# What is new with Koide sum rules?

by arivero
Tags: koide, rules
 Astronomy Sci Advisor PF Gold P: 22,678 MTd2 points to the 2012 paper of Jerzy Kocik: http://arxiv.org/abs/1201.2067 which cites the exquisitely-titled 2005 paper of Rivero Gsponer: [11] A. Rivero and A. Gsponer, The strange formula of Dr. Koide, http://arxiv.org/abs/hep-ph/0505220 This is turning out to be a class act. The script resembles a Gothic novel that Isak Dinesen might have dreamed up.
PF Gold
P: 2,883
 Quote by marcus 1654 was a great year for particle physics! Descartes – in his 1654 letter to the princess of Bohemia, Elizabeth II – showed that the curvatures of four mutually tangent circles (reciprocal of radii), say a,b,c,d, satisfy the following “Descartes’s formula”...Nature is showing us she can be completely weird. Or maybe the word is witty.
Ah, also Bruce Schechter, in a comment in June of 2008 to http://dorigo.wordpress.com/2007/02/...vant-exercise/ , noticed the similarity, but did not suggest any way to generalize to 2/3. Jerzy Kocik has done an interesting extension here.
 P: 746 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 VEVs-squared 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 e-mu-tau/s-c-b relation that Alejandro found. Though let's note that that relation also resembles the u-s-c/s-c-b relation in the "zeroth-order" or "primordial" version of the extended Koide relations, as described in his paper; so there may be something more complicated than a Georgi-Jarlskog "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 quark-lepton complementarity for the mixing angles, see comment #16) are obtained.
P: 1,906
 Quote by atyy You guys are famous - someone's been reading your posts! http://motls.blogspot.com/2012/01/co...real.html#more
This post assures me why I canceled my rss subscription of his blog.
PF Gold
P: 2,883
 Quote by atyy You guys are famous - someone's been reading your posts! http://motls.blogspot.com/2012/01/co...real.html#more
In his first answer to Lubos, ohwilleke makes a remark that, while known in the papers, is not in this thread (it could be in old ones). Point is, you take the experimental values of top and bottom
mtop=172.9
mb=4.19
and then use Koide equation to produce charm and strange.
mc=((sqrt(mtop)+sqrt(mb))*(2-sqrt(3)*sqrt(1+2*sqrt(mtop*mb)/(sqrt(mtop)+sqrt(mb))^2)))^2
ms=((sqrt(mc)+sqrt(mb))*(2-sqrt(3)*sqrt(1+2*sqrt(mc*mb)/(sqrt(mc)+sqrt(mb))^2)))^2
Then you can consider to compare with the experimental value of lepton sum
leptons=0.000511+0.105659+1.77668
and then a quotient which is 1-sigma compatible with an integer appears:
$${{m_c+m_s+m_b} \over {m_e+m_\mu+ m_\tau}} = 2.995 (\pm 0.04 approx)$$
 P: 407 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
PF Gold
P: 2,883
 Quote by suprised Unfortunately the following site is in german so can't be understood by anyone
Be sure that I understand the point in the website, it is very clear, and it is really no different that the decimal system itself. Note that the program allows each number to generate six different ones via powers, plus some allowance for pi and 2 and its powers, so it has a lot of available combinations to cover ten thousand numbers (four digits). That is, assuming that really you have checked the page. Because it seems that you people do not actually read the posts...

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:

$${(\sqrt z + \sqrt x + \sqrt y)^2 \over (z+x+y) } = \frac 3 2$$

and we have solved for z

$$z=f(x,y)=\left[ ( \sqrt x +\sqrt y )\left(2- \sqrt{3+6 {\sqrt{xy} \over (\sqrt x+\sqrt y)^2}}\right) \right]^2$$

It was known since 1981 that this formula, for y=1.77668 and x=0.105659 gets f(x,y)=0.000510, ie, that $n_e \equiv f(m_\tau,m_\mu)$ was equal to the physical $m_e$. 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 $m_t$ and $m_b$, and iterating down four times to produce six particles, the total spectrum does not fare bad neither.
$$n_c \equiv f(m_t,m_b) \approx m_c$$ $$n_s \equiv f(m_b,n_c) \approx m_s$$ $$n_u \equiv f(n_c,n_s) \approx 0$$ $$n_d \equiv f(n_s,n_u) \approx m_d$$
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) $(m_b + n_c + n_s) / (m_e + m_\mu + m_\tau)= 3$
2) The phase angle to built the triplet $(m_b , n_c , n_s)$ is about 3 times the phase angle of the triplet $(m_e , m_\mu , m_\tau)$

