# What is new with Koide sum rules?

Gold Member
Ok, given that now (from this thread ) we suspect that D-branes are a thing, lets give other view to the "D-Branes + seesaw" formula of #80:
$$m_{k} = {R'^2 \over M_0 \alpha'^2} (n_k + {\theta_{k^i} - \theta_{k^f} \over 2 \pi})^2= m_0 (n_k + \lambda_k)^2$$

where $\lambda_k$ is the distance between the two branes and $n_k$ an extra wrapping that the open string can perform. This is a free adaptation of Johnson's primer hep-th/0007170: and the idea of see-sawing is not justified anywhere but lets say that it is just to test how compatible Koide formula is. Remember that the standard formula is $m_0(1+\mu_k)^2$, with conditions $\sum \mu_k = 0, \sum (\mu_k^2 - 1) =0$. Here, for the special case of three parallel branes, we have from the construction that $\lambda_3 = \lambda_2+\lambda_1$. I am not sure of how freely we can change the sign of a distance between branes; the formula apparently allows for it but it would need more discussion, so lets simply to investigate the case for wrapping $n_3=0$, so that we can freely flip $\lambda_3 \to - \lambda_3$ and grant $\lambda_3 + \lambda_2+\lambda_1 =0$ and lets put the other two wrappings at the same level n=1. With this, Koide formula should be
$$\frac 32 = { (n_1+n_2 + n_3)^2 \over n_1^2 + n_2^2 +n_3^2 + 2 \sum n_k \lambda_k + \sum \lambda_k^2} ={ 2 \over 1 + (\lambda_1+\lambda_2) + \lambda_1^2 + \lambda_2^2 + \lambda_1 \lambda_2}$$
with the extra condition that the distances are normalized to be all of them less than 1. The parametrization is a lot more inconvenient that the one we are used to, but solutions are not strange.
For instance if $\lambda_1=\lambda_2$ we have
$$\frac 32 = { 2 \over 1 + 2 \lambda_1 + 3 \lambda_1^2 }$$
and $\lambda_1= -\frac 13 + {\sqrt 2 \over 3}$
$$m_1=m_0 (1 + \lambda_1)^2= (\frac 23 + \frac {\sqrt 2 }3)^2 \\ m_2=m_0 (1 + \lambda_2)^2= (\frac 23 + \frac {\sqrt 2 }3)^2 \\ m_3=m_0 (0 - \lambda_1-\lambda_2)^2= 4 (-\frac 13 + \frac {\sqrt 2 }3)^2$$

So it seems doable but, well, the question of the winding numbers and the sign of the non-integer part needs more detail. Of course, without an argument to choose winding of the strings and distance between the branes they are attached to, it is just a change of variables.

@arivero has often mentioned the similarity of muon and pion masses, and also occasionally that of the tauon to the heavy-light charmed mesons; the idea being that the leptons are in some sense superpartners of mesons.

Now I am wondering if one could deduce a few quark masses, or even quark Koide relations, by assuming (1) Koide relation for charged leptons, (2) a barely broken meson-lepton supersymmetry, and (3) some form of "super GIM mechanism".

The GIM mechanism is a partial cancellation of amplitudes which allowed the existence and approximate mass of the charm quark to be predicted from the properties of kaons. The existence of charm, completing a second generation, explained the absence of flavor-changing neutral currents (FCNCs), and the specific mass explained a finetuning of kaon decays in which a virtual charm quark appears.

The lack of FCNCs is one of the built-in virtues of the standard model, that encourage some of us to think that it may be true to very high energies. Once you add new heavy particles, you need a feature like "minimal flavor violation" in order to preserve the lack of FCNCs, which in the SM comes automatically from anomaly cancellation. Supersymmetry specifically needs a "super GIM mechanism" (and there are one or two candidates for what that could be).

Now return to the hypotheses (1-3) above. By (1), we have the known masses of muon and tauon. Can we obtain the muon-pion and tauon-charm coincidences, through (2) and (3) respectively?

ohwilleke
Gold Member
Here is a nifty new little paper:

Phenomenological formula for CKM matrix and physical interpretation
Kohzo Nishida
(Submitted on 3 Aug 2017)
We propose a phenomenological formula relating the Cabibbo--Kobayashi--Masukawa matrix VCKM and quark masses in the form (md‾‾‾√ms‾‾‾√mb‾‾‾√)∝(mu‾‾‾√mc‾‾‾√mt‾‾‾√)VCKM. The results of the proposed formula are in agreement with the experimental data. Under the constraint of the formula, we show that the invariant amplitude of the charged current weak interactions is maximized.
Subjects: High Energy Physics - Phenomenology (hep-ph)
Cite as: arXiv:1708.01110 [hep-ph]
(or arXiv:1708.01110v1 [hep-ph] for this version)

• nikkkom
If they are onto something, then neutrino mass ratios can be inferred from PMNS matrix?

ohwilleke
Gold Member
If they are onto something, then neutrino mass ratios can be inferred from PMNS matrix?
The Nishida paper does not address leptons and the PMNS matrix. It is limited to the quarks and CKM matrix and is purely a phenomenological relationship. In general, the angles in the PMNS matrix parameterization are much bigger than those in the CKM matrix, which would imply neutrino masses closer to each other proportionately, and that is certainly what we see. But, I have no idea if the relationship is as precise.

