What is new with Koide sum rules?

[arivero]

Ok, given that now (from this thread ) we suspect that D-branes are a thing, let's 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 let's 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 let's 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 let's put the other two wrappings at the same level n=1. With this, Koide formula should be
<br /> \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}<br /> ={ 2 \over 1 + (\lambda_1+\lambda_2) + \lambda_1^2 + \lambda_2^2 + \lambda_1 \lambda_2}<br />
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
<br /> \frac 32 = { 2 \over 1 + 2 \lambda_1 + 3 \lambda_1^2 }<br />
and \lambda_1= -\frac 13 + {\sqrt 2 \over 3}
<br /> m_1=m_0 (1 + \lambda_1)^2= (\frac 23 + \frac {\sqrt 2 }3)^2 \\<br /> m_2=m_0 (1 + \lambda_2)^2= (\frac 23 + \frac {\sqrt 2 }3)^2 \\<br /> 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.

 

[mitchell porter]

@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]

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.
Comments: 6 pages, no figures
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]

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]

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.

Comments: 7 pages, no figure
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.
Comments: 24 pages, 2 figures
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)

 

[nikkkom]

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]

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).

 

[nikkkom]

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

 

[nikkkom]

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

(edit: corrected link)

 

[nikkkom]

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

#!/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:

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

 

[ohwilleke]

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

#!/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:

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.

 

[nikkkom]

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]

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.

 

[mitchell porter]

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]

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 anyone of the three into anyone 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.

 

[mfb]

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]

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]

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.

 

[mitchell porter]

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]

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]

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.
Comments: 5 pages, no figure
Subjects: High Energy Physics - Phenomenology (hep-ph)
Cite as: arXiv:1711.03221 [hep-ph]
(or arXiv:1711.03221v1 [hep-ph] for this version)

 

[mitchell porter]

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.

 

[arivero]

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.

 

[mitchell porter]

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.

 

[mitchell porter]

Some basic remarks on obtaining the Koide relation, and its generalizations, via string theory.

In the standard model, elementary masses and mixings come from yukawa couplings between two chiral fermions and the Higgs field. This is also the case in string theory realizations of the standard model. A sketch of how it works for intersecting branes may be seen in figure 1 (page 3) here. Each distinct species of elementary particle is found at a specific location in the extra dimensions, at a point where two branes intersect; the particle itself is an open string between the two branes.

The left-handed fermion is at one location, the right-handed fermion at another location, the Higgs boson is at a third location. The yukawa coupling is a three-way interaction in which e.g. a string jumps from the left-handed site to the right-handed site, by stretching out until it joins with a Higgs string. The probability amplitude for this to happen is dominated by the worldsheet with the minimum surface area, which is the triangle in the picture.

To a first approximation, the amplitude equals exp(-area). So if you know the mass matrices you want, this is a way to picture the stringy geometry that you need: the Higgs boson will be localized somewhere in the extra dimensions, the elementary chiral fermions will be scattered around it, and the distances and angles must be such that the areas of these triangles are - ln(yukawa).

But you can't just say that you want the strings to be located at specific points, and then just place them there. Or at least, you can't do that in any stringy model that I ever heard of. In real string theory, you'll have an overall geometry for the extra dimensions, and then the branes will occupy hypersurfaces in that geometry, and all the geometric parameters (the moduli) are dynamical. They will settle into a state of lowest energy, and that will determine the relative locations of everything... Perhaps this could be avoided if the background geometry were hyperbolic and rigid, or if numerous branes form a dense mesh so that there's always an intersection point near where you want your particles to be located. But I am not aware of any brane model where that can be done.

The masses and mixings present certain patterns or possible patterns, that might guide you in constructing such a brane geometry. But if we take Koide seriously, there's a very special and precise pattern present, specifically in the masses of the charged leptons. In Koide's field-theoretic models, he introduces extra fields, "yukawaons", which enter into the yukawa coupling, in order to produce his relation.

