Abstract
We show how the formulations of low x physics involving Wilson line operators can be fully rewritten into an infinite twist TMD or GTMD framework, respectively for inclusive and exclusive observables. This leads to a perfect match between low x physics and moderate x formulations of QCD in terms of GTMDs, TMDs, GPDs or PDFs. We derive the BFKL limit as a kinematic limit and argue that beyond the Wandzura-Wilczek approximation, 3-body and 4-body unintegrated PFDs should be taken into account even in this regime. Finally we analyse how saturation should be understood as 3 distinct effects: saturation through non-linearities in the evolution equations at small x, saturation through multiple interactions with slow gluons as TMD gauge links, and saturation as the enhancement of genuine twist corrections.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
I. Balitsky, Operator expansion for high-energy scattering, Nucl. Phys.B 463 (1996) 99 [hep-ph/9509348] [INSPIRE].
I. Balitsky, Factorization for high-energy scattering, Phys. Rev. Lett.81 (1998) 2024 [hep-ph/9807434] [INSPIRE].
I. Balitsky, Factorization and high-energy effective action, Phys. Rev .D 60 (1999) 014020 [hep-ph/9812311] [INSPIRE].
T. Altinoluk, N. Armesto, G. Beuf, M. Martinez and C.A. Salgado, Next-to-eikonal corrections in the CGC: gluon production and spin asymmetries in pA collisions, JHEP07 (2014) 068 [arXiv:1404.2219] [INSPIRE].
T. Altinoluk, N. Armesto, G. Beuf and A. Moscoso, Next-to-next-to-eikonal corrections in the CGC, JHEP01 (2016) 114 [arXiv:1505.01400] [INSPIRE].
T. Altinoluk and A. Dumitru, Particle production in high-energy collisions beyond the shockwave limit, Phys. Rev.D 94 (2016) 074032 [arXiv:1512.00279] [INSPIRE].
I. Balitsky and A. Tarasov, Rapidity evolution of gluon TMD from low to mod erate x, JHEP10 (2015) 017 [arXiv:1505.02151] [INSPIRE].
I. Balitsky and A. Tarasov, Gluon TMD in particle production from low to moderate x, JHEP06 (2016) 164 [arXiv:1603.06548] [INSPIRE].
I. Balitsky and A. Tarasov, Higher-twist corrections to gluon TMD factorization, JHEP07 (2017) 095 [arXiv:1706. 01415] [INSPIRE].
I. Balitsky and A. Tarasov, Power corrections to TMD factorization for Z-boson production, JHEP05 (2018) 150 [arXiv:1712.09389] [INSPIRE].
G.A. Chirilli, Sub-eikonal corrections to scattering amplitudes at high energy, JHEP01 (2019) 118 [arXiv:1807.11435] [INSPIRE].
Y.V. Kovchegov, D. Pitonyak and M.D. Sievert, Helicity evolution at small-x, JHEP01 (2016) 072 [Erratum ibid.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].
F. Dominguez, B.-W. Xiao and F. Yuan, krfactorization for hard processes in nuclei, Phys. Rev. Lett.106 (2011) 022301 [arXiv:1009.2141] [INSPIRE].
F. Dominguez, C. Marquet, B.-W. Xiao and F. Yuan, Universality of unintegrat ed gluon distributions at small x, Phys. Rev.D 83 (2011) 105005 [arXiv:1101.0715] [INSPIRE].
T. Altinoluk, R. Boussarie, C. Marquet and P. Taels, TMD factorization for dijets +photon production from the dilute-dense CGC framework, JHEP07 (2019) 079 [arXiv: 1810.11273] [INSPIRE].
T. Altinoluk, R. Boussarie and P. Kotko, Interplay of the CGC and TMD frameworks to all orders in kinematic twist, JHEP05 (2019) 156 [arXiv:1901.01175] [INSPIRE].
A. Metz and J. Zhou, Distribution of linearly polarized gluons inside a large nucleus, Phys. Rev.D 84 (2011) 051503 [arXiv:1105.1991] [INSPIRE].
