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Muons and tauons as KK modes?

  1. Jul 3, 2013 #1
    Muons and tauons as KK modes??

    Hello, everybody:

    I have (what I think is) a silly question. We all know that μ and τ particles are just as electrons, but with larger mass. Could it be that they are heavier Kaluza-Klein modes of the e in a higher dimensional theory, where the extra dimension is compactified?

    I know I'm talking without mentioning a specific model (I don't know any), but I wonder if it is plausible or if there is a physical/mathematical argument against such a thing.

  2. jcsd
  3. Jul 3, 2013 #2


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    It's an interesting question. The KK modes of a scalar field have masses given, in appropriate units, by roughly

    $$ m_n^2 = m_0^2 + \left(\frac{n}{R}\right)^2,$$

    where ##m_0## is the mass of the field in the higher dimensional theory, and ##R## is a characteristic length scale of the of the compact dimensions. For example, if there is one circular extra dimension, ##R## is the radius, but in some more complicated example it would be related to an appropriate root of the volume.

    In your model, ##m_0 \sim m_e## and

    $$ m_\mu^2 = m_e^2 + \frac{1}{R^2},$$


    $$R \sim \frac{1}{106~\mathrm{MeV}} \sim 2 \cdot 10^{-15} ~\mathrm{m}.$$

    So the extra dimensions would have to be of femtometer size, which is the same order of the size of the proton. Such a large scale extra dimension would have been seen in experiments and, in fact, the present limits on extra dimensions is that they have to be at least ##10^{-4}## times smaller than this value (##\sim 1 ~\mathrm{TeV}## in terms of energy scales).

    There is an easy way to see that such large extra dimensions would be ruled out. Namely the other particles of the SM would have to have KK partners, so, for example, we would need a particle with the same quantum numbers as the photon with a mass almost equal to the muon mass. There is no such particle.
  4. Jul 4, 2013 #3


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    For leptons, there should be particles with 2*muonmass-electronmass, 3*muonmass-2*electronmass and so on. Those particles are missing, and the tau does not even fit in that pattern.

    In addition, I think the decays of the muon do not fit to that hypothesis.
  5. Jul 4, 2013 #4
    That'll do!! Thanks!! It didn't occur to me to look for the size of the extra dimension...

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