# What is new with Koide sum rules?

1. Jan 24, 2017

### mitchell porter

Koide and Nishiura's latest (it came out today) contains new numerology.

In their model, each fermion family e (e,μ,τ), u (u,c,t), and d (d,s,b) gets its masses as eigenvalues of a matrix $Z (1 + b_f X)^{-1} Z$, mutiplied by a mass scale $m_{0f}$, where
$$Z = \frac 1 {\sqrt{m_e +m_μ + m_τ}} \begin{pmatrix} \sqrt{m_e} & 0 & 0 \\ 0 & \sqrt{m_μ} & 0 \\ 0 & 0 & \sqrt{m_τ} \end{pmatrix}$$
$$X = \frac 1 3 \begin{pmatrix} 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end{pmatrix}$$
and $b_f$ is a free parameter. f indicates the family, e, u, or d, and $b_f$ and $m_{0f}$ take different values according to the family.

For the e family, $b_e$ is just zero, so the matrix is $Z^2$, and it just gives the (e,μ,τ) masses by construction. For the d family, $b_d$ is a random-looking number. But for the u family, we have
$$b_u = -1.011$$
$$\frac {m_{0u}} {m_{0e}} = 3.121$$
i.e. very close to the integer values, -1 and 3.

All that is from section 4.1 (page 12). The model itself is a seesaw as displayed in section 2.1 (page 5 forward). There is no explanation for the values of those numbers.

2. Jan 24, 2017

### arivero

It looks very much as the discarded corrections for the lepton sector.

For b=-1 the matrix is singular, so perhaps it is just a signal of how massive the top quark is.

3. May 15, 2017

### mitchell porter

A few times in this thread (e.g. #129, #132), @arivero mentions that in a first approximation to his "waterfall" mass ansatz, the mass of the bottom quark comes out as exactly 12 times the constituent quark mass. Earlier this month, in a speculative new framework for hadron masses and systematics, I ran across the claim that the mass of the rho meson is sqrt(6) times the constituent quark mass (see footnote 6, bottom of page 6). This seems to be derivable within the framework of a so-called "gauged quark-level linear sigma model". The original linear sigma model is a famous 1960 creation of Gell-Mann and Levy, with nucleon, pion, and sigma meson fields. The quark-level linear sigma model employs quark fields in the place of the nucleon fields, and the gauged version of this adds vector mesons as well. This is, more or less, the framework I was citing in #134.

What I want to observe here, is that this first approximation to the bottom quark mass, as 12 times the constituent quark mass, could be expressed as 2 . sqrt(6) . sqrt(6) times the constituent quark mass. Furthermore, all these quantities (in the extended LSM framework) come from couplings to QCD condensates, where the value of the coupling is an exact quantity determined nonperturbatively. Also, the double appearance of one factor, is reminiscent of how Koide's yukawaon models work. So maybe this is a hint regarding how to realize the waterfall.

Last edited: May 15, 2017
4. Jun 6, 2017

### mitchell porter

The aspect of the waterfall that has confounded and tantalized me the most, is that it alternates between charge +2/3 and charge -1/3, whereas the original Koide triple is all charge +1. The original Koide triple naturally fits a family symmetry like Z3 or U(3), but these alternating triples don't; I haven't seen anything like them in the literature. The closest thing I have seen, are radiative models of fermion mass in which only the top couples directly to the Higgs, and then the lighter fermions get their masses through top loops; the lighter the particle, the higher the order at which the top appears.

I do think it is probably possible to construct a field theory in which a waterfall of Koide triples is produced, I just worry that we would only be able to do it in a very complicated and artificial way (by engineering an appropriate potential for the SM yukawa couplings, through the introduction of new charges, symmetries, and fields). That's the confounding part, that makes me doubt the worth of the enterprise. But the tantalizing part is when I remember that the principle of its construction is simple. Yes, it is at odds with the sort of mechanisms that are familiar in BSM theory, but perhaps we just haven't found the right mechanism. So I thought I would review a few of the possibilities that exist.

