What is new with Koide sum rules?

[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.

 

[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.

 

[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.

 

[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.

 

[member 11137]

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.

 

[arivero]

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.)

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:
$$\small 97 - 56 \sqrt 3 \leftarrow (0,[7-4 \sqrt 3 , 1), 16-8 \sqrt 3] \rightarrow 151 -28 {\sqrt 3}$$

 

[ohwilleke]

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?

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 anyone 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.

 

[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.

 

[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.

 

[mitchell porter]

I have been very skeptical of Koide relations for mesons. In fact I still am. But I guess that, if we are looking for QCD-like mechanisms, it does make sense to consider whether any of these hadronic Koide triples e.g. have an explanation in which Foot's vector appears. I don't know if you could work towards that by considering the cyclic basis for SU(3), which uses circulants.

Also, a search for diquark sum rules turned up this paper, which has mass formulas for heavy-light diquarks. If the sBootstrap is on the right path, then there is some sort of mapping between quarks and diquarks, and quark mass formulas may resemble diquark mass formulas. In the original bootstrap calculation for the rho meson mass (informal description), the rho meson is modeled as a bound state of pions, and the pion as a bound state of a pion and a rho meson. Given my remarks in #148, I might look for a "sbootstrap calculation" in which the bottom quark is dual to a light diquark (two light quarks), and a light quark is dual to a heavy-light diquark (a bottom quark plus a light quark)...

 

[arivero]

Ok, given that now (from this thread ) we suspect that D-branes are a thing, let's give other view to the "D-Branes + seesaw" formula of #80:
$$m_{k} = {R'^2 \over M_0 \alpha'^2} (n_k + {\theta_{k^i} - \theta_{k^f} \over 2 \pi})^2= m_0 (n_k + \lambda_k)^2$$

where $\lambda_k$ is the distance between the two branes and $n_k$ an extra wrapping that the open string can perform. This is a free adaptation of Johnson's primer hep-th/0007170:

and the idea of see-sawing is not justified anywhere but let's say that it is just to test how compatible Koide formula is. Remember that the standard formula is $m_0(1+\mu_k)^2$, with conditions $\sum \mu_k = 0, \sum (\mu_k^2 - 1) =0$. Here, for the special case of three parallel branes, we have from the construction that $\lambda_3 = \lambda_2+\lambda_1$. I am not sure of how freely we can change the sign of a distance between branes; the formula apparently allows for it but it would need more discussion, so let's simply to investigate the case for wrapping $n_3=0$, so that we can freely flip $\lambda_3 \to - \lambda_3$ and grant $\lambda_3 + \lambda_2+\lambda_1 =0$ and let's put the other two wrappings at the same level n=1. With this, Koide formula should be
$$\frac 32 = { (n_1+n_2 + n_3)^2 \over n_1^2 + n_2^2 +n_3^2 + 2 \sum n_k \lambda_k + \sum \lambda_k^2} ={ 2 \over 1 + (\lambda_1+\lambda_2) + \lambda_1^2 + \lambda_2^2 + \lambda_1 \lambda_2}$$
with the extra condition that the distances are normalized to be all of them less than 1. The parametrization is a lot more inconvenient that the one we are used to, but solutions are not strange.
For instance if $\lambda_1=\lambda_2$ we have
$$\frac 32 = { 2 \over 1 + 2 \lambda_1 + 3 \lambda_1^2 }$$
and $\lambda_1= -\frac 13 + {\sqrt 2 \over 3}$
$$m_1=m_0 (1 + \lambda_1)^2= (\frac 23 + \frac {\sqrt 2 }3)^2 \\ m_2=m_0 (1 + \lambda_2)^2= (\frac 23 + \frac {\sqrt 2 }3)^2 \\ m_3=m_0 (0 - \lambda_1-\lambda_2)^2= 4 (-\frac 13 + \frac {\sqrt 2 }3)^2$$

So it seems doable but, well, the question of the winding numbers and the sign of the non-integer part needs more detail. Of course, without an argument to choose winding of the strings and distance between the branes they are attached to, it is just a change of variables.

