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I ## \mu~\to e~ \gamma ## decay width and neutrino propagator

  1. Feb 4, 2017 #1
    Hi all,

    I'm studying ## \mu \to e~ \gamma ## decay from cheng & Lie' book " gauge theory of elementary particles ". In Equation (13.84), he wrote the neutrino propagator
    ## \sum_i \Big ( \frac{U^{*}_{ei} U_{\mu i}}{(p+k)^2-m_i^2} \Big), ##
    (where the sum taken over neutrinos flavors) in the form:

    ##\sum_i U^{*}_{ei} U_{\mu i} \Big ( \frac{1}{(p+k)^2} + \frac{m_i^2}{[(p+k)^2]^2} + ...... \Big) ##
    ##= \sum_i \frac{U^{*}_{ei} U_{\mu i} m_i^2}{[(p+k)^2]^2} + ........##

    Do any one know how did he drive this ? Then he write that:

    the leading term vanishes, ## \sum_i U^{*}_{ei} U_{\mu i}##, reflecting the GIM cancellation mechanism.

    I can't get this statement ..


    Thanks,
     
  2. jcsd
  3. Feb 4, 2017 #2

    Orodruin

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    No, the sum should be taken over the neutrino masses. The flavour eigenstates do not have definite masses.

    Just expand the quotient for small ##m_i##.

    It follows directly from the unitarity of ##U##. It is the ##\mu##-##e## element of ##U U^\dagger = 1##.
     
  4. Feb 4, 2017 #3
    Hi, thanks for replying:

    Can you please write the general form, like for instance taylor expansion, it seems i'm not so good in math !

    Actually this still not clear for me , ##\sum_i U^{*}_{ei} U_{\mu i} ## is multiplied by the first term ## \frac{1}{(p+k)^2} ## as well as the second term ## \frac{m_i^2}{[(p+k)^2]^2} ##, so why the first one which has been ignored .. also what's GIM mechanism and he says due to this mechanism this first leading term vanishes ..
     
  5. Feb 4, 2017 #4

    Orodruin

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    $$
    \frac{1}{1+x} = 1 - x + x^2 + \ldots
    $$

    The first term ## \frac{1}{(p+k)^2} ## is independent of ##i## and can be taken out of the sum. The second term is not independent of ##i##.
     
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