Abstract
A scheme is proposed for the subtraction of soft and collinear divergences present in massless final state real emission phase space integrals. The scheme is based on a local slicing procedure which utilises the soft and collinear factorisation properties of amplitudes to produce universal counter-terms whose analytic integration is relatively simple. As a first application the scheme is applied to establish a general pole formula for final state real radiation at NLO and NNLO in Yang Mills theory for arbitrary multiplicities. All required counter-terms are evaluated to all orders in the dimensional regulator in terms of Γ — and pFq hypergeometric — functions. As a proof of principle the poles in the dimensional regulator of the H → gggg double real emission contribution to the H → gg decay rate are reproduced.
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References
T. Kinoshita, Mass singularities of Feynman amplitudes, J. Math. Phys. 3 (1962) 650 [INSPIRE].
T.D. Lee and M. Nauenberg, Degenerate systems and mass singularities, Phys. Rev. 133 (1964) B1549 [INSPIRE].
H.D. Politzer, Asymptotic freedom: an approach to strong interactions, Phys. Rept. 14 (1974) 129 [INSPIRE].
H. Georgi and H.D. Politzer, Electroproduction scaling in an asymptotically free theory of strong interactions, Phys. Rev. D 9 (1974) 416 [INSPIRE].
G. Altarelli and G. Parisi, Asymptotic freedom in parton language, Nucl. Phys. B 126 (1977) 298 [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] [hep-ph/9605323] [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. Frixione, Z. Kunszt and A. Signer, Three jet cross-sections to next-to-leading order, Nucl. Phys. B 467 (1996) 399 [hep-ph/9512328] [INSPIRE].
R. Frederix et al., Automation of next-to-leading order computations in QCD: the FKS subtraction, JHEP 10 (2009) 003 [arXiv:0908.4272] [INSPIRE].
Z. Nagy and D.E. Soper, General subtraction method for numerical calculation of one loop QCD matrix elements, JHEP 09 (2003) 055 [hep-ph/0308127] [INSPIRE].
T. Binoth and G. Heinrich, Numerical evaluation of phase space integrals by sector decomposition, Nucl. Phys. B 693 (2004) 134 [hep-ph/0402265] [INSPIRE].
C. Anastasiou, K. Melnikov and F. Petriello, A new method for real radiation at NNLO, Phys. Rev. D 69 (2004) 076010 [hep-ph/0311311] [INSPIRE].
M. Czakon, A novel subtraction scheme for double-real radiation at NNLO, Phys. Lett. B 693 (2010) 259 [arXiv:1005.0274] [INSPIRE].
R. Boughezal, K. Melnikov and F. Petriello, A subtraction scheme for NNLO computations, Phys. Rev. D 85 (2012) 034025 [arXiv:1111.7041] [INSPIRE].
F. Caola, K. Melnikov and R. Röntsch, Nested soft-collinear subtractions in NNLO QCD computations, Eur. Phys. J. C 77 (2017) 248 [arXiv:1702.01352] [INSPIRE].
M. Czakon, D. Heymes and A. Mitov, High-precision differential predictions for top-quark pairs at the LHC, Phys. Rev. Lett. 116 (2016) 082003 [arXiv:1511.00549] [INSPIRE].
R. Boughezal et al., Higgs boson production in association with a jet at next-to-next-to-leading order, Phys. Rev. Lett. 115 (2015) 082003 [arXiv:1504.07922] [INSPIRE].
S. Weinzierl, Subtraction terms at NNLO, JHEP 03 (2003) 062 [hep-ph/0302180] [INSPIRE].
S. Frixione and M. Grazzini, Subtraction at NNLO, JHEP 06 (2005) 010 [hep-ph/0411399] [INSPIRE].
C. Anastasiou et al., NNLO QCD corrections to pp → γ ∗ γ ∗ in the large N F limit, JHEP 02 (2015) 182 [arXiv:1408.4546] [INSPIRE].
C. Anastasiou, F. Herzog and A. Lazopoulos, On the factorization of overlapping singularities at NNLO, JHEP 03 (2011) 038 [arXiv:1011.4867] [INSPIRE].
A. Gehrmann-De Ridder, T. Gehrmann and E.W.N. Glover, Antenna subtraction at NNLO, JHEP 09 (2005) 056 [hep-ph/0505111] [INSPIRE].