 

[mitchell porter]

@arivero has often mentioned the similarity of muon and pion masses, and also occasionally that of the tauon to the heavy-light charmed mesons; the idea being that the leptons are in some sense superpartners of mesons.

Now I am wondering if one could deduce a few quark masses, or even quark Koide relations, by assuming (1) Koide relation for charged leptons, (2) a barely broken meson-lepton supersymmetry, and (3) some form of "super GIM mechanism".

The GIM mechanism is a partial cancellation of amplitudes which allowed the existence and approximate mass of the charm quark to be predicted from the properties of kaons. The existence of charm, completing a second generation, explained the absence of flavor-changing neutral currents (FCNCs), and the specific mass explained a finetuning of kaon decays in which a virtual charm quark appears.

The lack of FCNCs is one of the built-in virtues of the standard model, that encourage some of us to think that it may be true to very high energies. Once you add new heavy particles, you need a feature like "minimal flavor violation" in order to preserve the lack of FCNCs, which in the SM comes automatically from anomaly cancellation. Supersymmetry specifically needs a "super GIM mechanism" (and there are one or two candidates for what that could be).

Now return to the hypotheses (1-3) above. By (1), we have the known masses of muon and tauon. Can we obtain the muon-pion and tauon-charm coincidences, through (2) and (3) respectively?

 

[ohwilleke]

Here is a nifty new little paper:

Phenomenological formula for CKM matrix and physical interpretation
Kohzo Nishida
(Submitted on 3 Aug 2017)
We propose a phenomenological formula relating the Cabibbo--Kobayashi--Masukawa matrix VCKM and quark masses in the form (md‾‾‾√ms‾‾‾√mb‾‾‾√)∝(mu‾‾‾√mc‾‾‾√mt‾‾‾√)VCKM. The results of the proposed formula are in agreement with the experimental data. Under the constraint of the formula, we show that the invariant amplitude of the charged current weak interactions is maximized.
Comments: 6 pages, no figures
Subjects: High Energy Physics - Phenomenology (hep-ph)
Cite as: arXiv:1708.01110 [hep-ph]
(or arXiv:1708.01110v1 [hep-ph] for this version)

 

[nikkkom]

If they are onto something, then neutrino mass ratios can be inferred from PMNS matrix?

 

[ohwilleke]

If they are onto something, then neutrino mass ratios can be inferred from PMNS matrix?

The Nishida paper does not address leptons and the PMNS matrix. It is limited to the quarks and CKM matrix and is purely a phenomenological relationship. In general, the angles in the PMNS matrix parameterization are much bigger than those in the CKM matrix, which would imply neutrino masses closer to each other proportionately, and that is certainly what we see. But, I have no idea if the relationship is as precise.

The other thing about the Nishida paper is that it is a formulation with the square roots of masses and with the bare CKM matrix entries. But, neither of those are observables. You observe the mass and not the square root of the mass, and your observable in the CKM matrix is the square of the CKM matrix entry which gives a transition probability, and not the CKM matrix entry. Since both of the quantities in the Nishida paper are square roots of observable quantities, my intuition is that it ought to be possible to express the same relationship in terms of masses and squares of CKM matrix entries, i.e. the observable quantities. It may be that there is a technical reason that this won't work, but it is something that struck me reading it.

 

[ohwilleke]

Koide's latest paper:

Structure of Right-Handed Neutrino Mass Matrix
Yoshio Koide
(Submitted on 4 Aug 2017)
Recently, Nishiura and the author have proposed a unified quark-lepton mass matrix model under a family symmetry U(3)×U(3)′. The model can give excellent parameter-fitting to the observed quark and neutrino data. The model has a reasonable basis as far as the quark sector, but the form of the right-handed neutrino mass matrix MR does not have a theoretical grand, that is, it was nothing but a phenomenological assumption. In this paper, it is pointed out that the form of MR is originated in structure of neutrino mass matrix for (νi,Nα) where νi (i=1,2,3) and Nα (α=1,2,3) are U(3)-family and U(3)′-family triplets, respectively.