In #137 and #139, @ohwilleke said that the sequence resembles a cascade of W boson decays, and speculated that the Koide relation (for a consecutive triple in the sequence) might be due to a sort of equilibrium between the three quark fields involved. This makes me wonder: the radiative models of mass generation, in which the lighter quarks get their mass from top loops, usually involve new scalars. Could you have such a model, in which some of the scalars involved are the Goldstone bosons that arise from the Higgs field, and which are "eaten" by the W and Z bosons to become their spin-0 components?

Another underused representation of the Koide relation is the geometric representation due to Robert Foot. (It does make an appearance in @arivero's original paper introducing the waterfall.) The Koide formula holds if the 3-vector of square roots of masses forms an angle of 45 degrees with respect to the vector (1,1,1). For a set of six sqrt-masses, forming four overlapping triples, one may envisage a set of four such relationships in a six-dimensional space. This could be the basis of a potential for the yukawa couplings, or perhaps even a model with extra dimensions. The same goes for Jerzy Kocik's Cartesian formula mentioned by @Blackforest in #126.

A further possibility is that it might be done nonperturbatively with condensates, in a Nambu-Jona-Lasinio model. This would fit some recent thoughts (#134 and #143), and I wrote about it here. Finally, there's @arivero's own scenario, described in #131-#132.

5. Jun 6, 2017

### Blackforest

May I first thank you for your résumé on the topic and for re-mentioning an article on which I proposed to think about a few years ago? Being myself not a professional I get the biggest difficulty to meet the criteria allowing an intervention on your forums. This doesn’t impeach me to think about some of the topics which are discussed here.

Concerning the Koïde formula and the article related to an old Descartes theorem, I would like to add that diverse personal thoughts are pushing my intuition into a direction questioning the role of the tetrahedrons into that discussion. I don’t mean that tetrahedrons are paving our space-time (a view that would feet with the branch which is working on tetrahedral meshes [e.g.: Data structures for geometric and topological aspects of finite element in Progress In Electromagnetics Research, PIER 32, 151–169, 2001]).

No; referring to articles studying the altitudes of that object, I just wonder about some of their fascinating properties and ask myself in which way the trace-less quadratic forms of rank 3 which are sometimes associated with these platonic objects [e.g.: N.A. Court, Notes on the orthocentric tetrahedron, Amer. Math. Monthly 41 (1934) 499–502] may eventually be an alternative and useful tool for the description of the propagation of light?

Although my thoughts are seemingly a little bit off-topic, I would like to remark that four spheroids may deform the four faces of a tetrahedron and, because of that, be involved into a discussion roughly related to the Descartes theorem mentioned in the article at post 126.

I apologize if I have disturbed the forum with that piece of dream. Best regards.

6. Jun 6, 2017

### arivero

Well. actually its usage is very obvious in early Koide models, as first he imposes the "couplings" $z_1, z_2, z_3$ to be in a plane orthogonal to (1,1,1), then he asks them to be in a circle of some radious proportional to "coupling" $z_0$ and finally the composites are given masses proportional to $(z_i + z_0)^2$. In these models the idea is that $z^i$^2 is the "self coupling potential" of the preon with itself, while $2 z_a z_b$ is the "interaction energy". He already uses different radius for leptons and quarks, but I think that the (1,1,1) axis is always used. By the way, a lot of years ago it was suggested to try an angle around (1,1,0) for the up-quark tuple. In this sense, changing the main axis, Foot's representation is underused, yep.

Radiative models are always very tempting due to the relative regularity of the masses, very nicely spaced by quantities of the order of strong and em couplings. I think that this was remarked, for the s-c gap, by Gsponer when doing our colaboration, and anyway it catches the eye when one looks a log plot of the standard model:

(btw, remember that source for this kind of plot is available in github at https://gist.github.com/arivero/e74ad3848290845de5ca )

My point to favour a breaking instead of a radiative approach to the waterfall is that the scheme of two linked Koide tuples
$$(0,[1-{\sqrt 3 \over 2} , 1+{\sqrt 3 \over 2}), 4]$$ already covers a lot of the spectrum, as said above in https://www.physicsforums.com/threads/what-is-new-with-koide-sum-rules.551549/page-7#post-5049287 comment #132: we get (u,[s,c),b) in quarks as well as the original (e,mu,tau) tuple. But it can also be argued that the extremes, down and top, obtained by linking another two tuples left and right, become ugly:
$$13 - 15 {\sqrt 3 \over 2} \leftarrow (0,[1-{\sqrt 3 \over 2} , 1+{\sqrt 3 \over 2}), 4] \rightarrow 109 + 95 {\sqrt 3 \over 2}$$

