 #211
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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 Koidelike relationships in the diquark sector.
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 darkmatter 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 quarkdiquark 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 wellknown 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.
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 CabibboKobayashiMasukawa 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 (hepph)
Cite as: arXiv:1708.01110 [hepph]
(or arXiv:1708.01110v1 [hepph] 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.
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 impossibleseeming infrared relationship.
Nonetheless, the LHC results appear to be telling us that the world works in a different way.
Another approach does a bit better but is still hardly precision physics:The SU(3) YangMills matrix model coupled to fundamental fermions is an approximation of quantum chromodynamics (QCD) on a 3sphere of radius R. The spectrum of this matrix model Hamiltonian is estimated using standard variational methods, and is analyzed in the strong coupling limit. By employing a matching prescription to determine the dependence of the YangMills coupling constant g on R, we relate the asymptotic values of the energy eigenvalues in the R→∞ (flat space) limit to the masses of light hadrons. We find that the matrix model estimates the light hadron spectrum fairly accurately, with the light baryon masses falling within 10%, and most light meson masses falling within about 30% of their observed values.
PeiLin Yin, Chen Chen, Gastao Krein, Craig D. Roberts, Jorge Segovia, ShuSheng Xu "Masses of groundstate mesons and baryons, including those with heavy quarks" (March 1, 2019).Using a confining, symmetrypreserving regularisation of a vector×vector contact interaction, we compute the spectra of groundstate pseudoscalar and vector (fg¯) mesons, scalar and axialvector (fg) diquarks, and JP=1/2+,3/2+ (fgh) baryons, where f,g,h∈{u,d,s,c,b}. The diquark correlations are essentially dynamical and play a key role in formulating and solving the threevalencequark baryon problems. The baryon spectrum obtained from this largelyalgebraic approach reproduces the 22 known experimental masses with an accuracy of 2.9(2.4) %. It also possesses the richness of states typical of constituentquark models, predicting many heavyquark baryons not yet observed. This study indicates that diquark correlations are an important component of all baryons; and owing to the dynamical character of the diquarks, it is typically the lightest allowed diquark correlation which defines the most important component of a baryon's Faddeev amplitude.

A curious paper from China today, "A translational flavor symmetry in the mass terms of Dirac and Majorana fermions" by Zhizhong Xing. The symmetry in the title is a "discrete shift in flavor space" whose definition I don't understand, but it has two consequences of interest. First, it implies that in each triple that respects the symmetry, one mass should equal zero. This fits the "m_e = m_u = 0" version of the waterfall.
I didn't, but that shouldn't make you hesitate. All avenues need to be explored.Carl, about your old approaches to hadrons. Did you consider the Koide tuple of pi,D,B mesons? If not, why? If yes, do you remember where?
And I assume that the energy depends partly on the difference in values between neighboring points but that other quantum numbers are determined by the actual things stepping. Hence the effect of the phase change is to preserve all quantum numbers except for energy = mass.
So for the pi, I would look for Koide triplets among the pi and its excited states. And for me, they have to have the same quantum numbers. A possibility would be (pi, pi[1300], pi[1800]).
#!/usr/bin/env python
from math import sqrt
from itertools import permutations, combinations
p="""Particle ID(s) Mass (GeV) Errors (GeV) Width (GeV) Errors (GeV) Name Charges
211 1.3957039E01 +1.8E07 1.8E07 2.5284E17 +5.0E21 5.0E21 pi +
111 1.349768E01 +5.0E07 5.0E07 7.81E09 +1.2E10 1.2E10 pi 0
221 5.47862E01 +1.7E05 1.7E05 1.31E06 +5.0E08 5.0E08 eta 0
331 9.5778E01 +6.0E05 6.0E05 1.88E04 +6.0E06 6.0E06 eta'(958) 0
321 4.93677E01 +1.6E05 1.6E05 5.317E17 +9.0E20 9.0E20 K +
311 4.97611E01 +1.3E05 1.3E05 K 0
411 1.86966E+00 +5.0E05 5.0E05 6.33E13 +4.0E15 4.0E15 D +
421 1.86484E+00 +5.0E05 5.0E05 1.605E12 +6.0E15 6.0E15 D 0
431 1.96835E+00 +7.0E05 7.0E05 1.305E12 +1.0E14 1.0E14 D(s) +
521 5.27934E+00 +1.2E04 1.2E04 4.018E13 +1.0E15 1.0E15 B +
511 5.27965E+00 +1.2E04 1.2E04 4.333E13 +1.1E15 1.1E15 B 0
531 5.36688E+00 +1.4E04 1.4E04 4.342E13 +1.7E15 1.7E15 B(s) 0
541 6.27447E+00 +3.2E04 3.2E04 1.291E12 +2.3E14 2.3E14 B(c) +
441 2.9839E+00 +4.0E04 4.0E04 3.20E02 +7.0E04 7.0E04 eta(c)(1S) 0
443 3.096900E+00 +6.0E06 6.0E06 9.26E05 +1.7E06 1.7E06 J/psi(1S) 0
553 9.46030E+00 +2.6E04 2.6E04 5.40E05 +1.3E06 1.3E06 Upsilon(1S) 0
200553 1.03552E+01 +5.0E04 5.0E04 2.03E05 +1.9E06 1.9E06 Upsilon(3S) 0
"""
p.split("\n")[3].split()
m=dict()
for line in p.split("\n")[2:1]:
d=line.split()
print(d)
m[d[2]+d[1]]=float(d[1])
result=[]
for triplet in permutations(m,3):
a,b,c = triplet
a,b,c = m[a],m[b],m[c]
if b < c:
k=((a+b+c)/(sqrt(a)+sqrt(b)+sqrt(c))**2)
result.append([abs(k2/3),triplet,k,"++"]) #use Python3 for floats!
