# What is new with Koide sum rules?

Gold Member
I am detaching this to BSM because it is already getting too much parameters in the bag.

What has happened this year is that Werner Rodejohann and He Zhang, from the MPI in Heidelberg, proposed that the quark sector did not need to match triplets following weak isospin, and then empirically found that it was possible to build triplets choosing either the massive or the massless quarks. This was preprint http://arxiv.org/abs/1101.5525 and it is already published in Physics Letters B.

Later, two weeks ago, another researcher from the same institute veifyed the previous assertion and proposed a six quarks generalisation, in http://arxiv.org/abs/1111.0480

Then myself, answering to https://www.physicsforums.com/showthread.php?t=485458", checked that there was also a Koide triplet for the quarks of intermediate mass. I have not tried to find a link between this and the whole six quarks generalisation, but I found other interesting thing: that the mass constant AND the phase for the intermediate quarks is three times the one of the charged leptons. This seems to be a reflect of the limit when the mass of electron is zero, jointly with an orthogonality between the triplets of quarks and leptons in this limit: it implies a phase of 15 degrees for leptons and 45 degrees for quarks, so that 45+120+15=280. If besides orthogonality of Koide-Foot vectors we ask for equality of the masses (charm equal to tau, strange equal muon), the mass constant needs to be three too.

So,

with the premises
1. Top, Bottom, Charm have a Koide sum rule
2. Strange, Charm, Bottom have a Koide sum rule
3. Electron, Muon, Tau have a Koide sum rule
4. phase and mass of S-C-B are three times the phase and mass of e-mu-tau

And the input
• electron=0.510998910 \pm 0.000000013
• muon=105.6583668 \pm 0.0000038

The sum rules allow to calculate the following masses.
• tau=1776.96894(7) MeV
• strange=92.274758(3) MeV
• charm=1359.56428(5) MeV
• bottom=4197.57589(15) MeV
• top=173.263947(6) GeV

(Errors are just from the extreme plus and minus, actually they would be a bit smaller; most probably any fundamental theory for the sum rules should propose greater second order corrections.).

Furthermore, the mass unit for leptons is 0.313,856,4 GeV and then for intermediate quarks is 0.941,569 GeV. Very typical QCD masses.

Plus, some of the sum rules of the Heidelberg group(s) can be used to give diffrent estimates for up and down. Or it can be tryed from other triplets (eg, Marrni with top-up-down).

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Gold Member
phase and mass of S-C-B are three times the phase and mass of e-mu-tau
I suggest that this is the key, when considered in conjunction with Kartavtsev's generalization of the Koide relations. See equation 10 in his paper, and the paragraph beneath it: The formula works best when all six quarks are included at once, and similarly extending the original Koide relation to include the neutrinos will not reduce its validity, because the neutrino masses are so small. Your analogy between s-c-b and e-mu-tau is a clue to an even tighter mapping between Kartavtsev's formula for the quarks and the corresponding formula for the leptons. (I also suggest that the extra factor of 3 has to do with color - there are three times as many quarks as there are leptons, when color is taken into account - but it may take a while to implement that idea.)

Gold Member
In GUT, Howard Georgi and Cecilia Jarlskog discovered that it was possible to build mass relationships between the down sector and the leptons where generations could arbitrarily be equal, one third or three times the mass of the other. This was done with a ugly mix of Higgsess, but they conjectured thet the factor 3 was coming really from colour.

I am not sure about Kartavtsev formula, but yes it could be possible to explain the perturbation away 15 degrees via some renormalisation running. I think it is mostly an electromagnetic correction, something involving alpha and the quotient of (sum of) lepton and quark masses. But it is just a weak conjecture; it is easy to see QCD involved here, but electromagnetism is a different beast.

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qsa
Does that mean charge and mass are linked. What about the gravity and strong force in ADS.

Gold Member
Does that mean charge and mass are linked. What about the gravity and strong force in ADS.

It is an incentive, indeed. Or just plain KK compact in AdS.

Gold Member
This is what I think is going on. Before perturbations, there is at least a Koide triplet with a zero mass component and another one that is in the opossite side of Foot cone, so orthogonal to it.

That means a phase of 15 degrees (pi/12 radians) for the former and a phase of -45 degrees for the later:

$$m_k= M (1 +\sqrt 2 \cos({360k \over 3} +15))^2$$$$n_k= N (1 +\sqrt 2 \cos({360k \over 3} -45-120))^2$$

You can check orthogonality of the Koide-Foot vectors (roots of masses):
$$({3+\sqrt 3 \over 2}, 0, {3-\sqrt 3 \over 2})*({1-\sqrt 3 \over 2}, 2, {1+\sqrt 3 \over 2})=0$$

And the point is that the comparison of the mass tuples:
$$m=\left( 3 (1+ {\sqrt 3 \over 2}) M, 0 , 3 (1 - {\sqrt 3 \over 2}) M\right)$$$$n=\left( (1 - {\sqrt 3 \over 2}) N, 4N, (1+ {\sqrt 3 \over 2}) N\right)$$
makes very very tempting to set $N=3M$ And so we do.

