# What is new with Koide sum rules?

Gold Member
I was not aware that Baez did a recent note

https://johncarlosbaez.wordpress.com/2021/04/04/the-koide-formula/

Well, basically telling that he was not aware neither • ohwilleke
CarlB
Homework Helper
My new paper (out for review at Foundations of Physics) has a connection to Koide but it isn't mentioned: https://vixra.org/abs/2105.0146

There are a couple of relations. First, Marni Sheppeard recognized the way I redid the Koide formula as a Discrete Fourier Transform and supposed that what we needed was a Discrete Fourier Transform for a non Abelian (well she said non commutative) symmetry. My new paper is exactly about that.

And the paper generalizes the Dirac / Weyl equation to one with more interesting Pauli spin matrices. But the underlying symmetry is a point group which implies that space is a lattice. For this the paper cites Iwo Bialynicki-Birula's paper on the Weyl / Dirac equation on a lattice: https://arxiv.org/abs/hep-th/9304070

That paper shows that you can get the special-relativity compatible Weyl / Dirac equation on a cubic lattice of quantum cellular automata provided you use a specific formula for updating the cellular automata. That formula is given by his equations (10) thru (12). But if you work out those equations, you'll find that both his paper and my old Koide paper https://arxiv.org/abs/1006.3114 in its equation (11) are about making steps in the +-x, +-y, and +-z directions. Except that while his considers all possible signs, mine is about +x, +y and +z only. The result is that where my paper is dedicated to the (1,1,1) direction where, over the long term, you have equal steps in the +x, +y and +z direction, his paper shows how to generalize it to steps averaging in any direction. And this tells precisely how to interpret the extra group of size three needed in my new paper; just as in my old Koide paper, the group of size 3 corresponds to assigning a factor of exp(2i k pi / 3) to steps in three directions. Such an assignment can be done 3 ways and still have phases cancel over different paths with the same beginning and end.

Carl

• ohwilleke and arivero
Gold Member
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?

• ohwilleke and CarlB
CarlB
Homework Helper
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?
I didn't, but that shouldn't make you hesitate. All avenues need to be explored.

Not sure if I've ever written this down completely. My reasoning, based on the Spin Path Integral paper, is that the three generations form by a transformation on the phases of the path integrals when considered as steps in the +-x, +-y and +-z directions. That is, if we take those parts of the path integral and write them as three separate groups, Px, Py and Pz, then we can assemble the groups to make a propagator for some particle X. Now we multiply these groups individually by 1, w and w* where w^3 = 1 and check if they can still be assembled into a propagator. Recall that a propagator has to satisfy the QQ=Q relation in order to preserve particle identity and quantum numbers.

The underlying idea here is that the three charged leptons all have the same quantum numbers and therefore there are no superselection rules that forbid their mixing by superposition. And in fact, by the weak force, the charged leptons are indeed produced in superposition as indicated by the PMNS matrix (or the CKM matrix for the quarks). Indeed, once you produce a charged lepton by the weak force you can only determine which lepton it is by measuring its mass which may not be that easy. The idea therefore is that the Koide relationship is about quantum states that differ in mass and mass only.

Consider the paths that begin at a point p0 = (x0,y0,z0) and ends at some other point p1 = (x1,y1,z1). Such a path must have a number of +y and -y steps that sums to give y1-y0, that is,
y1-y0 = N(+y) - N(-y)
where "N" is the number of steps in a path that happens to go from p0 to p1. Changing to a new path with one extra +y and one extra -y leaves y1-y0 unchanged so it contributes to the same sum in the path integral sum but the phase rule will be unchanged by such a path change. Uh, let me explain that better. The idea is that whatever change we make to the +x legs we make the negative of that change to the -x legs and same for y and z legs. And the phase changes are cubed roots of unity.

The effect of the complex phases 1, w and w* is to change the phase of a path integral only for its +-y and +-z parts as the +-x parts take the 1. Consider the +-y parts. Such a path will get a phase of 2 pi/3 (N(+y) - N(-y)) = 2pi/3(x1-x0) so this does not depend on the path. Similar for the +-z steps. In particular, if the path happens to have y1-y0 a multiple of 3, and also z1-z0 a multiple of 3, the phase change causes no change in the path integrals and that part of the propagator is unchanged by the phase change. From this you can see that for any particular pair of beginning and ending points p0 and p1, the phase change will only have an effect of multiplying those paths by either 1, w or w* and therefore, the paths themselves will do whatever interference they would have done without the phase change only the final phase is changed. And from that you can see that the phase change preserves the path integral in that if the paths make a rational propagator (in the sense of unitarity and preserving whatever the particle identity is) before the phase change it also is rational afterwards. Only thing that has changed are the relative phases at neighboring points.

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, pi).

Unfortunately, the lower mass mesons are a bit of a mess because their masses are not at all sharp. The lack of sharp masses means that two different mesons (of the same quantum numbers) can often be close enough together that it is impossible to distinguish them (as they have the same quantum numbers and their masses overlap). Consequently, those labels like pi with mass 1300(100) cannot be trusted to give you a list of all the states, some are undoubtedly shared. And the unsharp masses give unsatisfying coincidences.

Where I would try the Koide rules first is for mesons with sharply defined masses. Conveniently, these will also give sharper equalities. These occur at the higher mass mesons. For example, the Upsilon[1s], Upsilon[2s], Upsilon[3s], Upsilon[4s], Upsilon, Upsilon and Upsilon are b b-bar mesons all with the same quantum numbers, i.e. the same I^G(J^PC) = 0-(1--). Their masses are quite accurately known, i.e. the Upsilon[1s] mass is 9460.30(.26) and can be included easily in Koide relationships. My guess is they occur in groups of 3+3 and that one of the Upsilons is misidentified, with one of the triplets corresponding to the charged leptons and the other to the neutrinos. In my papers, the difference between two pairs as in an up quark part and the corresponding down quark part (ie part = left handed etc), is one of a sign change. So they are closely related and maybe the corresponding change for mesons does not correspond to an obvious quantum number so they come in groups of 6. Or maybe they really can be excited so the 1s, 2s, 3s and 4s are parts of four different Koide triplets. I don't know.

If I were continuing to pursue this (and if I had more mes I would do just that), I would try to find meta relations between different sets of Koide triplets by looking for coincidences in the sharp high mass mesons and then see if they can organize the low mass mesons, that is, tell us which low mass mesons are superpositions of different mesons.

Carl

Gold Member
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, pi).

