×

Bootstrapping the QCD soft anomalous dimension. (English) Zbl 1382.81196

Summary: The soft anomalous dimension governs the infrared singularities of scattering amplitudes to all orders in perturbative quantum field theory, and is a crucial ingredient in both formal and phenomenological applications of non-abelian gauge theories. It has recently been computed at three-loop order for massless partons by explicit evaluation of all relevant Feynman diagrams. In this paper, we show how the same result can be obtained, up to an overall numerical factor, using a bootstrap procedure. We first give a geometrical argument for the fact that the result can be expressed in terms of single-valued harmonic polylogarithms. We then use symmetry considerations as well as known properties of scattering amplitudes in collinear and high-energy (Regge) limits to constrain an ansatz of basis functions. This is a highly non-trivial cross-check of the result, and our methods pave the way for greatly simplified higher-order calculations.

MSC:

81T60 Supersymmetric field theories in quantum mechanics
81U05 \(2\)-body potential quantum scattering theory
81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
81V05 Strong interaction, including quantum chromodynamics
81T50 Anomalies in quantum field theory
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
81T20 Quantum field theory on curved space or space-time backgrounds
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] A.M. Polyakov, Gauge Fields as Rings of Glue, Nucl. Phys.B 164 (1980) 171 [INSPIRE]. · doi:10.1016/0550-3213(80)90507-6
[2] I. Ya. Arefeva, Quantum contour field equations, Phys. Lett.93B (1980) 347 [INSPIRE].
[3] V.S. Dotsenko and S.N. Vergeles, Renormalizability of Phase Factors in the Nonabelian Gauge Theory, Nucl. Phys.B 169 (1980) 527 [INSPIRE]. · doi:10.1016/0550-3213(80)90103-0
[4] R.A. Brandt, F. Neri and M.-a. Sato, Renormalization of Loop Functions for All Loops, Phys. Rev.D 24 (1981) 879 [INSPIRE].
[5] 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]. · doi:10.1016/0370-2693(86)91439-5
[6] G.P. Korchemsky and A.V. Radyushkin, Infrared asymptotics of perturbative QCD: Renormalization properties of the Wilson loops in higher orders of perturbation theory, Sov. J. Nucl. Phys.44 (1986) 877 [INSPIRE].
[7] G.P. Korchemsky and A.V. Radyushkin, Renormalization of the Wilson Loops Beyond the Leading Order, Nucl. Phys.B 283 (1987) 342 [INSPIRE].
[8] A.H. Mueller, On the Asymptotic Behavior of the Sudakov Form-factor, Phys. Rev.D 20 (1979) 2037 [INSPIRE].
[9] J.C. Collins, Algorithm to Compute Corrections to the Sudakov Form-factor, Phys. Rev.D 22 (1980) 1478 [INSPIRE].
[10] A. Sen, Asymptotic Behavior of the Sudakov Form-Factor in QCD, Phys. Rev.D 24 (1981) 3281 [INSPIRE].
[11] A. Sen, Asymptotic Behavior of the Wide Angle On-Shell Quark Scattering Amplitudes in Nonabelian Gauge Theories, Phys. Rev.D 28 (1983) 860 [INSPIRE].
[12] J.G.M. Gatheral, Exponentiation of Eikonal Cross-sections in Nonabelian Gauge Theories, Phys. Lett.133B (1983) 90 [INSPIRE]. · doi:10.1016/0370-2693(83)90112-0
[13] J. Frenkel and J.C. Taylor, Nonabelian eikonal exponentiation, Nucl. Phys.B 246 (1984) 231 [INSPIRE]. · doi:10.1016/0550-3213(84)90294-3
[14] G.F. Sterman, Infrared divergences in perturbative QCD, AIP Conf. Proc.74 (1981) 22. · doi:10.1063/1.33099
[15] L. Magnea and G.F. Sterman, Analytic continuation of the Sudakov form-factor in QCD, Phys. Rev.D 42 (1990) 4222 [INSPIRE].