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] [INSPIRE].
J. Currie, E.W.N. Glover and S. Wells, Infrared structure at NNLO using antenna subtraction, JHEP 04 (2013) 066 [arXiv:1301.4693] [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].
J. Currie et al., Precise predictions for dijet production at the LHC, Phys. Rev. Lett. 119 (2017) 152001 [arXiv:1705.10271] [INSPIRE].
J. Cruz-Martinez et al., Second-order QCD effects in Higgs boson production through vector boson fusion, Phys. Lett. B 781 (2018) 672 [arXiv:1802.02445] [INSPIRE].
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] [INSPIRE].
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] [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].
V. Del Duca et al., Three-jet production in electron-positron collisions at next-to-next-to-leading order accuracy, Phys. Rev. Lett. 117 (2016) 152004 [arXiv:1603.08927] [INSPIRE].
W.T. Giele and E.W.N. Glover, Higher order corrections to jet cross-sections in e + e − annihilation, Phys. Rev. D 46 (1992) 1980 [INSPIRE].
K. Fabricius et al., Higher order perturbative QCD calculation of jet cross-sections in e + e − annihilation, Z. Phys. C 11 (1981) 315 [INSPIRE].
A. Gehrmann-De Ridder and E.W.N. Glover, A complete O(αα s) calculation of the photon + 1 jet rate in e + e − annihilation, Nucl. Phys. B 517 (1998) 269 [hep-ph/9707224] [INSPIRE].
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] [INSPIRE].
R. Boughezal et al., Higgs boson production in association with a jet at NNLO using jettiness subtraction, Phys. Lett. B 748 (2015) 5 [arXiv:1505.03893] [INSPIRE].
J. Gaunt, M. Stahlhofen, F.J. Tackmann and J.R. Walsh, N-jettiness subtractions for NNLO QCD calculations, JHEP 09 (2015) 058 [arXiv:1505.04794] [INSPIRE].
S. Catani et al., Diphoton production at hadron colliders: a fully-differential QCD calculation at NNLO, Phys. Rev. Lett. 108 (2012) 072001 [arXiv:1110.2375] [INSPIRE].
R. Boughezal, X. Liu and F. Petriello, W-boson plus jet differential distributions at NNLO in QCD, Phys. Rev. D 94 (2016) 113009 [arXiv:1602.06965] [INSPIRE].
J. Gao, C.S. Li and H.X. Zhu, Top quark decay at next-to-next-to leading order in QCD, Phys. Rev. Lett. 110 (2013) 042001 [arXiv:1210.2808] [INSPIRE].
J. Gao and H.X. Zhu, Electroweak prodution of top-quark pairs in e + e − annihilation at NNLO in QCD: the vector contributions, Phys. Rev. D 90 (2014) 114022 [arXiv:1408.5150] [INSPIRE].
I. Moult et al., N-jettiness subtractions for gg → H at subleading power, Phys. Rev. D 97 (2018) 014013 [arXiv:1710.03227] [INSPIRE].
R. Boughezal, A. Isgrò and F. Petriello, Next-to-leading-logarithmic power corrections for N -jettiness subtraction in color-singlet production, Phys. Rev. D 97 (2018) 076006 [arXiv:1802.00456] [INSPIRE].
M. Cacciari et al., Fully differential vector-boson-fusion Higgs production at next-to-next-to-leading order, Phys. Rev. Lett. 115 (2015) 082002 [arXiv:1506.02660] [INSPIRE].
J. Currie et al., N 3 LO corrections to jet production in deep inelastic scattering using the Projection-to-Born method, JHEP 05 (2018) 209 [arXiv:1803.09973] [INSPIRE].
F. Dulat, B. Mistlberger and A. Pelloni, Differential Higgs production at N 3 LO beyond threshold, JHEP 01 (2018) 145 [arXiv:1710.03016] [INSPIRE].
C. Anastasiou and K. Melnikov, Higgs boson production at hadron colliders in NNLO QCD, Nucl. Phys. B 646 (2002) 220 [hep-ph/0207004] [INSPIRE].
T.O. Eynck, E. Laenen, L. Phaf and S. Weinzierl, Comparison of phase space slicing and dipole subtraction methods for \( {\gamma}^{\ast}\to \overline{Q} \), Eur. Phys. J. C 23 (2002) 259 [hep-ph/0109246] [INSPIRE].