for doublet in combinations(m,2):
b,c = doublet
a,b,c = 0,m[b],m[c]
k=((a+b+c)/(sqrt(a)+sqrt(b)+sqrt(c))**2)
result.append([abs(k2.00/3),doublet,k,"0++"])
for triplet in combinations(m,3):
a,b,c = triplet
a,b,c = m[a],m[b],m[c]
k=((a+b+c)/(sqrt(a)+sqrt(b)+sqrt(c))**2)
result.append([abs(k2/3),triplet,k,"+++"]) #use Python3 for floats!
for doublet in permutations(m,2):
b,c = doublet
a,b,c = 0,m[b],m[c]
k=((a+b+c)/(sqrt(a)sqrt(b)+sqrt(c))**2)
result.append([abs(k2.00/3),doublet,k,"0+"])
result.sort()
for x in result:
print(f'{"".join(x[1]):<30}',"\t",x[2],x[3])
pi0D(s)+eta(c)(1S)0 0.6663837057987507 ++
pi0D+ 0.6661367723217316 0++
pi+D(s)+B+ 0.6673593324132077 ++
pi+D(s)+B0 0.6673602750393526 ++
pi0D0 0.6658883844469711 0++
pi+D(s)+B(s)0 0.667632095646159 ++
pi0D(s)+J/psi(1S)0 0.6654846261441458 ++
pi+D(s)+ 0.6678662052942218 0++
pi+pi0Upsilon(1S)0 0.6682566138237737 +++
pi0D(s)+B(c)+ 0.6683538259548514 ++
pi0D(s)+B(s)0 0.6647562935736748 ++
pi0D(s)+B0 0.6644668556345085 ++
pi0D(s)+B+ 0.6644658496344038 ++
pi+D(s)+J/psi(1S)0 0.6690366259117551 ++
pi0D+J/psi(1S)0 0.6698183238780191 ++
pi0D+B+ 0.6698242106935278 ++
pi0D+B0 0.6698253129918151 ++
pi+D(s)+eta(c)(1S)0 0.6699885664600504 ++
pi0D0J/psi(1S)0 0.6700415949749136 ++
pi0D0B+ 0.670097549041931 ++
pi0D0B0 0.6700986560032072 ++
pi0D+B(s)0 0.6701414512426769 ++
pi0D0B(s)0 0.6704160867174216 ++
pi+D+ 0.6629167285356727 0++
pi0D+eta(c)(1S)0 0.6706380841323579 ++
pi+D0 0.6626683899479252 0++
pi0D0eta(c)(1S)0 0.6708574511027877 ++
pi+D(s)+B(c)+ 0.6710682904109853 ++
pi0D(s)+ 0.6710861088843688 0++
pi+D+B+ 0.6727737789483542 ++
pi+D+B0 0.67277481621395 ++
pi+D0B+ 0.6730499585060149 ++
pi+D0B0 0.6730510003496702 ++
pi+D+B(s)0 0.6730728758703244 ++
eta0Upsilon(1S)0Upsilon(3S)0 0.6602536515686964 ++
pi+D0B(s)0 0.6733503288525908 ++
pi+D+J/psi(1S)0 0.673443593712495 ++
pi+D0J/psi(1S)0 0.6736705861236503 ++
pi0D+B(c)+ 0.6739753428121785 ++
pi0D0B(c)+ 0.674261376733939 ++
pi+D+eta(c)(1S)0 0.6743176053201272 ++
pi+D0eta(c)(1S)0 0.6745407648041061 ++
K+B(c)+ 0.6578600967642535 0++
K0B(c)+ 0.6570972277563282 0++
pi+D+B(c)+ 0.6767411895047183 ++
pi+D0B(c)+ 0.6770298238881567 ++
...
#!/usr/bin/env python
# coding: utf8
from math import sqrt
from itertools import permutations, combinations
p="""* Particle ID(s) Mass (GeV) Errors (GeV) Width (GeV) Errors (GeV) Name Charges
24 8.0379E+01 +1.2E02 1.2E02 2.08E+00 +4.0E02 4.0E02 W +
23 9.11876E+01 +2.1E03 2.1E03 2.4952E+00 +2.3E03 2.3E03 Z 0
25 1.2525E+02 +1.7E01 1.7E01 3.2E03 +2.8E03 2.2E03 H 0
11 5.109989461E04 +3.1E12 3.1E12 0.E+00 +0.0E+00 0.0E+00 e 
13 1.056583745E01 +2.4E09 2.4E09 2.9959837E19 +3.0E25 3.0E25 mu 
15 1.77686E+00 +1.2E04 1.2E04 2.267E12 +4.0E15 4.0E15 tau 
1 4.67E03 +0.5E03 0.2E03 d 1/3
2 2.16E03 +0.5E03 0.3E03 u +2/3
3 9.3E02 +1.1E02 5.0E03 s 1/3
4 1.27E+00 +2.0E02 2.0E02 c +2/3
5 4.180E+00 +3.0E02 2.0E02 b 1/3
6 1.725E+02 +7.0E01 7.0E01 1.42E+00 +1.9E01 1.5E01 t +2/3
211 1.3957039E01 +1.8E07 1.8E07 2.5284E17 +5.0E21 5.0E21 pi +
111 1.349768E01 +5.0E07 5.0E07 7.81E09 +1.2E10 1.2E10 pi 0
221 5.47862E01 +1.7E05 1.7E05 1.31E06 +5.0E08 5.0E08 eta 0
9000221 6.0E01 +2.0E01 2.0E01 4.5E01 +3.5E01 3.5E01 f(0)(500) 0
113 213 7.7526E01 +2.3E04 2.3E04 1.491E01 +8.0E04 8.0E04 rho(770) 0,+