Comments: 7 pages, no figure
Subjects: High Energy Physics - Phenomenology (hep-ph)
Cite as: arXiv:1708.01406 [hep-ph]
(or arXiv:1708.01406v1 [hep-ph] for this version)

This paper is complemented by another earlier this year in a more reflective mood which was mentioned only in passing earlier in this thread.

Sumino Model and My Personal View
Yoshio Koide
(Submitted on 8 Jan 2017)
There are two formulas for charged lepton mass relation: One is a formula (formula A) which was proposed based on a U(3) family model on 1982. The formula A will be satisfied only masses switched off all interactions except for U(3) family interactions. Other one (formula B) is an empirical formula which we have recognized after a report of the precise measurement of tau lepton mass, 1992. The formula B is excellently satisfied by pole masses of the charged leptons. However, this excellent agreement may be an accidental coincidence. Nevertheless, 2009, Sumino has paid attention to the formula B. He has proposed a family gauge boson model and thereby he has tried to understand why the formula B is so well satisfied with pole masses. In this talk, the following views are given: (i) What direction of flavor physics research is suggested by the formula A; (ii) How the Sumino model is misunderstood by people and what we should learn from his model; (iii) What is strategy of my recent work, U(3)×U(3)′ model.
Comments: 5 pages, Talk given at a mini-workshop on "quarks, leptons and family gauge bosons", Osaka, Japan, December 26-27, 2016
Subjects: High Energy Physics - Phenomenology (hep-ph)
Cite as: arXiv:1701.01921 [hep-ph]
(or arXiv:1701.01921v1 [hep-ph] for this version)
Submission history
From: Yoshio Koide [view email]
[v1] Sun, 8 Jan 2017 07:33:59 GMT (6kb)

His recent co-authored papers with Nishiura which he alludes to in this abstract to this paper are:

Flavon VEV Scales in U(3)×U(3)′ Model
Yoshio Koide, Hiroyuki Nishiura
(Submitted on 23 Jan 2017 (v1), last revised 27 Feb 2017 (this version, v2))
We have already proposed a quark and lepton mass matrix model based on U(3)×U(3)′ family symmetry as the so-called Yukawaon model, in which the U(3) symmetry is broken by VEVs of flavons (Φf) αi which are (3,3∗) of U(3)×U(3)′. The model has successfully provided the unified description of quark and lepton masses and mixings by using the observed charged lepton masses as only family-number dependent input parameters. Our next concern is scales of VEVs of the flavons. In the present paper, we estimate the magnitudes of the VEV scales of flavons of the model which is newly reconstructed without changing the previous phenomenological success of parameter fitting for masses and mixings of quarks and leptons. We estimate that VEVs of flavons with (8+1,1), (3,3∗), and (1,8+1) are of 25the orders of 10 TeV, 104 TeV, and 107 TeV, respectively.
Comments: 23 pages, 1 figure, Introduction revised, Appendix remuved
Subjects: High Energy Physics - Phenomenology (hep-ph)
DOI: 10.1142/S0217751X17500853
Cite as: arXiv:1701.06287 [hep-ph]
(or arXiv:1701.06287v2 [hep-ph] for this version)

