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Reconstruction of the dipole amplitude in the dipole picture as a mathematical inverse problem

H. Hänninen1,*, A. Kykkänen2, and H. Schlüter3

  • 1Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35, 40014 University of Jyväskylä, Finland
  • 2Department of Computational Applied Mathematics and Operations Research, Rice University, Houston, Texas, USA
  • 3Department of Mathematics and Statistics, University of Helsinki, P.O. Box 3, 00014 University of Helsinki, Finland

  • *Contact author: henri.j.hanninen@jyu.fi
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Phys. Rev. D 112, 094026 – Published 14 November, 2025

DOI: https://doi.org/10.1103/7fhf-7fp4

Abstract

We show that the inference problem of constraining the dipole amplitude with inclusive deep inelastic scattering data can be written into a discrete linear inverse problem, in an analogous manner as can be done for computed tomography. To this formulation of the problem, we apply standard inverse problems methods and algorithms to reconstruct known dipole amplitudes from simulated reduced cross section data with realistic precision. The main difference of this approach to previous works is that this implementation does not require any fit parametrization of the dipole amplitude. The freedom from parametrization also enables us for the first time to quantify the uncertainties of the inferred dipole amplitude in a novel more general framework. This mathematical approach to small-x phenomenology opens a path to parametrization bias-free inference of the dipole amplitude from HERA and Electron-Ion Collider data.

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References (145)

  1. S. Navas et al. (Particle Data Group Collaboration), Review of particle physics, Phys. Rev. D 110, 030001 (2024).
  2. M. Gell-Mann, The Eightfold Way: A Theory of strong interaction symmetry, Report No. CTSL-20, TID-12608, 1961.
  3. M. Gell-Mann, A schematic model of baryons and mesons, Phys. Lett. 8, 214 (1964).
  4. G. Zweig, An SU3 model for strong interaction symmetry and its breaking, Technical Report, CERN, Geneva, 1964.
  5. G. Zweig, An SU(3) Model for Strong Interaction Symmetry and its Breaking. Version 2 (Hadronic Press, Geneva, 1964), 2, pp. 22–101.
  6. Y. Ne’eman, Derivation of strong interactions from a gauge invariance, Nucl. Phys. 26, 222 (1961).
  7. J. D. Bjorken and S. L. Glashow, Elementary particles and SU(4), Phys. Lett. 11, 255 (1964).
  8. R. P. Feynman, Very high-energy collisions of hadrons, Phys. Rev. Lett. 23, 1415 (1969).
  9. E. D. Bloom et al., High-energy inelastic ep scattering at 6-degrees and 10-degrees, Phys. Rev. Lett. 23, 930 (1969).
  10. M. Breidenbach, J. I. Friedman, H. W. Kendall, E. D. Bloom, D. H. Coward, H. C. DeStaebler, J. Drees, L. W. Mo, and R. E. Taylor, Observed behavior of highly inelastic electron-proton scattering, Phys. Rev. Lett. 23, 935 (1969).
  11. F. Gross et al., 50 years of quantum chromodynamics, Eur. Phys. J. C 83, 1125 (2023).
  12. J. Greensite, An Introduction to The Confinement Problem (Springer, Cham, 2011), Vol. 821.
  13. A. Morreale and F. Salazar, Mining for gluon saturation at colliders, Universe 7, 312 (2021).
  14. A. Accardi et al., Electron ion collider: The next QCD frontier: Understanding the glue that binds us all, Eur. Phys. J. A 52, 268 (2016).
  15. R. Abdul Khalek et al., Science requirements and detector concepts for the electron-ion collider: EIC Yellow Report, Nucl. Phys. A1026, 122447 (2022).
  16. R. Abir et al., The case for an EIC theory alliance: Theoretical challenges of the EIC, arXiv:2305.14572.
  17. P. Agostini et al. (LHeC, FCC-he Study Group Collaborations), The large hadron–electron collider at the HL-LHC, J. Phys. G 48, 110501 (2021).
