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Muon decay lifetime

  1. May 14, 2008 #1

    I am inquiring if anyone here is qualified to numerically calculate the following equation:

    Fermi coupling constant and Muon decay lifetime: (ref. 1)
    [tex]\frac{G_F}{(\hbar c)^3} = \sqrt{\frac{192 \pi^3 \hbar}{(m_{\mu} c^2)^5 \tau_{\mu}}[/tex]

    Muon decay lifetime: (ref. 2)
    [tex]\tau_{\mu} = 2.197034 \cdot 10^{- 6} \; \text{s}[/tex]

    According to ref. 3, the Fermi coupling constant is:
    [tex]\frac{G_F}{(\hbar c)^3} = 1.166391 \cdot 10^{- 5} \; \text{GeV}^{- 2}[/tex]

    Muon decay width and lifetime: ???
    [tex]\Gamma_{\mu} = \frac{1}{\tau_{\mu}}[/tex]

    However, according to ref. 2, the muon decay width is:
    [tex]\Gamma_{\mu} = \frac{G_F^2 m_\mu^5}{192\pi^3} I \left(\frac{m_e^2}{m_\mu^2}\right)[/tex]

    [tex]I(x)=1-8x+12x^2ln\left(\frac{1}{x}\right)+8x^3-x^4[/tex]

    Also, Wikipedia ref. 2 does not explain what the [tex]I(x)[/tex] function is, or what [tex]x[/tex] represents.

    I presume that:
    [tex]I(x) = I \left(\frac{m_e^2}{m_\mu^2}\right) \; \; \; x = \frac{m_e^2}{m_\mu^2}[/tex]

    Muon decay width: (ref. 4)
    [tex]\Gamma_{\mu} = 3 \cdot 10^{- 19} \; \text{GeV}[/tex]

    key:
    [tex]G_F[/tex] - Fermi coupling constant
    [tex]m_{e}[/tex] - electron mass
    [tex]m_{\mu}[/tex] - muon mass

    Reference:
    Muon lifetime, pg. 4
    Muon decays - Wikipedia
    Physical constants - Wikipedia
    Beuty for Beginners, pg. 149
     
    Last edited: May 14, 2008
  2. jcsd
  3. May 14, 2008 #2

    malawi_glenn

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    I did muon calculation last week infact, however we did fermi contact approximation and assumed [tex] \frac{m_e^2}{m_\mu^2} << 1 [/tex].

    i.e. we assued [tex] I(\frac{m_e^2}{m_\mu^2}) = 1 [/tex]



    Just use mass of muon= [tex] m_{\mu} = 0.105658369 \text{GeV} [/tex] and
    [tex] G_F = 1.166 \cdot 10^{-5} \text{GeV} ^{-1} [/tex]

    Then convert the witdh [tex] \Gamma [/tex] into S.I units, i.e Joule

    Then, at last: [tex] \tau = \hbar / \Gamma [/tex]

    Good luck
     
  4. May 14, 2008 #3
    It is easy to plug in the values and check that the more refined calculation provides a very small correction. Besides, wikipedia does give the appropriate reference...
     
  5. May 14, 2008 #4

    malawi_glenn

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    yes, with all that, I obtained lifetime = 2.1888 * 10^-6 s
     
  6. May 14, 2008 #5

    Thanks malawi glenn and humanino for your collaboration!

    [tex]x = \frac{m_e^2}{m_\mu^2} << 1[/tex]

    Dimensionless x value obtained:
    [tex]x = \frac{m_e^2}{m_\mu^2} = \frac{(0.00051099891844 \; \text{GeV})^2}{(0.105658369 \; \text{GeV})^2} = 2.33901042277445 \cdot 10^{- 5} \ll 1[/tex]

    [tex]\boxed{x = 2.33901042277445 \cdot 10^{- 5}}[/tex]

    [tex]I(x) = 1 - 8x + 12x^2 ln \left( \frac{1}{x} \right)+ 8 x^3 - x^4[/tex]
    [tex]I \left( \frac{m_e^2}{m_\mu^2} \right) < 1[/tex]
    [tex]\boxed{I \left( \frac{m_e^2}{m_\mu^2} \right) = 0.999812949171918}[/tex]

    Reference:
    Electron - Wikipedia
    Muon - Wikipedia
     
  7. May 14, 2008 #6

    Unit key:
    [tex]\Gamma_{\mu} = \text{GeV}[/tex] - Muon decay width
    [tex]m_{e} = \text{GeV}[/tex] - Electron mass
    [tex]m_{\mu} = \text{GeV}[/tex] - Muon mass
    [tex]\tau_{\mu} = \text{s}[/tex] - Muon lifetime

    Wikipedia Muon lifetime:
    [tex]\tau_{\mu} = 2.197034 \cdot 10^{- 6} \; \text{s}[/tex]