[16] G.P. Korchemsky, On Near forward high-energy scattering in QCD, Phys. Lett.B 325 (1994) 459 [hep-ph/9311294] [INSPIRE].
[17] I.A. Korchemskaya and G.P. Korchemsky, Evolution equation for gluon Regge trajectory, Phys. Lett.B 387 (1996) 346 [hep-ph/9607229] [INSPIRE]. · Zbl 0961.11040
[18] 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]. · Zbl 1397.81355
[19] 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] [hep-ph/9605323] [INSPIRE].
[20] S. Catani, The Singular behavior of QCD amplitudes at two loop order, Phys. Lett.B 427 (1998) 161 [hep-ph/9802439] [INSPIRE]. · Zbl 1342.81676
[21] G.F. Sterman and M.E. Tejeda-Yeomans, Multiloop amplitudes and resummation, Phys. Lett.B 552 (2003) 48 [hep-ph/0210130] [INSPIRE]. · Zbl 1005.81519
[22] L.J. Dixon, L. Magnea and G.F. Sterman, Universal structure of subleading infrared poles in gauge theory amplitudes, JHEP08 (2008) 022 [arXiv:0805.3515] [INSPIRE]. · doi:10.1088/1126-6708/2008/08/022
[23] N. Kidonakis, G. Oderda and G.F. Sterman, Evolution of color exchange in QCD hard scattering, Nucl. Phys.B 531 (1998) 365 [hep-ph/9803241] [INSPIRE].
[24] 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]. · Zbl 1094.81559
[25] Yu. L. Dokshitzer and G. Marchesini, Soft gluons at large angles in hadron collisions, JHEP01 (2006) 007 [hep-ph/0509078] [INSPIRE].
[26] 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]. · Zbl 1294.81278
[27] E. Gardi and L. Magnea, Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP03 (2009) 079 [arXiv:0901.1091] [INSPIRE]. · doi:10.1088/1126-6708/2009/03/079
[28] T. Becher and M. Neubert, Infrared singularities of scattering amplitudes in perturbative QCD, Phys. Rev. Lett.102 (2009) 162001 [arXiv:0901.0722] [INSPIRE]. · doi:10.1103/PhysRevLett.102.162001
[29] T. Becher and M. Neubert, On the Structure of Infrared Singularities of Gauge-Theory Amplitudes, JHEP06 (2009) 081 [Erratum ibid.1311 (2013) 024] [arXiv:0903.1126] [INSPIRE]. · Zbl 1342.81350
[30] E. Gardi and L. Magnea, Infrared singularities in QCD amplitudes, Nuovo Cim.C32N5-6 (2009) 137 [arXiv:0908.3273] [INSPIRE].
[31] L.J. Dixon, Matter Dependence of the Three-Loop Soft Anomalous Dimension Matrix, Phys. Rev.D 79 (2009) 091501 [arXiv:0901.3414] [INSPIRE]. · Zbl 1342.81574
[32] L.J. Dixon, E. Gardi and L. Magnea, On soft singularities at three loops and beyond, JHEP02 (2010) 081 [arXiv:0910.3653] [INSPIRE]. · Zbl 1270.81217 · doi:10.1007/JHEP02(2010)081
[33] 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]. · Zbl 1306.81333
[34] 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, JHEP12 (2011) 021 [arXiv:1109.3581] [INSPIRE]. · Zbl 1306.81333 · doi:10.1007/JHEP12(2011)021
[35] S. Caron-Huot, When does the gluon reggeize?, JHEP05 (2015) 093 [arXiv:1309.6521] [INSPIRE]. · doi:10.1007/JHEP05(2015)093
[36] V. Ahrens, M. Neubert and L. Vernazza, Structure of Infrared Singularities of Gauge-Theory Amplitudes at Three and Four Loops, JHEP09 (2012) 138 [arXiv:1208.4847] [INSPIRE]. · doi:10.1007/JHEP09(2012)138
[37] S.G. Naculich, H. Nastase and H.J. Schnitzer, All-loop infrared-divergent behavior of most-subleading-color gauge-theory amplitudes, JHEP04 (2013) 114 [arXiv:1301.2234] [INSPIRE]. · Zbl 1342.81606 · doi:10.1007/JHEP04(2013)114
[38] O. Erdoğan and G. Sterman, Ultraviolet divergences and factorization for coordinate-space amplitudes, Phys. Rev.D 91 (2015) 065033 [arXiv:1411.4588] [INSPIRE].
