Some Dubious Numerology About The Set Of Fundamental Particles

The possibility of physics beyond the standard model is studied. The sole requirement of cancellation of the net zero point energy density between fermions and bosons or the requirement of Lorentz invariance of the zero point stress-energy tensor implies that particles beyond the standard model must exist. Some simple and minimal extensions of the standard model such as the two Higgs doublet model, right handed neutrinos, mirror symmetry and supersymmetry are studied. If, the net zero point energy density vanishes or if the zero point stress-energy tensor is Lorentz invariant, it is shown that none of the studied models of beyond the standard one can be possible extensions in their current forms.

Damian Ejlli, "Beyond the standard model with sum rules" (September 14, 2017).

The paper argues that there are three respects in which a weighted sum of terms related to fundamental fermions should equal a weighted set of terms related to fundamental bosons.

Each fundamental particle is assigned a "degeneracy factor" that serves as it weight.

Purportedly:

(1) The sum of the fermion degeneracy factor for each of the fundamental fermions should be equal to the sum of the boson degeneracy factor for each of the fundamental bosons.

(2) The sum of the fermion degeneracy factor times the square of the mass of each of the fundamental fermions should be equal to the sum of the boson degeneracy factor times the square of the mass of each of the fundamental bosons.

(3) The sum of the fermion degeneracy factor times the fourth power of the mass of each of the fundamental fermions should be equal to the sum of the boson degeneracy factor times the fourth power of the mass of each of the fundamental bosons.

The trouble is that except for some trivial cases that bear no similarity to reality, it appears that this will never be true. 

Naively, it appears to me that a sum of raw weights, squared masses with same weights, and fourth power masses with the same weights are never going to simultaneously balance, unless all of the fundamental particle masses are identical.

In that special case, the sum of the weights for the fermions equals the sum of the weights for the bosons, so if every particle on the fermion side has the same mass as every particle on the boson side, then mass squared on each side will be the same and mass to the fourth power on each side will be the same.

But, if the masses are different for each particle, as in real life, it isn't at all obvious that the weighted sum of mass squared can every be equal to the weighted sum of mass to the fourth, because the square of mass squared is not a linear transformation, but linear parity of masses terms must remain.

There is also reason to doubt the formula (1) for the weights, which was formulated in 1951 by Pauli, before second and third generation particles were known to exist, before quarks and gluons were discovered, before the modern graviton was conceived, and before neutrino mass was known to exist, is correct.

Each quark counts 12 points. Each charged lepton counts 4 points. A massive Dirac neutrino counts 4 points, while a massive Majorana neutrino or a massless neutrino counts 2 points. The W bosons count 6 points, the Z boson counts 3 points, the Higgs boson counts 1 point, the photon counts 2 points and gluons apparently count 2 points each for 8 flavor variations of gluon.

The fermion side apparently has 68 more points than the boson side. If massive Dirac neutrinos are assumed then each generation of fermions is worth 32 points, so the second and third generations are combined worth 64 points. If these higher generations were disregarded as distinct from the first generation, since they have the same quantum numbers and could be considered excited states, then the fermion side only leads by 4 points.

The basic point calculation, modified for color and the existence of distinct antiparticles is 2S+1 for massive particles and 2 for massless particles. But, both known massless particles are spin-1 and it could be that the formula for massless particles should actually be 2S, in which case a massless graviton would add 4 additional points to the boson side and balance (1).

Another way that the formula could balance if the second and third generations of fermions were disregarded would be the addition of a spin-3/2 gravitino singlet. But, while this can come close to balancing (2) and (3) with the right mass, the gravitino needs a mass of about 530 GeV to balance (2) and a mass of about 560 GeV to balance (3) (an approach that also ignores the fact that the higher generation fermion weights are ignored, although perhaps ignoring masses makes sense in an equation that doesn't include masses, but not in one that does include masses). Ignoring the graviton might actually be appropriate because it does not enter the stress-energy tensor in general relativity.

As far as I can tell, there is simply no way that both (2) and (3) can be simultaneously true in a non-trivial case, and empirically (2) is approximately true and not inconsistent with the evidence within existing error bars, only without any weighting. 

It seems more likely that the cancellation of the net zero point energy density between fermions and bosons or the requirement of Lorentz invariance of the zero point stress-energy tensor is in the first case not true, and in the second case ill defined or non-physical.