What is new with Koide sum rules?

[arivero]

Given that a lot of Koide stuff seems related (hat tip to de Vries and Brannen here) to this matrix

\begin{pmatrix}-\frac{2\,\mathrm{sin}\left( t\right) }{\sqrt{6}} & \frac{\mathrm{sin}\left( t\right) }{\sqrt{6}}+\frac{\mathrm{cos}\left( t\right) }{\sqrt{2}} & \frac{\mathrm{sin}\left( t\right) }{\sqrt{6}}-\frac{\mathrm{cos}\left( t\right) }{\sqrt{2}}\cr \frac{2\,\mathrm{cos}\left( t\right) }{\sqrt{6}} & \frac{\mathrm{sin}\left( t\right) }{\sqrt{2}}-\frac{\mathrm{cos}\left( t\right) }{\sqrt{6}} & -\frac{\mathrm{sin}\left( t\right) }{\sqrt{2}}-\frac{\mathrm{cos}\left( t\right) }{\sqrt{6}}\cr \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}} & \frac{1}{\sqrt{3}}\end{pmatrix}

I have setup a wxMaxima notebook to play with it. Not that I like Maxima, I used it in a VAX and it was already superseded by REDUCE when Mathematica come. But it comes with Ubuntu and has a graphical interface, which Reduce has not.

 

[mitchell porter]

Koide has a model of leptons with an S4 symmetry embedded in a flavor SU(3).

 

[mitchell porter]

Since this is a Koide thread we have to mention Zenczykowski's latest, though it is about "family triples", and not what I call the "sequential triples" of the waterfall... He's still building on the generalization of the e-mu-tau 2/9 parameter to u-c-t and d-s-b; he proposes that another parameter, which is just "1" for e-mu-tau, is also "1" for the quarks if you use Goffinet's concept of "pseudo-mass". If that's true it's a breakthrough, as well as a headache for the waterfall, because aren't we getting overloaded with too many relationships at once?

He mentions the usual problem, that these relations work best for low-energy masses. We've previously discussed Sumino's efforts to have family gauge bosons cancel out certain QED corrections, so that Koide's relation may be exact; but I was always curious about whether there might be some dual description of physics, in which, rather than thinking of the UV as fundamental, you thought of the physics as "IR + new degrees of freedom at a series of higher energies" - the idea being that the cause of Koide relations might be more transparent in this hypothetical "infrared first" formulation.

Well, I wonder if this paper by Davide Gaiotto (from January 2012) might be relevant: "Domain Walls for Two-Dimensional Renormalization Group Flows". "Renormalization Group domain walls are natural conformal interfaces between two CFTs related by an RG flow. The RG domain wall gives an exact relation between the operators in the UV and IR CFTs." It seems a tiny step towards what I had in mind.

 

[arivero]

If that's true it's a breakthrough, as well as a headache for the waterfall,

The waterfall could happily miss the last triple, d-u-s, in exchange by one of the "standard" ones, but d-u-s does a better prediction of the down mass that d-s-b.

A motivation to follow this track could be, put all the quarks in the faces of a cube, such that all the equations of the waterfall are the faces that meet in some vertex. You will notice that his cube has a property, that opposite faces have opposite weak isospin. You can also notice that we only need three equations to fix the faces.

One of the vertexes of this cube is DSB, and of course is opposite vertex has the faces of the up-type quarks. This is the only axis that does not correspond to a waterfall symmetry, and on other hand the DUS vertex is the only axis which is used in both extremes.

Going to discrete groups, S4 is the group of permutations of the four "Z3 axis" in a cube, while the subgroup S3 is contained in four not-very-different ways, each of them being the permutations that keep one of such axis invariant (you can exchange fully the vertex by the opposite, to implement the Z2 subgroup of S3).

 

[arivero]

The real problem of S4 is to know the physics content, the objects we are permuting. The suggestion of putting quarks (or leptons) in the faces of a cube is rarely seen in the literature.

About PZ and pseudomasses, I think it is not very different of the initial objections to Harari-Haut-Weyers, they also do a similar trick, or a trick that can be interpreted as taking only the diagonal of the undiagonalised mass matrix.