Yeah but if we agree that e mu and tau have the same quantum numbers, so happens with pi, D, B... they differ by flavour charges. So I got surprised you only went for excited states. At some point I checked for ground states and they are very Koidistic: with the surprising point that neutral mesons did a better Koide fit than charged mesons... but well, symmetry breaking everywhere.

• ohwilleke
Gold Member
Let me share some python3 code with only fundamental states of mesons (whole table can be copy pasted of https://pdg.lbl.gov/2021/html/computer_read.html if you want to do more general)

Python:
#!/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                          1.3957039E-01     +1.8E-07 -1.8E-07  2.5284E-17        +5.0E-21 -5.0E-21 pi                  +
111                          1.349768E-01      +5.0E-07 -5.0E-07  7.81E-09          +1.2E-10 -1.2E-10 pi                  0
221                          5.47862E-01       +1.7E-05 -1.7E-05  1.31E-06          +5.0E-08 -5.0E-08 eta                 0
331                          9.5778E-01        +6.0E-05 -6.0E-05  1.88E-04          +6.0E-06 -6.0E-06 eta'(958)           0
321                          4.93677E-01       +1.6E-05 -1.6E-05  5.317E-17         +9.0E-20 -9.0E-20 K                   +
311                          4.97611E-01       +1.3E-05 -1.3E-05                                      K                   0
411                          1.86966E+00       +5.0E-05 -5.0E-05  6.33E-13          +4.0E-15 -4.0E-15 D                   +
421                          1.86484E+00       +5.0E-05 -5.0E-05  1.605E-12         +6.0E-15 -6.0E-15 D                   0
431                          1.96835E+00       +7.0E-05 -7.0E-05  1.305E-12         +1.0E-14 -1.0E-14 D(s)                +
521                          5.27934E+00       +1.2E-04 -1.2E-04  4.018E-13         +1.0E-15 -1.0E-15 B                   +
511                          5.27965E+00       +1.2E-04 -1.2E-04  4.333E-13         +1.1E-15 -1.1E-15 B                   0
531                          5.36688E+00       +1.4E-04 -1.4E-04  4.342E-13         +1.7E-15 -1.7E-15 B(s)                0
541                          6.27447E+00       +3.2E-04 -3.2E-04  1.291E-12         +2.3E-14 -2.3E-14 B(c)                +
441                          2.9839E+00        +4.0E-04 -4.0E-04  3.20E-02          +7.0E-04 -7.0E-04 eta(c)(1S)          0
443                          3.096900E+00      +6.0E-06 -6.0E-06  9.26E-05          +1.7E-06 -1.7E-06 J/psi(1S)           0
553                          9.46030E+00       +2.6E-04 -2.6E-04  5.40E-05          +1.3E-06 -1.3E-06 Upsilon(1S)         0
200553                          1.03552E+01       +5.0E-04 -5.0E-04  2.03E-05          +1.9E-06 -1.9E-06 Upsilon(3S)         0
"""
p.split("\n").split()
m=dict()
for line in p.split("\n")[2:-1]:
d=line.split()
print(d)
m[d[-2]+d[-1]]=float(d)
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(k-2/3),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(k-2.00/3),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(k-2/3),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(k-2.00/3),doublet,k,"0-+"])
result.sort()
for x in result:
print(f'{"|".join(x):<30}',"\t",x,x)

looking at the output:

Code:
pi0|D(s)+|eta(c)(1S)0               0.6663837057987507 -++
pi0|D+                              0.6661367723217316 0++
pi+|D(s)+|B+                        0.6673593324132077 -++
pi+|D(s)+|B0                        0.6673602750393526 -++
pi0|D0                              0.6658883844469711 0++
pi+|D(s)+|B(s)0                     0.667632095646159 -++
pi0|D(s)+|J/psi(1S)0                0.6654846261441458 -++
pi+|D(s)+                           0.6678662052942218 0++
pi+|pi0|Upsilon(1S)0                0.6682566138237737 +++
pi0|D(s)+|B(c)+                     0.6683538259548514 -++
pi0|D(s)+|B(s)0                     0.6647562935736748 -++
pi0|D(s)+|B0                        0.6644668556345085 -++
pi0|D(s)+|B+                        0.6644658496344038 -++
pi+|D(s)+|J/psi(1S)0                0.6690366259117551 -++
pi0|D+|J/psi(1S)0                   0.6698183238780191 -++
pi0|D+|B+                           0.6698242106935278 -++
pi0|D+|B0                           0.6698253129918151 -++
pi+|D(s)+|eta(c)(1S)0               0.6699885664600504 -++
pi0|D0|J/psi(1S)0                   0.6700415949749136 -++
pi0|D0|B+                           0.670097549041931 -++
pi0|D0|B0                           0.6700986560032072 -++
pi0|D+|B(s)0                        0.6701414512426769 -++
pi0|D0|B(s)0                        0.6704160867174216 -++
pi+|D+                              0.6629167285356727 0++
pi0|D+|eta(c)(1S)0                  0.6706380841323579 -++
pi+|D0                              0.6626683899479252 0++
pi0|D0|eta(c)(1S)0                  0.6708574511027877 -++
pi+|D(s)+|B(c)+                     0.6710682904109853 -++
pi0|D(s)+                           0.6710861088843688 0++
pi+|D+|B+                           0.6727737789483542 -++
pi+|D+|B0                           0.67277481621395 -++
pi+|D0|B+                           0.6730499585060149 -++
pi+|D0|B0                           0.6730510003496702 -++
pi+|D+|B(s)0                        0.6730728758703244 -++
eta0|Upsilon(1S)0|Upsilon(3S)0      0.6602536515686964 -++
pi+|D0|B(s)0                        0.6733503288525908 -++
pi+|D+|J/psi(1S)0                   0.673443593712495 -++
pi+|D0|J/psi(1S)0                   0.6736705861236503 -++
pi0|D+|B(c)+                        0.6739753428121785 -++
pi0|D0|B(c)+                        0.674261376733939 -++
pi+|D+|eta(c)(1S)0                  0.6743176053201272 -++
pi+|D0|eta(c)(1S)0                  0.6745407648041061 -++
K+|B(c)+                            0.6578600967642535 0++
K0|B(c)+                            0.6570972277563282 0++
pi+|D+|B(c)+                        0.6767411895047183 -++
pi+|D0|B(c)+                        0.6770298238881567 -++
...

there are one or two problems: more noticeably, that tuples of charged and neutral seem to fare better. But on the other hand, neutrals, particularly etas, are known to be more mixed and it is not easy to decide which mass values should be use.

Last edited:
• ohwilleke
Gold Member
Addendum: python code to check all the known masses in the pdg listing.

