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Tau lepton branching ratios

  1. Apr 29, 2016 #1

    Fek

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    1. The problem statement, all variables and given/known data
    Branching ratios of tau lepton decay:
    hadrons 66%
    muon 17%
    electron 17%

    Use your knowledge of the decay of W bosons to justify these rates.

    2. Relevant equations


    3. The attempt at a solution

    The W+ boson can decay to any combination of quarks with +1 charge, but the CKM matrix suppresses many of these heavily, so lets just consider ud, us, cd, cs (where d/s are anti).

    BR (electron) = g^2(w) / (3*((2g^2(w) cos^2x + 2g^2(w)sin^2x) + 2g^2(w)
    = 1( 6 + 2) = 1/8

    where g^2(w) cos^2x is the matrix element squared for ud and and cs
    g^2(w)sin^2x is the matrix element squared for us and cd
    g^2(w) is the matrix element squared for any lepton.

    This doesn't give the right answer. Furthermore shouldn't density of states be very important given the fact that the lepton channel is 3 body, and in real life the tau decays into 3 or 4 particles most commonly?
     
  2. jcsd
  3. Apr 30, 2016 #2

    mfb

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    2016 Award

    Staff: Mentor

    You ignored tb for a good reason. There is something else you should ignore for the same reason.

    That is a smaller effect from hadronization.
     
  4. May 7, 2016 #3

    Fek

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    I'm afraid I just can't see it. I'v
    Many thanks for your reply. I can see I shouldn't have include the charm containing mesons as they are slightly to heavy. This brings the branching ratios to 50% hadrons and 25% for each lepton. Have I missed something else or is the remaining discrepancy justified by a density of states argument? Many thanks.
     
  5. May 7, 2016 #4

    mfb

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    What happened to the quark colors now? The previous calculation seemed to have those.
     
  6. May 8, 2016 #5

    Fek

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    Sorry made a silly mistake. So the BR should be 60% for hadrons and 20% for the electron
    Sorry made a silly mistake. BR should now be 60% for hadrons (3 * cos^2(x) + sin^2(x)) / (3 * (cos^2(x) + sin^2(x)) + 1 + 1 ) , and 20% for each lepton. Many thanks.
     
  7. May 8, 2016 #6

    mfb

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    Right, and that is close to the actual values. You can get an even better approximation if you take kaons into account (with up, not with charm).
     
  8. May 9, 2016 #7

    Fek

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    Have I not taken into account Kaons with the Cabibo factors (cos(x)) for non mixed states, and sin(x) for mixed states?
     
  9. May 9, 2016 #8

    mfb

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    Well, I don't understand your numerator. Why is there no 3 for the sin term?
    But sin and cos together should work, and give 20%, right. Okay, blame phase space for the rest.
     
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