For a basic M of 313.86 MeV, that means $m=(1757,0,126.1)$ and $n=(126.1,3766,1757)$. That should be the lepton masses tau,e,mu and the quark masses s,b,c before applying the small rotation (or perturbation).

And finally here comes the second guessing. We notice that also one phase is three times the other, and we guess (based on our previous empirical check ) that it is going to keep so, $\delta_q=3 \delta_l$. With this premise, we have rotated the lepton vector to fit experiment and then copied the phase to the quark sector. Perhaps it is not so; but in this way we have got to proceed with only two experimental inputs to fix all the other masses.

Ok, whatever, what we do is

1) input m_e and m_l into Koide sum rule, to get m_tau.
Code:
me=0.510998910
mmu=105.6583668
mtau=((sqrt(me)+sqrt(mmu))*(2+sqrt(3)*sqrt(1+2*sqrt(me*mmu)/(sqrt(me)+sqrt(mmu))^2)))^2
we get mtau: 1776.968... but this is no news, it is Koide 1981.

2) Use the lepton triplet to get the values of M and delta.
Code:
m=(me+mmu+mtau)/6
pi=4*a(1)
cos=(sqrt(me/m)-1)/sqrt(2)
tan=sqrt(1-cos^2)/cos
delta=pi+a(tan)-2*pi/3
We get delta about 2/9 (or and m about 313.8 MeV. Again, this is old news. But the mass is very reminiscent of QCD, and the point that in the next formula we multiply by three, getting the order of the proton mass (or neutron, or even approx eta'), is also curious.

3) Multiply these parameters as said, $3M, 3\delta$, and use them to build a quark triplet.
Code:
mc=3*m*(1+sqrt(2)*c(3*delta+4*pi/3))^2
ms=3*m*(1+sqrt(2)*c(3*delta+2*pi/3))^2
mb=3*m*(1+sqrt(2)*c(3*delta))^2
If you are going to check Koide, remember that with this phase, the value of sqrt(ms) is negative:
Code:
(sqrt(mb)-sqrt(ms)+sqrt(mc))^2/(mb+ms+mc)
1.50000000000000000002

4) use again Koide sum rule to get the mass of the top.
Code:
mtop=((sqrt(mc)+sqrt(mb))*(2+sqrt(3)*sqrt(1+2*sqrt(mc*mb)/(sqrt(mc)+sqrt(mb))^2)))^2

5) print your new outputs and check against http://pdglive.lbl.gov/listings1.brl?quickin=Y [Broken]
Code:
ms
92.27475468510853794238
mc
1359.56423480142772524333
mb
4197.57575183796073176386
mtop
173263.94170381397040438241

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Gold Member
Kartavtsev's

Hmm it seems we should call it Goffinet-Kartavtsev. It is also 3.56 in http://cp3.irmp.ucl.ac.be/upload/theses/phd/goffinet.pdf Goffinet was in one of the teams (Brannen was *the* other) trying Koide for neutrinos in the 2005.

Gold Member
$$m=\left( 3 (1+ {\sqrt 3 \over 2}) M, 0 , 3 (1 - {\sqrt 3 \over 2}) M\right)$$
By the way, this tuple in its version
$$m_1=0, {m_2 \over m_3}= {(2 - \sqrt 3 ) \over (2+ \sqrt 3) }$$
is also discussed in Rivero-Gsponer 2005, but it is at least as old as 1978, in a paper usually quoted by Koide: http://inspirehep.net/record/130343?ln=es
Quark Masses and Cabibbo Angles.
Haim Harari (Weizmann Inst.), Herve Haut, Jacques Weyers (Louvain U.).
Phys.Lett. B78 (1978) 459

Gold Member
5) print your new outputs and check against http://pdglive.lbl.gov/listings1.brl?quickin=Y [Broken]
Code:
ms
92.27475468510853794238
mc
1359.56423480142772524333
mb
4197.57575183796073176386
mtop
173263.94170381397040438241

Hmm, I forget to add, instead of pdf you can also try http://arxiv.org/abs/1109.2163

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Gold Member
In my opinion, this s-c-b relation is a big big clue about family symmetries.

Consider Koide's latest yukawaon model. It has U(3) x O(3) family symmetries, constructed to preserve the e-mu-tau relation at low energies. It's clearly hard work to make field-theory models with this property, but he's done it.