The other thing about the Nishida paper is that it is a formulation with the square roots of masses and with the bare CKM matrix entries. But, neither of those are observables. You observe the mass and not the square root of the mass, and your observable in the CKM matrix is the square of the CKM matrix entry which gives a transition probability, and not the CKM matrix entry. Since both of the quantities in the Nishida paper are square roots of observable quantities, my intuition is that it ought to be possible to express the same relationship in terms of masses and squares of CKM matrix entries, i.e. the observable quantities. It may be that there is a technical reason that this won't work, but it is something that struck me reading it.

ohwilleke
Gold Member
Koide's latest paper:

Structure of Right-Handed Neutrino Mass Matrix
Yoshio Koide
(Submitted on 4 Aug 2017)
Recently, Nishiura and the author have proposed a unified quark-lepton mass matrix model under a family symmetry U(3)×U(3)′. The model can give excellent parameter-fitting to the observed quark and neutrino data. The model has a reasonable basis as far as the quark sector, but the form of the right-handed neutrino mass matrix MR does not have a theoretical grand, that is, it was nothing but a phenomenological assumption. In this paper, it is pointed out that the form of MR is originated in structure of neutrino mass matrix for (νi,Nα) where νi (i=1,2,3) and Nα (α=1,2,3) are U(3)-family and U(3)′-family triplets, respectively.

Subjects: High Energy Physics - Phenomenology (hep-ph)
Cite as: arXiv:1708.01406 [hep-ph]
(or arXiv:1708.01406v1 [hep-ph] for this version)

This paper is complemented by another earlier this year in a more reflective mood which was mentioned only in passing earlier in this thread.

Sumino Model and My Personal View
Yoshio Koide
(Submitted on 8 Jan 2017)
There are two formulas for charged lepton mass relation: One is a formula (formula A) which was proposed based on a U(3) family model on 1982. The formula A will be satisfied only masses switched off all interactions except for U(3) family interactions. Other one (formula B) is an empirical formula which we have recognized after a report of the precise measurement of tau lepton mass, 1992. The formula B is excellently satisfied by pole masses of the charged leptons. However, this excellent agreement may be an accidental coincidence. Nevertheless, 2009, Sumino has paid attention to the formula B. He has proposed a family gauge boson model and thereby he has tried to understand why the formula B is so well satisfied with pole masses. In this talk, the following views are given: (i) What direction of flavor physics research is suggested by the formula A; (ii) How the Sumino model is misunderstood by people and what we should learn from his model; (iii) What is strategy of my recent work, U(3)×U(3)′ model.
Comments: 5 pages, Talk given at a mini-workshop on "quarks, leptons and family gauge bosons", Osaka, Japan, December 26-27, 2016
Subjects: High Energy Physics - Phenomenology (hep-ph)
Cite as: arXiv:1701.01921 [hep-ph]
(or arXiv:1701.01921v1 [hep-ph] for this version)
Submission history
From: Yoshio Koide [view email]
[v1] Sun, 8 Jan 2017 07:33:59 GMT (6kb)

His recent co-authored papers with Nishiura which he alludes to in this abstract to this paper are:

Flavon VEV Scales in U(3)×U(3)′ Model
Yoshio Koide, Hiroyuki Nishiura
(Submitted on 23 Jan 2017 (v1), last revised 27 Feb 2017 (this version, v2))
We have already proposed a quark and lepton mass matrix model based on U(3)×U(3)′ family symmetry as the so-called Yukawaon model, in which the U(3) symmetry is broken by VEVs of flavons (Φf) αi which are (3,3∗) of U(3)×U(3)′. The model has successfully provided the unified description of quark and lepton masses and mixings by using the observed charged lepton masses as only family-number dependent input parameters. Our next concern is scales of VEVs of the flavons. In the present paper, we estimate the magnitudes of the VEV scales of flavons of the model which is newly reconstructed without changing the previous phenomenological success of parameter fitting for masses and mixings of quarks and leptons. We estimate that VEVs of flavons with (8+1,1), (3,3∗), and (1,8+1) are of 25the orders of 10 TeV, 104 TeV, and 107 TeV, respectively.
Comments: 23 pages, 1 figure, Introduction revised, Appendix remuved
Subjects: High Energy Physics - Phenomenology (hep-ph)
DOI: 10.1142/S0217751X17500853
Cite as: arXiv:1701.06287 [hep-ph]
(or arXiv:1701.06287v2 [hep-ph] for this version)

