Abstract
We study the single soft-gluon current at two loops with two energetic partons in massless perturbative QCD, which describes, for example, the soft limit of the two-loop amplitude for gg → Hg. The results are presented as Laurent expansions in ϵ in D = 4 − 2ϵ spacetime dimension. We calculate the expansion to order ϵ 2 analytically, which is a necessary ingredient for Higgs production at hadron colliders at next-to-next-to-next-to-leading order in the soft-virtual approximation. We also give two-loop results of the single soft-gluon current in \( \mathcal{N}=4 \) Super-Yang-Mills theory, and find that it has uniform transcendentality. By iteration relation of splitting amplitudes, our calculations can determine the three-loop single soft-gluon current to order ϵ 0 in \( \mathcal{N}=4 \) Super-Yang-Mills theory in the limit of large N c .
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. Altarelli and G. Parisi, Asymptotic freedom in parton language, Nucl. Phys. B 126 (1977) 298 [INSPIRE].
A. Bassetto, M. Ciafaloni and G. Marchesini, Jet structure and infrared sensitive quantities in perturbative QCD, Phys. Rept. 100 (1983) 201 [INSPIRE].
F.A. Berends and W. Giele, Multiple soft gluon radiation in parton processes, Nucl. Phys. B 313 (1989) 595 [INSPIRE].
Y.L. Dokshitzer, V.A. Khoze, A.H. Mueller and S. Troian, Basics of perturbative QCD, Editions Frontières, Gif-sur-Yvette France (1991).
R.K. Ellis, W.J. Stirling and B. Webber, QCD and collider physics, Cambridge University Press, Cambridge U.K. (1996).
J.M. Campbell and E.N. Glover, Double unresolved approximations to multiparton scattering amplitudes, Nucl. Phys. B 527 (1998) 264 [hep-ph/9710255] [INSPIRE].
S. Catani and M. Grazzini, Infrared factorization of tree level QCD amplitudes at the next-to-next-to-leading order and beyond, Nucl. Phys. B 570 (2000) 287 [hep-ph/9908523] [INSPIRE].
Z. Bern and G. Chalmers, Factorization in one loop gauge theory, Nucl. Phys. B 447 (1995) 465 [hep-ph/9503236] [INSPIRE].
D.A. Kosower, All order collinear behavior in gauge theories, Nucl. Phys. B 552 (1999) 319 [hep-ph/9901201] [INSPIRE].
S. Catani and M. Grazzini, Collinear factorization and splitting functions for next-to-next-to-leading order QCD calculations, Phys. Lett. B 446 (1999) 143 [hep-ph/9810389] [INSPIRE].
V. Del Duca, A. Frizzo and F. Maltoni, Factorization of tree QCD amplitudes in the high-energy limit and in the collinear limit, Nucl. Phys. B 568 (2000) 211 [hep-ph/9909464] [INSPIRE].
Z. Bern, V. Del Duca and C.R. Schmidt, The infrared behavior of one loop gluon amplitudes at next-to-next-to-leading order, Phys. Lett. B 445 (1998) 168 [hep-ph/9810409] [INSPIRE].
S. Catani, The singular behavior of QCD amplitudes at two loop order, Phys. Lett. B 427 (1998) 161 [hep-ph/9802439] [INSPIRE].
Z. Bern, V. Del Duca, W.B. Kilgore and C.R. Schmidt, The infrared behavior of one loop QCD amplitudes at next-to-next-to leading order, Phys. Rev. D 60 (1999) 116001 [hep-ph/9903516] [INSPIRE].
S. Catani and M. Grazzini, The soft gluon current at one loop order, Nucl. Phys. B 591 (2000) 435 [hep-ph/0007142] [INSPIRE].
D.A. Kosower, All orders singular emission in gauge theories, Phys. Rev. Lett. 91 (2003) 061602 [hep-ph/0301069] [INSPIRE].
S. Catani, D. de Florian and G. Rodrigo, The triple collinear limit of one loop QCD amplitudes, Phys. Lett. B 586 (2004) 323 [hep-ph/0312067] [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, Two-loop g → gg splitting amplitudes in QCD, JHEP 08 (2004) 012 [hep-ph/0404293] [INSPIRE].