By the way, I note that the abbreviations 2/27 and 4/27 are first used by Sheppeard in her note 342

 

[mitchell porter]

The pseudomass adds up the contributions of all the mass eigenstates to one of the weak eigenstates. So it looks like "sequential" triples, like in the waterfall, apply to mass eigenstates, family triples (as in the original Koide formula) apply to weak eigenstates, and this wasn't noticed until recently because, for charged leptons, the weak eigenstates are the same as the mass eigenstates.

For the quarks, we can then think of the waterfall as the dominant chain of relationships, and then the mixing parameters encode the rotation away from waterfall mass values, required to produce family triples with 2n/27 phases.

For the leptons, perhaps the family triples dominate, and a waterfall is weak or nonexistent. (I'm still not clear on whether right-handed neutrinos could have masses of the order of the quarks, as in #81, and then give rise to the observed small masses via seesaw.)

p.s. Chris Quigg had a paper yesterday - "Beyond Confinement" - in which 2/27 shows up as the exponent in a relation between the top mass and the nucleon mass, in a unified theory!

 

[arivero]

I hated macsyma and I am transferring my hate to its free twin, maxima, but still this is interesting. I got again the waterfall while I was trying to solve for the full S4 symmetric set of Koide equations.

This is, I was trying to find mass values in the six faces of a cube such of for each vertex we have a Koide equation. The group of rotations of the cube is S4; imposing Koide explictly breaks the symmetry as it gives different values to different faces. Not a very convincing motivation, but ok to play a little bit.

Now, in maxima. You define
K(x,y,z):=x^2+y^2+z^2-4*(x*y+y*z+z*x);
so that
expand(K(x,y,z)*K(x,-y,z)*K(x,y,-z)*K(x,-y,-z));
is a degree 8 even polynomial on three variables;
we put all of it

Q(x,y,z):=z^8-28*y^2*z^6-28*x^2*z^6+198*y^4*z^4-1172*x^2*y^2*z^4+198*x^4*z^4-28*y^6*z^2-1172*x^2*y^4*z^2-1172*x^4*y^2*
z^2-28*x^6*z^2+y^8-28*x^2*y^6+198*x^4*y^4-28*x^6*y^2+x^8;

and now we can use maxima "eliminate" to do the equivalent, I guess, of Sylvester matrix.
factor(eliminate([Q(1,x1,x2),Q(1,x2,x3)],[x2])) shows two terms that just validate x^2=x3^2. So in my first step I also canceled these factors:

step1:factor(eliminate([Q(1,x1,x2),Q(1,x2,x3)],[x2]))/(x3-x1)^8/(x3+x1)^8;

In the next step, factor(eliminate([paso1[1],Q(1,x3,x4)],[x3])) happens to have as a factor the polynomial for Q(1,x1,x4) and then it trivially tell us that the whole system of vertexes (1,x1,x2),(1,x2,x3),(1,x3,x4),(1,x4,x1) has solutions. As we want to exhibe actually some solution, I cancel the factor first

step2:factor(eliminate([step1[1],Q(1,x3,x4)],[x3]))/factor(Q(1,x1,x4)^12);

and now I cross each of the factors against the extant equation Q(1,x1,x4)

for i:1 thru 16 do (
pol:part(part(step2[1],i),1),
sl:solve([pol,Q(1,x1,x4)],[x1,x4]),
for k:1 thru length(sl) do (
  s:ev([x1,x4],sl[k]),
  if featurep(s[1],real) and featurep(s[2],real) then 
     (s:abs(s),
     if s[1]>s[2] then s:[s[2],s[1]],
     print(s,float(s)), 
     )
  )
);

The process solves to:
four complete x1,x2,x3,x4,x1 cycles
two sequences x1,x2,x3,x4
two sequences x1,x2,x3

The solutions for the waterfall triplets cbt and bcs appear in the list, numerically as sqrt(t)=10.12, sqrt(b)=1.464, sqrt(s)=0.267, with sqrt(c)=1, in one of the not-closing sequences. I guess I need an expert on discrete groups in order to understand what is going on.

