Two-loop g → gg splitting amplitudes in QCD

Journal of High Energy Physics

Volumn 8, Issue 8, 2004, Pages 301-402

  Bern, Zvi ^a   Dixon, Lance J ^b   Kosower, David A ^c  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b SLAC NATIONAL ACCELERATOR LABORATORY   (United States)

^c CEA SACLAY   (France)

Author keywords

Deep Inelastic Scattering; Extended Supersymmetry; NLO Computations; QCD

Indexed keywords

EID: 23044432598     PISSN: 10298479     EISSN: 10298479     Source Type: Journal    
DOI: 10.1088/1126-6708/2004/08/012     Document Type: Article

References (131)

1. F.A. Berends, R. Kleiss, P. De Causmaecker, R. Gastmans and T.T. Wu, Single Bremsstrahlung processes in gauge theories, Phys. Lett. B 103 (1981) 124;
2. P. De Causmaecker, R. Gastmans, W. Troost and T.T. Wu, Helicity amplitudes for massless QED, Phys. Lett. B 105 (1981) 215;
3. J.F. Gunion and Z. Kunszt, Improved analytic techniques for tree graph calculations and the ggqq̄ lepton anti-lepton subprocess, Phys. Lett. B 161 (1985) 333;
4. Z. Xu, D.-H. Zhang and L. Chang, Helicity amplitudes for multiple Bremsstrahlung in massless nonabelian gauge theories, Nucl. Phys. B 291 (1987) 392.
5. J.E. Paton and H.-M. Chan, Generalized Veneziano model with isospin, Nucl. Phys. B 10 (1969) 516;
6. P. Cvitanović, P.G. Lauwers and P.N. Scharbach, Gauge invariance structure of quantum chromodynamics, Nucl. Phys. B 186 (1981) 165;
7. M.L. Mangano, The color structure of gluon emission, Nucl. Phys. B 309 (1988) 461.
8. F.A. Berends and W. Giele, The six-gluon process as an example of Weyl-van der Waerden spinor calculus, Nucl. Phys. B 294 (1987) 700.
9. M.L. Mangano, S.J. Parke and Z. Xu, Duality and multi-gluon scattering, Nucl. Phys. B 298 (1988) 653.
10. Z. Bern and D.A. Kosower, Color decomposition of one-loop amplitudes in gauge theories, Nucl. Phys. B 362 (1991) 389.
11. V. Del Duca, L.J. Dixon and F. Maltoni, New color decompositions for gauge amplitudes at tree and loop level, Nucl. Phys. B 571 (2000) 51 [hep-ph/9910563].
12. F.A. Berends and W.T. Giele, Recursive calculations for processes with n gluons, Nucl. Phys. B 306 (1988) 759;
13. D.A. Kosower, Light cone recurrence relations for QCD amplitudes, Nucl. Phys. B 335 (1990) 23;
14. G. Mahlon and T.-M. Yan, Multi-photon production at high-energies in the Standard Model, 1, Phys. Rev. D 47 (1993) 1776 [hep-ph/9210213];
15. G. Mahlon, T.-M. Yan and C. Dunn, Generalized gluon currents and applications in QCD, Phys. Rev. D 48 (1993) 1337 [hep-ph/9210212].
16. Z. Bern and D.A. Kosower, The computation of loop amplitudes in gauge theories, Nucl. Phys. B 379 (1992) 451.
17. E. Witten, Perturbative gauge theory as a string theory in twistor space, hep-th/0312171;
18. F. Cachazo, P. Svrcek and E. Witten, MHV vertices and tree amplitudes in gauge theory, hep-th/0403047.
19. 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].
20. 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].
21. Z. Bern and A.G. Morgan, Massive loop amplitudes from unitarity, Nucl. Phys. B 467 (1996) 479 [hep-ph/9511336].
22. 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].
23. Z. Bern, L.J. Dixon and D.A. Kosower, A two-loop four-gluon helicity amplitude in QCD, J. High Energy Phys. 01 (2000) 027 [hep-ph/0001001].
24. Z. Bern, A. De Freitas and L.J. Dixon, Two-loop helicity amplitudes for gluon-gluon scattering in QCD and supersymmetric Yang-Mills theory, J. High Energy Phys. 03 (2002) 018 [hep-ph/0201161].