and

Quark and Lepton Mass Matrices Described by Charged Lepton Masses
Yoshio Koide, Hiroyuki Nishiura
(Submitted on 28 Dec 2015 (v1), last revised 14 May 2016 (this version, v3))
Recently, we proposed a unified mass matrix model for quarks and leptons, in which, mass ratios and mixings of the quarks and neutrinos are described by using only the observed charged lepton mass values as family-number-dependent parameters and only six family-number-independent free parameters. In spite of quite few parameters, the model gives remarkable agreement with observed data (i.e. CKM mixing, PMNS mixing and mass ratios). Taking this phenomenological success seriously, we give a formulation of the so-called Yukawaon model in details from a theoretical aspect, especially for the construction of superpotentials and R charge assignments of fields. The model is considerably modified from the previous one, while the phenomenological success is kept unchanged.
Comments: 14 pages, no figure, accepted version by MPLAl
Subjects: High Energy Physics - Phenomenology (hep-ph)
DOI: 10.1142/S021773231650125X
Cite as: arXiv:1512.08386 [hep-ph]
(or arXiv:1512.08386v3 [hep-ph] for this version)
Submission history
From: Yoshio Koide [view email]
[v1] Mon, 28 Dec 2015 12:14:42 GMT (13kb)
[v2] Tue, 8 Mar 2016 23:39:40 GMT (13kb)
[v3] Sat, 14 May 2016 04:59:24 GMT (14kb)

and
Quark and Lepton Mass Matrix Model with Only Six Family-Independent Parameters
Yoshio Koide, Hiroyuki Nishiura
(Submitted on 19 Oct 2015 (v1), last revised 7 Dec 2015 (this version, v2))
We propose a unified mass matrix model for quarks and leptons, in which sixteen observables of mass ratios and mixings of the quarks and neutrinos are described by using no family number-dependent parameters except for the charged lepton masses and only six family number-independent free parameters. The model is constructed by extending the so-called "Yukawaon" model to a seesaw type model with the smallest number of possible family number-independent free parameters. As a result, once the six parameters is fixed by the quark mixing and the mass ratios of quarks and neutrinos, no free parameters are left in the lepton mixing matrix. The results are in excellent agreement with the neutrino mixing data. We predict δℓCP=−68∘ for the leptonic CP violating phase and ⟨m⟩≃21 meV for the effective Majorana neutrino mass.
Comments: 10 pages, 2 figures, accepted version of a rapid communication i PRD
Subjects: High Energy Physics - Phenomenology (hep-ph)
Journal reference: Phys. Rev. D 92, 111301 (2015)
DOI: 10.1103/PhysRevD.92.111301
Cite as: arXiv:1510.05370 [hep-ph]
(or arXiv:1510.05370v2 [hep-ph] for this version)
Submission history
From: Yoshio Koide [view email]
[v1] Mon, 19 Oct 2015 06:55:38 GMT (553kb)
[v2] Mon, 7 Dec 2015 02:54:31 GMT (553kb)

and

Origin of Hierarchical Structures of Quark and Lepton Mass Matrices
Yoshio Koide, Hiroyuki Nishiura
(Submitted on 17 Mar 2015)
It is shown that the so-called "Yukawaon" model can give a unified description of masses, mixing and CP violation parameters of quarks and leptons without using any hierarchical (family number-dependent) parameters besides the charged lepton masses. Here, we have introduced a phase matrix P=daig(eiϕ1,eiϕ2,eiϕ3) with the phase parameters (ϕ1,ϕ2,ϕ3) which are described in terms of family number-independent parameters, together with using only the charged lepton mass parameters as the family number-dependent parameters. In this paper, the CP violating phase parameters δqCP and δℓCP in the standard expression of VCKM and UPMNS are predicted as δqCP≃72∘ and δℓCP≃−76∘, respectively, i.e. δℓCP∼−δqCP.
Subjects: High Energy Physics - Phenomenology (hep-ph)
Journal reference: Phys.Rev. D 91, 116002 (2015)
DOI: 10.1103/PhysRevD.91.116002
Cite as: arXiv:1503.04900 [hep-ph]
(or arXiv:1503.04900v1 [hep-ph] for this version)
Submission history
From: Hiroyuki Nishiura [view email]
[v1] Tue, 17 Mar 2015 03:15:32 GMT (903kb)

Last edited:
The Nishida paper does not address leptons and the PMNS matrix. It is limited to the quarks and CKM matrix and is purely a phenomenological relationship.
Yes, I know that.
I meant, since PMNS matrix is the very same thing for leptons as CKM is for quarks.
Then by analogy with his observation for quarks, let's assume for leptons (sqrt(mass(vi))) = PMNS * (sqrt(mass(li))).
Modulo minus signs, this gives a way to calculate normed vector of square roots of neutrino masses.

ohwilleke
Gold Member
Yes, I know that.
I meant, since PMNS matrix is the very same thing for leptons as CKM is for quarks.
Then by analogy with his observation for quarks, let's assume for leptons (sqrt(mass(vi))) = PMNS * (sqrt(mass(li))).
Modulo minus signs, this gives a way to calculate normed vector of square roots of neutrino masses.
The trouble is that as you formulate it, this doesn't come remotely close. The charged lepton masses are roughly a million times larger than the neutrino masses and the PMNS matrix entries are of O(1).

