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Investigation of the ratio σrF2(Q2/s,Q2) in the momentum-space approach

G. R. Boroun*

  • *Contact author: boroun@razi.ac.ir

Phys. Rev. C 113, 045208 – Published 13 April, 2026

DOI: https://doi.org/10.1103/nc1k-vlc7

Abstract

I present a calculation of the ratio σrF2(x,Q2) in a momentum-space approach using the Block-Durand-Ha parametrization of the proton structure function F2(x,Q2). The results are compared with the H1 data and extended to high inelasticity. I also examine the ratio σrF2(Q2s,Q2) obtained at a fixed s and Q2 to the minimum value of x given by Q2/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 σrF2 can be applied in analyses of the CERN Large Hadron Collider and Future Circular Collider projects. The effect of adding a simple higher twist term of the form F2*H2/Q2 to the description of the ratio σrF2(Q2s,Q2) at low-x and low-Q2 values for comparison with the color dipole bounds and the HERA data is investigated.

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References (46)

  1. C. Adloff et al. (H1 Collaboration), Deep-inelastic inclusive ep scattering at low x and a determination of αs, Eur. Phys. J. C 21, 33 (2001).
  2. F. D. Aaron et al. (H1 and ZEUS Collaborations), Combined measurement and QCD analysis of the inclusive e±p scattering cross sections at HERA, J. High Energy Phys. 01 (2010) 109.
  3. H. Abramowicz et al. (H1 and ZEUS Collaborations), Combination of measurements of inclusive deep inelastic e±p scattering cross sections and QCD analysis of HERA data, Eur. Phys. J. C 75, 580 (2015).
  4. P. Agostini et al. (LHeC Collaboration and FCC-he Study Group), The Large Hadron–Electron Collider at the HL-LHC, J. Phys. G: Nucl. Part. Phys. 48, 110501 (2021).
  5. F. E. Taylor, Determination of FL at x=Q2/s with HERA data, Phys. Rev. D 111, 052001 (2025).
  6. T. Lappi, H. Mantysaari, H. Paukkunen, and M. Tevio, Evolution of structure functions in momentum space, Eur. Phys. J. C8484, (2024).
  7. M. M. Block, L. Durand, and P. Ha, Connection of the virtual γ*p cross section of ep deep inelastic scattering to real γp scattering, and the implications for νN and ep total cross sections, Phys. Rev. D 89, 094027 (2014).
  8. M. M. Block, L. Durand, and D. W. McKay, Analytic treatment of leading-order parton evolution equations: Theory and tests, Phys. Rev. D 79, 014031 (2009).
  9. M. M. Block, L. Durand, P. Ha, and D. W. McKay, Analytic solution to leading order coupled DGLAP evolution equations: A new perturbative QCD tool, Phys. Rev. D 83, 054009 (2011).
  10. M. M. Block, L. Durand, P. Ha, and D. W. McKay, Applications of the leading-order Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution equations to the combined HERA data on deep inelastic scattering, Phys. Rev. D 84, 094010 (2011).
  11. M. M. Block, L. Durand, P. Ha, and D. W. McKay, Implications of a Froissart bound saturation of γ*–p deep inelastic scattering. I. Quark distributions at ultra small x, Phys. Rev. D 88, 014006 (2013).
  12. G. R. Boroun and P. Ha, Decoupling of the structure functions in momentum space based on the Laplace transformation, Phys. Rev. D 109, 094037 (2024).
  13. G. R. Boroun and P. Ha, Reduced cross section and gluon distribution in a momentum-space approach, Phys. Rev. D 111, 034012 (2025).
  14. C. Ewerz et al., The new FL measurement from HERA and the dipole model, Phys. Lett. B 720, 181 (2013).
  15. C. Ewerz and O. Nachtmann, Bounds on ratios of DIS structure functions from the color dipole picture, Phys. Lett. B 648, 279 (2007).
  16. M. Niedziela and M. Praszalowicz, Acta Phys. Pol. B 46, 2018 (2015).
  17. M. Kuroda and D. Schildknecht, The energy dependence of the saturation scale in DIS at low x, Phys. Lett. B 618, 84 (2005).
  18. M. Kuroda and D. Schildknecht, The color dipole picture and the ratio of R(W2,Q2)=σL/σT, Phys. Lett. B 670, 129 (2008).
  19. M. Kuroda and D. Schildknecht, Color dipole picture of low-x DIS: Model-independent and model-dependent results, Phys. Rev. D 85, 094001 (2012).
