What is new with Koide sum rules?

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ftr

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Thanks ftr, I'm still there :smile:
Oh Good, I hate loosing unconventional talents.I see that you have been working very hard behind the scenes .Good luck and be strong:biggrin:
 

Hans de Vries

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I see that you have been working very hard behind the scenes .
Thanks, indeed, with many new insights.


Insights on the spinor level:

1) How to calculate all three spin-vectors
How to calculate all three spin-vectors ##s_x,~s_y## and ## s_z## of a spinor and how to do so with a single matrix multiplication. The sum of the three vectors is the total spin ##s##: T
he precessing spin 1/2 pointer.

2) A second triplet of spinor rotation generators
These generators rotate the spinor in its local reference frame instead of in world coordinates. This uncovers the (infinitesimal) rigid-body aspect of field theory with generators that rotate a spinor around its own three principle axis.

Insights on the fermion field level:

1) A single fermion field

The two light-like chiral components ##\xi_L## and ##\xi_R## each get two orthogonal polarization states, with the orientation of the states defined by spinors.
$$\mbox{Dirac field}~~
\left(\!

\begin{array}{c}
\xi_{_L} \\ \xi_{_R}
\end{array}
\!\right)
~~~~\Longrightarrow~~~~
\left(\!\!\!
\begin{array}{rc}
\xi_{_{L}} \\ \pm{\mathbf{\mathsf{i}}}_g\,\xi_{_{L}} \\ \pm~~\,\xi_{_{R}} \\ \pm{\mathbf{\mathsf{i}}}_g\,\xi_{_{R}}
\end{array}
\!\!\right)
~~\mbox{Unified Fermion field}$$

2) A Standard Model fermion generator.
All standard model fermions, three generations of leptons and quarks and their anti-particles are the eigen-vectors of a single generator with only the charge and its sign as input. All fermions obtained this way posses all the right electroweak properties corresponding with a ##\sin^2\theta_w## of 0.25

3) A single electroweak fermion Lagrangian.
The many different electroweak-fermion pieces of the Lagrangian can be replaced by:
$$\mathcal{L} ~~=~~ \bar{\psi}\,\check{m}\big(\,\gamma^\mu_{_0}\partial_\mu+\mathbf{U}-\check{m}\,\big)\,\psi,~~~~~~~~
\mathbf{U} ~=~\tfrac{\,g'}{\,2\,}\gamma^\mu_{_o}\gamma^5_{_o}Z_\mu + \tfrac{g}{2}\gamma^\mu_{_1}A_\mu + \tfrac{g}{2}\gamma^\mu_{_2}W^1_\mu + \tfrac{g}{2}\gamma^\mu_{_3}\gamma^5_{_o}W^2_\mu$$

4) A single bilinear field matrix
This matrix contains all bilinear field components as well as all source currents for all electroweak bosons. The matrix is calculated with a single matrix multiplication.

Insights on the electroweak boson level.

1) The fundamental representation of the electromagnetic field.
This representation uses the operator fields acting on the fermion field:
$$\begin{array}{lrcl}
\mbox{mass dimension 1:}~~~~ & \mathbf{A} &=& \gamma^\mu A_\mu \\
\mbox{mass dimension 2:}~~~~ & \mathbf{F} &=& \vec{K}\cdot\vec{E}-\vec{J}\cdot\vec{B} \\
\mbox{mass dimension 3:}~~~~ & \mathbf{J}\, &=& \gamma^\mu~j_\mu \\
\end{array}$$
We now obtain the fundamental covariant description of the electromagnetic field:
$$/\!\!\! \partial\mathbf{A} = \mathbf{F}~~~~~~ ~~~/\!\!\!\partial\mathbf{F} = \mathbf{J}$$
In the first step we have applied the conservation law ##\partial_\mu A^\mu\!=\!0## on the diagonal and the second step involves all four of Maxwell's laws, the inhomogeneous ##\partial_\mu F^{\mu\nu}\!=\!j^\nu## as well as the homogeneous ##~\partial_\mu\! *\!\!F^{\mu\nu}\!=\!0##.