I omitted normalization. He uses normalized vectors, they are length 1. His formula can give relative masses of neutrinos.

In general I think the CKM matrix tells you that transitions are more likely to occur within the same generation rather than the other...
In "What is new with Koide sum rules?" thread, ohwilleke spotted and posted a link to this recent paper which might be a start in understanding what CKM matrix is all about and why it has precisely these values:

Here is a nifty new little paper:
Phenomenological formula for CKM matrix and physical interpretation

Kohzo Nishida

(Submitted on 3 Aug 2017)

We propose a phenomenological formula relating the Cabibbo--Kobayashi--Masukawa matrix VCKM and quark masses in the form (md‾‾‾√ms‾‾‾√mb‾‾‾√)∝(mu‾‾‾√mc‾‾‾√mt‾‾‾√)VCKM. The results of the proposed formula are in agreement with the experimental data. Under the constraint of the formula, we show that the invariant amplitude of the charged current weak interactions is maximized.
https://arxiv.org/abs/1708.01110

Last edited:
Just for gigs, I created this abomination to check whether it actually works out

Code:
#!/usr/bin/python
import math
from math import sqrt

# experimental quark masses, MeV
md=4.7
mu=2.2
ms=96
mc=1280
mb=4180
mt=173500

print "Square roots of experimental quark masses, MeV"
rmd=sqrt(md)
rmu=sqrt(mu)
rms=sqrt(ms)
rmc=sqrt(mc)
rmb=sqrt(mb)
rmt=sqrt(mt)
print "dn: %.4f %.4f %.4f" % (rmd,rms,rmb)
print "up: %.4f %.4f %.4f" % (rmu,rmc,rmt)

print "Unit-length vectors built of those (divided by norm)"
nu=sqrt(mu+mc+mt)
nd=sqrt(md+ms+mb)
nmd=rmd/nd
nmu=rmu/nu
nms=rms/nd
nmc=rmc/nu
nmb=rmb/nd
nmt=rmt/nu
print "dn: %.4f %.4f %.4f" % (nmd,nms,nmb)
print "up: %.4f %.4f %.4f" % (nmu,nmc,nmt)

print "Kohzo Nishida says that (normed_sqrt_up_masses) = CKM * (normed_sqrt_down_masses)"
c11=0.97435  ;c12=-0.2287 ;c13=0.005641
c21=0.2286   ;c22=0.9712  ;c23=-0.06700
c31=0.009846 ;c32=0.06652 ;c33=0.9977
u1 = c11*nmd + c12*nms + c13*nmb
u2 = c21*nmd + c22*nms + c23*nmb
u3 = c31*nmd + c32*nms + c33*nmb
print "rc: %.4f %.4f %.4f" % (u1,u2,u3)
Output:

Code:
Square roots of experimental quark masses, MeV
dn: 2.1679 9.7980 64.6529
up: 1.4832 35.7771 416.5333
Unit-length vectors built of those (divided by norm)
dn: 0.0331 0.1498 0.9882
up: 0.0035 0.0856 0.9963
Kohzo Nishida says that (normed_sqrt_up_masses) = CKM * (normed_sqrt_down_masses)
rc: 0.0036 0.0868 0.9962

• arivero
ohwilleke
Gold Member
Just for gigs, I created this abomination to check whether it actually works out

Code:
#!/usr/bin/python
import math
from math import sqrt

# experimental quark masses, MeV
md=4.7
mu=2.2
ms=96
mc=1280
mb=4180
mt=173500

print "Square roots of experimental quark masses, MeV"
rmd=sqrt(md)
rmu=sqrt(mu)
rms=sqrt(ms)
rmc=sqrt(mc)
rmb=sqrt(mb)
rmt=sqrt(mt)
print "dn: %.4f %.4f %.4f" % (rmd,rms,rmb)
print "up: %.4f %.4f %.4f" % (rmu,rmc,rmt)

print "Unit-length vectors built of those (divided by norm)"
nu=sqrt(mu+mc+mt)
nd=sqrt(md+ms+mb)
nmd=rmd/nd
nmu=rmu/nu
nms=rms/nd
nmc=rmc/nu
nmb=rmb/nd
nmt=rmt/nu
print "dn: %.4f %.4f %.4f" % (nmd,nms,nmb)
print "up: %.4f %.4f %.4f" % (nmu,nmc,nmt)

print "Kohzo Nishida says that (normed_sqrt_up_masses) = CKM * (normed_sqrt_down_masses)"
c11=0.97435  ;c12=-0.2287 ;c13=0.005641
c21=0.2286   ;c22=0.9712  ;c23=-0.06700
c31=0.009846 ;c32=0.06652 ;c33=0.9977
u1 = c11*nmd + c12*nms + c13*nmb
u2 = c21*nmd + c22*nms + c23*nmb
u3 = c31*nmd + c32*nms + c33*nmb
print "rc: %.4f %.4f %.4f" % (u1,u2,u3)
Output:

