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What is new with Koide sum rules? 
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#19
Jan1112, 12:05 PM

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MTd2 points to the 2012 paper of Jerzy Kocik:
http://arxiv.org/abs/1201.2067 which cites the exquisitelytitled 2005 paper of Rivero Gsponer: [11] A. Rivero and A. Gsponer, The strange formula of Dr. Koide, http://arxiv.org/abs/hepph/0505220 This is turning out to be a class act. The script resembles a Gothic novel that Isak Dinesen might have dreamed up. 


#20
Jan1112, 06:45 PM

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#21
Jan1412, 06:23 AM

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Since we have another thread discussing E6 grand unification, I will point out the work of Berthold Stech.
In my comment #14, I said that an "obvious" way to make a model for these extended Koide relations, would be to extend the standard model with a new scalar sector of "flavons", whose VEVssquared determine the Yukawa couplings, and with a gauged family symmetry that protects the Koide relations, as suggested by Yukinari Sumino. (The Koide relations among the flavon VEVs would result from a flavon potential.) So it's very interesting that Stech's E6 models more or less resemble this picture. The Yukawas come from flavon VEVs, and there's a flavor symmetry. Especially interesting is that his lightest Higgs is at about 123 GeV! Stech's models definitely do not produce Koide relations at present. In particular, I can't think of any model ever that implies the peculiar emutau/scb relation that Alejandro found. Though let's note that that relation also resembles the usc/scb relation in the "zerothorder" or "primordial" version of the extended Koide relations, as described in his paper; so there may be something more complicated than a GeorgiJarlskog "multiplication by three" at work here. (Another quantitative issue to investigate is whether all six lepton masses, neutrinos as well as charged leptons, can be arranged into a set of four chained Koide triplets like the quarks, or whether the leptons naturally fall into two disjoint triplets, this being an aspect of how they differ from the quarks.) But the "peculiar" relation should be seen simply as a challenge: come up with a flavon potential and a new symmetry which produces it. And Stech's E6 framework looks worth investigating, though the minimal way to proceed would be just to add flavons (and maybe more Higgses) to the standard model until the extended Koide relations (and quarklepton complementarity for the mixing angles, see comment #16) are obtained. 


#22
Jan1712, 12:20 AM

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You guys are famous  someone's been reading your posts! http://motls.blogspot.com/2012/01/co...real.html#more



#23
Jan1712, 04:34 AM

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#24
Jan1712, 10:37 AM

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[tex]{{m_c+m_s+m_b} \over {m_e+m_\mu+ m_\tau}} = 2.995 (\pm 0.04 approx) [/tex] 


#25
Jan1712, 12:41 PM

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Guys, wake up  science simply doesn't work in this way.
Unfortunately the following site is in german so can't be understood by anyone, but it's kind of a healthy medecine. It shows how to obtain the natural constants out of 5 given numbers (like your birthday, the hight of the pyramides in inches, etc), up to an accuracy of 10^4. And vice versa, you can fit any given number by the natural constants: http://www.hars.de/misz/rado.html 


#26
Jan1712, 02:09 PM

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Here, it is not about getting multiple quantities with different formulae, it is the puzzle that we are getting multiple quantities with a single formula. Namely, we have taken Koide equation for a z,x,y triplet: [tex] {(\sqrt z + \sqrt x + \sqrt y)^2 \over (z+x+y) } = \frac 3 2 [/tex] and we have solved for z [tex]z=f(x,y)=\left[ ( \sqrt x +\sqrt y )\left(2 \sqrt{3+6 {\sqrt{xy} \over (\sqrt x+\sqrt y)^2}}\right) \right]^2[/tex] It was known since 1981 that this formula, for y=1.77668 and x=0.105659 gets f(x,y)=0.000510, ie, that [itex]n_e \equiv f(m_\tau,m_\mu)[/itex] was equal to the physical [itex]m_e[/itex]. Up to now, one can live with this and appeal to GIGO arguments, garbage it garbage out, disregarding the point that the equation was actually found from physical models. There is a lot of physical models, some of them could hit in a random equation. NOW, the new observation is that taking as input [itex]m_t[/itex] and [itex]m_b[/itex], and iterating down four times to produce six particles, the total spectrum does not fare bad neither. [tex]n_c \equiv f(m_t,m_b) \approx m_c[/tex] [tex]n_s \equiv f(m_b,n_c) \approx m_s[/tex] [tex]n_u \equiv f(n_c,n_s) \approx 0 [/tex] [tex]n_d \equiv f(n_s,n_u) \approx m_d[/tex] So we have verified that the Koide equation also does a decent work in the quark ladder. Not a different equation. Nor different parameter. Nor different powers. The SAME equation. Just 30 years later. Still, it can be argued that charm and strange have a very broad range of values in the experimental sector, some of them even arguable up to definition of the concept. Thus, we have looked for comparison between the quark and lepton spectrum and found that: 1) [itex] (m_b + n_c + n_s) / (m_e + m_\mu + m_\tau)= 3 [/itex] 2) The phase angle to built the triplet [itex] (m_b , n_c , n_s)[/itex] is about 3 times the phase angle of the triplet [itex](m_e , m_\mu , m_\tau)[/itex] Both 1 and 2 can be described telling that the triplets, in its square root form, are almost orthogonal when ordered in the cone around (1,1,1). We can either keep 1 and 2 as a verification of the values of charm and strange, and stop here, or use it as extra postulates to produce all the masses from only two values. 