In terms of string theory, it's possible that the Koide relation, if it can be produced at all, might be due solely to a special symmetry of the compact geometry and the location of branes within it - that might be enough to induce the mass relation. Or, there might be extra string states involved - the worldsheet may trace out an n-gon with n>3. A further interesting possibility is that virtual branes may be involved - branes that wrap some closed hypersurface in the compact geometry, with which the strings interact; a kind of vacuum polarization. It would be interesting indeed if yukawaons were associated with such "Euclidean branes".

(I will also mention again that a Koide relation among pole masses seems to require still further interactions that produce special cancellations, like the family gauge bosons introduced by Sumino. All the mechanisms mentioned above are also potentially relevant here.)

How about the generalization of the Koide relation which initiated this thread, the waterfall of quark triplets introduced by @arivero in arXiv:1111.7232? Unlike the original Koide relation, there is still no field-theoretic implementation of the full waterfall, because the triples include quarks with different hypercharges, and that's just more difficult to do. But all my comments still apply, and the paper contains some remarks on the geometry of the mass vectors involved, which, who knows, might be directly relevant to a stringy implementation.

There's one more notable phenomenon, and that is the appearance of mass scales from QCD - 313 MeV, 939 MeV - in some of these Koide triples, when they are expressed using Carl Brannen's method. 939 MeV is the nucleon mass and it has been obtained from lattice QCD, but I am not aware of any simplified heuristic explanation of where that number comes from, that QCD theorists would agree with. In a few places in this thread, I have posted about papers which do purport to give a field-theoretic derivation of these quantities (Schumacher in #134, Gorsky et al in #136). The holographic QCD of Sakai and Sugimoto also gives a framework (from string theory rather than field theory) in which the nucleon mass can be obtained, once all the parameters of the brane geometry have been specified.

If the QCD scales do appear in the extended Koide relations for a reason, and not just by chance, I think it has to be because there is some QCD-like theory underlying the standard model. There have been many proposals for what this could be, as has been documented throughout the thread on "the wrong turn of string theory". Presumably one should then look for a stringy implementation of QCD mechanisms like those just described, and then rerun the previous arguments about yukawa couplings on top of that.

 

[arivero]

An anonymous edition in the wikipedia, deleted because it did not provide sources, points out that Koide equation amounts to say that the square roots x_n={\sqrt {m_{n}}} are the three solutions of a cubic equation
ax^{3}+bx^{2}+cx+d=0
when b^{2}=6ac

This idea is along the line of writting Koide formula as (x_1^2 + x_2^2 + x _3^2) - 4 (x_1 x_2 + x_2 x_3 + x_3 x_1) =0 A point that Goffinet already exploited to build his quartic equation.

I was wondering, one can always multiply the cubic by ax^{3}-bx^{2}+cx-d, can we? If so, we shoud have also
a^2 m^3+(2 a c-b^2) m^2+(c^2-2 b d) m-d^2 = 0

 

[lpetrich]

I agree, bare plus corrections seems the best approach, and in fact it is the usual approach to calculate the decay width. But I am intrigued really about the size of phase space, and more particularly about which is the maximum energy that the neutrino pair can carry. In principle is is a measurable quantity. Is it 105.6583668 - 0.510998910, i.e, m_\mu(m_\mu) - m_e(m_e) (?), or is it m_\mu(m_\mu) - m_e(m_\mu)?

Since they are both free particles, the electron's and the muon's masses are both on-shell masses (pole masses): $m_e(m_e)$ and $m_\mu(m_\mu)$.

This points to a more serious problem with Koide's mass formula. How well does it hold up at electroweak-unification energy scales or GUT energy scales?

 

[mitchell porter]

This points to a more serious problem with Koide's mass formula. How well does it hold up at electroweak-unification energy scales or GUT energy scales?