Obviously, if a very similar relation for s-c-b holds, then that should have enormous ramifications for the structure of a yukawaon model. In fact, as Kartavtsev remarks, it's problematic to have just one half of the b-t doublet in the formula - which is why I think the six-particle formula might be fundamental, but perhaps with some secondary, constraining sub-relation that connects s-c-b.

Anyway, I think the obvious thing to do is to try to modify the yukawaon model so as to obtain the s-c-b relation. I also think it would make a lot of sense to combine it with the Georgi-Jarlskog relation, which in its original form was also achieved in an SU(5) theory, such as Koide works with in his paper above.

Alternatively, one can go the route of Carl Brannen, and just reconstruct the whole of quantum field theory around the clue provided by Koide's formula. But for now I think I will stick with the yukawaon approach.

Gold Member
In my opinion, this s-c-b relation is a big big clue about family symmetries.

Indeed. I have done a first surview of the early theories, who aimed to calculate the Cabibbo angle and occasionaly met some mass formula, such as the one from Harari et al. All of them proceed by putting a discrete symmetry but most of them do not use the standard model but the Left-Right symmetric model. They put the symmetry in the R part, then they break this SU(2)_R. It makes sense, as then a up quark is linked not only with a down_L but also with a bottom_R, and then the mass pattern needs some more levels to accommodate everything.

By the way, are we two the only persons reading the thread? It is good to exchange and archive ideas (I am finding now in PF some valuables from six years ago) but it should me nice if other readers have some input, or just a wave and a hello. In order to give other persons an entry point, let me coalesce all the bc -l code in a single cut-paste block:

Code:
pi=4*a(1)
me=0.510998910
mmu=105.6583668
mtau=((sqrt(me)+sqrt(mmu))*(2+sqrt(3)*sqrt(1+2*sqrt(me*mmu)/(sqrt(me)+sqrt(mmu))^2)))^2
m=(me+mmu+mtau)/6
cos=(sqrt(me/m)-1)/sqrt(2)
tan=sqrt(1-cos^2)/cos
delta=pi+a(tan)-2*pi/3
mc=3*m*(1+sqrt(2)*c(3*delta+4*pi/3))^2
ms=3*m*(1+sqrt(2)*c(3*delta+2*pi/3))^2
mb=3*m*(1+sqrt(2)*c(3*delta))^2
mtop=((sqrt(mc)+sqrt(mb))*(2+sqrt(3)*sqrt(1+2*sqrt(mc*mb)/(sqrt(mc)+sqrt(mb))^2)))^2

I'd be glad if someone uploads some equivalent maxima, macsima or symbolic algebra whatever code.

And of course, there is a pending puzzle: to explain the phase of the triplet charm-bottom-top

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Jimster41
Gold Member
The new research in this thread has been reported in http://vixra.org/abs/1111.0062 and http://arxiv.org/abs/1111.7232 Due to the interference of the holding process plus Thanksgiving day, vixra has been substantially faster in this case! Plus, the comments feature in vixra can be useful, if you want to point out missing references, you can do it there.

Gold Member
Wojciech Krolikowski, who (like Koide) found a formula for the charged lepton masses, has now extended it to all six quarks as well, in http://arxiv.org/abs/1201.1251. I am still working towards an explanation of Alejandro's formulas (which I consider a massive breakthrough) and feel like I hardly have room in my head for this new paper as well, but I'm sure that feeling will pass...

Gold Member
I have been thinking about this for weeks now and I have lots of ideas, but nothing decisive, so it's time to talk.

First, let's consider what a "standard" approach to a discovery like this is. If someone guesses a pattern in the masses and mixing angles, the answer usually involves some combination of multiple Higgses, "flavons" whose VEVs contribute to the Yukawas, and family symmetries (usually discrete).

Let us suppose provisionally that the quantity appearing in the Koide relation is a VEV (and that the corresponding Yukawa is the square of this VEV). It seems that some of these VEVs are negative, thus the "minus sqrt mass" term appearing in s-c-b (and in Brannen's neutrino triplet).

For the quarks we then have a set of six quantities, which to a first approximation satisfy the Koide relation in four sets of three (dus, usc, scb, cbt). The Koide relation only holds well for scb and cbt, but there is some evidence that the actual values for dus and usc are highly perturbed away from a "primordial" set of mass values which includes m_u = 0. Another aspect of this perturbation is that the primordial Koide phase for the scb triplet is 45 degrees, but the real value is 2/3 radians.