2) A single electroweak boson field
As given in the Lagrangian above. Note that each electroweak boson has its own set of gamma matrices.
$$\mathbf{U} ~=~\tfrac{\,g'}{\,2\,}\gamma^\mu_{_o}\gamma^5_{_o}Z_\mu + \tfrac{g}{2}\gamma^\mu_{_1}A_\mu + \tfrac{g}{2}\gamma^\mu_{_2}W^1_\mu + \tfrac{g}{2}\gamma^\mu_{_3}\gamma^5_{_o}W^2_\mu$$


The documents, mathematica files and the stand alone matlab executable are available here,
but look at the video for the best introduction.
 
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A paper today on "String Landscape and Fermion Masses". They guess at the statistical distribution of fermion masses in string vacua, and then argue that the standard model fermions satisfy their hypothesis. Normally I don't have much interest in papers like this, since they prove so little. I would much rather see progress in calculating masses for individual vacua.

However, there's an oddity here. They model the distribution of quark masses, and then the distribution of charged lepton masses, using a two-parameter "Weibull distribution". The parameters are a shape parameter k and a (mass) scale parameter l. They find (equation 3.6), "surprisingly", that the two distributions have the same shape parameter, to three decimal places, so differing only by mass scale. Is this circumstantial evidence that a similar mechanism (e.g. @arivero's waterfall) is behind both sets of yukawas?
 

arivero

Gold Member
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using a two-parameter "Weibull distribution". The parameters are a shape parameter k and a (mass) scale parameter l. They find (equation 3.6), "surprisingly", that the two distributions have the same shape parameter, to three decimal places, so differing only by mass scale. Is this circumstantial evidence that a similar mechanism (e.g. @arivero's waterfall) is behind both sets of yukawas?
Hmm, the main property Weibull distribution is that you can integrate it, so perhaps they are just seeing some exponential fitting. As for the coincidence of shape... How are they "fitting" the distribution anyway? max likelihood? for a sample of six points?
 

arivero

Gold Member
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Hmm, I can not reproduce the fit, perhaps because of precision or rounding errors, with scipy. I have no idea how the authors are using chi-square test and p-values in the paper, so I go with KS test.

Code:
Python 3.6.5 (default, Mar 31 2018, 19:45:04) [GCC] on linux
Type "help", "copyright", "credits" or "license" for more information.
>>> import scipy.stats as s
>>> import numpy as np
>>> def printStats(data,fit):
...     nnlf=s.weibull_min.nnlf(fit,np.array(data))
...     ks=s.stats.kstest(np.array(data),'weibull_min',fit)
...     print("Fit:",fit)
...     print("negloglikelihood",nnlf)
...     print(ks)
...
>>> data=[2.3,4.8,95,1275,4180,173210]
>>> printStats(data,s.weibull_min.fit(data, floc=0))
Fit: (0.26861598701150763, 0, 2288.475995797873)
negloglikelihood 51.591787735494115
KstestResult(statistic=0.15963622669415056, pvalue=0.9979920390593924)
>>> data=[0.511,106,1777]
>>> printStats(data,s.weibull_min.fit(data, floc=0))
Fit: (0.37366611506161873, 0, 229.48782534013557)
negloglikelihood 19.233771988350043
KstestResult(statistic=0.23629696537671507, pvalue=0.996122995979272)
>>>
Anyway even if scipy adjusts to 0.373 for leptons, their fit is not bad neither, lets fix the parameter and see
Code:
>>> printStats(data,s.weibull_min.fit(data, floc=0,f0=0.26861598701150763))
Fit: (0.26861598701150763, 0, 163.62855309410182)
negloglikelihood 19.44374499168725
KstestResult(statistic=0.25597858377056465, pvalue=0.9893658166203932)
The fit in this case reproduce the scale they found, 194. I wonder if what happens is that their fitter takes as starting point the value of the previous fit, or something so. Also, if we add the three leptons to the quark sector, so that
data=[0.511,106,1777,2.3,4.8,95,1275,4180,173210]
the fit is still
Code:
Fit: (0.2698428583536703, 0, 1156.8564935786583)
negloglikelihood 71.49265190220518
KstestResult(statistic=0.14728900912921583, pvalue=0.9897758037009418)
Thus telling that the same random distribution can of course generate values for the lepton sector. Unsurprising.


