Abstract
In high energy proton-nucleus collisions, the single- and double-inclusive soft gluon productions at the leading order have been calculated and phenomenologically studied in various approaches for many years. These studies do not take into account the saturation and multiple rescatterings in the field of the proton. The first saturation correction to these leading order results (the terms that are enhanced by the combination \( {\alpha}_s^2{\mu}^2 \), where μ2 is the proton’s color charge squared per unit transverse area) has not been completely derived despite recent attempts using a diagrammatic approach. This paper is the first in a series of papers towards analytically completing the first saturation correction to physical observables in high energy proton-nucleus collisions. Our approach is to analytically solve the classical Yang-Mills equations in the dilute-dense regime using the Color Glass Condensate effective theory and compute physical observables constructed from classical gluon fields. In the current paper, the Yang-Mills equations are solved perturbatively in the field of the dilute object (the proton). Next-to-leading order and next-to-next-to-leading order analytic solutions are explicitly constructed. A systematic way to obtain all higher order analytic solutions is outlined.
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C. Gale, S. Jeon and B. Schenke, Hydrodynamic Modeling of Heavy-Ion Collisions, Int. J. Mod. Phys. A 28 (2013) 1340011 [arXiv:1301.5893] [INSPIRE].
U. Heinz and R. Snellings, Collective flow and viscosity in relativistic heavy-ion collisions, Ann. Rev. Nucl. Part. Sci. 63 (2013) 123 [arXiv:1301.2826] [INSPIRE].
B. Schenke, C. Shen and P. Tribedy, Hybrid Color Glass Condensate and hydrodynamic description of the Relativistic Heavy Ion Collider small system scan, Phys. Lett. B 803 (2020) 135322 [arXiv:1908.06212] [INSPIRE].
R. Baier, A. H. Mueller, D. Schiff and D. T. Son, ’Bottom up’ thermalization in heavy ion collisions, Phys. Lett. B 502 (2001) 51 [hep-ph/0009237] [INSPIRE].
A. Kurkela and Y. Zhu, Isotropization and hydrodynamization in weakly coupled heavy-ion collisions, Phys. Rev. Lett. 115 (2015) 182301 [arXiv:1506.06647] [INSPIRE].
A. Kurkela, A. Mazeliauskas, J.-F. Paquet, S. Schlichting and D. Teaney, Matching the Nonequilibrium Initial Stage of Heavy Ion Collisions to Hydrodynamics with QCD Kinetic Theory, Phys. Rev. Lett. 122 (2019) 122302 [arXiv:1805.01604] [INSPIRE].
B. Wu and Y. V. Kovchegov, Time-dependent observables in heavy ion collisions. Part I. Setting up the formalism, JHEP 03 (2018) 158 [arXiv:1709.02866] [INSPIRE].
Y. V. Kovchegov and B. Wu, Time-dependent observables in heavy ion collisions. Part II. In search of pressure isotropization in the φ4 theory, JHEP 03 (2018) 157 [arXiv:1709.02868] [INSPIRE].
A. Krasnitz and R. Venugopalan, The Initial energy density of gluons produced in very high-energy nuclear collisions, Phys. Rev. Lett. 84 (2000) 4309 [hep-ph/9909203] [INSPIRE].
A. Krasnitz and R. Venugopalan, The Initial gluon multiplicity in heavy ion collisions, Phys. Rev. Lett. 86 (2001) 1717 [hep-ph/0007108] [INSPIRE].
B. Schenke, P. Tribedy and R. Venugopalan, Fluctuating Glasma initial conditions and flow in heavy ion collisions, Phys. Rev. Lett. 108 (2012) 252301 [arXiv:1202.6646] [INSPIRE].
C. Gale, S. Jeon, B. Schenke, P. Tribedy and R. Venugopalan, Event-by-event anisotropic flow in heavy-ion collisions from combined Yang-Mills and viscous fluid dynamics, Phys. Rev. Lett. 110 (2013) 012302 [arXiv:1209.6330] [INSPIRE].
K. Dusling, W. Li and B. Schenke, Novel collective phenomena in high-energy proton-proton and proton-nucleus collisions, Int. J. Mod. Phys. E 25 (2016) 1630002 [arXiv:1509.07939] [INSPIRE].
J. L. Nagle and W. A. Zajc, Small System Collectivity in Relativistic Hadronic and Nuclear Collisions, Ann. Rev. Nucl. Part. Sci. 68 (2018) 211 [arXiv:1801.03477] [INSPIRE].
Y. V. Kovchegov and A. H. Mueller, Gluon production in current nucleus and nucleon - nucleus collisions in a quasiclassical approximation, Nucl. Phys. B 529 (1998) 451 [hep-ph/9802440] [INSPIRE].
