Applicability of parton model at low factorization scale Consider scattering process at partonic level occuring at low factorization scale $\mu^{2}\lesssim 1\text{ GeV}^{2}$ (for example, with the "hard" process $2\to 1$ with low mass produced particle, with the factorization scale chosen to be the squared mass of this particle). Using parton distribution functions generated by package LHAPDF, I obtain suppressed (in comparison with production at larger factorization scale) but non-zero result for the cross-section. 
My question is whether the parton model (and therefore parton distributuon function) is well-defined at $\mu^{2}\lesssim 1\text{ GeV}^{2}$. Could you help me please?
 A: I'm a little confused by what you mean by a $2\rightarrow1$ process, because one cannot have a $2\rightarrow1$ process for which all particles are on-shell (the usual assumption when doing perturbative QCD calculations).  But, assuming the formulae you've used don't necessarily break down at any particular scale (which is common), the parton model doesn't make sense when the scales are $\sim\Lambda_{QCD}$.  In particular, you should expect large corrections from higher orders in $\alpha_s((\mu/\Lambda_{QCD})^2)$.
A: The interacting probe with momentum transfer $Q$ has resolving capabilities $\sim 1/Q$ of the hadronic target so, in particular, if $Q$ is of the order of $1 \,\text{GeV}  \sim m_H$ then the resolution is at most of the order of the characteristic size of the hadron. We therefore do not really see the partonic structure of the proton, perhaps at best its valence content. At such energy resolutions one cannot justify the applicability of the parton model. 
Typically the parton model is to be understood within a (collinear) factorisation in which hard scattering matrix elements $C_i$ are convoluted with the PDFs $F_i$ to ascertain observables such as structure functions, cross sections etc. One usually writes e.g $$\mathcal M \sim  \sum_i C_i \otimes F_i + \mathcal O \left(\frac{\Lambda_{\text{QCD}}^2}{Q^2}\right)$$ with the higher twist terms neglected because of their suppression due to the relatively large scale $Q^2$. In your case, if $Q \sim \Lambda_{\text{QCD}}$ then these terms are $\mathcal O(1)$ and the factorisation into this nice sector split is spoiled.
