Abstract
The high-energy evolution of Wilson line operators, which at leading order is described by the Balitsky-JIMWLK equations, receives large radiative corrections en-hanced by single and double collinear logarithms at next-to-leading order and beyond. We propose a method for resumming such logarithmic corrections to all orders, at the level of the Langevin formulation of the JIMWLK equation. The ensuing, collinearly-improved Langevin equation features generalized Wilson line operators, which depend not only upon rapidity (the logarithm of the longitudinal momentum), but also upon the transverse size of the color neutral projectile to which the Wilson lines belong. This additional scale dependence is built up during the evolution, via the condition that the successive emissions of soft gluons be ordered in time. The presence of this transverse scale in the Langevin equation furthermore allows for the resummation of the one-loop running coupling corrections.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
E. Iancu, A. Leonidov and L. McLerran, The color glass condensate: an introduction, hep-ph/0202270 [INSPIRE].
E. Iancu and R. Venugopalan, The color glass condensate and high-energy scattering in QCD, hep-ph/0303204 [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].
I. Balitsky, Operator expansion for high-energy scattering, Nucl. Phys. B 463 (1996) 99 [hep-ph/9509348] [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].
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. 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. 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].
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].
A. Dumitru, J. Jalilian-Marian, T. Lappi, B. Schenke and R. Venugopalan, Renormalization group evolution of multi-gluon correlators in high energy QCD, Phys. Lett. B 706 (2011) 219 [arXiv:1108.4764] [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].
S. Schlichting and B. Schenke, The shape of the proton at high energies, Phys. Lett. B 739 (2014) 313 [arXiv:1407.8458] [INSPIRE].
B. Schenke and S. Schlichting, 3D glasma initial state for relativistic heavy ion collisions, arXiv:1605.07158 [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].
E. Iancu, K. Itakura and L. McLerran, A Gaussian effective theory for gluon saturation, Nucl. Phys. A 724 (2003) 181 [hep-ph/0212123] [INSPIRE].
J.P. Blaizot, F. Gelis and R. Venugopalan, High-energy pA collisions in the color glass condensate approach. 2. Quark production, Nucl. Phys. A 743 (2004) 57 [hep-ph/0402257] [INSPIRE].
Y.V. Kovchegov, J. Kuokkanen, K. Rummukainen and H. Weigert, Subleading-N c corrections in non-linear small-x evolution, Nucl. Phys. A 823 (2009) 47 [arXiv:0812.3238] [INSPIRE].
C. Marquet and H. Weigert, New observables to test the color glass condensate beyond the large-N c limit, Nucl. Phys. A 843 (2010) 68 [arXiv:1003.0813] [INSPIRE].
F. Dominguez, C. Marquet, B.-W. Xiao and F. Yuan, Universality of unintegrated gluon distributions at small x, Phys. Rev. D 83 (2011) 105005 [arXiv:1101.0715] [INSPIRE].
E. Iancu and D.N. Triantafyllopoulos, Higher-point correlations from the JIMWLK evolution, JHEP 11 (2011) 105 [arXiv:1109.0302] [INSPIRE].
E. Iancu and D.N. Triantafyllopoulos, JIMWLK evolution in the gaussian approximation, JHEP 04 (2012) 025 [arXiv:1112.1104] [INSPIRE].
T. Lappi, A. Ramnath, K. Rummukainen and H. Weigert, JIMWLK evolution of the odderon, arXiv:1606.00551 [INSPIRE].
D.N. Triantafyllopoulos, The energy dependence of the saturation momentum from RG improved BFKL evolution, Nucl. Phys. B 648 (2003) 293 [hep-ph/0209121] [INSPIRE].
Y.V. Kovchegov and H. Weigert, Quark loop contribution to BFKL evolution: running coupling and leading-N f NLO intercept, Nucl. Phys. A 789 (2007) 260 [hep-ph/0612071] [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].
I. Balitsky, Quark contribution to the small-x evolution of color dipole, Phys. Rev. D 75 (2007) 014001 [hep-ph/0609105] [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].
E. Avsar, A.M. Stasto, D.N. Triantafyllopoulos and D. Zaslavsky, Next-to-leading and resummed BFKL evolution with saturation boundary, JHEP 10 (2011) 138 [arXiv:1107.1252] [INSPIRE].
I. Balitsky and G.A. Chirilli, Rapidity evolution of Wilson lines at the next-to-leading order, Phys. Rev. D 88 (2013) 111501 [arXiv:1309.7644] [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].
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]
V.S. Fadin, M.I. Kotsky and R. Fiore, Gluon Reggeization in QCD in the next-to-leading order, Phys. Lett. B 359 (1995) 181 [INSPIRE].
V.S. Fadin, M.I. Kotsky and L.N. Lipatov, One-loop correction to the BFKL kernel from two gluon production, Phys. Lett. B 415 (1997) 97 [INSPIRE].
G. Camici and M. Ciafaloni, Non-Abelian q q contributions to small x anomalous dimensions, Phys. Lett. B 386 (1996) 341 [hep-ph/9606427] [INSPIRE].
