Implications of Nishida's mass/CKM observation

• I
https://arxiv.org/abs/1708.01110
"Phenomenological formula for CKM matrix and its physical interpretation"

This paper was mentioned in "Koide sum rules" thread, but not discussed much. But to me, this looks like a big deal.

Here's the "numerology":

Square roots of experimental quark masses, MeV
down: 2.1679 9.7980 64.6529
up: 1.4832 35.7771 416.5333

Unit-length vectors built of those (IOW: divided by norm)
down: 0.0331 0.1498 0.9882
up: 0.0035 0.0856 0.9963

CKM matrix:
0.97435 -0.2287 0.005641
0.2286 0.9712 -0.06700
0.009846 0.06652 0.9977

Kohzo Nishida says that (normed_sqrt_up_masses) = CKM * (normed_sqrt_down_masses)
0.0036 0.0868 0.9962

Match with experimentally known value for "down mass vector" is eerily good.

Square roots of masses of quarks seems to be definitely linked to CKM matrix. And again, like in Koide rule, we have square roots of masses. Hmmm.

I'm no specialist (and thus would like specialists to chime in), but to me this says "mass" is a square of something, or some complex value multiplied by its complex conjugate.

What theories can give something like this?

ohwilleke

In Koide's yukawaon models, the yukawa couplings are proportional to the square of the vev of a new scalar field, the yukawaon field. This comes about because the interaction term, which in the standard model is just left-handed chiral fermion times Higgs field times right-handed chiral fermion, here also has two yukawaon factors. Ultimately the structure of the term, as in the standard model, is due to the charges of the fields; in the yukawaon models, there's a new flavor-dependent charge, gauged by the family bosons.

Over the years we have occasionally taken an interest in the Gell-Mann-Oakes-Renner relation, according to which the square of a scalar meson's mass is proportional to a sum of quark masses. The point is that you then have a meson mass proportional to a square root of quark masses, and then if you had supersymmetry, the mass of the mesino superpartner of the meson, another fermion, might also be related to those sqrt-masses.

In Koide's work, he usually assumes that yukawaons are only directly relevant for the charged fermions (electron, muon, tau), and that the yukawas for the other fermions are built up from more elaborate combinations of those same yukawaons. Nishida's paper might be regarded as evidence that the yukawas of the other families are determined in the same way (e.g. they have their own yukawaons). It also suggests that we should revisit Zenczykowski.

ohwilleke
ohwilleke
Gold Member
https://arxiv.org/abs/1708.01110
"Phenomenological formula for CKM matrix and its physical interpretation"

This paper was mentioned in "Koide sum rules" thread, but not discussed much. But to me, this looks like a big deal.

Here's the "numerology":

Square roots of experimental quark masses, MeV
down: 2.1679 9.7980 64.6529
up: 1.4832 35.7771 416.5333

Unit-length vectors built of those (IOW: divided by norm)
down: 0.0331 0.1498 0.9882
up: 0.0035 0.0856 0.9963

CKM matrix:
0.97435 -0.2287 0.005641
0.2286 0.9712 -0.06700
0.009846 0.06652 0.9977

Kohzo Nishida says that (normed_sqrt_up_masses) = CKM * (normed_sqrt_down_masses)
0.0036 0.0868 0.9962

Match with experimentally known value for "down mass vector" is eerily good.

Square roots of masses of quarks seems to be definitely linked to CKM matrix. And again, like in Koide rule, we have square roots of masses. Hmmm.

I'm no specialist (and thus would like specialists to chime in), but to me this says "mass" is a square of something, or some complex value multiplied by its complex conjugate.

What theories can give something like this?

Nice summary. I think that you can get to this if the factor that is driving the relative values in the mass matrix are the W boson transitions between them which give rise to the quark masses dynamically. Thus, the up quark masses are basically a weighted average of the down quark masses with weights proportional to which down quarks are most likely to transition to which up quarks.

For example, if you try to do a Koide triple relationship of the up quark, the down quark and the strange quark, you get a prediction that the up quark mass is virtually zero. But, if you adjust that prediction for the fact that a very heavy bottom quark has a small probability of transitioning to an up quark via a W boson interaction, then you get an adjustment that brings it all right on the money.

In this regard the new paper points out this connection to W boson interactions:

In this paper we modify the previously proposed formula to fit the latest experimental data, and we show that the invariant amplitude of the charged current weak interactions is maximized under the constraint of the formula.

It then goes on to make a phenomenological prediction in a non-static case:

If we assume that the quark mixing angles are dynamic parameters, the variation of the Lagrangian is . . . . . Thus, we conclude that the origin of the quark mixing is the most likely configuration of scattering. The conclusion predicts that the value of the CKM matrix elements has momentum dependence because the mixing angle to maximize M of (23) depends on the momentum of each particle. Then the formula e (u) = V e (d) is a zeroth approximation. In the first approximation, we will need to replace the components q m (q) i of e (q) by q E (q) i + m (q) i .

Moreover, if you can get the relative masses of the fundamental fermions, you can also get the absolute mass scale of the collection of them from the very suggestive phenomenological relationship that the sum of the square of the masses of the fundamental particles of the Standard Model equals the square of the Higgs vev.