    Muon decay width:
    [tex]\Gamma_{\mu} = \frac{\hbar}{10^{9} e \tau_{\mu}} = \frac{G_F^2 m_{\mu}^5}{192 \pi^3} I \left( \frac{m_e^2}{m_\mu^2} \right) [/tex]
    [tex]e[/tex] - electron charge magnitude

    Muon decay width with leptonic correction term:
    [tex]\boxed{\Gamma_{\mu} = 3.00867837568648 \cdot 10^{- 19} \; \text{GeV}}[/tex]

    Fermi coupling constant:
    [tex]\boxed{G_F = \sqrt{ \frac{192 \pi^3 \hbar}{10^{9} e m_{\mu}^5 \tau_{\mu} I \left( \frac{m_e^2}{m_\mu^2} \right) }}} [/tex]

    Solution for Fermi coupling constant with Wikipedia Electron and Muon mass and Muon lifetime and leptonic correction term:
    [tex]\boxed{G_F = 1.16391365532758 \cdot 10^{- 5} \; \text{GeV}^{- 2}}[/tex]

    Wikipedia Fermi coupling constant:
    [tex]\boxed{G_F = 1.166391 \cdot 10^{- 5} \; \text{GeV}^{- 2}}[/tex]

    Reference:
    Muon - Wikipedia
    Physical constants - Wikipedia
     
    Last edited: May 14, 2008
  8. May 14, 2008 #7

    Muon lifetime:
    [tex]\boxed{\tau_{\mu} = \frac{192 \pi^3 \hbar}{10^{9} e G_F^2 m_{\mu}^5 I \left( \frac{m_e^2}{m_\mu^2} \right)}}[/tex]

    [tex]\boxed{\tau_{\mu} = 2.19703403501795 \cdot 10^{- 6} \; \text{s}}[/tex]

    Wikipedia Muon lifetime:
    [tex]\boxed{\tau_{\mu} = 2.197034 \cdot 10^{- 6} \; \text{s}}[/tex]

    Reference:
    Muon - Wikipedia
    Physical constants - Wikipedia
     
    Last edited: May 14, 2008
  9. May 15, 2008 #8

    malawi_glenn

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    Wery good! Now do the contribution from second order feynman amplitudes =D
     
  10. May 15, 2008 #9

    [tex]\Gamma_{\mu} = \frac{G_F^2 m_{\mu}^5}{192 \pi^3} I \left( \frac{m_e^2}{m_\mu^2} \right) = \alpha_w^2 \frac{m_{\mu}^5}{m_W^4}[/tex]

    key:
    [tex]\alpha_w[/tex] - electroweak fine structure constant
    [tex]m_W = 80.398 \; \text{GeV}[/tex] - W Boson mass

    Electroweak fine structure constant:
    [tex]\boxed{\alpha_w = G_F m_W^2 \sqrt{\frac{I \left( \frac{m_e^2}{m_\mu^2} \right)}{192 \pi^3}}}[/tex]

    [tex]\boxed{\alpha_w = 9.77054112064435 \cdot 10^{- 4}}[/tex]

    key:
    [tex]\alpha_s = 1[/tex] - strong fine structure constant
    [tex]m_p = 0.9382720298 \; \text{GeV}[/tex] - Proton mass
    [tex]m_X[/tex] - X Boson mass
    [tex]\Gamma_p[/tex] - Proton decay width
    [tex]\tau_p = 3.1536 \cdot 10^{42} \; \text{s} \; \; \; (10^{35} \; \text{years})[/tex] - Super-Kamiokande Proton decay lifetime

    [tex]\Gamma_p = \frac{\hbar}{10^{9} e \tau_p} = \alpha_s^2 \frac{m_p^5}{m_X^4}[/tex]

    [tex]\boxed{\Gamma_p = 2.08717693773387 \cdot 10^{- 67} \; \text{GeV}}[/tex]

    X Boson mass:
    [tex]\boxed{m_X = \left( \frac{10^9 e t_p m_p^5 \alpha_s^2}{\hbar} \right)^{\frac{1}{4}}}[/tex]

    [tex]\boxed{m_X = 4.32037202924731 \cdot 10^{16} \; \text{GeV}}[/tex]

    Reference:
    Beauty for Beginners, pg. 149
    Proton decay - Wikipedia
    W and Z bosons - Wikipedia
    X and Y bosons - Wikipedia
    Electronuclear force - Wikipedia
    Grand unification theory - Wikipedia
    Big Bang Expansion, Fundamental Forces - hyperphysics

     
    Last edited: May 15, 2008
  11. May 15, 2008 #10

    malawi_glenn

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    What are you doing?

    "It is a strong interaction" is my signature for all my posts:P
     
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