That will depend on what happens at intermediate scales. In the past ten years, Koide and his collaborators have considered many variations on the theme that the mass formula is exact at some high scale, and is somewhat preserved at lower scales by a version of Sumino's mechanism, in which the bosons of a gauged flavor symmetry cancel a major electromagnetic contribution to the running. According to this paradigm, even when the Sumino mechanism is included, one has to regard the precision with which the formula works for the pole masses, as partly coincidental.

To be a little more specific: Sumino said that there would be a unification of electroweak and the flavor symmetry at around 10^3 TeV, and predicted that the next decimal place of the tau lepton pole mass would deviate from the formula. Koide has modified Sumino's theory in ways that imply larger corrections at low scales (and thus the formula's success when applied to the pole masses is more of a coincidence in these theories), but has retained the idea that the new gauge bosons have masses of around 10^3 TeV.

Meanwhile, one could guess that the pole masses are the important quantities after all, but then some wholly new perspective or mechanism is needed. We do have the concept of an infrared fixed point; maybe there's some nonperturbative perspective that mixes UV and IR in which it makes sense; but right now these models by Koide and friends are the only ones that address this problem.

 

[arivero]

How compatible could it be a composite Higgs with GUT? One could explain Koide coincidente, the other could explein coupling coincidence.

 

[arivero]

Hmm, I should avoid to type from the phone. Well, anyway, the point was that perhaps GUT scale is not relevant for Koide. It is amusing that the main argument that we have (had?) for GUT is another numerical coincidence, the one of the coupling constants, but there was nothing about coincidence of yukawas... at most, variations on the theme of Jarslkog and Georgi https://en.wikipedia.org/wiki/Georgi–Jarlskog_mass_relation.

Another problem for quarks is that the pole mass is not directly measurable. Worse, Koide formula seems to work better with MSbar masses. Taking as input 4.18 and 1.28 GeV, Koide formula predicts 168.9 GeV for the top quark, while taking the pole masses 4.78 and 1.67 the prediction goes off to 203.2 GeV. (we nail it with intermediate mixes, eg input 4.18 and 1.37 predicts 173.3). Note that we now suspect that the MSbar mass of the top has a very noticeable EW contribution; Jegerlehner says that it actually counterweights the QCD contribution.

 

[ohwilleke]

A new Koide paper:

Parameter-Independent Quark Mass Relation in the U(3)×U(3)′ Model

Yoshio Koide, Hiroyuki Nishiura

(Submitted on 18 May 2018)

Recently, we have proposed a quark mass matrix model based on U(3)×U(3)′ family symmetry, in which up- and down-quark mass matrices Mu and Md are described only by complex parameters au and ad, respectively. When we use charged lepton masses as additional input values, we can successfully obtain predictions for quark masses and Cabibbo-Kobayashi-Maskawa mixing. Since we have only one complex parameter aq for each mass matrix Mq, we can obtain a parameter-independent mass relation by using three equations for Tr[Hq], Tr[HqHq] and detHq, where Hq≡MqM†q (q=u,d). In this paper, we investigate its parameter-independent feature of the quark mass relation in the model.

 

[ohwilleke]

Koide considers the possibility that his charged lepton rule could be a function of SUSY physics. https://arxiv.org/abs/1805.09533

The observed charged lepton masses satisfy the relations K≡(me+mμ+mτ)/(me‾‾‾√+mμ‾‾‾√+mτ‾‾‾√)2=2/3 and κ≡memμmτ‾‾‾‾‾‾‾‾√/(me‾‾‾√+mμ‾‾‾√+mτ‾‾‾√)3=1/486 with great accuracy. These parameters are given as K=(Tr[ΦΦ])/(Tr[Φ])2and κ=detΦ/(Tr[Φ])3 if the charged lepton masses mei are given by mei∝∑kΦ kiΦ ik where Φ is a U(3)-family nonet scalar. Simple scalar potential forms to realize the relations have been already proposed in non-supersymmetric scenarios, but the potential forms are not stable against the renormalization group effects. In this paper, we examine supersymmetric scenarios to make the parameters K and κ stable against the effects, and show possible simple superpotential forms for the relations.