PS. wow, now I notice that Q surely is the trick that Goffinet uses to avoid the square roots somewhere in his thesis.

 

[mitchell porter]

Exciting progress!

wow, now I notice that Q surely is the trick that Goffinet uses to avoid the square roots somewhere in his thesis.

http://cp3.irmp.ucl.ac.be/upload/theses/phd/goffinet.pdf (section 3.2.1).

I guess I need an expert on discrete groups in order to understand what is going on.

A further step would be to look for an S4-symmetric potential where these solutions are the local minima.

 

[arivero]

http://cp3.irmp.ucl.ac.be/upload/theses/phd/goffinet.pdf (section 3.2.1).[/QUOTE]

Indeed it is the same polynomial.

z^8-28*y^2*z^6-28*x^2*z^6+198*y^4*z^4-1172*x^2*y^2*z^4+198*x^4*z^4-28*y^6*z^2-1172*x^2*y^4*z^2-1172*x^4*y^2*
z^2-28*x^6*z^2+y^8-28*x^2*y^6+198*x^4*y^4-28*x^6*y^2+x^8;

  4         3         3        2  2             2        2  2       3
(%o3) z  - 28 y z  - 28 x z  + 198 y  z  - 1172 x y z  + 198 x  z  - 28 y  z
           2           2           3      4         3        2  2       3
 - 1172 x y  z - 1172 x  y z - 28 x  z + y  - 28 x y  + 198 x  y  - 28 x  y
    4
 + x

The minor improvement is that here we are sure that it is an ·"if and only if" relationship; Goffinet, in the text, was worried that the squaring could be introducing spureous solutions. As we have shown that this poly decomposes exactly in the product of the four possible sign combinations of Koide equation, now we are in position to grant that every solution of Goffinet's matrix version (3.30) of the equation is really a Koide solution.

Let me copy here this equation 3.30, setting the determinant of M as a function of the traces in M and M^2:

|M| = {2 \over 3 * 32^2} {(7 (Tr M)^2 - 8 Tr M^2)^2 \over Tr M}

 

[arivero]

The following Mathematica code will help to find the solutions to the S4 symmetric Koide system.

If this post is the only one you are going to read, remember that we are organising three generations in opposite faces of a cube, and each corner must agree with Koide equation. A way to solve this is to fix one face, say to unity, and then check to four corners of this face.

Please use the code line-by-line; it is listed here without EOF separators! Also, please verify that you select the non common factor in each step, the order could change between versions of Mathematica (this is done with version 9.0 in the free trial period)

K[u_, v_, t_] := u u + v v + t t - 4 (u v + v t + t u)
G[m1_, m2_, m3_] =FullSimplify[K[Sqrt[m1], Sqrt[m2], Sqrt[m3]] K[-Sqrt[m1], Sqrt[m2], Sqrt[m3]] K[ Sqrt[m1], -Sqrt[m2], Sqrt[m3]] K[Sqrt[m1], Sqrt[m2], -Sqrt[m3]]]
Expand[G[1, a, b]]
step1 = FactorList[Resultant[G[1, x, y2], G[1, y2, x2], y2]]
Resultant[step1[[2, 1]], G[1, x2, y], x2]
step2a = FactorList[Resultant[step1[[3, 1]], G[1, x2, y], x2]]
step2b = FactorList[Resultant[step1[[4, 1]], G[1, x2, y], x2]]
step2 = {step2a[[2, 1]], step2a[[4, 1]], step2b[[3, 1]], 
   step2b[[4, 1]]};
step2[[1]]
s1 = N[Solve[{step2[[1]] == 0, G[1, y, x] == 0, x >= 0, y >= 0, 
    x >= y}, {x, y}], 8]
s2 = N[Solve[{step2[[2]] == 0, G[1, y, x] == 0, x >= 0, y >= 0, 
    x >= y}, {x, y}], 8]
s3 = N[Solve[{step2[[3]] == 0, G[1, y, x] == 0, x >= 0, y >= 0, 
    x >= y}, {x, y}], 8]
s4 = N[Solve[{step2[[4]] == 0, G[1, y, x] == 0, x >= 0, y >= 0, 
    x >= y}, {x, y}], 8]
sol = Join[s1, s2, s3, s4];
{1/x*174.1, y/x*174.1} /. sol