S. Badger and E.N. Glover, Two loop splitting functions in QCD, JHEP 07 (2004) 040 [hep-ph/0405236] [INSPIRE].
A. Gehrmann-De Ridder, T. Gehrmann, E. Glover and G. Heinrich, Infrared structure of e + e − → 3 jets at NNLO, JHEP 11 (2007) 058 [arXiv:0710.0346] [INSPIRE].
I. Bierenbaum, M. Czakon and A. Mitov, The singular behavior of one-loop massive QCD amplitudes with one external soft gluon, Nucl. Phys. B 856 (2012) 228 [arXiv:1107.4384] [INSPIRE].
S. Catani, D. de Florian and G. Rodrigo, Space-like (versus time-like) collinear limits in QCD: is factorization violated?, JHEP 07 (2012) 026 [arXiv:1112.4405] [INSPIRE].
J. Currie, E.W.N. Glover and S. Wells, Infrared structure at NNLO using antenna subtraction, JHEP 04 (2013) 066 [arXiv:1301.4693] [INSPIRE].
I. Feige and M.D. Schwartz, An on-shell approach to factorization, Phys. Rev. D 88 (2013) 065021 [arXiv:1306.6341] [INSPIRE].
A. Gehrmann-De Ridder, T. Gehrmann and E.N. Glover, Antenna subtraction at NNLO, JHEP 09 (2005) 056 [hep-ph/0505111] [INSPIRE].
G. Somogyi, Z. Trócsányi and V. Del Duca, Matching of singly- and doubly-unresolved limits of tree-level QCD squared matrix elements, JHEP 06 (2005) 024 [hep-ph/0502226] [INSPIRE].
M. Czakon, A novel subtraction scheme for double-real radiation at NNLO, Phys. Lett. B 693 (2010) 259 [arXiv:1005.0274] [INSPIRE].
P. Bärnreuther, M. Czakon and A. Mitov, Percent level precision physics at the Tevatron: first genuine NNLO QCD corrections to \( q\overline{q}\to t\overline{t}+X \), Phys. Rev. Lett. 109 (2012) 132001 [arXiv:1204.5201] [INSPIRE].
M. Czakon, P. Fiedler and A. Mitov, The total top quark pair production cross-section at hadron colliders through \( O\left( {alpha_S^4} \right) \), Phys. Rev. Lett. 110 (2013) 252004 [arXiv:1303.6254] [INSPIRE].
A. Gehrmann-De Ridder, T. Gehrmann, E. Glover and J. Pires, Second order QCD corrections to jet production at hadron colliders: the all-gluon contribution, Phys. Rev. Lett. 110 (2013) 162003 [arXiv:1301.7310] [INSPIRE].
R. Boughezal, F. Caola, K. Melnikov, F. Petriello and M. Schulze, Higgs boson production in association with a jet at next-to-next-to-leading order in perturbative QCD, JHEP 06 (2013) 072 [arXiv:1302.6216] [INSPIRE].
ATLAS collaboration, Observation of a new particle in the search for the standard model Higgs boson with the ATLAS detector at the LHC, Phys. Lett. B 716 (2012) 1 [arXiv:1207.7214] [INSPIRE].
CMS collaboration, Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC, Phys. Lett. B 716 (2012) 30 [arXiv:1207.7235] [INSPIRE].
R.V. Harlander and W.B. Kilgore, Next-to-next-to-leading order Higgs production at hadron colliders, Phys. Rev. Lett. 88 (2002) 201801 [hep-ph/0201206] [INSPIRE].
C. Anastasiou and K. Melnikov, Higgs boson production at hadron colliders in NNLO QCD, Nucl. Phys. B 646 (2002) 220 [hep-ph/0207004] [INSPIRE].
V. Ravindran, J. Smith and W.L. van Neerven, NNLO corrections to the total cross-section for Higgs boson production in hadron hadron collisions, Nucl. Phys. B 665 (2003) 325 [hep-ph/0302135] [INSPIRE].
S. Catani and M. Grazzini, An NNLO subtraction formalism in hadron collisions and its application to Higgs boson production at the LHC, Phys. Rev. Lett. 98 (2007) 222002 [hep-ph/0703012] [INSPIRE].