223 7.8266E01 +1.3E04 1.3E04 8.68E03 +1.3E04 1.3E04 omega(782) 0
331 9.5778E01 +6.0E05 6.0E05 1.88E04 +6.0E06 6.0E06 eta'(958) 0
9010221 9.90E01 +2.0E02 2.0E02 6.E02 +5.0E02 5.0E02 f(0)(980) 0
9000111 9000211 9.80E01 +2.0E02 2.0E02 7.5E02 +2.5E02 2.5E02 a(0)(980) 0,+
333 1.019461E+00 +1.6E05 1.6E05 4.249E03 +1.3E05 1.3E05 phi(1020) 0
10223 1.166E+00 +6.0E03 6.0E03 3.75E01 +3.5E02 3.5E02 h(1)(1170) 0
10113 10213 1.2295E+00 +3.2E03 3.2E03 1.42E01 +9.0E03 9.0E03 b(1)(1235) 0,+
20113 20213 1.23E+00 +4.0E02 4.0E02 4.2E01 +1.8E01 1.8E01 a(1)(1260) 0,+
225 1.2755E+00 +8.0E04 8.0E04 1.867E01 +2.2E03 2.5E03 f(2)(1270) 0
20223 1.2819E+00 +5.0E04 5.0E04 2.27E02 +1.1E03 1.1E03 f(1)(1285) 0
100221 1.294E+00 +4.0E03 4.0E03 5.5E02 +5.0E03 5.0E03 eta(1295) 0
100111 100211 1.30E+00 +1.0E01 1.0E01 4.0E01 +2.0E01 2.0E01 pi(1300) 0,+
115 215 1.3182E+00 +6.0E04 6.0E04 1.07E01 +5.0E03 5.0E03 a(2)(1320) 0,+
10221 1.35E+00 +1.5E01 1.5E01 3.5E01 +1.5E01 1.5E01 f(0)(1370) 0
9000113 9000213 1.354E+00 +2.5E02 2.5E02 3.30E01 +3.5E02 3.5E02 pi(1)(1400) 0,+
9020221 1.4088E+00 +2.0E03 2.0E03 5.01E02 +2.6E03 2.6E03 eta(1405) 0
10333 1.416E+00 +8.0E03 8.0E03 9.0E02 +1.5E02 1.5E02 h(1)(1415) 0
20333 1.4263E+00 +9.0E04 9.0E04 5.45E02 +2.6E03 2.6E03 f(1)(1420) 0
1000223 1.41E+00 +6.0E02 6.0E02 2.9E01 +1.9E01 1.9E01 omega(1420) 0
10111 10211 1.474E+00 +1.9E02 1.9E02 2.65E01 +1.3E02 1.3E02 a(0)(1450) 0,+
100113 100213 1.465E+00 +2.5E02 2.5E02 4.0E01 +6.0E02 6.0E02 rho(1450) 0,+
100331 1.475E+00 +4.0E03 4.0E03 9.0E02 +9.0E03 9.0E03 eta(1475) 0
9030221 1.506E+00 +6.0E03 6.0E03 1.12E01 +9.0E03 9.0E03 f(0)(1500) 0
335 1.5174E+00 +2.5E03 2.5E03 8.6E02 +5.0E03 5.0E03 f(2)'(1525) 0
9010113 9010213 1.661E+00 +1.5E02 1.1E02 2.4E01 +5.0E02 5.0E02 pi(1)(1600) 0,+
9020113 9020213 1.655E+00 +1.6E02 1.6E02 2.5E01 +4.0E02 4.0E02 a(1)(1640) 0,+
10225 1.617E+00 +5.0E03 5.0E03 1.81E01 +1.1E02 1.1E02 eta(2)(1645) 0
30223 1.670001E+00 +3.0E02 3.0E02 3.15E01 +3.5E02 3.5E02 omega(1650) 0
227 1.667E+00 +4.0E03 4.0E03 1.68E01 +1.0E02 1.0E02 omega(3)(1670) 0
10115 10215 1.6706E+00 +2.9E03 1.2E03 2.58E01 +8.0E03 9.0E03 pi(2)(1670) 0,+
100333 1.680E+00 +2.0E02 2.0E02 1.5E01 +5.0E02 5.0E02 phi(1680) 0
117 217 1.6888E+00 +2.1E03 2.1E03 1.61E01 +1.0E02 1.0E02 rho(3)(1690) 0,+
30113 30213 1.720E+00 +2.0E02 2.0E02 2.5E01 +1.0E01 1.0E01 rho(1700) 0,+
9000115 9000215 1.70E+00 +4.0E02 4.0E02 2.7E01 +6.0E02 6.0E02 a(2)(1700) 0,+
10331 1.704E+00 +1.2E02 1.2E02 1.23E01 +1.8E02 1.8E02 f(0)(1710) 0
9010111 9010211 1.810E+00 +9.0E03 1.1E02 2.15E01 +7.0E03 8.0E03 pi(1800) 0,+
337 1.854E+00 +7.0E03 7.0E03 8.7E02 +2.8E02 2.3E02 phi(3)(1850) 0
9050225 1.936E+00 +1.2E02 1.2E02 4.64E01 +2.4E02 2.4E02 f(2)(1950) 0
119 219 1.967E+00 +1.6E02 1.6E02 3.24E01 +1.5E02 1.8E02 a(4)(1970) 0,+
9060225 2.01E+00 +6.0E02 8.0E02 2.0E01 +6.0E02 6.0E02 f(2)(2010) 0
229 2.018E+00 +1.1E02 1.1E02 2.37E01 +1.8E02 1.8E02 f(4)(2050) 0
9080225 2.297E+00 +2.8E02 2.8E02 1.5E01 +4.0E02 4.0E02 f(2)(2300) 0
9090225 2.35E+00 +5.0E02 4.0E02 3.2E01 +7.0E02 6.0E02 f(2)(2340) 0
321 4.93677E01 +1.6E05 1.6E05 5.317E17 +9.0E20 9.0E20 K +
311 4.97611E01 +1.3E05 1.3E05 K 0
9000311 9000321 8.45E01 +1.7E02 1.7E02 4.68E01 +3.0E02 3.0E02 K(0)*(700) 0,+