B. Z. Kopeliovich, A. V. Tarasov and A. Schafer, Bremsstrahlung of a quark propagating through a nucleus, Phys. Rev. C 59 (1999) 1609 [hep-ph/9808378] [INSPIRE].
A. Kovner and U. A. Wiedemann, Eikonal evolution and gluon radiation, Phys. Rev. D 64 (2001) 114002 [hep-ph/0106240] [INSPIRE].
A. Dumitru and L. D. McLerran, How protons shatter colored glass, Nucl. Phys. A 700 (2002) 492 [hep-ph/0105268] [INSPIRE].
J. P. Blaizot, F. Gelis and R. Venugopalan, High-energy pA collisions in the color glass condensate approach. 1. Gluon production and the Cronin effect, Nucl. Phys. A 743 (2004) 13 [hep-ph/0402256] [INSPIRE].
I. Balitsky, Scattering of shock waves in QCD, Phys. Rev. D 70 (2004) 114030 [hep-ph/0409314] [INSPIRE].
G. A. Chirilli, Y. V. Kovchegov and D. E. Wertepny, Classical Gluon Production Amplitude for Nucleus-Nucleus Collisions: First Saturation Correction in the Projectile, JHEP 03 (2015) 015 [arXiv:1501.03106] [INSPIRE].
L. McLerran and V. V. Skokov, Odd Azimuthal Anisotropy of the Glasma for pA Scattering, Nucl. Phys. A 959 (2017) 83 [arXiv:1611.09870] [INSPIRE].
Y. V. Kovchegov and V. V. Skokov, How classical gluon fields generate odd azimuthal harmonics for the two-gluon correlation function in high-energy collisions, Phys. Rev. D 97 (2018) 094021 [arXiv:1802.08166] [INSPIRE].
I. Balitsky, Operator expansion for high-energy scattering, Nucl. Phys. B 463 (1996) 99 [hep-ph/9509348] [INSPIRE].
I. Balitsky, Factorization and high-energy effective action, Phys. Rev. D 60 (1999) 014020 [hep-ph/9812311] [INSPIRE].
I. Balitsky, Effective field theory for the small x evolution, Phys. Lett. B 518 (2001) 235 [hep-ph/0105334] [INSPIRE].
J. Jalilian-Marian, A. Kovner, A. Leonidov and H. Weigert, The BFKL equation from the Wilson renormalization group, Nucl. Phys. B 504 (1997) 415 [hep-ph/9701284] [INSPIRE].
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 (1998) 014014 [hep-ph/9706377] [INSPIRE].
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 (1998) 014015 [hep-ph/9709432] [INSPIRE].
E. Iancu, A. Leonidov and L. D. McLerran, The Renormalization group equation for the color glass condensate, Phys. Lett. B 510 (2001) 133 [hep-ph/0102009] [INSPIRE].
E. Iancu, A. Leonidov and L. D. McLerran, Nonlinear gluon evolution in the color glass condensate. 1, Nucl. Phys. A 692 (2001) 583 [hep-ph/0011241] [INSPIRE].
E. Ferreiro, E. Iancu, A. Leonidov and L. McLerran, Nonlinear gluon evolution in the color glass condensate. 2, Nucl. Phys. A 703 (2002) 489 [hep-ph/0109115] [INSPIRE].
H. Weigert, Unitarity at small Bjorken x, Nucl. Phys. A 703 (2002) 823 [hep-ph/0004044] [INSPIRE].
L. D. McLerran and R. Venugopalan, Computing quark and gluon distribution functions for very large nuclei, Phys. Rev. D 49 (1994) 2233 [hep-ph/9309289] [INSPIRE].
L. D. McLerran and R. Venugopalan, Gluon distribution functions for very large nuclei at small transverse momentum, Phys. Rev. D 49 (1994) 3352 [hep-ph/9311205] [INSPIRE].
T. Lappi, Wilson line correlator in the MV model: Relating the glasma to deep inelastic scattering, Eur. Phys. J. C 55 (2008) 285 [arXiv:0711.3039] [INSPIRE].
Y. V. Kovchegov, Quantum structure of the nonAbelian Weizsacker-Williams field for a very large nucleus, Phys. Rev. D 55 (1997) 5445 [hep-ph/9701229] [INSPIRE].
S. Schlichting and V. V. Skokov, Saturation corrections to dilute-dense particle production and azimuthal correlations in the Color Glass Condensate, Phys. Lett. B 806 (2020) 135511 [arXiv:1910.12496] [INSPIRE].