E. Akcakaya, A. Schafer and J. Zhou, A zimuthal asymmetries for quark pair production in pA collisions, Phys. Rev.D 87 (2013) 054010 [arXiv: 1208.4965] [INSPIRE].
A. Dumitru and V. Skokov, cos(4𝜑) azimuthal anisotropy in small-x DIS dijet production beyond the leading power TMD limit, Phys. Rev.D 94 (2016) 014030 [arXiv:1605.02739] [INSPIRE].
C. Marquet, E. Petreska and C. Roiesnel, Transverse-momentum-dependent gluon distributions from JIMWLK evolution, JHEP10 (2016) 065 [arXiv:1608.02577] [INSPIRE].
D. Boer, P.J. Mulders, J. Zhou and Y.-J. Zhou, Suppression of maximal linear gluon polarization in angular asymmetries, JHEP10 (2017) 196 [arXiv:1702.08195] [INSPIRE].
C. Marquet, C. Roiesnel and P. Taels, Linearly polarized small-x gluons in forward heavy-quark pair production, Phys. Rev.D 97 (2018) 014004 [arXiv:1710.05698] [INSPIRE].
E. Petreska, TMD gluon distributions at small x in the CGC theory, Int. J. Mod . Phys.E 27 (2018) 1830003 [arXiv:1804.04981] [INSPIRE].
Y. Hatta, B.-W. Xiao and F. Yuan, Probing the small-x gluon tomography in correlated hard diffractive dijet production in deep inelastic scattering, Phys. Rev. Lett.116 (2016) 202301 [arXiv: 1601.01585] [INSPIRE].
R. Boussarie, Y. Hatta, B.-W. Xiao and F. Yuan, Probing the Weizsäcker- Williams gluon Wigner distribution inpp collisions, Phys. Rev.D 98 (2018) 074015 [arXiv:1807.08697] [INSPIRE].
Y. Hatta, B.-W. Xiao and F. Yuan, Gluon tomography from deeply virtual Compton scattering at small-x, Phys. Rev.D 95 (2017) 114026 [arXiv:1703 . 02085] [INSPIRE].
A.V. Belitsky, X.-D. Ji and F. Yuan, Quark imaging in the proton via quantum phase space distributions, Phys. Rev.D 69 (2004) 074014 [hep-ph/0307383] [INSPIRE].
C. Loree and B. Pasquini, Quark Wigner distributions and orbital angular momentum, Phys. Rev.D 84 (2011) 014015 [arXiv:1106.0139] [INSPIRE].
E.A. Kuraev, L.N. Lipatov and V.S. Fadin, The Pomeranchuk singularity in non-Abelian gauge theories, Sov. Phys. JETP45 (1977) 199 [Zh. Eksp. Tear. 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].
A.H. Mueller, Small x behavior and parton saturation: a QCD model, Nucl. Ph ys.B 335 (1990) 115 [INSPIRE].
A.H. Mueller, Soft gluons in the infinite momentum wave function and the BFKL Pomeron, Nucl. Ph ys.B 415 (1994) 373 [INSPIRE].
A.H. Mueller, Unitarity and the BFKL Pomeron, Nucl. Phys.B 437 (1995) 107 [hep-ph/9408245] [INSPIRE].
L.D. McLerran and R. Venugopalan, Computing quark and gluon distribution functions for very large nuclei, Phy s. Rev.D 49 (1994) 2233 [hep-ph/9309289] [INSPIRE].
L.D. McLerran and R. Venugopalan, Gluon distribution functions for very large nuclei at small transv erse 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].
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. 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].
A. Kovner and J.G. Milhano, Vector potential versus color charge density in low x evolution, Phys. Rev.D 61 (2000) 014012 [hep-ph/9904420] [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].
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, Nonlinear gluon evolution in the color glass condensate. l, 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].
L.V. Gribov, E.M. Levin and M.G. Ryskin, Semihard processes in QCD, Phys. Rept.100 (1983) 1 [INSPIRE].
A. Dumitru, A. Hayashigaki and J. Jalilian-Marian, The color glass condensat e and hadron production in the forward region, Nucl. Phys.A 765 (2006) 464 [hep-ph/0506308] [INSPIRE].
