# 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.

## 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!!!

Homework Helper
Gold Member

## 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.

## 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.