You can be intrigued that the solutions are more detailed than a simultaneus Solve[] of the system of four equations G[1, x, y2]==0, G[1, y2, x2]==0, G[1, x2, y]==0, G[1, y, x]==0. I am intrigued too. It seems that some of the particular solutions found by the Resultant method are embedded inside a continuous spectrum of solutions, and then the Solve method avoids listing them twice. I wished to know more on the relationship between resultants and continuous solutions.

It is amusing that the first triplet of the list is Rodejohann-Zhang triplet,
{1, {x -> 102.50258, y -> 2.1435935}}. Scale it times 174.1/102.50 and you get 1.69849, 174.1, 3.64088

My own triplet appears later, as it is generated by the last polynomial... it is
1, {x -> 2.1435935, y -> 0.071796770}. Use the same scale factor than before, and you get 1.69849, 3.64088, 0.12195

Both triplets are of the kind that becomes hidden in the continuous under Solve.

I am not sure about why this resolvent method does not find solutions with a zero, for instance 1.69849, 0.12195, 0. They can be searched by starting from G[0, x, y2]==0, G[0, y2, x2]==0, G[0, x2, y]==0, G[0, y, x]==0

IN:  Solve[{G[0, x, y2] == 0, G[0, y2, x2] == 0, G[0, x2, y] == 0, 
  G[0, y, x] == 0, x == 0.12195, G[r, x, y2] == 0, G[r, y2, x2] == 0, 
  G[r, x2, y] == 0, G[r, y, x] == 0, y >= y2}, {x, y, y2, x2, r}]

OUT: 
{x -> 0.12195, y -> 0.00875562, y2 -> 0.00875562, x2 -> 0.000628625,   r -> 0},
{x -> 0.12195, y -> 0.00875562, y2 -> 0.00875562,  x2 -> 0.12195, r -> 0}, 
{x -> 0.12195, y -> 0.00875562,  y2 -> 0.00875562, x2 -> 0.12195, r -> 0.261411},
{x -> 0.12195, y -> 0.00875562, y2 -> 0.00875562, x2 -> 0.12195,  r -> 3.13693}, 
{x -> 0.12195, y -> 1.69854, y2 -> 0.00875562,  x2 -> 0.12195, r -> 0}, 
{x -> 0.12195, y -> 1.69854, y2 -> 1.69854,  x2 -> 0.12195, r -> 0}, 
{x -> 0.12195, y -> 1.69854, y2 -> 1.69854,  x2 -> 0.12195, r -> 3.64099}, 
{x -> 0.12195, y -> 1.69854,  y2 -> 1.69854, x2 -> 0.12195, r -> 43.6919}, 
{x -> 0.12195,  y -> 1.69854, y2 -> 1.69854, x2 -> 23.6577, r -> 0}}

EDIT: ok, a faster recipe could be

pols:factor(eliminate([G(1,x,a),G(1,a,y)],[a]))/(y-x)^4$
f1:ev(part(pols[1],1),[y=x])$
float(sol1:solve([f1,G(1,a,x)],[x,a]));

the output has the following positive solutions:
[x = 29.85640584694755, a = 0.12453316162267], 
[x = 29.85640584694755, a = 650.4292237442922], 
[x = 29.85640646055102, a = 199.4974226119286], 
[x = 29.85640646055102, a = 13.92820323027551]
[x = 2.143593539448983, a = 102.5025773880714], 
[x = 2.143593539448983, a = 0.071796769724491], 

and thus
(%i58) mc
;
(%o58)                              1.69854
(%i59) mc * 2.143593;
(%o59)                           3.64097845422
(%i60) mc *  0.07179;
(%o60)                           0.1219381866
(%i61) mc * 102.50257;
(%o61)                          174.1047152478

 

[mitchell porter]

|M| = {2 \over 3 * 32^2} {(7 (Tr M)^2 - 8 Tr M^2)^2 \over Tr M}

You could also: start with a 6x6 mass matrix including fictitious "up-down yukawas" as I have suggested, impose Goffinet's property on each of the four 3x3 blocks on the diagonal, and on the two larger blocks as in #82, and then finally impose a "checkerboard texture" in which all the "up-down yukawas" are set to zero, as in #73... and then see if the two larger blocks ever resemble the actual yukawa matrices.