[39] T. Gehrmann, E.W.N. Glover, T. Huber, N. Ikizlerli and C. Studerus, Calculation of the quark and gluon form factors to three loops in QCD, JHEP06 (2010) 094 [arXiv:1004.3653] [INSPIRE]. · Zbl 1288.81146 · doi:10.1007/JHEP06(2010)094
[40] N. Kidonakis, Two-loop soft anomalous dimensions and NNLL resummation for heavy quark production, Phys. Rev. Lett.102 (2009) 232003 [arXiv:0903.2561] [INSPIRE]. · doi:10.1103/PhysRevLett.102.232003
[41] A. Mitov, G.F. Sterman and I. Sung, The Massive Soft Anomalous Dimension Matrix at Two Loops, Phys. Rev.D 79 (2009) 094015 [arXiv:0903.3241] [INSPIRE].
[42] T. Becher and M. Neubert, Infrared singularities of QCD amplitudes with massive partons, Phys. Rev.D 79 (2009) 125004 [Erratum ibid.D 80 (2009) 109901] [arXiv:0904.1021] [INSPIRE]. · Zbl 1331.81304
[43] 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]. · Zbl 1203.81165
[44] M. Czakon, A. Mitov and G.F. Sterman, Threshold Resummation for Top-Pair Hadroproduction to Next-to-Next-to-Leading Log, Phys. Rev.D 80 (2009) 074017 [arXiv:0907.1790] [INSPIRE]. · Zbl 0409.16011
[45] A. Ferroglia, M. Neubert, B.D. Pecjak and L.L. Yang, Two-loop divergences of massive scattering amplitudes in non-abelian gauge theories, JHEP11 (2009) 062 [arXiv:0908.3676] [INSPIRE]. · doi:10.1088/1126-6708/2009/11/062
[46] J.-y. Chiu, A. Fuhrer, R. Kelley and A.V. Manohar, Factorization Structure of Gauge Theory Amplitudes and Application to Hard Scattering Processes at the LHC, Phys. Rev.D 80 (2009) 094013 [arXiv:0909.0012] [INSPIRE].
[47] A. Mitov, G.F. Sterman and I. Sung, Computation of the Soft Anomalous Dimension Matrix in Coordinate Space, Phys. Rev.D 82 (2010) 034020 [arXiv:1005.4646] [INSPIRE].