313 8.9555E01 +2.0E04 2.0E04 4.73E02 +5.0E04 5.0E04 K*(892) 0
323 8.9167E01 +2.6E04 2.6E04 5.14E02 +8.0E04 8.0E04 K*(892) +
323 8.955E01 +8.0E04 8.0E04 4.62E02 +1.3E03 1.3E03 K*(892) +
10313 10323 1.253E+00 +7.0E03 7.0E03 9.0E02 +2.0E02 2.0E02 K(1)(1270) 0,+
20313 20323 1.403E+00 +7.0E03 7.0E03 1.74E01 +1.3E02 1.3E02 K(1)(1400) 0,+
100313 100323 1.414E+00 +1.5E02 1.5E02 2.32E01 +2.1E02 2.1E02 K*(1410) 0,+
10311 10321 1.43E+00 +5.0E02 5.0E02 2.7E01 +8.0E02 8.0E02 K(0)*(1430) 0,+
315 1.4324E+00 +1.3E03 1.3E03 1.09E01 +5.0E03 5.0E03 K(2)*(1430) 0
325 1.4273E+00 +1.5E03 1.5E03 1.000E01 +2.1E03 2.1E03 K(2)*(1430) +
9000313 9000323 1.67E+00 +5.0E02 5.0E02 1.6E01 +5.0E02 5.0E02 K(1)(1650) 0,+
30313 30323 1.718E+00 +1.8E02 1.8E02 3.2E01 +1.1E01 1.1E01 K*(1680) 0,+
10315 10325 1.773E+00 +8.0E03 8.0E03 1.86E01 +1.4E02 1.4E02 K(2)(1770) 0,+
317 327 1.779E+00 +8.0E03 8.0E03 1.61E01 +1.7E02 1.7E02 K(3)*(1780) 0,+
20315 20325 1.819E+00 +1.2E02 1.2E02 2.64E01 +3.4E02 3.4E02 K(2)(1820) 0,+
9010315 9010325 1.99E+00 +6.0E02 5.0E02 3.49E01 +5.0E02 3.0E02 K(2)*(1980) 0,+
319 329 2.048E+00 +8.0E03 9.0E03 1.99E01 +2.7E02 1.9E02 K(4)*(2045) 0,+
411 1.86966E+00 +5.0E05 5.0E05 6.33E13 +4.0E15 4.0E15 D +
421 1.86484E+00 +5.0E05 5.0E05 1.605E12 +6.0E15 6.0E15 D 0
423 2.00685E+00 +5.0E05 5.0E05 D*(2007) 0
413 2.01026E+00 +5.0E05 5.0E05 8.34E05 +1.8E06 1.8E06 D*(2010) +
10421 10411 2.343E+00 +1.0E02 1.0E02 2.29E01 +1.6E02 1.6E02 D(0)*(2300) 0,+
10423 10413 2.4221E+00 +6.0E04 6.0E04 3.13E02 +1.9E03 1.9E03 D(1)(2420) 0,+
20423 2.412E+00 +9.0E03 9.0E03 3.14E01 +2.9E02 2.9E02 D(1)(2430) 0
425 415 2.4611E+00 +7.0E04 8.0E04 4.73E02 +8.0E04 8.0E04 D(2)*(2460) 0,+
431 1.96835E+00 +7.0E05 7.0E05 1.305E12 +1.0E14 1.0E14 D(s) +
433 2.1122E+00 +4.0E04 4.0E04 D(s)* +
10431 2.3178E+00 +5.0E04 5.0E04 D(s0)*(2317) +
20433 2.4595E+00 +6.0E04 6.0E04 D(s1)(2460) +
10433 2.53511E+00 +6.0E05 6.0E05 9.2E04 +5.0E05 5.0E05 D(s1)(2536) +
435 2.5691E+00 +8.0E04 8.0E04 1.69E02 +7.0E04 7.0E04 D(s2)*(2573) +
521 5.27934E+00 +1.2E04 1.2E04 4.018E13 +1.0E15 1.0E15 B +
511 5.27965E+00 +1.2E04 1.2E04 4.333E13 +1.1E15 1.1E15 B 0
513 523 5.32470E+00 +2.1E04 2.1E04 B* 0,+
515 5.7395E+00 +7.0E04 7.0E04 2.42E02 +1.7E03 1.7E03 B(2)*(5747) 0
525 5.7372E+00 +7.0E04 7.0E04 2.0E02 +5.0E03 5.0E03 B(2)*(5747) +
531 5.36688E+00 +1.4E04 1.4E04 4.342E13 +1.7E15 1.7E15 B(s) 0
533 5.4154E+00 +1.8E03 1.5E03 B(s)* 0
535 5.83986E+00 +1.2E04 1.2E04 1.49E03 +2.7E04 2.7E04 B(s2)*(5840) 0
541 6.27447E+00 +3.2E04 3.2E04 1.291E12 +2.3E14 2.3E14 B(c) +
441 2.9839E+00 +4.0E04 4.0E04 3.20E02 +7.0E04 7.0E04 eta(c)(1S) 0
443 3.096900E+00 +6.0E06 6.0E06 9.26E05 +1.7E06 1.7E06 J/psi(1S) 0
10441 3.41471E+00 +3.0E04 3.0E04 1.08E02 +6.0E04 6.0E04 chi(c0)(1P) 0
20443 3.51067E+00 +5.0E05 5.0E05 8.4E04 +4.0E05 4.0E05 chi(c1)(1P) 0
10443 3.52538E+00 +1.1E04 1.1E04 7.E04 +4.0E04 4.0E04 h(c)(1P) 0
445 3.55617E+00 +7.0E05 7.0E05 1.97E03 +9.0E05 9.0E05 chi(c2)(1P) 0
100441 3.6375E+00 +1.1E03 1.1E03 1.13E02 +3.2E03 2.9E03 eta(c)(2S) 0
100443 3.68610E+00 +6.0E05 6.0E05 2.94E04 +8.0E06 8.0E06 psi(2S) 0
30443 3.7737E+00 +4.0E04 4.0E04 2.72E02 +1.0E03 1.0E03 psi(3770) 0
100445 3.9225E+00 +1.0E03 1.0E03 3.52E02 +2.2E03 2.2E03 chi(c2)(3930) 0
