Abstract
We consider the singular behaviour of QCD scattering amplitudes in kinematical configurations where two or more momenta of the external partons become collinear. At the tree level, this behaviour is known to be controlled by factorization formulae in which the singular collinear factor is universal (process independent). We show that this strict (process-independent) factorization is not valid at one-loop and higher-loop orders in the case of the collinear limit in space-like regions (e.g., collinear radiation from initial-state partons). We introduce a generalized version of all-order collinear factorization, in which the space-like singular factors retain some dependence on the momentum and colour charge of the non-collinear partons. We present explicit results on one-loop and two-loop amplitudes for both the two-parton and multiparton collinear limits. At the level of squared amplitudes and, more generally, cross sections in hadron-hadron collisions, the violation of strict collinear factorization has implications on the non-abelian structure of logarithmically-enhanced terms in perturbative calculations (starting from the next-to-next-to-leading order) and on various factorization issues of mass singularities (starting from the next-to-next-to-next-to-leading order).
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References
S. Catani, The Singular behavior of QCD amplitudes at two loop order, Phys. Lett. B 427 (1998) 161 [hep-ph/9802439] [INSPIRE].
G.F. Sterman and M.E. Tejeda-Yeomans, Multiloop amplitudes and resummation, Phys. Lett. B 552 (2003) 48 [hep-ph/0210130] [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].
L.J. Dixon, L. Magnea and G.F. Sterman, Universal structure of subleading infrared poles in gauge theory amplitudes, JHEP 08 (2008) 022 [arXiv:0805.3515] [INSPIRE].
T. Becher and M. Neubert, Infrared singularities of scattering amplitudes in perturbative QCD, Phys. Rev. Lett. 102 (2009) 162001 [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].
T. Becher and M. Neubert, On the Structure of Infrared Singularities of Gauge-Theory Amplitudes, JHEP 06 (2009) 081 [arXiv:0903.1126] [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].
A. Bassetto, M. Ciafaloni and G. Marchesini, Jet Structure and Infrared Sensitive Quantities in Perturbative QCD, Phys. Rept. 100 (1983) 201 [INSPIRE].
Yu.L. Dokshitzer, V.A. Khoze, A.H. Mueller and S.I. Troian, Basics of Perturbative QCD, Editions Frontières, Gif-sur-Yvette (1991).
F.A. Berends and W. Giele, Multiple Soft Gluon Radiation in Parton Processes, Nucl. Phys. B 313 (1989) 595 [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, 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].
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].
S. Badger and E.N. Glover, Two loop splitting functions in QCD, JHEP 07 (2004) 040 [hep-ph/0405236] [INSPIRE].
G. Altarelli and G. Parisi, Asymptotic Freedom in Parton Language, Nucl. Phys. B 126 (1977) 298 [INSPIRE].
F.A. Berends and W. Giele, Recursive Calculations for Processes with n Gluons, Nucl. Phys. B 306 (1988) 759 [INSPIRE].
M.L. Mangano and S.J. Parke, Multiparton amplitudes in gauge theories, Phys. Rept. 200 (1991) 301 [hep-th/0509223] [INSPIRE].
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, 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].
T. Birthwright, E.N. Glover, V. Khoze and P. Marquard, Multi-gluon collinear limits from MHV diagrams, JHEP 05 (2005) 013 [hep-ph/0503063] [INSPIRE].
T. Birthwright, E.N. Glover, V. Khoze and P. Marquard, Collinear limits in QCD from MHV rules, JHEP 07 (2005) 068 [hep-ph/0505219] [INSPIRE].
Z. Bern, G. Chalmers, L.J. Dixon and D.A. Kosower, One loop N gluon amplitudes with maximal helicity violation via collinear limits, Phys. Rev. Lett. 72 (1994) 2134 [hep-ph/9312333] [INSPIRE].
Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226] [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 and P. Uwer, One loop splitting amplitudes in gauge theory, Nucl. Phys. B 563 (1999) 477 [hep-ph/9903515] [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].
D.A. Kosower, All order collinear behavior in gauge theories, Nucl. Phys. B 552 (1999) 319 [hep-ph/9901201] [INSPIRE].
J.C. Collins, D.E. Soper and G.F. Sterman, Factorization of Hard Processes in QCD, in A.H. Mueller ed., Perturbative Quantum Chromodynamics, Adv. Ser. Direct. High Energy Phys. 5 (1988) 1, World Scientific, Singapore (1989) [hep-ph/0409313] [INSPIRE].