Code:
Square roots of experimental quark masses, MeV
dn: 2.1679 9.7980 64.6529
up: 1.4832 35.7771 416.5333
Unit-length vectors built of those (divided by norm)
dn: 0.0331 0.1498 0.9882
up: 0.0035 0.0856 0.9963
Kohzo Nishida says that (normed_sqrt_up_masses) = CKM * (normed_sqrt_down_masses)
rc: 0.0036 0.0868 0.9962
If you really wanted to go large, you could do a crude Monte Carlo error analysis by having a program do the same thing for every combination of +1 sigma, the best fit vale and -1 sigma, for all of the input values based on the Particle Data Group error bars.

• arivero
That would amount to reimplementing CKM Fitter code from scratch :)

I assume CKM Fitter people, if not already grappling with the possibilities opened by this paper, to do so Real Soon.

ohwilleke
Gold Member
The latest measurement of the tau lepton mass again confirms the original Koide's rule. Koide's rule, a formula proposed in 1981, six years after the tau lepton was discovered, when its mass was known much less accurately, predicts the mass of the tau lepton based upon the mass of the electron and the muon. This prediction using current electron and muon mass measurements is:

1776.96894 ± 0.00007 MeV/c^2.

The uncertainty is entirely due to uncertainty in the electron and muon mass measurements. The low uncertainty in the Koide's rule prediction reflects the fact that the electron and muon mass have been measured much more precisely than the tau lepton mass.

The latest measurement from BESIII, which is the most precise single experimental measurement to date is:

1776.91 ± 0.12 + 0.10/− 0.13 MeV/c^2 (the combined error is ± 0.17).

This result is 0.06 MeV less than the Koide's rule prediction which is consistent to less than one-half of a standard deviation of experimental uncertainty from the predicted value.

The new result is closer to the Koide's rule prediction than the Particle Data Group (PDG) value for 2016 which is:

1776.83 ± 0.12 MeV/c^2

The PDG value is within about 1.2 standard deviations of the Koide's rule prediction. This new result will probably push the next PDG value closer to the Koide's rule prediction.

Koide's rule is one of the most accurate phenomenological hypotheses in existence which has no Standard Model theoretical explanation, although given the precision to which it is true, there is almost certainly some explanation for this correspondence based upon new physics beyond (or really "within") the Standard Model.

Something that troubles me, is that every explanation we have for the Koide formula seems to be at odds with Higgs criticality, in that the latter suggests that physics is just standard model up to high scales, whereas the explanations for Koide involve new physics at low scales. See Koide's remarks from January. He says one may think of the formula as holding approximately among running masses, or exactly among pole masses. If we focus just on well-defined field theories that have been written out, they all involve new physics (e.g. Koide's yukawaon fields, the vevs of which contribute to the SM yukawas). In the case of the Sumino mechanism for the pole masses, there are family gauge bosons which are supposed to show up by 104 TeV, i.e. 107 GeV. If we focus just on the yukawaons... Koide seems to have argued that new physics should show around 1012 GeV. I would be a little happier with that, it's in the vicinity of the lowest-scale explanations of Higgs criticality.

But for this reason, I also wonder if we could do with a new, infrared perspective on the Higgs mechanism. The most recent paper by Arkani-Hamed et al actually provides such a perspective, but only for gauge boson mass, not for fermion mass.

ohwilleke
Gold Member
Something that troubles me, is that every explanation we have for the Koide formula seems to be at odds with Higgs criticality, in that the latter suggests that physics is just standard model up to high scales, whereas the explanations for Koide involve new physics at low scales. See Koide's remarks from January. He says one may think of the formula as holding approximately among running masses, or exactly among pole masses. If we focus just on well-defined field theories that have been written out, they all involve new physics (e.g. Koide's yukawaon fields, the vevs of which contribute to the SM yukawas). In the case of the Sumino mechanism for the pole masses, there are family gauge bosons which are supposed to show up by 104 TeV, i.e. 107 GeV. If we focus just on the yukawaons... Koide seems to have argued that new physics should show around 1012 GeV. I would be a little happier with that, it's in the vicinity of the lowest-scale explanations of Higgs criticality.

