Abstract
We provide a complete and detailed study of the high-energy limit of four-parton scattering amplitudes in QCD, giving explicit results at two loops and higher orders, and going beyond next-to-leading logarithmic (NLL) accuracy. Building upon recent results, we use the techniques of infrared factorization to investigate the failure of the simplest form of Regge factorization, starting at next-to-next-to-leading logarithmic accuracy (NNLL) in ln(s/|t|). We provide detailed accounts and explicit expressions for the terms responsible for this breaking in the case of two-loop and three-loop quark and gluon amplitudes in QCD; in particular, we recover and explain a known non-logarithmic double-pole contribution at two-loops, and we compute all non-factorizing single-logarithmic singular contributions at three loops. Conversely, we use high-energy factorization to show that the hard functions of infrared factorization vanish in d = 4 to all orders in the coupling, up to NLL accuracy in ln(s/|t|). This provides clear evidence for the infrared origin of high-energy logarithms. Finally, we extend earlier studies to t-channel exchanges of color representations beyond the octet, which enables us to give predictions based on the dipole formula for single-pole NLL contributions at three and four loops.
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I.I. Balitsky, L.N. Lipatov and V.S. Fadin, Regge processes in non-Abelian gauge theories (in Russian), in Leningrad 1979, Proceedings, Physics Of Elementary Particles, Leningrad Russia (1979), pg. 109 [INSPIRE].
V.S. Fadin, R. Fiore, M.G. Kozlov and A.V. Reznichenko, Proof of the multi-Regge form of QCD amplitudes with gluon exchanges in the NLA, Phys. Lett. B 639 (2006) 74 [hep-ph/0602006] [INSPIRE].
V.S. Fadin and L.N. Lipatov, Radiative corrections to QCD scattering amplitudes in a multi-Regge kinematics, Nucl. Phys. B 406 (1993) 259 [INSPIRE].
V. Del Duca and E.W.N. Glover, The high-energy limit of QCD at two loops, JHEP 10 (2001) 035 [hep-ph/0109028] [INSPIRE].
P.D.B. Collins, An introduction to Regge theory and high-energy physics, Cambridge University Press, Cambridge U.K. (1977) [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].
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].
G.P. Korchemsky, On near forward high-energy scattering in QCD, Phys. Lett. B 325 (1994) 459 [hep-ph/9311294] [INSPIRE].
I.A. Korchemskaya and G.P. Korchemsky, High-energy scattering in QCD and cross singularities of Wilson loops, Nucl. Phys. B 437 (1995) 127 [hep-ph/9409446] [INSPIRE].
I.A. Korchemskaya and G.P. Korchemsky, Evolution equation for gluon Regge trajectory, Phys. Lett. B 387 (1996) 346 [hep-ph/9607229] [INSPIRE].
V. Del Duca, G. Falcioni, L. Magnea and L. Vernazza, High-energy QCD amplitudes at two loops and beyond, Phys. Lett. B 732 (2014) 233 [arXiv:1311.0304] [INSPIRE].
V. Del Duca, G. Falcioni, L. Magnea and L. Vernazza, Beyond Reggeization for two- and three-loop QCD amplitudes, PoS(RADCOR 2013)046 [arXiv:1312.5098] [INSPIRE].
I. Balitsky, Factorization and high-energy effective action, Phys. Rev. D 60 (1999) 014020 [hep-ph/9812311] [INSPIRE].
I. Balitsky, High-energy QCD and Wilson lines, in At the frontier of particle physics, vol. 2, M. Shifman ed., World Scientific, Singapore (2001), pg. 1237 [hep-ph/0101042] [INSPIRE].
T. Kucs, Scattering amplitudes in high-energy QCD, Phys. Rev. D 69 (2004) 054016 [hep-ph/0307141] [INSPIRE].
S. Caron-Huot, When does the gluon reggeize?, arXiv:1309.6521 [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 [Erratum ibid. 04 (2014) 112] [hep-ph/0304168] [INSPIRE].
