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We discuss the upper limit, $k_{\text{max}}$, of the transverse-momentumintegration performed in the $k_t$-factorization formula. Based on explicitcalculations in the Yukawa theory and the study of seminal papers, we arguethat $k_{\text{max}}$ is equal to the factorization scale $\mu_F$ used tofactorize the cross section into an off-shell hard coefficient and a universalfactor. There is consequently a relation between $k_{\text{max}}$ and thedefinition of unintegrated parton densities (UPDFs). The use of an inconsistentrelation leads potentially to the overestimation of the cross section, whichhas been observed, e.g., in D-meson production. One of our conclusions is thatUPDFs 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 significantlyabove may result in the overestimation of the cross section. On the opposite,for UPDFs related to the collinear ones by an integration of the transversemomentum up to infinity, any $k_{\text{max}}^2 > Q^2$ is fine.

Examination of $k_t$-factorization in a Yukawa theory