Mass dimensions and weak interaction

Question

The Fermi constant has a mass dimension of $-2$ and a value of $10^{-5}GeV^{-2}$.

How can I infer from this information that the mass scale of the weak interaction is about $10^2 GeV$?

Answer 1

The Fermi constant has a mass dimension of -2

The Fermi constant, $G_F$ itself has (by defintion) the dimension "energy times volume".

You are apparently asking about the "reduced" quantity $\frac{G_F}{(\hbar~c)^3}$,
(which is considered to be more practical in some applications) and
which has the dimension of "inverse energy squared";
symbolically: $$\frac{G_F}{(\hbar~c)^3} = (S_E)^{(-2)} = \frac{1}{S_E^2} = \frac{1}{(S_m~c^2)^2},$$

for the corresponding value $S_E$ of "energy scale", or the corresponding value $S_m$ of "mass scale".

and a value of $10^{-5}~GeV^{-2}$.

(Approximately, of course.)

How can I infer from this information that the mass scale of the weak interaction is about $10^{2}~GeV$ ?

Well, just rearrange the above formula; i.e. solve for $S_E$:

$$S_E = \frac{1}{\sqrt{\frac{G_F}{(\hbar~c)^3}}},$$

and insert the (approximate) value of Fermi constant:

$$S_E = \frac{1}{\sqrt{10^{-5}~GeV^{-2}}} = \sqrt{10^{5}~GeV^{2}} = \sqrt{10^{5}}~\text{GeV} \approx 300~\text{GeV}.$$

That's of course only a rough "order of magnitude" approximation of $10^{2}~\text{GeV}$ as "mass scale of the weak interaction".

In the Wikipedia page linked above you also find the relation

$$\frac{G_F}{(\hbar~c)^3} = \frac{\sqrt{2}}{8}~\frac{g^2}{(m_W~c^2)^2},$$

in terms of $m_W$, the mass of an actual particle (the $W$ boson) which "actually mediates charged weak interactions". The experimentally determined value of $$m_W~c^2 \approx 80\text{GeV} $$ is (incidentally) considerably closer to $10^{2}~\text{GeV}$.