# What is new with Koide sum rules?

1. Nov 4, 2018

### ftr

Oh Good, I hate loosing unconventional talents.I see that you have been working very hard behind the scenes .Good luck and be strong

2. Nov 6, 2018

### Hans de Vries

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.

Last edited: Nov 6, 2018
3. Feb 18, 2019

### mitchell porter

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?

4. Feb 19, 2019

### arivero

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?

5. Feb 19, 2019

### arivero

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 (Text):

Python 3.6.5 (default, Mar 31 2018, 19:45:04) [GCC] on linux
>>> 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 (Text):

>>> 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 (Text):

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 (Text):

>>> 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.

6. Feb 20, 2019

### mitchell porter

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.)

Last edited: Feb 20, 2019
7. Feb 20, 2019

### arivero

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.

8. Feb 20, 2019

### mitchell porter

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.

9. Feb 20, 2019

### arivero

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.

10. Feb 20, 2019

### arivero

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 (Text):

>>> 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)
GofResult(statistic=0.25987976933243573, pvalue=0.9716940635456661)
>>> fit=(0.26861598701150763, 0, 163.62855309410182)
>>> w=s.weibull_min(*fit)