In the collinear factorisation equation for a QCD cross-section, schematically
$$
\sigma\_{AB\to X} 
=
f^A\_a \otimes \hat{\sigma}\_{ab\to X} \otimes f^B\_b
,
$$
we essentially convolute the probability of producing a final-state $X$ given initial-state partons $a$ and $b$ with the probability (ok, number) density of finding $a$ in hadron $A$ and $b$ in hadron $B$ with the requisite initial-state momenta, expressed through the parton distribution functions $f^A\_a$ and $f^B\_b$.

Is there a intuitive way of understanding why it's ok to adopt this almost-classical approach, of multiplying probabilities, rather than working with probability amplitudes as is ubiquitous elsewhere in QFT?