Abstract
Various conventional running coupling prescriptions reproducing β0-dependent terms of NLO JIMWLK are reviewed and found to be theoretically inconsistent: the JIMWLK evolution Hamiltonian with running coupling violates the requirement of positive semidefiniteness. This requirement appears to be tightly related to the possibility of having a Langevin formulation for the evolution.
We also review the scheme that attributes a part of β0-dependent terms to the DGLAP evolution of the projectile. The remaining β0-dependent contributions sum up into so-called “daughter dipole” prescription, which leads to a manifestly positive semidefinite Hamiltonian.
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References
L.N. Lipatov, The Bare Pomeron in Quantum Chromodynamics, Sov. Phys. JETP 63 (1986) 904 [INSPIRE].
L.V. Gribov, E.M. Levin and M.G. Ryskin, Semihard Processes in QCD, Phys. Rept. 100 (1983) 1 [INSPIRE].
I.I. Balitsky and L.N. Lipatov, The Pomeranchuk Singularity in Quantum Chromodynamics, Sov. J. Nucl. Phys. 28 (1978) 822 [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 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].
A. Kovner, J.G. Milhano and H. Weigert, Relating different approaches to nonlinear QCD evolution at finite gluon density, Phys. Rev. D 62 (2000) 114005 [hep-ph/0004014] [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. Balitsky, Operator expansion for high-energy scattering, Nucl. Phys. B 463 (1996) 99 [hep-ph/9509348] [INSPIRE].
I. Balitsky, Operator expansion for diffractive high-energy scattering, AIP Conf. Proc. 407 (1997) 953 [hep-ph/9706411] [INSPIRE].
Y.V. Kovchegov, Small x F(2) structure function of a nucleus including multiple pomeron exchanges, Phys. Rev. D 60 (1999) 034008 [hep-ph/9901281] [INSPIRE].
I. Balitsky and G.A. Chirilli, Next-to-leading order evolution of color dipoles, Phys. Rev. D 77 (2008) 014019 [arXiv:0710.4330] [INSPIRE].
J.-P. Blaizot, E. Iancu and H. Weigert, Nonlinear gluon evolution in path integral form, Nucl. Phys. A 713 (2003) 441 [hep-ph/0206279] [INSPIRE].
K. Rummukainen and H. Weigert, Universal features of JIMWLK and BK evolution at small x, Nucl. Phys. A 739 (2004) 183 [hep-ph/0309306] [INSPIRE].
T. Lappi, Gluon spectrum in the glasma from JIMWLK evolution, Phys. Lett. B 703 (2011) 325 [arXiv:1105.5511] [INSPIRE].
T. Lappi and H. Mäntysaari, On the running coupling in the JIMWLK equation, Eur. Phys. J. C 73 (2013) 2307 [arXiv:1212.4825] [INSPIRE].
A. Kovner, M. Lublinsky and Y. Mulian, Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner evolution at next to leading order, Phys. Rev. D 89 (2014) 061704 [arXiv:1310.0378] [INSPIRE].
M. Lublinsky and Y. Mulian, High Energy QCD at NLO: from light-cone wave function to JIMWLK evolution, JHEP 05 (2017) 097 [arXiv:1610.03453] [INSPIRE].
L. Dai and M. Lublinsky, NLO JIMWLK evolution with massive quarks, JHEP 07 (2022) 093 [arXiv:2203.13695] [INSPIRE].
T. Lappi and H. Mäntysaari, Direct numerical solution of the coordinate space Balitsky-Kovchegov equation at next to leading order, Phys. Rev. D 91 (2015) 074016 [arXiv:1502.02400] [INSPIRE].
T. Lappi and H. Mäntysaari, Next-to-leading order Balitsky-Kovchegov equation with resummation, Phys. Rev. D 93 (2016) 094004 [arXiv:1601.06598] [INSPIRE].
G. Beuf, H. Hänninen, T. Lappi and H. Mäntysaari, Color Glass Condensate at next-to-leading order meets HERA data, Phys. Rev. D 102 (2020) 074028 [arXiv:2007.01645] [INSPIRE].
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 (2023) 192301 [arXiv:2211.03504] [INSPIRE].
G.P. Salam, An introduction to leading and next-to-leading BFKL, Acta Phys. Polon. B 30 (1999) 3679 [hep-ph/9910492] [INSPIRE].
K. Kutak and A.M. Stasto, Unintegrated gluon distribution from modified BK equation, Eur. Phys. J. C 41 (2005) 343 [hep-ph/0408117] [INSPIRE].
L. Motyka and A.M. Stasto, Exact kinematics in the small x evolution of the color dipole and gluon cascade, Phys. Rev. D 79 (2009) 085016 [arXiv:0901.4949] [INSPIRE].
G. Beuf, Improving the kinematics for low-x QCD evolution equations in coordinate space, Phys. Rev. D 89 (2014) 074039 [arXiv:1401.0313] [INSPIRE].
E. Iancu et al., Resumming double logarithms in the QCD evolution of color dipoles, Phys. Lett. B 744 (2015) 293 [arXiv:1502.05642] [INSPIRE].