Amusingly, we can indeed find the same k parameter in the two fits if we allow to move the origin of the quark sector
Code:
>>> data=[2.3,4.8,95,1275,4180,173210]
>>> printStats(data,s.weibull_min.fit(data))
Fit: (0.37359275206555403, 2.2999999999999994, 39837.607589227395)
negloglikelihood 30.744667740180212
KstestResult(statistic=0.48342279946216715, pvalue=0.08187510735420012)
but then same freedom in lepton sector goes to a different fit too.
 
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perhaps they are just seeing some exponential fitting
The paradigm of Tye et al is something like: We consider a landscape of string vacua in which vacua are indexed by fluxes (and other properties), and we suppose that the flux values are sampled from a uniform distribution. But the yukawas depend on the fluxes in an "anti-natural" way (Lubos's word), such that uniformly distributed fluxes translate into Weibull-distributed yukawas (distribution divergently peaked at zero). "Related distributions" at Wikipedia shows how a uniformly distributed variable can be mapped to an exponentially distributed variable, and then to a Weibull distribution.

Optimistically, we could construct a refined version of the paradigm in which we aim to get the sbootstrap from an SO(32) flux mini-landscape, and then the Koide waterfall ansatz from that. In section 3 of Tye et al, they talk about the (unspecified) functional dependence of yukawas on fluxes. One could add an intermediate dependence e.g. on Brannen's Koide parameters (phase and mass scale), and the number of sequentially chained Koide triplets. By treating the Brannen parameters as random variables that depend upon randomly distributed flux values, one can then study how the resulting masses are distributed, and what kind of dependency on the fluxes would make Tye et al's scenario work out.

(It is still mysterious why the lepton "waterfall", consisting of just one triplet, and the quark waterfall, consisting of four triplets, would have the same Weibull shape, but this might be clarified with further study. Since Weibull involves a bias towards low values, one would be looking at how the low end of the waterfall behaves. Is the Weibull fit so loose that a Brannen phase of 2/9, as for e,mu,tau, and a phase of 2/3, as for b,c,s, produce roughly the same behavior? Or maybe there's something about applying that Georgi-Jarlskog-like factor of 3 to both Brannen phase and Brannen mass, at the same time, which preserves Weibull shape? These are concrete questions that could actually be answered.)
 
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arivero

Gold Member
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It is still mysterious why the lepton "waterfall", consisting of just one triplet, and the quark waterfall, consisting of four triplets, would have the same Weibull shape, but this might be clarified with further study
I am disappointed that the fit algorithm in scipy fails to produce the same shape... I wonder how they are doing the fit, if R or some manual code, of different precision. The use of chi square points to some ad-hoc code; after all, the point of the Weibull distribution is that it has an exact and very simple cdf,
, and then it is very easy to calculate matchings even by hand. On the other hand, that could mean that they have found some analytic result and misinterpreted it as a probabilistic parameter.

The paper was not designed, I think, to give exact proportions, but to convey the message that even if you claim that yukawas are random, your theory should tell what the random distribution is, and statistical test for the likeliness of "living in this vacuum" can incorporate the information of the actual values of the yukawa couplings. And indeed is a good counter against the naive concept of equaling naturalness to likeliness.
 
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I now suspect that they simply decided apriori that shape should be the same. In the introduction to part 3, they say "Once dynamics introduces a new scale... it will fix l, while k is unchanged"; and in 3.2 they say colored and colorless particles fit this paradigm. So I think they just did some kind of joint fit, deliberately assuming (or aiming for) a common k value.