Two problems: first, the SM yukawas are complex-valued and underdetermined by the experimental data (PDG). One would need to decide if the elements of the matrix M are the SM yukawas or secondary quantities derived from them. Second, the larger blocks are there in order to produce family Koide triplets, as in Zenczykowski; but Z's Koide triplets are made of Goffinet's pseudomasses, which are obtained by applying the CKM matrix to a vector of masses. It's not clear to me whether or not the larger blocks should be transformed somehow, before the Goffinet property is imposed.

 

[arivero]

Let me copy here this equation 3.30, setting the determinant of M as a function of the traces in M and M^2:

|M| = {2 \over 3 * 32^2} {(7 (Tr M)^2 - 8 Tr M^2)^2 \over Tr M}

Just a thinking... Koide masses are fixed by an angle theta and a mass M_0, which is proportional to the trace. So if we suplement the above equation with the already know Tr M = 6 M_0, it takes a look very much as a the terms one usually sees in generalised Higgs potentials.

0 = {1 \over 1536} (252 m_0^2 - 8 Tr M^2)^2 - 6 m_0 Det(M)

 

[mitchell porter]

There is a very phenomenological paper from Koide (and colleague Ishida) today. It seems to be the first paper that talks about adapting Sumino's mechanism to the quarks.

But let's take a step back. Koide found his formula 30 years ago. Koide has proposed a number of field-theoretic models to explain it; so have a few other people (actually, who else has made a proper field-theoretic model, apart from Ernest Ma?). All QFT models of the relation have the problem that there should be deviations from the formula, because of quantum corrections, but empirically it is exact within error.

Yukinari Sumino was the first person to develop a model in which the corrections are cancelled. It's a little complicated, but it involves a family symmetry that is gauged and then spontaneously broken. The heavy family gauge bosons do the cancelling of the corrections coming from QED.

Koide and Yamagarbagea adapted Sumino's mechanism to supersymmetry. The present paper does not mention supersymmetry, but it does assume the modified version of Sumino's mechanism (in which the mass hierarchy of the family gauge bosons is inverted, compared to Sumino's original version).

Koide and Ishida's inspiration is a tiny aberration in the data for B meson decays. I still haven't digested the paper, but they seem to say at the end that, naively, even a Sumino meson shouldn't be able to produce the dip (that may be there, or which may go away with more data). But there could be some enhancement, and, importantly for them, if the dip is due to their family bosons, then a corresponding dip will not appear in another particular measurement.

From my perspective, this paper runs ahead of theory, because we still have no field-theoretic model of any of the generalized Koide relations for quarks, let alone adaptations of the Sumino mechanism for such models. Koide's own recent BSM work generally assumes that there's a nonet of scalars whose VEVs are diag(√m_e,√m_μ,√m_τ), and then he builds mass matrices for all the SM fermions out of couplings to these. It is from within this theoretical context that he will have guessed at the quark couplings with the Sumino mesons.

Since the quarks have their own Koide relations, it seems very unlikely that their masses are produced in the manner of Koide's recent models. Still, it's always useful to have papers that go "too far ahead" - in this case, trying to interpret a known anomaly as a signal of quark-sector Sumino mesons! Thinking about how the ideas in the paper work, may help those of us still struggling to find an approach to "Koide for quarks".