9000443 4.0390E+00 +1.0E03 1.0E03 8.0E02 +1.0E02 1.0E02 psi(4040) 0
9010443 4.191E+00 +5.0E03 5.0E03 7.0E02 +1.0E02 1.0E02 psi(4160) 0
9020443 4.421E+00 +4.0E03 4.0E03 6.2E02 +2.0E02 2.0E02 psi(4415) 0
553 9.46030E+00 +2.6E04 2.6E04 5.40E05 +1.3E06 1.3E06 Upsilon(1S) 0
10551 9.8594E+00 +5.0E04 5.0E04 chi(b0)(1P) 0
20553 9.8928E+00 +4.0E04 4.0E04 chi(b1)(1P) 0
10553 9.8993E+00 +8.0E04 8.0E04 h(b)(1P) 0
555 9.9122E+00 +4.0E04 4.0E04 chi(b2)(1P) 0
100553 1.002326E+01 +3.1E04 3.1E04 3.20E05 +2.6E06 2.6E06 Upsilon(2S) 0
20555 1.01637E+01 +1.4E03 1.4E03 Upsilon(2)(1D) 0
110551 1.02325E+01 +6.0E04 6.0E04 chi(b0)(2P) 0
120553 1.02555E+01 +5.0E04 5.0E04 chi(b1)(2P) 0
100555 1.02686E+01 +5.0E04 5.0E04 chi(b2)(2P) 0
200553 1.03552E+01 +5.0E04 5.0E04 2.03E05 +1.9E06 1.9E06 Upsilon(3S) 0
300553 1.05794E+01 +1.2E03 1.2E03 2.05E02 +2.5E03 2.5E03 Upsilon(4S) 0
9000553 1.08852E+01 +2.6E03 1.6E03 3.7E02 +4.0E03 4.0E03 Upsilon(10860) 0
9010553 1.1000E+01 +4.0E03 4.0E03 2.4E02 +8.0E03 6.0E03 Upsilon(11020) 0
2212 9.38272081E01 +6.0E09 6.0E09 0.E+00 +0.0E+00 0.0E+00 p +
2112 9.39565413E01 +6.0E09 6.0E09 7.485E28 +5.0E31 5.0E31 n 0
12112 12212 1.440E+00 +3.0E02 3.0E02 3.5E01 +1.0E01 1.0E01 N(1440) 0,+
1214 2124 1.515E+00 +5.0E03 5.0E03 1.10E01 +1.0E02 1.0E02 N(1520) 0,+
22112 22212 1.530E+00 +1.5E02 1.5E02 1.50E01 +2.5E02 2.5E02 N(1535) 0,+
32112 32212 1.650E+00 +1.5E02 1.5E02 1.25E01 +2.5E02 2.5E02 N(1650) 0,+
2116 2216 1.675E+00 +5.0E03 1.0E02 1.45E01 +1.5E02 1.5E02 N(1675) 0,+
12116 12216 1.685E+00 +5.0E03 5.0E03 1.20E01 +1.0E02 5.0E03 N(1680) 0,+
21214 22124 1.72E+00 +8.0E02 7.0E02 2.0E01 +1.0E01 1.0E01 N(1700) 0,+
42112 42212 1.710E+00 +3.0E02 3.0E02 1.4E01 +6.0E02 6.0E02 N(1710) 0,+
31214 32124 1.720E+00 +3.0E02 4.0E02 2.5E01 +1.5E01 1.0E01 N(1720) 0,+
1218 2128 2.18E+00 +4.0E02 4.0E02 4.0E01 +1.0E01 1.0E01 N(2190) 0,+
1114 2114 2214 2224 1.2320E+00 +2.0E03 2.0E03 1.170E01 +3.0E03 3.0E03 Delta(1232) ,0,+,++
31114 32114 32214 32224 1.57E+00 +7.0E02 7.0E02 2.5E01 +5.0E02 5.0E02 Delta(1600) ,0,+,++
1112 1212 2122 2222 1.610E+00 +2.0E02 2.0E02 1.30E01 +2.0E02 2.0E02 Delta(1620) ,0,+,++
11114 12114 12214 12224 1.710E+00 +2.0E02 2.0E02 3.0E01 +8.0E02 8.0E02 Delta(1700) ,0,+,++
11112 11212 12122 12222 1.860E+00 +6.0E02 2.0E02 2.5E01 +7.0E02 7.0E02 Delta(1900) ,0,+,++
1116 1216 2126 2226 1.880E+00 +3.0E02 2.5E02 3.3E01 +7.0E02 6.0E02 Delta(1905) ,0,+,++
21112 21212 22122 22222 1.90E+00 +5.0E02 5.0E02 3.0E01 +1.0E01 1.0E01 Delta(1910) ,0,+,++
21114 22114 22214 22224 1.92E+00 +5.0E02 5.0E02 3.0E01 +6.0E02 6.0E02 Delta(1920) ,0,+,++
11116 11216 12126 12226 1.95E+00 +5.0E02 5.0E02 3.0E01 +1.0E01 1.0E01 Delta(1930) ,0,+,++
1118 2118 2218 2228 1.930E+00 +2.0E02 1.5E02 2.8E01 +5.0E02 5.0E02 Delta(1950) ,0,+,++
3122 1.115683E+00 +6.0E06 6.0E06 2.501E15 +1.9E17 1.9E17 Lambda 0
13122 1.4051E+00 +1.3E03 1.0E03 5.05E02 +2.0E03 2.0E03 Lambda(1405) 0
3124 1.5190E+00 +1.0E03 1.0E03 1.60E02 +1.0E03 1.0E03 Lambda(1520) 0
23122 1.600E+00 +3.0E02 3.0E02 2.0E01 +5.0E02 5.0E02 Lambda(1600) 0
33122 1.674E+00 +4.0E03 4.0E03 3.0E02 +5.0E03 5.0E03 Lambda(1670) 0
13124 1.690E+00 +5.0E03 5.0E03 7.0E02 +1.0E02 1.0E02 Lambda(1690) 0
43122 1.80E+00 +5.0E02 5.0E02 2.0E01 +5.0E02 5.0E02 Lambda(1800) 0