[48] E. Gardi, From Webs to Polylogarithms, JHEP04 (2014) 044 [arXiv:1310.5268] [INSPIRE]. · doi:10.1007/JHEP04(2014)044
[49] G. Falcioni, E. Gardi, M. Harley, L. Magnea and C.D. White, Multiple Gluon Exchange Webs, JHEP10 (2014) 10 [arXiv:1407.3477] [INSPIRE]. · doi:10.1007/JHEP10(2014)010
[50] J.M. Henn, A.V. Smirnov and V.A. Smirnov, Analytic results for planar three-loop four-point integrals from a Knizhnik-Zamolodchikov equation, JHEP07 (2013) 128 [arXiv:1306.2799] [INSPIRE]. · Zbl 1342.81352 · doi:10.1007/JHEP07(2013)128
[51] M. Dukes, E. Gardi, H. McAslan, D.J. Scott and C.D. White, Webs and Posets, JHEP01 (2014) 024 [arXiv:1310.3127] [INSPIRE]. · Zbl 1333.81442 · doi:10.1007/JHEP01(2014)024
[52] M. Dukes, E. Gardi, E. Steingrimsson and C.D. White, Web worlds, web-colouring matrices and web-mixing matrices, J. Comb. Theory Ser.A 120 (2013) 1012 [arXiv:1301.6576] [INSPIRE]. · Zbl 1277.05155
[53] E. Gardi, E. Laenen, G. Stavenga and C.D. White, Webs in multiparton scattering using the replica trick, JHEP11 (2010) 155 [arXiv:1008.0098] [INSPIRE]. · Zbl 1294.81278 · doi:10.1007/JHEP11(2010)155
[54] E. Gardi and C.D. White, General properties of multiparton webs: Proofs from combinatorics, JHEP03 (2011) 079 [arXiv:1102.0756] [INSPIRE]. · Zbl 1301.81297 · doi:10.1007/JHEP03(2011)079
[55] E. Gardi, J.M. Smillie and C.D. White, On the renormalization of multiparton webs, JHEP09 (2011) 114 [arXiv:1108.1357] [INSPIRE]. · Zbl 1301.81296 · doi:10.1007/JHEP09(2011)114
[56] E. Gardi, J.M. Smillie and C.D. White, The Non-Abelian Exponentiation theorem for multiple Wilson lines, JHEP06 (2013) 088 [arXiv:1304.7040] [INSPIRE]. · Zbl 1342.81676 · doi:10.1007/JHEP06(2013)088
[57] E. Laenen, G. Stavenga and C.D. White, Path integral approach to eikonal and next-to-eikonal exponentiation, JHEP03 (2009) 054 [arXiv:0811.2067] [INSPIRE]. · doi:10.1088/1126-6708/2009/03/054
[58] Ø. Almelid, C. Duhr and E. Gardi, Three-loop corrections to the soft anomalous dimension in multileg scattering, Phys. Rev. Lett.117 (2016) 172002 [arXiv:1507.00047] [INSPIRE]. · doi:10.1103/PhysRevLett.117.172002
[59] F.C.S. Brown, Single-valued multiple polylogarithms in one variable, C.R. Acad. Sci. ParisSer. I (2004) 338. · Zbl 1048.11053
[60] E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys.A 15 (2000) 725 [hep-ph/9905237] [INSPIRE]. · Zbl 0951.33003
[61] L.J. Dixon, C. Duhr and J. Pennington, Single-valued harmonic polylogarithms and the multi-Regge limit, JHEP10 (2012) 074 [arXiv:1207.0186] [INSPIRE]. · doi:10.1007/JHEP10(2012)074
[62] L.J. Dixon, J.M. Drummond and J.M. Henn, Bootstrapping the three-loop hexagon, JHEP11 (2011) 023 [arXiv:1108.4461] [INSPIRE]. · Zbl 1306.81092 · doi:10.1007/JHEP11(2011)023
[63] L.J. Dixon, J.M. Drummond and J.M. Henn, Analytic result for the two-loop six-point NMHV amplitude in N = 4 super Yang-Mills theory, JHEP01 (2012) 024 [arXiv:1111.1704] [INSPIRE]. · Zbl 1306.81093 · doi:10.1007/JHEP01(2012)024
[64] L.J. Dixon, J.M. Drummond, M. von Hippel and J. Pennington, Hexagon functions and the three-loop remainder function, JHEP12 (2013) 049 [arXiv:1308.2276] [INSPIRE]. · Zbl 1342.81159 · doi:10.1007/JHEP12(2013)049
