Muon decay lifetime

  • Thread starter Orion1
  • Start date
  • #1
970
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"
 
Last edited by a moderator:

Answers and Replies

  • #2
malawi_glenn
Science Advisor
Homework Helper
4,786
22
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
2,425
7
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
malawi_glenn
Science Advisor
Homework Helper
4,786
22
yes, with all that, I obtained lifetime = 2.1888 * 10^-6 s
 
  • #5
970
3

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]
 
Last edited by a moderator:
  • #6
970
3

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]
 
Last edited by a moderator:
  • #7
970
3

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]
 
Last edited by a moderator:
  • #8
malawi_glenn
Science Advisor
Homework Helper
4,786
22
Wery good! Now do the contribution from second order feynman amplitudes =D
 
  • #9
970
3

[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!
 
Last edited by a moderator:
  • #10
malawi_glenn
Science Advisor
Homework Helper
4,786
22
What are you doing?

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

Related Threads on Muon decay lifetime

  • Last Post
Replies
4
Views
2K
  • Last Post
Replies
10
Views
4K
  • Last Post
Replies
5
Views
4K
  • Last Post
Replies
0
Views
2K
  • Last Post
Replies
15
Views
6K
Replies
8
Views
1K
  • Last Post
Replies
3
Views
3K
  • Last Post
Replies
5
Views
2K
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
1
Views
637
Top