53122 1.79E+00 +5.0E02 5.0E02 1.1E01 +6.0E02 6.0E02 Lambda(1810) 0
3126 1.820E+00 +5.0E03 5.0E03 8.0E02 +1.0E02 1.0E02 Lambda(1820) 0
13126 1.825E+00 +5.0E03 5.0E03 9.0E02 +3.0E02 3.0E02 Lambda(1830) 0
23124 1.890E+00 +2.0E02 2.0E02 1.2E01 +4.0E02 4.0E02 Lambda(1890) 0
3128 2.100E+00 +1.0E02 1.0E02 2.0E01 +5.0E02 1.0E01 Lambda(2100) 0
23126 2.09E+00 +4.0E02 4.0E02 2.5E01 +5.0E02 5.0E02 Lambda(2110) 0
3222 1.18937E+00 +7.0E05 7.0E05 8.209E15 +2.7E17 2.7E17 Sigma +
3212 1.192642E+00 +2.4E05 2.4E05 8.9E06 +9.0E07 8.0E07 Sigma 0
3112 1.197449E+00 +3.0E05 3.0E05 4.450E15 +3.2E17 3.2E17 Sigma 
3114 1.3872E+00 +5.0E04 5.0E04 3.94E02 +2.1E03 2.1E03 Sigma(1385) 
3214 1.3837E+00 +1.0E03 1.0E03 3.6E02 +5.0E03 5.0E03 Sigma(1385) 0
3224 1.38280E+00 +3.5E04 3.5E04 3.60E02 +7.0E04 7.0E04 Sigma(1385) +
13112 13212 13222 1.660E+00 +2.0E02 2.0E02 2.0E01 +1.0E01 1.0E01 Sigma(1660) ,0,+
13114 13214 13224 1.675E+00 +1.0E02 1.0E02 7.0E02 +3.0E02 3.0E02 Sigma(1670) ,0,+
23112 23212 23222 1.75E+00 +5.0E02 5.0E02 1.5E01 +5.0E02 5.0E02 Sigma(1750) ,0,+
3116 3216 3226 1.775E+00 +5.0E03 5.0E03 1.20E01 +1.5E02 1.5E02 Sigma(1775) ,0,+
23114 23214 23224 1.91E+00 +4.0E02 4.0E02 2.2E01 +8.0E02 7.0E02 Sigma(1910) ,0,+
13116 13216 13226 1.915E+00 +2.0E02 1.5E02 1.2E01 +4.0E02 4.0E02 Sigma(1915) ,0,+
3118 3218 3228 2.030E+00 +1.0E02 5.0E03 1.80E01 +2.0E02 3.0E02 Sigma(2030) ,0,+
3322 1.31486E+00 +2.0E04 2.0E04 2.27E15 +7.0E17 7.0E17 Xi 0
3312 1.32171E+00 +7.0E05 7.0E05 4.02E15 +4.0E17 4.0E17 Xi 
3314 1.5350E+00 +6.0E04 6.0E04 9.9E03 +1.7E03 1.9E03 Xi(1530) 
3324 1.53180E+00 +3.2E04 3.2E04 9.1E03 +5.0E04 5.0E04 Xi(1530) 0
203312 203322 1.690E+00 +1.0E02 1.0E02 Xi(1690) ,0
13314 13324 1.823E+00 +5.0E03 5.0E03 2.4E02 +1.5E02 1.0E02 Xi(1820) ,0
103316 103326 1.950E+00 +1.5E02 1.5E02 6.0E02 +2.0E02 2.0E02 Xi(1950) ,0
203316 203326 2.025E+00 +5.0E03 5.0E03 2.0E02 +1.5E02 5.0E03 Xi(2030) ,0
3334 1.67245E+00 +2.9E04 2.9E04 8.02E15 +1.1E16 1.1E16 Omega 
203338 2.252E+00 +9.0E03 9.0E03 5.5E02 +1.8E02 1.8E02 Omega(2250) 
4122 2.28646E+00 +1.4E04 1.4E04 3.25E12 +5.0E14 5.0E14 Lambda(c) +
14122 2.59225E+00 +2.8E04 2.8E04 2.6E03 +6.0E04 6.0E04 Lambda(c)(2595) +
104122 2.62811E+00 +1.9E04 1.9E04 Lambda(c)(2625) +
204126 2.88163E+00 +2.4E04 2.4E04 5.6E03 +8.0E04 6.0E04 Lambda(c)(2880) +
4112 2.45375E+00 +1.4E04 1.4E04 1.83E03 +1.1E04 1.9E04 Sigma(c)(2455) 0
4212 2.4529E+00 +4.0E04 4.0E04 Sigma(c)(2455) +
4222 2.45397E+00 +1.4E04 1.4E04 1.89E03 +9.0E05 1.8E04 Sigma(c)(2455) ++
4114 2.51848E+00 +2.0E04 2.0E04 1.53E02 +4.0E04 5.0E04 Sigma(c)(2520) 0
4214 2.5175E+00 +2.3E03 2.3E03 Sigma(c)(2520) +
4224 2.51841E+00 +2.1E04 1.9E04 1.478E02 +3.0E04 4.0E04 Sigma(c)(2520) ++
4232 2.46771E+00 +2.3E04 2.3E04 1.444E12 +1.7E14 1.7E14 Xi(c) +
4132 2.47044E+00 +2.8E04 2.8E04 4.31E12 +1.8E13 1.8E13 Xi(c) 0
4322 2.5782E+00 +5.0E04 5.0E04 Xi(c)' +
4312 2.5787E+00 +5.0E04 5.0E04 Xi(c)' 0
4314 2.64616E+00 +2.5E04 2.5E04 2.35E03 +2.2E04 2.2E04 Xi(c)(2645) 0
4324 2.64510E+00 +3.0E04 3.0E04 2.14E03 +1.9E04 1.9E04 Xi(c)(2645) +
104314 2.7939E+00 +5.0E04 5.0E04 1.00E02 +1.1E03 1.1E03 Xi(c)(2790) 0
104324 2.7919E+00 +5.0E04 5.0E04 8.9E03 +1.0E03 1.0E03 Xi(c)(2790) +
104312 2.81979E+00 +3.0E04 3.0E04 2.54E03 +2.5E04 2.5E04 Xi(c)(2815) 0