 

[arivero]

A few months ago I sketched a report on the topic of predictions from Koide equation that could be more readable for people used to PhysRev latex format. It is here: http://es.scribd.com/doc/157932274/Koide-equations-for-quark-mass-triplets

 

[arivero]

Also, perhaps all the thing about sqrt(M) is a red herring. We could just contemplate a correction "susy-like" going only up to order two,

M_i = (1 + \lambda_i + \lambda_i^2) M

and then Koide eq is the system Tr \lambda = 0, Tr (\lambda^2) = Tr(1)

 

[mitchell porter]

Maybe I'm stupid but I don't understand any of those equations. What matrices are M, M_i, \lambda_i? What is the \lambda in the final equations?

edit: Let me guess... The first quantities are all scalars. M is a Koide-Brannen mass scale, M_i is the mass of the ith member of the corresponding Koide triple, and \lambda is a matrix with the \lambda_is as eigenvalues??

 

[arivero]

edit: Let me guess... The first quantities are all scalars. M is a Koide-Brannen mass scale, M_i is the mass of the ith member of the corresponding Koide triple, and \lambda is a matrix with the \lambda_is as eigenvalues??

Ok, Trace was a bit of pedantry. Instead, say
\lambda_1+ \lambda_2 +\lambda_3 =0
\lambda_1^2 +\lambda_2^2 +\lambda_3^2 =3
And I have forgot a factor 2, have I? It should be
M_i=M (1 + 2\lambda_i + \lambda_i^2) = M (1 + \lambda_i)^2

Well, perhaps the importance of sqrt(M) is not a redherring, at all.

 

[mitchell porter]

There have been two new "yukawaon" papers.

Koide and Nishiura have made a substantial technical change, in order to make the family-symmetry interactions of the SM fermions anomaly-free (previously, new fields had to be introduced just to cancel the anomalies).

Aulakh and Khosa produced "Grand Yukawonification", one of the few papers not by Koide that even mentions the yukawaon models. Actually their philosophy is rather different. If I am reading it correctly, this is a susy SO(10) model, in which GUT symmetry breaking is achieved by some very high-dimensional representations (e.g. a Higgs with 126 components), and then some of these Higgs components are gauged under an SO(3) family symmetry, and the yukawas come from their VEVs.

It would be edifying to compare and contrast what they do, and what Koide does. They call theirs a top-down approach, as opposed to Koide's bottom-up approach. Koide introduced new yukawaon fields and a new scale for family symmetry breaking; they just put to work some of the components of the GUT Higgs, and the GUT scale is also the family scale.

Also, it seems to me that their approach has something in common with the 1990 paper by Koide which was the first step towards yukawaons (for a very brief history, see this talk). In subsequent work, the SM yukawa terms are produced by operators coupling SM fermions, the usual SM Higgs, and the yukawaon VEVs, but in this paper from 1990, the masses come from direct couplings between SM fermions and yukawaon VEVs (I think). And this seems to be what Aulakh and Khosa are doing. The downside is that they are not explaining the Koide formula (or any of its generalizations)...

 

[mitchell porter]

A new Koide paper from New Zealand, "Model for inner structure and mass spectrum of charged leptons" by Vladimir Kruglov. The precise logic of the paper eludes me so far, but it contains many new ansatze, and may therefore be of value even if the overall framework is flawed.

 

[ftr]

A new Koide paper from New Zealand, "Model for inner structure and mass spectrum of charged leptons" by Vladimir Kruglov. The precise logic of the paper eludes me so far, but it contains many new ansatze, and may therefore be of value even if the overall framework is flawed.

I don't have the time to elaborate, but the paper should be looked upon as seminal. The logic and overall idea is correct, however the detail and the ansatz are probably somewhat wrong at short range.

P.S. and since you are from the same area, can you invite him to participate in the thread!

 

[MTd2]

P.S. and since you are from the same area, can you invite him to participate in the thread!

I think you should trust the Mercator projection to infer distances!

 

[ftr]

I think you should trust the Mercator projection to infer distances!

what's two hours flight. Take midpoint between their mass centers, then draw a circle with 4000 km radius, you will only see two countries. That is how close they are! Even their flags look the same.

 

[phyzguy]

I don't have the time to elaborate, but the paper should be looked upon as seminal. The logic and overall idea is correct, however the detail and the ansatz are probably somewhat wrong at short range.

P.S. and since you are from the same area, can you invite him to participate in the thread!