  18. D. P. Anderle et al., Electron-ion collider in China, Front. Phys. (Beijing) 16, 64701 (2021).
  19. G. Uhlmann, Inside Out: Inverse Problems and Applications (Cambridge University Press, Cambridge, England, 2003).
  20. W. Lassell, 1. Observations of Le Verrier’s Planet, In the Meridian, Mon. Not. R. Astron. Soc. 7, 154 (1846).
  21. G. B. Airy, Account of some circumstances historically connected with the discovery of the planet exterior to Uranus, Mon. Not. R. Astron. Soc. 7, 121 (1846).
  22. E. Adams and J. C., III. An explanation of the observed irregularities in the motion of uranus, on the hypothesis of disturbance caused by a more distant planet; with a determination of the mass, orbit, and position of the disturbing body, Mon. Not. R. Astron. Soc. 7, 149 (1846).
  23. P. Argoul, Overview of inverse problems, DEA. Parameter Identification in Civil Engineering, Ecole Nationale des Ponts et chaussées (2012), https://cel.hal.science/cel-00781172v1.
  24. P. C. Hansen, J. Jørgensen, and W. R. B. Lionheart, Computed Tomography: Algorithms, Insight, and Just Enough Theory (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2021).
  25. M. Salo, Inverse problems for elliptic PDE, 2023. Lectures in Calderón type inverse problems for elliptic PDE, University of Jyväskylä. https://users.jyu.fi/~salomi/lecturenotes/Beijing_notes_v1.pdf.
  26. M. Salo, Applications of microlocal analysis to inverse problems, arXiv:1908.03041.
  27. T. M. Buzug, Computed Tomography: From Photon Statistics to Modern Cone-Beam CT (Springer, Berlin, Heidelberg, 2009).
  28. S. Deans, The Radon Transform and Some of Its Applications, A Wiley-Interscience Publication (Wiley, New York, 1983).
  29. A. C. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (Society for Industrial and Applied Mathematics, Philadelphia, 2001).
  30. P. C. Hansen and J. S. Jørgensen, Air tools II: Algebraic iterative reconstruction methods, improved implementation, Num. Algorith. 79, 107 (2017).
  31. C. Casuga, M. Karhunen, and H. Mäntysaari, Inferring the initial condition for the Balitsky-Kovchegov equation, Phys. Rev. D 109, 054018 (2024).
  32. C. Casuga, H. Hänninen, and H. Mäntysaari, Initial condition for the Balitsky-Kovchegov equation at next-to-leading order, Phys. Rev. D 112, 034003 (2025).
  33. G. Beuf, H. Hänninen, T. Lappi, and H. Mäntysaari, Color glass condensate at next-to-leading order meets HERA data, Phys. Rev. D 102, 074028 (2020).
  34. H. Hänninen, Deep inelastic scattering in the dipole picture at next-to-leading order, Ph.D. thesis, Jyvaskyla University, Jyvaskyla University, 2021, arXiv:2112.08818.
  35. H. Hänninen, H. Mäntysaari, R. Paatelainen, and J. Penttala, Proton structure functions at next-to-leading order in the dipole picture with massive quarks, Phys. Rev. Lett. 130, 192301 (2023).
  36. R. Abdul Khalek, J. J. Ethier, J. Rojo, and G. van Weelden, nNNPDF2.0: Quark flavor separation in nuclei from LHC data, J. High Energy Phys. 09 (2020) 183.
  37. L. Del Debbio, T. Giani, and M. Wilson, Bayesian approach to inverse problems: An application to NNPDF closure testing, Eur. Phys. J. C 82, 330 (2022).
  38. R. D. Ball, A. Candido, J. Cruz-Martinez, S. Forte, T. Giani, F. Hekhorn, K. Kudashkin, G. Magni, and J. Rojo (NNPDF Collaboration), Evidence for intrinsic charm quarks in the proton, Nature (London) 608, 483 (2022).
  39. R. D. Ball et al. (NNPDF Collaboration), Determination of the theory uncertainties from missing higher orders on NNLO parton distributions with percent accuracy, Eur. Phys. J. C 84, 517 (2024).