104322 2.81651E+00 +2.5E04 2.5E04 2.43E03 +2.6E04 2.6E04 Xi(c)(2815) +
4332 2.6952E+00 +1.7E03 1.7E03 2.46E12 +2.6E13 2.2E13 Omega(c) 0
4334 2.7659E+00 +2.0E03 2.0E03 Omega(c)(2770) 0
5122 5.61960E+00 +1.7E04 1.7E04 4.475E13 +2.7E15 2.7E15 Lambda(b) 0
5112 5.81564E+00 +2.7E04 2.7E04 5.3E03 +5.0E04 5.0E04 Sigma(b) 
5222 5.81056E+00 +2.5E04 2.5E04 5.0E03 +5.0E04 5.0E04 Sigma(b) +
5114 5.83474E+00 +3.0E04 3.0E04 1.04E02 +8.0E04 8.0E04 Sigma(b)* 
5224 5.83032E+00 +2.7E04 2.7E04 9.4E03 +5.0E04 5.0E04 Sigma(b)* +
5132 5.7970E+00 +6.0E04 6.0E04 4.19E13 +1.1E14 1.1E14 Xi(b) 
5232 5.7919E+00 +5.0E04 5.0E04 4.45E13 +9.0E15 9.0E15 Xi(b) 0
5332 6.0461E+00 +1.7E03 1.7E03 4.0E13 +5.0E14 4.0E14 Omega(b) 
"""
p.split("\n")[3].split()
m=dict()
for line in p.split("\n")[1:1]:
d=line[32:].split()
m[d[2]+d[1]]=float(d[0])
total=0
class limlist(list):
def append(self,e):
global total
total = total + 1
if e[0]/0.6666666666666666 < 0.015:
return super().append(e)
result=limlist()
for triplet in permutations(m,3):
a,b,c = triplet
a,b,c = m[a],m[b],m[c]
if b < c:
try:
k=((a+b+c)/(sqrt(a)+sqrt(b)+sqrt(c))**2)
result.append([abs(k2/3),triplet,k,"++"]) #use Python3 for floats!
except:
print(triplet)
for doublet in combinations(m,2):
b,c = doublet
a,b,c = 0,m[b],m[c]
try:
k=((a+b+c)/(sqrt(a)+sqrt(b)+sqrt(c))**2)
result.append([abs(k2.00/3),doublet,k,"0++"])
except:
print(doublet)
for triplet in combinations(m,3):
a,b,c = triplet
a,b,c = m[a],m[b],m[c]
k=((a+b+c)/(sqrt(a)+sqrt(b)+sqrt(c))**2)
result.append([abs(k2/3),triplet,k,"+++"]) #use Python3 for floats!
for doublet in permutations(m,2):
b,c = doublet
a,b,c = 0,m[b],m[c]
try:
if b < c:
k=((a+b+c)/(sqrt(a)sqrt(b)+sqrt(c))**2)
result.append([abs(k2.00/3),doublet,k,"0+"])
except:
print(doublet)
print(len(result),total, len(result)/total)
result.sort()
for x in result:
print(f'{"".join(x[1]):<40}',"\t",x[2],x[3])
Z0phi(1020)0K*(1680)0,+ 0.6666643553495888 +++
emutau 0.6666605124107325 +++
u+2/3pi0Sigma 0.6663665393367705 ++
d1/3pi+p+ 0.6670577228854672 ++
W+eta(1475)0p+ 0.6661180018623803 +++
eK+B(c)+ 0.6672821110571778 ++
pi+D(s)+B+ 0.6673593324132077 ++
pi+D(s)+B0 0.6673602750393526 ++
pi0D0 0.6658883844469711 0++
Z0D0n0 0.6655131325199812 +++
pi+D(s)+ 0.6678662052942218 0++
H0pi+B(c)+ 0.665088269066683 +++
u+2/3s1/3D+ 0.6650850283126151 +++
Z0eB(c)+ 0.6682520792016591 +++
Z0taun0 0.668553309238286 +++
s1/3c+2/3 0.6647669161994931 0++
pi0D(s)+B0 0.6644668556345085 ++
pi0D(s)+B+ 0.6644658496344038 ++
t+2/3B+p+ 0.6644561729935445 +++
t+2/3B0p+ 0.6644518594228188 +++
c+2/3b1/3t+2/3 0.6693241713057847 +++
pi+D(s)*+B(c)+ 0.6635394131136364 ++
pi0D+B+ 0.6698242106935278 ++
pi0D0B0 0.6700986560032072 ++
u+2/3s1/3D(s)+ 0.6704131350152354 +++
pi0D0B(s)0 0.6704160867174216 ++
pi+D+ 0.6629167285356727 0++
s1/3c+2/3b1/3 0.6745970625597004 ++
pi+D+B+ 0.6727737789483542 ++
It would help me if you gave the quark composition, with superpositions, of the mesons that the Koide equation relates. Of course the problem with looking for coincidences among a list of states without a restriction of this sort is that the statistics have to be adjusted for the fact that any list of random numbers will have coincidences and the longer the list the more and better coincidences.
