- Open Access
Exclusive production off a dilute proton within a refined hotspot description
Phys. Rev. D 112, 114006 – Published 1 December, 2025
DOI: https://doi.org/10.1103/rqhc-dfsf
Abstract
We revisit the calculation of exclusive production in electron-proton scattering within the QCD dipole model, employing a refined description of the proton in terms of gluonic hot spots in the dilute regime. In contrast to earlier studies, we incorporate key missing elements; the event-by-event fluctuations in both the number of hot spots and their local saturation scales, as well as the first relativistic corrections to the wave function. We perform a Bayesian analysis to constrain the model parameters using HERA data. These enhancements significantly improve the agreement with HERA data. Saturation scale fluctuations are found to be more important than fluctuations in the number of hot spots. Including the first relativistic correction is found to be important, especially in order to describe the observed power-law behavior in the incoherent cross section at large .
Physics Subject Headings (PhySH)
Article Text
Supplemental Material
References (70)
- A. J. Baltz, G. Baur, D. Denterria, L. Frankfurt, F. Gelis, V. Guzey, K. Hencken, Y. Kharlov, M. Klasen, and S. Klein, The physics of ultraperipheral collisions at the LHC, Phys. Rep. 458, 1 (2008).
- S. Klein et al., New opportunities at the photon energy frontier, arXiv:2009.03838.
- 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).
- A. Dumitru and T. Stebel, Multiquark matrix elements in the proton and three gluon exchange for exclusive production in photon-proton diffractive scattering, Phys. Rev. D 99, 094038 (2019).
- A. Dumitru and R. Paatelainen, Sub-femtometer scale color charge fluctuations in a proton made of three quarks and a gluon, Phys. Rev. D 103, 034026 (2021); 108, 059901(E) (2023).
- A. Dumitru, H. Mäntysaari, and R. Paatelainen, Color charge correlations in the proton at NLO: Beyond geometry based intuition, Phys. Lett. B 820, 136560 (2021).
- V. N. Gribov and L. N. Lipatov, Deep inelastic e p scattering in perturbation theory, Sov. J. Nucl. Phys. 15, 438 (1972).
- L. N. Lipatov, The parton model and perturbation theory, Yad. Fiz. 20, 181 (1974).
- G. Altarelli and G. Parisi, Asymptotic freedom in Parton language, Nucl. Phys. B126, 298 (1977).
- Y. L. Dokshitzer, Calculation of the structure functions for deep inelastic scattering and annihilation by perturbation theory in quantum chromodynamics, Sov. Phys. JETP 46, 641 (1977), https://inspirehep.net/literature/3074621.
- L. N. Lipatov, Reggeization of the vector meson and the vacuum singularity in non-Abelian gauge theories, Sov. J. Nucl. Phys. 23, 338 (1976).
- E. A. Kuraev, L. N. Lipatov, and V. S. Fadin, Multi—Reggeon processes in the Yang-Mills theory, Sov. Phys. JETP 44, 443 (1976), https://inspirehep.net/literature/116034.
- E. A. Kuraev, L. N. Lipatov, and V. S. Fadin, The Pomeranchuk singularity in non-Abelian gauge theories, Sov. Phys. JETP 45, 199 (1977), .https://inspirehep.net/literature/123995
- I. I. Balitsky and L. N. Lipatov, The Pomeranchuk singularity in quantum chromodynamics, Sov. J. Nucl. Phys. 28, 822 (1978).
- J. L. Albacete and A. Soto-Ontoso, Hot spots and the hollowness of proton–proton interactions at high energies, Phys. Lett. B 770, 149 (2017).
- J. L. Albacete, H. Petersen, and A. Soto-Ontoso, Symmetric cumulants as a probe of the proton substructure at LHC energies, Phys. Lett. B 778, 128 (2018).
- H. Mäntysaari and B. Schenke, Evidence of strong proton shape fluctuations from incoherent diffraction, Phys. Rev. Lett. 117, 052301 (2016).
- H. Mäntysaari and B. Schenke, Revealing proton shape fluctuations with incoherent diffraction at high energy, Phys. Rev. D 94, 034042 (2016).
- H. Mäntysaari and B. Schenke, Probing subnucleon scale fluctuations in ultraperipheral heavy ion collisions, Phys. Lett. B 772, 832 (2017).