 

arivero

Gold Member
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I now suspect that they simply decided apriori that shape should be the same. In the introduction to part 3, they say "Once dynamics introduces a new scale... it will fix l, while k is unchanged"; and in 3.2 they say colored and colorless particles fit this paradigm. So I think they just did some kind of joint fit, deliberately assuming (or aiming for) a common k value.
That was my suspicion too, as I can at leat get the same k if I do the fit with quarks... but then it is very puzzling that they claim chi^2=1 for leptons in 3.6. Again, I have no idea how do they calculate the chi coefficient.
 

arivero

Gold Member
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A remark: for the Anderson-Darling test statistics, the fit fixing k=0.269 seems to have better p-value in lepton sector that the direct fit from scipy.
Code:
>>> import scipy.stats as s
>>> data=[0.511,106,1777]
>>> fit=(0.37366611506161873, 0, 229.48782534013557)
>>> from skgof import ks_test, cvm_test, ad_test
>>> w=s.weibull_min(*fit)
>>> ad_test(data,w)
GofResult(statistic=0.25987976933243573, pvalue=0.9716940635456661)
>>> fit=(0.26861598701150763, 0, 163.62855309410182)
>>> w=s.weibull_min(*fit)
>>> ad_test(data,w)
GofResult(statistic=0.22716618686611634, pvalue=0.9893423546344761)
So the question of how has the coincidence happened depends on knowing how they are optimizing the parameters.
 

arivero

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Lets quote here also the paper Mitchell mentioned elsewhere, https://arxiv.org/abs/1903.00160, as it could be useful to anyone looking for Koide-like relationships in the diquark sector.
 
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A paper by Goldman and Stephenson today, promotes the idea that the standard model mass matrices can be obtained by "democratic" yukawa couplings that all have the same value, plus small perturbations.

The reason is as follows. Suppose we have a 3x3 matrix in which all matrix entries are the same (e.g. they could all be equal to 1). You can diagonalize this matrix, by multiplying by a "tribimaximal" matrix. The resulting matrix will be diag(m,0,0) for some m. But for quarks and charged leptons, we have that the third generation is much more massive than the first two. So in all cases, the mass matrix can be approximated by a matrix of the form diag(m,0,0).

Goldman and Stephenson perform an inverse tribimaximal transformation on the quark mass matrices in order to show just how close to democratic they are (eqn 6 and 7), and they show that, for a particular parametrization, the deviations from democracy are small (equation 11)... the largest of these perturbations is still just .02, so if a model can be found, it can be analyzed perturbatively. They proposed in a previous paper that these perturbations might arise from interactions with dark-matter sterile neutrinos, but they don't provide a model. The other potentially significant thing they observe, is that some of the perturbation parameters need to be complex, so they propose that this is where CP violation comes from (section IV B).

They call their idea Higgs Universality, since the idea is that to a first approximation, the coupling of all fermions to the Higgs is the same.

They don't present a model. However, I will point out that recent work by Koide and Nishiura (mentioned, e.g., at #141 in this thread) to some extent is such a model. Koide and Nishiura have a universal ansatz for the mass matrices, which involves contributions from the democratic matrix, the unit matrix, and a matrix diag(√e,√μ,√τ). Ironically, however, for the charged leptons, the contribution from the democratic matrix is zero. This is ironic, not only because Goldman and Stephenson assert (calculations promised for a future paper) that the charged lepton masses can also be obtained by a small perturbation of a democratic matrix; but Koide himself obtained them that way, in earlier work!

If I look at the history of Koide's attempts to explain his own formula, I see three kinds of model. First, the preon model where he first obtained it. Second, the democratic model. Third, the perturbed democratic model with Nishiura. It is my understanding that @arivero's sbootstrap was partly inspired by the preon model, perhaps because some of the preons can be paired up in a fashion reminiscent of quark-diquark supersymmetry. (This should be compared with Risto Raitio's approach to supersymmetric preons.) It would be intriguing if one could close the circle of Koide's models, and obtain the "perturbed democratic model" by having democratically interacting preons mix with their own composites - the latter providing the "√e,√μ,√τ" perturbation.