and

Quark and Lepton Mass Matrices Described by Charged Lepton Masses
Yoshio Koide, Hiroyuki Nishiura
(Submitted on 28 Dec 2015 (v1), last revised 14 May 2016 (this version, v3))
Recently, we proposed a unified mass matrix model for quarks and leptons, in which, mass ratios and mixings of the quarks and neutrinos are described by using only the observed charged lepton mass values as family-number-dependent parameters and only six family-number-independent free parameters. In spite of quite few parameters, the model gives remarkable agreement with observed data (i.e. CKM mixing, PMNS mixing and mass ratios). Taking this phenomenological success seriously, we give a formulation of the so-called Yukawaon model in details from a theoretical aspect, especially for the construction of superpotentials and R charge assignments of fields. The model is considerably modified from the previous one, while the phenomenological success is kept unchanged.
Comments: 14 pages, no figure, accepted version by MPLAl
Subjects: High Energy Physics - Phenomenology (hep-ph)
DOI: 10.1142/S021773231650125X
Cite as: arXiv:1512.08386 [hep-ph]
(or arXiv:1512.08386v3 [hep-ph] for this version)
Submission history
From: Yoshio Koide [view email]
[v1] Mon, 28 Dec 2015 12:14:42 GMT (13kb)
[v2] Tue, 8 Mar 2016 23:39:40 GMT (13kb)
[v3] Sat, 14 May 2016 04:59:24 GMT (14kb)

and
Quark and Lepton Mass Matrix Model with Only Six Family-Independent Parameters
Yoshio Koide, Hiroyuki Nishiura
(Submitted on 19 Oct 2015 (v1), last revised 7 Dec 2015 (this version, v2))
We propose a unified mass matrix model for quarks and leptons, in which sixteen observables of mass ratios and mixings of the quarks and neutrinos are described by using no family number-dependent parameters except for the charged lepton masses and only six family number-independent free parameters. The model is constructed by extending the so-called "Yukawaon" model to a seesaw type model with the smallest number of possible family number-independent free parameters. As a result, once the six parameters is fixed by the quark mixing and the mass ratios of quarks and neutrinos, no free parameters are left in the lepton mixing matrix. The results are in excellent agreement with the neutrino mixing data. We predict δℓCP=−68∘ for the leptonic CP violating phase and ⟨m⟩≃21 meV for the effective Majorana neutrino mass.
Comments: 10 pages, 2 figures, accepted version of a rapid communication i PRD
Subjects: High Energy Physics - Phenomenology (hep-ph)
Journal reference: Phys. Rev. D 92, 111301 (2015)
DOI: 10.1103/PhysRevD.92.111301
Cite as: arXiv:1510.05370 [hep-ph]
(or arXiv:1510.05370v2 [hep-ph] for this version)
Submission history
From: Yoshio Koide [view email]
[v1] Mon, 19 Oct 2015 06:55:38 GMT (553kb)
[v2] Mon, 7 Dec 2015 02:54:31 GMT (553kb)

and

Origin of Hierarchical Structures of Quark and Lepton Mass Matrices
Yoshio Koide, Hiroyuki Nishiura
(Submitted on 17 Mar 2015)
It is shown that the so-called "Yukawaon" model can give a unified description of masses, mixing and CP violation parameters of quarks and leptons without using any hierarchical (family number-dependent) parameters besides the charged lepton masses. Here, we have introduced a phase matrix P=daig(eiϕ1,eiϕ2,eiϕ3) with the phase parameters (ϕ1,ϕ2,ϕ3) which are described in terms of family number-independent parameters, together with using only the charged lepton mass parameters as the family number-dependent parameters. In this paper, the CP violating phase parameters δqCP and δℓCP in the standard expression of VCKM and UPMNS are predicted as δqCP≃72∘ and δℓCP≃−76∘, respectively, i.e. δℓCP∼−δqCP.
Comments: 24 pages, 2 figures
Subjects: High Energy Physics - Phenomenology (hep-ph)
Journal reference: Phys.Rev. D 91, 116002 (2015)
DOI: 10.1103/PhysRevD.91.116002
Cite as: arXiv:1503.04900 [hep-ph]
(or arXiv:1503.04900v1 [hep-ph] for this version)
Submission history
From: Hiroyuki Nishiura [view email]
[v1] Tue, 17 Mar 2015 03:15:32 GMT (903kb)

 

[nikkkom]

The Nishida paper does not address leptons and the PMNS matrix. It is limited to the quarks and CKM matrix and is purely a phenomenological relationship.