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.
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PF Gold
P: 2,606
 Quote by arivero Point is, you take the experimental values of top and bottom mtop=172.9 mb=4.19 and then use Koide equation to produce charm and strange.
I was surprised that Lubos didn't explicitly point out what seems to me to be the biggest flaw with all of the Koide formulas, which is that no one bothers accounting for running quark masses. What you're quoting above are the quark masses at their own pole, namely $m_q(m_q)$. No physical theory is going to relate quark masses at scales that differ by 1.5 orders of magnitude.

Unfortunately, the most convenient reference for RG results of running masses is a nearly 20 year old paper by Koide himself, hep-ph/9410270, but Table VI is at least illustrative. Your above values give $m_t/m_b\sim 40$, while

$$m_t(1~\mathrm{GeV})\sim 420~\mathrm{GeV},~~~m_b(1~\mathrm{GeV})\sim 7~\mathrm{GeV},~~~m_t/m_b (1~\mathrm{GeV})\sim 60.$$

Using the formula from arivero's post #24, these values give $m_c(1~\mathrm{GeV})\sim 5.5~\mathrm{GeV},$ 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 Koide-type relation.
P: 746
 Quote by fzero 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 Koide-type relation.
We won't really know until someone finds a class of models which naively imply Alejandro's formula, and then performs an analysis like that in Sumino, http://arxiv.org/abs/0812.2103. As things stand, the situation is consistent with there being some sort of new symmetry which is visible only obscurely for the quarks, but which stands out sharply for the charged leptons.
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PF Gold
P: 2,606
 Quote by mitchell porter We won't really know until someone finds a class of models which naively imply Alejandro's formula, and then performs an analysis like that in Sumino, http://arxiv.org/abs/0812.2103. As things stand, the situation is consistent with there being some sort of new symmetry which is visible only obscurely for the quarks, but which stands out sharply for the charged leptons.
That Sumino paper is bizarre. He starts with the Koide formula, which holds empirically for the pole masses and says that he wants it to be valid for the masses defined at some high-energy scale. What I'm saying is that the Koide formula as written there does not hold at any particular scale with the same accuracy as it does for the pole masses. While he's trying to do something that makes sense (write a formula that relates masses defined at the same scale), there doesn't seem to be much reason to cling to Koide's formula that we already know doesn't work away from the pole masses.

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 $m_e$ and $m_W$, it's unlikely that such a mechanism exists.
 P: 746 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.
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PF Gold
P: 2,606
 Quote by mitchell porter 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.
Let's denote the pole masses by $m_i(m_i)$. The Koide result is that

$$\frac{ \sqrt{m_e(m_e)} + \sqrt{m_\mu(m_\mu)} +\sqrt{m_\tau(m_\tau)}}{\sqrt{m_e(m_e) + m_\mu(m_\mu) +m_\tau(m_\tau) }} = \sqrt{\frac{3}{2} } \pm 10^{-5} .$$

Now $m_i(E)$ definitely runs with energy and we know this because it's been measured. What I understood is that, in Sumino's model, the one-loop corrections to

$$r(E) = \frac{ \sqrt{m_e(E)} + \sqrt{m_\mu(E)} +\sqrt{m_\tau(E)}}{\sqrt{m_e(E) + m_\mu(E) +m_\tau(E) }}$$

cancel.

