Longitudinal Structure Function at the Limit $x=Q^2/s$

TL;DR

This work analyzes the longitudinal structure function $F_L(x,Q^2)$ at the kinematic limit $x_{min}=Q^2/s$ using an expansion method within the color dipole framework at next-to-leading order, for both nucleons and nuclei. By expressing $F_L$ in terms of singlet quark and gluon distributions and expanding around a reference point, the authors highlight gluon-dominated dynamics at small $x$ and employ BDH/DL parametrizations along with GBW/BGK-inspired dipole models to incorporate nonlinear effects and heavy-quark mass rescaling $\tilde{x}=x(1+4m_f^2/Q^2)$. They compare predictions with HERA data, finding good agreement at moderate to large $Q^2$ and demonstrating that nonlinear saturation corrections in nuclei become significant at low $Q^2$, with the ratio $F_L^A/(A F_L^p)$ serving as a diagnostic of nonlinear low-$x$ dynamics. The results also encompass deuterium data from JLab and provide analytic Appendix expressions for practical approximations, offering predictions for future electron-ion colliders and insights into nuclear gluon distributions at small $x$.

Abstract

The longitudinal structure function for nucleons and nuclei is considered at fixed $\sqrt{s}$ and $Q^2$ to the minimum value of $x$ given by $Q^2/s$. This is done using the expansion method and color dipole model in the next-to-leading order approximation. The extracted longitudinal structure functions were consistent with HERA hepdata [ https://www.hepdata.net/record/ins1377206] and the determination of $F_{L}$ at $x=Q^2/s$ [Frank E.Taylor, Phys. Rev. D {\bf111}, 052001 (2025)] at moderate and large $Q^2$ values. The results, consistent with the dipole picture at low $Q^2$ values, show that the longitudinal structure function is small as expected due to the transverse polarization of the exchanged photon and the strong suppression of the dominant gluon component. Nonlinear corrections to the nuclear longitudinal structure function at low values of $x$ and $Q^2$ are also considered. These results may enhance the deep inelastic scattering neutral current data in future colliders at low $x$ and low $Q^2$. The longitudinal structure functions for deuterium at low four-momentum transfer squared, $Q^2<1~\mathrm{GeV}^2$ at the LO and NLO approximations are determined and compared with the JLab E00-002 data.

Figures (9)

• Figure 1: Ratios of $\eta_{a}(\frac{Q^2}{s},Q^2)$ and $\xi_{a}(\frac{Q^2}{s},Q^2)$ at $\sqrt{s}=318~\mathrm{GeV}$ as a function of $Q^2$ with $a=0, 0.5$ and $0.666$ according to Eqs. (\ref{['FLA0_eq']}), (\ref{['FL4_eq']}) and (\ref{['FLA6_eq']}), respectively .
• Figure 2: The DIS longitudinal structure functions at LO (black solid curve) and NLO (green dashed curve) approximations are shown as a function of $Q^2$ for $\sqrt{s}=318~\mathrm{GeV}$. These results consider the charm effect in the rescaling of the Bjorken variable $x$. The coefficients are based on the results from Fit 1 in Table I. The HERA combined H1 and ZEUS reduced cross section datasets (squares-blue and circles-red) and their uncertainties are shown.
• Figure 3: The same as Fig.1 at $\sqrt{s}=300~\mathrm{GeV}$.
• Figure 4: The DIS longitudinal structure functions without rescaling (black solid curve) and with rescaling (green dashed curve) approximations at the NLO approximation are shown as a function of $Q^2$ for $\sqrt{s}=318~\mathrm{GeV}$. The coefficients are based on the results from Fit 1 in Table I. The HERA combined H1 and ZEUS reduced cross section datasets (squares-blue and circles-red) and their uncertainties are shown.
• Figure 5: The DIS longitudinal structure functions at the NLO approximation are shown by applying the active flavors ($n_{f}=3$, green dashed curve), ($n_{f}=4$, black solid curve) and ($n_{f}=5$, brown dashed-dot curve) as a function of $Q^2$ for $\sqrt{s}=318~\mathrm{GeV}$ based on the results in Table I and QCD cut-off. The HERA combined H1 and ZEUS reduced cross section datasets (squares-blue and circles-red) and their uncertainties are shown.