Let $\sigma$ be a cross-section for the collision of two protons as given by
$$ \sigma = \intop_0^1 \mathrm{d}x_1 \intop_0^1 \mathrm{d}x_2 \, \sum_{a,b}f_a(x_1,Q^2) f_b(x_2,Q^2) \frac{1}{2\hat{s}} \intop \mathrm{d}\Phi ~ \overline{\left|M_{ab}\right|^2}, $$
where $f_i$ are the parton density functions, $\Phi$ is the available phase-space and $M_{ab}$ ist the (let's say tree-level) Matrix element of the partonic process.
What I fail to understand is which of the integrands can be understood as probability densities in the sense that the integral over the whole domain equals $1$.
I have for example always thought of $M_{ab}$ as a probability amplitude, so $\overline{\left|M_{ab}\right|^2}$ would be a probability density. But in which variables? It can't be the momenta of the particles or else the integral over the full phase-space would always be $1$.
Also $f_i(x_1,Q^2)$ is said to give the probabilty of finding parton $i$ with momentum fraction $x_1$ inside the proton at scale $Q$, but does that mean that
$$ \intop_0^1 \mathrm{d}x_1 f_i(x_1,Q^2) = 1? $$
Finally people usually say that $\sigma$ represents the probabilty of a process taking place, but since it has a dimension ($\text{GeV}^{-2}$), this can't be true in the mathematical sense.
Can anyone help me connect the dots here?
