Abstract
Within the framework of local analytic sector subtraction, we present the full analytic integration of double-real and real-virtual local infrared counterterms that enter NNLO QCD computations with any number of massless final-state partons. We show that a careful choice of phase-space mappings leads to simple analytic results, including non-singular terms, that can be obtained with conventional integration techniques.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. Heinrich, Collider Physics at the Precision Frontier, arXiv:2009.00516 [INSPIRE].
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].
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].
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. 510 (1998) 503] [hep-ph/9605323] [INSPIRE].
S. Frixione and M. Grazzini, Subtraction at NNLO, JHEP 06 (2005) 010 [hep-ph/0411399] [INSPIRE].
A. Gehrmann-De Ridder, T. Gehrmann and E.W.N. Glover, Antenna subtraction at NNLO, JHEP 09 (2005) 056 [hep-ph/0505111] [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].
A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover and G. Heinrich, Jet rates in electron-positron annihilation at \( O\left({\alpha}_s^3\right) \) in QCD, Phys. Rev. Lett. 100 (2008) 172001 [arXiv:0802.0813] [INSPIRE].
M. Czakon, A novel subtraction scheme for double-real radiation at NNLO, Phys. Lett. B 693 (2010) 259 [arXiv:1005.0274] [INSPIRE].
M. Czakon, P. Fiedler and A. Mitov, Total Top-Quark Pair-Production Cross Section at Hadron Colliders Through \( O\left({\alpha}_S^4\right) \), Phys. Rev. Lett. 110 (2013) 252004 [arXiv:1303.6254] [INSPIRE].
M. Czakon and D. Heymes, Four-dimensional formulation of the sector-improved residue subtraction scheme, Nucl. Phys. B 890 (2014) 152 [arXiv:1408.2500] [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].
V. Del Duca, C. Duhr, G. Somogyi, F. Tramontano and Z. Trócsányi, Higgs boson decay into b-quarks at NNLO accuracy, JHEP 04 (2015) 036 [arXiv:1501.07226] [INSPIRE].
A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover, A. Huss and T.A. Morgan, Precise QCD predictions for the production of a Z boson in association with a hadronic jet, Phys. Rev. Lett. 117 (2016) 022001 [arXiv:1507.02850] [INSPIRE].
V. Del Duca, C. Duhr, A. Kardos, G. Somogyi and Z. Trócsányi, 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].
V. Del Duca et al., Jet production in the CoLoRFulNNLO method: event shapes in electron-positron collisions, Phys. Rev. D 94 (2016) 074019 [arXiv:1606.03453] [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].
J. Currie, A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover, A. Huss and J. Pires, Precise predictions for dijet production at the LHC, Phys. Rev. Lett. 119 (2017) 152001 [arXiv:1705.10271] [INSPIRE].
F. Caola, K. Melnikov and R. Röntsch, Analytic results for decays of color singlets to gg and \( q\overline{q} \) final states at NNLO QCD with the nested soft-collinear subtraction scheme, Eur. Phys. J. C 79 (2019) 1013 [arXiv:1907.05398] [INSPIRE].
F. Caola, K. Melnikov and R. Röntsch, Analytic results for color-singlet production at NNLO QCD with the nested soft-collinear subtraction scheme, Eur. Phys. J. C 79 (2019) 386 [arXiv:1902.02081] [INSPIRE].
H.A. Chawdhry, M.L. Czakon, A. Mitov and R. Poncelet, NNLO QCD corrections to three-photon production at the LHC, JHEP 02 (2020) 057 [arXiv:1911.00479] [INSPIRE].
M. Czakon, A. van Hameren, A. Mitov and R. Poncelet, Single-jet inclusive rates with exact color at \( \mathcal{O}\left({\alpha}_s^4\right) \), JHEP 10 (2019) 262 [arXiv:1907.12911] [INSPIRE].
A. Gehrmann-De Ridder, T. Gehrmann, E.W.N. Glover, A. Huss and J. Pires, Triple Differential Dijet Cross Section at the LHC, Phys. Rev. Lett. 123 (2019) 102001 [arXiv:1905.09047] [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].
W.T. Giele, E.W.N. Glover and D.A. Kosower, Higher order corrections to jet cross-sections in hadron colliders, Nucl. Phys. B 403 (1993) 633 [hep-ph/9302225] [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].
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] [INSPIRE].
R. Boughezal, K. Melnikov and F. Petriello, A subtraction scheme for NNLO computations, Phys. Rev. D 85 (2012) 034025 [arXiv:1111.7041] [INSPIRE].
R. Bonciani, S. Catani, M. Grazzini, H. Sargsyan and A. Torre, The qT subtraction method for top quark production at hadron colliders, Eur. Phys. J. C 75 (2015) 581 [arXiv:1508.03585] [INSPIRE].
R. Boughezal, C. Focke, X. Liu and F. Petriello, W-boson production in association with a jet at next-to-next-to-leading order in perturbative QCD, Phys. Rev. Lett. 115 (2015) 062002 [arXiv:1504.02131] [INSPIRE].