T. Altinoluk and A. Kovner, Particle production at high energy and large transverse momentum – ‘the hybrid formalism’ revisited, Phys. Rev.D 83 (2011) 105004 [arXiv: 1102 .5327] [INSPIRE].
G.A. Chirilli, B.-W. Xiao and F. Yuan, One-loop factorization for inclusive hadron production in pA collisions in the saturation formalism, Phys. Rev. Lett.108 (2012) 122301 [arXiv: 1112 .1061] [INSPIRE].
G.A. Chirilli, B.-W. Xiao and F. Yuan, Inclusive hadron productions in pA collisions, Phys. Rev.D 86 (2012) 054005 [arXiv:1203. 6139] [INSPIRE].
A.M. Stasto, B.-W. Xiao and D. Zaslavsky, Towards the test of saturation physics beyond leading logarithm, Phys. Rev. Lett.112 (2014) 012302 [arXiv: 1307.4057] [INSPIRE].
A.M. Sta.Sto, B.-W. Xiao, F. Yuan and D. Zaslavsky, Matching collinear and small x factorization calculations for inclusive hadron production in pA collisions, Phys. Rev.D 90 (2014) 014047 [arXiv:1405.6311] [INSPIRE].
T. Altinoluk, N. Armesto, G. Beuf, A. Kovner and M. Lublinsky, Single-inclusive particle production in proton-nucleus collisions at next-to-leading order in the hybrid formalism, Phys. Rev.D 91 (2015) 094016 [arXiv:1411.2869] [INSPIRE].
K. Watanabe, B.-W. Xiao, F. Yuan and D. Zaslavsky, Implementing the exact kinematical constraint in the saturation formalism, Phys. Rev.D 92 (2015) 034026 [arXiv: 1505.05183] [INSPIRE].
B. Ducloue, T. Lappi and Y. Zhu, Single inclusive forward hadron production at next-to-leading order, Phys. Rev.D 93 (2016) 114016 [arXiv:1604.00225] [INSPIRE].
E. Iancu, A.H. Mueller and D.N. Triantafyllopoulos, CGC factorization for forward particle production in proton-nucleus collisions at next-to-leading order, JHEP12 (2016) 041 [arXiv: 1608.05293] [INSPIRE].
B. Ducloué et al., Use of a running coupling in the NLO calculation of forward hadron production, Phys. Rev.D 97 (2018) 054020 [arXiv: 1712.07480] [INSPIRE].
B. Ducloué, T. Lappi andY. Zhu, Implementation of NLO high energy factorization in single inclusive forward hadron production, Phys. Rev .D 95 (2017) 114007 [arXiv:1703.04962] [INSPIRE].
P. Kotko, K. Kutak, C. Marquet, E. Petreska, S. Sapeta and A. van Hameren, Improved TMD factorization for forward dijet production in dilute-dense hadronic collisions, JHEP09 (2015) 106 [arXiv:1503.03421] [INSPIRE].
A. van Hameren, P. Kotko, K. Kutak, C. Marquet, E. Petreska and S. Sapeta, Forward di-jet production in p+Pb collisions in the small-x improved TMD factorization framework, JHEP12 (2016) 034 [Erratum ibid.02 (2019) 158] [arXiv:1607.03121] [INSPIRE].
A.H. Mueller and H. Navelet, An inclusive minijet cross-section and the bare Pomeron in QCD, Nucl. Phys.B 282 (1987) 727 [INSPIRE].
P. Agostini, T. Altinoluk and N. Armesto, Non-eikonal corrections to multi-particle production in the color glass condensate, Eur. Phys. J.C 79 (2019) 600 [arXiv :1902.04483] [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, Re summing 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].
Y. Hatta and E. Iancu, Collinearly improved JIMWLK evolution in Langevin form, JHEP08 (2016) 083 [arXiv:1606.03269] [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: 1902.07930
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, 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 licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Altinoluk, T., Boussarie, R. Low x physics as an infinite twist (G)TMD framework: unravelling the origins of saturation. J. High Energ. Phys. 2019, 208 (2019). https://doi.org/10.1007/JHEP10(2019)208
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2019)208