Speaking of supersymmetry, the study of the supermathematics of Grassmann, Berezin, etc, has given me a new perspective on where the problematic phase of 2/9, discovered by @CarlB, could come from (see e.g. #173 in this thread). Phases that are rational multiples of π are much more natural. I had previously noticed that the well-known expansion of π/4 as 1 - 1/3 + 1/5 - ... contains a 2/3 in its first two terms, so if the analogous expansion for π/12 were somehow truncated there, one could obtain 2/9. The only problem was that I couldn't think of a good reason for such a truncation. One just had to construct a model with a π/12 phase and hope, perhaps, that it approximated Carl's ansatz well enough.

However - that expansion can be obtained as a Taylor series in x, for x=1. Meanwhile, for a grassmann number θ, θ^2 (and all higher powers) equals zero, because of anticommutativity: ab=-ba, so θ.θ = -θ.θ = 0. So, what if you took a Taylor series for x=1, and superanalytically continued it to x=θ...? All powers of x equal to x^2 or higher, will drop out. Unfortunately, 1/3 or 1/9 doesn't naturally show up as the coefficient of x, but rather as the coefficient of x^3, and I haven't thought of a sensible way to associate it with x^1.
 

ohwilleke

Gold Member
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A paper by Goldman and Stephenson today, promotes the idea that the standard model mass matrices can be obtained by "democratic" yukawa couplings that all have the same value, plus small perturbations.

The reason is as follows. Suppose we have a 3x3 matrix in which all matrix entries are the same (e.g. they could all be equal to 1). You can diagonalize this matrix, by multiplying by a "tribimaximal" matrix. The resulting matrix will be diag(m,0,0) for some m. But for quarks and charged leptons, we have that the third generation is much more massive than the first two. So in all cases, the mass matrix can be approximated by a matrix of the form diag(m,0,0).

Goldman and Stephenson perform an inverse tribimaximal transformation on the quark mass matrices in order to show just how close to democratic they are (eqn 6 and 7), and they show that, for a particular parametrization, the deviations from democracy are small (equation 11)... the largest of these perturbations is still just .02, so if a model can be found, it can be analyzed perturbatively. They proposed in a previous paper that these perturbations might arise from interactions with dark-matter sterile neutrinos, but they don't provide a model. The other potentially significant thing they observe, is that some of the perturbation parameters need to be complex, so they propose that this is where CP violation comes from (section IV B).

They call their idea Higgs Universality, since the idea is that to a first approximation, the coupling of all fermions to the Higgs is the same.

They don't present a model. However, I will point out that recent work by Koide and Nishiura (mentioned, e.g., at #141 in this thread) to some extent is such a model. Koide and Nishiura have a universal ansatz for the mass matrices, which involves contributions from the democratic matrix, the unit matrix, and a matrix diag(√e,√μ,√τ). Ironically, however, for the charged leptons, the contribution from the democratic matrix is zero. This is ironic, not only because Goldman and Stephenson assert (calculations promised for a future paper) that the charged lepton masses can also be obtained by a small perturbation of a democratic matrix; but Koide himself obtained them that way, in earlier work!

If I look at the history of Koide's attempts to explain his own formula, I see three kinds of model. First, the preon model where he first obtained it. Second, the democratic model. Third, the perturbed democratic model with Nishiura. It is my understanding that @arivero's sbootstrap was partly inspired by the preon model, perhaps because some of the preons can be paired up in a fashion reminiscent of quark-diquark supersymmetry. (This should be compared with Risto Raitio's approach to supersymmetric preons.) It would be intriguing if one could close the circle of Koide's models, and obtain the "perturbed democratic model" by having democratically interacting preons mix with their own composites - the latter providing the "√e,√μ,√τ" perturbation.

