Abstract
We calculate single-logarithmic corrections to the small-x flavor-singlet helicity evolution equations derived recently [1,2,3] in the double-logarithmic approximation. The new single-logarithmic part of the evolution kernel sums up powers of αs ln(1/x), which are an important correction to the dominant powers of αs ln2(1/x) summed up by the double-logarithmic kernel from [1,2,3] at small values of Bjorken x and with αs the strong coupling constant. The single-logarithmic terms arise separately from either the longitudinal or transverse momentum integrals. Consequently, the evolution equations we derive employing the light-cone perturbation theory simultaneously include the small-x evolution kernel and the leading-order polarized DGLAP splitting functions. We further enhance the equations by calculating the running coupling corrections to the kernel.
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References
Y.V. Kovchegov, D. Pitonyak and M.D. Sievert, Helicity Evolution at Small-x, JHEP 01 (2016) 072 [Erratum JHEP 10 (2016) 148] [arXiv:1511.06737] [INSPIRE].
Y.V. Kovchegov, D. Pitonyak and M.D. Sievert, Helicity Evolution at Small x: Flavor Singlet and Non-Singlet Observables, Phys. Rev. D 95 (2017) 014033 [arXiv:1610.06197] [INSPIRE].
Y.V. Kovchegov and M.D. Sievert, Small-x Helicity Evolution: an Operator Treatment, Phys. Rev. D 99 (2019) 054032 [arXiv:1808.09010] [INSPIRE].
T. Altinoluk, N. Armesto, G. Beuf, M. Martínez and C.A. Salgado, Next-to-eikonal corrections in the CGC: gluon production and spin asymmetries in pA collisions, JHEP 07 (2014) 068 [arXiv:1404.2219] [INSPIRE].
Y. Hatta, Y. Nakagawa, F. Yuan, Y. Zhao and B. Xiao, Gluon orbital angular momentum at small-x, Phys. Rev. D 95 (2017) 114032 [arXiv:1612.02445] [INSPIRE].
Y.V. Kovchegov, D. Pitonyak and M.D. Sievert, Small-x asymptotics of the quark helicity distribution, Phys. Rev. Lett. 118 (2017) 052001 [arXiv:1610.06188] [INSPIRE].
Y.V. Kovchegov, D. Pitonyak and M.D. Sievert, Small-x Asymptotics of the Quark Helicity Distribution: Analytic Results, Phys. Lett. B 772 (2017) 136 [arXiv:1703.05809] [INSPIRE].
Y.V. Kovchegov, D. Pitonyak and M.D. Sievert, Small-x Asymptotics of the Gluon Helicity Distribution, JHEP 10 (2017) 198 [arXiv:1706.04236] [INSPIRE].
G.A. Chirilli, Sub-eikonal corrections to scattering amplitudes at high energy, JHEP 01 (2019) 118 [arXiv:1807.11435] [INSPIRE].
Y.V. Kovchegov, Orbital Angular Momentum at Small x, JHEP 03 (2019) 174 [arXiv:1901.07453] [INSPIRE].
R. Boussarie, Y. Hatta and F. Yuan, Proton Spin Structure at Small-x, Phys. Lett. B 797 (2019) 134817 [arXiv:1904.02693] [INSPIRE].
F. Cougoulic and Y.V. Kovchegov, Helicity-dependent generalization of the JIMWLK evolution, Phys. Rev. D 100 (2019) 114020 [arXiv:1910.04268] [INSPIRE].
Y.V. Kovchegov and Y. Tawabutr, Helicity at Small x: Oscillations Generated by Bringing Back the Quarks, JHEP 08 (2020) 014 [arXiv:2005.07285] [INSPIRE].
F. Cougoulic and Y.V. Kovchegov, Helicity-dependent extension of the McLerran-Venugopalan model, Nucl. Phys. A 1004 (2020) 122051 [arXiv:2005.14688] [INSPIRE].
T. Altinoluk, G. Beuf, A. Czajka and A. Tymowska, Quarks at next-to-eikonal accuracy in the CGC: Forward quark-nucleus scattering, Phys. Rev. D 104 (2021) 014019 [arXiv:2012.03886] [INSPIRE].