25. V.A. Smirnov, Analytical result for dimensionally regularized massless on-shell double box, Phys. Lett. B 460 (1999) 397 [hep-ph/9905323].
26. J.B. Tausk, Non-planar massless two-loop Feynman diagrams with four on-shell legs, Phys. Lett. B 469 (1999) 225 [hep-ph/9909506].
27. V.A. Smirnov and O.L. Veretin, Analytical results for dimensionally regularized massless on-shell double boxes with arbitrary indices and numerators, Nucl. Phys. B 566 (2000) 469 [hep-ph/9907385].
28. C. Anastasiou, T. Gehrmann, C. Oleari, E. Remiddi and J.B. Tausk, The tensor reduction and master integrals of the two-loop massless crossed box with light-like legs, Nucl. Phys. B 580 (2000) 577 [hep-ph/0003261].
29. T. Gehrmann and E. Remiddi, Differential equations for two-loop four-point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329].
30. 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];
31. Two-loop master integrals for γ* → 3 jets: the non-planar topologies, Nucl. Phys. B 601 (2001) 287 [hep-ph/0101124].
32. T. Gehrmann and E. Remiddi, Analytic continuation of massless two-loop four-point functions, Nucl. Phys. B 640 (2002) 379 [hep-ph/0207020].
33. S. Laporta, High-precision calculation of multi-loop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033].
34. S. Moch, P. Uwer and S. Weinzierl, Nested sums, expansion of transcendental functions and multi-scale multi-loop integrals, J. Math. Phys. 43 (2002) 3363 [hep-ph/0110083].
35. Z. Bern, L.J. Dixon and A. Ghinculov, Two-loop correction to Bhabha scattering, Phys. Rev. D 63 (2001) 053007 [hep-ph/0010075].
36. C. Anastasiou, E.W.N. Glover, C. Oleari and M.E. Tejeda-Yeomans, Two-loop QCD corrections to qq̄ → q′q̄′, Nucl. Phys. B 601 (2001) 318 [hep-ph/0010212];
37. Two-loop QCD corrections to qq̄ → qq̄, Nucl Phys. B 601 (2001) 341 [hep-ph/0011094].
38. 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].
39. 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].
40. Z. Bern, A. De Freitas and L.J. Dixon, Two-loop amplitudes for gluon fusion into two photons, J. High Energy Phys. 09 (2001) 037 [hep-ph/0109078].
41. Z. Bern, A. De Freitas, L.J. Dixon, A. Ghinculov and H.L. Wong, QCD and QED corrections to light-by-light scattering, J. High Energy Phys. 11 (2001) 031 [hep-ph/0109079].
42. C. Anastasiou, E.W.N. Glover and M.E. Tejeda-Yeomans, Two-loop QED and QCD corrections to massless fermion-boson scattering, Nucl. Phys. B 629 (2002) 255 [hep-ph/0201274].
43. T. Binoth, E.W.N. Glover, P. Marquard and J.J. van der Bij, Two-loop corrections to light-by-light scattering in supersymmetric QED, J. High Energy Phys. 05 (2002) 060 [hep-ph/0202266].
44. 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, J. High Energy Phys. 06 (2003) 028 [hep-ph/0304168].
45. E.W.N. Glover and M.E. Tejeda-Yeomans, Two-loop QCD helicity amplitudes for massless quark massless gauge-boson scattering, J. High Energy Phys. 06 (2003) 033 [hep-ph/0304169].
46. E.W.N. Glover, Two-loop QCD helicity amplitudes for massless quark-quark scattering, J. High Energy Phys. 04 (2004) 021 [hep-ph/0401119].
47. - → 3 jets, Nucl. Phys. B 627 (2002) 107 [hep-ph/0112081].
48. - → 3 jets, Nucl. Phys. B 642 (2002) 227 [hep-ph/0206067];
49. f contribution, Acta Phys. Polon. B33 (2002) 2921 [hep-ph/0207167].
50. S. Catani, The singular behaviour of QCD amplitudes at two-loop order, Phys. Lett. B 427 (1998) 161 [hep-ph/9802439].