Yes, I know that.
I meant, since PMNS matrix is the very same thing for leptons as CKM is for quarks.
Then by analogy with his observation for quarks, let's assume for leptons (sqrt(mass(vi))) = PMNS * (sqrt(mass(li))).
Modulo minus signs, this gives a way to calculate normed vector of square roots of neutrino masses.

 

[ohwilleke]

Yes, I know that.
I meant, since PMNS matrix is the very same thing for leptons as CKM is for quarks.
Then by analogy with his observation for quarks, let's assume for leptons (sqrt(mass(vi))) = PMNS * (sqrt(mass(li))).
Modulo minus signs, this gives a way to calculate normed vector of square roots of neutrino masses.

The trouble is that as you formulate it, this doesn't come remotely close. The charged lepton masses are roughly a million times larger than the neutrino masses and the PMNS matrix entries are of O(1).

 

[nikkkom]

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

 

[nikkkom]

In general I think the CKM matrix tells you that transitions are more likely to occur within the same generation rather than the other...

In "What is new with Koide sum rules?" thread, ohwilleke spotted and posted a link to this recent paper which might be a start in understanding what CKM matrix is all about and why it has precisely these values:

Here is a nifty new little paper:
Phenomenological formula for CKM matrix and physical interpretation

Kohzo Nishida

(Submitted on 3 Aug 2017)

We propose a phenomenological formula relating the Cabibbo--Kobayashi--Masukawa matrix VCKM and quark masses in the form (md‾‾‾√ms‾‾‾√mb‾‾‾√)∝(mu‾‾‾√mc‾‾‾√mt‾‾‾√)VCKM. The results of the proposed formula are in agreement with the experimental data. Under the constraint of the formula, we show that the invariant amplitude of the charged current weak interactions is maximized.

https://arxiv.org/abs/1708.01110

(edit: corrected link)

 

[nikkkom]

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

#!/usr/bin/python
import math
from math import sqrt

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

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

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

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

Output:

Square roots of experimental quark masses, MeV
dn: 2.1679 9.7980 64.6529
up: 1.4832 35.7771 416.5333
Unit-length vectors built of those (divided by norm)
dn: 0.0331 0.1498 0.9882
up: 0.0035 0.0856 0.9963
Kohzo Nishida says that (normed_sqrt_up_masses) = CKM * (normed_sqrt_down_masses)
rc: 0.0036 0.0868 0.9962

 

[ohwilleke]

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

#!/usr/bin/python
import math
from math import sqrt

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

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

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

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

Output:

Square roots of experimental quark masses, MeV
dn: 2.1679 9.7980 64.6529
up: 1.4832 35.7771 416.5333
Unit-length vectors built of those (divided by norm)
dn: 0.0331 0.1498 0.9882
up: 0.0035 0.0856 0.9963
Kohzo Nishida says that (normed_sqrt_up_masses) = CKM * (normed_sqrt_down_masses)
rc: 0.0036 0.0868 0.9962

If you really wanted to go large, you could do a crude Monte Carlo error analysis by having a program do the same thing for every combination of +1 sigma, the best fit vale and -1 sigma, for all of the input values based on the Particle Data Group error bars.

 

[nikkkom]

That would amount to reimplementing CKM Fitter code from scratch :)

I assume CKM Fitter people, if not already grappling with the possibilities opened by this paper, to do so Real Soon.

 

[ohwilleke]

The latest measurement of the tau lepton mass again confirms the original Koide's rule. Koide's rule, a formula proposed in 1981, six years after the tau lepton was discovered, when its mass was known much less accurately, predicts the mass of the tau lepton based upon the mass of the electron and the muon. This prediction using current electron and muon mass measurements is:

1776.96894 ± 0.00007 MeV/c^2.