A. De Freitas and Z. Bern, Two-loop helicity amplitudes for quark-quark scattering in QCD and gluino-gluino scattering in supersymmetric Yang-Mills theory, JHEP 09 (2004) 039 [hep-ph/0409007] [INSPIRE].
E.W.N. Glover and M.E. Tejeda-Yeomans, Two loop QCD helicity amplitudes for massless quark massless gauge boson scattering, JHEP 06 (2003) 033 [hep-ph/0304169] [INSPIRE].
E.W.N. Glover, Two loop QCD helicity amplitudes for massless quark quark scattering, JHEP 04 (2004) 021 [hep-ph/0401119] [INSPIRE].
M. Beneke, P. Falgari and C. Schwinn, Soft radiation in heavy-particle pair production: all-order colour structure and two-loop anomalous dimension, Nucl. Phys. B 828 (2010) 69 [arXiv:0907.1443] [INSPIRE].
W. Beenakker et al., Towards NNLL resummation: hard matching coefficients for squark and gluino hadroproduction, JHEP 10 (2013) 120 [arXiv:1304.6354] [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. 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].
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].
I. Feige and M.D. Schwartz, Hard-soft-collinear factorization to all orders, Phys. Rev. D 90 (2014) 105020 [arXiv:1403.6472] [INSPIRE].
L. Magnea and G.F. Sterman, Analytic continuation of the Sudakov form-factor in QCD, Phys. Rev. D 42 (1990) 4222 [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 [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 [Erratum ibid. 11 (2013) 024] [arXiv:0903.1126] [INSPIRE].
E. Gardi and L. Magnea, Infrared singularities in QCD amplitudes, Nuovo Cim. C32N5-6 (2009) 137 [arXiv:0908.3273] [INSPIRE].
G.P. Korchemsky and A.V. Radyushkin, Loop space formalism and renormalization group for the infrared asymptotics of QCD, Phys. Lett. B 171 (1986) 459 [INSPIRE].
G.P. Korchemsky and A.V. Radyushkin, Renormalization of the Wilson loops beyond the leading order, Nucl. Phys. B 283 (1987) 342 [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].
V. Ahrens, M. Neubert and L. Vernazza, Structure of infrared singularities of gauge-theory amplitudes at three and four loops, JHEP 09 (2012) 138 [arXiv:1208.4847] [INSPIRE].
E. Gardi, From webs to polylogarithms, JHEP 04 (2014) 044 [arXiv:1310.5268] [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. Grozin, J.M. Henn, G.P. Korchemsky and P. Marquard, The three-loop cusp anomalous dimension in QCD, arXiv:1409.0023 [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].
L. Magnea, Analytic resummation for the quark form-factor in QCD, Nucl. Phys. B 593 (2001) 269 [hep-ph/0006255] [INSPIRE].
V. Del Duca and C.R. Schmidt, Virtual next-to-leading corrections to the impact factors in the high-energy limit, Phys. Rev. D 57 (1998) 4069 [hep-ph/9711309] [INSPIRE].
L. Tyburski, Reggeization of the fermion-fermion scattering amplitude in non-Abelian gauge theories, Phys. Rev. D 13 (1976) 1107 [INSPIRE].
V.S. Fadin, E.A. Kuraev and L.N. Lipatov, On the Pomeranchuk singularity in asymptotically free theories, Phys. Lett. B 60 (1975) 50 [INSPIRE].
L.N. Lipatov, Reggeization of the vector meson and the vacuum singularity in non-Abelian gauge theories, Sov. J. Nucl. Phys. 23 (1976) 338 [Yad. Fiz. 23 (1976) 642] [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 [Erratum ibid. 45 (1977) 199] [Zh. Eksp. Teor. Fiz. 71 (1976)840] [INSPIRE].
