Examination of kt-factorization in a Yukawa theory

We discuss the upper limit, kmax, of the transverse-momentum integration performed in the kt-factorization formula. Based on explicit calculations in the Yukawa theory and the study of seminal papers, we argue that kmax is equal to the factorization scale μF used to factorize the cross section into an off-shell hard coefficient and a universal factor. There is consequently a relation between kmax 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 [1]. One of our conclusions is that UPDFs related to collinear PDFs by an integration up to μ ∼ Q, where Q is the hard scale and μ the scale in the collinear PDFs, imply that kmax2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {k}_{\textrm{max}}^2 $$\end{document} ∼ Q2. 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 kmax2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {k}_{\textrm{max}}^2 $$\end{document}> Q2 is fine.