Calculating Fermi Coupling Constant (muon)

[tryingtolearn1]

Homework Statement

Fermi Coupling Constant (muon)

Relevant Equations

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

I am trying to determine the Fermi Coupling Constant which is measured to be $1.1663787 *10^{-5}\text{Ge}V^{-2}$. The formula for Fermi is $\frac{G_F}{(\hbar c)^3}=\sqrt{\frac{\hbar}{\tau_\mu}\cdot\frac{192\pi^3}{(m_\mu c^2)^5}},$ where $m_\mu$ is the mass of a muon which is $\approx 0.105$GeV, $c$ is the speed of light $\approx 3*10^8$m/s, $\hbar$ is reduced plank constant which is $\hbar\approx 6.582\cdot 10^{-25}\,\rm{GeV}\cdot\rm{s}$ and $\tau_\mu$ is the mean decay time of a muon which is $\approx 2.2*10^{-6} s$. Now plugging all these values into the formula gives $$\frac{G_F}{(\hbar c)^3}=\sqrt{\frac{\hbar}{\tau_\mu}\cdot\frac{192\pi^3}{(m_\mu c^2)^5}}$$
$$=\sqrt{\frac{6.582*10^{-25}*192*\pi^3}{2.2*10^{-6}*(0.105*(3*10^8)^2)^5}}\approx 4.83157 × 10^{-48}$$ which is totally off from the expected value of $1.1663787 *10^{-5}\text{Ge}V^{-2}$. Not sure what I am doing wrong?

 

[TSny]

$m_\mu$ is the mass of a muon which is $\approx 0.105$GeV

Consider why the mass of the muon is expressed in units of energy.

 

[tryingtolearn1]

Consider why the mass of the muon is expressed in units of energy.

Ops I meant the mass of the muon is $\approx 0.105GeV/c^2$ but even with that unit correction I still get a value that is way off.

 

[TSny]

Ops I meant the mass of the muon is $\approx 0.105GeV/c^2$ but even with that unit correction I still get a value that is way off.

Did you change anything in your first attempt shown below?

$$\frac{G_F}{(\hbar c)^3}=\sqrt{\frac{6.582*10^{-25}*192*\pi^3}{2.2*10^{-6}*(0.105*(3*10^8)^2)^5}}\approx 4.83157 × 10^{-48}$$

If so, can you show your new calculation? It would help if you show all units for all the numbers in your calculation. See if the units cancel to give the desired units for the answer.

 

[TSny]

the mass of the muon is $\approx 0.105GeV/c^2$

Using this value for m, what is the value of mc² in units of GeV?

 

[tryingtolearn1]

Did you change anything in your first attempt shown below?

If so, can you show your new calculation? It would help if you show all units for all the numbers in your calculation. See if the units cancel to give the desired units for the answer.

My new calculation using only dimensional analysis is $$\sqrt{\frac{GeV\cdot s}{s}\cdot\frac{1}{[(\frac{GeV\cdot s}{c^2}) \cdot c^2]^5}}=GeV^{-2}\cdot s^{-5/2}$$ which has an extra $s^{-5/2}$.

 

[tryingtolearn1]

Using this value for m, what is the value of mc² in units of GeV?

Actually I figured out my issue. I was include an extra factor of $c^5$. Inside the square root it should have been $.105^5$ instead of $(.105*(3*10^8)^2)^5$. Ty