[65] 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, JHEP06 (2014) 116 [arXiv:1402.3300] [INSPIRE]. · Zbl 1333.81238 · doi:10.1007/JHEP06(2014)116
[66] L.J. Dixon and M. von Hippel, Bootstrapping an NMHV amplitude through three loops, JHEP10 (2014) 065 [arXiv:1408.1505] [INSPIRE]. · doi:10.1007/JHEP10(2014)065
[67] L.J. Dixon, M. von Hippel and A.J. McLeod, The four-loop six-gluon NMHV ratio function, JHEP01 (2016) 053 [arXiv:1509.08127] [INSPIRE]. · doi:10.1007/JHEP01(2016)053
[68] S. Caron-Huot, L.J. Dixon, A. McLeod and M. von Hippel, Bootstrapping a Five-Loop Amplitude Using Steinmann Relations, Phys. Rev. Lett.117 (2016) 241601 [arXiv:1609.00669] [INSPIRE]. · doi:10.1103/PhysRevLett.117.241601
[69] J.M. Drummond, G. Papathanasiou and M. Spradlin, A Symbol of Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon, JHEP03 (2015) 072 [arXiv:1412.3763] [INSPIRE]. · doi:10.1007/JHEP03(2015)072
[70] L.J. Dixon, J. Drummond, T. Harrington, A.J. McLeod, G. Papathanasiou and M. Spradlin, Heptagons from the Steinmann Cluster Bootstrap, JHEP02 (2017) 137 [arXiv:1612.08976] [INSPIRE]. · Zbl 1377.81197 · doi:10.1007/JHEP02(2017)137
[71] 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]. · Zbl 1360.81293 · doi:10.1016/j.physletb.2014.03.033
[72] V. Del Duca, G. Falcioni, L. Magnea and L. Vernazza, Analyzing high-energy factorization beyond next-to-leading logarithmic accuracy, JHEP02 (2015) 029 [arXiv:1409.8330] [INSPIRE]. · doi:10.1007/JHEP02(2015)029
[73] S. Caron-Huot, E. Gardi and L. Vernazza, Two-parton scattering in the high-energy limit, JHEP06 (2017) 016 [arXiv:1701.05241] [INSPIRE]. · Zbl 1380.81390 · doi:10.1007/JHEP06(2017)016
[74] E. Gardi, Ø. Almelid and C. Duhr, Long-distance singularities in multi-leg scattering amplitudes, PoSLL2016 (2016) 058 [arXiv:1606.05697] [INSPIRE]. · Zbl 1360.81293
[75] 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]. · Zbl 1109.81374
[76] 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]. · Zbl 1388.81323
[77] S. Moch, J.A.M. Vermaseren and A. Vogt, Three-loop results for quark and gluon form-factors, Phys. Lett.B 625 (2005) 245 [hep-ph/0508055] [INSPIRE].
[78] R.H. Boels, T. Huber and G. Yang, The four-loop non-planar cusp anomalous dimension in N = 4 SYM,arXiv:1705.03444[INSPIRE]. · Zbl 1301.81296
[79] F.C.S. Brown, Multiple zeta values and periods of moduli spaces M0,n, Annales Sci. Ecole Norm. Sup.42 (2009) 371 [math/0606419] [INSPIRE].
[80] K.T. Chen, Iterated path integrals, Bull. Amer. Math. Soc.83 (1977) 831. · Zbl 0389.58001 · doi:10.1090/S0002-9904-1977-14320-6
[81] A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett.5 (1998) 497 [arXiv:1105.2076] [INSPIRE]. · Zbl 0961.11040 · doi:10.4310/MRL.1998.v5.n4.a7
[82] A.B. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry, Duke Math. J.128 (2005) 209. · Zbl 1095.11036 · doi:10.1215/S0012-7094-04-12822-2
[83] A.B. Goncharov, A simple construction of Grassmannian polylogarithms, arXiv:0908.2238 [INSPIRE]. · Zbl 1360.11077
[84] C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP10 (2012) 075 [arXiv:1110.0458] [INSPIRE]. · Zbl 1397.81355 · doi:10.1007/JHEP10(2012)075
