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Natural units

  1. Feb 24, 2016 #1
    1. The problem statement, all variables and given/known data

    In natural units, the inverse lifetime of the muon is given by

    ##\tau^{-1}=\frac{G_{F}^{2}m^{5}}{192 \pi^{3}}##,

    where ##m## is the muon mass, ##106\ \text{MeV}##. What is the dimension of ##G_{F}## in natural units? Put in the factors of ##\hbar## and ##c## so that the equation can be interpreted in conventional units as well. From this, find the lifetime in seconds if ##G_{F}=1.166 \times 10^{-11}## in ##\text{MeV}## units.

    2. Relevant equations

    3. The attempt at a solution

    The dimension of ##\tau^{-1}## is ##\text{M}## in natural units. Therefore,

    ##[\tau^{-1}]=[G_{F}]^{2}\ [m]^{5}##

    ##\text{M} = [G_{F}]^{2}\ \text{M}^{5}##

    ##[G_{F}] = \text{M}^{-2}##.

    Therefore, the dimension of ##G_{F}## is ##\text{M}^{-2}##.

    Am I correct so far?
     
  2. jcsd
  3. Feb 24, 2016 #2
  4. Feb 28, 2016 #3

    nrqed

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    Yes, you are correct.
     
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