Koide Mass Formula for Neutrinos

[arivero]

Just for fun, let me start from the neutron mass and the pion mass.

939.565346 ± 0.000023 MeV
134.9766 ± 0.0006 MeV

We take Koide fundamental mass: 939.565346 / 3=313.188449
(It is also possible to use (2 N - P)/3 = 313.62 )

and for the phase we solve
313.188449 ( 1 + sqrt(2) cos(phase_mu))^2 = 134.9766

which is 1.81615071 o
So
phase_e=1.81615071 + 2*pi/3=3.91054581
phase_tau=1.81615071 - 2*pi/3=-0.278244391=6.00494092

and then mass of electron 0.0833034615 and
mass of tau 1 744.07079

No bad for the tau, but horrible for the electron.

What happens, probably, is that the mass of the pion is not a pure QCD object, it has got mixings and corrections. But perhaps the same could be predicated of the leptonic part. It is interesting to remember that m_e=0 directly "predicts", via phase_e= 3/4 pi, that
m_muon=125 MeV and m_tau=1753 MeV

 

[CarlB]

No bad for the tau, but horrible for the electron.

What happens, probably, is that the mass of the pion is not a pure QCD object, it has got mixings and corrections.

Of course the thing I don't like about this is that if you expect Koide formulas to operate between two leptons and a meson you open up a great deal of numerology.

That paper I wrote on Koide formulas for hadrons was based on the assumption that the Koide formula applied to excitations of quantum objects bound by color, where the excitations happen to be such that the usual quantum numbers (angular momentum) are left unchanged.

When I get some time, I plan on rewriting the paper as a statement on the relationship between the old string theory (for hadrons) and the new string theory (for leptons). That is, if both these objects are naturally modeled by string theories, we should expect to see echoes of the lepton generation structure in the hadrons. And since it's pretty clear that the generations come in 3s, so should the hadron excitations. But for this to work, I've also got to follow the assumption that the hadron excitations share quantum numbers in a manner similar to how the three generations share quantum numbers.

 

[arivero]

Of course the thing I don't like about this is that if you expect Koide formulas to operate between two leptons and a meson you open up a great deal of numerology.

No, note I used phase_mu and not phase_pi in the notation . I was assuming unbroken supersymmetry between the pion and the muon. Of course it fails.

On other hand, taking the fundamental mass of Koide equation equal to the constituient mass of the quark and just setting the electron mass to zero works fine to predict muon and tau.

Some texts refer to the constituient mass as the "renormalised quark mass", I am not sure if this interpretation is right, but it makes sense to think that the mass of the electron should be equal to the mass of the quark, and that this equality is preserved in a subtle way in Koide equations.

 

[arivero]

A new notation
{\overline \sqrt m}^2= {1 \over 2} \bar m

 

[CarlB]

I can't figure out that notation, maybe it's my browser (Fireforx).

Meanwhile, in my quest to get into grad school, I just got back my results for the physics GRE test; I maxed it out, 990 out of 990. If nothing else, it means that my applications will be carefully read.

 

[arivero]

I can't figure out that notation, maybe it's my browser (Fireforx).

Meanwhile, in my quest to get into grad school, I just got back my results for the physics GRE test; I maxed it out, 990 out of 990. If nothing else, it means that my applications will be carefully read.

Hmm do you need the GRE for graduate too? I thought it was for undergraduate? While it is important in any case, it is not the same to be in the 1% top sample of all the physics undergrads that the 1% of all the physics grads. Absolute congratulations, of course.

The notation is:

square root of a n-tuple (term by term)
average of the tuple (thus a single term)
square of it

equal to

one half of
average of the same n-tuple.

If instead of one-half, we put unity, then the only solution is the degenerated one. The -arguable- advantage of the notation is that it hiddens the number of generations.

 

[CarlB]

Okay I got it. Actually I end up using a simplified notation like this when I do computations that involve of lot of these.

The GRE is required for students wishing to enter physics graduate school in most schools in the US. Students typically take it while a senior. It's supposed to cover only undergraduate work. A 990 is around the 95% percentile this year, according to the voice I heard over the phone but I understand that that sometimes changes. To get a good score mostly means that you know a wide variety of basic physics and don't make a lot of mistakes in a timed test. In any case, I maxed it out so at least it means that they can't reject me from graduate school because of a low GRE.

 

[arivero]

A 990 is around the 95% percentile this year, according to the voice I heard over the phone but I understand that that sometimes changes.

Funny, I had expected them to calibrate each year, to set 500=50%, 990=99%. Ok, of course, not gaussian shape.

 

[CarlB]

They recalibrate the tests so as to maintain an approximately equivalent scores but they are scaled linearly. They give the percentages separately.

The high scores are dominated by international students as the best of the international students outnumber the best of the US students. But for money reasons, US schools give preference to US students. A plot of "highest ranked" institution making an offer of admission versus Physics GRE score is given here:
http://www.physicsgre.com/viewtopic.php?f=1&t=3474#p29723

It's not necessary to get a 990 to get into the top schools but it helps.

 

[arivero]

Koide [Y. Koide, Lett. Nuov. Cim., 34 (1982), 201]:
tan \theta_c = \left(\frac{\sqrt{m_\mu}-\sqrt{m_e}}{2\sqrt{m_\tau}}-\sqrt{m_\mu}-\sqrt{m_e}}\right)^{1/3}

Hmm there are some typos in the LaTeX source. From Phys. Rev. Lett. 47, 1241–1243 (1981), it is

tan \theta_c = \frac{\sqrt 3 (\sqrt{m_\mu}-\sqrt{m_e})}{2\sqrt{m_\tau}-\sqrt{m_\mu}-\sqrt{m_e}}

I guess that this is a manip of the same formula used for the phase delta_1 here, isn't it? Actually I think it is more elegant to see it as a phase that as Cabibbo angle, but it could be interesting to review the argument of Koide to tell that it is the Cabibbo angle

Also, I found that Koide and Oneda did some use of the same kind of formulae for mesons here
http://ptp.ipap.jp/link?PTP/81/199/
http://prd.aps.org/abstract/PRD/v36/i3/p815_1