[85] F. Brown, On the decomposition of motivic multiple zeta values, arXiv:1102.1310 [INSPIRE]. · Zbl 1321.11087
[86] C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP08 (2012) 043 [arXiv:1203.0454] [INSPIRE]. · Zbl 1397.16028 · doi:10.1007/JHEP08(2012)043
[87] F.C.S. Brown, Notes on motivic periods, arXiv:1512.06410 [INSPIRE]. · Zbl 1390.14024
[88] D. Gaiotto, J. Maldacena, A. Sever and P. Vieira, Pulling the straps of polygons, JHEP12 (2011) 011 [arXiv:1102.0062] [INSPIRE]. · Zbl 1306.81153 · doi:10.1007/JHEP12(2011)011
[89] S. Abreu, R. Britto and H. Grönqvist, Cuts and coproducts of massive triangle diagrams, JHEP07 (2015) 111 [arXiv:1504.00206] [INSPIRE]. · Zbl 1388.83151 · doi:10.1007/JHEP07(2015)111
[90] F. Chavez and C. Duhr, Three-mass triangle integrals and single-valued polylogarithms, JHEP11 (2012) 114 [arXiv:1209.2722] [INSPIRE]. · Zbl 1397.81071 · doi:10.1007/JHEP11(2012)114
[91] O. Schnetz, Graphical functions and single-valued multiple polylogarithms, Commun. Num. Theor. Phys.08 (2014) 589 [arXiv:1302.6445] [INSPIRE]. · Zbl 1320.81075 · doi:10.4310/CNTP.2014.v8.n4.a1
[92] J. Drummond, C. Duhr, B. Eden, P. Heslop, J. Pennington and V.A. Smirnov, Leading singularities and off-shell conformal integrals, JHEP08 (2013) 133 [arXiv:1303.6909] [INSPIRE]. · Zbl 1342.81574 · doi:10.1007/JHEP08(2013)133
[93] V. Del Duca, L.J. Dixon, C. Duhr and J. Pennington, The BFKL equation, Mueller-Navelet jets and single-valued harmonic polylogarithms, JHEP02 (2014) 086 [arXiv:1309.6647] [INSPIRE]. · Zbl 1333.81418 · doi:10.1007/JHEP02(2014)086
[94] V. Del Duca et al., Multi-Regge kinematics and the moduli space of Riemann spheres with marked points, JHEP08 (2016) 152 [arXiv:1606.08807] [INSPIRE]. · Zbl 1390.81627 · doi:10.1007/JHEP08(2016)152
[95] J. Broedel, M. Sprenger and A. Torres Orjuela, Towards single-valued polylogarithms in two variables for the seven-point remainder function in multi-Regge-kinematics, Nucl. Phys.B 915 (2017) 394 [arXiv:1606.08411] [INSPIRE]. · Zbl 1354.81036 · doi:10.1016/j.nuclphysb.2016.12.016
[96] O. Schnetz, Numbers and Functions in Quantum Field Theory, arXiv:1606.08598 [INSPIRE]. · Zbl 1217.05110
[97] V. Del Duca, C. Duhr, R. Marzucca and B. Verbeek, The analytic structure and the transcendental weight of the BFKL ladder at NLL accuracy, arXiv:1705.10163 [INSPIRE]. · Zbl 1383.81289
[98] F.C.S. Brown, Single-valued hyperlogarithms and unipotent differential equations, http://www.ihes.fr/ brown/RHpaper5.pdf.
[99] F. Brown, Single-valued Motivic Periods and Multiple Zeta Values, SIGMA2 (2014) e25 [arXiv:1309.5309] [INSPIRE]. · Zbl 1377.11099
[100] Y.-T. Chien, M.D. Schwartz, D. Simmons-Duffin and I.W. Stewart, Jet Physics from Static Charges in AdS, Phys. Rev.D 85 (2012) 045010 [arXiv:1109.6010] [INSPIRE]. · Zbl 1333.81442
[101] C. Cheung, A. de la Fuente and R. Sundrum, 4D scattering amplitudes and asymptotic symmetries from 2D CFT, JHEP01 (2017) 112 [arXiv:1609.00732] [INSPIRE]. · Zbl 1373.81319 · doi:10.1007/JHEP01(2017)112
[102] Ø. Almelid, The Three-Loop Soft Anomalous Dimension of Massless Multileg Scattering, Ph.D. Thesis, University of Edinburgh, Edinburgh U.K. (2016). · Zbl 1301.81297
[103] Ø. Almelid, C. Duhr and E. Gardi, in preparation.