Why would you think this is seminal? There is a model for an electron, with arbitrary constants adjusted so that the mass comes out right. It is stated that excited states can represent the muon and tau, but no attempt is made to calculate the muon and tau masses. At a very minimum, I would expect it to show the correct ratios of the lepton masses, but it doesn't do that. There is also no attempt to explain why there are only three solutions to the eigenvalue equations. Also, there are statements made about how the scalar field (Theta) corresponds to the Higgs field, but no backing for those statements that I can see. Please explain why you think this is important work. What am I missing?

 

[ftr]

Why would you think this is seminal? There is a model for an electron, with arbitrary constants adjusted so that the mass comes out right. It is stated that excited states can represent the muon and tau, but no attempt is made to calculate the muon and tau masses. At a very minimum, I would expect it to show the correct ratios of the lepton masses, but it doesn't do that. There is also no attempt to explain why there are only three solutions to the eigenvalue equations. Also, there are statements made about how the scalar field (Theta) corresponds to the Higgs field, but no backing for those statements that I can see. Please explain why you think this is important work. What am I missing?

I agree with all your objections(I have already stated some of it), actually these were my own questions, and that is why I asked if the author can participate. However, the concept Is familiar to me because I have thought about it independently and I was surprised to find that the concept was thought about by Dirac(references) and even Lorentz long time ago.

http://web.archive.org/web/20041224...i.nl/physis/HistoricPaper/Dirac/Dirac1962.pdf

But to discuss in detail I do need the time.

 

[mitchell porter]

phyzguy is right about this paper, it doesn't derive the Koide relation at all, and offers no evidence that the proposed model works.

It could be regarded as a soliton model, in which there is a spherically symmetric spinor wave coupled to a similar wave in a scalar field, with a peak of charge and mass density in some central region, which then drops away with distance. The excited states presumably have many hills and valleys surrounding the central peak, like excited states of the simple harmonic oscillator. The author doesn't even try to calculate the energies of the excited states, but just says (page 5, end of part IV) that the first two excited states will correspond to the muon and tauon.

Because these soliton-like objects are spherically symmetric, the author may hope for a convergence with the ideas of Gerald Rosen, who has written a number of papers trying to obtain various modified versions of Brannen's formula from models of particles as dynamical two-dimensional surfaces like Dirac's membrane. (Kruglov cites two of these papers, more can be found in http://home.comcast.net/~gerald-rosen/publications.htm.) In this soliton model, the two-dimensional surface might be the surface of maximum charge density or maximum mass density.

There's lots missing from Kruglov's paper - not just calculations for the "muon" and "tauon" states (it may actually be possible to falsify the model as presented, by doing those calculations) - but also the interaction of these "electrons" - e.g. do they scatter like real electrons? Kruglov's theory seems quite simple - QED with a mass term for the electron (i.e. not one obtained via Higgs mechanism), plus the extra scalar - and it may be a QFT equation that has already been thoroughly analyzed...

Then there are all the experimental results and theoretical arguments against the idea of the electron having internal structure and against muon and tauon as excited states of that internal structure - perhaps someone could dig up the details of this, which I admit I only know as a talking point.

In M-theory phenomenological models (like the G2-MSSM of Kane et al), the particles are two-dimensional membranes (M2-branes), but the membranes are Planck-scale in size, and the particle generations don't correspond to excitations of the membranes, but have some other origin, e.g. each generation comes from branes "stuck" to a different singular point in the Kaluza-Klein space.

 

[arivero]

A cross reference to the thread on 14 dimensions:

Given that this S^8 is a kind of connection, maybe we are talking about the connection which is working like a momentum space. Well, I am saying that this intersection to form some kind of koide relation. Well, this relation will of null rays will satisfy a koide relation.

Not sure if MTd2 is going to ellaborate on it, or if we should expect some more detailed paper. Meanwhile, I return to my tomb and my silent rest.

 

[MTd2]

Not sure if MTd2 is going to ellaborate on it, or if we should expect some more detailed paper. Meanwhile, I return to my tomb and my silent rest.

