Finding the Dimension and Lifetime of G_F in Natural Units

[spaghetti3451]

Homework Statement

In natural units, the inverse lifetime of the muon is given by

$\tau^{-1}=\frac{G_{F}^{2}m^{5}}{192 \pi^{3}}$,

where $m$ is the muon mass, $106\ \text{MeV}$. What is the dimension of $G_{F}$ in natural units? Put in the factors of $\hbar$ and $c$ so that the equation can be interpreted in conventional units as well. From this, find the lifetime in seconds if $G_{F}=1.166 \times 10^{-11}$ in $\text{MeV}$ units.

Homework Equations

The Attempt at a Solution

The dimension of $\tau^{-1}$ is $\text{M}$ in natural units. Therefore,

$[\tau^{-1}]=[G_{F}]^{2}\ [m]^{5}$

$\text{M} = [G_{F}]^{2}\ \text{M}^{5}$

$[G_{F}] = \text{M}^{-2}$.

Therefore, the dimension of $G_{F}$ is $\text{M}^{-2}$.

Am I correct so far?

 

[spaghetti3451]

bumpp!

 

[nrqed]

Homework Statement

In natural units, the inverse lifetime of the muon is given by

$\tau^{-1}=\frac{G_{F}^{2}m^{5}}{192 \pi^{3}}$,

where $m$ is the muon mass, $106\ \text{MeV}$. What is the dimension of $G_{F}$ in natural units? Put in the factors of $\hbar$ and $c$ so that the equation can be interpreted in conventional units as well. From this, find the lifetime in seconds if $G_{F}=1.166 \times 10^{-11}$ in $\text{MeV}$ units.

Homework Equations

The Attempt at a Solution

The dimension of $\tau^{-1}$ is $\text{M}$ in natural units. Therefore,

$[\tau^{-1}]=[G_{F}]^{2}\ [m]^{5}$

$\text{M} = [G_{F}]^{2}\ \text{M}^{5}$

$[G_{F}] = \text{M}^{-2}$.

Therefore, the dimension of $G_{F}$ is $\text{M}^{-2}$.

Am I correct so far?

Yes, you are correct.