Not considering error bands in the sorting. It could be used to generate some histograms, statistics and, adding the error bands, scores of the most common values of Koide formula and get some insight about the likeliness of getting a coincidence.

Python:
#!/usr/bin/env python
# coding: utf-8
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        +1.2E-02 -1.2E-02  2.08E+00          +4.0E-02 -4.0E-02 W                   +
23                          9.11876E+01       +2.1E-03 -2.1E-03  2.4952E+00        +2.3E-03 -2.3E-03 Z                   0
25                          1.2525E+02        +1.7E-01 -1.7E-01  3.2E-03           +2.8E-03 -2.2E-03 H                   0
11                          5.109989461E-04   +3.1E-12 -3.1E-12  0.E+00            +0.0E+00 -0.0E+00 e                   -
13                          1.056583745E-01   +2.4E-09 -2.4E-09  2.9959837E-19     +3.0E-25 -3.0E-25 mu                  -
15                          1.77686E+00       +1.2E-04 -1.2E-04  2.267E-12         +4.0E-15 -4.0E-15 tau                 -
1                          4.67E-03          +0.5E-03 -0.2E-03                                      d                -1/3
2                          2.16E-03          +0.5E-03 -0.3E-03                                      u                +2/3
3                          9.3E-02           +1.1E-02 -5.0E-03                                      s                -1/3
4                          1.27E+00          +2.0E-02 -2.0E-02                                      c                +2/3
5                          4.180E+00         +3.0E-02 -2.0E-02                                      b                -1/3
6                          1.725E+02         +7.0E-01 -7.0E-01  1.42E+00          +1.9E-01 -1.5E-01 t                +2/3
211                          1.3957039E-01     +1.8E-07 -1.8E-07  2.5284E-17        +5.0E-21 -5.0E-21 pi                  +
111                          1.349768E-01      +5.0E-07 -5.0E-07  7.81E-09          +1.2E-10 -1.2E-10 pi                  0
221                          5.47862E-01       +1.7E-05 -1.7E-05  1.31E-06          +5.0E-08 -5.0E-08 eta                 0
9000221                          6.0E-01           +2.0E-01 -2.0E-01  4.5E-01           +3.5E-01 -3.5E-01 f(0)(500)           0
113     213                  7.7526E-01        +2.3E-04 -2.3E-04  1.491E-01         +8.0E-04 -8.0E-04 rho(770)          0,+
223                          7.8266E-01        +1.3E-04 -1.3E-04  8.68E-03          +1.3E-04 -1.3E-04 omega(782)          0
331                          9.5778E-01        +6.0E-05 -6.0E-05  1.88E-04          +6.0E-06 -6.0E-06 eta'(958)           0
9010221                          9.90E-01          +2.0E-02 -2.0E-02  6.E-02            +5.0E-02 -5.0E-02 f(0)(980)           0
9000111 9000211                  9.80E-01          +2.0E-02 -2.0E-02  7.5E-02           +2.5E-02 -2.5E-02 a(0)(980)         0,+
333                          1.019461E+00      +1.6E-05 -1.6E-05  4.249E-03         +1.3E-05 -1.3E-05 phi(1020)           0
10223                          1.166E+00         +6.0E-03 -6.0E-03  3.75E-01          +3.5E-02 -3.5E-02 h(1)(1170)          0
10113   10213                  1.2295E+00        +3.2E-03 -3.2E-03  1.42E-01          +9.0E-03 -9.0E-03 b(1)(1235)        0,+
20113   20213                  1.23E+00          +4.0E-02 -4.0E-02  4.2E-01           +1.8E-01 -1.8E-01 a(1)(1260)        0,+
225                          1.2755E+00        +8.0E-04 -8.0E-04  1.867E-01         +2.2E-03 -2.5E-03 f(2)(1270)          0
20223                          1.2819E+00        +5.0E-04 -5.0E-04  2.27E-02          +1.1E-03 -1.1E-03 f(1)(1285)          0
100221                          1.294E+00         +4.0E-03 -4.0E-03  5.5E-02           +5.0E-03 -5.0E-03 eta(1295)           0
100111  100211                  1.30E+00          +1.0E-01 -1.0E-01  4.0E-01           +2.0E-01 -2.0E-01 pi(1300)          0,+
115     215                  1.3182E+00        +6.0E-04 -6.0E-04  1.07E-01          +5.0E-03 -5.0E-03 a(2)(1320)        0,+
10221                          1.35E+00          +1.5E-01 -1.5E-01  3.5E-01           +1.5E-01 -1.5E-01 f(0)(1370)          0
9000113 9000213                  1.354E+00         +2.5E-02 -2.5E-02  3.30E-01          +3.5E-02 -3.5E-02 pi(1)(1400)       0,+
9020221                          1.4088E+00        +2.0E-03 -2.0E-03  5.01E-02          +2.6E-03 -2.6E-03 eta(1405)           0
10333                          1.416E+00         +8.0E-03 -8.0E-03  9.0E-02           +1.5E-02 -1.5E-02 h(1)(1415)          0
20333                          1.4263E+00        +9.0E-04 -9.0E-04  5.45E-02          +2.6E-03 -2.6E-03 f(1)(1420)          0
1000223                          1.41E+00          +6.0E-02 -6.0E-02  2.9E-01           +1.9E-01 -1.9E-01 omega(1420)         0
10111   10211                  1.474E+00         +1.9E-02 -1.9E-02  2.65E-01          +1.3E-02 -1.3E-02 a(0)(1450)        0,+
100113  100213                  1.465E+00         +2.5E-02 -2.5E-02  4.0E-01           +6.0E-02 -6.0E-02 rho(1450)         0,+
100331                          1.475E+00         +4.0E-03 -4.0E-03  9.0E-02           +9.0E-03 -9.0E-03 eta(1475)           0
9030221                          1.506E+00         +6.0E-03 -6.0E-03  1.12E-01          +9.0E-03 -9.0E-03 f(0)(1500)          0
335                          1.5174E+00        +2.5E-03 -2.5E-03  8.6E-02           +5.0E-03 -5.0E-03 f(2)'(1525)         0
9010113 9010213                  1.661E+00         +1.5E-02 -1.1E-02  2.4E-01           +5.0E-02 -5.0E-02 pi(1)(1600)       0,+
9020113 9020213                  1.655E+00         +1.6E-02 -1.6E-02  2.5E-01           +4.0E-02 -4.0E-02 a(1)(1640)        0,+
10225                          1.617E+00         +5.0E-03 -5.0E-03  1.81E-01          +1.1E-02 -1.1E-02 eta(2)(1645)        0
30223                          1.670001E+00         +3.0E-02 -3.0E-02  3.15E-01          +3.5E-02 -3.5E-02 omega(1650)         0
227                          1.667E+00         +4.0E-03 -4.0E-03  1.68E-01          +1.0E-02 -1.0E-02 omega(3)(1670)      0
10115   10215                  1.6706E+00        +2.9E-03 -1.2E-03  2.58E-01          +8.0E-03 -9.0E-03 pi(2)(1670)       0,+
100333                          1.680E+00         +2.0E-02 -2.0E-02  1.5E-01           +5.0E-02 -5.0E-02 phi(1680)           0
117     217                  1.6888E+00        +2.1E-03 -2.1E-03  1.61E-01          +1.0E-02 -1.0E-02 rho(3)(1690)      0,+
30113   30213                  1.720E+00         +2.0E-02 -2.0E-02  2.5E-01           +1.0E-01 -1.0E-01 rho(1700)         0,+
9000115 9000215                  1.70E+00          +4.0E-02 -4.0E-02  2.7E-01           +6.0E-02 -6.0E-02 a(2)(1700)        0,+
10331                          1.704E+00         +1.2E-02 -1.2E-02  1.23E-01          +1.8E-02 -1.8E-02 f(0)(1710)          0
9010111 9010211                  1.810E+00         +9.0E-03 -1.1E-02  2.15E-01          +7.0E-03 -8.0E-03 pi(1800)          0,+
