Local unitarity: cutting raised propagators and localising renormalisation. (English) Zbl 1534.81094
Summary: The Local Unitarity (LU) representation of differential cross-sections locally realises the cancellations of infrared singularities predicted by the Kinoshita-Lee-Nauenberg theorem. In this work we solve the two remaining challenges to enable practical higher-loop computations within the LU formalism. The first concerns the generalisation of the LU representation to graphs with raised propagators. The solution to this problem results in a generalisation of distributional Cutkosky rules. The second concerns the regularisation of ultraviolet and spurious soft singularities, solved using a fully automated and local renormalisation procedure based on Bogoliubov’s \(R\)-operation. We detail an all-order construction for the hybrid \(\overline{\mathrm{MS}}\) and On-Shell scheme whose only analytic input is single-scale vacuum diagrams. We validate this novel technology by providing (semi-)inclusive results for two multi-leg processes at NLO, study limits of individual supergraphs up to N3LO and present the first physical NNLO cross-sections computed fully numerically in momentum-space, namely for the processes \(\gamma^\ast\rightarrow jj\) and \(\gamma^\ast\to t\bar{t}\).
MSC:
| 81T15 | Perturbative methods of renormalization applied to problems in quantum field theory |
| 81Q30 | Feynman integrals and graphs; applications of algebraic topology and algebraic geometry |
| 81T18 | Feynman diagrams |
| 81V35 | Nuclear physics |
Keywords:
automation; higher-order perturbative calculations; renormalization and regularization; scattering amplitudesSoftware:
MultivariateApart; FMFT; DiffExp; AMFlow; FastJet; Kira; CutTools; pySecDec; FiniteFlow; DiffSharp; FireFly; COLLIER; Ninja; LiteRedReferences:
| [1] | Czakon, M.; Heymes, D., Four-dimensional formulation of the sector-improved residue subtraction scheme, Nucl. Phys. B, 890, 152 (2014) · Zbl 1326.81237 · doi:10.1016/j.nuclphysb.2014.11.006 |
| [2] | Campbell, JM; Ellis, RK; Seth, S., Non-local slicing approaches for NNLO QCD in MCFM, JHEP, 06, 002 (2022) · doi:10.1007/JHEP06(2022)002 |
| [3] | Asteriadis, K.; Caola, F.; Melnikov, K.; Röntsch, R., Analytic results for deep-inelastic scattering at NNLO QCD with the nested soft-collinear subtraction scheme, Eur. Phys. J. C, 80, 8 (2020) · doi:10.1140/epjc/s10052-019-7567-9 |
| [4] | Gehrmann-De Ridder, A.; Gehrmann, T.; Glover, EWN, Antenna subtraction at NNLO, JHEP, 09, 056 (2005) · doi:10.1088/1126-6708/2005/09/056 |
| [5] | Currie, J.; Glover, EWN; Wells, S., Infrared structure at NNLO using antenna subtraction, JHEP, 04, 066 (2013) · doi:10.1007/JHEP04(2013)066 |
| [6] | G. Somogyi and Z. Trócsányi, A new subtraction scheme for computing QCD jet cross sections at next-to-leading order accuracy, hep-ph/0609041 [INSPIRE]. · Zbl 1214.81293 |
| [7] | Somogyi, G., Subtraction with hadronic initial states at NLO: an NNLO-compatible scheme, JHEP, 05, 016 (2009) · doi:10.1088/1126-6708/2009/05/016 |
| [8] | Caola, F.; Melnikov, K.; Röntsch, R., Nested soft-collinear subtractions in NNLO QCD computations, Eur. Phys. J. C, 77, 248 (2017) · doi:10.1140/epjc/s10052-017-4774-0 |
| [9] | 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]. |