R. Boughezal, F. Caola, K. Melnikov, F. Petriello and M. Schulze, 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].
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].
M. Grazzini, S. Kallweit and M. Wiesemann, Fully differential NNLO computations with MATRIX, Eur. Phys. J. C 78 (2018) 537 [arXiv:1711.06631] [INSPIRE].
S. Catani, S. Devoto, M. Grazzini, S. Kallweit and J. Mazzitelli, Top-quark pair production at the LHC: Fully differential QCD predictions at NNLO, JHEP 07 (2019) 100 [arXiv:1906.06535] [INSPIRE].
S. Kallweit, V. Sotnikov and M. Wiesemann, Triphoton production at hadron colliders in NNLO QCD, Phys. Lett. B 812 (2021) 136013 [arXiv:2010.04681] [INSPIRE].
M. Cacciari, F.A. Dreyer, A. Karlberg, G.P. Salam and G. Zanderighi, Fully Differential Vector-Boson-Fusion Higgs Production at Next-to-Next-to-Leading Order, Phys. Rev. Lett. 115 (2015) 082002 [Erratum ibid. 120 (2018) 139901] [arXiv:1506.02660] [INSPIRE].
F. Herzog, Geometric IR subtraction for final state real radiation, JHEP 08 (2018) 006 [arXiv:1804.07949] [INSPIRE].
F.A. Dreyer and A. Karlberg, Fully differential Vector-Boson Fusion Higgs Pair Production at Next-to-Next-to-Leading Order, Phys. Rev. D 99 (2019) 074028 [arXiv:1811.07918] [INSPIRE].
L. Magnea, E. Maina, G. Pelliccioli, C. Signorile-Signorile, P. Torrielli and S. Uccirati, Factorisation and Subtraction beyond NLO, JHEP 12 (2018) 062 [arXiv:1809.05444] [INSPIRE].
C. Anastasiou and G. Sterman, Removing infrared divergences from two-loop integrals, JHEP 07 (2019) 056 [arXiv:1812.03753] [INSPIRE].
Y. Ma, A Forest Formula to Subtract Infrared Singularities in Amplitudes for Wide-angle Scattering, JHEP 05 (2020) 012 [arXiv:1910.11304] [INSPIRE].
C. Anastasiou, R. Haindl, G. Sterman, Z. Yang and M. Zeng, Locally finite two-loop amplitudes for off-shell multi-photon production in electron-positron annihilation, arXiv:2008.12293 [INSPIRE].
S. Catani, T. Gleisberg, F. Krauss, G. Rodrigo and J.-C. Winter, From loops to trees by-passing Feynman’s theorem, JHEP 09 (2008) 065 [arXiv:0804.3170] [INSPIRE].
I. Bierenbaum, S. Catani, P. Draggiotis and G. Rodrigo, A Tree-Loop Duality Relation at Two Loops and Beyond, JHEP 10 (2010) 073 [arXiv:1007.0194] [INSPIRE].
G.F.R. Sborlini, F. Driencourt-Mangin and G. Rodrigo, Four-dimensional unsubtraction with massive particles, JHEP 10 (2016) 162 [arXiv:1608.01584] [INSPIRE].
G.F.R. Sborlini, F. Driencourt-Mangin, R. Hernandez-Pinto and G. Rodrigo, Four-dimensional unsubtraction from the loop-tree duality, JHEP 08 (2016) 160 [arXiv:1604.06699] [INSPIRE].
Z. Capatti, V. Hirschi, D. Kermanschah, A. Pelloni and B. Ruijl, Numerical Loop-Tree Duality: contour deformation and subtraction, JHEP 04 (2020) 096 [arXiv:1912.09291] [INSPIRE].
Z. Capatti, V. Hirschi, A. Pelloni and B. Ruijl, Local Unitarity: a representation of differential cross-sections that is locally free of infrared singularities at any order, arXiv:2010.01068 [INSPIRE].
F.A. Dreyer and A. Karlberg, Vector-Boson Fusion Higgs Production at Three Loops in QCD, Phys. Rev. Lett. 117 (2016) 072001 [arXiv:1606.00840] [INSPIRE].
L. Cieri, X. Chen, T. Gehrmann, E.W.N. Glover and A. Huss, Higgs boson production at the LHC using the qT subtraction formalism at N3LO QCD, JHEP 02 (2019) 096 [arXiv:1807.11501] [INSPIRE].
J. Currie, T. Gehrmann, E.W.N. Glover, A. Huss, J. Niehues and A. Vogt, N3LO corrections to jet production in deep inelastic scattering using the Projection-to-Born method, JHEP 05 (2018) 209 [arXiv:1803.09973] [INSPIRE].
F.A. Dreyer and A. Karlberg, Vector-Boson Fusion Higgs Pair Production at N3LO, Phys. Rev. D 98 (2018) 114016 [arXiv:1811.07906] [INSPIRE].