In his "yukawaon" papers, Koide obtains VEV relations from supersymmetric vacuum conditions. So that is one way to get a set of four chained Koide triplets - construct a superpotential which implies Koide VEV relations for the four sets of three. (It would also be good to do this without supersymmetry.)

However, it's clear (from the relation between e-mu-tau and s-c-b) that the important parametrization of the Koide relation is the one (due to Carl Brannen?) featuring a mass scale and a phase. These parameters don't stand forth in Koide's constructions. Since Brannen uses circulant matrices, perhaps we should therefore be interested in models like Stephen Adler's multi-Higgs models with Z_3 symmetry, where there are three or six Higgs doublets, and where there are circulant (or "retrocirculant") mass matrices.

Another thing we can learn from Koide is the importance of the Sumino mechanism. The running of the masses ought to spoil the original Koide relation for the charged leptons, but it remains exact. Sumino suggested that the bosons of a gauged family symmetry could cancel the electromagnetic radiative corrections which would otherwise spoil the relation.

Koide's latest yukawaon models are SU(5)-compliant supersymmetric models in which the Koide relation for the charged leptons comes from SUSY vacuum conditions, and in which the Sumino family symmetry exists and is gauged. So one way forward is to follow his lead: look for a basic explanation of these extended Koide relations - perhaps using Adler-Brannen circulant mass matrices, perhaps using a version of Georgi-Jarlskog to explain the factor of 3 difference between s-c-b and e-mu-tau - and then use the Sumino mechanism to protect the relations (though it's not yet clear whether the quark mass relations are exact enough to need protection).

However, this still leaves one more clue unused - the appearance of QCD mass scales in the Brannen parametrization of the Koide formula. This leads me to think in terms of holographic QCD and Alejandro's own "sBootstrap".

The basic paradigm of holographic QCD is that you have a stack of color branes and a stack of flavor branes that intersect. A gluon is a string between color branes, a quark is a string between a flavor brane and a color brane, a meson is a string between flavor branes, and a baryon is a brane instanton connected to multiple flavor branes by strings.

The sBootstrap is a combinatorial construction in which all the SM fermions are made from pairings of the five "light" quarks ("light" here means lighter than the top quark). Leptons are made from mesonlike pairings, quarks from diquarklike pairings. Since holographic QCD contains fermionic meson strings ("mesinos"), an hQCD implementation of the sBootstrap would say that the leptons are mesinos. The situation for the quarks is less satisfactory; but one might imagine that there is some mixing between quark strings and fermionic "diquarkinos".

Top-down holographic QCD constructions (Sakai-Sugimoto is the best known) so far don't resemble QCD exactly. For one example, they are usually studied in the large-N limit, N being the number of colors, whereas reality involves N=3. But also, the spectrum has extra stuff not seen in reality. The fermionic mesons already mentioned are one of these trouble spots.

However, if we expect the leptons to come from the mesino sector, then the trouble becomes a virtue. We might look for a hQCD model that contains the whole standard model via the sBootstrap. (Or we might look for a more conventional string model which nonetheless realizes the leptons in this fashion.)

How is this relevant to explaining the extended Koide relations? The point is that it offers an avenue whereby QCD mass scales may show up in lepton mass formulae, since the leptons would just be the mesinos of some SQCD-like theory.

Gold Member
actual values for dus and usc are highly perturbed away from a "primordial" set of mass values which includes m_u = 0.

Were this to happen for the triplet down-up-strange, we could expect that the perturbation term is,
$$\delta m_a = { m_b m_c \over M}$$
with M fixed, the same for the three equations. Traditionally it was expected to come from instanton, or tunneling between similar states, if you prefer. If seems that lattice QCD has ruled out this term, but it is unclear.

Note that if m_u = 0, only the up quark gets a correction. This was the expected way to solve the "CP problem" (or was it the "strong CP problem"? whatever)

For d,u,s= (5.3, 0.036, 92), all units in MeV, it is more or less the same, and we can set M to 185 MeV to get

$$m_u= 0.036 + { 92 \cdot 5.3 \over M} = 0.036 + { 487.6 \over M} = 2.67 MeV$$
$$m_d= 5.3 + { 0.036 \cdot 92 \over M} = 5.3 + { 3.312 \over M} = 5.32 MeV$$
$$m_s= 92 + { 0.036 \cdot 5.3 \over M} = 92 + { 0.1908 \over M} = 92 MeV$$

but at the price of an extra free parameter M. Not bad, because it is about M=185 MeV, so still expected from QCD, chiral scale, etc... There is a wide range to choose without violating the experimental constraints. But it is still an extra parameter.