337                          1.854E+00         +7.0E-03 -7.0E-03  8.7E-02           +2.8E-02 -2.3E-02 phi(3)(1850)        0
9050225                          1.936E+00         +1.2E-02 -1.2E-02  4.64E-01          +2.4E-02 -2.4E-02 f(2)(1950)          0
119     219                  1.967E+00         +1.6E-02 -1.6E-02  3.24E-01          +1.5E-02 -1.8E-02 a(4)(1970)        0,+
9060225                          2.01E+00          +6.0E-02 -8.0E-02  2.0E-01           +6.0E-02 -6.0E-02 f(2)(2010)          0
229                          2.018E+00         +1.1E-02 -1.1E-02  2.37E-01          +1.8E-02 -1.8E-02 f(4)(2050)          0
9080225                          2.297E+00         +2.8E-02 -2.8E-02  1.5E-01           +4.0E-02 -4.0E-02 f(2)(2300)          0
9090225                          2.35E+00          +5.0E-02 -4.0E-02  3.2E-01           +7.0E-02 -6.0E-02 f(2)(2340)          0
321                          4.93677E-01       +1.6E-05 -1.6E-05  5.317E-17         +9.0E-20 -9.0E-20 K                   +
311                          4.97611E-01       +1.3E-05 -1.3E-05                                      K                   0
9000311 9000321                  8.45E-01          +1.7E-02 -1.7E-02  4.68E-01          +3.0E-02 -3.0E-02 K(0)*(700)        0,+
313                          8.9555E-01        +2.0E-04 -2.0E-04  4.73E-02          +5.0E-04 -5.0E-04 K*(892)             0
323                          8.9167E-01        +2.6E-04 -2.6E-04  5.14E-02          +8.0E-04 -8.0E-04 K*(892)             +
323                          8.955E-01         +8.0E-04 -8.0E-04  4.62E-02          +1.3E-03 -1.3E-03 K*(892)             +
10313   10323                  1.253E+00         +7.0E-03 -7.0E-03  9.0E-02           +2.0E-02 -2.0E-02 K(1)(1270)        0,+
20313   20323                  1.403E+00         +7.0E-03 -7.0E-03  1.74E-01          +1.3E-02 -1.3E-02 K(1)(1400)        0,+
100313  100323                  1.414E+00         +1.5E-02 -1.5E-02  2.32E-01          +2.1E-02 -2.1E-02 K*(1410)          0,+
10311   10321                  1.43E+00          +5.0E-02 -5.0E-02  2.7E-01           +8.0E-02 -8.0E-02 K(0)*(1430)       0,+
315                          1.4324E+00        +1.3E-03 -1.3E-03  1.09E-01          +5.0E-03 -5.0E-03 K(2)*(1430)         0
325                          1.4273E+00        +1.5E-03 -1.5E-03  1.000E-01         +2.1E-03 -2.1E-03 K(2)*(1430)         +
9000313 9000323                  1.67E+00          +5.0E-02 -5.0E-02  1.6E-01           +5.0E-02 -5.0E-02 K(1)(1650)        0,+
30313   30323                  1.718E+00         +1.8E-02 -1.8E-02  3.2E-01           +1.1E-01 -1.1E-01 K*(1680)          0,+
10315   10325                  1.773E+00         +8.0E-03 -8.0E-03  1.86E-01          +1.4E-02 -1.4E-02 K(2)(1770)        0,+
317     327                  1.779E+00         +8.0E-03 -8.0E-03  1.61E-01          +1.7E-02 -1.7E-02 K(3)*(1780)       0,+
20315   20325                  1.819E+00         +1.2E-02 -1.2E-02  2.64E-01          +3.4E-02 -3.4E-02 K(2)(1820)        0,+
9010315 9010325                  1.99E+00          +6.0E-02 -5.0E-02  3.49E-01          +5.0E-02 -3.0E-02 K(2)*(1980)       0,+
319     329                  2.048E+00         +8.0E-03 -9.0E-03  1.99E-01          +2.7E-02 -1.9E-02 K(4)*(2045)       0,+
411                          1.86966E+00       +5.0E-05 -5.0E-05  6.33E-13          +4.0E-15 -4.0E-15 D                   +
421                          1.86484E+00       +5.0E-05 -5.0E-05  1.605E-12         +6.0E-15 -6.0E-15 D                   0
423                          2.00685E+00       +5.0E-05 -5.0E-05                                      D*(2007)            0
413                          2.01026E+00       +5.0E-05 -5.0E-05  8.34E-05          +1.8E-06 -1.8E-06 D*(2010)            +
10421   10411                  2.343E+00         +1.0E-02 -1.0E-02  2.29E-01          +1.6E-02 -1.6E-02 D(0)*(2300)       0,+
10423   10413                  2.4221E+00        +6.0E-04 -6.0E-04  3.13E-02          +1.9E-03 -1.9E-03 D(1)(2420)        0,+
20423                          2.412E+00         +9.0E-03 -9.0E-03  3.14E-01          +2.9E-02 -2.9E-02 D(1)(2430)          0
425     415                  2.4611E+00        +7.0E-04 -8.0E-04  4.73E-02          +8.0E-04 -8.0E-04 D(2)*(2460)       0,+
431                          1.96835E+00       +7.0E-05 -7.0E-05  1.305E-12         +1.0E-14 -1.0E-14 D(s)                +
433                          2.1122E+00        +4.0E-04 -4.0E-04                                      D(s)*               +
10431                          2.3178E+00        +5.0E-04 -5.0E-04                                      D(s0)*(2317)        +
20433                          2.4595E+00        +6.0E-04 -6.0E-04                                      D(s1)(2460)         +
10433                          2.53511E+00       +6.0E-05 -6.0E-05  9.2E-04           +5.0E-05 -5.0E-05 D(s1)(2536)         +
435                          2.5691E+00        +8.0E-04 -8.0E-04  1.69E-02          +7.0E-04 -7.0E-04 D(s2)*(2573)        +
521                          5.27934E+00       +1.2E-04 -1.2E-04  4.018E-13         +1.0E-15 -1.0E-15 B                   +
511                          5.27965E+00       +1.2E-04 -1.2E-04  4.333E-13         +1.1E-15 -1.1E-15 B                   0
513     523                  5.32470E+00       +2.1E-04 -2.1E-04                                      B*                0,+
515                          5.7395E+00        +7.0E-04 -7.0E-04  2.42E-02          +1.7E-03 -1.7E-03 B(2)*(5747)         0
525                          5.7372E+00        +7.0E-04 -7.0E-04  2.0E-02           +5.0E-03 -5.0E-03 B(2)*(5747)         +
531                          5.36688E+00       +1.4E-04 -1.4E-04  4.342E-13         +1.7E-15 -1.7E-15 B(s)                0
533                          5.4154E+00        +1.8E-03 -1.5E-03                                      B(s)*               0
535                          5.83986E+00       +1.2E-04 -1.2E-04  1.49E-03          +2.7E-04 -2.7E-04 B(s2)*(5840)        0
541                          6.27447E+00       +3.2E-04 -3.2E-04  1.291E-12         +2.3E-14 -2.3E-14 B(c)                +
441                          2.9839E+00        +4.0E-04 -4.0E-04  3.20E-02          +7.0E-04 -7.0E-04 eta(c)(1S)          0
443                          3.096900E+00      +6.0E-06 -6.0E-06  9.26E-05          +1.7E-06 -1.7E-06 J/psi(1S)           0
10441                          3.41471E+00       +3.0E-04 -3.0E-04  1.08E-02          +6.0E-04 -6.0E-04 chi(c0)(1P)         0
20443                          3.51067E+00       +5.0E-05 -5.0E-05  8.4E-04           +4.0E-05 -4.0E-05 chi(c1)(1P)         0
10443                          3.52538E+00       +1.1E-04 -1.1E-04  7.E-04            +4.0E-04 -4.0E-04 h(c)(1P)            0
445                          3.55617E+00       +7.0E-05 -7.0E-05  1.97E-03          +9.0E-05 -9.0E-05 chi(c2)(1P)         0
100441                          3.6375E+00        +1.1E-03 -1.1E-03  1.13E-02          +3.2E-03 -2.9E-03 eta(c)(2S)          0