However, we know from running the pole masses in the first relation that $r(E)$ differs from $\sqrt{3/2}$ by one part in $10^{-3}$ (stated below eq (2) in Sumino). Now, since $m_e(m_e)/m_\tau(m_\tau) \sim 3 \cdot 10^{-4}$, at this level of precision, we might as well just drop the terms with $m_e$ from $r(E)$. The Koide relation really doesn't convincingly extend to the electron and is just some numerology involving $\mu$ and $\tau$. The situation for the up quarks is even worse since $m_u/m_t$ is much, much smaller than the experimental uncertainty in the top mass.
P: 746
 Quote by fzero Now $m_i(E)$ definitely runs with energy and we know this because it's been measured.
I think the point is that in a theory with "Koide symmetry" (i.e. whatever it is that produces the Koide relation) but not "Sumino family symmetry", the $m_i(m_i)$ start out at a set of values which don't satisfy Koide symmetry. The additional Sumino family symmetry adjusts the RG trajectory so that the pole masses do satisfy Koide symmetry. But that doesn't mean that Koide symmetry is exact for the running masses at low energies; it only becomes exact above the family-symmetry unification scale. The masses do not satisfy the symmetry at any single value of E below that scale; but the relation happens to hold for the pole masses at their different scales.

I still haven't verified this! But I believe this is how it's supposed to work.
PF Gold
P: 2,883
 Quote by fzero corrections between $m_e$ and $m_W$
By the way, according for instance table IV of http://arxiv.org/pdf/hep-ph/0601031v2 or section F of http://prd.aps.org/abstract/PRD/v46/i9/p3945_1 (where, note, the wrong measured value for tau is still used), all the damage to Koide relation for leptons is done already when moving electron and muon up to the GeV scale. From 1 GeV up to any high energy (without GUT), the mismatch keeps about 1.0017 - 1.0019, i.e. about the 0.2% of "error".

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.
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PF Gold
P: 2,606
 Quote by mitchell porter I think the point is that in a theory with "Koide symmetry" (i.e. whatever it is that produces the Koide relation) but not "Sumino family symmetry", the $m_i(m_i)$ start out at a set of values which don't satisfy Koide symmetry. The additional Sumino family symmetry adjusts the RG trajectory so that the pole masses do satisfy Koide symmetry. But that doesn't mean that Koide symmetry is exact for the running masses at low energies; it only becomes exact above the family-symmetry unification scale. The masses do not satisfy the symmetry at any single value of E below that scale; but the relation happens to hold for the pole masses at their different scales. I still haven't verified this! But I believe this is how it's supposed to work.
After reading a bit more, I see that the paper is saying that he can engineer $r(\Lambda)$ to have the right value and this is treated as an initial condition for the EFT. He makes remarks saying that the running of $r(\mu)$ is formally protected but notes that the physical argument breaks down below $\Lambda$.

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 $r(\mu)$ to the corresponding ratio of pole masses. It still seems that he wants to fix the masses in $r(\Lambda)$ to match the ratios between the pole masses. This is precisely what I'm saying is completely unphysical.

 Quote by arivero By the way, according for instance table IV of http://arxiv.org/pdf/hep-ph/0601031v2 or section F of http://prd.aps.org/abstract/PRD/v46/i9/p3945_1 (where, note, the wrong measured value for tau is still used), all the damage to Koide relation for leptons is done already when moving electron and muon up to the GeV scale. From 1 GeV up to any high energy (without GUT), the mismatch keeps about 1.0017 - 1.0019, i.e. about the 0.2% of "error".
Assuming that the calculations in the paper are correct, this is very useful to illustrate my point. However the authors obviously reach the wrong conclusions. They seem to think that $k-1$ being different from zero at a larger degree than the ratio $m_1/m_3$ or the experimental uncertainty $\Delta m_3/m_3$ still implies that "Koide's relation is a universal result." This is not a scientific conclusion, we require a higher standard.

 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.
The pole masses are obviously the right ones in processes such as particle production at threshold. For decay processes, I believe the right criterion is that the process has to make sense in the rest frame of the decaying particle. Therefore the pole mass of A is the right one to use and any running of the B mass is a small contribution to the kinematics of the final state.
PF Gold
P: 2,883
 Quote by fzero The pole masses are obviously the right ones in processes such as particle production at threshold. For decay processes, I believe the right criterion is that the process has to make sense in the rest frame of the decaying particle. Therefore the pole mass of A is the right one to use and any running of the B mass is a small contribution to the kinematics of the final state.
I am not sure. Consider a decay muon to electron plus a pair neutrino antineutrinos, as usual. As it is possible that the electron is left in the same rest frame that the initial muon, I could say that the energy available for the neutrino pair is the difference of pole masses of muon and electron, not the muon pole mass minus the renormalised electron mass at muon scale. I think I should had put more care when I attended to the undergraduate lectures, twenty years ago.

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