Help calculating the Fermi coupling constant from the muon lifetime

[martinhiggs]

Homework Statement

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

Homework Equations

$$G_F=\sqrt{\frac{192.pi^{3}}{\tau.m_{\mu}}}$$

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!

 

[афк]

Mass of the muon as 0.105GeV/c²

You are missing the 1/c² in the mass of the muon.

 

[martinhiggs]

I get an even larger result if I divide by c^2...

 

[афк]

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".

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

Are you using the following formula for the muon decay?

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

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

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

$$\hbar\approx 6.582\cdot 10^{-25}\,\rm{GeV}\cdot\rm{s}$$

If you now insert your value for the muon life time, you should get a reasonable result.

 

[martinhiggs]

Ah yes, got it! Thank you for your help, I've spent all morning trying to work it out! :)