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Gold Member
I have references to most of the recent Koide and quark-lepton complementarity papers in a series of posts here: http://dispatchesfromturtleisland.blogspot.com/search?q=koide

One is notable for suggesting a nearly massless up quark while being spot on for the other quark masses, which if true, might help explain strong CP invariance: http://arxiv.org/abs/hep-lat/0112029 and light neutrinos http://arxiv.org/PS_cache/hep-th/pdf/0608/0608053v1.pdf There are definitional issues that go into the current operational definition of up quark mass, http://arxiv.org/PS_cache/hep-ph/pdf/0312/0312225v2.pdf which are pertinent to this question.

There is a very detailed exploration of Q-L complementarity relations in phenomenology here: http://cp3.irmp.ucl.ac.be/upload/theses/phd/goffinet.pdf This was first proposed in 1990: http://prd.aps.org/abstract/PRD/v41/i11/p3502_1 by Foot and Lew. Other citations to related points here: http://dispatchesfromturtleisland.blogspot.com/2011/11/musings-on-higgs-boson-coupling.html QLC without a parameterization specific formulation was sketched out at http://arxiv.org/abs/1112.2371 MINOS seems to be hinting at some corroboration of this hypothesis: http://web.mit.edu/panic11/talks/th...20/whitehead/947-0-MINOS_PANIC_2011July28.pdf

Gold Member
Dearly Missed
A new paper that cites Brannen and Arivero:

http://arxiv.org/PS_cache/arxiv/pdf/1201/1201.2067v1.pdf

He finds a natural geometric set up in which he finds all masses, including quarks, but putting all of the 6 together!

1654 was a great year for particle physics!

Descartes – in his 1654 letter to the princess of Bohemia, Elizabeth II – showed that the curvatures of four mutually tangent circles (reciprocal of radii), say a,b,c,d, satisfy the following “Descartes’s formula”...​

Nature is showing us she can be completely weird. Or maybe the word is witty.

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Gold Member
Dearly Missed
MTd2 points to the 2012 paper of Jerzy Kocik:
http://arxiv.org/abs/1201.2067

which cites the exquisitely-titled 2005 paper of Rivero Gsponer:
[11] A. Rivero and A. Gsponer, The strange formula of Dr. Koide, http://arxiv.org/abs/hep-ph/0505220

This is turning out to be a class act. The script resembles a Gothic novel that Isak Dinesen might have dreamed up.

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Gold Member
1654 was a great year for particle physics!

Descartes – in his 1654 letter to the princess of Bohemia, Elizabeth II – showed that the curvatures of four mutually tangent circles (reciprocal of radii), say a,b,c,d, satisfy the following “Descartes’s formula”...​

Nature is showing us she can be completely weird. Or maybe the word is witty.

Ah, also Bruce Schechter, in a comment in June of 2008 to http://dorigo.wordpress.com/2007/02/07/short-but-not-irrelevant-exercise/ , noticed the similarity, but did not suggest any way to generalize to 2/3. Jerzy Kocik has done an interesting extension here.

Gold Member
Since we have another thread discussing E6 grand unification, I will point out the work of Berthold Stech.

In my comment #14, I said that an "obvious" way to make a model for these extended Koide relations, would be to extend the standard model with a new scalar sector of "flavons", whose VEVs-squared determine the Yukawa couplings, and with a gauged family symmetry that protects the Koide relations, as suggested by Yukinari Sumino. (The Koide relations among the flavon VEVs would result from a flavon potential.)

So it's very interesting that Stech's E6 models more or less resemble this picture. The Yukawas come from flavon VEVs, and there's a flavor symmetry. Especially interesting is that his lightest Higgs is at about 123 GeV!

Stech's models definitely do not produce Koide relations at present. In particular, I can't think of any model ever that implies the peculiar e-mu-tau/s-c-b relation that Alejandro found. Though let's note that that relation also resembles the u-s-c/s-c-b relation in the "zeroth-order" or "primordial" version of the extended Koide relations, as described in his paper; so there may be something more complicated than a Georgi-Jarlskog "multiplication by three" at work here. (Another quantitative issue to investigate is whether all six lepton masses, neutrinos as well as charged leptons, can be arranged into a set of four chained Koide triplets like the quarks, or whether the leptons naturally fall into two disjoint triplets, this being an aspect of how they differ from the quarks.)

But the "peculiar" relation should be seen simply as a challenge: come up with a flavon potential and a new symmetry which produces it. And Stech's E6 framework looks worth investigating, though the minimal way to proceed would be just to add flavons (and maybe more Higgses) to the standard model until the extended Koide relations (and quark-lepton complementarity for the mixing angles, see comment #16) are obtained.

