Examination of $k_t$-factorization in a Yukawa theory

We discuss the upper limit, $k_{\text{max}}$, of the transverse-momentum integration performed in the $k_t$-factorization formula. Based on explicit calculations in the Yukawa theory and the study of seminal papers, we argue that $k_{\text{max}}$ is equal to the factorization scale $\mu_F$ used to factorize the cross section into an off-shell hard coefficient and a universal factor. There is consequently a relation between $k_{\text{max}}$ and the definition of unintegrated parton densities (UPDFs). The use of an inconsistent relation leads potentially to the overestimation of the cross section, which has been observed, e.g., in D-meson production. One of our conclusions is that UPDFs related to collinear PDFs by an integration up to $\mu\sim Q$, where $Q$ is the hard scale and $\mu$ the scale in the collinear PDFs, imply that $k_{\text{max}}^2 \sim Q^2$. Integrating the transverse-momentum significantly above may result in the overestimation of the cross section. On the opposite, for UPDFs related to the collinear ones by an integration of the transverse momentum up to infinity, any $k_{\text{max}}^2 > Q^2$ is fine.