51. F.A. Berends and W.T. Giele, Multiple soft gluon radiation in parton processes, Nucl. Phys. B 313 (1989) 595.
52. J.M. Campbell and E.W.N. Glover, Double unresolved approximations to multiparton scattering amplitudes, Nucl. Phys. B 527 (1998) 264 [hep-ph/9710255].
53. 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];
54. 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].
55. 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];
56. 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].
57. D.A. Kosower and P. Uwer, One-loop splitting amplitudes in gauge theory, Nucl. Phys. B 563 (1999) 477 [hep-ph/9903515].
58. S. Weinzierl, Subtraction terms for one-loop amplitudes with one unresolved parton, J. High Energy Phys. 07 (2003) 052 [hep-ph/0306248].
59. G. Altarelli and G. Parisi, Asymptotic freedom in parton language, Nucl. Phys. B 126 (1977) 298.
60. S. Moch, J.A.M. Vermaseren and A. Vogt, Non-singlet structure functions at three loops: fermionic contributions, Nucl. Phys. B 646 (2002) 181 [hep-ph/0209100].
61. S. Moch, J.A.M. Vermaseren and A. Vogt, The three-loop splitting functions in QCD: the non-singlet case, Nucl. Phys. B 688 (2004) 101 [hep-ph/0403192];
62. A. Vogt, S. Moch and J.A.M. Vermaseren, The three-loop splitting functions in QCD: the singlet case, Nucl. Phys. B 691 (2004) 129 [hep-ph/0404111].
63. W.L. van Neerven and A. Vogt, NNLO evolution of deep-inelastic structure functions: the non-singlet case, Nucl. Phys. B 568 (2000) 263 [hep-ph/9907472];
64. NNLO evolution of deep-inelastic structure functions: the singlet case, Nucl. Phys. B 588 (2000) 345 [hep-ph/0006154];
65. Improved approximations for the three-loop splitting functions in QCD, Phys. Lett. B 490 (2000) 111 [hep-ph/0007362].
66. D.A. Kosower and P. Uwer, One-loop splitting amplitudes and the Altarelli-Parisi kernels, in proceedings of International Europhysics Conference on High-Energy Physics (EPS-HEP 99), Tampere, Finland 1999;
67. D.A. Kosower and P. Uwer, Evolution kernels from splitting amplitudes, Nucl. Phys. B 674 (2003) 365 [hep-ph/0307031].
68. 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].
69. J.A. Minahan and K. Zarembo, The Bethe-ansatz for N = 4 super Yang-Mills, J. High Energy Phys. 03 (2003) 013 [hep-th/0212208];
70. N. Beisert, S. Frolov, M. Staudacher and A.A. Tseytlin, Precision spectroscopy of AdS/CFT, J. High Energy Phys. 10 (2003) 037 [hep-th/0308117], and references therein.
71. A.V. Kotikov, L.N. Lipatov, A.I. Onishchenko and V.N. Velizhanin, Three-loop universal anomalous dimension of the Wilson operators in N = 4 SUSY Yang-Mills model, Phys. Lett. B 595 (2004) 521 [hep-th/0404092].
72. C. Anastasiou, Z. Bern, L.J. Dixon and D.A. Kosower, Planar amplitudes in maximally supersymmetric Yang-Mills theory, Phys. Rev. Lett. 91 (2003) 251602 [hep-th/0309040].
73. Z. Bern, L.J. Dixon, M. Perelstein and J.S. Rozowsky, Multi-leg one-loop gravity amplitudes from gauge theory, Nucl. Phys. B 546 (1999) 423 [hep-th/9811140].
74. J.C. Collins, D.E. Soper and G. Sterman, All order factorization for Drell-Yan cross-sections, Phys. Lett. B 134 (1984) 263;
75. Factorization for short distance hadron-hadron scattering, Nucl. Phys. B 261 (1985) 104;
76. Factorization of hard processes in quantum chromodynamics, in Perturbative quantum chromodynamics, A.H. Mueller ed., World Scientific Singapore 1989.
