What is the mathematical expression for the Higgs boson coupling constant? I have been searching around and cannot get an expression for the Higg's coupling constant.
By 'coupling constant', I mean for the strong force $$\alpha_S=\frac{{g_S}^2}{4 \pi \hbar c}\approx 0.1\tag{1}$$
and for the weak force
$$\alpha_W=\frac{{g_W}^2}{4 \pi \hbar c}\approx 0.034\tag{2}$$
and for the EM force
$$\alpha_{EM}=\frac{{e}^2}{4 \pi \epsilon_0\hbar c}\approx 0.0073$$
Where in $(1)$ and $(2)$, $g_S$ and $g_W$ is the vertex strength in the Feynman $\mathrm{diagram^{\chi}}$:

Does a similar expression exist for the Higgs boson?

$\chi$ - Image from ICL dept. of physics.
 A: The Higgs boson does not have a separate coupling to the rest of the particles in the table of elementary particles.  The magnitude of the coupling is a combination of the weak and the electromagnetic coupling constant, depending on the reaction measured . There are efforts to quantify this from experimental data.

We estimate the expected precision at a multi-TeV muon collider for measuring the Higgs boson couplings with electroweak gauge bosons, HVV and HHVV (V=W±,Z), as well as the trilinear Higgs self-coupling HHH.

A: This article notes that particles can exert a force on one another via the Higgs field and assigns a dimensionless coupling constant (of the same form as the examples given):
\begin{equation}
\alpha_{Higgs} = \left ( \frac{mc^2}{4 \pi v} \right )^2
\end{equation}
where $v$ is the vacuum expectation value of the Higgs field. Note that this force extremely short range due to the mass of the Higgs boson.
In the standard model, the Yukawa couplings of each of the fermions to the Higgs field are independent inputs. Therefore, the value of the expression above will depend on the particle you consider.