 

[mitchell porter]

While Strings 2018 convened in Okinawa, Koide gave a talk at Osaka University (PDF) reviewing very succinctly the nature of his relation, the contribution of Sumino, and the very latest theoretical ideas.

 

[ohwilleke]

While Strings 2018 convened in Okinawa, Koide gave a talk at Osaka University (PDF) reviewing very succinctly the nature of his relation, the contribution of Sumino, and the very latest theoretical ideas.

Thanks. The presentation is a riot! Such humor and humility.

 

[ohwilleke]

<Moderator's note: twitter link removed: too much advertising and inappropriate source.>

I didn't know that Twitter links were categorically forbidden, even top flight newspapers use them now and a lot of worthwhile discussion among experts in the field also occurs by Twitter before it ends up being published if it is published at all. Surely there must be some appropriate way to note where other people are discussing an idea. The link isn't being used as a source of authority in this case, it is being used as a link to a discussion elsewhere, in much the same way that someone might link to another Physics Forum thread or a link to leaked information about an imminent announcement.

A skeptical lot. I don't think they give sufficient credit to the fact that Koide's rule was proposed in 1981 when it was a poor fit to the tau mass which has consistently improved for 37 years of increased precision in measurement (even from 2012 to 2018), or to the fact that the number of significant digits of match is high and consistent to MOE with data when it wasn't built to match existing data.

But, credit to them for getting to a lot of the key related articles quickly (Descarte's circle and quark mass relations) and hitting on some key points quickly.

-1 for the guy saying that 0.999999... is not equal to 1.

Is there merit to the analytic expression they reference? How accurate is it? How old is it?

Also, the other bit of numerology with the analytical expressions of the lepton masses in terms of the fine structure constant and pi was interesting.
<Moderator's note: twitter link removed: too much advertising and inappropriate source.>

If I knew Twitter links were forbidden across the board, I would have included a more direct sourcing by clicking through to the references therein and the references within the referenced material. It is a bit irksome not to know that in advance and have to recreate a reference. I would also urge the Mods to reconsider a category ban on Twitter links as a matter of moderation policy, and to make it more clear if it is to be a policy. Mostly I was simply trying to save myself the tedium of trying to type it a formula accurately using LaTeX.

The interesting series of formulas are for the ratio of the muon mass to the electron mass, of the tau mass to the muon mass, and of the tau mass to the electron mass which are compared using 1998 CODATA and PDG sources.

There are three expressions shared by the three formulas:

A = 1-4pi(alpha^2)
B = 1 + (alpha/2)
C = 1 + 2pi*(alpha/2) = 1+ pi*alpha

The muon mass/electron mass formula is (1/(2*pi*alpha²))^2/3*(C/B)

It purports to have a difference of 1 in the 7th significant digit from the PDG value.

The tau mass/muon mass formula is (1/2*alpha)^2/3*(B/A)

It purports to match a 5 significant digit PDG value.

The tau mass/electron mass formula is (1/4pi*alpha³)^2/3*(C/A)

It purports to have a difference of 1 in the 5th significant digit from the PDG value.

For what it is worth, I haven't confirmed the calculations or the referenced CODATA and PDG constants.

PDG for the tau mass is 1776.82 +/ 0.12 MeV

Koide's prediction for the tau mass is 1776.968921 +/- 0.000158 MeV

This formula predicts a tau mass of 1776.896635 MeV, which is about 0.07 MeV less than the Koide prediction, although there might be some rounding error issues and I don't have a MOE for the formula number. I used the five significant digit estimate of the tau mass to electron mass ratio in the illustration, so a difference in the sixth significant digit could be simply rounding error.

What to make of Dirac's 1937 Conjecture?