100443                          3.68610E+00       +6.0E-05 -6.0E-05  2.94E-04          +8.0E-06 -8.0E-06 psi(2S)             0
30443                          3.7737E+00        +4.0E-04 -4.0E-04  2.72E-02          +1.0E-03 -1.0E-03 psi(3770)           0
100445                          3.9225E+00        +1.0E-03 -1.0E-03  3.52E-02          +2.2E-03 -2.2E-03 chi(c2)(3930)       0
9000443                          4.0390E+00        +1.0E-03 -1.0E-03  8.0E-02           +1.0E-02 -1.0E-02 psi(4040)           0
9010443                          4.191E+00         +5.0E-03 -5.0E-03  7.0E-02           +1.0E-02 -1.0E-02 psi(4160)           0
9020443                          4.421E+00         +4.0E-03 -4.0E-03  6.2E-02           +2.0E-02 -2.0E-02 psi(4415)           0
553                          9.46030E+00       +2.6E-04 -2.6E-04  5.40E-05          +1.3E-06 -1.3E-06 Upsilon(1S)         0
10551                          9.8594E+00        +5.0E-04 -5.0E-04                                      chi(b0)(1P)         0
20553                          9.8928E+00        +4.0E-04 -4.0E-04                                      chi(b1)(1P)         0
10553                          9.8993E+00        +8.0E-04 -8.0E-04                                      h(b)(1P)            0
555                          9.9122E+00        +4.0E-04 -4.0E-04                                      chi(b2)(1P)         0
100553                          1.002326E+01      +3.1E-04 -3.1E-04  3.20E-05          +2.6E-06 -2.6E-06 Upsilon(2S)         0
20555                          1.01637E+01       +1.4E-03 -1.4E-03                                      Upsilon(2)(1D)      0
110551                          1.02325E+01       +6.0E-04 -6.0E-04                                      chi(b0)(2P)         0
120553                          1.02555E+01       +5.0E-04 -5.0E-04                                      chi(b1)(2P)         0
100555                          1.02686E+01       +5.0E-04 -5.0E-04                                      chi(b2)(2P)         0
200553                          1.03552E+01       +5.0E-04 -5.0E-04  2.03E-05          +1.9E-06 -1.9E-06 Upsilon(3S)         0
300553                          1.05794E+01       +1.2E-03 -1.2E-03  2.05E-02          +2.5E-03 -2.5E-03 Upsilon(4S)         0
9000553                          1.08852E+01       +2.6E-03 -1.6E-03  3.7E-02           +4.0E-03 -4.0E-03 Upsilon(10860)      0
9010553                          1.1000E+01        +4.0E-03 -4.0E-03  2.4E-02           +8.0E-03 -6.0E-03 Upsilon(11020)      0
2212                          9.38272081E-01    +6.0E-09 -6.0E-09  0.E+00            +0.0E+00 -0.0E+00 p                   +
2112                          9.39565413E-01    +6.0E-09 -6.0E-09  7.485E-28         +5.0E-31 -5.0E-31 n                   0
12112   12212                  1.440E+00         +3.0E-02 -3.0E-02  3.5E-01           +1.0E-01 -1.0E-01 N(1440)           0,+
1214    2124                  1.515E+00         +5.0E-03 -5.0E-03  1.10E-01          +1.0E-02 -1.0E-02 N(1520)           0,+
22112   22212                  1.530E+00         +1.5E-02 -1.5E-02  1.50E-01          +2.5E-02 -2.5E-02 N(1535)           0,+
32112   32212                  1.650E+00         +1.5E-02 -1.5E-02  1.25E-01          +2.5E-02 -2.5E-02 N(1650)           0,+
2116    2216                  1.675E+00         +5.0E-03 -1.0E-02  1.45E-01          +1.5E-02 -1.5E-02 N(1675)           0,+
12116   12216                  1.685E+00         +5.0E-03 -5.0E-03  1.20E-01          +1.0E-02 -5.0E-03 N(1680)           0,+
21214   22124                  1.72E+00          +8.0E-02 -7.0E-02  2.0E-01           +1.0E-01 -1.0E-01 N(1700)           0,+
42112   42212                  1.710E+00         +3.0E-02 -3.0E-02  1.4E-01           +6.0E-02 -6.0E-02 N(1710)           0,+
31214   32124                  1.720E+00         +3.0E-02 -4.0E-02  2.5E-01           +1.5E-01 -1.0E-01 N(1720)           0,+
1218    2128                  2.18E+00          +4.0E-02 -4.0E-02  4.0E-01           +1.0E-01 -1.0E-01 N(2190)           0,+
1114    2114    2214    2224  1.2320E+00        +2.0E-03 -2.0E-03  1.170E-01         +3.0E-03 -3.0E-03 Delta(1232)  -,0,+,++
31114   32114   32214   32224  1.57E+00          +7.0E-02 -7.0E-02  2.5E-01           +5.0E-02 -5.0E-02 Delta(1600)  -,0,+,++
1112    1212    2122    2222  1.610E+00         +2.0E-02 -2.0E-02  1.30E-01          +2.0E-02 -2.0E-02 Delta(1620)  -,0,+,++
11114   12114   12214   12224  1.710E+00         +2.0E-02 -2.0E-02  3.0E-01           +8.0E-02 -8.0E-02 Delta(1700)  -,0,+,++
11112   11212   12122   12222  1.860E+00         +6.0E-02 -2.0E-02  2.5E-01           +7.0E-02 -7.0E-02 Delta(1900)  -,0,+,++
1116    1216    2126    2226  1.880E+00         +3.0E-02 -2.5E-02  3.3E-01           +7.0E-02 -6.0E-02 Delta(1905)  -,0,+,++
21112   21212   22122   22222  1.90E+00          +5.0E-02 -5.0E-02  3.0E-01           +1.0E-01 -1.0E-01 Delta(1910)  -,0,+,++
21114   22114   22214   22224  1.92E+00          +5.0E-02 -5.0E-02  3.0E-01           +6.0E-02 -6.0E-02 Delta(1920)  -,0,+,++
11116   11216   12126   12226  1.95E+00          +5.0E-02 -5.0E-02  3.0E-01           +1.0E-01 -1.0E-01 Delta(1930)  -,0,+,++
1118    2118    2218    2228  1.930E+00         +2.0E-02 -1.5E-02  2.8E-01           +5.0E-02 -5.0E-02 Delta(1950)  -,0,+,++
3122                          1.115683E+00      +6.0E-06 -6.0E-06  2.501E-15         +1.9E-17 -1.9E-17 Lambda              0
13122                          1.4051E+00        +1.3E-03 -1.0E-03  5.05E-02          +2.0E-03 -2.0E-03 Lambda(1405)        0
3124                          1.5190E+00        +1.0E-03 -1.0E-03  1.60E-02          +1.0E-03 -1.0E-03 Lambda(1520)        0
23122                          1.600E+00         +3.0E-02 -3.0E-02  2.0E-01           +5.0E-02 -5.0E-02 Lambda(1600)        0
33122                          1.674E+00         +4.0E-03 -4.0E-03  3.0E-02           +5.0E-03 -5.0E-03 Lambda(1670)        0
13124                          1.690E+00         +5.0E-03 -5.0E-03  7.0E-02           +1.0E-02 -1.0E-02 Lambda(1690)        0
43122                          1.80E+00          +5.0E-02 -5.0E-02  2.0E-01           +5.0E-02 -5.0E-02 Lambda(1800)        0
53122                          1.79E+00          +5.0E-02 -5.0E-02  1.1E-01           +6.0E-02 -6.0E-02 Lambda(1810)        0
3126                          1.820E+00         +5.0E-03 -5.0E-03  8.0E-02           +1.0E-02 -1.0E-02 Lambda(1820)        0
13126                          1.825E+00         +5.0E-03 -5.0E-03  9.0E-02           +3.0E-02 -3.0E-02 Lambda(1830)        0
23124                          1.890E+00         +2.0E-02 -2.0E-02  1.2E-01           +4.0E-02 -4.0E-02 Lambda(1890)        0
3128                          2.100E+00         +1.0E-02 -1.0E-02  2.0E-01           +5.0E-02 -1.0E-01 Lambda(2100)        0
23126                          2.09E+00          +4.0E-02 -4.0E-02  2.5E-01           +5.0E-02 -5.0E-02 Lambda(2110)        0
3222                          1.18937E+00       +7.0E-05 -7.0E-05  8.209E-15         +2.7E-17 -2.7E-17 Sigma               +