| [10] | Boughezal, R.; Focke, C.; Liu, X.; Petriello, F., W -boson production in association with a jet at next-to-next-to-leading order in perturbative QCD, Phys. Rev. Lett., 115 (2015) · doi:10.1103/PhysRevLett.115.062002 |
| [11] | Torres Bobadilla, WJ, May the four be with you: novel IR-subtraction methods to tackle NNLO calculations, Eur. Phys. J. C, 81, 250 (2021) · doi:10.1140/epjc/s10052-021-08996-y |
| [12] | Ossola, G.; Papadopoulos, CG; Pittau, R., CutTools: a program implementing the OPP reduction method to compute one-loop amplitudes, JHEP, 03, 042 (2008) · doi:10.1088/1126-6708/2008/03/042 |
| [13] | Denner, A.; Dittmaier, S.; Hofer, L., Collier: a fortran-based Complex One-Loop LIbrary in Extended Regularizations, Comput. Phys. Commun., 212, 220 (2017) · Zbl 1376.81070 · doi:10.1016/j.cpc.2016.10.013 |
| [14] | Peraro, T., Ninja: automated integrand reduction via Laurent expansion for one-loop amplitudes, Comput. Phys. Commun., 185, 2771 (2014) · Zbl 1360.81021 · doi:10.1016/j.cpc.2014.06.017 |
| [15] | Hirschi, V.; Peraro, T., Tensor integrand reduction via Laurent expansion, JHEP, 06, 060 (2016) · Zbl 1388.81006 · doi:10.1007/JHEP06(2016)060 |
| [16] | von Manteuffel, A.; Schabinger, RM, A novel approach to integration by parts reduction, Phys. Lett. B, 744, 101 (2015) · Zbl 1330.81151 · doi:10.1016/j.physletb.2015.03.029 |
| [17] | Peraro, T., FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs, JHEP, 07, 031 (2019) · doi:10.1007/JHEP07(2019)031 |
| [18] | Klappert, J.; Lange, F., Reconstructing rational functions with FireFly, Comput. Phys. Commun., 247 (2020) · Zbl 1509.68342 · doi:10.1016/j.cpc.2019.106951 |
| [19] | J. Klappert, F. Lange, P. Maierhöfer and J. Usovitsch, Integral reduction with Kira 2.0 and finite field methods, Comput. Phys. Commun.266 (2021) 108024 [arXiv:2008.06494] [INSPIRE]. · Zbl 1523.81078 |
| [20] | Heller, M.; von Manteuffel, A., MultivariateApart: generalized partial fractions, Comput. Phys. Commun., 271 (2022) · Zbl 1524.26024 · doi:10.1016/j.cpc.2021.108174 |
| [21] | Caffo, M.; Czyz, H.; Laporta, S.; Remiddi, E., The master differential equations for the two loop sunrise selfmass amplitudes, Nuovo Cim. A, 111, 365 (1998) |
| [22] | Czakon, ML; Niggetiedt, M., Exact quark-mass dependence of the Higgs-gluon form factor at three loops in QCD, JHEP, 05, 149 (2020) · doi:10.1007/JHEP05(2020)149 |
| [23] | Hidding, M., DiffExp, a Mathematica package for computing Feynman integrals in terms of one-dimensional series expansions, Comput. Phys. Commun., 269 (2021) · Zbl 1518.65150 · doi:10.1016/j.cpc.2021.108125 |
| [24] | Moriello, F., Generalised power series expansions for the elliptic planar families of Higgs + jet production at two loops, JHEP, 01, 150 (2020) · doi:10.1007/JHEP01(2020)150 |
| [25] | X. Liu and Y.-Q. Ma, AMFlow: a Mathematica package for Feynman integrals computation via auxiliary mass flow, arXiv:2201.11669 [INSPIRE]. |
| [26] | Anastasiou, C.; Sterman, G., Removing infrared divergences from two-loop integrals, JHEP, 07, 056 (2019) · doi:10.1007/JHEP07(2019)056 |
| [27] | Anastasiou, C.; Haindl, R.; Sterman, G.; Yang, Z.; Zeng, M., Locally finite two-loop amplitudes for off-shell multi-photon production in electron-positron annihilation, JHEP, 04, 222 (2021) · doi:10.1007/JHEP04(2021)222 |