The uncertainty is entirely due to uncertainty in the electron and muon mass measurements. The low uncertainty in the Koide's rule prediction reflects the fact that the electron and muon mass have been measured much more precisely than the tau lepton mass.

The latest measurement from BESIII, which is the most precise single experimental measurement to date is:

1776.91 ± 0.12 + 0.10/− 0.13 MeV/c^2 (the combined error is ± 0.17).

This result is 0.06 MeV less than the Koide's rule prediction which is consistent to less than one-half of a standard deviation of experimental uncertainty from the predicted value.

The new result is closer to the Koide's rule prediction than the Particle Data Group (PDG) value for 2016 which is:

1776.83 ± 0.12 MeV/c^2

The PDG value is within about 1.2 standard deviations of the Koide's rule prediction. This new result will probably push the next PDG value closer to the Koide's rule prediction.

Koide's rule is one of the most accurate phenomenological hypotheses in existence which has no Standard Model theoretical explanation, although given the precision to which it is true, there is almost certainly some explanation for this correspondence based upon new physics beyond (or really "within") the Standard Model.

 

[mitchell porter]

Something that troubles me, is that every explanation we have for the Koide formula seems to be at odds with Higgs criticality, in that the latter suggests that physics is just standard model up to high scales, whereas the explanations for Koide involve new physics at low scales. See Koide's remarks from January. He says one may think of the formula as holding approximately among running masses, or exactly among pole masses. If we focus just on well-defined field theories that have been written out, they all involve new physics (e.g. Koide's yukawaon fields, the vevs of which contribute to the SM yukawas). In the case of the Sumino mechanism for the pole masses, there are family gauge bosons which are supposed to show up by 10⁴ TeV, i.e. 10⁷ GeV. If we focus just on the yukawaons... Koide seems to have argued that new physics should show around 10¹² GeV. I would be a little happier with that, it's in the vicinity of the lowest-scale explanations of Higgs criticality.

But for this reason, I also wonder if we could do with a new, infrared perspective on the Higgs mechanism. The most recent paper by Arkani-Hamed et al actually provides such a perspective, but only for gauge boson mass, not for fermion mass.

 

[ohwilleke]

Something that troubles me, is that every explanation we have for the Koide formula seems to be at odds with Higgs criticality, in that the latter suggests that physics is just standard model up to high scales, whereas the explanations for Koide involve new physics at low scales. See Koide's remarks from January. He says one may think of the formula as holding approximately among running masses, or exactly among pole masses. If we focus just on well-defined field theories that have been written out, they all involve new physics (e.g. Koide's yukawaon fields, the vevs of which contribute to the SM yukawas). In the case of the Sumino mechanism for the pole masses, there are family gauge bosons which are supposed to show up by 10⁴ TeV, i.e. 10⁷ GeV. If we focus just on the yukawaons... Koide seems to have argued that new physics should show around 10¹² GeV. I would be a little happier with that, it's in the vicinity of the lowest-scale explanations of Higgs criticality.

But for this reason, I also wonder if we could do with a new, infrared perspective on the Higgs mechanism. The most recent paper by Arkani-Hamed et al actually provides such a perspective, but only for gauge boson mass, not for fermion mass.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

 

[mfb]

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

The new arxiv submission is just a review.

 

[ohwilleke]

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

The new arxiv submission is just a review.

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

 

[mfb]

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

 

[mitchell porter]

I am not very impressed with the yukawaon approach, or Sumino's mechanism

But the calculations at the core of how they work, could actually give substance to the kind of bootstrap you suggest. Look at Sumino 2008. Section 4, equation 36, you have a potential-energy function for a nine-component scalar. With a few extra conditions, it has a minimum very close to the square-root-masses of the charged leptons. Meanwhile, section 3 (e.g. figure 4) describes how massive gauge bosons can cancel the QED loop effects that would spoil the Koide relation for the pole masses.

