# What is new with Koide sum rules?

1. 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.

2. 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
3. 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?

4. 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.

5. 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.

6. Mar 13, 2018

### 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.

Last edited: Mar 13, 2018
7. May 21, 2018 at 1:15 PM

### ohwilleke

A new Koide paper:

8. May 25, 2018 at 3:02 PM

### ohwilleke

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