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

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. The effect of adding a simple higher twist term of the form $F_{2}{\ast}H_{2}/Q^2$ to the description of the ratio $\frac{σ_{r}}{F_{2}}(\frac{Q^2}{s}, Q^2)$ at low-$x$ and low-$Q^2$ values for comparison with the color dipole bounds and the HERA data is investigated.

💡 Research Summary

This paper presents a detailed investigation of the ratio between the reduced deep-inelastic scattering (DIS) cross-section (σ_r) and the proton structure function F2, denoted as σ_r/F2, utilizing a momentum-space approach. The primary goal is to compute this ratio directly from a parameterization of the measurable structure function F2, thereby bypassing dependencies on unobservable parton distribution functions (PDFs) and factorization schemes.

The authors employ the Block-Durand-Ha (BDH) parameterization for F2(x, Q^2), which is known for its accurate description of experimental data across a wide range of x and Q^2, particularly at low x. The core methodological innovation involves using a Laplace transform technique to solve the QCD evolution equation that relates F2 to the longitudinal structure function F_L. This process yields a closed-form integral expression (Eq. 13) for the ratio σ_r/F2 solely in terms of the parameterized F2 and the strong coupling constant α_s.

The calculated ratio is first validated against published H1 collaboration data from HERA for various Q^2 values (1.5 to 45 GeV^2) and moderate inelasticity y. The results show excellent agreement with the data, confirming the robustness of the momentum-space approach within the standard kinematic region.

The study then extends the analysis to the kinematic limit of maximum inelasticity, where y = 1 and x reaches its minimum value x_min = Q^2/s at a fixed center-of-mass energy (√s = 318 GeV). In this limit, the ratio simplifies to 1 - F_L/F2. The predictions from the derived formula (Eq. 16) are compared with a dedicated HERA data analysis for this specific limit and with predictions from color dipole models (CDM), specifically the BGK and IP-Sat models. The results align well with both the data and the theoretical bounds set by the CDM (where σ_r/F2 is expected to be between 2/3 and 8/11), demonstrating the consistency of the approach with established non-perturbative QCD frameworks.

Furthermore, the paper explores the impact of higher-twist (HT) corrections, which become significant at low Q^2. A phenomenological HT term of the form F2 * H2/Q^2 is added to the leading-twist F2 description. Using an HT coefficient H2 ≈ 0.12 GeV^2, as suggested by other phenomenological fits, the agreement between the theoretical prediction and the data at low Q^2 values is improved. This indicates the necessity of accounting for power corrections in this kinematic domain.

In conclusion, the research successfully establishes that the momentum-space approach combined with the BDH parameterization provides a reliable and theoretically clean framework for calculating the σ_r/F2 ratio. It accurately reproduces existing HERA data, makes predictions for high-inelasticity regions where data is scarce, and shows consistency with color dipole model expectations. The method, including the option to incorporate higher-twist effects, is shown to be a valuable tool for analyzing future data from high-energy collider experiments like the Large Hadron Electron Collider (LHeC), the Electron-Ion Collider (EIC), and Future Circular Colliders (FCC), where probing extreme low-x and high-Q^2 kinematics will be crucial.