3212                          1.192642E+00      +2.4E-05 -2.4E-05  8.9E-06           +9.0E-07 -8.0E-07 Sigma               0
3112                          1.197449E+00      +3.0E-05 -3.0E-05  4.450E-15         +3.2E-17 -3.2E-17 Sigma               -
3114                          1.3872E+00        +5.0E-04 -5.0E-04  3.94E-02          +2.1E-03 -2.1E-03 Sigma(1385)         -
3214                          1.3837E+00        +1.0E-03 -1.0E-03  3.6E-02           +5.0E-03 -5.0E-03 Sigma(1385)         0
3224                          1.38280E+00       +3.5E-04 -3.5E-04  3.60E-02          +7.0E-04 -7.0E-04 Sigma(1385)         +
13112   13212   13222          1.660E+00         +2.0E-02 -2.0E-02  2.0E-01           +1.0E-01 -1.0E-01 Sigma(1660)     -,0,+
13114   13214   13224          1.675E+00         +1.0E-02 -1.0E-02  7.0E-02           +3.0E-02 -3.0E-02 Sigma(1670)     -,0,+
23112   23212   23222          1.75E+00          +5.0E-02 -5.0E-02  1.5E-01           +5.0E-02 -5.0E-02 Sigma(1750)     -,0,+
3116    3216    3226          1.775E+00         +5.0E-03 -5.0E-03  1.20E-01          +1.5E-02 -1.5E-02 Sigma(1775)     -,0,+
23114   23214   23224          1.91E+00          +4.0E-02 -4.0E-02  2.2E-01           +8.0E-02 -7.0E-02 Sigma(1910)     -,0,+
13116   13216   13226          1.915E+00         +2.0E-02 -1.5E-02  1.2E-01           +4.0E-02 -4.0E-02 Sigma(1915)     -,0,+
3118    3218    3228          2.030E+00         +1.0E-02 -5.0E-03  1.80E-01          +2.0E-02 -3.0E-02 Sigma(2030)     -,0,+
3322                          1.31486E+00       +2.0E-04 -2.0E-04  2.27E-15          +7.0E-17 -7.0E-17 Xi                  0
3312                          1.32171E+00       +7.0E-05 -7.0E-05  4.02E-15          +4.0E-17 -4.0E-17 Xi                  -
3314                          1.5350E+00        +6.0E-04 -6.0E-04  9.9E-03           +1.7E-03 -1.9E-03 Xi(1530)            -
3324                          1.53180E+00       +3.2E-04 -3.2E-04  9.1E-03           +5.0E-04 -5.0E-04 Xi(1530)            0
203312  203322                  1.690E+00         +1.0E-02 -1.0E-02                                      Xi(1690)          -,0
13314   13324                  1.823E+00         +5.0E-03 -5.0E-03  2.4E-02           +1.5E-02 -1.0E-02 Xi(1820)          -,0
103316  103326                  1.950E+00         +1.5E-02 -1.5E-02  6.0E-02           +2.0E-02 -2.0E-02 Xi(1950)          -,0
203316  203326                  2.025E+00         +5.0E-03 -5.0E-03  2.0E-02           +1.5E-02 -5.0E-03 Xi(2030)          -,0
3334                          1.67245E+00       +2.9E-04 -2.9E-04  8.02E-15          +1.1E-16 -1.1E-16 Omega               -
203338                          2.252E+00         +9.0E-03 -9.0E-03  5.5E-02           +1.8E-02 -1.8E-02 Omega(2250)         -
4122                          2.28646E+00       +1.4E-04 -1.4E-04  3.25E-12          +5.0E-14 -5.0E-14 Lambda(c)           +
14122                          2.59225E+00       +2.8E-04 -2.8E-04  2.6E-03           +6.0E-04 -6.0E-04 Lambda(c)(2595)     +
104122                          2.62811E+00       +1.9E-04 -1.9E-04                                      Lambda(c)(2625)     +
204126                          2.88163E+00       +2.4E-04 -2.4E-04  5.6E-03           +8.0E-04 -6.0E-04 Lambda(c)(2880)     +
4112                          2.45375E+00       +1.4E-04 -1.4E-04  1.83E-03          +1.1E-04 -1.9E-04 Sigma(c)(2455)      0
4212                          2.4529E+00        +4.0E-04 -4.0E-04                                      Sigma(c)(2455)      +
4222                          2.45397E+00       +1.4E-04 -1.4E-04  1.89E-03          +9.0E-05 -1.8E-04 Sigma(c)(2455)     ++
4114                          2.51848E+00       +2.0E-04 -2.0E-04  1.53E-02          +4.0E-04 -5.0E-04 Sigma(c)(2520)      0
4214                          2.5175E+00        +2.3E-03 -2.3E-03                                      Sigma(c)(2520)      +
4224                          2.51841E+00       +2.1E-04 -1.9E-04  1.478E-02         +3.0E-04 -4.0E-04 Sigma(c)(2520)     ++
4232                          2.46771E+00       +2.3E-04 -2.3E-04  1.444E-12         +1.7E-14 -1.7E-14 Xi(c)               +
4132                          2.47044E+00       +2.8E-04 -2.8E-04  4.31E-12          +1.8E-13 -1.8E-13 Xi(c)               0
4322                          2.5782E+00        +5.0E-04 -5.0E-04                                      Xi(c)'              +
4312                          2.5787E+00        +5.0E-04 -5.0E-04                                      Xi(c)'              0
4314                          2.64616E+00       +2.5E-04 -2.5E-04  2.35E-03          +2.2E-04 -2.2E-04 Xi(c)(2645)         0
4324                          2.64510E+00       +3.0E-04 -3.0E-04  2.14E-03          +1.9E-04 -1.9E-04 Xi(c)(2645)         +
104314                          2.7939E+00        +5.0E-04 -5.0E-04  1.00E-02          +1.1E-03 -1.1E-03 Xi(c)(2790)         0
104324                          2.7919E+00        +5.0E-04 -5.0E-04  8.9E-03           +1.0E-03 -1.0E-03 Xi(c)(2790)         +
104312                          2.81979E+00       +3.0E-04 -3.0E-04  2.54E-03          +2.5E-04 -2.5E-04 Xi(c)(2815)         0
104322                          2.81651E+00       +2.5E-04 -2.5E-04  2.43E-03          +2.6E-04 -2.6E-04 Xi(c)(2815)         +
4332                          2.6952E+00        +1.7E-03 -1.7E-03  2.46E-12          +2.6E-13 -2.2E-13 Omega(c)            0
4334                          2.7659E+00        +2.0E-03 -2.0E-03                                      Omega(c)(2770)      0
5122                          5.61960E+00       +1.7E-04 -1.7E-04  4.475E-13         +2.7E-15 -2.7E-15 Lambda(b)           0
5112                          5.81564E+00       +2.7E-04 -2.7E-04  5.3E-03           +5.0E-04 -5.0E-04 Sigma(b)            -
5222                          5.81056E+00       +2.5E-04 -2.5E-04  5.0E-03           +5.0E-04 -5.0E-04 Sigma(b)            +
5114                          5.83474E+00       +3.0E-04 -3.0E-04  1.04E-02          +8.0E-04 -8.0E-04 Sigma(b)*           -
5224                          5.83032E+00       +2.7E-04 -2.7E-04  9.4E-03           +5.0E-04 -5.0E-04 Sigma(b)*           +
5132                          5.7970E+00        +6.0E-04 -6.0E-04  4.19E-13          +1.1E-14 -1.1E-14 Xi(b)               -
5232                          5.7919E+00        +5.0E-04 -5.0E-04  4.45E-13          +9.0E-15 -9.0E-15 Xi(b)               0
5332                          6.0461E+00        +1.7E-03 -1.7E-03  4.0E-13           +5.0E-14 -4.0E-14 Omega(b)            -
"""
p.split("\n").split()
m=dict()
for line in p.split("\n")[1:-1]:
d=line[32:].split()
m[d[-2]+d[-1]]=float(d)
total=0
class limlist(list):
def append(self,e):
global total
total = total + 1
if e/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(k-2/3),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(k-2.00/3),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(k-2/3),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(k-2.00/3),doublet,k,"0-+"])
except:
print(doublet)