77. Z. Bern and G. Chalmers, Factorization in one-loop gauge theory, Nucl. Phys. B 447 (1995) 465 [hep-ph/9503236].
78. A. Bassetto, G. Nardelli and R. Soldati, Yang-Mills theories in algebraic noncovariant gauges: canonical quantization and renormalization, World Scientific, Singapore 1991;
79. G. Leibbrandt, Noncovariant gauges: quantization of Yang-Mills and Chern-Simons theory in axial-type gauges, World Scientific, Singapore 1994.
80. A. Andrasi and J.C. Taylor, Are axial gauges useful in perturbative QCD?, Nucl. Phys. B 310 (1988) 222.
81. L.J. Dixon and M. Schreiber, in progress.
82. F.V. Tkachov, A theorem on analytical calculability of four loop renormalization group functions, Phys. Lett. B 100 (1981) 65;
83. K.G. Chetyrkin and F.V. Tkachov, Integration by parts: the algorithm to calculate beta functions in 4 loops, Nucl. Phys. B 192 (1981) 159.
84. C. Anastasiou and A. Lazopoulos, Automatic integral reduction for higher order perturbative calculations, J. High Energy Phys. 07 (2004) 046 [hep-ph/0404258].
85. G. Sterman and M.E. Tejeda-Yeomans, Multi-loop amplitudes and resummation, Phys. Lett. B 552 (2003) 48 [hep-ph/0210130].
86. A. Koukoutsakis, Higgs bosons and QCD jets at two loops, Ph.D. thesis, University of Durham 2003.
87. 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].
88. 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].
89. D.A. Kosower, All-order collinear behavior in gauge theories, Nucl. Phys. B 552 (1999) 319 [hep-ph/9901201].
90. M.L. Mangano and S.J. Parke, Multiparton amplitudes in gauge theories, Phys. Rept. 200 (1991) 301;
91. L.J. Dixon, Calculating scattering amplitudes efficiently, hep-ph/9601359.
92. S.J. Parke and T.R. Taylor, An amplitude for n-gluon scattering, Phys. Rev. Lett. 56 (1986) 2459.
93. G. 't Hooft and M.J.G. Veltman, Regularization and renormalization of gauge fields, Nucl. Phys. B 44 (1972) 189.
94. Z. Bern, A. De Freitas, L.J. Dixon and H.L. Wong, Supersymmetric regularization, two-loop QCD amplitudes and coupling shifts, Phys. Rev. D 66 (2002) 085002 [hep-ph/0202271].
95. W. Siegel, Supersymmetric dimensional regularization via dimensional reduction, Phys. Lett. B 84 (1979) 193;
96. D.M. Capper, D.R.T. Jones and P. van Nieuwenhuizen, Regularization by dimensional reduction of supersymmetric and nonsupersymmetric gauge theories, Nucl. Phys. B 167 (1980) 479;
97. I. Jack, D.R.T. Jones and K.L. Roberts, Equivalence of dimensional reduction and dimensional regularization, Z. Physik C 63 (1994) 151 [hep-ph/9401349].
98. Z. Bern, J.S. Rozowsky and B. Yan, Two-loop four-gluon amplitudes in N = 4 super-Yang-Mills, Phys. Lett. B 401 (1997) 273 [hep-ph/9702424];
99. Z. Bern, L.J. Dixon, D.C. Dunbar, M. Perelstein and J.S. Rozowsky, On the relationship between Yang-Mills theory and gravity and its implication for ultraviolet divergences, Nucl. Phys. B 530 (1998) 401 [hep-th/9802162].
100. J.-L. Gervais and A. Neveu, Feynman rules for massive gauge fields with dual diagram topology, Nucl. Phys. B 46 (1972) 381.
101. R.K. Ellis, H. Georgi, M. Machacek, H.D. Politzer and G.G. Ross, Factorization and the parton model in QCD, Phys. Lett. B 78 (1978) 281;
102. Perturbation theory and the parton model in QCD, Nucl. Phys. B 152 (1979) 285.
103. G. Curci, W. Furmanski and R. Petronzio, Evolution of parton densities beyond leading order: the nonsinglet case, Nucl. Phys. B 175 (1980) 27.
104. W. Furmanski and R. Petronzio, Singlet parton densities beyond leading order, Phys. Lett. B 97 (1980) 437.