On other hand, radiative jumps could be the answer to the use of tuples with different charges, and it is not really incompatible with a "breaking" view. But the main problem, to me, is not that we have different charges, but that they need to be at the same time in different tuples; this makes the preon a very hard pill to swallow. It is true that we have also the two koide tuples (0, [pi, D), B]) which are obviously composites but here QCD makes some magic to set the pion mass as about the same size that the strange quark; honestly we could have expected the equivalent pair of tuples to be (pi,[K,D),B]

Edit: same line, but scaling, looks a bit less ugly, but only a little bit:
$$\small 97 - 56 \sqrt 3 \leftarrow (0,[7-4 \sqrt 3 , 1), 16-8 \sqrt 3] \rightarrow 151 -28 {\sqrt 3}$$

Last edited: Jun 6, 2017
7. Jun 7, 2017

### ohwilleke

Just to recap the extended Koide's rule for quarks notion a bit. The idea would be that the quark masses arise dynamically from W boson interactions. Each "target" quark flavor's mass would be a blend of the three other different quark masses that the target quark could transform into via W boson interactions, weighted in some manner by the relative likelihood of each W boson transformation in the CKM matrix. Thus, no additional boson is necessary for the mass generation mechanism. A similar mechanism would apply, at least, to the charged leptons and probably to the entire lepton sector, giving rise to Koide's rule for charged leptons.

Notably, all Standard Model fundamental particles that have mass have weak force interactions with W and Z bosons, while none of the Standard Model fundamental particles that lack mass (the gluon and the photon) have weak force interactions with W and Z bosons. This scheme is also, incidentally, an argument for the non-existence of massive right handed neutrinos, since right handed neutrinos do not interact with W and Z bosons via the weak force, so even if they exist as fundamental particles, they should not have any mass.

In cases where almost all interactions of the target quark flavor can be attributed to just two of the three possible source quarks, Koide's formula is a very good approximation of the relative masses in a triple, e.g., in the case of the t-b-c triple where the b quark is the target and the t and c quarks are the source quarks. The likelihood of a b-u transformation via a W boson under the CKM matrix is so low that it can be disregarded in a first order approximation.

Empirically, a linear interpolation from a three quark Koide triple estimate of a quark mass, to one that considers all three source quark masses, can be achieved by multiplying the probability o the omitted source quark mass by the probability of a CKM transformation to the omitted source quark.

For example, the Koide waterfall applied naively give you a mass of zero or very nearly zero for an up quark derived from a u-d-s triple, which also is the dominant source of W boson transformations from a u quark to another quark flavor, energy conservation permitting. This is significantly different from the actual measured mass of the u quark on the order of 2 Mev. But, if you multiply the square of the CKM element for an up quark-bottom quark transformation via a W boson which represents the probability of such an event by the mass of the bottom quark, you get a value much closer to the experimentally measured value of the mass of the up quark.

More generally, the Koide triples for quarks which are least accurate when compared to experiment are those for which the square of the CKM entry for the W boson transformation from the target quark to the omitted source quark is greatest.

Of course, to get the entire quark mass matrix you need to simultaneously solve six sets of equations that set for a relationship between each of the six target quark masses relative to the three source quark masses for each at the same time, and ideally, you would do so in a manner (probably inspired by the geometric interpretation of Koide's rule) that is an exact non-linear formulation, rather than a non-linear formulation that approximates the target quark mass from two of the three source quarks and then adjusts the result with a linear approximation for the third source quark.

If one could make this work, you could derive the relative masses of the six quarks entirely from the four CKM matrix parameters and one mass constant (either the mass of any one of the quarks or a mass scale for the quark sector as a whole).

This hypothesis also assumes that the CKM matrix is logically prior to the fundamental fermion masses in the SM, which the structure of the CKM matrix tends to support. It is pretty much impossible to go the other direction and derive the quark mass matrix from the CKM matrix because there isn't a big enough difference between the entries for up-like quarks and down-like quarks of the same generation. The Wolfenstein parameterization likewise favors an understanding of the CKM matrix that treats generations as distinct units rather than individual quarks.