E. Iancu et al., Collinearly-improved BK evolution meets the HERA data, Phys. Lett. B 750 (2015) 643 [arXiv:1507.03651] [INSPIRE].
Y. Hatta and E. Iancu, Collinearly improved JIMWLK evolution in Langevin form, JHEP 08 (2016) 083 [arXiv:1606.03269] [INSPIRE].
B. Ducloué et al., Non-linear evolution in QCD at high-energy beyond leading order, JHEP 04 (2019) 081 [arXiv:1902.06637] [INSPIRE].
E. Gotsman, E. Levin, M. Lublinsky and U. Maor, Towards a new global QCD analysis: low x DIS data from nonlinear evolution, Eur. Phys. J. C 27 (2003) 411 [hep-ph/0209074] [INSPIRE].
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 (2009) 034031 [arXiv:0902.1112] [INSPIRE].
J.L. Albacete et al., AAMQS: a non-linear QCD analysis of new HERA data at small-x including heavy quarks, Eur. Phys. J. C 71 (2011) 1705 [arXiv:1012.4408] [INSPIRE].
J.L. Albacete and C. Marquet, Single Inclusive Hadron Production at RHIC and the LHC from the Color Glass Condensate, Phys. Lett. B 687 (2010) 174 [arXiv:1001.1378] [INSPIRE].
J. Kuokkanen, K. Rummukainen and H. Weigert, HERA-Data in the Light of Small x Evolution with State of the Art NLO Input, Nucl. Phys. A 875 (2012) 29 [arXiv:1108.1867] [INSPIRE].
J.L. Albacete, A. Dumitru, H. Fujii and Y. Nara, CGC predictions for p + Pb collisions at the LHC, Nucl. Phys. A 897 (2013) 1 [arXiv:1209.2001] [INSPIRE].
T. Lappi and H. Mäntysaari, Single inclusive particle production at high energy from HERA data to proton-nucleus collisions, Phys. Rev. D 88 (2013) 114020 [arXiv:1309.6963] [INSPIRE].
J.L. Albacete, Resummation of double collinear logs in BK evolution versus HERA data, Nucl. Phys. A 957 (2017) 71 [arXiv:1507.07120] [INSPIRE].
B. Ducloué, E. Iancu, G. Soyez and D.N. Triantafyllopoulos, HERA data and collinearly-improved BK dynamics, Phys. Lett. B 803 (2020) 135305 [arXiv:1912.09196] [INSPIRE].
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].
A. Kovner, M. Lublinsky, V.V. Skokov and Z. Zhao, Not all that is β0 is β-function: the DGLAP resummation and the running coupling in NLO JIMWLK, arXiv:2308.15545 [INSPIRE].
A. Kovner, High energy evolution: the wave function point of view, Acta Phys. Polon. B 36 (2005) 3551 [hep-ph/0508232] [INSPIRE].
A. Kovner, M. Lublinsky and Y. Mulian, NLO JIMWLK evolution unabridged, JHEP 08 (2014) 114 [arXiv:1405.0418] [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].
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].
S. Catani, B.R. Webber and G. Marchesini, QCD coherent branching and semiinclusive processes at large x, Nucl. Phys. B 349 (1991) 635 [INSPIRE].
Y.L. Dokshitzer, V.A. Khoze and S.I. Troian, Specific features of heavy quark production. LPHD approach to heavy particle spectra, Phys. Rev. D 53 (1996) 89 [hep-ph/9506425] [INSPIRE].
G. Beuf, Dipole factorization for DIS at NLO: combining the \(q\overline{q }\) and \(q\overline{q }g\) contributions, Phys. Rev. D 96 (2017) 074033 [arXiv:1708.06557] [INSPIRE].
Acknowledgments
We thank I. Balitsky, A. Dumitru, A. Kovner, and Yu. Kovchegov for stimulating discussions. V.S. thanks V. Kazakov for illuminating discussions.
This material is based upon work supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics through the Contract No. DE-SC0020081 (V.S.) and the Saturated Glue (SURGE) Topical Collaboration (V.S.). M.L. and V.S. got support from the Binational Science Foundation under grant #2022132. M.L. is supported by the Binational Science Foundation grant #2021789 and by the ISF grant #910/23. GB is supported in part by the National Science Centre (Poland) under the research grant no 2020/38/E/ST2/00122 (SONATA BIS 10). This work has been performed in the framework of the MSCA RISE 823947 “Heavy ion collisions: collectivity and precision in saturation physics” (HIEIC).
We thank ExtreMe Matter Institute (EMMI), Physics Department of Muenster University (A. Andronic), Ben-Gurion University of the Negev, National Centre for Nuclear Research, Institute for Nuclear Theory at University of Washington, and ECT* for their support and hospitality during various stages of completing this project.
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Altinoluk, T., Beuf, G., Lublinsky, M. et al. On running coupling in the JIMWLK evolution and its Langevin formulation. J. High Energ. Phys. 2024, 131 (2024). https://doi.org/10.1007/JHEP03(2024)131
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DOI: https://doi.org/10.1007/JHEP03(2024)131