But for this reason, I also wonder if we could do with a new, infrared perspective on the Higgs mechanism. The most recent paper by Arkani-Hamed et al actually provides such a perspective, but only for gauge boson mass, not for fermion mass.
Simply relying on new physics, in and of itself, isn't very troubling because this is an area where new physics wouldn't contract the Standard Model, it would merely fill in a gap where the Standard Model provides no explanation and instead resorts to determining the values of constants experimentally with a theory.

Furthermore, I would say that of people who are familiar with the Standard Model almost nobody thinks that the values of the Standard Model experimentally measured constants are really arbitrary. Feynman said so in QED and a couple of his other books. I've seen at least a couple of other big name physicists reiterate that hypothesis, although I don't have references readily at hand. A few folks think that there is no deeper theory, and many don't think about the issue at all, but the vast majority of people who understand it believe in their heart of hearts that there is a deeper structure with some mechanism out there to find that we just haven't yet grasped.

But, the trick is how to come up with BSM physics that doesn't contradict the SM and reasonable inferences from it to explain these constants. How can we construct new physics to explain the Standard Model constant values in some sector that doesn't screw up anything else?

The go to explanation the last two times we had a jumble of constants that needed to be explained - the Periodic Table and the Particle Zoo, ended up being resolved with preon-like theories the cut through a mass of fundamental constants by showing that they were derived from a smaller number of more fundamental components. And, one could conceive of a theory that could do that - I've seen just one reasonably successful effect at doing so by a Russian theoretical physicist, V. N. Yershov - but the LHC bounds on compositeness (which admittedly have some model dependence) are very, very stiff. Preons wouldn't screw anything else up, although they might require a new boson to carry an "ultra-strong force" that binds the preons.

I am not very impressed with the yukawaon approach, or Sumino's mechanism. They are baroque and not very well motivated and, as you note, involve low scale new physics where it is hard to believe that we could have missed anything so profound.

As you know, I am on record as thinking that Koide's rule and the quark mass hierarchy emerge dynamically through a mechanism mediated by the W boson, which is very clean in the case of the charged leptons with only three masses to balance and a situation where a W boson can turn any one of the three into any one of the remaining two (conservation of mass-energy permitting). The situation is messier with the quarks where any given quark can by transformed via the W into one of three other kinds of quarks (but not five other kinds of quarks in one hop), and where there is not a quark equivalent to lepton universality due to the structure of the CKM matrix.

In this analysis, the Higgs vev is out there setting the overall scale of the fundamental fermion and boson masses, the Higgs boson mass is perhaps most easily understood as a gap filling process of elimination result after all other fundamental boson masses have been set, and the W boson plays a key role in divvying up the overall mass allowed to the fermion sector among the responsive quarks, and separately among the respective charged leptons (and perhaps among the neutrinos as well - hard to know), maybe it even plays a role in divvying up the overall mass allowed to the fermion sector between quarks and leptons (as suggested in some extended Koide rule analysis).

That description, of course, is in some ways heuristic. It still needs to produce a model in which the Higgs boson couples to each fundamental particle of the Standard Model (except photons, gluons and possibly also except neutrinos), in proportion to the rest mass of each, so the focuses on the Higgs yukawas and the the W boson interactions respectively have to both be true to some extent in any theory, it is just a matter of which perspective provides "the most information for free" which is what good theories do.

Humans like to impute motives to processes even when they are in equilibrium and interdependent. We like to say either that the Higgs boson causes fundamental particle masses, or the the W boson does, or that fundamental particle masses are tied to their self-interaction plus an excitation factor for higher generations, or what have you.

But, these anthropomorphic imputations of cause and effect and motive may be basically category errors in the same way that it really isn't accurate to say that the length of the hypotenuse of a right triangle is caused by the length of its other two sides. Yes, there is an equation that relates the length of the three sides of a right triangle to each other, and yes, knowing any two, you can determine the third, but it isn't really correct to say that there are lines of causation that run in any particular direction (or alternatively, you could say that the lines of causation run both ways and are mutual). I suspect that the relationships between the Standard Model constants is going to be something like that which is just the kind of equation that Koide's rule involves.

Of course, this dynamic balancing hypothesis I've suggested is hardly the only possible way to skin the cat. (Is it not PC to say that anymore?).

Indeed, from the point of view of natural philosophy and just good hypothesis generation, one way to identify a really good comprehensive and unified theory is that its predictions are overdetermined such that there are multiple independent ways to accomplish the same result that must necessarily all be true for the theory to hold together.