G. Camici and M. Ciafaloni, Irreducible part of the next-to-leading BFKL kernel, Phys. Lett. B 412 (1997) 396 [Erratum ibid. B 417 (1998) 390] [hep-ph/9707390] [INSPIRE].
V.S. Fadin and L.N. Lipatov, BFKL Pomeron in the next-to-leading approximation, Phys. Lett. B 429 (1998) 127 [hep-ph/9802290] [INSPIRE].
M. Ciafaloni and G. Camici, Energy scale(s) and next-to-leading BFKL equation, Phys. Lett. B 430 (1998) 349 [hep-ph/9803389] [INSPIRE].
L.N. Lipatov, Reggeization of the vector meson and the vacuum singularity in non-Abelian gauge theories, Sov. J. Nucl. Phys. 23 (1976) 338 [Yad. Fiz. 23 (1976) 642] [INSPIRE].
E.A. Kuraev, L.N. Lipatov and V.S. Fadin, The Pomeranchuk singularity in non-Abelian 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].
J. Kwiecinski, A.D. Martin and A.M. Stasto, A unified BFKL and GLAP description of F 2 data, Phys. Rev. D 56 (1997) 3991 [hep-ph/9703445] [INSPIRE].
G.P. Salam, A resummation of large subleading corrections at small x, JHEP 07 (1998) 019 [hep-ph/9806482] [INSPIRE].
M. Ciafaloni, D. Colferai and G.P. Salam, Renormalization group improved small x equation, Phys. Rev. D 60 (1999) 114036 [hep-ph/9905566] [INSPIRE].
G. Altarelli, R.D. Ball and S. Forte, Resummation of singlet parton evolution at small x, Nucl. Phys. B 575 (2000) 313 [hep-ph/9911273] [INSPIRE].
M. Ciafaloni, D. Colferai, G.P. Salam and A.M. Stasto, Renormalization group improved small x Green’s function, Phys. Rev. D 68 (2003) 114003 [hep-ph/0307188] [INSPIRE].
A. Sabio Vera, An ‘all-poles’ approximation to collinear resummations in the Regge limit of perturbative QCD, Nucl. Phys. B 722 (2005) 65 [hep-ph/0505128] [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, J.D. Madrigal, A.H. Mueller, G. Soyez and D.N. Triantafyllopoulos, Resumming double logarithms in the QCD evolution of color dipoles, Phys. Lett. B 744 (2015) 293 [arXiv:1502.05642] [INSPIRE].
E. Iancu, J.D. Madrigal, A.H. Mueller, G. Soyez and D.N. Triantafyllopoulos, Collinearly-improved BK evolution meets the HERA data, Phys. Lett. B 750 (2015) 643 [arXiv:1507.03651] [INSPIRE].
M. Ciafaloni, Coherence effects in initial jets at small Q 2 /s, Nucl. Phys. B 296 (1988) 49 [INSPIRE].
B. Andersson, G. Gustafson and J. Samuelsson, The linked dipole chain model for DIS, Nucl. Phys. B 467 (1996) 443 [INSPIRE].
J. Kwiecinski, A.D. Martin and P.J. Sutton, Constraints on gluon evolution at small x, Z. Phys. C 71 (1996) 585 [hep-ph/9602320] [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].
E. Avsar and E. Iancu, CCFM evolution with unitarity corrections, Nucl. Phys. A 829 (2009) 31 [arXiv:0906.2683] [INSPIRE].
J.L. Albacete, Resummation of double collinear logs in BK evolution versus HERA data, arXiv:1507.07120 [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].
A. Banfi, G. Marchesini and G. Smye, Away from jet energy flow, JHEP 08 (2002) 006 [hep-ph/0206076] [INSPIRE].
H. Weigert, Nonglobal jet evolution at finite N c , Nucl. Phys. B 685 (2004) 321 [hep-ph/0312050] [INSPIRE].
Y. Hatta and T. Ueda, Resummation of non-global logarithms at finite N c , Nucl. Phys. B 874 (2013) 808 [arXiv:1304.6930] [INSPIRE].
D. Neill, The edge of jets and subleading non-global logs, arXiv:1508.07568 [INSPIRE].
A.H. Mueller, Soft gluons in the infinite momentum wave function and the BFKL Pomeron, Nucl. Phys. B 415 (1994) 373 [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].
Y.V. Kovchegov and E. Levin, Quantum chromodynamics at high energy, Cambridge University Press, Cambridge U.K. (2012).
E. Iancu and D.N. Triantafyllopoulos, JIMWLK evolution for multi-particle production in Langevin form, JHEP 11 (2013) 067 [arXiv:1307.1559] [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].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1606.03269
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Hatta, Y., Iancu, E. Collinearly improved JIMWLK evolution in Langevin form. J. High Energ. Phys. 2016, 83 (2016). https://doi.org/10.1007/JHEP08(2016)083
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2016)083