A. Kovner and M. Lublinsky, Angular and long range rapidity correlations in particle production at high energy, Int. J. Mod. Phys. E 22 (2013) 1330001 [arXiv:1211.1928] [INSPIRE].
Y. V. Kovchegov and D. E. Wertepny, Long-Range Rapidity Correlations in Heavy-Light Ion Collisions, Nucl. Phys. A 906 (2013) 50 [arXiv:1212.1195] [INSPIRE].
I. Balitsky, High-energy effective action from scattering of QCD shock waves, Phys. Rev. D 72 (2005) 074027 [hep-ph/0507237] [INSPIRE].
Y. Hatta, E. Iancu, L. McLerran, A. Stasto and D. N. Triantafyllopoulos, Effective Hamiltonian for QCD evolution at high energy, Nucl. Phys. A 764 (2006) 423 [hep-ph/0504182] [INSPIRE].
A. Kovner, E. Levin, M. Li and M. Lublinsky, Reggeon Field Theory and Self Duality: Making Ends Meet, JHEP 10 (2020) 185 [arXiv:2007.12132] [INSPIRE].
A. Krasnitz, Y. Nara and R. Venugopalan, Gluon production in the color glass condensate model of collisions of ultrarelativistic finite nuclei, Nucl. Phys. A 717 (2003) 268 [hep-ph/0209269] [INSPIRE].
T. Lappi, Production of gluons in the classical field model for heavy ion collisions, Phys. Rev. C 67 (2003) 054903 [hep-ph/0303076] [INSPIRE].
B. Schenke, S. Schlichting and R. Venugopalan, Azimuthal anisotropies in p+Pb collisions from classical Yang-Mills dynamics, Phys. Lett. B 747 (2015) 76 [arXiv:1502.01331] [INSPIRE].
L. D. McLerran, The Color glass condensate and small x physics: Four lectures, in Lecture Notes in Physics 583, Springer (2002), pp. 291–334 [hep-ph/0104285] [INSPIRE].
E. Iancu and R. Venugopalan, The Color glass condensate and high-energy scattering in QCD, in Quark-gluon plasma 3 , R.C. Hwa and X.-N. Wang eds., World Scientific (2004), pp. 249–363 [hep-ph/0303204] [INSPIRE].
Y. V. Kovchegov and E. Levin, Quantum chromodynamics at high energy, in Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology 33, Cambridge University Press, Cambridge U.K. (2012).
A. Kovner, L. D. McLerran and H. Weigert, Gluon production from nonAbelian Weizsacker-Williams fields in nucleus-nucleus collisions, Phys. Rev. D 52 (1995) 6231 [hep-ph/9502289] [INSPIRE].
M. Gyulassy and L. D. McLerran, Yang-Mills radiation in ultrarelativistic nuclear collisions, Phys. Rev. C 56 (1997) 2219 [nucl-th/9704034] [INSPIRE].
J. P. Blaizot, T. Lappi and Y. Mehtar-Tani, On the gluon spectrum in the glasma, Nucl. Phys. A 846 (2010) 63 [arXiv:1005.0955] [INSPIRE].
J. Berges, K. Boguslavski, S. Schlichting and R. Venugopalan, Universal attractor in a highly occupied non-Abelian plasma, Phys. Rev. D 89 (2014) 114007 [arXiv:1311.3005] [INSPIRE].
M. Li and V. V. Skokov, First saturation correction in high energy proton-nucleus collisions. Part II. Single inclusive semi-hard gluon production, JHEP 06 (2021) 141 [arXiv:2104.01879] [INSPIRE].
R. J. Fries, J. I. Kapusta and Y. Li, Near-fields and initial energy density in the color glass condensate model, nucl-th/0604054 [INSPIRE].
G. Chen, R. J. Fries, J. I. Kapusta and Y. Li, Early Time Dynamics of Gluon Fields in High Energy Nuclear Collisions, Phys. Rev. C 92 (2015) 064912 [arXiv:1507.03524] [INSPIRE].
A. L. Dixon and W. L. Ferrar, Integrals for the Product of Two Bessel Functions (II), Q. J. Math. os-4 (1933) 297.
G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd edition, Cambridge University Press, Cambridge U.K. (1995).
M. Abramowitz and I. Stegun, Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables, 0009-revised edition, Dover Publications (1965).
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Li, M., Skokov, V.V. First saturation correction in high energy proton-nucleus collisions. Part I. Time evolution of classical Yang-Mills fields beyond leading order. J. High Energ. Phys. 2021, 140 (2021). https://doi.org/10.1007/JHEP06(2021)140
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DOI: https://doi.org/10.1007/JHEP06(2021)140