#27
Jan1712, 07:32 PM

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Unfortunately, the most convenient reference for RG results of running masses is a nearly 20 year old paper by Koide himself, hepph/9410270, but Table VI is at least illustrative. Your above values give [itex]m_t/m_b\sim 40[/itex], while [tex]m_t(1~\mathrm{GeV})\sim 420~\mathrm{GeV},~~~m_b(1~\mathrm{GeV})\sim 7~\mathrm{GeV},~~~m_t/m_b (1~\mathrm{GeV})\sim 60.[/tex] Using the formula from arivero's post #24, these values give [itex]m_c(1~\mathrm{GeV})\sim 5.5~\mathrm{GeV},[/itex] which is about 3.6 times the correct value. I don't believe that things are going to get better at any other mass scale or by using any more modern results for the RG equations, so there's no reason to believe that there's any deep significance to any Koidetype relation. 


#28
Jan1712, 08:25 PM

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#29
Jan1712, 10:00 PM

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I'm also very skeptical that you can cancel the RG correction to that combination of masses without leaving a trace of the new physics in the running of, e.g., the electron mass on its own. Since there's no evidence for any supression of QED radiative corrections between [itex]m_e[/itex] and [itex]m_W[/itex], it's unlikely that such a mechanism exists. 


#30
Jan1812, 12:50 AM

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The idea seems to be that the Koide relation holds exactly at high energies, and it also holds for the pole masses, because the corrections due to the family gauge bosons cancel the QED corrections for each charged lepton, at its own mass scale. Above that scale, the mass will just run normally as in the SM, until the scale where electroweak unifies with the family force (100s or 1000s of TEVs), at which point the Koide relation becomes manifest again.
But I'm just telling you how I think it's supposed to work, I'm still getting my head around the details. 


#31
Jan1812, 01:19 AM

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[tex] \frac{ \sqrt{m_e(m_e)} + \sqrt{m_\mu(m_\mu)} +\sqrt{m_\tau(m_\tau)}}{\sqrt{m_e(m_e) + m_\mu(m_\mu) +m_\tau(m_\tau) }} = \sqrt{\frac{3}{2} } \pm 10^{5} .[/tex] Now [itex]m_i(E)[/itex] definitely runs with energy and we know this because it's been measured. What I understood is that, in Sumino's model, the oneloop corrections to [tex] r(E) = \frac{ \sqrt{m_e(E)} + \sqrt{m_\mu(E)} +\sqrt{m_\tau(E)}}{\sqrt{m_e(E) + m_\mu(E) +m_\tau(E) }} [/tex] cancel. However, we know from running the pole masses in the first relation that [itex]r(E)[/itex] differs from [itex]\sqrt{3/2}[/itex] by one part in [itex]10^{3}[/itex] (stated below eq (2) in Sumino). Now, since [itex]m_e(m_e)/m_\tau(m_\tau) \sim 3 \cdot 10^{4}[/itex], at this level of precision, we might as well just drop the terms with [itex]m_e[/itex] from [itex]r(E)[/itex]. The Koide relation really doesn't convincingly extend to the electron and is just some numerology involving [itex]\mu[/itex] and [itex]\tau[/itex]. The situation for the up quarks is even worse since [itex]m_u/m_t[/itex] is much, much smaller than the experimental uncertainty in the top mass. 


#32
Jan1812, 02:04 AM

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I still haven't verified this! But I believe this is how it's supposed to work. 


#33
Jan1812, 05:47 AM

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I wonder, is there some context where pole masses are more relevant that running masses? For instance, when we compare two masses to decide if the particle A can decay to particle B, are we supposed to compare pole masses, or to run the mass of B to the A scale, or run the mass of A to the B scale? I'd expect the two later procedures to be equivalent, but given that stability is about the total balance of energy, perhaps the former procedure is more relevant. 


#34
Jan1812, 04:59 PM

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The troubling part is that he seems to be pushing for some running of masses below this scale that is not at all like what actually happens. We know that the ratios of the pole masses are not the same as the ratios of the running masses at observable energies. There's no calculation in the paper that uses real physics to explain how to get from [itex]r(\mu)[/itex] to the corresponding ratio of pole masses. It still seems that he wants to fix the masses in [itex]r(\Lambda)[/itex] to match the ratios between the pole masses. This is precisely what I'm saying is completely unphysical. 


#35
Jan1812, 07:00 PM

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Of course it is irrelevant for the experimental results, the running of electron fro .511 to 105 is surely negligible. 


#36
Jan1812, 07:43 PM

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I think that any simplification (like just using effective mass parameters) probably leaves too much physics out that is of the same degree of importance. 


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