  40. S. Munier, A. M. Stasto, and A. H. Mueller, Impact parameter dependent S matrix for dipole proton scattering from diffractive meson electroproduction, Nucl. Phys. B603, 427 (2001).
  41. K. J. Golec-Biernat and M. Wusthoff, Saturation effects in deep inelastic scattering at low Q2 and its implications on diffraction, Phys. Rev. D 59, 014017 (1998).
  42. J. Bartels, K. J. Golec-Biernat, and H. Kowalski, A modification of the saturation model: DGLAP evolution, Phys. Rev. D 66, 014001 (2002).
  43. H. Kowalski and D. Teaney, An impact parameter dipole saturation model, Phys. Rev. D 68, 114005 (2003).
  44. E. Iancu, K. Itakura, and S. Munier, Saturation and BFKL dynamics in the HERA data at small x, Phys. Lett. B 590, 199 (2004).
  45. H. Kowalski, L. Motyka, and G. Watt, Exclusive diffractive processes at HERA within the dipole picture, Phys. Rev. D 74, 074016 (2006).
  46. G. Watt and H. Kowalski, Impact parameter dependent colour glass condensate dipole model, Phys. Rev. D 78, 014016 (2008).
  47. A. H. Rezaeian, M. Siddikov, M. Van de Klundert, and R. Venugopalan, Analysis of combined HERA data in the impact-parameter dependent saturation model, Phys. Rev. D 87, 034002 (2013).
  48. H. Mäntysaari and P. Zurita, In depth analysis of the combined HERA data in the dipole models with and without saturation, Phys. Rev. D 98, 036002 (2018).
  49. J. L. Albacete, N. Armesto, J. G. Milhano, and C. A. Salgado, Non-linear QCD meets data: A global analysis of lepton-proton scattering with running coupling BK evolution, Phys. Rev. D 80, 034031 (2009).
  50. J. L. Albacete, N. Armesto, J. G. Milhano, P. Quiroga-Arias, and C. A. Salgado, AAMQS: A non-linear QCD analysis of new HERA data at small-x including heavy quarks, Eur. Phys. J. C 71, 1705 (2011).
  51. T. Lappi and H. Mäntysaari, Single inclusive particle production at high energy from HERA data to proton-nucleus collisions, Phys. Rev. D 88, 114020 (2013).
  52. B. Ducloué, E. Iancu, G. Soyez, and D. N. Triantafyllopoulos, HERA data and collinearly-improved BK dynamics, Phys. Lett. B 803, 135305 (2020).
  53. G. Soyez, Saturation QCD predictions with heavy quarks at HERA, Phys. Lett. B 655, 32 (2007).
  54. J. Berger and A. M. Stasto, Small x nonlinear evolution with impact parameter and the structure function data, Phys. Rev. D 84, 094022 (2011).
  55. A. H. Rezaeian and I. Schmidt, Impact-parameter dependent color glass condensate dipole model and new combined HERA data, Phys. Rev. D 88, 074016 (2013).
  56. D. Bendova, J. Cepila, J. G. Contreras, and M. Matas, Solution to the Balitsky-Kovchegov equation with the collinearly improved kernel including impact-parameter dependence, Phys. Rev. D 100, 054015 (2019).
  57. L. D. McLerran and R. Venugopalan, Computing quark and gluon distribution functions for very large nuclei, Phys. Rev. D 49, 2233 (1994).
  58. G. Beuf, Dipole factorization for DIS at NLO: Loop correction to the γT,L*→qq¯ light-front wave functions, Phys. Rev. D 94, 054016 (2016).
  59. G. Beuf, Dipole factorization for DIS at NLO: Combining the qq¯ and qq¯g contributions, Phys. Rev. D 96, 074033 (2017).
  60. H. Hänninen, T. Lappi, and R. Paatelainen, One-loop corrections to light cone wave functions: The dipole picture DIS cross section, Ann. Phys. (Amsterdam) 393, 358 (2018).
