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Homework Help: Help calculating the Fermi coupling constant from the muon lifetime

  1. Dec 9, 2009 #1
    1. The problem statement, all variables and given/known data

    I have to find the Fermi Coupling Constant using my measurement of the muon lifetime. My measurement is 2.1786x10^-6s

    2. Relevant equations

    [tex]G_F=\sqrt{\frac{192.pi^{3}}{\tau.m_{\mu}}}[/tex]

    3. The attempt at a solution

    I tried plugging in the numbers that I have and I get ~495,000ish which obviously cannot be right as the value of the Fermi Coupling Constant is:

    1.166 37x10-5 GeV-2

    I used lifetime as 0.0000021786s
    Mass of the muon as 0.105GeV

    I think I have to put in factors of hbar and c somewhere, or change units or something, but it's driving me crazy and I can't figure it out!
     
  2. jcsd
  3. Dec 9, 2009 #2
    You are missing the 1/c² in the mass of the muon.
     
  4. Dec 9, 2009 #3
    I get an even larger result if I divide by c^2...
     
  5. Dec 9, 2009 #4
    Indeed, a few h's and c's are missing. I looked up the value for the http://physics.nist.gov/cgi-bin/cuu/Value?gf".

    [tex]\frac{G_F}{(\hbar\,c)^3}=1.16637\cdot 10^{-5}\,(\rm{GeV})^{-2}[/tex]

    Are you using the following formula for the muon decay?

    [tex]\Gamma_{\mu}=\frac{\hbar}{\tau_\mu}\approx\frac{G_F^2}{192\pi^3(\hbar c)^6}\cdot(m_\mu c^2)^5[/tex]

    Apart from SI/natural unit conversion troubles, the muon mass gets a different exponent compared to your formula.

    [tex]\Rightarrow \frac{G_F}{(\hbar c)^3}=\sqrt{\frac{\hbar}{\tau_\mu}\cdot\frac{192\pi^3}{(m_\mu c^2)^5}}[/tex]

    [tex]\hbar\approx 6.582\cdot 10^{-25}\,\rm{GeV}\cdot\rm{s}[/tex]

    If you now insert your value for the muon life time, you should get a reasonable result.
     
    Last edited by a moderator: Apr 24, 2017
  6. Dec 9, 2009 #5
    Ah yes, got it! Thank you for your help, I've spent all morning trying to work it out! :)
     
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