print(len(result),total, len(result)/total)
result.sort()
for x in result:
print(f'{"|".join(x):<40}',"\t",x,x)

Some interesting tuples (and a lot of mad mixes, selected for illustration)

Code:
Z0|phi(1020)0|K*(1680)0,+                     0.6666643553495888 +++
e-|mu-|tau-                                   0.6666605124107325 +++
u+2/3|pi0|Sigma-                              0.6663665393367705 -++
d-1/3|pi+|p+                                  0.6670577228854672 -++
W+|eta(1475)0|p+                              0.6661180018623803 +++
e-|K+|B(c)+                                   0.6672821110571778 -++
pi+|D(s)+|B+                                  0.6673593324132077 -++
pi+|D(s)+|B0                                  0.6673602750393526 -++
pi0|D0                                        0.6658883844469711 0++
Z0|D0|n0                                      0.6655131325199812 +++
pi+|D(s)+                                     0.6678662052942218 0++
H0|pi+|B(c)+                                  0.665088269066683 +++
u+2/3|s-1/3|D+                                0.6650850283126151 +++
Z0|e-|B(c)+                                   0.6682520792016591 +++
Z0|tau-|n0                                    0.668553309238286 +++
s-1/3|c+2/3                                   0.6647669161994931 0++
pi0|D(s)+|B0                                  0.6644668556345085 -++
pi0|D(s)+|B+                                  0.6644658496344038 -++
t+2/3|B+|p+                                   0.6644561729935445 +++
t+2/3|B0|p+                                   0.6644518594228188 +++
c+2/3|b-1/3|t+2/3                             0.6693241713057847 +++
pi+|D(s)*+|B(c)+                              0.6635394131136364 -++
pi0|D+|B+                                     0.6698242106935278 -++
pi0|D0|B0                                     0.6700986560032072 -++
u+2/3|s-1/3|D(s)+                             0.6704131350152354 +++
pi0|D0|B(s)0                                  0.6704160867174216 -++
pi+|D+                                        0.6629167285356727 0++
s-1/3|c+2/3|b-1/3                             0.6745970625597004 -++
pi+|D+|B+                                     0.6727737789483542 -++