  61. G. Beuf, T. Lappi, and R. Paatelainen, Massive quarks in NLO dipole factorization for DIS: Longitudinal photon, Phys. Rev. D 104, 056032 (2021).
  62. G. Beuf, T. Lappi, and R. Paatelainen, Massive quarks at one loop in the dipole picture of deep inelastic scattering, Phys. Rev. Lett. 129, 072001 (2022).
  63. G. Beuf, T. Lappi, and R. Paatelainen, Massive quarks in NLO dipole factorization for DIS: Transverse photon, Phys. Rev. D 106, 034013 (2022).
  64. E. Iancu, A. H. Mueller, and D. N. Triantafyllopoulos, CGC factorization for forward particle production in proton-nucleus collisions at next-to-leading order, J. High Energy Phys. 12 (2016) 041.
  65. B. Ducloué, T. Lappi, and Y. Zhu, Implementation of NLO high energy factorization in single inclusive forward hadron production, Phys. Rev. D 95, 114007 (2017).
  66. A. M. Stasto, B.-W. Xiao, and D. Zaslavsky, Towards the test of saturation physics beyond leading logarithm, Phys. Rev. Lett. 112, 012302 (2014).
  67. A. M. Staśto, B.-W. Xiao, F. Yuan, and D. Zaslavsky, Matching collinear and small x factorization calculations for inclusive hadron production in pA collisions, Phys. Rev. D 90, 014047 (2014).
  68. T. Altinoluk, N. Armesto, G. Beuf, A. Kovner, and M. Lublinsky, Single-inclusive particle production in proton-nucleus collisions at next-to-leading order in the hybrid formalism, Phys. Rev. D 91, 094016 (2015).
  69. Z.-B. Kang, I. Vitev, and H. Xing, Next-to-leading order forward hadron production in the small-x regime: Rapidity factorization, Phys. Rev. Lett. 113, 062002 (2014).
  70. T. Altinoluk, N. Armesto, G. Beuf, A. Kovner, and M. Lublinsky, Heavy quarks in proton-nucleus collisions—the hybrid formalism, Phys. Rev. D 93, 054049 (2016).
  71. R. Boussarie, A. V. Grabovsky, L. Szymanowski, and S. Wallon, On the one loop γ(*)→qq¯ impact factor and the exclusive diffractive cross sections for the production of two or three jets, J. High Energy Phys. 11 (2016) 149.
  72. R. Boussarie, A. V. Grabovsky, D. Y. Ivanov, L. Szymanowski, and S. Wallon, Next-to-leading order computation of exclusive diffractive light vector meson production in a saturation framework, Phys. Rev. Lett. 119, 072002 (2017).
  73. H. Mäntysaari and J. Penttala, Exclusive heavy vector meson production at next-to-leading order in the dipole picture, Phys. Lett. B 823, 136723 (2021).
  74. H. Mäntysaari and J. Penttala, Complete calculation of exclusive heavy vector meson production at next-to-leading order in the dipole picture, J. High Energy Phys. 08 (2022) 247.
  75. G. Beuf, H. Hänninen, T. Lappi, Y. Mulian, and H. Mäntysaari, Diffractive deep inelastic scattering at NLO in the dipole picture: The qqg contribution, Phys. Rev. D 106, 094014 (2022).
  76. G. Beuf, T. Lappi, H. Mäntysaari, R. Paatelainen, and J. Penttala, Diffractive deep inelastic scattering at NLO in the dipole picture, J. High Energy Phys. 05 (2024) 024.
  77. T. Altinoluk, G. Beuf, A. Czajka, and A. Tymowska, DIS dijet production at next-to-eikonal accuracy in the CGC, Phys. Rev. D 107, 074016 (2023).
  78. T. Altinoluk, N. Armesto, and G. Beuf, Probing quark transverse momentum distributions in the color glass condensate: Quark-gluon dijets in deep inelastic scattering at next-to-eikonal accuracy, Phys. Rev. D 108, 074023 (2023).