In other words, for example, there really ought to be more than one more or less independent ways to determine the Higgs boson mass from first principles in really good theory. So: (1) maybe one way to determine the Higgs boson mass is to start at a GUT scale where it has a boundary mass value of zero in a metastable universe and track its beta function back to its pole mass (also here) and (2) another way ought to be to start with half the of the square of the Higgs vev and then subtract out the square of the W and Z boson masses and take the square root, and (3) another way ought to be with the fine tuned kind of calculations that give rise to the "hierarchy problem", and (4) maybe another looks at the relationship between the top quark mass, the W boson mass and the Higgs boson mass in electroweak theory, and (5) another might look to self-interactions via fundamental forces (also here) as establishing the first generation and fundamental boson masses and come up with a way of seeing the second and third generations as the only possible mathematically consistent excitations of first generation masses derived from self-interactions (somewhat along the same lines is this global mass trend line), and (6) another might start with half of the Higgs vev as a "tree level" value of the "bare" Higgs boson mass and make high loop corrections (something similar is found here) and (7) maybe there is a deeper theory that gives significance to the fact that the measured Higgs boson mass is very nearly the mass that minimizes the second loop corrections necessary to convert the mass of a gauge boson from an MS scheme to a pole mass scheme, (8) maybe there is something related to the fact that the Higgs boson mass appears to maximize its decay rate into photons, and (9) maybe there ought to be some other way as well that starts with constraints particular to massive spin-0 even parity objects in general using the kind of methodology in the paper below then limits that parameter space using measured values of the Standard Model coupling constants and maybe a gravitational coupling constants such that any quark mass (since quarks interact with all three Standard Model forces plus gravity) could be used to fix its value subject to those constraints.

"Magically," maybe all nine of those methods might produce the same Higgs boson mass prediction despite not having obvious derivations from each other. The idea is not that any of (1) to (9) are actually correct descriptions of the real world source of the Higgs boson mass, but to illustrate what a correct overdetermined theory might "feel" like.

There might be nine independent correct ways to come up with a particular fundamental mass that all have to be true for the theory to hold together making these values the only possible one that a consistent TOE that adhere to a handful of elementary axioms could have, in sort of the polar opposite of a many universes scenario where every physical constant is basically random input into some Creator God's computer simulation and we just ended up living in one of them.

In particular, I do think that at least some of the approaches to an overdetermined Higgs mechanism may indeed involve something that make sense on an infrared scale, rather than relying on new particles or forces at a UV scale as so much of the published work tends to do.

Relations like L & CP and Koide's rule and the fact that the Higgs mass is such that it doesn't require UV completion to be unitary and analytic up to the GUT scale and the fact that the top quark width fits the SM prediction as do the Higgs boson branching fractions and the electron g-2 all point to a conclusion that the SM is or very nearly is a complete set of fundamental particles.

Even the muon g-2 discrepancy is pretty small - the measured value and the computed one (0.0011659209 versus 0.0011659180) are identical down to one part per 1,000,000, so there can't be that many missing particles contributing loops that are missing from the Standard Model computation. We are talking about a discrepancy of 29 * 10^-10 in the value. Maybe that difference really is three sigma (and not just a case were somebody has underestimated the one of the systemic errors in the measurement by a factor O(1) or O(10) or so) and something that points at BSM physics, but it sure doesn't feel like we are on the brink of discovering myriad new BSM particles in the UV as null search after null search at the LHC seem to confirm.

Too many of the process me measure in HEP are sensitive to the global content of the model (including the UV part to very high scales given the precision of our measurements) because of the way that so many of the observables are functions of all possible ways that something could happen for us to be missing something really big while we fail to see BSM effects almost anywhere while doing lots and lots and lots of experimental confirmations of every conceivable kind.

Also FWIW, the latter paper that you reference (79 pages long) has the following abstract:

Scattering Amplitudes For All Masses and Spins
Nima Arkani-Hamed, Tzu-Chen Huang, Yu-tin Huang
(Submitted on 14 Sep 2017)
We introduce a formalism for describing four-dimensional scattering amplitudes for particles of any mass and spin. This naturally extends the familiar spinor-helicity formalism for massless particles to one where these variables carry an extra SU(2) little group index for massive particles, with the amplitudes for spin S particles transforming as symmetric rank 2S tensors. We systematically characterise all possible three particle amplitudes compatible with Poincare symmetry. Unitarity, in the form of consistent factorization, imposes algebraic conditions that can be used to construct all possible four-particle tree amplitudes. This also gives us a convenient basis in which to expand all possible four-particle amplitudes in terms of what can be called "spinning polynomials". Many general results of quantum field theory follow the analysis of four-particle scattering, ranging from the set of all possible consistent theories for massless particles, to spin-statistics, and the Weinberg-Witten theorem. We also find a transparent understanding for why massive particles of sufficiently high spin can not be "elementary". The Higgs and Super-Higgs mechanisms are naturally discovered as an infrared unification of many disparate helicity amplitudes into a smaller number of massive amplitudes, with a simple understanding for why this can't be extended to Higgsing for gravitons. We illustrate a number of applications of the formalism at one-loop, giving few-line computations of the electron (g-2) as well as the beta function and rational terms in QCD. "Off-shell" observables like correlation functions and form-factors can be thought of as scattering amplitudes with external "probe" particles of general mass and spin, so all these objects--amplitudes, form factors and correlators, can be studied from a common on-shell perspective.

