$\bar{\rm MS}$ parton densities with NNLO heavy flavor matching conditions

TL;DR

This paper develops a complete set of $\overline{\rm MS}$ parton densities across three, four, and five flavors by implementing NNLO heavy-flavor matching at the charm and bottom thresholds. It employs a two-loop operator matrix element framework to generate the 4- and 5-flavor densities from a 3-flavor baseline and evolves them using NLO Altarelli-Parisi splitting functions in $x$-space. The study highlights significant effects of the NNLO matching near threshold regions and finds persistent, sizeable impacts at small $x$ even for large $\mu^2$, with roughly 10% differences between 3- and 5-flavor densities at high scales. These results have direct implications for precision phenomenology at HERA and the modeling of heavy-quark contributions in DIS.

Abstract

A study is made of charm and bottom flavor matching conditions on parton densities. Starting from an $\bar{\rm MS}$ three-flavor density set, where the scale $μ< m_c$, and using the recently derived two-loop matching conditions, we provide a new set of four-flavor parton densities where $ m_c \le μ< m_b$ and five-flavor densities where $μ\ge m_b$. The effect of the next-to-next-to-leading order matching conditions on the evolution equations is important for scales just above the transition regions. This includes the small $x$ and small $Q^2$ domain studied by the H1 and ZEUS experiments at HERA. At small $x$ the effects of the matching conditions never die away even for large $μ^2$.