105. R.K. Ellis and W. Vogelsang, The evolution of parton distributions beyond leading order: the singlet case, hep-ph/9602356.
106. S. Mandelstam, Light-cone superspace and the ultraviolet finiteness of the N = 4 model, Nucl. Phys. B 213 (1983) 149.
107. G. Heinrich and Z. Kunszt, Two-loop anomalous dimension in light-cone gauge with Mandelstam-Leibbrandt prescription, Nucl. Phys. B 519 (1998) 405 [hep-ph/9708334];
108. A. Bassetto, G. Heinrich, Z. Kunszt and W. Vogelsang, The light-cone gauge and the calculation of the two-loop splitting functions, Phys. Rev. D 58 (1998) 094020 [hep-ph/9805283].
109. L.D. Landau, On analytic properties of vertex parts in quantum field theory, Nucl. Phys. 13 (1959) 181;
110. S. Mandelstam, Determination of the pion-nucleon scattering amplitude from dispersion relations and unitarity. General theory, Phys. Rev. 112 (1958) 1344;
111. Analytic properties of transition amplitudes in perturbation theory, Phys. Rev. 115 (1959) 1741;
112. R.E. Cutkosky, Singularities and discontinuities of Feynman amplitudes, J. Math. Phys. 1 (1960) 429.
113. M.E. Peskin and D.V. Schroeder, An introduction to quantum field theory, Addison-Wesley, Menlo Park 1995.
114. W.L. van Neerven, Dimensional regularization of mass and infrared singularities in two-loop on-shell vertex functions, Nucl. Phys. B 268 (1986) 453.
115. Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys. B 412 (1994) 751 [hep-ph/9306240];
116. J.M. Campbell, E.W.N. Glover and D. J. Miller, The one-loop QCD corrections for γ* → qq̄gg, Phys. Lett. B 409 (1997) 503 [hep-ph/9706297];
117. T. Binoth, J.P. Guillet and G. Heinrich, Reduction formalism for dimensionally regulated one-loop N-point integrals, Nucl. Phys. B 572 (2000) 361 [hep-ph/9911342].
118. J.C. Collins, Renormalization: an introduction to the renormalization group and the operator-product expansion, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge 1984.
119. D.A. Kosower, Multiple singular emission in gauge theories, Phys. Rev. D 67 (2003) 116003 [hep-ph/0212097];
120. S. Weinzierl, Subtraction terms at NNLO, J. High Energy Phys. 03 (2003) 062 [hep-ph/0302180].
121. L. Lewin, Dilogarithms and associated functions, Macdonald, London 1958.
122. E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237].
123. C. Anastasiou, E.W.N. Glover and C. Oleari, The two-loop scalar and tensor pentabox graph with light-like legs, Nucl. Phys. B 575 (2000) 416 [hep-ph/9912251], erratum ibid. B585 (2000) 763.
124. MAPLE, http://www.maplesoft.com.
125. J.A.M. Vermaseren, New features of FORM, math-ph/0010025.
126. R.J. Gonsalves, Dimensionally regularized two-loop on-shell quark form-factor, Phys. Rev. D 28 (1983) 1542.
127. M.T. Grisaru, H.N. Pendleton and P. van Nieuwenhuizen, Supergravity and the S matrix, Phys. Rev. D 15 (1977) 996;
128. M.T. Grisaru and H.N. Pendleton, Some properties of scattering amplitudes in supersymmetric theories, Nucl. Phys. B 124 (1977) 81;
129. S.J. Parke and T.R. Taylor, Perturbative QCD utilizing extended super symmetry, Phys. Lett. B 157 (1985) 81, erratum ibid. 174B (1985) 465.
130. Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One-loop self-dual and N = 4 super Yang-Mills, Phys. Lett. B 394 (1997) 105 [hep-th/9611127].
131. S. Catani and M. Grazzini, The soft-gluon current at one-loop order, Nucl. Phys. B 591 (2000) 435 [hep-ph/0007142].