S. Bloch and D. Kreimer, Mixed Hodge structures and renormalization in physics, Commun. Num. Theor. Phys. 2 (2008) 637 [arXiv:0804.4399] [INSPIRE].
F. Brown, Feynman amplitudes, coaction principle and cosmic Galois group, Commun. Num. Theor. Phys. 11 (2017) 453 [arXiv:1512.06409] [INSPIRE].
G.F. Sterman, Mass divergences in annihilation processes. 1. Origin and nature of divergences in cut vacuum polarization diagrams, Phys. Rev. D 17 (1978) 2773 [INSPIRE].
G.F. Sterman, Mass divergences in annihilation processes. 2. Cancellation of divergences in cut vacuum polarization diagrams, Phys. Rev. D 17 (1978) 2789 [INSPIRE].
J. Collins, Foundations of perturbative QCD, Cambridge University Press, Cambridge U.K. (2013).
T. Hahn, CUBA: a library for multidimensional numerical integration, Comput. Phys. Commun. 168 (2005) 78 [hep-ph/0404043] [INSPIRE].
W. Zimmermann, Convergence of Bogolyubov’s method of renormalization in momentum space, Commun. Math. Phys. 15 (1969) 208 [INSPIRE].
B. Humpert and W.L. van Neerven, Graphical mass factorization, Phys. Lett. B 102 (1981) 426.
V.A. Smirnov and K.G. Chetyrkin, R∗ operation in the minimal subtraction scheme, Theor. Math. Phys. 63 (1985) 462 [INSPIRE].
F. Herzog, Zimmermann’s forest formula, infrared divergences and the QCD β-function, Nucl. Phys. B 926 (2018) 370 [arXiv:1711.06121] [INSPIRE].
J.C. Collins and D.E. Soper, Back-to-back jets in QCD, Nucl. Phys. B 193 (1981) 381 [Erratum ibid. B 213 (1983) 545] [INSPIRE].
C. Anastasiou, C. Duhr, F. Dulat and B. Mistlberger, Soft triple-real radiation for Higgs production at N3LO, JHEP 07 (2013) 003 [arXiv:1302.4379] [INSPIRE].
A.V. Smirnov, Algorithm FIRE — Feynman Integral REduction, JHEP 10 (2008) 107 [arXiv:0807.3243] [INSPIRE].
A.V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun. 189 (2015) 182 [arXiv:1408.2372] [INSPIRE].
C. Anastasiou and A. Lazopoulos, Automatic integral reduction for higher order perturbative calculations, JHEP 07 (2004) 046 [hep-ph/0404258] [INSPIRE].
C. Anastasiou et al., NNLO phase space master integrals for two-to-one inclusive cross sections in dimensional regularization, JHEP 11 (2012) 062 [arXiv:1208.3130] [INSPIRE].
C. Anastasiou et al., Soft expansion of double-real-virtual corrections to Higgs production at N 3 LO, JHEP 08 (2015) 051 [arXiv:1505.04110] [INSPIRE].
M. Ritzmann and W.J. Waalewijn, Fragmentation in jets at NNLO, Phys. Rev. D 90 (2014) 054029 [arXiv:1407.3272] [INSPIRE].
T. Huber and D. Maître, HypExp: a Mathematica package for expanding hypergeometric functions around integer-valued parameters, Comput. Phys. Commun. 175 (2006) 122 [hep-ph/0507094] [INSPIRE].
A. Gehrmann-De Ridder, T. Gehrmann and G. Heinrich, Four particle phase space integrals in massless QCD, Nucl. Phys. B 682 (2004) 265 [hep-ph/0311276] [INSPIRE].
S. Bühler, F. Herzog, A. Lazopoulos and R. Müller, The fully differential hadronic production of a Higgs boson via bottom quark fusion at NNLO, JHEP 07 (2012) 115 [arXiv:1204.4415] [INSPIRE].
S. Catani, The singular behavior of QCD amplitudes at two loop order, Phys. Lett. B 427 (1998) 161 [hep-ph/9802439] [INSPIRE].
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Herzog, F. Geometric IR subtraction for final state real radiation. J. High Energ. Phys. 2018, 6 (2018). https://doi.org/10.1007/JHEP08(2018)006
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DOI: https://doi.org/10.1007/JHEP08(2018)006