non-dark mass in the universe. Thus, this distinction between the Higgs v.e.v. and the Higgs boson mass would have the added bonus of suggesting some mysterious connection between the strong force and the electroweak force that could point towards a way towards a GUT.
Phenomenological Formulas For The Lonely Top Quark Mass
It is also interesting to observe that there are a variety of formulas to which one can appeal as a special case to find the top quark mass, many of which, of course, have no real good theoretical motivation. For example, the square root of the top quark mass less the square root of the W boson mass equals the bottom quark mass, which suggests that formula t+W-2(tW)^1/2=b^2 is correct to within the limits of experimental accuracy. One can imagine this kind of equation popping out in some way as the leading order derived somehow from the Feynman diagram for a top quark decaying into a bottom quark, with the diagrams weighed somehow by their transition probabilities, since top to bottom quark decay via W boson emission from the top quark is the dominant mode of decay for top quarks to the near exclusion of all other terms (Only 0.02% of top quark decays are to charm or up rather than bottom quarks quarks, while 5.2% of bottom quarks transition to top or up quarks rather than charm quarks, energy permitting).
Incidentally, the natural next iterations of this equation (at least at the leading order term values) not true or even close for b+W-2(bW)^1/2=c^2, and for c+W-2(cW)^1/2=s^2, so this is special pleading to some extent.
What would the Higgs v.e.v. be if you did the sum of squares of masses of a type to the power of spin? You'd get a bit less than 600 GeV.
But, one can also imagine some sort of weighting of mass matrix values with CKM and PMNS matrix values, with top quark mass being virtually removed from the equation because its amplitude to remain as a top quark rather than a bottom quark is so trivally low, while the other particles far more regularly transition into each other.
The Weak Sector and Strong Sector Compared
If the Higgs v.e.v. and Higgs boson mass are in some sense superpositions of the masses of all of the fundamental particles using rules similar to those involved in weak force calculations, then this suggsts that quark-lepton complementarity is related in some deep way to the electroweak force from whose CKM and PMNS matrixes it is gleaned, while having little or nothing to do with the strong force (although the weak force does recognize different colored quarks as equal and distinct fundamental particles on a par with each other probability-wise, rather than mere versions of the same single thing).
Indeed, one way to think about quark-lepton complementarity is to see the four parameters of the CKM matrix whose three theta angles correspond to the three dimensions of space and whose CP violation phase corresponds to the dimension of time, which can be express as a four dimensional unit vector with a complentary PMNS unit vector to it. This notion has the happy feature of representing the relationship of the CKM matrix and PMNS matrix to each other in the background independent way preferred by general relativity, and since the CKM/PMNS matrixes appear to have some deep relationships to rest mass, and since rest mass is one of the core elements of general relativity (indeed, the Newtonian formulation of gravity that ignores all non-rest mass terms in the stress-energy matrix is an extremely good approximation of reality over a broad range of circumstances), this suggests an avenue of attack to linking the two.
It also heightens the intuition that the three generations of fermions are a weak force thing (as transitions between generations are mediated by the weak force), rather than a strong force thing - since the strong force treats all quarks of the same color essentially the same.
Less impressively, there are both four electroweak bosons (apart from the Higgs) and four kinds of fermions, each with a different electromagnetic charge. But, it would, again, provide a distinction between a correspondence that the electroweak force seems to play a part in, and one that doesn't obviously relate to the strong force.
And, of course, there are no demonstrated circumstances in which the strong force is observed to be CP violating, even though there are natural terms in the QCD equations to permit this to happen. The obliviousness of the strong force to electromagnetic charge could be at play here and it is worth remarking that hadrons and anti-hadrons appear to have the same masses, even though most of their masses arise from the strong force in the conventional account, rather than the rest mass of the component fermions.
Musings On Realistic Four Color QCD Models
Query if there is some sort of Color Charge-Parity near symmetry relationship analogous to the electroweak charge-parity relationship that is broken in a similar way in chiral strong force interactions. After all, there are, for example, red and anti-red colors for quarks and gluons that can be reversed with a reversal of parity.
It is also worth noting that one way to see the eight gluons is as a 2*2*2 toggle of three orthogonal unit vectors in space, but not time. Indeed, this heuristic is one of the better ways to understand why there are eight rather than nine (RGB*RGB) kinds of gluons. Viewed this way, the necessity that a hardron or gluon be color neutral is another way of saying that the vector sum of its color charges, which can be expressed in a space-like background independent way, must sum to zero. But, why in a general relativistic world are there not sixteen gluons with a color charge toggle forward and backward in time as well?
Indeed, it isn't obvious to me that color charge has to be anything other than polarization, because I'm not aware of any way yet devised to directly measure gluon polarization, since gluons are always confined and can't be measured directly so far, although experiments to measure gluon polarization have been proposed.
Given that the natural scale of time-like distances is so much greater than the natural scale of space-like distances for human beings, and that gluons like photons are massless, one can imagine that the time-like fourth component of color charge is suppressed in much the same way that the time-like polarity components of photons are suppressed even though QED equations consider 4-part rather than the day to day functional 2-part polarizations of photons, with two components cancelling out (see, e.g. Richard P. Feynnman, QED at pages 120-123), but might have some physical implications in extreme conditions or another class of interactions.
In the QED equation, the extra two components of polarization are particular relevant to virtual photons, for example, between electrons and protons in atoms), so one might imagine by crude analogy, that a missing fourth color charge might be mostly pertinent to the exchange of virtual gluons, for example, between quarks within hadrons, that we tweak by using an effective coupling constant of the strong force when we should be using a bare value perhaps.
It might be easier to reconile a four color strong force with the time-like color strongly suppressed with background invariant general relativity.
It also might very well be that if one used the running of the bare coupling constant of four color strong force with the timelike color suppressed, rather than the effective coupling constant of the three color strong force, that one might be able to achieve the holy grail of GUT theorists, a GUT scale energy level where the strong, weak and electroweak running coupling constants converge to exactly the same value, which, by hypothesis, would be precisely the same as the energy level where the electromagnetic and weak force running coupling constants coincide (without the nuisance of a menagerie of supersymmetric particles - we'd have the same old six quark (but in four colors each instead of three), the same old six leptons, the same old four electroweak bosons, and sixteen rather than eight kinds of gluons that are otherwise the same in all respects.
Anyway, it is a nifty idea that would be simply in principle to check out even if it might be harder to actually do once one got down to really cruncking equations and putting numbers into lattice models and beta functions for the strong force coupling constant.