LHC Higgs Cross Section Working Group collaboration, S. Dittmaier et al., Handbook of LHC Higgs cross sections: 1. Inclusive observables, arXiv:1101.0593 [INSPIRE].
R. Boughezal, K. Melnikov and F. Petriello, A subtraction scheme for NNLO computations, Phys. Rev. D 85 (2012) 034025 [arXiv:1111.7041] [INSPIRE].
L. Garland, T. Gehrmann, E.N. Glover, A. Koukoutsakis and E. Remiddi, The two loop QCD matrix element for e + e − → 3 jets, Nucl. Phys. B 627 (2002) 107 [hep-ph/0112081] [INSPIRE].
L. Garland, T. Gehrmann, E.N. Glover, A. Koukoutsakis and E. Remiddi, Two loop QCD helicity amplitudes for e + e − → 3 jets, Nucl. Phys. B 642 (2002) 227 [hep-ph/0206067] [INSPIRE].
T. Gehrmann, M. Jaquier, E. Glover and A. Koukoutsakis, Two-loop QCD corrections to the helicity amplitudes for H → 3 partons, JHEP 02 (2012) 056 [arXiv:1112.3554] [INSPIRE].
Z. Bern, L.J. Dixon and V.A. Smirnov, Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond, Phys. Rev. D 72 (2005) 085001 [hep-th/0505205] [INSPIRE].
M. Spradlin, A. Volovich and C. Wen, Three-loop leading singularities and BDS ansatz for five particles, Phys. Rev. D 78 (2008) 085025 [arXiv:0808.1054] [INSPIRE].
J. Gatheral, Exponentiation of eikonal cross-sections in nonabelian gauge theories, Phys. Lett. B 133 (1983) 90 [INSPIRE].
J. Frenkel and J. Taylor, Nonabelian eikonal exponentiation, Nucl. Phys. B 246 (1984) 231 [INSPIRE].
S. Catani, D. de Florian and M. Grazzini, Higgs production in hadron collisions: soft and virtual QCD corrections at NNLO, JHEP 05 (2001) 025 [hep-ph/0102227] [INSPIRE].
R.V. Harlander and W.B. Kilgore, Soft and virtual corrections to proton proton → H + x at NNLO, Phys. Rev. D 64 (2001) 013015 [hep-ph/0102241] [INSPIRE].
D. de Florian and J. Mazzitelli, A next-to-next-to-leading order calculation of soft-virtual cross sections, JHEP 12 (2012) 088 [arXiv:1209.0673] [INSPIRE].
C.W. Bauer, S. Fleming and M.E. Luke, Summing Sudakov logarithms in B → X(sγ) 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, D. Pirjol and I.W. Stewart, Soft collinear factorization in effective field theory, Phys. Rev. D 65 (2002) 054022 [hep-ph/0109045] [INSPIRE].
M. Beneke, A. Chapovsky, M. Diehl and T. Feldmann, Soft collinear effective theory and heavy to light currents beyond leading power, Nucl. Phys. B 643 (2002) 431 [hep-ph/0206152] [INSPIRE].
P. Nogueira, Automatic Feynman graph generation, J. Comput. Phys. 105 (1993) 279.
K. Chetyrkin and F. Tkachov, Integration by parts: the algorithm to calculate β-functions in 4 loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].
F. Tkachov, A theorem on analytical calculability of four loop renormalization group functions, Phys. Lett. B 100 (1981) 65 [INSPIRE].
A. Smirnov, Algorithm FIRE — Feynman Integral REduction, JHEP 10 (2008) 107 [arXiv:0807.3243] [INSPIRE].
S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].
R. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
Z. Bern and D.A. Kosower, The computation of loop amplitudes in gauge theories, Nucl. Phys. B 379 (1992) 451 [INSPIRE].
Z. Bern, A. De Freitas, L.J. Dixon and H. Wong, Supersymmetric regularization, two loop QCD amplitudes and coupling shifts, Phys. Rev. D 66 (2002) 085002 [hep-ph/0202271] [INSPIRE].
