Abstract
We perform the integration of all iterated singly-unresolved subtraction terms, as defined in ref. [1], over the two-particle factorized phase space. We also sum over the unresolved parton flavours. The final result can be written as a convolution (in colour space) of the Born cross section and an insertion operator. We spell out the insertion operator in terms of 24 basic integrals that are defined explicitly. We compute the coefficients of the Laurent expansion of these integrals in two different ways, with the method of Mellin-Barnes representations and sector decomposition. Finally, we present the Laurent-expansion of the full insertion operator for the specific examples of electron-positron annihilation into two and three jets.
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G. Somogyi, Z. Trócsányi and V. Del Duca, A subtraction scheme for computing QCD jet cross sections at NNLO: regularization of doubly-real emissions, JHEP 01 (2007) 070 [hep-ph/0609042] [SPIRES].
G. Somogyi and Z. Trócsányi, A subtraction scheme for computing QCD jet cross sections at NNLO: regularization of real-virtual emission, JHEP 01 (2007) 052 [hep-ph/0609043] [SPIRES].
G. Somogyi and Z. Trócsányi, A new subtraction scheme for computing QCD jet cross sections at next-to-leading order accuracy, hep-ph/0609041 [SPIRES].
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] [SPIRES].
S. Weinzierl, Subtraction terms at NNLO, JHEP 03 (2003) 062 [hep-ph/0302180] [SPIRES].
S. Weinzierl, Subtraction terms for one-loop amplitudes with one unresolved parton, JHEP 07 (2003) 052 [hep-ph/0306248] [SPIRES].
S. Frixione and M. Grazzini, Subtraction at NNLO, JHEP 06 (2005) 010 [hep-ph/0411399] [SPIRES].
A. Gehrmann-De Ridder, T. Gehrmann and E.W.N. Glover, Antenna Subtraction at NNLO, JHEP 09 (2005) 056 [hep-ph/0505111] [SPIRES].
A. Daleo, T. Gehrmann and D. Maître, Antenna subtraction with hadronic initial states, JHEP 04 (2007) 016 [hep-ph/0612257] [SPIRES].
A. Daleo, A. Gehrmann-De Ridder, T. Gehrmann and G. Luisoni, Antenna subtraction at NNLO with hadronic initial states: initial-final configurations, JHEP 01 (2010) 118 [arXiv:0912.0374] [SPIRES].
E.W. Nigel Glover and J. Pires, Antenna subtraction for gluon scattering at NNLO, JHEP 06 (2010) 096 [arXiv:1003.2824] [SPIRES].
M. Czakon, A novel subtraction scheme for double-real radiation at NNLO, Phys. Lett. B 693 (2010) 259 [arXiv:1005.0274] [SPIRES].
A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover and G. Heinrich, Infrared structure of e + e − → 3 jets at NNLO, JHEP 11 (2007) 058 [arXiv:0710.0346] [SPIRES].
A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover and G. Heinrich, Jet rates in electron-positron annihilation at O(α s 3) in QCD, Phys. Rev. Lett. 100 (2008) 172001 [arXiv:0802.0813] [SPIRES].
S. Weinzierl, NNLO corrections to 3-jet observables in electron-positron annihilation, Phys. Rev. Lett. 101 (2008) 162001 [arXiv:0807.3241] [SPIRES].
S. Weinzierl, The infrared structure of e + e − → 3 jets at NNLO reloaded, JHEP 07 (2009) 009 [arXiv:0904.1145] [SPIRES].
A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover and G. Heinrich, Second-order QCD corrections to the thrust distribution, Phys. Rev. Lett. 99 (2007) 132002 [arXiv:0707.1285] [SPIRES].
A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover and G. Heinrich, NNLO corrections to event shapes in e + e − annihilation, JHEP 12 (2007) 094 [arXiv:0711.4711] [SPIRES].
A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover and G. Heinrich, NNLO moments of event shapes in e + e − annihilation, JHEP 05 (2009) 106 [arXiv:0903.4658] [SPIRES].
S. Weinzierl, Event shapes and jet rates in electron-positron annihilation at NNLO, JHEP 06 (2009) 041 [arXiv:0904.1077] [SPIRES].
S. Weinzierl, Moments of event shapes in electron-positron annihilation at NNLO, Phys. Rev. D 80 (2009) 094018 [arXiv:0909.5056] [SPIRES].
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] [SPIRES].
S. Catani, G. Ferrera and M. Grazzini, W boson production at hadron colliders: the lepton charge asymmetry in NNLO QCD, JHEP 05 (2010) 006 [arXiv:1002.3115] [SPIRES].
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] [SPIRES].
G. Heinrich, Sector decomposition, Int. J. Mod. Phys. A 23 (2008) 1457 [arXiv:0803.4177] [SPIRES].
Z. Nagy, G. Somogyi and Z. Trócsányi, Separation of soft and collinear infrared limits of QCD squared matrix elements, hep-ph/0702273 [SPIRES].
G. Somogyi, Subtraction with hadronic initial states: an NNLO- compatible scheme, JHEP 05 (2009) 016 [arXiv:0903.1218] [SPIRES].
A.V. Kotikov, Differential equation method: The Calculation of N point Feynman diagrams, Phys. Lett. B 267 (1991) 123 [SPIRES].
E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim. A 11 (1997) 1435 [hep-th/9711188] [SPIRES].
V.A. Smirnov, Analytical result for dimensionally regularized massless on-shell double box, Phys. Lett. B 460 (1999) 397 [hep-ph/9905323] [SPIRES].
J.B. Tausk, Non-planar massless two-loop Feynman diagrams with four on-shell legs, Phys. Lett. B 469 (1999) 225 [hep-ph/9909506] [SPIRES].
V.A. Smirnov, Evaluating Feynman integrals, Springer Tracts Mod. Phys. 211 (2004) 1 [SPIRES].
M. Czakon, Automatized analytic continuation of Mellin-Barnes integrals, Comput. Phys. Commun. 175 (2006) 559 [hep-ph/0511200] [SPIRES].
J. Gluza, K. Kajda and T. Riemann, AMBRE - a Mathematica package for the construction of Mellin-Barnes representations for Feynman integrals, Comput. Phys. Commun. 177 (2007) 879 [arXiv:0704.2423] [SPIRES].
G. Somogyi and Z. Trócsányi, A subtraction scheme for computing QCD jet cross sections at NNLO: integrating the subtraction terms I, JHEP 08 (2008) 042 [arXiv:0807.0509] [SPIRES].
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] [SPIRES].
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] [SPIRES].
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] [SPIRES].
Z. Nagy and Z. Trócsányi, Group independent color decomposition of next-to-leading order matrix elements for e + e − → four partons, Phys. Lett. B 414 (1997) 187 [hep-ph/9708342] [SPIRES].
W.L. van Neerven, Dimensional regularization of mass and infrared singularities in two loop on-shell vertex functions, Nucl. Phys. B 268 (1986) 453 [SPIRES].
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Bolzoni, P., Somogyi, G. & Trócsányi, Z. A subtraction scheme for computing QCD jet cross sections at NNLO: integrating the iterated singly-unresolved subtraction terms. J. High Energ. Phys. 2011, 59 (2011). https://doi.org/10.1007/JHEP01(2011)059
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DOI: https://doi.org/10.1007/JHEP01(2011)059