- J. Cepila, J. G. Contreras, and J. D. Tapia Takaki, Energy dependence of dissociative photoproduction as a signature of gluon saturation at the LHC, Phys. Lett. B 766, 186 (2017).
- J. Cepila, J. G. Contreras, and M. Krelina, Coherent and incoherent photonuclear production in an energy-dependent hot-spot model, Phys. Rev. C 97, 024901 (2018).
- H. Mäntysaari and B. Schenke, Confronting impact parameter dependent JIMWLK evolution with HERA data, Phys. Rev. D 98, 034013 (2018).
- M. C. Traini and J.-P. Blaizot, Diffractive incoherent vector meson production off protons: A quark model approach to gluon fluctuation effects, Eur. Phys. J. C 79, 327 (2019).
- A. Kumar and T. Toll, Investigating the structure of gluon fluctuations in the proton with incoherent diffraction at HERA, Eur. Phys. J. C 82, 837 (2022).
- A. Kumar and T. Toll, Energy dependence of the proton geometry in exclusive vector meson production, Phys. Rev. D 105, 114011 (2022).
- S. Demirci, T. Lappi, and S. Schlichting, Hot spots and gluon field fluctuations as causes of eccentricity in small systems, Phys. Rev. D 103, 094025 (2021).
- S. Demirci, T. Lappi, and S. Schlichting, Proton hot spots and exclusive vector meson production, Phys. Rev. D 106, 074025 (2022).
- H. Mäntysaari, B. Schenke, C. Shen, and W. Zhao, Bayesian inference of the fluctuating proton shape, Phys. Lett. B 833, 137348 (2022).
- B. Schenke and R. Venugopalan, Eccentric protons? Sensitivity of flow to system size and shape in , and collisions, Phys. Rev. Lett. 113, 102301 (2014).
- H. Mäntysaari, B. Schenke, C. Shen, and P. Tribedy, Imprints of fluctuating proton shapes on flow in proton-lead collisions at the LHC, Phys. Lett. B 772, 681 (2017).
- J. S. Moreland, J. E. Bernhard, and S. A. Bass, Bayesian calibration of a hybrid nuclear collision model using and Pb-Pb data at energies available at the CERN Large Hadron Collider, Phys. Rev. C 101, 024911 (2020).
- H. Mäntysaari, Review of proton and nuclear shape fluctuations at high energy, Rep. Prog. Phys. 83, 082201 (2020).
- E. Iancu and R. Venugopalan, The color glass condensate and high-energy scattering in QCD, in Quark—Gluon Plasma 3, edited by R. C. Hwa and X.-N. Wang (World Scientific, Singapore, 2003), pp. 249–3363.
- L. D. McLerran and R. Venugopalan, Gluon distribution functions for very large nuclei at small transverse momentum, Phys. Rev. D 49, 3352 (1994).
- L. D. McLerran and R. Venugopalan, Computing quark and gluon distribution functions for very large nuclei, Phys. Rev. D 49, 2233 (1994).
- L. D. McLerran and R. Venugopalan, Green’s functions in the color field of a large nucleus, Phys. Rev. D 50, 2225 (1994).
- I. Bautista, A. Fernandez Tellez, and M. Hentschinski, BFKL evolution and the growth with energy of exclusive and photoproduction cross sections, Phys. Rev. D 94, 054002 (2016).
- J. Penttala and C. Royon, Gluon saturation effects in exclusive heavy vector meson photoproduction, Phys. Lett. B 864, 139394 (2025).
- 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).
- J. E. Parkkila, A. Onnerstad, S. F. Taghavi, C. Mordasini, A. Bilandzic, M. Virta, and D. J. Kim, New constraints for QCD matter from improved Bayesian parameter estimation in heavy-ion collisions at LHC, Phys. Lett. B 835, 137485 (2022).
- D. Everett et al. (JETSCAPE Collaboration), Multisystem Bayesian constraints on the transport coefficients of QCD matter, Phys. Rev. C 103, 054904 (2021).
- C. Casuga, M. Karhunen, and H. Mäntysaari, Inferring the initial condition for the Balitsky-Kovchegov equation, Phys. Rev. D 109, 054018 (2024).