C. Bomhof, P. Mulders and F. Pijlman, Gauge link structure in quark-quark correlators in hard processes, Phys. Lett. B 596 (2004) 277 [hep-ph/0406099] [INSPIRE].
J. Collins and J.-W. Qiu, k T factorization is violated in production of high-transverse-momentum particles in hadron-hadron collisions, Phys. Rev. D 75 (2007) 114014 [arXiv:0705.2141] [INSPIRE].
J. Collins, 2-soft-gluon exchange and factorization breaking, arXiv:0708.4410 [INSPIRE].
A. Bacchetta, C. Bomhof, P. Mulders and F. Pijlman, Single spin asymmetries in hadron-hadron collisions, Phys. Rev. D 72 (2005) 034030 [hep-ph/0505268] [INSPIRE].
C. Bomhof, P. Mulders and F. Pijlman, The Construction of gauge-links in arbitrary hard processes, Eur. Phys. J. C 47 (2006) 147 [hep-ph/0601171] [INSPIRE].
W. Vogelsang and F. Yuan, Hadronic Dijet Imbalance and Transverse-Momentum Dependent Parton Distributions, Phys. Rev. D 76 (2007) 094013 [arXiv:0708.4398] [INSPIRE].
T.C. Rogers and P.J. Mulders, No Generalized TMD-Factorization in Hadro-Production of High Transverse Momentum Hadrons, Phys. Rev. D 81 (2010) 094006 [arXiv:1001.2977] [INSPIRE].
J.R. Forshaw, A. Kyrieleis and M.H. Seymour, Super-leading logarithms in non-global observables in QCD, JHEP 08 (2006) 059 [hep-ph/0604094] [INSPIRE].
J. Forshaw, A. Kyrieleis and M.H. Seymour, Super-leading logarithms in non-global observables in QCD: Colour basis independent calculation, JHEP 09 (2008) 128 [arXiv:0808.1269] [INSPIRE].
J. Keates and M.H. Seymour, Super-leading logarithms in non-global observables in QCD: Fixed order calculation, JHEP 04 (2009) 040 [arXiv:0902.0477] [INSPIRE].
M.H. Seymour, Breakdown of Coherence?, arXiv:0710.2733 [INSPIRE], in proceedings of 12 th International Conference on Elastic and Diffractive Scattering: Forward Physics and QCD, J. Bartels, K. Borras, M. Diehl and H. Jung eds., Hamburg, Germany, 21–25 May 2007, pg. 507 DESY-PROC-2007-02 [arXiv:0712.3633].
J. Forshaw and M.H. Seymour, Soft gluons and superleading logarithms in QCD, Nucl. Phys. Proc. Suppl. 191 (2009) 257 [arXiv:0901.3037] [INSPIRE].
S. Catani and M.H. Seymour, The Dipole formalism for the calculation of QCD jet cross-sections at next-to-leading order, Phys. Lett. B 378 (1996) 287 [hep-ph/9602277] [INSPIRE].
S. Catani and M.H. Seymour, A General algorithm for calculating jet cross-sections in NLO QCD, Nucl. Phys. B 485 (1997) 291 [Erratum ibid. B 510 (1998) 503-504] [hep-ph/9605323] [INSPIRE].
L. Lewin, Dilogarithms and Associated Functions, Macdonald, London (1958).
L. Lewin, Polylogarithms and Associated Functions, North Holland, New York (1981).
J.P. Ralston and B. Pire, Oscillatory scale breaking and the Chromo-Coulomb phase shift, Phys. Rev. Lett. 49 (1982) 1605 [INSPIRE].
B. Pire and J.P. Ralston, Fixed angle elastic scattering and the Chromo-Coulomb phase shift, Phys. Lett. B 117 (1982) 233 [INSPIRE].
S. Catani, M. Ciafaloni and G. Marchesini, Noncancelling infrared divergences in QCD coherent state, Nucl. Phys. B 264 (1986) 588 [INSPIRE].
R. Bonciani, S. Catani, M.L. Mangano and P. Nason, Sudakov resummation of multiparton QCD cross-sections, Phys. Lett. B 575 (2003) 268 [hep-ph/0307035] [INSPIRE].
S.M. Aybat and G.F. Sterman, Soft-Gluon Cancellation, Phases and Factorization with Initial-State Partons, Phys. Lett. B 671 (2009) 46 [arXiv:0811.0246] [INSPIRE].