e  mu  tau  0.6666605  +++ 
pi+  D(s)+  B+  0.6673593  ++ 
pi0  D0  0.6658883  0++  
pi+  D(s)+  0.6678662  0++ 
c  b  t  0.669324  +++ 
s  c  b  0.674597  ++ 
s  c  0.664767  0++ 
And I do not forget your mixed leptonquark tuple for the first generation.To get anything approaching meaningful matches for quarks you need to use an tbcsud array alternating up type and down type to fit the waterfall of decays. And, to make it really fit well, I think you need to think of a primary set of transitions up and down in mass, and then adjust for other possible transitions.
e=dict()
for line in p.split("\n")[1:1]:
d=line[32:].split()
e[d[2]+d[1]]=[float(d[1]),float(d[2])]
import numpy as np
def rmass(mass,errors):
scale =  errors[1]/errors[0]
base = np.random.normal(mass, errors[0], 1000)
if scale==1:
calc = base
else:
calc = np.where(base < mass, base*scale, base)
return np.where(calc < 0, 0, calc)
def rkoide(triplet,signs):
if signs[0]=="0":
a=0
mb,mc = map(m.get,triplet)
eb,ec = map(e.get,triplet)
b,c = rmass(mb,eb), rmass(mc,ec)
sign = 0
else:
ma,mb,mc = map(m.get,triplet)
ea,eb,ec = map(e.get,triplet)
a,b,c = rmass(ma,ea), rmass(mb,eb), rmass(mc,ec)
sign = +1 if signs[0]=="+" else 1
koide=(a+b+c)/np.square(sign*np.sqrt(a)+np.sqrt(b)+np.sqrt(c))
return np.mean(koide), np.std(koide ) #, np.std(koide, ddof=1)
%%time
for x in result:
k,std=rkoide(x[1],x[3])
x[0]=max(abs(k+std2/3),abs(max(kstd,0)2/3))
#print(f'{"".join(x[1]):<40}',"\t{:.8f} + {:.8f}".format(abs(k2/3),std))
I like the idea of having a hypothesis testing comparison of Koide predicted values v. other theories advanced in the literature, or one could do Chisquares for all of the hypotheses as a nonparametric statistic.The basic objective is to make a graph where the significance of the relation is obviously above chance. Another way of doing that might be to make a graph with modified particle masses and show that the graph with the real particle masses is a lot better (but again, with this you have to avoid the selection bias so you need to include all particle masses or something like that). What I've done is to try automatic algorithms on sets of 6 masses for excitations of heavy mesons with random numbers instead of actual meson masses. I got some good data but not enough to really bother about.
we observe that the top mass series attains its smallest term at the eighth order in perturbation theory, far beyond the fourloop order currently known. On the other hand, the bottom series reaches its minimal term at this order, while the charm series starts to diverge from the twoloop order, which renders the charm pole mass of limited use for phenomenology. From a pragmatic point of view, the minimal term represents the ultimate accuracy beyond which the purely perturbative use of the pole quark mass ceases to be meaningful.
I review the structure of the leading infrared renormalon divergence of the relation between the pole mass and the MS⎯⎯⎯⎯⎯⎯⎯⎯⎯ mass of a heavy quark, with applications to the top, bottom and charm quark. That the pole quark mass definition must be abandoned in precision computations is a wellknown consequence of the rapidly diverging series. The definitions and physics motivations of several leading renormalonfree, shortdistance mass definitions suitable for processes involving nearly onshell heavy quarks are discussed.
The citation of this paper is particular provocative. https://link.springer.com/article/10.1140/epjc/s1005201639903 the abstract says:More divulgation. Now Ethan in Forbes blog https://www.forbes.com/sites/starts...ngbeyondthestandardmodel/?sh=134a79273ac0
Two empirical formulas for the lepton and quark masses (i.e. Kartavtsev’s extended Koide formulas), 𝐾𝑙=(∑𝑙𝑚𝑙)/(∑𝑙𝑚𝑙‾‾‾√)2=2/3Kl=(∑lml)/(∑lml)2=2/3 and 𝐾𝑞=(∑𝑞𝑚𝑞)/(∑𝑞𝑚𝑞‾‾‾√)2=2/3Kq=(∑qmq)/(∑qmq)2=2/3, are explored in this paper. For the lepton sector, we show that 𝐾𝑙=2/3Kl=2/3, only if the uncertainty of the tauon mass is relaxed to about 2𝜎2σ confidence level, and the neutrino masses can consequently be extracted with the current experimental data. For the quark sector, the extended Koide formula should only be applied to the running quark masses, and 𝐾𝑞Kq is found to be rather insensitive to the renormalization effects in a large range of energy scales from GeV to 10121012 GeV. We find that 𝐾𝑞Kq is always slightly larger than 2/3, but the discrepancy is merely about 5 %.