Gold Member

In his first answer to Lubos, ohwilleke makes a remark that, while known in the papers, is not in this thread (it could be in old ones). Point is, you take the experimental values of top and bottom
Code:
mtop=172.9
mb=4.19
and then use Koide equation to produce charm and strange.
Code:
mc=((sqrt(mtop)+sqrt(mb))*(2-sqrt(3)*sqrt(1+2*sqrt(mtop*mb)/(sqrt(mtop)+sqrt(mb))^2)))^2
ms=((sqrt(mc)+sqrt(mb))*(2-sqrt(3)*sqrt(1+2*sqrt(mc*mb)/(sqrt(mc)+sqrt(mb))^2)))^2
Then you can consider to compare with the experimental value of lepton sum
Code:
leptons=0.000511+0.105659+1.77668
and then a quotient which is 1-sigma compatible with an integer appears:
$${{m_c+m_s+m_b} \over {m_e+m_\mu+ m_\tau}} = 2.995 (\pm 0.04 approx)$$

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suprised
Guys, wake up - science simply doesn't work in this way.

Unfortunately the following site is in german so can't be understood by anyone, but it's kind of a healthy medecine. It shows how to obtain the natural constants out of 5 given numbers (like your birthday, the hight of the pyramides in inches, etc), up to an accuracy of 10^-4. And vice versa, you can fit any given number by the natural constants:

Gold Member
Unfortunately the following site is in german so can't be understood by anyone

Be sure that I understand the point in the website, it is very clear, and it is really no different that the decimal system itself. Note that the program allows each number to generate six different ones via powers, plus some allowance for pi and 2 and its powers, so it has a lot of available combinations to cover ten thousand numbers (four digits). That is, assuming that really you have checked the page. Because it seems that you people do not actually read the posts...

Here, it is not about getting multiple quantities with different formulae, it is the puzzle that we are getting multiple quantities with a single formula.

Namely, we have taken Koide equation for a z,x,y triplet:

$${(\sqrt z + \sqrt x + \sqrt y)^2 \over (z+x+y) } = \frac 3 2$$

and we have solved for z

$$z=f(x,y)=\left[ ( \sqrt x +\sqrt y )\left(2- \sqrt{3+6 {\sqrt{xy} \over (\sqrt x+\sqrt y)^2}}\right) \right]^2$$

It was known since 1981 that this formula, for y=1.77668 and x=0.105659 gets f(x,y)=0.000510, ie, that $n_e \equiv f(m_\tau,m_\mu)$ was equal to the physical $m_e$. Up to now, one can live with this and appeal to GIGO arguments, garbage it garbage out, disregarding the point that the equation was actually found from physical models. There is a lot of physical models, some of them could hit in a random equation.

NOW, the new observation is that taking as input $m_t$ and $m_b$, and iterating down four times to produce six particles, the total spectrum does not fare bad neither.
$$n_c \equiv f(m_t,m_b) \approx m_c$$ $$n_s \equiv f(m_b,n_c) \approx m_s$$ $$n_u \equiv f(n_c,n_s) \approx 0$$ $$n_d \equiv f(n_s,n_u) \approx m_d$$
So we have verified that the Koide equation also does a decent work in the quark ladder. Not a different equation. Nor different parameter. Nor different powers. The SAME equation. Just 30 years later.

Still, it can be argued that charm and strange have a very broad range of values in the experimental sector, some of them even arguable up to definition of the concept. Thus, we have looked for comparison between the quark and lepton spectrum and found that:

1) $(m_b + n_c + n_s) / (m_e + m_\mu + m_\tau)= 3$
2) The phase angle to built the triplet $(m_b , n_c , n_s)$ is about 3 times the phase angle of the triplet $(m_e , m_\mu , m_\tau)$

Both 1 and 2 can be described telling that the triplets, in its square root form, are almost orthogonal when ordered in the cone around (1,1,1).

We can either keep 1 and 2 as a verification of the values of charm and strange, and stop here, or use it as extra postulates to produce all the masses from only two values.

Homework Helper
Gold Member
Point is, you take the experimental values of top and bottom
Code:
mtop=172.9
mb=4.19
and then use Koide equation to produce charm and strange.

I was surprised that Lubos didn't explicitly point out what seems to me to be the biggest flaw with all of the Koide formulas, which is that no one bothers accounting for running quark masses. What you're quoting above are the quark masses at their own pole, namely $m_q(m_q)$. No physical theory is going to relate quark masses at scales that differ by 1.5 orders of magnitude.

Unfortunately, the most convenient reference for RG results of running masses is a nearly 20 year old paper by Koide himself, hep-ph/9410270, but Table VI is at least illustrative. Your above values give $m_t/m_b\sim 40$, while

$$m_t(1~\mathrm{GeV})\sim 420~\mathrm{GeV},~~~m_b(1~\mathrm{GeV})\sim 7~\mathrm{GeV},~~~m_t/m_b (1~\mathrm{GeV})\sim 60.$$

Using the formula from arivero's post #24, these values give $m_c(1~\mathrm{GeV})\sim 5.5~\mathrm{GeV},$ which is about 3.6 times the correct value.