G.A. Chirilli, High-energy operator product expansion at sub-eikonal level, JHEP 06 (2021) 096 [arXiv:2101.12744] [INSPIRE].
Jefferson Lab Angular Momentum collaboration, First analysis of world polarized DIS data with small-x helicity evolution, Phys. Rev. D 104 (2021) L031501 [arXiv:2102.06159] [INSPIRE].
C.A. Aidala, S.D. Bass, D. Hasch and G.K. Mallot, The Spin Structure of the Nucleon, Rev. Mod. Phys. 85 (2013) 655 [arXiv:1209.2803] [INSPIRE].
R.L. Jaffe and A. Manohar, The g1 Problem: Fact and Fantasy on the Spin of the Proton, Nucl. Phys. B 337 (1990) 509 [INSPIRE].
X.-D. Ji, Gauge-Invariant Decomposition of Nucleon Spin, Phys. Rev. Lett. 78 (1997) 610 [hep-ph/9603249] [INSPIRE].
E. Leader and C. Lorcé, The angular momentum controversy: What’s it all about and does it matter?, Phys. Rept. 541 (2014) 163 [arXiv:1309.4235] [INSPIRE].
A. Accardi et al., Electron Ion Collider: The Next QCD Frontier: Understanding the glue that binds us all, Eur. Phys. J. A 52 (2016) 268 [arXiv:1212.1701] [INSPIRE].
E.C. Aschenauer et al., The RHIC Spin Program: Achievements and Future Opportunities, arXiv:1304.0079 [INSPIRE].
E.-C. Aschenauer et al., The RHIC SPIN Program: Achievements and Future Opportunities, arXiv:1501.01220 [INSPIRE].
D. Boer et al., Gluons and the quark sea at high energies: Distributions, polarization, tomography, arXiv:1108.1713 [INSPIRE].
A. Prokudin, Y. Hatta, Y.V. Kovchegov and C. Marquet eds., Proceedings of the Probing Nucleons and Nuclei in High Energy Collisions: Dedicated to the Physics of the Electron Ion Collider, Seattle, WA, U.S.A., 1 October–16 November 2018, World Scientific (2020) [arXiv:2002.12333] [INSPIRE].
R. Abdul Khalek et al., Science Requirements and Detector Concepts for the Electron-Ion Collider: EIC Yellow Report, arXiv:2103.05419 [INSPIRE].
J. Bartels, B.I. Ermolaev and M.G. Ryskin, Nonsinglet contributions to the structure function g1 at small-x, Z. Phys. C 70 (1996) 273 [hep-ph/9507271] [INSPIRE].
J. Bartels, B.I. Ermolaev and M.G. Ryskin, Flavor singlet contribution to the structure function g1 at small x, Z. Phys. C 72 (1996) 627 [hep-ph/9603204] [INSPIRE].
J. Blümlein and A. Vogt, On the behavior of nonsinglet structure functions at small x, Phys. Lett. B 370 (1996) 149 [hep-ph/9510410] [INSPIRE].
J. Blümlein and A. Vogt, The Singlet contribution to the structure function g1(x, Q2) at small x, Phys. Lett. B 386 (1996) 350 [hep-ph/9606254] [INSPIRE].
R. Kirschner and L.N. Lipatov, Double Logarithmic Asymptotics and Regge Singularities of Quark Amplitudes with Flavor Exchange, Nucl. Phys. B 213 (1983) 122 [INSPIRE].
R. Kirschner, Reggeon interactions in perturbative QCD, Z. Phys. C 65 (1995) 505 [hep-th/9407085] [INSPIRE].
R. Kirschner, Regge asymptotics of scattering with flavor exchange in QCD, Z. Phys. C 67 (1995) 459 [hep-th/9404158] [INSPIRE].
S. Griffiths and D.A. Ross, Studying the perturbative Reggeon, Eur. Phys. J. C 12 (2000) 277 [hep-ph/9906550] [INSPIRE].
E.A. Kuraev, L.N. Lipatov and V.S. Fadin, The Pomeranchuk Singularity in NonAbelian Gauge Theories, Sov. Phys. JETP 45 (1977) 199 [Zh. Eksp. Teor. Fiz. 72 (1977) 377] [INSPIRE].