A.L. Mason, Radiation gauge calculation of high-energy scattering amplitudes, Nucl. Phys. B 120 (1977) 275 [INSPIRE].
H. Cheng and C.Y. Lo, High-energy amplitudes of Yang-Mills theory in arbitrary perturbative orders. 1, Phys. Rev. D 15 (1977) 2959 [INSPIRE].
E.A. Kuraev, L.N. Lipatov and V.S. Fadin, The Pomeranchuk singularity in non-abelian gauge theories, Sov. Phys. JETP 45 (1977) 199 [Zh. Eksp. Teor. Fiz. 72 (1977) 377] [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].
V.S. Fadin, M.I. Kotsky and R. Fiore, Gluon Reggeization in QCD in the next-to-leading order, Phys. Lett. B 359 (1995) 181 [INSPIRE].
V.S. Fadin, R. Fiore and M.I. Kotsky, Gluon Regge trajectory in the two loop approximation, Phys. Lett. B 387 (1996) 593 [hep-ph/9605357] [INSPIRE].
V.S. Fadin, R. Fiore and A. Quartarolo, Reggeization of quark quark scattering amplitude in QCD, Phys. Rev. D 53 (1996) 2729 [hep-ph/9506432] [INSPIRE].
J. Blumlein, V. Ravindran and W.L. van Neerven, On the gluon Regge trajectory in O(α 2 S ), Phys. Rev. D 58 (1998) 091502 [hep-ph/9806357] [INSPIRE].
G. Parisi, Summing large perturbative corrections in QCD, Phys. Lett. B 90 (1980) 295 [INSPIRE].
G.F. Sterman, Summation of large corrections to short distance hadronic cross-sections, Nucl. Phys. B 281 (1987) 310 [INSPIRE].
T.O. Eynck, E. Laenen and L. Magnea, Exponentiation of the Drell-Yan cross-section near partonic threshold in the DIS and MS-bar schemes, JHEP 06 (2003) 057 [hep-ph/0305179] [INSPIRE].
V. Ahrens, T. Becher, M. Neubert and L.L. Yang, Origin of the large perturbative corrections to Higgs production at hadron colliders, Phys. Rev. D 79 (2009) 033013 [arXiv:0808.3008] [INSPIRE].
A. Broggio, A. Ferroglia, B.D. Pecjak and Z. Zhang, NNLO hard functions in massless QCD, JHEP 12 (2014) 005 [arXiv:1409.5294] [INSPIRE].
V. Del Duca, C. Duhr and V.A. Smirnov, An analytic result for the two-loop hexagon Wilson loop in N = 4 SYM, JHEP 03 (2010) 099 [arXiv:0911.5332] [INSPIRE].
L.J. Dixon, J.M. Drummond, C. Duhr and J. Pennington, The four-loop remainder function and multi-Regge behavior at NNLLA in planar N = 4 super-Yang-Mills theory, JHEP 06 (2014) 116 [arXiv:1402.3300] [INSPIRE].
V. Del Duca, L.J. Dixon, C. Duhr and J. Pennington, The BFKL equation, Mueller-Navelet jets and single-valued harmonic polylogarithms, JHEP 02 (2014) 086 [arXiv:1309.6647] [INSPIRE].
J.R. Andersen and J.M. Smillie, Multiple jets at the LHC with high energy jets, JHEP 06 (2011) 010 [arXiv:1101.5394] [INSPIRE].
S. Moch, J.A.M. Vermaseren and A. Vogt, The three loop splitting functions in QCD: the nonsinglet case, Nucl. Phys. B 688 (2004) 101 [hep-ph/0403192] [INSPIRE].
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Del Duca, V., Falcioni, G., Magnea, L. et al. Analyzing high-energy factorization beyond next-to-leading logarithmic accuracy. J. High Energ. Phys. 2015, 29 (2015). https://doi.org/10.1007/JHEP02(2015)029
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DOI: https://doi.org/10.1007/JHEP02(2015)029