Speaking of supersymmetry, the study of the supermathematics of Grassmann, Berezin, etc, has given me a new perspective on where the problematic phase of 2/9, discovered by @CarlB, could come from (see e.g. #173 in this thread). Phases that are rational multiples of π are much more natural. I had previously noticed that the well-known expansion of π/4 as 1 - 1/3 + 1/5 - ... contains a 2/3 in its first two terms, so if the analogous expansion for π/12 were somehow truncated there, one could obtain 2/9. The only problem was that I couldn't think of a good reason for such a truncation. One just had to construct a model with a π/12 phase and hope, perhaps, that it approximated Carl's ansatz well enough.

However - that expansion can be obtained as a Taylor series in x, for x=1. Meanwhile, for a grassmann number θ, θ^2 (and all higher powers) equals zero, because of anticommutativity: ab=-ba, so θ.θ = -θ.θ = 0. So, what if you took a Taylor series for x=1, and superanalytically continued it to x=θ...? All powers of x equal to x^2 or higher, will drop out. Unfortunately, 1/3 or 1/9 doesn't naturally show up as the coefficient of x, but rather as the coefficient of x^3, and I haven't thought of a sensible way to associate it with x^1.
FWIW, my hypothesis is that Koide-type relationships and the mass hierarchy general arises because (1) the CKM matrix is logically prior to the mass matrix, and (2) the mass matrix represents a dynamic balancing of the mass of each particle of one type, with each of the particles it could transition to via the W boson, adjusted for transition probabilities, in a simultaneous equation that covers and balances all transitions at once.

Charged lepton transitions are approximately democratic, because the neutrino contribution is so small. The CKM matrix is predominantly one factor equal to the likelihood of a first to second generation transition and a second factor equal to a second to third generation transition, with the probability of a first to third generation transition equal to the product of the two probabilities. CP violation and differences between up-type and down-type quarks in transition probabilities are second or lower order effects.
 

CarlB

Science Advisor
Homework Helper
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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)
Has anyone checked this with the square root neutrino masses, one of which is negative? If not, I'm inclined to do it myself.

His formula seems very natural; he's proposing that V is a unitary matrix that takes a real unit vector to a real unit vector. Just what you expect a unitary matrix to do.

The next thing to note is that V is taking a real vector to a real vector. The usual definition of the CKM and MNS matrices allows one to multiply any row or column by a complex phase; this doesn't change the matrix. But the restriction that the matrix take this real vector to that real vector defines these arbitrary complex phases. For example, if you multiply any row of V by a complex phase you can see that his formula will be broken as it won't preserve real vectors.

It is a fact that any unitary matrix can be put, (typically in 4 ways for 3x3 unitary matrices, see equation (56) of http://vixra.org/pdf/1511.0083v1.pdf for the CKM matrix) into a form where the complex phases are defined by requiring that all rows and columns sum to 1, called "magic" in the literature in reference to magic squares. This is an additional requirement to the fact that the sum of the absolute squares of a row or column of a unitary matrix are one.

That any 3x3 unitary matrix can be put into magic form was proved by Gibbs: http://vixra.org/abs/0907.0002 When one puts a unitary matrix into that form, the vector (1,1,1) becomes an eigenvector with eigenvalue 1. My intuition suggests that these are related problems. And that also implies to me that there will be four solutions to these sorts of problems (for a typical random unitary matrix, and an infinite number of solutions for special cases).
 
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CarlB

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Has anyone checked this with the square root neutrino masses, one of which is negative? If not, I'm inclined to do it myself.
I just realized a fairly large problem with how I was thinking of this. For the 2x2 case, his formula is providing two complex equations (i.e. real equations which imply that the imaginary part is zero) which is 4 real restrictions. That happens to match the number of real degrees of freedom in a 2x2 unitary matrix so it determines the answer.

But for the 3x3 case he's only providing three complex equations which gives six restrictions but 3x3 unitary matrices have 9 degrees of freedom. Five of those nine are arbitrary complex phases multiplying rows or columns and the other four determine the probabilities. So he's got enough to decide all the complex phases (which look to be restricted indeed by the real nature of the vectors) and one of the 4 restrictions on probabilities. Rereading the article, he notes this saying "Our basic postulate is to interpret one of the unitary matrices,V" where the emphasis could be on "one of".

