Abstract
We present the generating function approach to the perturbative exponentiation of correlators of a product of Wilson lines and loops. The exponentiated expression is presented in closed form as an algebraic function of correlators of known operators, which can be seen as a generating function for web diagrams. The expression is naturally split onto two parts: the exponentiation kernel, which accumulates all non-trivial information about web diagrams, and the defect of exponentiation, which reconstructs the matrix exponent and is a function of the exponentiation kernel. The detailed comparison of the presented approach with existing approaches to exponentiation is presented as well. We also give examples of calculations within the generating function exponentiation, namely, we consider different configurations of light-like Wilson lines in the multi-gluon-exchange-webs (MGEW) approximation. Within this approximation the corresponding correlators can be calculated exactly at any order of perturbative expansion by only algebraic manipulations. The MGEW approximation shows violation of the dipole formula for infrared singularities at three-loop order.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D.R. Yennie, S.C. Frautschi and H. Suura, The infrared divergence phenomena and high-energy processes, Annals Phys. 13 (1961) 379 [INSPIRE].
G.F. Sterman, Infrared Divergences in Perturbative QCD, AIP Conf. Proc. 74 (1981) 22 [INSPIRE].
J.G.M. Gatheral, Exponentiation of Eikonal Cross-sections in Nonabelian Gauge Theories, Phys. Lett. B 133 (1983) 90 [INSPIRE].
J. Frenkel and J.C. Taylor, Nonabelian Eikonal Exponentiation, Nucl. Phys. B 246 (1984) 231 [INSPIRE].
E. Laenen, G. Stavenga and C.D. White, Path integral approach to eikonal and next-to-eikonal exponentiation, JHEP 03 (2009) 054 [arXiv:0811.2067] [INSPIRE].
E. Gardi, E. Laenen, G. Stavenga and C.D. White, Webs in multiparton scattering using the replica trick, JHEP 11 (2010) 155 [arXiv:1008.0098] [INSPIRE].
A. Mitov, G.F. Sterman and I. Sung, Diagrammatic Exponentiation for Products of Wilson Lines, Phys. Rev. D 82 (2010) 096010 [arXiv:1008.0099] [INSPIRE].
E. Gardi, J.M. Smillie and C.D. White, The Non-Abelian Exponentiation theorem for multiple Wilson lines, JHEP 06 (2013) 088 [arXiv:1304.7040] [INSPIRE].
A.A. Vladimirov, Generating function for web diagrams, Phys. Rev. D 90 (2014) 066007 [arXiv:1406.6253] [INSPIRE].
E. Gardi and C.D. White, General properties of multiparton webs: Proofs from combinatorics, JHEP 03 (2011) 079 [arXiv:1102.0756] [INSPIRE].
G. Falcioni, E. Gardi, M. Harley, L. Magnea and C.D. White, Multiple Gluon Exchange Webs, JHEP 10 (2014) 010 [arXiv:1407.3477] [INSPIRE].
A.N. Vasil’ev, The field theoretic renormalization group in critical behavior theory and stochastic dynamics, Chapman and Hall/CRC, Boca Raton U.S.A. (2004).
A.V. Belitsky, Two loop renormalization of Wilson loop for Drell-Yan production, Phys. Lett. B 442 (1998) 307 [hep-ph/9808389] [INSPIRE].
V.E. Nazaikinskii, V.E. Shatalov and B. Yu. Sternin, Methods of noncommutative analysis: theory and applications, New York: Walter de Gruyter, Berlin Germany (1996).
A.V. Belitsky, X. Ji and F. Yuan, Final state interactions and gauge invariant parton distributions, Nucl. Phys. B 656 (2003) 165 [hep-ph/0208038] [INSPIRE].
A. Grozin, J.M. Henn, G.P. Korchemsky and P. Marquard, Three Loop Cusp Anomalous Dimension in QCD, Phys. Rev. Lett. 114 (2015) 062006 [arXiv:1409.0023] [INSPIRE].
D. Knauss and K. Scharnhorst, Two Loop Renormalization of Nonsmooth String Operators in Yang-Mills Theory, Annalen Phys. 41 (1984) 331 [INSPIRE].
I.A. Korchemskaya and G.P. Korchemsky, On lightlike Wilson loops, Phys. Lett. B 287 (1992) 169 [INSPIRE].
G.P. Korchemsky and A.V. Radyushkin, Renormalization of the Wilson Loops Beyond the Leading Order, Nucl. Phys. B 283 (1987) 342 [INSPIRE].
O. Erdoğan and G.F. Sterman, Gauge Theory Webs and Surfaces, Phys. Rev. D 91 (2015) 016003 [arXiv:1112.4564] [INSPIRE].
J.M. Henn and T. Huber, The four-loop cusp anomalous dimension in \( \mathcal{N}=4 \) super Yang-Mills and analytic integration techniques for Wilson line integrals, JHEP 09 (2013) 147 [arXiv:1304.6418] [INSPIRE].
E. Gardi, From Webs to Polylogarithms, JHEP 04 (2014) 044 [arXiv:1310.5268] [INSPIRE].
M. Sjödahl, ColorMath — A package for color summed calculations in SU(N c ), Eur. Phys. J. C 73 (2013) 2310 [arXiv:1211.2099] [INSPIRE].
S.M. Aybat, L.J. Dixon and G.F. Sterman, The Two-loop anomalous dimension matrix for soft gluon exchange, Phys. Rev. Lett. 97 (2006) 072001 [hep-ph/0606254] [INSPIRE].
S.M. Aybat, L.J. Dixon and G.F. Sterman, The Two-loop soft anomalous dimension matrix and resummation at next-to-next-to leading pole, Phys. Rev. D 74 (2006) 074004 [hep-ph/0607309] [INSPIRE].
T. Becher and M. Neubert, Infrared singularities of scattering amplitudes in perturbative QCD, Phys. Rev. Lett. 102 (2009) 162001 [Erratum ibid. 111 (2013) 199905] [arXiv:0901.0722] [INSPIRE].
E. Gardi and L. Magnea, Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP 03 (2009) 079 [arXiv:0901.1091] [INSPIRE].
L.J. Dixon, E. Gardi and L. Magnea, On soft singularities at three loops and beyond, JHEP 02 (2010) 081 [arXiv:0910.3653] [INSPIRE].
L. Magnea, Progress on the infrared structure of multi-particle gauge theory amplitudes, PoS(LL2014)073 [arXiv:1408.0682] [INSPIRE].
E. Gardi, J.M. Smillie and C.D. White, On the renormalization of multiparton webs, JHEP 09 (2011) 114 [arXiv:1108.1357] [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: 1501.03316
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
Vladimirov, A.A. Exponentiation for products of Wilson lines within the generating function approach. J. High Energ. Phys. 2015, 120 (2015). https://doi.org/10.1007/JHEP06(2015)120
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP06(2015)120