Sumino goes on to introduce multiple new scalars and big new gauge groups, in order to implement these ideas. But the core of it really is that the sqrt-masses minimize some energy function, and that electromagnetic running is countered by some kind of flavor physics.

 

[ohwilleke]

But the calculations at the core of how they work, could actually give substance to the kind of bootstrap you suggest. Look at Sumino 2008. Section 4, equation 36, you have a potential-energy function for a nine-component scalar. With a few extra conditions, it has a minimum very close to the square-root-masses of the charged leptons. Meanwhile, section 3 (e.g. figure 4) describes how massive gauge bosons can cancel the QED loop effects that would spoil the Koide relation for the pole masses.

Sumino goes on to introduce multiple new scalars and big new gauge groups, in order to implement these ideas. But the core of it really is that the sqrt-masses minimize some energy function, and that electromagnetic running is countered by some kind of flavor physics.

Good point. Sometimes people can get the right result even if the mechanism to explain its origin is wrong. Similarly, both dark matter and modified gravity theories can explain galactic rotation, but one of those mechanisms is wrong.

 

[ohwilleke]

Another Formula for the Charged Lepton Masses
Yoshio Koide
(Submitted on 9 Nov 2017)
A charged lepton mass formula (me+mμ+mτ)/(me‾‾‾√+mμ‾‾‾√+mτ‾‾‾√)2=2/3 is well-known. Since we can, in general, have two relations for three quantities, we may also expect another relation for the charged lepton masses. Then, the relation will be expressed by a form of memμmτ‾‾‾‾‾‾‾‾√/(me‾‾‾√+mμ‾‾‾√+mτ‾‾‾√)3. According to this conjecture, a scalar potential model is speculated.
Comments: 5 pages, no figure
Subjects: High Energy Physics - Phenomenology (hep-ph)
Cite as: arXiv:1711.03221 [hep-ph]
(or arXiv:1711.03221v1 [hep-ph] for this version)

 

[mitchell porter]

The standard Koide formula says that "K", a function of the sqrt-masses, equals 2/3. Koide derived this formula from a scalar potential with U(3) family symmetry in 1990. In this latest paper, he defines a new function of the sqrt-masses, "kappa", extends the 1990 potential, and chooses rational values for a few coefficients, in order to predict that "kappa" equals 1/486. The actual value of "kappa" is 1/486.663.

Both K and kappa are scale-invariant, in that they remain unchanged if all the masses are multiplied by a constant. This means that the charged lepton masses are fully determined if one specifies K, kappa, and a mass scale. Here I am reminded of Brannen's formula, which appears in Koide 2007, equations 3.2, 3.3, 3.6, 3.9. The angle 2/9 radians is usually regarded as the key parameter, and has no known field-theoretic derivation.

The quantity 2/9 does actually appear in Koide's latest paper (equation 16), but nothing like Brannen's trigonometric formulas. One could see if they are hiding somewhere in the algebra. Another place to check would be Sumino's potential that I mentioned here, in comment #170.

 

[arivero]

How does he address the existence of other tuples?. Particularly the one existing in the literature previously to his research, (0, m_d, m_s)? For this one, the value of the new parameter is just zero.

 

[mitchell porter]

How does he address the existence of other tuples?. Particularly the one existing in the literature previously to his research, (0, m_d, m_s)? For this one, the value of the new parameter is just zero.

Bearing in mind that in these papers, the masses are determined by the vev of a new, matrix-valued scalar field, we can say that a massless electron requires that the determinant of the vev matrix is zero. (See equation 20 in the 1990 paper.) I do not know if such a case can be obtained through e.g. a choice of coefficients for the scalar potential, as simple as that which Koide now exhibits.

The Haut-Harari-Weyers triple of up, down, strange, includes quarks of different charges, so it requires something more than just the square root of a yukawa matrix. In their original paper, it is just an accident that the Koide relation is satisfied, but one could look for an extended model in which there's a reason.