How is the Fermi coupling constant related to the muon decay lifetime?

In summary, we discussed the calculation of the Fermi coupling constant and muon decay lifetime, as well as the muon decay width and lifetime. We also looked at the I(x) function and its significance in the calculation. Additionally, we explored the contribution from second order Feynman amplitudes and the electroweak fine structure constant. Lastly, we discussed the proton decay width and X boson mass in relation to the strong interaction.
  • #1
Orion1
973
3

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:
http://www.physics.union.edu/images/summer06/pochedley.pdf" [Broken]
http://en.wikipedia.org/wiki/Muon" [Broken]
http://en.wikipedia.org/wiki/Physical_constant" [Broken]
http://books.google.com/books?id=-S...=M5VYRBiseTeT87rr7tjglfO6AAo&hl=en#PPA149,M1"
 
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  • #2
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
 
  • #3
malawi_glenn said:
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]
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...
 
  • #4
yes, with all that, I obtained lifetime = 2.1888 * 10^-6 s
 
  • #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:
http://en.wikipedia.org/wiki/Electron" [Broken]
http://en.wikipedia.org/wiki/Muon" [Broken]
 
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  • #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:
http://en.wikipedia.org/wiki/Muon" [Broken]
http://en.wikipedia.org/wiki/Physical_constant" [Broken]
 
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  • #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:
http://en.wikipedia.org/wiki/Muon" [Broken]
http://en.wikipedia.org/wiki/Physical_constant" [Broken]
 
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  • #8
Wery good! Now do the contribution from second order feynman amplitudes =D
 
  • #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:
http://books.google.com/books?id=-S...=M5VYRBiseTeT87rr7tjglfO6AAo&hl=en#PPA149,M1"
http://en.wikipedia.org/wiki/Proton_decay" [Broken]
http://en.wikipedia.org/wiki/W_and_Z_bosons" [Broken]
http://en.wikipedia.org/wiki/X_and_Y_bosons" [Broken]
http://en.wikipedia.org/wiki/Electronuclear_force" [Broken]
http://en.wikipedia.org/wiki/Grand_unification_theory" [Broken]
http://hyperphysics.phy-astr.gsu.edu/hbase/astro/unify.html#c1"

malawi_glenn said:
It is a strong interaction!
 
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  • #10
What are you doing?

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

What is muon decay lifetime?

Muon decay lifetime is the average amount of time it takes for a muon particle to decay into other particles.

Why is muon decay lifetime important?

Muon decay lifetime is important because it provides insight into the fundamental properties of particles and can help us better understand the behavior of the universe.

What affects the muon decay lifetime?

The muon decay lifetime is affected by factors such as temperature, magnetic fields, and the presence of other particles.

How is the muon decay lifetime measured?

The muon decay lifetime can be measured using particle accelerators or by observing natural muon decay in cosmic rays. Scientists also use mathematical models to predict and verify the decay lifetime.

What can the muon decay lifetime tell us about the muon particle?

The muon decay lifetime can tell us about the stability and interactions of the muon particle, as well as its mass and charge. It can also provide evidence for new physics beyond the Standard Model.

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