T. Gehrmann and E. Remiddi, Two loop master integrals for γ * → 3 jets: the planar topologies, Nucl. Phys. B 601 (2001) 248 [hep-ph/0008287] [INSPIRE].
M. Czakon, Automatized analytic continuation of Mellin-Barnes integrals, Comput. Phys. Commun. 175 (2006) 559 [hep-ph/0511200] [INSPIRE].
A. Kotikov, L. Lipatov, A. Onishchenko and V. Velizhanin, Three loop universal anomalous dimension of the Wilson operators in N = 4 SUSY Yang-Mills model, Phys. Lett. B 595 (2004) 521 [Erratum ibid. B 632 (2006) 754-756] [hep-th/0404092] [INSPIRE].
C. Anastasiou, Z. Bern, L.J. Dixon and D. Kosower, Planar amplitudes in maximally supersymmetric Yang-Mills theory, Phys. Rev. Lett. 91 (2003) 251602 [hep-th/0309040] [INSPIRE].
Z. Bern, M. Czakon, L.J. Dixon, D.A. Kosower and V.A. Smirnov, The four-loop planar amplitude and cusp anomalous dimension in maximally supersymmetric Yang-Mills theory, Phys. Rev. D 75 (2007) 085010 [hep-th/0610248] [INSPIRE].
A. Pak, M. Rogal and M. Steinhauser, Production of scalar and pseudo-scalar Higgs bosons to next-to-next-to-leading order at hadron colliders, JHEP 09 (2011) 088 [arXiv:1107.3391] [INSPIRE].
C. Anastasiou, S. Buehler, C. Duhr and F. Herzog, NNLO phase space master integrals for two-to-one inclusive cross sections in dimensional regularization, JHEP 11 (2012) 062 [arXiv:1208.3130] [INSPIRE].
M. Höschele, J. Hoff, A. Pak, M. Steinhauser and T. Ueda, Higgs boson production at the LHC: NNLO partonic cross sections through order ϵ and convolutions with splitting functions to N 3 LO, Phys. Lett. B 721 (2013) 244 [arXiv:1211.6559] [INSPIRE].
C. Anastasiou, C. Duhr, F. Dulat and B. Mistlberger, Soft triple-real radiation for Higgs production at N3LO, JHEP 07 (2013) 003 [arXiv:1302.4379] [INSPIRE].
R.D. Ball, M. Bonvini, S. Forte, S. Marzani and G. Ridolfi, Higgs production in gluon fusion beyond NNLO, Nucl. Phys. B 874 (2013) 746 [arXiv:1303.3590] [INSPIRE].
S. Buehler and A. Lazopoulos, Scale dependence and collinear subtraction terms for Higgs production in gluon fusion at N3LO, JHEP 10 (2013) 096 [arXiv:1306.2223] [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].
C. Duhr and T. Gehrmann, The two-loop soft current in dimensional regularization, arXiv:1309.4393 [INSPIRE].
H.R.P. Ferguson and D.H. Bailey, A polynomial time, numerically stble integer relationalgorithm, RNR-91-032 (1991).
H.R.P. Ferguson, D.H. Bailey and S. Arno, Analysis of PSLQ, an integer relation finding algorithm, Math. Comput. 68 (1999) 351.
J. Vermaseren, Harmonic sums, Mellin transforms and integrals, Int. J. Mod. Phys. A 14 (1999) 2037 [hep-ph/9806280] [INSPIRE].
S. Moch and P. Uwer, XSummer: transcendental functions and symbolic summation in form, Comput. Phys. Commun. 174 (2006) 759 [math-ph/0508008] [INSPIRE].
A. Smirnov, V. Smirnov and M. Tentyukov, FIESTA 2: parallelizeable multiloop numerical calculations, Comput. Phys. Commun. 182 (2011) 790 [arXiv:0912.0158] [INSPIRE].
T. Binoth and G. Heinrich, An automatized algorithm to compute infrared divergent multiloop integrals, Nucl. Phys. B 585 (2000) 741 [hep-ph/0004013] [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1309.4391
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Li, Y., Zhu, H.X. Single soft gluon emission at two loops. J. High Energ. Phys. 2013, 80 (2013). https://doi.org/10.1007/JHEP11(2013)080
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP11(2013)080