I am waiting for Kneemo to say something. I am like shooting in the dark... Well, not completely. I am doing some numerology and showing some mathematical stuff and see if there is a useful correlation.

 

[mitchell porter]

We have discussed the possibility of a massless up quark several times in this thread (#51, #60, #78). There are two papers which touch on this idea today.

First, Dvali et al, "On How Neutrino Protects the Axion". This cites an older paper of Dvali's, "Three-Form Gauging of axion Symmetries and Gravity", in which it is said (page 12) that "at low energies, the QCD Lagrangian contains a massless three-form field", and that both axion and massless-up-quark solutions to the strong CP problem can be understood as a Higgsing of this three-form.

According to remarks on page 13, in the case of the massless up quark, it's the eta-prime meson which Higgses the three-form.

The other paper today is "Charge Quantization and the Standard Model from the CP2 and CP3 Nonlinear σ-Models" by Hellerman et al. This paper is part of a research program aiming to get charge quantization without grand unification. Instead of embedding the whole SM gauge group in a larger simple group, as in a GUT, part of the SM gauge group is identified as the locally gauged part of a CPn global symmetry in a "nonlinear sigma model". So in each case, SU(3)c x SU(2)L x U(1)Y is split into two parts, which we could call the unembedded and the embedded part. In the CP1 model, SU(3)c x SU(2)L is unembedded, and U(1)Y is embedded into CP1. In the CP2 model, SU(3)c is unembedded, and SU(2)L x U(1)Y is embedded into CP2. In the CP3 model, SU(2)L is unembedded, and SU(3)c x U(1)Y is embedded into CP3. Models employing CP4 and higher are mathematically possible, but their phenomenological viability is not discussed.

On page 4 of today's paper in this series, we read that in the CP3 model, there is a Goldstone boson with the quantum numbers of an up squark, and that if the supersymmetric CP3 model were considered, there could be a massless up quark. It's also stated that in the CP2 model, it might be possible to get the SM Higgs from the corresponding Goldstone boson, and that it could dovetail with the proposal to explain the Higgs mass as arising from ultra-high-energy boundary conditions (e.g. as in Shaposhnikov-Wetterich, though the present proposal has nothing to do with asymptotic safety).

In my opinion, these extra phenomenological twists should be regarded as a bit untested and opportunistic. The key idea in the papers of Hellerman et al is that these NLSMs provide an alternative to the GUT explanation of charge quantization. They develop that idea, and then they note that there might be a way to incorporate these older ideas (massless up as solution to strong CP, high-scale boundary conditions as reason for Higgs boson mass) into their scheme, but this latter part of their work is still sketchy.

 

[kneemo]

I am waiting for Kneemo to say something. I am like shooting in the dark... Well, not completely. I am doing some numerology and showing some mathematical stuff and see if there is a useful correlation.

What's curious is the Yukawaon model that Koide uses to derive the Koide relation. He starts with some superpotential and assumes some SUSY vacuum conditions (∂W=0) to get a cubic equation that leads to the Koide relation for the leptons.

In supergravity, critical points of a superpotential ∂W=0 correspond to attractor points of the scalar field trajectories that localize on a black hole horizon. In D=4, the attractor mechanism is obeyed for black hole solutions with non-vanishing quartic invariant I≠0. These are rank four solutions that come in three canonical forms under E7:

a) k(1,(-1,-1,-1))
b) k(1,(1,1,-1))
c) k(1,(1,1,1))

where k > 0 and (1,1,-1), for example, corresponds to a diagonalized 3x3 Hermitian matrix with eigenvalues (1,1,-1) that the reduced structure group E6 can in general act on.

Families b) and c) resemble the forms of the eigenvalues found in the neutrino and lepton Koide relations.

If the Yukawaon model derivation of the Koide relation is really the result of an attractor mechanism, this would explain what the Yukawaon really is and why the situation isn't as messy as it could be. In essence, the Koide relation would just be the result of moduli (complex scalar fields) being stabilized on a microscopic black hole event horizon.

 

[MTd2]

Hmm, my last post in this thread should go in the other thread! But well, I will think about this anyway.