  79. T. Altinoluk, G. Beuf, and S. Mulani, Forward parton-nucleus scattering at next-to-eikonal accuracy in the color glass condensate, Phys. Rev. D 111, 034028 (2025).
  80. T. Altinoluk, G. Beuf, E. Blanco, and S. Mulani, Quark TMDs from back-to-back dijet production at forward rapidities in pA collisions beyond eikonal accuracy in the CGC, J. High Energy Phys. 06 (2025) 097.
  81. T. Altinoluk, G. Beuf, A. Czajka, and C. Marquet, Back-to-back dijet production in DIS at next-to-eikonal accuracy and twist-3 gluon TMDs, Phys. Rev. D 111, 014010 (2025).
  82. Y. Hatta, B.-W. Xiao, and F. Yuan, Semi-inclusive diffractive deep inelastic scattering at small x, Phys. Rev. D 106, 094015 (2022).
  83. P. Agostini, T. Altinoluk, and N. Armesto, Next-to-eikonal corrections to dijet production in deep inelastic scattering in the dilute limit of the color glass condensate, J. High Energy Phys. 07 (2024) 137.
  84. M. Fucilla, A. V. Grabovsky, E. Li, L. Szymanowski, and S. Wallon, NLO computation of diffractive di-hadron production in a saturation framework, J. High Energy Phys. 03 (2023) 159.
  85. M. Fucilla, A. Grabovsky, E. Li, L. Szymanowski, and S. Wallon, Diffractive single hadron production in a saturation framework at the NLO, J. High Energy Phys. 02 (2024) 165.
  86. G. Beuf, H. Hänninen, T. Lappi, and H. Mäntysaari, Color Glass Condensate at next-to-leading order meets HERA data (software) (2020), https://github.com/hejajama/nlobkfitter.
  87. G. Beuf, H. Hänninen, T. Lappi, and H. Mäntysaari, Color glass condensate at next-to-leading order meets HERA data (data) (2020), 10.5281/zenodo.4229269.
  88. J. Hadamard, Sur les problèmes aux dérivés partielles et leur signification physique, Princeton Univ. Bull. 13, 49 (1902).
  89. A. H. Mueller, Soft gluons in the infinite momentum wave function and the BFKL pomeron, Nucl. Phys. B415, 373 (1994).
  90. A. H. Mueller, Unitarity and the BFKL pomeron, Nucl. Phys. B437, 107 (1995).
  91. A. H. Mueller and B. Patel, Single and double BFKL pomeron exchange and a dipole picture of high-energy hard processes, Nucl. Phys. B425, 471 (1994).
  92. Z. Chen and A. H. Mueller, The dipole picture of high-energy scattering, the BFKL equation and many gluon compound states, Nucl. Phys. B451, 579 (1995).
  93. H. Abramowicz et al. (H1, ZEUS Collaboration), 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).
  94. H. Abramowicz et al. (H1, ZEUS Collaboration), Combination and QCD analysis of charm and beauty production cross-section measurements in deep inelastic ep scattering at HERA, Eur. Phys. J. C 78, 473 (2018).
  95. F. D. Aaron et al. (H1, ZEUS Collaboration), Combined measurement and QCD analysis of the inclusive e±p scattering cross sections at HERA, J. High Energy Phys. 01 (2010) 109.
  96. M. E. Peskin and D. V. Schroeder, An Introduction to Quantum Field Theory (Addison-Wesley, Reading, USA, 1995), p. 842.
  97. J. B. Kogut and D. E. Soper, Quantum electrodynamics in the infinite momentum frame, Phys. Rev. D 1, 2901 (1970).
  98. J. D. Bjorken, J. B. Kogut, and D. E. Soper, Quantum electrodynamics at infinite momentum: Scattering from an external field, Phys. Rev. D 3, 1382 (1971).
  99. G. P. Lepage and S. J. Brodsky, Exclusive processes in perturbative quantum chromodynamics, Phys. Rev. D 22, 2157 (1980).
