# Consistent treatment of charm production in higher-orders at tree-level within $k_T$-factorization approach

## TL;DR

This paper develops a consistent method to incorporate higher-order corrections into $k_T$-factorization calculations of charm production, improving agreement with experimental data and addressing double-counting issues.

## Contribution

It introduces a novel prescription merging LO, NLO, and NNLO matrix elements in $k_T$-factorization, and proposes a simple method to avoid double-counting of higher-order contributions.

## Key findings

- KMR uPDF provides good description of charm data at LO.
- Including higher-orders with KMR uPDF aligns results with standard calculations.
- PB uPDF underestimates data, but improves with higher-order inclusion.

## Abstract

We discuss production of $c \bar c$-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 (NNLO) matrix elements. For the first time we include in this framework 2 $\to$ 3 and 2 $\to$ 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 $\to$ 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.