C.W. Bauer, B.O. Lange and G. Ovanesyan, On Glauber modes in Soft-Collinear Effective Theory, JHEP 07 (2011) 077 [arXiv:1010.1027] [INSPIRE].
V. Del Duca, C. Duhr, E. Gardi, L. Magnea and C.D. White, An infrared approach to Reggeization, Phys. Rev. D 85 (2012) 071104 [arXiv:1108.5947] [INSPIRE].
V. Del Duca, C. Duhr, E. Gardi, L. Magnea and C.D. White, The Infrared structure of gauge theory amplitudes in the high-energy limit, JHEP 12 (2011) 021 [arXiv:1109.3581] [INSPIRE].
Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys. B 435 (1995) 59 [hep-ph/9409265] [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, Progress in one loop QCD computations, Ann. Rev. Nucl. Part. Sci. 46 (1996) 109 [hep-ph/9602280] [INSPIRE].
Z. Kunszt, A. Signer and Z. Trócsányi, One loop helicity amplitudes for all 2 → 2 processes in QCD and N = 1 supersymmetric Yang-Mills theory, Nucl. Phys. B 411 (1994) 397 [hep-ph/9305239] [INSPIRE].
S. Catani, M.H. Seymour and Z. Trócsányi, Regularization scheme independence and unitarity in QCD cross-sections, Phys. Rev. D 55 (1997) 6819 [hep-ph/9610553] [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].
W. Giele and E.N. Glover, Higher order corrections to jet cross-sections in e + e − annihilation, Phys. Rev. D 46 (1992) 1980 [INSPIRE].
Z. Kunszt, A. Signer and Z. Trócsányi, Singular terms of helicity amplitudes at one loop in QCD and the soft limit of the cross-sections of multiparton processes, Nucl. Phys. B 420 (1994) 550 [hep-ph/9401294] [INSPIRE].
C. Anastasiou, E.N. Glover, C. Oleari and M. Tejeda-Yeomans, Two-loop QCD corrections to the scattering of massless distinct quarks, Nucl. Phys. B 601 (2001) 318 [hep-ph/0010212] [INSPIRE].
C. Anastasiou, E.N. Glover, C. Oleari and M. Tejeda-Yeomans, Two loop QCD corrections to massless identical quark scattering, Nucl. Phys. B 601 (2001) 341 [hep-ph/0011094] [INSPIRE].
C. Anastasiou, E.N. Glover, C. Oleari and M. Tejeda-Yeomans, Two loop QCD corrections to massless quark gluon scattering, Nucl. Phys. B 605 (2001) 486 [hep-ph/0101304] [INSPIRE].
E.N. Glover, C. Oleari and M. Tejeda-Yeomans, Two loop QCD corrections to gluon-gluon scattering, Nucl. Phys. B 605 (2001) 467 [hep-ph/0102201] [INSPIRE].
Z. Bern, A. De Freitas and L.J. Dixon, Two loop helicity amplitudes for gluon-gluon scattering in QCD and supersymmetric Yang-Mills theory, JHEP 03 (2002) 018 [hep-ph/0201161] [INSPIRE].
Z. Bern, A. De Freitas and L.J. Dixon, Two loop helicity amplitudes for quark gluon scattering in QCD and gluino gluon scattering in supersymmetric Yang-Mills theory, JHEP 06 (2003) 028 [hep-ph/0304168] [INSPIRE].
A. Sen, Asymptotic Behavior of the Wide Angle On-Shell Quark Scattering Amplitudes in Nonabelian Gauge Theories, Phys. Rev. D 28 (1983) 860 [INSPIRE].
S. Catani and M. Ciafaloni, Generalized coherent state for soft gluon emission, Nucl. Phys. B 249 (1985) 301 [INSPIRE].
J. Botts and G.F. Sterman, Hard Elastic Scattering in QCD: Leading Behavior, Nucl. Phys. B 325 (1989) 62 [INSPIRE].
S. Weinzierl, The Infrared structure of e + e − → 3 jets at NNLO reloaded, JHEP 07 (2009) 009 [arXiv:0904.1145] [INSPIRE].
A. Gehrmann-De Ridder, T. Gehrmann and E.N. Glover, Antenna subtraction at NNLO, JHEP 09 (2005) 056 [hep-ph/0505111] [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].
S. Catani and M. Grazzini, QCD transverse-momentum resummation in gluon fusion processes, Nucl. Phys. B 845 (2011) 297 [arXiv:1011.3918] [INSPIRE].