The Koide triple formula works so well in the charged lepton sector because any given charged lepton target particle has only two source particles (the other two charged leptons) rather than three, and because any contributions from W boson interactions between neutrinos and charged leptons is negligible since all of the neutrino masses are on the order of 1,000,00+ times smaller than any of the charged lepton masses, so the linear interpolation of additional elements which is material in the case of the quark masses in much smaller than the precision of experimental mass measurement accuracy in the case of the charged leptons.

Last edited: Jun 7, 2017
8. Jun 12, 2017

### mitchell porter

Today it occurred to me, what if the rho meson were somehow the mediator in these radiative cascades? That can't be literally true, but first let me present the argument. Again, it goes back to the fact that in @arivero's first approximation to the quark masses, the "unperturbed waterfall", the bottom quark mass is twelve times the constituent quark mass.

As described in Schumacher 2014, page 4, one paradigm for explaining the constituent quark mass is very similar to the Higgs mechanism for fermion mass: there's a vev, and a coupling to that vev. It's there in equation 5. The coupling is pion-pion-sigma meson, and the vev is, I guess, the chiral condensate. I don't quite get how it is supposed to work - maybe the quark emits a sigma meson, which couples to two pions from the chiral condensate??

Now it so happens that the pion-pion-rho meson coupling is about 6 (see e.g. Delbourgo & Scadron, equations 29 and 34, where it comes out as 2π). So we have that the bottom quark mass is something like the constituent quark mass, times the pion-pion-rho coupling, times 2. Schumacher already explained the constituent quark as something like a bare quark, coupling via sigma meson to two pions from the chiral condensate. Now suppose we have a sigma meson condensate as well. A sigma meson is often modeled as two quarks and two antiquarks, so it can be decomposed into two pions in two ways. Could the bottom quark be like a constituent quark that then couples to a sigma meson condensate, in those two ways, via rho meson - with its mass thereby picking up a further factor of 6+6, i.e. 12??

More precisely, one should imagine a rho-like light vector meson. The real point is that the bottom quark would be the unexpected origin of the waterfall, the flavor that couples directly to the mass-giving condensates, with all the other quarks getting their masses from loop effects, some sort of radiative equilibrium, etc. The top quark, the traditional source of radiative cascade, is a little peculiar in this bottom-centric picture, because it's far more massive than the bottom. So some further idea about mechanism may be required; and in grand unified theories, it is common to regard top and bottom together as having a special status with respect to mass. In any case, the facts are (1) top, bottom, charm are a Koide triple (2) bottom mass apparently has the right magnitude, to be obtained from strong couplings to QCD condensates.

A few more remarks. There are tantalizing similarities between QCD vector mesons like the rho, and the electroweak gauge bosons. The QCD vector mesons have at least a formal resemblance to an emergent gauge symmetry that has been higgsed. In holographic QCD, they are the Kaluza-Klein reduction of higher-dimensional flavor gauge bosons. In the sbootstrap, one might hope to see the W and Z emerge along with the leptons, in one of these ways... Meanwhile, one may try to implement the waterfall through methods like those of Cabo or Zubkov - by positing four-quark interactions whose couplings are fixed nonperturbatively, by consistency arguments.

9. Jun 14, 2017

### arivero

Question, given the current trend of events in D and B mesons... does anyone knows if Brannen finished his inspection of Koide-like relationships in hadron spectroscopy? I think he published some attempt on excited states for mesons having similar quark composition, but I do not remember if he extended to different compositions, say (Pi, B, D) or similar tuples.

Edit: remember that by the coincidence between piom and muon, and D (or charm) with tau, we have a pseudotuple (0, pion, D) and then also the "scb" one: $${(-\sqrt M_{\pi^-} + \sqrt M_{D^-} + \sqrt M_{B^-})^2 \over M_{\pi^-} + M_{D^-} + M_{B^-}} = 1.486...$$ and so variations of these oscillate around $\frac 32$ above or below without any particular pattern, as far as I can see. Normally we do not look at these ones except as part of some self-consistency, but who knows, perhaps they have some role in decay puzzles.

Last edited: Jun 15, 2017