So, with the masses of the W, Z, Higgs and the value of the Higgs vev, the assumption that the neutrino masses are nearly zero, and the electron and muon masses, you can derive using this paper, Koide's charged lepton rule, and the square of the Higgs vev relationship, all six of the quark masses and the tau lepton mass, stripping out seven Standard Model parameters and equally important, shedding some light on why they take the values that they do.

A similar conjecture was presented by one of the same authors in 1996, but the quality of the input data is much better now, and the formula was refined, and with it has come a better fit. In that article the authors explained that:

In the standard model the Higgs mechanism generates the fermion masses in the forms of mass matrices and the KM matrix is given by the product of unitary matrices diagonalizing the up and down quark mass matrices. In this note the KM matrix is derived as a unitary matrix relating the mass vectors in the generation space. To give a theoretical foundation to our formula for the KM matrix in Eq.(14), we must reformulate the mass vector description in the ordinary mass matrix scheme of the standard model. In fact it is possible to show that nonhermitian quark mass matrices being diagonalizable by a unitary matrix and a unit matrix lead to the mass vector description under a simple additional condition. . . .

In this way we have found a mechanism which relates the mass vector description to the ordinary mass matrix formalism in the standard model. For such a mechanism to work, the quark mass matrices must have the form Mα = U α† L Mdiag α , (α = u, d) and the generation vector space must have a direction of anisotropy specified by the e vector in Eq.(33)1 . In this connection it is worthwhile to mention that Foot [19] introduced a generation space where the vectors consist of the square roots of quark masses and the e vector plays a special role. In such a vector space he found an interesting geometrical interpretation for Koide’s lepton mass formula [4,20]. Esposito and Santorelli [21] extended the Foot method to the quark masses and the Dirac masses of neutrinos.

Last edited:
I think that you can get to this if the factor that is driving the relative values in the mass matrix are the W boson transitions between them which give rise to the quark masses dynamically.

Yes, I also feel that in the as-yet unknown theory, CKM matrix "produces" quark masses, not the other way around. For one, masses can't give us the complete CKM matrix, it has a complex phase, but inference from masses only gives you a real-valued matrix.

ohwilleke
ohwilleke
Gold Member
Yes, I also feel that in the as-yet unknown theory, CKM matrix "produces" quark masses, not the other way around. For one, masses can't give us the complete CKM matrix, it has a complex phase, but inference from masses only gives you a real-valued matrix.

A few more relevant observations about the CKM matrix that suggest that it and not the mass matrix is primary, because generational differences seem to be primary.

The Wolfenstein parameterization of the CKM matrix (articulated first in L. Wolfenstein, "Parametrization of the Kobayashi-Maskawa Matrix", 51 (21) Physical Review Letters 1945 (1983)) emphasizes the extent to which the probability of a quark flavor change when it emits a W boson depends upon a change in quark family.

* In the Wolfenstein parameterization, "lambda" is roughly 0.02257 (and is another way of stating the Cabibbo angle), "A" is roughly 0.814, and the "p-in" CP violating term is about 0.135 minus 0.349i.

* A transition from the first generation to the second generation (or visa versa) happens with a probability of lambda squared (about 5.07%-5.08%).

* A transition from the second generation to the third generation (or visa verse) happens with a probability of about A squared times lambda to the fourth power (about 0.16%-0.17%).

* A transition from the first generation to the third generation (or visa versa) happens with a probability roughly equal to the probability of a transition from the first generation to the second generation, multiplied by the probability of a transition from the second generation to the third generation, times an adjustment in the form of a complex number an absolute value of a magnitude on the O(1) that includes a CP violating phase. In all, a first to third generation (or visa versa) quark family transition happens with a probability of about 0.0012% to 0.0075%.

* The probability of a second to third generation transition is consistent with two-thirds (between 62.7% and 69.7% with a best fit value of 66.3%) of the square of the probability of a first to second generation transition. It isn't impossible that this is really a formula constant in the theory derived with algebra from a deeper theory, rather than a physically measured constant "A".

* The probability that a quark will remain in the same quark generation is equal to one minus the probability that it will change generations (about 94.9202% in the first generation, about 94.7585 in the second generation, and about 99.8293% in the third generation).

The Wolfenstein parameterization emphasizes that the slight percentage differences between the probability of CKM matrix entry Vcb and Vts, between Vcd and Vus, flows mostly from compensating from other entries in that row, such as the considerably more significant (roughly 6-1) differences between the tiny Vtd and Vub. There are, in fact, more entries in the CKM matrix with the two CP violating parameters than are shown in a simplified version of the Wolfenstein parameterization, in the Standard Model CKM matrix, but those effects are tiny on a percentage basis relative to the magnitude of the other, much larger CKM matrix entries.

Of course, the decay of a quark at rest is mass-energy conservation barred from becoming a heavier quark. It can only decay to the lighter quark (something that is possible for all types of quarks except up quarks). Only quarks with sufficient momentum can transition to higher mass quarks.

The point that the Wolfenstein parameterization underscores is that the CKM matrix derives largely from crossing one of two (or both) of the fundamental fermion families, not just from the particular quarks involved in the transition.

I also wonder if the CP violating parameter of the CKM matrix really has the same source as the other parameters of the CKM matrix, or if the CKM matrix just ends up conflating the two when the actually have sources in separate mechanisms in a deeper theory, because the observational evidence doesn't provide an easy way to disentangle the two effects.