I don't believe that things are going to get better at any other mass scale or by using any more modern results for the RG equations, so there's no reason to believe that there's any deep significance to any Koide-type relation.

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I don't believe that things are going to get better at any other mass scale or by using any more modern results for the RG equations, so there's no reason to believe that there's any deep significance to any Koide-type relation.
We won't really know until someone finds a class of models which naively imply Alejandro's formula, and then performs an analysis like that in Sumino, http://arxiv.org/abs/0812.2103. As things stand, the situation is consistent with there being some sort of new symmetry which is visible only obscurely for the quarks, but which stands out sharply for the charged leptons.

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We won't really know until someone finds a class of models which naively imply Alejandro's formula, and then performs an analysis like that in Sumino, http://arxiv.org/abs/0812.2103. As things stand, the situation is consistent with there being some sort of new symmetry which is visible only obscurely for the quarks, but which stands out sharply for the charged leptons.

That Sumino paper is bizarre. He starts with the Koide formula, which holds empirically for the pole masses and says that he wants it to be valid for the masses defined at some high-energy scale. What I'm saying is that the Koide formula as written there does not hold at any particular scale with the same accuracy as it does for the pole masses. While he's trying to do something that makes sense (write a formula that relates masses defined at the same scale), there doesn't seem to be much reason to cling to Koide's formula that we already know doesn't work away from the pole masses.

I'm also very skeptical that you can cancel the RG correction to that combination of masses without leaving a trace of the new physics in the running of, e.g., the electron mass on its own. Since there's no evidence for any supression of QED radiative corrections between $m_e$ and $m_W$, it's unlikely that such a mechanism exists.

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The idea seems to be that the Koide relation holds exactly at high energies, and it also holds for the pole masses, because the corrections due to the family gauge bosons cancel the QED corrections for each charged lepton, at its own mass scale. Above that scale, the mass will just run normally as in the SM, until the scale where electroweak unifies with the family force (100s or 1000s of TEVs), at which point the Koide relation becomes manifest again.

But I'm just telling you how I think it's supposed to work, I'm still getting my head around the details.

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The idea seems to be that the Koide relation holds exactly at high energies, and it also holds for the pole masses, because the corrections due to the family gauge bosons cancel the QED corrections for each charged lepton, at its own mass scale. Above that scale, the mass will just run normally as in the SM, until the scale where electroweak unifies with the family force (100s or 1000s of TEVs), at which point the Koide relation becomes manifest again.

But I'm just telling you how I think it's supposed to work, I'm still getting my head around the details.

Let's denote the pole masses by $m_i(m_i)$. The Koide result is that

$$\frac{ \sqrt{m_e(m_e)} + \sqrt{m_\mu(m_\mu)} +\sqrt{m_\tau(m_\tau)}}{\sqrt{m_e(m_e) + m_\mu(m_\mu) +m_\tau(m_\tau) }} = \sqrt{\frac{3}{2} } \pm 10^{-5} .$$

Now $m_i(E)$ definitely runs with energy and we know this because it's been measured. What I understood is that, in Sumino's model, the one-loop corrections to

$$r(E) = \frac{ \sqrt{m_e(E)} + \sqrt{m_\mu(E)} +\sqrt{m_\tau(E)}}{\sqrt{m_e(E) + m_\mu(E) +m_\tau(E) }}$$

cancel.

However, we know from running the pole masses in the first relation that $r(E)$ differs from $\sqrt{3/2}$ by one part in $10^{-3}$ (stated below eq (2) in Sumino). Now, since $m_e(m_e)/m_\tau(m_\tau) \sim 3 \cdot 10^{-4}$, at this level of precision, we might as well just drop the terms with $m_e$ from $r(E)$. The Koide relation really doesn't convincingly extend to the electron and is just some numerology involving $\mu$ and $\tau$. The situation for the up quarks is even worse since $m_u/m_t$ is much, much smaller than the experimental uncertainty in the top mass.

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Now $m_i(E)$ definitely runs with energy and we know this because it's been measured.
I think the point is that in a theory with "Koide symmetry" (i.e. whatever it is that produces the Koide relation) but not "Sumino family symmetry", the $m_i(m_i)$ start out at a set of values which don't satisfy Koide symmetry. The additional Sumino family symmetry adjusts the RG trajectory so that the pole masses do satisfy Koide symmetry. But that doesn't mean that Koide symmetry is exact for the running masses at low energies; it only becomes exact above the family-symmetry unification scale. The masses do not satisfy the symmetry at any single value of E below that scale; but the relation happens to hold for the pole masses at their different scales.

