Consistent treatment of charm production in higher-orders at tree-level within kT-factorization approach

We discuss production of c (c) over bar -pairs within k(T)-factorization approach (off-shell initial partons) with unintegrated parton distribution functions (uPDFs). We present a consistent prescription which merges the standard leading-order (LO) k(T)-factorization calculations for this process with tree-level next-to-leading order (NLO) and next-to-next-to-leading order matrix elements. For the first time we include in this framework 2 -> 3 and 2 -> 4 processes with extra partonic emissions for single particle distributions as well as for correlation observables. The use of the KMR uPDF leads to a good description of the existing charm (D-meson) data already at the leading-order. On the other hand, a new parton-branching (PB) uPDF strongly underestimates the same experimental data. A direct inclusion of the higher-orders at tree-level leads to an overestimation of the data, especially for the KMR uPDF. This suggests a significant double-counting. We propose a simple method how to avoid the double-counting. Our procedure leads to a much better description of the experimental data when including the higher-order contributions. Then with the KMR uPDF we get similar results (both for single particle and correlation observables) as for the standard calculations of the 2 -> 2 processes. For the PB uPDF inclusion of the higher-orders considerably improves description of the experimental data. We conclude that the LO calculation with the KMR uPDF effectively includes the higher-orders which is not the case for the PB uPDF. The phenomenological analysis presented here is first attempt to study the charm cross section at the LHC differentially beyond the NLO.