  20. M. Kuroda and D. Schildknecht, Color dipole picture of deep inelastic scattering, revisited, Int. J. Mod. Phys. A 31, 1650157 (2016).
  21. N. N. Nikolaev and B. G. Zakharov, On determination of the large-1x gluon distribution at HERA, Phys. Lett. B 332, 184 (1994).
  22. N. N. Nikolaev and B. G. Zakharov, Colour transparency and scaling properties of nuclear shadowing in deep inelastic scattering, Z. Phys. C: Part. Fields 49, 607 (1991).
  23. N. N. Nikolaev and B. G. Zakharov, Pomeron structure function and diffraction dissociation of virtual photons in perturbative QCD, Z. Phys. C: Part. Fields 53, 331 (1992).
  24. G. R. Boroun and B. Rezaei, An evaluation of the proton structure functions F2 and FL at small x, Phys. Lett. B 816, 136274 (2021).
  25. G. R. Boroun, M. Kuroda, and D. Schildknecht, The proton gluon distribution from the color dipole picture, Eur. Phys. J. Plus 140, 1149 (2025).
  26. G. R. Boroun and B. Rezaei, Color dipole model bounds with the gluon-gluon recombination correction, Phys. Rev. C 103, 065202 (2021).
  27. G. R. Boroun and B. Rezaei, Ratio of the structure functions and the color dipole model bound, Nucl. Phys. A 990, 244 (2019).
  28. G. R. Boroun, Physical limits in the color-dipole model bounds, Eur. Phys. J. A 57, 219 (2021).
  29. G. R. Boroun, Heavy quark structure functions from unifying the color dipole picture and double asymptotic scaling approaches, Phys. Rev. D 109, 054012 (2024).
  30. A. Accardi et al., Electron-ion collider: The next QCD frontier, Eur. Phys. J. A 52, 268 (2016).
  31. R. Abdul Khalek et al., Science requirements and detector concepts for the electron-ion collider, Nucl. Phys. A 1026, 122447 (2022).
  32. L. P. Kaptari et al., Extracting the longitudinal structure function FL(x,Q2) at small x from a Froissart-bounded parametrization of F2(x,Q2), Phys. Rev. D 99, 096019 (2019).
  33. V. Andreev et al. (H1 Collaboration), Measurement of inclusive ep cross sections at high Q2 at s=225 and 252 GeV and of the longitudinal proton structure function FL at HERA, Eur. Phys. J. C 74, 2814 (2014).
  34. M. A. G. Aivazis et al., Leptoproduction of heavy quarks. II. A unified QCD formulation of charged and neutral current processes from fixed-target to collider energies, Phys. Rev. D 50, 3102 (1994).
  35. D. A. Fagundes and M. V. T. Machado, Asymptotic gluon density within the color dipole picture in light of HERA high-precision data, Phys. Rev. D 107, 014004 (2023).
  36. H. Kowalski and D. Teaney, Impact parameter dipole saturation model, Phys. Rev. D 68, 114005 (2003).
  37. J. Bartels, K. J. Golec-Biernat, and H. Kowalski, Modification of the saturation model: Dokshitzer-Gribov-Lipatov-Altarelli-Parisi evolution, Phys. Rev. D 66, 014001 (2002).
  38. J. Bartels, K. Golec-Biernat, and L. Motyka, Twist expansion of the nucleon structure functions, F2, and FL, in the DGLAP improved saturation model, Phys. Rev. D 81, 054017 (2010).
  39. J. Bartels, K. Golec-Biernat, and K. Peters, An estimate of higher twist at small xB and low Q2 based upon a saturation model, Eur. Phys. J. C 17, 121 (2000).
  40. L. Motyka and M. Sadzikowski, Twist decomposition of non-linear effects in Balitsky–Kovchegov evolution of proton structure functions, Eur. Phys. J. C 83, 1062 (2023).
  41. L. Motyka and M. Sadzikowski, Twist decomposition of proton structure from BFKL and BK amplitudes, Acta Phys. Pol. B 45, 2079 (2014).
  42. A. M. Cooper-Sarkar, PoS DIS2016, 013 (2016).
  43. I. Abt et al., Study of HERA ep data at low Q2 and low xBj and the need for higher-twist corrections to standard perturbative QCD fits, Phys. Rev. D 94, 034032 (2016).
  44. F. D. Aaron et al. (H1 Collaboration), Measurement of the inclusive ep scattering cross section at low Q2 and x at HERA, Eur. Phys. J. C 63, 625 (2009).
  45. H. Khanpour, A. Mirjalili, and S. Atashbar Tehrani, Analytic derivation of the next-to-leading order proton structure function F2p(x,Q2) based on the Laplace transformation, Phys. Rev. C 95, 035201 (2017).
  46. R. D. Ball, A. Chiefa, and R. Stegeman, Parton distributions with higher twist and jet power corrections, arXiv:2511.14387.

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