I still haven't verified this! But I believe this is how it's supposed to work.

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corrections between $m_e$ and $m_W$
By the way, according for instance table IV of http://arxiv.org/pdf/hep-ph/0601031v2 or section F of http://prd.aps.org/abstract/PRD/v46/i9/p3945_1 (where, note, the wrong measured value for tau is still used), all the damage to Koide relation for leptons is done already when moving electron and muon up to the GeV scale. From 1 GeV up to any high energy (without GUT), the mismatch keeps about 1.0017 - 1.0019, i.e. about the 0.2% of "error".

I wonder, is there some context where pole masses are more relevant that running masses? For instance, when we compare two masses to decide if the particle A can decay to particle B, are we supposed to compare pole masses, or to run the mass of B to the A scale, or run the mass of A to the B scale? I'd expect the two later procedures to be equivalent, but given that stability is about the total balance of energy, perhaps the former procedure is more relevant.

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I think the point is that in a theory with "Koide symmetry" (i.e. whatever it is that produces the Koide relation) but not "Sumino family symmetry", the $m_i(m_i)$ start out at a set of values which don't satisfy Koide symmetry. The additional Sumino family symmetry adjusts the RG trajectory so that the pole masses do satisfy Koide symmetry. But that doesn't mean that Koide symmetry is exact for the running masses at low energies; it only becomes exact above the family-symmetry unification scale. The masses do not satisfy the symmetry at any single value of E below that scale; but the relation happens to hold for the pole masses at their different scales.

I still haven't verified this! But I believe this is how it's supposed to work.

After reading a bit more, I see that the paper is saying that he can engineer $r(\Lambda)$ to have the right value and this is treated as an initial condition for the EFT. He makes remarks saying that the running of $r(\mu)$ is formally protected but notes that the physical argument breaks down below $\Lambda$.

The troubling part is that he seems to be pushing for some running of masses below this scale that is not at all like what actually happens. We know that the ratios of the pole masses are not the same as the ratios of the running masses at observable energies. There's no calculation in the paper that uses real physics to explain how to get from $r(\mu)$ to the corresponding ratio of pole masses. It still seems that he wants to fix the masses in $r(\Lambda)$ to match the ratios between the pole masses. This is precisely what I'm saying is completely unphysical.

By the way, according for instance table IV of http://arxiv.org/pdf/hep-ph/0601031v2 or section F of http://prd.aps.org/abstract/PRD/v46/i9/p3945_1 (where, note, the wrong measured value for tau is still used), all the damage to Koide relation for leptons is done already when moving electron and muon up to the GeV scale. From 1 GeV up to any high energy (without GUT), the mismatch keeps about 1.0017 - 1.0019, i.e. about the 0.2% of "error".

Assuming that the calculations in the paper are correct, this is very useful to illustrate my point. However the authors obviously reach the wrong conclusions. They seem to think that $k-1$ being different from zero at a larger degree than the ratio $m_1/m_3$ or the experimental uncertainty $\Delta m_3/m_3$ still implies that "Koide's relation is a universal result." This is not a scientific conclusion, we require a higher standard.

I wonder, is there some context where pole masses are more relevant that running masses? For instance, when we compare two masses to decide if the particle A can decay to particle B, are we supposed to compare pole masses, or to run the mass of B to the A scale, or run the mass of A to the B scale? I'd expect the two later procedures to be equivalent, but given that stability is about the total balance of energy, perhaps the former procedure is more relevant.

The pole masses are obviously the right ones in processes such as particle production at threshold. For decay processes, I believe the right criterion is that the process has to make sense in the rest frame of the decaying particle. Therefore the pole mass of A is the right one to use and any running of the B mass is a small contribution to the kinematics of the final state.

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The pole masses are obviously the right ones in processes such as particle production at threshold. For decay processes, I believe the right criterion is that the process has to make sense in the rest frame of the decaying particle. Therefore the pole mass of A is the right one to use and any running of the B mass is a small contribution to the kinematics of the final state.

I am not sure. Consider a decay muon to electron plus a pair neutrino antineutrinos, as usual. As it is possible that the electron is left in the same rest frame that the initial muon, I could say that the energy available for the neutrino pair is the difference of pole masses of muon and electron, not the muon pole mass minus the renormalised electron mass at muon scale. I think I should had put more care when I attended to the undergraduate lectures, twenty years ago.

Of course it is irrelevant for the experimental results, the running of electron fro .511 to 105 is surely negligible.