L. Magnea, E. Maina, G. Pelliccioli, C. Signorile-Signorile, P. Torrielli and S. Uccirati, Local analytic sector subtraction at NNLO, JHEP 12 (2018) 107 [Erratum ibid. 06 (2019) 013] [arXiv:1806.09570] [INSPIRE].
A. Gehrmann-De Ridder, Radiative corrections to the photon +1 jet rate at LEP, Ph.D. thesis, Duhram University, Duhram U.K. (1997).
A. Gehrmann-De Ridder, T. Gehrmann and E.W.N. Glover, Radiative corrections to the photon + 1 jet rate at LEP, Phys. Lett. B 414 (1997) 354 [hep-ph/9705305] [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].
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].
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] [INSPIRE].
V. Del Duca, G. Somogyi and Z. Trócsányi, Integration of collinear-type doubly unresolved counterterms in NNLO jet cross sections, JHEP 06 (2013) 079 [arXiv:1301.3504] [INSPIRE].
F. Caola, M. Delto, H. Frellesvig and K. Melnikov, The double-soft integral for an arbitrary angle between hard radiators, Eur. Phys. J. C 78 (2018) 687 [arXiv:1807.05835] [INSPIRE].
M. Delto and K. Melnikov, Integrated triple-collinear counter-terms for the nested soft-collinear subtraction scheme, JHEP 05 (2019) 148 [arXiv:1901.05213] [INSPIRE].
V. Del Duca, N. Deutschmann and S. Lionetti, Momentum mappings for subtractions at higher orders in QCD, JHEP 12 (2019) 129 [arXiv:1910.01024] [INSPIRE].
A. Bassetto, M. Ciafaloni and G. Marchesini, Jet Structure and Infrared Sensitive Quantities in Perturbative QCD, Phys. Rept. 100 (1983) 201 [INSPIRE].
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] [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].
K.G. Chetyrkin and F.V. Tkachov, Integration by Parts: The Algorithm to Calculate β-functions in 4 Loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].
S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].
V. Smirnov, Feynman integral calculus, Springer, New York U.S.A. (2006).
J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].
J.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys. A 48 (2015) 153001 [arXiv:1412.2296] [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].
C. Anastasiou, L.J. Dixon, K. Melnikov and F. Petriello, Dilepton rapidity distribution in the Drell-Yan process at NNLO in QCD, Phys. Rev. Lett. 91 (2003) 182002 [hep-ph/0306192] [INSPIRE].
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] [INSPIRE].
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] [INSPIRE].
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].
D.A. Kosower, All order collinear behavior in gauge theories, Nucl. Phys. B 552 (1999) 319 [hep-ph/9901201] [INSPIRE].
D.A. Kosower and P. Uwer, One loop splitting amplitudes in gauge theory, Nucl. Phys. B 563 (1999) 477 [hep-ph/9903515] [INSPIRE].
S. Catani and M. Grazzini, The soft gluon current at one loop order, Nucl. Phys. B 591 (2000) 435 [hep-ph/0007142] [INSPIRE].
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] [INSPIRE].
S. Catani, The Singular behavior of QCD amplitudes at two loop order, Phys. Lett. B 427 (1998) 161 [hep-ph/9802439] [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].
E. Gardi and L. Magnea, Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP 03 (2009) 079 [arXiv:0901.1091] [INSPIRE].
E. Gardi and L. Magnea, Infrared singularities in QCD amplitudes, Frascati Phys. Ser. 50 (2010) 137 [arXiv:0908.3273] [INSPIRE].
T. Becher and M. Neubert, Infrared singularities of scattering amplitudes in perturbative QCD, Phys. Rev. Lett. 102 (2009) 162001 [Erratum ibid. 111 (2013) 199905] [arXiv:0901.0722] [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].
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].
T. Huber and D. Maître, HypExp 2, Expanding Hypergeometric Functions about Half-Integer Parameters, Comput. Phys. Commun. 178 (2008) 755 [arXiv:0708.2443] [INSPIRE].
T. Binoth and G. Heinrich, An automatized algorithm to compute infrared divergent multiloop integrals, Nucl. Phys. B 585 (2000) 741 [hep-ph/0004013] [INSPIRE].
G. Heinrich, Sector Decomposition, Int. J. Mod. Phys. A 23 (2008) 1457 [arXiv:0803.4177] [INSPIRE].
S. Borowka et al., pySecDec: a toolbox for the numerical evaluation of multi-scale integrals, Comput. Phys. Commun. 222 (2018) 313 [arXiv:1703.09692] [INSPIRE].
E. Byckling and K. Kajantie, Particle Kinematics, University of Jyvaskyla, Jyvaskyla Finland (1971).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2010.14493
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Magnea, L., Pelliccioli, G., Signorile-Signorile, C. et al. Analytic integration of soft and collinear radiation in factorised QCD cross sections at NNLO. J. High Energ. Phys. 2021, 37 (2021). https://doi.org/10.1007/JHEP02(2021)037
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2021)037