Abstract
We derive a factorization theorem that allows for resummation of small-x logarithms by exploiting Glauber operators in the soft collinear effective field theory. Our analysis is carried out for the hadronic tensor Wμν in deep inelastic scattering, and leads to the definition of a new gauge invariant soft function Sμν that describes quark and gluon emission in the central region. This soft function provides a new framework for extending resummed calculations for coefficient functions to higher logarithmic orders. Our factorization also defines impact factors by universal collinear functions that are process independent, for instance being identical in small-x DIS and Drell-Yan.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
L.V. Gribov, E.M. Levin and M.G. Ryskin, Semihard Processes in QCD, Phys. Rept. 100 (1983) 1 [INSPIRE].
L.N. Lipatov, Reggeization of the Vector Meson and the Vacuum Singularity in Nonabelian Gauge Theories, Sov. J. Nucl. Phys. 23 (1976) 338 [INSPIRE].
E.A. Kuraev, L.N. Lipatov and V.S. Fadin, Multi-Reggeon Processes in the Yang-Mills Theory, Sov. Phys. JETP 44 (1976) 443 [INSPIRE].
E.A. Kuraev, L.N. Lipatov and V.S. Fadin, The Pomeranchuk Singularity in Nonabelian Gauge Theories, Sov. Phys. JETP 45 (1977) 199 [INSPIRE].
I.I. Balitsky and L.N. Lipatov, The Pomeranchuk Singularity in Quantum Chromodynamics, Sov. J. Nucl. Phys. 28 (1978) 822 [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, 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, Green’s functions in the color field of a large nucleus, Phys. Rev. D 50 (1994) 2225 [hep-ph/9402335] [INSPIRE].
I. Balitsky, Operator expansion for high-energy scattering, Nucl. Phys. B 463 (1996) 99 [hep-ph/9509348] [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].
J. Jalilian-Marian, A. Kovner, L.D. McLerran and H. Weigert, The Intrinsic glue distribution at very small x, Phys. Rev. D 55 (1997) 5414 [hep-ph/9606337] [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].
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].
S. Catani and F. Hautmann, High-energy factorization and small x deep inelastic scattering beyond leading order, Nucl. Phys. B 427 (1994) 475 [hep-ph/9405388] [INSPIRE].
M. Ciafaloni and D. Colferai, The BFKL equation at next-to-leading level and beyond, Phys. Lett. B 452 (1999) 372 [hep-ph/9812366] [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].
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].
M. Ciafaloni, D. Colferai, G.P. Salam and A.M. Stasto, The Gluon splitting function at moderately small x, Phys. Lett. B 587 (2004) 87 [hep-ph/0311325] [INSPIRE].
M. Ciafaloni et al., Extending QCD perturbation theory to higher energies, Phys. Lett. B 576 (2003) 143 [hep-ph/0305254] [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].
G. Altarelli, R.D. Ball and S. Forte, Factorization and resummation of small x scaling violations with running coupling, Nucl. Phys. B 621 (2002) 359 [hep-ph/0109178] [INSPIRE].
G. Altarelli, R.D. Ball and S. Forte, An Anomalous dimension for small x evolution, Nucl. Phys. B 674 (2003) 459 [hep-ph/0306156] [INSPIRE].
G. Altarelli, R.D. Ball and S. Forte, Perturbatively stable resummed small x evolution kernels, Nucl. Phys. B 742 (2006) 1 [hep-ph/0512237] [INSPIRE].
S. Marzani, R.D. Ball, P. Falgari and S. Forte, BFKL at next-to-next-to-leading order, Nucl. Phys. B 783 (2007) 143 [arXiv:0704.2404] [INSPIRE].
R.S. Thorne, The Running coupling BFKL anomalous dimensions and splitting functions, Phys. Rev. D 64 (2001) 074005 [hep-ph/0103210] [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].
J. Blümlein, QCD evolution of structure functions at small x, Lect. Notes Phys. 546 (2000) 42 [hep-ph/9909449] [INSPIRE].
C.W. Bauer, S. Fleming and M.E. Luke, Summing Sudakov logarithms in B → Xsγ in effective field theory., Phys. Rev. D 63 (2000) 014006 [hep-ph/0005275] [INSPIRE].
C.W. Bauer, S. Fleming, D. Pirjol and I.W. Stewart, An Effective field theory for collinear and soft gluons: Heavy to light decays, Phys. Rev. D 63 (2001) 114020 [hep-ph/0011336] [INSPIRE].
C.W. Bauer and I.W. Stewart, Invariant operators in collinear effective theory, Phys. Lett. B 516 (2001) 134 [hep-ph/0107001] [INSPIRE].
C.W. Bauer, D. Pirjol and I.W. Stewart, Soft collinear factorization in effective field theory, Phys. Rev. D 65 (2002) 054022 [hep-ph/0109045] [INSPIRE].
C.W. Bauer et al., Hard scattering factorization from effective field theory, Phys. Rev. D 66 (2002) 014017 [hep-ph/0202088] [INSPIRE].
I.Z. Rothstein and I.W. Stewart, An Effective Field Theory for Forward Scattering and Factorization Violation, JHEP 08 (2016) 025 [arXiv:1601.04695] [INSPIRE].
M. Ciafaloni and D. Colferai, K factorization and impact factors at next-to-leading level, Nucl. Phys. B 538 (1999) 187 [hep-ph/9806350] [INSPIRE].
J. Blümlein, The Theory of Deeply Inelastic Scattering, Prog. Part. Nucl. Phys. 69 (2013) 28 [arXiv:1208.6087] [INSPIRE].
