Thursday, October 21, 2010

Theory Update 11

Recall that back in July we were looking at Koide fits for the quark masses, and it was noted that the quark equivalent of $r = \sqrt{2}$ (the parameter for the leptons) is probably around $1.76$. Today, in a comment on the previous post, someone called Dave (who I hope will soon better identify himself) has shown us (see diagram below) how simple geometry predicts a value of

$r = 1.7602838$

since the leptons lie on a circle with equation $x^2 + y^2 = r^2$ and so do the (up) quarks, and then we inscribe the blue lepton triangle inside the red quark triangle. The lepton triangle side has length $2.4494987 = \sqrt{6}$, while the quark triangle side has length $3.048901024$. This now allows us to make accurate predictions for the quark masses!

21 comments:

  1. Not to mention the CKM matrix ... recall that the (offset by 1) Koide eigenvalues could be expressed in terms of a cosine and sine, when we wrote them out with mixing matrix coefficients. So that is what these $x$ and $y$ parameters are about.

    Ok, now there must be several billion people on the planet who could figure this out ... except maybe the string theorists.

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  2. So plugging this $1.7603$ into the Koide formula, and choosing a suitable scale, we get the masses:

    1. up quarks, at a phase of $2/27$:
    $17159.144$, $0.202$, $124.905$

    2. down quarks, at a phase of $4/27$:
    $4282.462$, $5.201$, $71.639$

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  3. Note that for both leptons and quarks, the triangle side length $L$ obeys

    $(L/r)^2 = 3$

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  4. So the bigger triangle is completely defined by the ansatz of phases: sending the $2/9$ for leptons to $2/27$ for the quarks. Presumably we could keep dividing angles by $3$ and come up with an infinite sequence of Koide eigenvalue triplets this way, with the $r$ factor always determined by inscribing.

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  5. Good to have a clear prediction. Wikipedia gives the recent bounds on masses (see this). The most recent estimates for u and d quark masses are 5 and 20 MeV.

    I assume that the order of masses is t,u, c and b,d, s and unit is GeV.


    For d the prediction would be at the lower boundary of 2-15 MeV range. For s 71.639 MeV is below 100-130 MeV. For b prediction would be in the required range.

    For U type quarks the predictions would be by order of magnitude too small as compared to the lower bounds.

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  6. Using the known relationship:

    Mt/sqrt2=(cosOw+sinOw)Mz
    (Ow is Wienberg angle)

    to scale the Top mass, we have:

    2.05MeV 1.27GeV 174.55GeV

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  7. Cool, Dave. Yes, I was a bit rough with the scale choice above. And I think the down quark radius should be reselected from the phase choice. It was the down quarks that did not quite fit the 1.76 before.

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  8. Actually, no, the down quarks work best at $1.76$, which comes from the phase $\theta = 2/27$ in the relation

    $r = \sqrt{2} \sin (5 \pi/6 - 2/9 + \theta) / \sin (\pi/6)$

    Then the best Koide fits give us:

    1. up quarks at $\mu = 22990$,
    $174.553$ GeV, $1.271$ GeV, $2.054$ MeV

    2. down quarks at $\theta = 4/27$, $\mu = 635$,
    $4771$ MeV, $79.8$ MeV, $5.8$ MeV

    and the down quark set really pushes the experimental limits, as before.

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  9. ... or maybe we should try to put the down quarks on the big triangle?

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  10. One other possibility for the down quarks is to set $r$ at the average of $r(2/27) = 1.76$ and $r(2/9) = \sqrt{2}$. Then at $\mu = 710$ and $\theta = 5/54$ the triplet is:

    $4728$ MeV, $80.57$ MeV, $4.85$ MeV

    At least the experimental bounds really are testing these different scenarios.

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  11. compare to the mass of top quark.

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  12. Oh, I didn't bother actually looking up the latest top quark estimates. I'm sure the experimentalists can do that, and fix up the Koide fits if necessary.

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  13. So according to PDG, we just need to down scale $\mu$ a little bit.

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  14. Actually, who was the first one suggesting the triangle presentation of Koide's angles, and in which paper or blog entry? I guess it was Carl, but I am not sure, it is the first time I see it.

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  15. Ok, it is http://www.brannenworks.com/koidehadrons.pdf formula 46

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  16. Ok, I see what puzzles me of the triangle interpretation: that there is not a clear role for the sqrt(2) in the radius of the circle, while in Foot's interpretation is the mark of an angle of 45 degrees with the unbroken "halfmass vector".

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  17. Alejandro, I believe it was Hans de Vries who first pointed the triangle out, some years ago. And yes, there are many ways to look at the $\sqrt{2}$ geometrically ... but two dimensions at a time, please!

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  18. After some thought, yes, it could be that the 45 degrees angle is a redherring. The "angle with 1,1,1", sqrt(2/3), has really two different pieces: the sqrt(3) comes from the number of families, and the sqrt(2), is the size of the symmetry breaking, as it appears very naturally in eq 46 of Brannen's.

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  19. Oh, and in the mixing matrix, the parameter $1$ gives an angle of 45 degrees. The $\sqrt{2}$ instead gives an angle of $0.615$ radians, whatever that means.

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  20. Ah, the $0.615$ seems to come from the inverse sine of $1/ \sqrt{3}$. That links the $2$ and the $3$ then.

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  21. That is, the number of generations ($= 3$) gives a sine value, and if we use this angle with a tan function (like in my mixing matrices) then it gives the $\sqrt{2}$.

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