Dirac's conjecture on the electron radius v. size of the universe being roughly the same as the fine structure constant v. Newton's constant is also intriguing.
<Moderator's note: twitter link removed: too much advertising and inappropriate source.>

The conjecture called the Dirac Large Numbers Hypothesis is discussed at Wikipedia here: https://en.wikipedia.org/wiki/Dirac_large_numbers_hypothesis

An analysis that explores the same thing with a bit more clear language is here: http://www.jgiesen.de/astro/stars/diracnumber.htm

A 2017 preprint with eight citations discusses it here: https://arxiv.org/pdf/1707.07563.pdf

A 2013 paper revised in 2015 analyzes it here: http://pragtec.com/physique/download/Large_Numbers_Hypothesis_of_Dirac_de.php

A 2003 paper touches on it at https://www.jstor.org/stable/41134170?seq=1#page_scan_tab_contents

I didn't know that twitter links were categorically forbidden and would purge the ads if I knew how. It seemed a convenient way to link to an academically explored idea. Also, without the link the latest insights of very notable commentator, and mathematical physicist Baez are harder to present. If the latest commentary of leading scientists on scientific issues isn't acceptable to reference, it should be. Is it permissible to cut and paste a post from a Twitter thread by someone like Baez?

Baez notes that even though this coincidence holds at the moment, that we have enough data to know that the magnitude of Newton's constant has not changed that dramatically over the history of the universe.

Neutrino Mass and Koide?

By the way - do you have links to any of the Koide-ish neutrino mass papers? The mass measurements are quite a bit more constrained that they were then (with normal hierarchy strongly favored, some sense of the CP violating phase, pretty accurate relative mass differences and a fairly tight sum of three neutrino masses cap) so it would be interesting to compare. Plugging in all of those constraints you get:

Mv1 0-7.6 meV
Mv2 8.42-16.1 meV
Mv3 56.92-66.2 meV

The CP violating phase seems to be centered around -pi.

Which is more information than it seems because most of the Mv2 an Mv3 mass ranges are perfectly correlated with the Mv1 mass range.

One ought to be able to look at the Koide-ish neutrino mass papers (which flip a +/- sign IIRC) and numerically run through the allowed range for Mv1 to see what the best fit is and use that to make a prediction for all three absolute neutrino masses.

Never mind, found it: http://brannenworks.com/MASSES.pdf It puts a negative sign in front of the square root of Mv1 in the denominator and comes up with:

m1 = 0.000383462480(38) eV
m2 = 0.00891348724(79) eV
m3 = 0.0507118044(45) eV (I think this maybe an error in the original as it doesn't seem to be consistent with the Mv3 squared - Mv2 squared value predicted, I think it should be 0.05962528 . . .).

m₂² − m₁² = 7.930321129(141) × 10⁻⁵ eV² ------ PDG Value 7.53±0.18 (a 2.22 sigma difference - i.e. a modest tension)
m₃² − m₂ ²= 2.49223685(44) × 10⁻³ eV² ------ PDG Value 2.51±0.05 (less than 1 sigma different)

There is no value of Mv1 which can make the Koide formula without a sign flip work. I tried to reproduce his calculation and came up with Mv1 of 0.31 meV using current PDG numbers for the M1-M2 and M2-M3 mass gaps, which isn't far off from Brannen's estimate.

 

[nikkkom]

Also, the other bit of numerology with the analytical expressions of the lepton masses in terms of the fine structure constant and pi was interesting.
<Moderator's note: twitter link removed: too much advertising and inappropriate source.>

PDG for the tau mass is 1776.82 +/ 0.12 MeV

Koide's prediction for the tau mass is 1776.968921 +/- 0.000158 MeV

This formula predicts a tau mass of 1776.896635 MeV, which is about 0.07 MeV less than the Koide prediction

I looked closely at Mills and his "hydrino" paper. Mills is a fraudster. I assume a deliberate one. Elaborate one, too - you need to look rather closely to find blatant inconsistencies in his formulas, but when I found a place where he said "this quantity needs to be imaginary, so just insert 'i' multiplier here", it is a dead giveaway. No actual honest scientist would ever do that. If by the logic of your theory something has to be imaginary, it must come out imaginary from the math. Inserting multipliers where you need them is nonsense.