[104] 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]. · Zbl 1288.81146
[105] V. Del Duca, C. Duhr and V.A. Smirnov, An Analytic Result for the Two-Loop Hexagon Wilson Loop in N = 4 SYM, JHEP03 (2010) 099 [arXiv:0911.5332] [INSPIRE]. · Zbl 1271.81104 · doi:10.1007/JHEP03(2010)099
[106] V. Del Duca, C. Duhr and V.A. Smirnov, The Two-Loop Hexagon Wilson Loop in N = 4 SYM, JHEP05 (2010) 084 [arXiv:1003.1702] [INSPIRE]. · Zbl 1287.81080 · doi:10.1007/JHEP05(2010)084
[107] A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical Polylogarithms for Amplitudes and Wilson Loops, Phys. Rev. Lett.105 (2010) 151605 [arXiv:1006.5703] [INSPIRE]. · doi:10.1103/PhysRevLett.105.151605
[108] J. Golden and M. Spradlin, An analytic result for the two-loop seven-point MHV amplitude \[inN=4 \mathcal{N}=4\] SYM, JHEP08 (2014) 154 [arXiv:1406.2055] [INSPIRE].
[109] J.M. Henn and B. Mistlberger, Four-Gluon Scattering at Three Loops, Infrared Structure and the Regge Limit, Phys. Rev. Lett.117 (2016) 171601 [arXiv:1608.00850] [INSPIRE]. · doi:10.1103/PhysRevLett.117.171601
[110] N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Local Integrals for Planar Scattering Amplitudes, JHEP06 (2012) 125 [arXiv:1012.6032] [INSPIRE]. · Zbl 1348.81339 · doi:10.1007/JHEP06(2012)125
[111] N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, Scattering Amplitudes and the Positive Grassmannian, Cambridge University Press, Cambridge U.K. (2016). · Zbl 1365.81004
[112] Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz and J. Trnka, Logarithmic Singularities and Maximally Supersymmetric Amplitudes, JHEP06 (2015) 202 [arXiv:1412.8584] [INSPIRE]. · Zbl 1388.81136 · doi:10.1007/JHEP06(2015)202
[113] Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz and J. Trnka, Evidence for a Nonplanar Amplituhedron, JHEP06 (2016) 098 [arXiv:1512.08591] [INSPIRE]. · Zbl 1388.81908 · doi:10.1007/JHEP06(2016)098
[114] Z. Bern, J.J.M. Carrasco and H. Johansson, New Relations for Gauge-Theory Amplitudes, Phys. Rev.D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].
[115] D.E. Radford, A Natural Ring Basis for the Shuffle Algebra and an Application to Group Schemes, J. Algebra58 (1979) 432. · Zbl 0409.16011 · doi:10.1016/0021-8693(79)90171-6
[116] R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The analytic S-matrix, Cambridge University Press, Cambridge U.K. (1966). · Zbl 0139.46204
[117] I. Balitsky, High-energy QCD and Wilson lines, hep-ph/0101042 [INSPIRE]. · Zbl 1025.81503
[118] 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]. · Zbl 1360.81293
[119] S. Caron-Huot and M. Herranen, High-energy evolution to three loops, arXiv:1604.07417 [INSPIRE]. · Zbl 1387.81262
[120] C. Anastasiou, E.W.N. Glover, C. Oleari and M.E. Tejeda-Yeomans, Two-loop QCD corrections to the scattering of massless distinct quarks, Nucl. Phys.B 601 (2001) 318 [hep-ph/0010212] [INSPIRE].
[121] C. Anastasiou, E.W.N. Glover, C. Oleari and M.E. Tejeda-Yeomans, Two loop QCD corrections to massless identical quark scattering, Nucl. Phys.B 601 (2001) 341 [hep-ph/0011094] [INSPIRE].