Last edited:
• ohwilleke
CarlB
Homework Helper
The Koide formula is a relation among changes in generation. It works for the charged leptons where the triplet is (e,mu,tau), one from each generation. It doesn't work for either of the quark sets (d,s,b) and (u,c,t) so it would be a huge miracle to me if it worked for mesons that differ only in their valence quark content. (That said, for the quark Koide equations see the Piotr ̇Zenczykowski papers i.e. https://arxiv.org/abs/1301.4143 There the coincidence that is applied to the quarks is about the phase 2/9 used for the charged leptons changing into 2/3 and 1/3 that when used for the two quark sets. Accordingly, I'd be more interested in coincidences involving that phase and I don't think anyone has looked for them.)

A triplet of mesons that differ by generation would be a triplet that is different from having d, s or b valence quark(s) or (u,c,t), that is, something like a triplet (d/d, s/s, b/b) if we transform both of the quarks or (d/d, d/s, d/b) if we keep the first constant at d. The whole problem is made more difficult by the fact that the mesons do not have precise quark composition but instead are superpositions. For example, pi0 is partly d/d and partly u/u.

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. This contributes to why the particle people like 5 sigma statistics but that's after the theoretical justification for the coincidence search not before they put every known meson into the hopper of the woodchipper.

• arivero
Gold Member
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.

By the way, this is one of the amazing things of the original Koide Tuple: that even after you put all the random combinations in, it stands out, even without considering error bands of each mass (to consider error bars, one could generate for each tuple one dozen or so of "mass measurements", and consider average and standard deviation when pondering the "quality" of the tuple)

As for quark content, now that I have generated the full list I am a bit amazed that the "Koide tuple" pi+,Ds+,B+ and the asociated "HHW tuple" pi+,Ds+ are better than the neutrals and better than the pi+,D+,B+.

 e- mu- tau- 0.66666 +++ pi+ D(s)+ B+ 0.667359 -++ pi0 D0 0.665888 0++ pi+ D(s)+ 0.667866 0++

So the best quark content seems to be (ud), (cs), (ub). I am not surprised because the main theme in generations is mixing, so getting isolated generations for the quark sector is not so desirable as it is for charged leptons.

On other hand, charged pi,D,B, quark content (ud) (uc) (ub) gives a Koide tuple with ratio 0.6727. Poor, but perhaps tolerable. With neutrals, pi,D,B, quark content (uu+dd), (cd+dc), (bd+db) gives a ratio 0.6701

For reference, the quark tuples, with pdg masses, are:

 c b t 0.669324 +++ s c b 0.674597 -++ s c 0.664767 0++

Worse that the tuples with mesons, but they -except for the top- are calculated, no measured, masses.

Last edited:
• CarlB and ohwilleke
ohwilleke
Gold Member
To get anything approaching meaningful matches for quarks you need to use an t-b-c-s-u-d 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.

• CarlB
Gold Member
To get anything approaching meaningful matches for quarks you need to use an t-b-c-s-u-d 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.
And I do not forget your mixed lepton-quark tuple for the first generation.

But I am not thinking in term of transitions but of broken symmetries.

• CarlB and ohwilleke
CarlB
Homework Helper
As far as proving significance, if you go to the trouble of programming the whole thing, it might be useful to make a chart showing non standard values of the Koide parameter. Then a "hit" would be a value at 2/3 that is significantly higher than the background. From that you ought to be able to get an estimate of the sigma. For a reasonably high sigma it will be a chart where it is totally obvious that 2/3 is a magic number.

This is just an ill-thought out idea. 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.

• ohwilleke and arivero
Gold Member
I ran some extra ideas and... well, the only positive thing is that the charged lepton tuple is always in the first positions.

For reference, let me include here the python algo to produce gaussian error statistics for each tuple. The code generates a sample of 1000 random masses distributed gaussian and then it averages the result:

Code:
e=dict()
for line in p.split("\n")[1:-1]:
d=line[32:].split()
e[d[-2]+d[-1]]=[float(d),float(d)]

import numpy as np
def rmass(mass,errors):
scale = - errors/errors
base  = np.random.normal(mass, errors, 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":
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=="+" 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,x)
x=max(abs(k+std-2/3),abs(max(k-std,0)-2/3))
#print(f'{"|".join(x):<40}',"\t{:.8f} +- {:.8f}".format(abs(k-2/3),std))

• CarlB
ohwilleke
Gold Member
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.
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 Chi-squares for all of the hypotheses as a non-parametric statistic.

• CarlB
CarlB
• 