His mass formulas you link to are probably constructed by trying combinations of fine structure constant, pi, and various powers of them until a "match" is "found". E.g. multiplying by (1-alpha) fudges your result by ~0,9% down. Multiplying by sqrt(1-alpha) fudges your result by ~0,3% down. Divisions fudge it up, etc. This way a "formula" for any value may be constructed.

 

[ohwilleke]

I looked closely at Mills and his "hydrino" paper. Mills is a fraudster. I assume a deliberate one. Elaborate one, too - you need to look rather closely to find blatant inconsistencies in his formulas, but when I found a place where he said "this quantity needs to be imaginary, so just insert 'i' multiplier here", it is a dead giveaway. No actual honest scientist would ever do that. If by the logic of your theory something has to be imaginary, it must come out imaginary from the math. Inserting multipliers where you need them is nonsense.

His mass formulas you link to are probably constructed by trying combinations of fine structure constant, pi, and various powers of them until a "match" is "found". E.g. multiplying by (1-alpha) fudges your result by ~0,9% down. Multiplying by sqrt(1-alpha) fudges your result by ~0,3% down. Divisions fudge it up, etc. This way a "formula" for any value may be constructed.

On further review this is a 1998 formula from a rather disreputable source but may very well still hold.

I don't know anything about Mills personally, and honestly don't expect that his GUT theory is right. But, I think his lepton mass formulas are interesting even though they may very well be numerology and no more. Looking at ways that physical quantities can be closely approximated often adds insight, even if the phenomenological formula has no basis in underlying theory that has been established yet.

Even if he formula is nothing more than tinkering, the number of significant digits of agreement achieved with three fairly simple looking formulas (part of which is a common factor for all three) with only one physical constant and one only one common transcendental number is still an admirable counterfeit.

It is also proof of concept that it is possible that a first principles formula that simple that did explain the quantities from a theoretical basis using only coupling constants could exist, even if it turns out that this isn't the one that is actually supported by a coherent theory. There are a great many quantities for which this is not possible even in principle.

Along the same lines, suppose that MOND is false that that we discover actual dark matter particles tomorrow. Any dark matter theory still needs to explain how it produces the very tight and simple phenomenological relationship between rotation curves and the distribution of baryonic matter in the universe that it does by some other means. The counterfeit or trial and error hypothesis can shed light on some feature of the true theory that makes it work.

 

[ftr]

suppose I give you this formula
proton electron mass ratio =3*(9/2)*(1/alpha-1) -1/3= 1836.152655 using codata for alpha
= 1836.1526734 using (1/alpha =137.036005 very close to average of codata and neutron Compton wave experiments base precision qed tests).

Can you say that this might have a physical basis or this is just a fluke. Is it possible to give probabilities for such and similar formulas.

 

[arivero]

Hmm we are going to complete a cycle, are we?. Please remember that our interest on Koide formula happened while examining different combinations of alpha and masses, in the thread https://www.physicsforums.com/threads/all-the-lepton-masses-from-g-pi-e.46055/

 

[arivero]

Hmm we are going to complete a cycle, are we?. Please remember that our interest on Koide formula happened while examining different combinations of alpha and masses, in the thread https://www.physicsforums.com/threads/all-the-lepton-masses-from-g-pi-e.46055/

Are some of these relationships linked to koide formula? Can not tell. Perhaps the most promising, to me, is the mass of proton compared with the sum of electron, muon and tau. Three confined quarks vs three free leptons.

