Charm Production in Deep Inelastic Scattering from Threshold to High $Q^{2}

TL;DR

This paper develops and implements a generalized MSbar (GM-VFNS) ACOT framework to describe charm production in deep inelastic scattering across the full energy range from threshold to high $Q^2$. By combining a 3-flavor scheme at low scales with a 4-flavor scheme at high scales and applying matched transition conditions, it delivers a complete order-$ abla$ calculation with Monte Carlo implementation, reproducing inclusive $F_2^c$ data and enabling differential distributions under kinematic cuts. The results show that the 4-flavor NLO calculation, with resummed mass logarithms into the charm PDF, yields good agreement with HERA data and offers computational efficiency over the conventional 3-flavor NNLO approach for inclusive quantities; however, differential distributions benefit from extending to the next order. The framework also accommodates semi-inclusive charm production with fragmentation functions, illustrating a path toward extracting gluon and charm distributions and investigating possible intrinsic charm components in the nucleon.

Abstract

Charm final states in deep inelastic scattering constitute $\sim 25%$ of the inclusive cross-section at small $x$ as measured at HERA. These data can reveal important information on the charm and gluon structure of the nucleon if they are interpreted in a consistent perturbative QCD framework which is valid over the entire energy range from threshold to the high energy limit. We describe in detail how this can be carried out order-by-order in PQCD in the generalized \msbar formalism of Collins (generally known as the ACOT approach), and demonstrate the inherent smooth transition from the 3-flavor to the 4-flavor scheme in a complete order $α_s$ calculation, using a Monte Carlo implementation of this formalism. This calculation is accurate to the same order as the conventional NLO $F_2$ calculation in the limit $\frac{Q}{m_c} >> 1$. It includes the resummed large logarithm contributions of the 3-flavor scheme (generally known in this context as the fixed-flavor-number or FFN scheme) to all orders of $α_s\ln(m_c^2/Q^2)$. For the inclusive structure function, comparison with recent HERA data and the existing FFN calculation reveals that the relatively simple order-$α_s$ (NLO) 4-flavor ($m_c \neq 0$) calculation can, in practice, be extended to rather low energy scales, yielding good agreement with data over the full measured $Q^2$ range. The Monte Carlo implementation also allows the calculation of differential distributions with relevant kinematic cuts. Comparisons with available HERA data show qualitative agreement; however, they also indicate the need to extend the calculation to the next order to obtain better description of the differential distributions.

Figures (5)

• Figure 1: Schematic illustration of the uncertainty bands (hence the relative merits) of the 3-flavor and 4-flavor scheme calculations of the charm contribution to the inclusive structure function -- and how they can be naturally merged into a composite scheme.
• Figure 2: The generic inclusive lepto-production process. We are primarily interesting in final states which contain some heavy quarks, in particular charm.
• Figure 5: Comparison of results of the present order $\alpha_s$ generalized $\overline{{\rm MS}}$ 4-flavor calculation to that of order $\alpha_s^2$ 3-flavor calculations for: (a) $x=0.01$; and (b) $x=0.0001$. $F_2^c$ is plotted against $Q$. The uncertainties of the calculation are represented by a band bounded by two curves obtained with the scale choices $\mu=c \sqrt{Q^2 + m_c^2}$; $c=0.5, 2$.
• Figure 6: Order $\alpha_s$ 4-flavor general mass calculation compared to the recent ZEUS data on $F_2^c$. The calculation uses the CTEQ5HQ parton distributions with $m_c=1.3 GeV$.
• Figure 7: Four differential distributions for $D^{\ast }$ production at HERA, calculated using the NLO $\left( \alpha _{s}\right)$ generalized $\overline{{\rm MS}}$ 4-flavor formalism and $m_c=1.3 GeV$. The two curves on each plot results from two choices of the scale constant $c=0.5,1$. The data points are results from the ZEUS 96-97 run ZEUScharm97.