Saturday, November 6, 2010

Theory Update 19

Alejandro Rivero reminds us that the physical meaning of the scale factor in Koide triplets should also be considered. Let us recall the scales for lepton and quark triplets. Fitting charged leptons to the eigenvalue formula

$\sqrt{m_i} = \mu (1 + r \cos (\theta + 2 \pi i/n))$

at $r = \sqrt{2}$ and $\theta = 2/9$, we find a natural scale factor of $\mu = 313.8$ MeV, which is very close to the dynamical (chiral limit) quark mass $m_{N} /3 = 313$ MeV, where $m_N$ is the neutron mass. The up quark Koide triplet requires a scale factor (at $r = 1.76$) of $\mu = 22.85$ GeV. This corresponds to a natural mass of

$M = 3 \times 22.85 = 68.6$ GeV

which is still mysterious, but does appear (for instance) as a pseudoscalar boson mass in little fairy models. It also corresponds to an outdated estimate of the top quark mass, based on a formula

$\sum (m_{i})^2 = (m_{W})^{2} (\frac{3}{2} + \frac{3}{4 C^2})$

where $m_W$ is the W boson mass and $C = \cos \theta_{W}$ uses the Weinberg angle. Alejandro was hoping for a Koide scale of $246$ GeV, which is a little over $3 m_W$. This may be obtained with a lepton scale (in GeV) defined by

$\frac{246.9}{2} = (\frac{3}{2} (m_W)^2 + \frac{2}{3} (m_Z)^2)^{0.5}$

So the two natural Koide scales exhibit a physical complementarity between leptons and quarks, as expected.


  1. In other words, Veltman is saying that the $246$ GeV comes from the sum of squares of all fermion masses. So if we have all the fermion masses, we get the $246$ GeV, whether or not there are fairy fields. Now the correction I made to the RHS of the mass relation corresponds to taking six times $m_W$, three times $m_Z$ and then a fairy factor of $- 1/3$ times $m_Z$, which is after all the favoured mass for the fairy field.

  2. Also, Mz*sin(0w)/313.86MeV=137.09

    and exp(-4pi*alpha) = .9123777

    (4pi*alpha is natural SQUARE of charge unit in Heaviside-Lorentz units, where c=uo=eo=Zo=hbar=1)

    Upon further defining the lepton scale mass (in kg) as a new unit, then multiplying by c, gives a natural unit of momentum, p.

    It turns out that e^2/p^2 = .912388

    (e is just elem. chg in SI units)

    This would assume charge is momentum, as it is in the force-current analogy between electrical and mechanical systems, which preserves the topology between them. See


  3. Hi Dave! Note that you can use dollar signs around the math for latex formatting here. So hbar becomes $\hbar$ when you put a backslash before the hbar bit.

    I like your approximation for $\alpha$ in terms of the lepton scale, because it is close to my favourite approximation:

    $\sqrt{\alpha^{-1}} = 4 \textrm{cosh} (2 \pi / 2 + \phi)$

    where $\phi$ is the golden ratio.

  4. Actually, with my $\alpha$ it is better to say

    $m_Z \sin \theta = 313.85 \times \alpha^{-1}$

    for $\theta$ such that $(\sin \theta)^2$ is close to $2/9$, not the Weinberg angle.

  5. ... which would define the $Z$ boson mass in terms of the golden ratio $\phi$, the lepton scale and the number $2/9$. Hmmm.

  6. And my $\alpha$ may be viewed as a Hopf link Jones invariant associated to a quantum dimension given by $\phi$, as I noted some years ago.

  7. In other words, we have the simple formula

    $m_Z = 313.8 \times \alpha^{-1} (3/ \sqrt{2})$ MeV

  8. There is a general observation that between the four fundamental scales, electron, muon (and pion), tau (and proton) and Z/W, the two separations electron-muon and tau-Z seem to be of order alfa. Before the discovery of the third generation, some mainstream work was done trying to generate the mass of the electron as a order alpha variation of the one of the muon.

  9. I suposse that it was known by numerologists the fact that the mass of the nucleon was approximated by the average of the three generations of leptons (and thus the mass of the quark by 1/3 of it), but probably it was disregarded as a coincidence even by the maddest crackpot. At least I have not seen evidence of its use in any paper.

    I think now that it is remarkable by two reasons. First, that it is a 99.8% match, as good as the one between the top quark mass (times $\sqrt 2$) and the electroweak scale (the 246 GeV). And it deserves a second though because, as Kea has explained, it appears as a fundamental quantity in Koide equation. In fact it is explicit if you write Koide's as a "comparision of averages",

    ${\bar \sqrt m}^2= {1 \over 2} \bar m$

  10. Alejandro, I'm afraid that only the inline commands work well in the comments. Sorry about that. You can just use $1/2$, for instance.

    And yes, the Koide relations certainly lend credence to these simple mass rules. If we could get a better understanding the low $E$ running behaviour in the Koide framework, we could further cut down the SM parameters. Note that the Weinberg angle is usually given in terms of

    $x = (\sin \theta_{W})^2 = 0.23119$

    which matches one of the CKM parameters. This is probably not a coincidence either, since both $x$ and the CKM entries appear in the branching ratios for Z and W bosons.


Note: Only a member of this blog may post a comment.