P.M. Nadolsky, C. Balázs, E.L. Berger and C.-P. Yuan, Gluon-gluon contributions to the production of continuum diphoton pairs at hadron colliders, Phys. Rev. D 76 (2007) 013008 [hep-ph/0702003] [INSPIRE].
D. de Florian and M. Grazzini, The Structure of large logarithmic corrections at small transverse momentum in hadronic collisions, Nucl. Phys. B 616 (2001) 247 [hep-ph/0108273] [INSPIRE].
T. Gehrmann and P.F. Monni, Antenna subtraction at NNLO with hadronic initial states: real-virtual initial-initial configurations, JHEP 12 (2011) 049 [arXiv:1107.4037] [INSPIRE].
C. Anastasiou, K. Melnikov and F. Petriello, Fully differential Higgs boson production and the di-photon signal through next-to-next-to-leading order, Nucl. Phys. B 724 (2005) 197 [hep-ph/0501130] [INSPIRE].
C. Anastasiou, F. Herzog and A. Lazopoulos, The Fully differential decay rate of a Higgs boson to bottom-quarks at NNLO in QCD, JHEP 03 (2012) 035 [arXiv:1110.2368] [INSPIRE].
U. Aglietti, V. Del Duca, C. Duhr, G. Somogyi and Z. Trócsányi, Analytic integration of real-virtual counterterms in NNLO jet cross sections. I., JHEP 09 (2008) 107 [arXiv:0807.0514] [INSPIRE].
P. Bolzoni, S.-O. Moch, G. Somogyi and Z. Trócsányi, Analytic integration of real-virtual counterterms in NNLO jet cross sections. II., JHEP 08 (2009) 079 [arXiv:0905.4390] [INSPIRE].
P. Bolzoni, G. Somogyi and Z. Trócsányi, A subtraction scheme for computing QCD jet cross sections at NNLO: integrating the iterated singly-unresolved subtraction terms, JHEP 01 (2011) 059 [arXiv:1011.1909] [INSPIRE].
S. Catani, L. Cieri, G. Ferrera, D. de Florian and M. Grazzini, Vector boson production at hadron colliders: A Fully exclusive QCD calculation at NNLO, Phys. Rev. Lett. 103 (2009) 082001 [arXiv:0903.2120] [INSPIRE].
S. Catani, L. Cieri, D. de Florian, G. Ferrera and M. Grazzini, Diphoton production at hadron colliders: a fully-differential QCD calculation at NNLO, Phys. Rev. Lett. 108 (2012) 072001 [arXiv:1110.2375] [INSPIRE].
S. Weinzierl, Moments of event shapes in electron-positron annihilation at NNLO, Phys. Rev. D 80 (2009) 094018 [arXiv:0909.5056] [INSPIRE].
M. Czakon, Double-real radiation in hadronic top quark pair production as a proof of a certain concept, Nucl. Phys. B 849 (2011) 250 [arXiv:1101.0642] [INSPIRE].
R. Boughezal, K. Melnikov and F. Petriello, A subtraction scheme for NNLO computations, Phys. Rev. D 85 (2012) 034025 [arXiv:1111.7041] [INSPIRE].
A. Gehrmann-De Ridder, E. Glover and J. Pires, Real-Virtual corrections for gluon scattering at NNLO, JHEP 02 (2012) 141 [arXiv:1112.3613] [INSPIRE].
J.C. Collins, D.E. Soper and G.F. Sterman, Transverse Momentum Distribution in Drell-Yan Pair and W and Z Boson Production, Nucl. Phys. B 250 (1985) 199 [INSPIRE].
Z. Bern and D.A. Kosower, Color decomposition of one loop amplitudes in gauge theories, Nucl. Phys. B 362 (1991) 389 [INSPIRE].
J.R. Forshaw, M.H. Seymour and A. Siodmok, private communication.
J.R. Forshaw, M.H. Seymour and A. Siodmok, On the breaking of collinear factorization in QCD, arXiv:1206.6363 [INSPIRE].
J.R. Forshaw, Coulomb gluon, talk presented at the Workshop on Event Generators and Resummation, DESY Hamburg, 29 May – 1 June 2012 [https://indico.desy.de/contributionDisplay.py?contribId=3&confId=5230].
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Catani, S., de Florian, D. & Rodrigo, G. Space-like (vs. time-like) collinear limits in QCD: is factorization violated?. J. High Energ. Phys. 2012, 26 (2012). https://doi.org/10.1007/JHEP07(2012)026
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DOI: https://doi.org/10.1007/JHEP07(2012)026