[122] C. Anastasiou, E.W.N. Glover, C. Oleari and M.E. Tejeda-Yeomans, Two loop QCD corrections to massless quark gluon scattering, Nucl. Phys.B 605 (2001) 486 [hep-ph/0101304] [INSPIRE]. · Zbl 1271.81104
[123] E.W.N. Glover, C. Oleari and M.E. Tejeda-Yeomans, Two loop QCD corrections to gluon-gluon scattering, Nucl. Phys.B 605 (2001) 467 [hep-ph/0102201] [INSPIRE]. · Zbl 1270.81217
[124] Z. Bern, A. De Freitas and L.J. Dixon, Two loop helicity amplitudes for gluon-gluon scattering in QCD and supersymmetric Yang-Mills theory, JHEP03 (2002) 018 [hep-ph/0201161] [INSPIRE]. · Zbl 1095.11036
[125] 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, JHEP09 (2004) 039 [hep-ph/0409007] [INSPIRE].
[126] 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].
[127] S.G. Naculich, H. Nastase and H.J. Schnitzer, Two-loop graviton scattering relation and IR behavior in N = 8 supergravity, Nucl. Phys.B 805 (2008) 40 [arXiv:0805.2347] [INSPIRE]. · Zbl 1190.83096 · doi:10.1016/j.nuclphysb.2008.07.001
[128] S.G. Naculich, H. Nastase and H.J. Schnitzer, Subleading-color contributions to gluon-gluon scattering in N = 4 SYM theory and relations to N = 8 supergravity, JHEP11 (2008) 018 [arXiv:0809.0376] [INSPIRE]. · doi:10.1088/1126-6708/2008/11/018
[129] S.G. Naculich and H.J. Schnitzer, IR divergences and Regge limits of subleading-color contributions to the four-gluon amplitude in N = 4 SYM Theory, JHEP10 (2009) 048 [arXiv:0907.1895] [INSPIRE]. · doi:10.1088/1126-6708/2009/10/048
[130] S. Catani, D. de Florian and G. Rodrigo, Space-like (versus time-like) collinear limits in QCD: Is factorization violated?, JHEP07 (2012) 026 [arXiv:1112.4405] [INSPIRE]. · doi:10.1007/JHEP07(2012)026
[131] J.R. Forshaw, M.H. Seymour and A. Siodmok, On the Breaking of Collinear Factorization in QCD, JHEP11 (2012) 066 [arXiv:1206.6363] [INSPIRE]. · doi:10.1007/JHEP11(2012)066
[132] 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].
[133] D.A. Kosower, All order collinear behavior in gauge theories, Nucl. Phys.B 552 (1999) 319 [hep-ph/9901201] [INSPIRE]. · Zbl 1397.81428
[134] I. Feige and M.D. Schwartz, Hard-Soft-Collinear Factorization to All Orders, Phys. Rev.D 90 (2014) 105020 [arXiv:1403.6472] [INSPIRE].
[135] 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].
[136] C. Duhr, E. Gardi and S. Jaskiewicz, in preparation.
[137] S. Catani and M. Grazzini, The soft gluon current at one loop order, Nucl. Phys.B 591 (2000) 435 [hep-ph/0007142] [INSPIRE].
[138] C. Duhr and T. Gehrmann, The two-loop soft current in dimensional regularization, Phys. Lett.B 727 (2013) 452 [arXiv:1309.4393] [INSPIRE]. · Zbl 1331.81304 · doi:10.1016/j.physletb.2013.10.063
[139] Y. Li and H.X. Zhu, Single soft gluon emission at two loops, JHEP11 (2013) 080 [arXiv:1309.4391] [INSPIRE]. · doi:10.1007/JHEP11(2013)080
[140] L.J. Dixon and H.X. Zhu, Soft factorization of gauge theory amplitudes beyond one loop, talk at Amplitudes 2015, Zurich Switzerland (2015), https://amp15.itp.phys.ethz.ch/talks/Zhu.pdf. · Zbl 1331.81304
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.