What is new with Koide sum rules?

1. Aug 9, 2017

nikkkom

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

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

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

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

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

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

Code (Text):

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

2. Aug 9, 2017

ohwilleke

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.

3. Aug 10, 2017

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.

4. Sep 28, 2017

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.

5. Sep 28, 2017

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.

6. Sep 28, 2017

ohwilleke

Simply relying on new physics, in and of itself, isn't very troubling because this is an area where new physics wouldn't contract the Standard Model, it would merely fill in a gap where the Standard Model provides no explanation and instead resorts to determining the values of constants experimentally with a theory.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Last edited: Sep 28, 2017
7. Sep 29, 2017

Staff: Mentor

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

The new arxiv submission is just a review.

8. Sep 30, 2017

ohwilleke

Good catch. I'm surprised how few tau mass measurements there have been. Only one in the last eight years.

9. Sep 30, 2017

Staff: Mentor

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

10. Oct 5, 2017

mitchell porter

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.

11. Oct 5, 2017

ohwilleke

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.

12. Nov 10, 2017

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.
Subjects: High Energy Physics - Phenomenology (hep-ph)
Cite as: arXiv:1711.03221 [hep-ph]
(or arXiv:1711.03221v1 [hep-ph] for this version)

13. Nov 11, 2017

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.

14. Nov 13, 2017

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.

15. Nov 14, 2017

mitchell porter

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.

16. Jan 20, 2018

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.

17. Mar 9, 2018

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
$$a^2 m^3+(2 a c-b^2) m^2+(c^2-2 b d) m-d^2 = 0$$

Last edited: Mar 9, 2018
18. Mar 10, 2018

lpetrich

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?

19. Mar 11, 2018

mitchell porter

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.

20. Mar 11, 2018

arivero

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