  100. S. J. Brodsky, H.-C. Pauli, and S. S. Pinsky, Quantum chromodynamics and other field theories on the light cone, Phys. Rep. 301, 299 (1998).
  101. Y. V. Kovchegov and E. Levin, Quantum Chromodynamics at High Energy (Oxford University Press, New York, 2013), Vol. 33.
  102. E. Iancu and R. Venugopalan, The color glass condensate and high-energy scattering in QCD, in Quark—Gluon Plasma 3 (World Scientific, Singapore, 2004), 3, Chap. 6.
  103. H. Weigert, Evolution at small xBj: The color glass condensate, Prog. Part. Nucl. Phys. 55, 461 (2005).
  104. F. Gelis, E. Iancu, J. Jalilian-Marian, and R. Venugopalan, The color glass condensate, Annu. Rev. Nucl. Part. Sci. 60, 463 (2010).
  105. J. L. Albacete and C. Marquet, Gluon saturation and initial conditions for relativistic heavy ion collisions, Prog. Part. Nucl. Phys. 76, 1 (2014).
  106. J.-P. Blaizot, High gluon densities in heavy ion collisions, Rep. Prog. Phys. 80, 032301 (2017).
  107. N. N. Nikolaev and B. G. Zakharov, Color transparency and scaling properties of nuclear shadowing in deep inelastic scattering, Z. Phys. C 49, 607 (1991).
  108. H. G. Dosch, T. Gousset, G. Kulzinger, and H. J. Pirner, Vector meson leptoproduction and nonperturbative gluon fluctuations in QCD, Phys. Rev. D 55, 2602 (1997).
  109. C. Marquet, A unified description of diffractive deep inelastic scattering with saturation, Phys. Rev. D 76, 094017 (2007).
  110. I. Balitsky, Operator expansion for high-energy scattering, Nucl. Phys. B463, 99 (1996).
  111. Y. V. Kovchegov, Unitarization of the BFKL pomeron on a nucleus, Phys. Rev. D 61, 074018 (2000).
  112. Y. V. Kovchegov, Small x F2 structure function of a nucleus including multiple pomeron exchanges, Phys. Rev. D 60, 034008 (1999).
  113. J. Penttala, Color-glass condensate beyond the Gaussian approximation, arXiv:2507.18711.
  114. H. Mäntysaari and B. Schenke, Confronting impact parameter dependent JIMWLK evolution with HERA data, Phys. Rev. D 98, 034013 (2018).
  115. A. Caldwell and H. Kowalski in, Proceedings of the 13th International Conference on Elastic and Diffractive Scattering (Blois Workshop): Moving Forward into the LHC Era (2009), 9, pp. 190–192, arXiv:0909.1254.
  116. P. C. Hansen, Regularization tools: A Matlab package for analysis and solution of discrete ill-posed problems, Numerical Algorithms 6, 1 (1994).
  117. P. C. Hansen, Regularization tools version 4.0 for Matlab 7.3, Numerical Algorithms 46, 189 (2007).
  118. J. L. Mueller and S. Siltanen, Linear and Nonlinear Inverse Problems with Practical Applications (Society for Industrial and Applied Mathematics, Philadelphia, PA, 2012).
  119. A. N. Tikhonov, On the stability of inverse problems, Dokl. Akad. Nauk SSSR 39, 195 (1943).
  120. A. N. Tikhonov, Solution of incorrectly formulated problems and the regularization method, Dokl. Akad. Nauk SSSR 151, 501 (1963).
  121. A. N. Tikhonov and V. Y. Arsenin, Solution of Ill-Posed Problems (Winston & Sons, Washington, 1977).
  122. D. L. Phillips, A technique for the numerical solution of certain integral equations of the first kind, J. ACM 9, 84 (1962).
  123. A. E. Hoerl, Application of ridge analysis to regression problems, Chem. Eng. Prog. 58, 54 (1962).
  124. G. Cimmino, Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari, La Ricerca scientifica : rivista del Consiglio nazionale delle ricerche XVI, 326 (1938).