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mfb
Mentor
This mass measurement is from 2014. The PDG average includes this measurement already, see this list.

The new arxiv submission is just a review.

• ohwilleke
ohwilleke
Gold Member
This mass measurement is from 2014. The PDG average includes this measurement already, see this list.

The new arxiv submission is just a review.
Good catch. I'm surprised how few tau mass measurements there have been. Only one in the last eight years.

mfb
Mentor
It is challenging to measure, just a few experiments had the ability to get a good estimate and most of them published their final measurement already.

I am not very impressed with the yukawaon approach, or Sumino's mechanism
But the calculations at the core of how they work, could actually give substance to the kind of bootstrap you suggest. Look at Sumino 2008. Section 4, equation 36, you have a potential-energy function for a nine-component scalar. With a few extra conditions, it has a minimum very close to the square-root-masses of the charged leptons. Meanwhile, section 3 (e.g. figure 4) describes how massive gauge bosons can cancel the QED loop effects that would spoil the Koide relation for the pole masses.

Sumino goes on to introduce multiple new scalars and big new gauge groups, in order to implement these ideas. But the core of it really is that the sqrt-masses minimize some energy function, and that electromagnetic running is countered by some kind of flavor physics.

ohwilleke
Gold Member
But the calculations at the core of how they work, could actually give substance to the kind of bootstrap you suggest. Look at Sumino 2008. Section 4, equation 36, you have a potential-energy function for a nine-component scalar. With a few extra conditions, it has a minimum very close to the square-root-masses of the charged leptons. Meanwhile, section 3 (e.g. figure 4) describes how massive gauge bosons can cancel the QED loop effects that would spoil the Koide relation for the pole masses.

Sumino goes on to introduce multiple new scalars and big new gauge groups, in order to implement these ideas. But the core of it really is that the sqrt-masses minimize some energy function, and that electromagnetic running is countered by some kind of flavor physics.
Good point. Sometimes people can get the right result even if the mechanism to explain its origin is wrong. Similarly, both dark matter and modified gravity theories can explain galactic rotation, but one of those mechanisms is wrong.

ohwilleke
Gold Member
Another Formula for the Charged Lepton Masses
Yoshio Koide
(Submitted on 9 Nov 2017)
A charged lepton mass formula (me+mμ+mτ)/(me‾‾‾√+mμ‾‾‾√+mτ‾‾‾√)2=2/3 is well-known. Since we can, in general, have two relations for three quantities, we may also expect another relation for the charged lepton masses. Then, the relation will be expressed by a form of memμmτ‾‾‾‾‾‾‾‾√/(me‾‾‾√+mμ‾‾‾√+mτ‾‾‾√)3. According to this conjecture, a scalar potential model is speculated.
Subjects: High Energy Physics - Phenomenology (hep-ph)
Cite as: arXiv:1711.03221 [hep-ph]
(or arXiv:1711.03221v1 [hep-ph] for this version)

The standard Koide formula says that "K", a function of the sqrt-masses, equals 2/3. Koide derived this formula from a scalar potential with U(3) family symmetry in 1990. In this latest paper, he defines a new function of the sqrt-masses, "kappa", extends the 1990 potential, and chooses rational values for a few coefficients, in order to predict that "kappa" equals 1/486. The actual value of "kappa" is 1/486.663.

Both K and kappa are scale-invariant, in that they remain unchanged if all the masses are multiplied by a constant. This means that the charged lepton masses are fully determined if one specifies K, kappa, and a mass scale. Here I am reminded of Brannen's formula, which appears in Koide 2007, equations 3.2, 3.3, 3.6, 3.9. The angle 2/9 radians is usually regarded as the key parameter, and has no known field-theoretic derivation.

The quantity 2/9 does actually appear in Koide's latest paper (equation 16), but nothing like Brannen's trigonometric formulas. One could see if they are hiding somewhere in the algebra. Another place to check would be Sumino's potential that I mentioned here, in comment #170.

Gold Member
How does he address the existence of other tuples?. Particularly the one existing in the literature previously to his research, (0, m_d, m_s)? For this one, the value of the new parameter is just zero.

How does he address the existence of other tuples?. Particularly the one existing in the literature previously to his research, (0, m_d, m_s)? For this one, the value of the new parameter is just zero.
Bearing in mind that in these papers, the masses are determined by the vev of a new, matrix-valued scalar field, we can say that a massless electron requires that the determinant of the vev matrix is zero. (See equation 20 in the 1990 paper.) I do not know if such a case can be obtained through e.g. a choice of coefficients for the scalar potential, as simple as that which Koide now exhibits.

The Haut-Harari-Weyers triple of up, down, strange, includes quarks of different charges, so it requires something more than just the square root of a yukawa matrix. In their original paper, it is just an accident that the Koide relation is satisfied, but one could look for an extended model in which there's a reason.