A.V. Manohar, An Introduction to spin dependent deep inelastic scattering, in the proceedings of the Lake Louise Winter Institute: Symmetry and Spin in the Standard Model, Lake Louise Canada, February 23–29 (1992) [hep-ph/9204208] [INSPIRE].
S. Moch and J.A.M. Vermaseren, Deep inelastic structure functions at two loops, Nucl. Phys. B 573 (2000) 853 [hep-ph/9912355] [INSPIRE].
J.A.M. Vermaseren, A. Vogt and S. Moch, The Third-order QCD corrections to deep-inelastic scattering by photon exchange, Nucl. Phys. B 724 (2005) 3 [hep-ph/0504242] [INSPIRE].
S.D. Drell, D.J. Levy and T.-M. Yan, A Field Theoretic Model for electron-Nucleon Deep Inelastic Scattering, Phys. Rev. Lett. 22 (1969) 744 [INSPIRE].
D.J. Gross and S.B. Treiman, Light cone structure of current commutators in the gluon quark model, Phys. Rev. D 4 (1971) 1059 [INSPIRE].
R.A. Brandt and G. Preparata, Operator product expansions near the light cone, Nucl. Phys. B 27 (1971) 541 [INSPIRE].
N.H. Christ, B. Hasslacher and A.H. Mueller, Light cone behavior of perturbation theory, Phys. Rev. D 6 (1972) 3543 [INSPIRE].
H. Georgi and H.D. Politzer, Electroproduction scaling in an asymptotically free theory of strong interactions, Phys. Rev. D 9 (1974) 416 [INSPIRE].
T. Jaroszewicz, Infrared Divergences and Regge Behavior in QCD, Acta Phys. Polon. B 11 (1980) 965 [INSPIRE].
M. Ciafaloni and D. Colferai, Dimensional regularisation and factorisation schemes in the BFKL equation at subleading level, JHEP 09 (2005) 069 [hep-ph/0507106] [INSPIRE].
M. Ciafaloni, D. Colferai, G.P. Salam and A.M. Stasto, Minimal subtraction vs. physical factorisation schemes in small-x QCD, Phys. Lett. B 635 (2006) 320 [hep-ph/0601200] [INSPIRE].
A.V. Manohar and I.W. Stewart, The Zero-Bin and Mode Factorization in Quantum Field Theory, Phys. Rev. D 76 (2007) 074002 [hep-ph/0605001] [INSPIRE].
J.-Y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, The Rapidity Renormalization Group, Phys. Rev. Lett. 108 (2012) 151601 [arXiv:1104.0881] [INSPIRE].
J.-Y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, A Formalism for the Systematic Treatment of Rapidity Logarithms in Quantum Field Theory, JHEP 05 (2012) 084 [arXiv:1202.0814] [INSPIRE].
I. Balitsky and G.A. Chirilli, Photon impact factor and kT-factorization for DIS in the next-to-leading order, Phys. Rev. D 87 (2013) 014013 [arXiv:1207.3844] [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, NLO JIMWLK evolution unabridged, JHEP 08 (2014) 114 [arXiv:1405.0418] [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, Dipole factorization for DIS at NLO: Loop correction to the \( {\gamma}_{T,L}^{\ast } \) → \( q\overline{q} \) light-front wave functions, Phys. Rev. D 94 (2016) 054016 [arXiv:1606.00777] [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].
K. Roy and R. Venugopalan, NLO impact factor for inclusive photon+dijet production in e + A DIS at small x, Phys. Rev. D 101 (2020) 034028 [arXiv:1911.04530] [INSPIRE].
F. Bergabo and J. Jalilian-Marian, Single inclusive hadron production in DIS at small x: next to leading order corrections, JHEP 01 (2023) 095 [arXiv:2210.03208] [INSPIRE].
I. Moult, S. Raman, G. Ridgway and I.W. Stewart, Anomalous dimensions from soft Regge constants, JHEP 05 (2023) 025 [arXiv:2207.02859] [INSPIRE].
S. Caron-Huot, When does the gluon reggeize?, JHEP 05 (2015) 093 [arXiv:1309.6521] [INSPIRE].
R. Boussarie et al., TMD Handbook, arXiv:2304.03302 [INSPIRE].
A. Pathak, A new form of QCD coherence for multiple soft emissions using Glauber-SCET, JHEP 06 (2022) 118 [arXiv:2108.13440] [INSPIRE].
Acknowledgments
We would like to thank Ira Rothstein for many conversations and early collaboration. We also like to thank Simone Marzani for colloboration at early stages. D.N. would like to thank Ian Moult for conservations regarding the use of the Glauber Lagrangian in SCET. We also thank Kyle Lee for helpful comments on the manuscript. D.N. was supported by the Department of Energy under Contract DE-AC52-06NA25396 at LANL and through the LANL/LDRD Program. AP acknowledges support from DESY (Hamburg, Germany), a member of the Helmholtz Association HGF. AP was previously a member of the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics, which is supported by the UK Science and Technology Facilities Council (STFC) under grant number ST/T001038/1, and at University of Vienna was supported by FWF Austrian Science Fund under the Project No. P28535-N27. IS was supported by the U.S. Department of Energy, Office of Science, Office of Nuclear Physics, from DE-SC0011090 and by the Simons Foundation through the Investigator grant 327942.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2303.13710
Rights and permissions
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.
About this article
Cite this article
Neill, D., Pathak, A. & Stewart, I.W. Small-x factorization from effective field theory. J. High Energ. Phys. 2023, 89 (2023). https://doi.org/10.1007/JHEP09(2023)089
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP09(2023)089