Investigation of the ratio $\frac{σ_{r}}{F_{2}}(Q^2/s,Q^2)$ in the momentum-space approach

TL;DR

The paper addresses computing the ratio $\frac{\sigma_r}{F_2}$ in a momentum-space framework using the Block-Durand-Ha (BDH) parameterization of $F_2$. It derives a direct momentum-space expression for the ratio involving $F_2^{BDH}$ and its $Q^2$-evolution, incorporating the derivative $\frac{dF_2^{BDH}}{d\ln Q^2}$ and Wilson coefficients with a charm-threshold rescaling $\chi=x\left(1+\frac{4m_c^2}{Q^2}\right)$. The BDH-based results are benchmarked against H1 and HERA data and are shown to satisfy color-dipole model bounds, while remaining consistent with BGK and IP-Sat model predictions, indicating robustness of the approach for high-energy collider analyses. The study demonstrates that a momentum-space, BDH-informed treatment can provide reliable DIS structure-function ratios relevant for LHC and future FCC investigations, without relying on PDFs or scheme-dependent inputs.

Abstract

We present a calculation of the ratio $\frac{σ_{r}}{F_{2}}(x, Q^2)$ in momentum-space approach using the Block-Durand-Ha (BDH) parameterization of the proton structure function $F_{2}(x,Q^2)$. The results are compared with H1 data and extended to high inelasticity. We also examine the ratio $\frac{σ_{r}}{F_{2}}(\frac{Q^2}{s}, Q^2)$ obtained at a fixed $\sqrt{s}$ and $Q^2$ to the minimum value of $x$ given by $Q^2/s$, comparing them with both the HERA data and the color dipole model bounds. These results and comparisons with HERA data demonstrate that the suggested method for the ratio $\frac{σ_{r}}{F_{2}}$ can be applied in analyses of the Large Hadron Collider and Future Circular Collider projects.

Figures (2)

• Figure 1: The extracted ratio $\frac{\sigma_{r}}{F_{2}}$ (blue curves) from the parametrization methods is compared with the H1 data (red circles) H1. The total errors, which account for both the reduced cross sections and the structure functions, are included. The error bands (cyan curves) of the ratio $\frac{\sigma_{r}}{F_{2}}$ correspond to the uncertainty in the parameterization of $F_{2}$ and $F_{L}$ as shown in Table I.
• Figure 2: We plot the ratio $\frac{\sigma_{r}}{F_{2}}(\frac{Q^2}{s}, Q^2)$ as a function of $Q^2$ at $y=1$ for the HERA NC ep inclusive scattering data with $\sqrt{s}=318~\mathrm{GeV}$. The blue curves are extracted and compared with the results in Table VIII of Ref.Taylor (red circles), as well as the BGK and IP-Sat models (yellow squares). The error bands correspond to the uncertainty in the parameterization of $F_{2}$ in Martin1. The dipole upper bounds are represented by dashed and dashed-dot lines corresponding to $\rho=1$ and $\frac{4}{3}$ in the CDM.