  125. S. Kaczmarz, Angenäherte auflösung von systemen linearer gleichungen, Bull. Int. l’Acad. Pol. Sci. Lett. 35, 355 (1937).
  126. R. Gordon, R. Bender, and G. T. Herman, Algebraic reconstruction techniques (art) for three-dimensional electron microscopy and x-ray photography, J. Theor. Biol. 29, 471 (1970).
  127. D. Calvetti and E. Somersalo, Introduction to Bayesian Scientific Computing: Ten Lectures on Subjective Computing (Springer, New York, 2007), Vol. 2.
  128. P. C. Hansen, Discrete Inverse Problems: Insight and Algorithms (Society for Industrial and Applied Mathematics, Philadelphia, 2010).
  129. H. A. Schlüter, Iterative tomographic reconstruction with priorconditioning, technical report, Technical University of Denmark, Department of Applied Mathematics and Technology, 2017.
  130. H. Engl, Discrepancy principles for tikhonov regularization of ill-posed problems leading to optimal convergence rates, J. Optim. Theory Appl. 52, 209 (1987).
  131. P. C. Hansen, The L-curve and its use in the numerical treatment of inverse problems, in Computational Inverse Problems in Electrocardiography (WIT Press, Ashurst Lodge, Ashurst, Southampton, 2000), pp. 119–142.
  132. G. H. Golub, M. Heath, and G. Wahba, Generalized cross-validation as a method for choosing a good ridge parameter, Technometrics 21, 215 (1979).
  133. B. Ducloué, H. Hänninen, T. Lappi, and Y. Zhu, Deep inelastic scattering in the dipole picture at next-to-leading order, Phys. Rev. D 96, 094017 (2017).
  134. A. Wirgin, The inverse crime, arXiv:math-ph/0401050.
  135. V. Barone and E. Predazzi, High-Energy Particle Diffraction, Texts and Monographs in Physics Vol. 565 (Springer-Verlag, Berlin Heidelberg, 2002).
  136. F. D. Aaron et al. (H1, ZEUS Collaboration), Combined inclusive diffractive cross sections measured with forward proton spectrometers in deep inelastic ep scattering at HERA, Eur. Phys. J. C 72, 2175 (2012).
  137. J. Jalilian-Marian, A. Kovner, L. D. McLerran, and H. Weigert, The Intrinsic glue distribution at very small x, Phys. Rev. D 55, 5414 (1997).
  138. J. Jalilian-Marian, A. Kovner, A. Leonidov, and H. Weigert, The BFKL equation from the Wilson renormalization group, Nucl. Phys. B504, 415 (1997).
  139. J. Jalilian-Marian, A. Kovner, and H. Weigert, The Wilson renormalization group for low x physics: Gluon evolution at finite parton density, Phys. Rev. D 59, 014015 (1998).
  140. J. Jalilian-Marian, A. Kovner, A. Leonidov, and H. Weigert, The Wilson renormalization group for low x physics: Towards the high density regime, Phys. Rev. D 59, 014014 (1998).
  141. E. Iancu and L. D. McLerran, Saturation and universality in QCD at small x, Phys. Lett. B 510, 145 (2001).
  142. E. Iancu, A. Leonidov, and L. D. McLerran, Nonlinear gluon evolution in the color glass condensate. 1., Nucl. Phys. A692, 583 (2001).
  143. E. Ferreiro, E. Iancu, A. Leonidov, and L. McLerran, Nonlinear gluon evolution in the color glass condensate. 2., Nucl. Phys. A703, 489 (2002).
  144. E. Iancu, A. Leonidov, and L. D. McLerran, The renormalization group equation for the color glass condensate, Phys. Lett. B 510, 133 (2001).
  145. H. Hänninen, A. Kykkänen, and H. Schlüter, Reconstruction of the dipole amplitude in the dipole picture as a mathematical inverse problem (software) (2025), https://github.com/hhannine/inversedipole/tree/arXiv-2509.05005.

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Phys. Rev. D 112, 094026– Published 14 November, 2025
  • Received 9 September 2025
  • Accepted 29 October 2025
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