I.I. Balitsky and L.N. Lipatov, The Pomeranchuk Singularity in Quantum Chromodynamics, Sov. J. Nucl. Phys. 28 (1978) 822 [Yad. Fiz. 28 (1978) 1597] [INSPIRE].
I.I. Balitsky, Operator expansion for high-energy scattering, Nucl. Phys. B 463 (1996) 99 [hep-ph/9509348] [INSPIRE].
I.I. Balitsky, Factorization and high-energy effective action, Phys. Rev. D 60 (1999) 014020 [hep-ph/9812311] [INSPIRE].
Y.V. Kovchegov, Small x F2 structure function of a nucleus including multiple Pomeron exchanges, Phys. Rev. D 60 (1999) 034008 [hep-ph/9901281] [INSPIRE].
Y.V. Kovchegov, Unitarization of the BFKL Pomeron on a nucleus, Phys. Rev. D 61 (2000) 074018 [hep-ph/9905214] [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].
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].
H. Weigert, Unitarity at small Bjorken x, Nucl. Phys. A 703 (2002) 823 [hep-ph/0004044] [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].
I.I. Balitsky, Quark contribution to the small-x evolution of color dipole, Phys. Rev. D 75 (2007) 014001 [hep-ph/0609105] [INSPIRE].
Y.V. Kovchegov and H. Weigert, Triumvirate of Running Couplings in Small-x Evolution, Nucl. Phys. A 784 (2007) 188 [hep-ph/0609090] [INSPIRE].
E. Gardi, J. Kuokkanen, K. Rummukainen and H. Weigert, Running coupling and power corrections in nonlinear evolution at the high-energy limit, Nucl. Phys. A 784 (2007) 282 [hep-ph/0609087] [INSPIRE].
Y.V. Kovchegov and H. Weigert, Quark loop contribution to BFKL evolution: Running coupling and leading-Nf NLO intercept, Nucl. Phys. A 789 (2007) 260 [hep-ph/0612071] [INSPIRE].
K. Itakura, Y.V. Kovchegov, L. McLerran and D. Teaney, Baryon stopping and valence quark distribution at small x, Nucl. Phys. A 730 (2004) 160 [hep-ph/0305332] [INSPIRE].
L.V. Gribov, E.M. Levin and M.G. Ryskin, Semihard Processes in QCD, Phys. Rept. 100 (1983) 1 [INSPIRE].
E. Iancu and R. Venugopalan, The Color glass condensate and high-energy scattering in QCD, in Quark-gluon plasma 4, R.C. Hwa and X.-N. Wang eds., World Scientific (2003), pp. 249–3363 [hep-ph/0303204] [INSPIRE].
H. Weigert, Evolution at small xbj: The Color glass condensate, Prog. Part. Nucl. Phys. 55 (2005) 461 [hep-ph/0501087] [INSPIRE].
J. Jalilian-Marian and Y.V. Kovchegov, Saturation physics and deuteron-Gold collisions at RHIC, Prog. Part. Nucl. Phys. 56 (2006) 104 [hep-ph/0505052] [INSPIRE].
F. Gelis, E. Iancu, J. Jalilian-Marian and R. Venugopalan, The Color Glass Condensate, Ann. Rev. Nucl. Part. Sci. 60 (2010) 463 [arXiv:1002.0333] [INSPIRE].
J.L. Albacete and C. Marquet, Gluon saturation and initial conditions for relativistic heavy ion collisions, Prog. Part. Nucl. Phys. 76 (2014) 1 [arXiv:1401.4866] [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) [INSPIRE].
V.N. Gribov and L.N. Lipatov, Deep inelastic e p scattering in perturbation theory, Sov. J. Nucl. Phys. 15 (1972) 438 [Yad. Fiz. 15 (1972) 781] [INSPIRE].
G. Altarelli and G. Parisi, Asymptotic Freedom in Parton Language, Nucl. Phys. B 126 (1977) 298 [INSPIRE].
Y.L. Dokshitzer, Calculation of the Structure Functions for Deep Inelastic Scattering and e+e− Annihilation by Perturbation Theory in Quantum Chromodynamics, Sov. Phys. JETP 46 (1977) 641 [Zh. Eksp. Teor. Fiz. 73 (1977) 1216] [INSPIRE].
