Heavy-quark contributions to the polarized DIS structure functions at NLO in the ACOT scheme

Abstract

This study explores the heavy-quark contributions to polarized structure functions in deep-inelastic scattering at next-to-leading order. The structure functions $g_1$, $g_4$, $g_5$, $g_6$, and $g_7$ are computed, while $g_2$ and $g_3$ are excluded due to the higher-twist suppression. The calculations are performed within the ACOT renormalization scheme, which ensures theoretical consistency across kinematic regions where heavy quarks transition from being dynamically produced to fully active degrees of freedom. By incorporating heavy-flavor contributions at next-to-leading-order, this work provides deeper insights into their role in polarized structure functions and the spin-dependent dynamics of QCD. Both analytical results and their numerical implementation are presented.

Figures (5)

• Figure 1: Real-emission diagrams for the quark-scattering channel $B^*(q) + Q_1(k_1) \to Q_2(k_2) + g(p_2)$ at $\mathcal{O}(\alpha_s)$. Gluon radiation off the incoming and outgoing heavy-quark lines generates the polarized partonic structure functions $\hat{g}_i^{\rm QS}$ with full dependence on the heavy-quark masses $m_1$ and $m_2$.
• Figure 2: One-loop vertex correction to the quark-boson vertex in the quark-scattering process $B^*(q) + Q_1(k_1) \to Q_2(k_2)$. The calculation is performed in the on-shell renormalization scheme, where self-energy insertions on the external heavy-quark legs are canceled by the corresponding mass and wave-function counterterms. Together with the real-emission diagrams in \ref{['fig:Real Emission Diagram']}, this contribution renders the NLO polarized structure functions infrared finite.
• Figure 3: Representative real-emission diagrams for boson-gluon fusion, $B^*(q) + g(p) \to Q_1(k_1) + Q_2(k_2)$, at $\mathcal{O}(\alpha_s)$. These diagrams contribute to the polarized partonic structure functions $\hat{g}_i^{\rm GF}$ and provide the gluon-initiated component of the NLO coefficient functions entering the ACOT scheme.
• Figure 4: Polarized structure functions $x g_1$, $x g_4$, and $x g_5$ in the ACOT scheme for NC and CC DIS. Results are shown at LO (dashed) and NLO (solid) for representative scales $Q^2 \in \{2,\,10,\,100,\,1000\}~\mathrm{GeV}^2$. The lower panels display the ratio $\text{NLO}/\text{LO}$, highlighting the impact of QCD corrections.
• Figure 5: Heatmaps of the relative deviation $K = g_i / g_i^{\rm ZM} - 1$. The color scale indicates the percentage deviation in the $(x,Q^2)$ plane. The dashed line marks the bottom-quark threshold $Q^2 \simeq m_b^2$.