- C. Alexa et al. (H1 Collaboration), Elastic and proton-dissociative photoproduction of mesons at HERA, Eur. Phys. J. C 73, 2466 (2013).
- H. Kowalski, L. Motyka, and G. Watt, Exclusive diffractive processes at HERA within the dipole picture, Phys. Rev. D 74, 074016 (2006).
- Y. Hatta, B.-W. Xiao, and F. Yuan, Gluon tomography from deeply virtual Compton scattering at small-x, Phys. Rev. D 95, 114026 (2017).
- M. L. Good and W. D. Walker, Diffraction disssociation of beam particles, Phys. Rev. 120, 1857 (1960).
- H. I. Miettinen and J. Pumplin, Diffraction scattering and the parton structure of hadrons, Phys. Rev. D 18, 1696 (1978).
- A. Caldwell and H. Kowalski, Investigating the gluonic structure of nuclei via scattering, Phys. Rev. C 81, 025203 (2010).
- S. R. Klein, Challenges to the Good-Walker paradigm in coherent and incoherent photoproduction, Phys. Rev. C 107, 055203 (2023).
- L. McLerran and P. Tribedy, Intrinsic fluctuations of the proton saturation momentum scale in high multiplicity p + p collisions, Nucl. Phys. A945, 216 (2016).
- A. Bzdak and K. Dusling, Probing proton fluctuations with asymmetric rapidity correlations, Phys. Rev. C 93, 031901 (2016).
- 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).
- 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).
- J. Jalilian-Marian, A. Kovner, A. Leonidov, and H. Weigert, The BFKL equation from the Wilson renormalization group, Nucl. Phys. B504, 415 (1997).
- E. Iancu, A. Leonidov, and L. D. McLerran, Nonlinear gluon evolution in the color glass condensate. 1, Nucl. Phys. A692, 583 (2001).
- E. Ferreiro, E. Iancu, A. Leonidov, and L. McLerran, Nonlinear gluon evolution in the color glass condensate. II, Nucl. Phys. A703, 489 (2002).
- E. Iancu, A. Leonidov, and L. D. McLerran, The renormalization group equation for the color glass condensate, Phys. Lett. B 510, 133 (2001).
- E. Iancu and L. D. McLerran, Saturation and universality in QCD at small x, Phys. Lett. B 510, 145 (2001).
- I. Balitsky, Operator expansion for high-energy scattering, Nucl. Phys. B463, 99 (1996).
- Y. V. Kovchegov, Small x structure function of a nucleus including multiple Pomeron exchanges, Phys. Rev. D 60, 034008 (1999).
- 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).
- Y. V. Kovchegov and E. Levin, Quantum Chromodynamics at High Energy, Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology (Cambridge University Press, Cambridge, England, 2012).
- A.-K. Angelopoulou, A. D. Le, and S. Munier, Scattering from an external field in quantum chromodynamics at high energies: From foundations to interdisciplinary connections, SciPost Phys. Lect. Notes 92, 1 (2025).
- T. Lappi, H. Mäntysaari, and J. Penttala, Relativistic corrections to the vector meson light front wave function, Phys. Rev. D 102, 054020 (2020).
- C. K. Williams and C. E. Rasmussen, Gaussian Processes for Machine Learning (MIT Press Cambridge, MA, 2006), Vol. 2.
- J. Goodman and J. Weare, Ensemble samplers with affine invariance, Commun. Appl. Math. Comput. Sci. 5, 65 (2010).
- D. Foreman-Mackey, D. W. Hogg, D. Lang, and J. Goodman, emcee: The MCMC hammer, Publ. Astron. Soc. Pac. 125, 306 (2013).
- G. T. Bodwin, H. S. Chung, D. Kang, J. Lee, and C. Yu, Improved determination of color-singlet nonrelativistic QCD matrix elements for S-wave charmonium, Phys. Rev. D 77, 094017 (2008).
- A. Aktas et al. (H1 Collaboration), Diffractive photoproduction of mesons with large momentum transfer at HERA, Phys. Lett. B 568, 205 (2003).
- See Supplemental Material at http://link.aps.org/supplemental/10.1103/rqhc-dfsf for posterior distributions for model parameters.