G.P. Lepage and S.J. Brodsky, Exclusive Processes in Perturbative Quantum Chromodynamics, Phys. Rev. D 22 (1980) 2157 [INSPIRE].
S.J. Brodsky, H.-C. Pauli and S.S. Pinsky, Quantum chromodynamics and other field theories on the light cone, Phys. Rept. 301 (1998) 299 [hep-ph/9705477] [INSPIRE].
B. Lampe and E. Reya, Spin physics and polarized structure functions, Phys. Rept. 332 (2000) 1 [hep-ph/9810270] [INSPIRE].
R. Mertig and W.L. van Neerven, The Calculation of the two loop spin splitting functions \( {P}_{ij}^{(1)} \)(x), Z. Phys. C 70 (1996) 637 [hep-ph/9506451] [INSPIRE].
S. Moch, J.A.M. Vermaseren and A. Vogt, The Three-Loop Splitting Functions in QCD: The Helicity-Dependent Case, Nucl. Phys. B 889 (2014) 351 [arXiv:1409.5131] [INSPIRE].
F. Cougoulic, Y.V. Kovchegov, A. Tarasov and Y. Tawabutr, in preparation (2022).
A.H. Mueller, Soft gluons in the infinite momentum wave function and the BFKL Pomeron, Nucl. Phys. B 415 (1994) 373 [INSPIRE].
A.H. Mueller and B. Patel, Single and double BFKL Pomeron exchange and a dipole picture of high-energy hard processes, Nucl. Phys. B 425 (1994) 471 [hep-ph/9403256] [INSPIRE].
A.H. Mueller, Unitarity and the BFKL Pomeron, Nucl. Phys. B 437 (1995) 107 [hep-ph/9408245] [INSPIRE].
S.J. Brodsky, G.P. Lepage and P.B. Mackenzie, On the Elimination of Scale Ambiguities in Perturbative Quantum Chromodynamics, Phys. Rev. D 28 (1983) 228 [INSPIRE].
Y.L. Dokshitzer and D.V. Shirkov, On Exact account of heavy quark thresholds in hard processes, Z. Phys. C 67 (1995) 449 [INSPIRE].
J.L. Albacete and Y.V. Kovchegov, Solving high energy evolution equation including running coupling corrections, Phys. Rev. D 75 (2007) 125021 [arXiv:0704.0612] [INSPIRE].
G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].
Y.V. Kovchegov, L. Szymanowski and S. Wallon, Perturbative odderon in the dipole model, Phys. Lett. B 586 (2004) 267 [hep-ph/0309281] [INSPIRE].
Y. Hatta, E. Iancu, K. Itakura and L. McLerran, Odderon in the color glass condensate, Nucl. Phys. A 760 (2005) 172 [hep-ph/0501171] [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].
L.D. McLerran and R. Venugopalan, Green’s functions in the color field of a large nucleus, Phys. Rev. D 50 (1994) 2225 [hep-ph/9402335] [INSPIRE].
A.H. Mueller, Small x Behavior and Parton Saturation: A QCD Model, Nucl. Phys. B 335 (1990) 115 [INSPIRE].
Y.V. Kovchegov, NonAbelian Weizsacker-Williams field and a two-dimensional effective color charge density for a very large nucleus, Phys. Rev. D 54 (1996) 5463 [hep-ph/9605446] [INSPIRE].
G. Veneziano, Some Aspects of a Unified Approach to Gauge, Dual and Gribov Theories, Nucl. Phys. B 117 (1976) 519 [INSPIRE].
B.I. Ermolaev, M. Greco and S.I. Troyan, Running coupling effects for the singlet structure function g1 at small x, Phys. Lett. B 579 (2004) 321 [hep-ph/0307128] [INSPIRE].
Y.V. Kovchegov and H. Weigert, Collinear Singularities and Running Coupling Corrections to Gluon Production in CGC, Nucl. Phys. A 807 (2008) 158 [arXiv:0712.3732] [INSPIRE].
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Kovchegov, Y.V., Tarasov, A. & Tawabutr, Y. Helicity evolution at small x: the single-logarithmic contribution. J. High Energ. Phys. 2022, 184 (2022). https://doi.org/10.1007/JHEP03(2022)184
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DOI: https://doi.org/10.1007/JHEP03(2022)184