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

[Orion1]

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

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

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

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

Muon decay width and lifetime: ?
$$\Gamma_{\mu} = \frac{1}{\tau_{\mu}}$$

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

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

Also, Wikipedia ref. 2 does not explain what the $$I(x)$$ function is, or what $$x$$ represents.

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

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

key:
$$G_F$$ - Fermi coupling constant
$$m_{e}$$ - electron mass
$$m_{\mu}$$ - muon mass

Reference:
http://www.physics.union.edu/images/summer06/pochedley.pdf"
http://en.wikipedia.org/wiki/Muon"
http://en.wikipedia.org/wiki/Physical_constant"
http://books.google.com/books?id=-S...=M5VYRBiseTeT87rr7tjglfO6AAo&hl=en#PPA149,M1"

 

[malawi_glenn]

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

i.e. we assued $$I(\frac{m_e^2}{m_\mu^2}) = 1$$



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

Then convert the witdh $$\Gamma$$ into S.I units, i.e Joule

Then, at last: $$\tau = \hbar / \Gamma$$

Good luck

 

[humanino]

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

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...

 

[malawi_glenn]

yes, with all that, I obtained lifetime = 2.1888 * 10^-6 s

 

[Orion1]

Thanks malawi glenn and humanino for your collaboration!

$$x = \frac{m_e^2}{m_\mu^2} << 1$$

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

$$\boxed{x = 2.33901042277445 \cdot 10^{- 5}}$$

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

Reference:
http://en.wikipedia.org/wiki/Electron"
http://en.wikipedia.org/wiki/Muon"

 

[Orion1]

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

Wikipedia Muon lifetime:
$$\tau_{\mu} = 2.197034 \cdot 10^{- 6} \; \text{s}$$

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

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

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

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

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

Reference:
http://en.wikipedia.org/wiki/Muon"
http://en.wikipedia.org/wiki/Physical_constant"

 

[Orion1]

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

$$\boxed{\tau_{\mu} = 2.19703403501795 \cdot 10^{- 6} \; \text{s}}$$

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

Reference:
http://en.wikipedia.org/wiki/Muon"
http://en.wikipedia.org/wiki/Physical_constant"

 

[malawi_glenn]

Wery good! Now do the contribution from second order feynman amplitudes =D

 

[Orion1]

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

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

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

$$\boxed{\alpha_w = 9.77054112064435 \cdot 10^{- 4}}$$

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

$$\Gamma_p = \frac{\hbar}{10^{9} e \tau_p} = \alpha_s^2 \frac{m_p^5}{m_X^4}$$

$$\boxed{\Gamma_p = 2.08717693773387 \cdot 10^{- 67} \; \text{GeV}}$$

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

$$\boxed{m_X = 4.32037202924731 \cdot 10^{16} \; \text{GeV}}$$

Reference:
http://books.google.com/books?id=-S...=M5VYRBiseTeT87rr7tjglfO6AAo&hl=en#PPA149,M1"
http://en.wikipedia.org/wiki/Proton_decay"
http://en.wikipedia.org/wiki/W_and_Z_bosons"
http://en.wikipedia.org/wiki/X_and_Y_bosons"
http://en.wikipedia.org/wiki/Electronuclear_force"
http://en.wikipedia.org/wiki/Grand_unification_theory"
http://hyperphysics.phy-astr.gsu.edu/hbase/astro/unify.html#c1"

It is a strong interaction!

 

[malawi_glenn]

What are you doing?

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