 

[mfb]

Using numbers from 1 to 9 plus e, pi and alpha, five different operations (+-*/^) and the option to take square roots, we have at least 10 options per operation. Even taking into account that multiple expressions can have the same result you would expect more than one additional significant figure added per operation. I count 7 in the above calculation plus one initial value. We would expect that we can get 8 significant figures just by random chance. And, surprise (?), we get 8 significant figures agreeing with measurements.

$\frac{e^8-10}{\phi} \approx 1836.153015$ - 6 significant figures (or 7 if we round) with just 3 operations.

 

[ftr]

6 significant figures (or 7 if we round) with just 3 operations.

Ok, but relating two fundamental constants with simple numbers seems to be much more stringent, doesn't it.

 

[mfb]

Ok, but relating two fundamental constants with simple numbers seems to be much more stringent, doesn't it.

You want the fine structure constant in?
$\displaystyle \frac{e^8-10(1+\alpha^2)}{\phi}\approx 1836.152686028$, an 8-digit approximation of 1836.15267389(17).

9 is not simpler than 8 and 10, an exponential is not very unnatural, and the golden ratio is always nice.

 

[ftr]

Ok. still some expression look more simpler and "natural" than others. see post #238

https://www.physicsforums.com/threads/all-the-lepton-masses-from-g-pi-e.46055/page-10

But anyway all this is useless unless backed up by a clear derivation.

 

[ftr]

BTW does anybody know the whereabouts of Hans de Vries. Or why he drop out.

 

[mfb]

Ok. still some expression look more simpler and "natural" than others. see post #238

https://www.physicsforums.com/threads/all-the-lepton-masses-from-g-pi-e.46055/page-10

But anyway all this is useless unless backed up by a clear derivation.

I get 31.8 bits for 3*(9/2)*(1/alpha-1) -1/3 counting one bit for the 1 in "-1" and ld(5) for alpha. The approximation is good for 26.5 bits, worse than expected.
I get 33.8 bits for (e^8-10(1+alpha^2))/phi again counting the 1 as one bit and e and phi as ld(5). The approximation is good for 27.2 bits, similarly worse.
I get 20.7 bits for (e^8-10)/phi. The approximation is good for 22.4 bits.

The last one is the only one that beats the algorithm from @Hans de Vries you referenced. phi is too exotic? Okay, give it ld(20), then we are still at 22.7 bits for 22.4 bits, or equality.

 

[ftr]

Although the bit calculation can be close, however, there are other considerations. For example the relation between the fundamental constants is very strong, i.e. one is the major bulk that makes the other in my equation (indicating a possible physics), in yours it affects the digits beyond the accuracy anyway, that is a very weak relation. Moreover, due to this consequence one constant is very sensitive to the accuracy you choose for the other(experimentally varying somewhat), hence the bit analysis accuracy problem. Also, if you reverse the formula, mine looks good, yours looks like ugly duckling .

 

[mitchell porter]

"Origin of fermion generations from extended noncommutative geometry" by Hefu Yu and Bo-Qiang Ma not only claims to get three generations by extending a noncommutative standard model for one generation, but the Koide relation too.

I have not yet tried to follow their constructions, but here is some of what they say. They extend the usual spectral triple (A,H,D) to (A,H,D,J,gamma) (eqn 25). They also allow fields (?) to be quaternion-valued. They consider two sets of basis quaternions, I, J, K and I', J', K'. They have two conditions on the second set (eqns 87 and 88) which together imply a Koide-like relation (eqn 89). The moduli squared of I', J', K' show up in the mass matrices (eqns 96-99) and this implies Koide relations for each family (eqns 100-101). They acknowledge that the Koide relation is not perfect for the quarks but they think it is close enough.

There is definitely handwaving here. They have probably grafted something akin to the Foot vector condition, onto the noncommutative standard model, in a quite artificial way. But we can't be sure of that, without dissecting their argument more thoroughly.

 

[Hans de Vries]

BTW does anybody know the whereabouts of Hans de Vries. Or why he drop out.

Thanks ftr, I'm still there