In calculating V, he uses a standard parameterization for the unitary matrices which I think are quite ugly when compared to mine.
 

ohwilleke

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I have updated the comparison of the extended Koide's formula for quarks of @arivero with updated data including the FLAG 19 report quark masses and the PDG 2018 data to see how this looks relative to when this was originally proposed in 2011. I have also updated the comparison of this new data to the LC & P hypothesis.
 
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On Dr Koide's personal website, there is a July 6 update in Japanese (you may need to switch your browser's encoding to see it properly), in which he expresses regret for not updating the website more often, and says that he has been in poor health for the past two years. Physics research is the "driving force" of his life, he seems to have received a new research grant, and he links to a copy of a recent research report (again in Japanese) summarizing his latest papers.

In this thread we have regularly entertained various generalizations of Koide's formula, and explanations of varying eccentricity. But let me try to say something about it from a "high orthodox" perspective. I compare it to Balmer's formula for the hydrogen spectrum. Today we can understand that formula in terms of quantum mechanics; but it's the particle masses, in particular, which lack explanation.

Despite a growth in pop-science skepticism about it, string theory is still the most promising framework for explaining the unexplained parameters of the standard model (I did say this would be taking an orthodox perspective!), e.g. as arising from the compactification geometry in various ways. Since string phenomenologists are always looking for ways to narrow the range of vacua that they need to consider, Koide's formula could be an excellent clue.

However, there are reasons why it is neglected. Most importantly, it is an exact relation among pole masses, whereas the renormalization group leads us to expect exact relations only among running masses at high scales. This reason for neglect is independent of string theory, it represents the "common sense" of quantum field theorists.

Nonetheless, this is not the final word. An infrared fixed point can impose relations among infrared quantities. Also - this is not as well understood, but may be significant - when quantum gravity is taken into account, there can be unusual relations between UV and IR of a quantum field theory. And in general, QFT still contains many hidden complexities. One might therefore take the attitude that the counterintuitive nature of Koide's formula - counterintuitive with respect to field theorist's common sense - is a further clue, about what needs to be investigated. One should directly investigate what would have to be true, for a theory to exhibit just this kind of unlikely or impossible-seeming infrared relationship.

(And fortunately, some of that necessary work may already have been done by Yukinari Sumino. Koide himself has also written a few papers approaching his formula from the perspective that the exactness for pole masses is just a coincidence, and that its approximate validity for the running masses is the real fact to be explained; so that kind of approach is possible too.)

Another thing which I think may hinder a successful investigation of Koide's formula by orthodoxy - and here I begin to deviate from orthodoxy, but in ways that many others have also begun to do - is the common assumption of naturalness of masses, achieved by TeV-scale supersymmetry. Such an assumption is not logically incompatible with Koide's formula - Koide has written many papers in which a supersymmetric framework is assumed. But usually (though not always) it doesn't add much to the explanation, it is instead there because he is a competent particle physicist and knows that it is a good and useful theoretical framework.

Nonetheless, the LHC results appear to be telling us that the world works in a different way. No partners (super or otherwise) have been found, and the Higgs mass is close to critical. So new paradigms like the relaxion (and, in string theory, nonsupersymmetric phenomenology) are slowly growing in popularity. Combining this with the peculiar infrared exactness of the Koide formula, suggests to me that one should be trying to explain the formula, in the context of some such new paradigm. Of course this makes life harder for the serious theorist; but it is a logical conclusion.
 

ohwilleke

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1,471
375
One might therefore take the attitude that the counterintuitive nature of Koide's formula - counterintuitive with respect to field theorist's common sense - is a further clue, about what needs to be investigated. One should directly investigate what would have to be true, for a theory to exhibit just this kind of unlikely or impossible-seeming infrared relationship.
Yes. It's a bit like MOND. It may be a phenomenological relationship not grounded in theory, but any theory has to reproduce it because it compactly describes the evidence.

Nonetheless, the LHC results appear to be telling us that the world works in a different way.
What an extraordinarily delicate way to express that sentiment.
 
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