| [28] | Gong, W.; Nagy, Z.; Soper, DE, Direct numerical integration of one-loop Feynman diagrams for N -photon amplitudes, Phys. Rev. D, 79 (2009) · doi:10.1103/PhysRevD.79.033005 |
| [29] | Becker, S.; Weinzierl, S., Direct numerical integration for multi-loop integrals, Eur. Phys. J. C, 73, 2321 (2013) · doi:10.1140/epjc/s10052-013-2321-1 |
| [30] | Buchta, S.; Chachamis, G.; Draggiotis, P.; Rodrigo, G., Numerical implementation of the loop-tree duality method, Eur. Phys. J. C, 77, 274 (2017) · doi:10.1140/epjc/s10052-017-4833-6 |
| [31] | Z. Capatti, V. Hirschi, D. Kermanschah, A. Pelloni and B. Ruijl, Manifestly causal loop-tree duality, arXiv:2009.05509 [INSPIRE]. · Zbl 1436.81144 |
| [32] | Capatti, Z.; Hirschi, V.; Kermanschah, D.; Pelloni, A.; Ruijl, B., Numerical loop-tree duality: contour deformation and subtraction, JHEP, 04, 096 (2020) · Zbl 1436.81144 · doi:10.1007/JHEP04(2020)096 |
| [33] | C. Anastasiou and K. Melnikov, Higgs boson production at hadron colliders in NNLO QCD, Nucl. Phys. B646 (2002) 220 [hep-ph/0207004] [INSPIRE]. |
| [34] | C. Anastasiou and K. Melnikov, Pseudoscalar Higgs boson production at hadron colliders in NNLO QCD, Phys. Rev. D67 (2003) 037501 [hep-ph/0208115] [INSPIRE]. |
| [35] | Anastasiou, C.; Duhr, C.; Dulat, F.; Herzog, F.; Mistlberger, B., Higgs boson gluon-fusion production in QCD at three loops, Phys. Rev. Lett., 114 (2015) · doi:10.1103/PhysRevLett.114.212001 |
| [36] | Duhr, C.; Dulat, F.; Mistlberger, B., Higgs boson production in bottom-quark fusion to third order in the strong coupling, Phys. Rev. Lett., 125 (2020) · doi:10.1103/PhysRevLett.125.051804 |
| [37] | Anastasiou, C., High precision determination of the gluon fusion Higgs boson cross-section at the LHC, JHEP, 05, 058 (2016) · doi:10.1007/JHEP05(2016)058 |
| [38] | Mistlberger, B., Higgs boson production at hadron colliders at N^3LO in QCD, JHEP, 05, 028 (2018) · doi:10.1007/JHEP05(2018)028 |
| [39] | Chen, L-B; Li, HT; Shao, H-S; Wang, J., Higgs boson pair production via gluon fusion at N^3LO in QCD, Phys. Lett. B, 803 (2020) · doi:10.1016/j.physletb.2020.135292 |
| [40] | Duhr, C.; Dulat, F.; Mistlberger, B., Charged current Drell-Yan production at N^3LO, JHEP, 11, 143 (2020) · doi:10.1007/JHEP11(2020)143 |
| [41] | Duhr, C.; Mistlberger, B., Lepton-pair production at hadron colliders at N^3LO in QCD, JHEP, 03, 116 (2022) · doi:10.1007/JHEP03(2022)116 |
| [42] | Chen, X.; Gehrmann, T.; Glover, EWN; Huss, A.; Mistlberger, B.; Pelloni, A., Fully differential Higgs boson production to third order in QCD, Phys. Rev. Lett., 127 (2021) · doi:10.1103/PhysRevLett.127.072002 |
| [43] | T. Kinoshita, Mass singularities of Feynman amplitudes, J. Math. Phys.3 (1962) 650 [INSPIRE]. · Zbl 0118.44501 |
| [44] | T.D. Lee and M. Nauenberg, Degenerate systems and mass singularities, Phys. Rev.133 (1964) B1549 [INSPIRE]. |
| [45] | F. Bloch and A. Nordsieck, Note on the radiation field of the electron, Phys. Rev.52 (1937) 54 [INSPIRE]. · JFM 63.1393.02 |
| [46] | Capatti, Z.; Hirschi, V.; Pelloni, A.; Ruijl, B., Local unitarity: a representation of differential cross-sections that is locally free of infrared singularities at any order, JHEP, 04, 104 (2021) · doi:10.1007/JHEP04(2021)104 |
| [47] | Capatti, Z., Local unitarity, SciPost Phys. Proc., 7, 024 (2022) · doi:10.21468/SciPostPhysProc.7.024 |
| [48] | Bierenbaum, I.; Buchta, S.; Draggiotis, P.; Malamos, I.; Rodrigo, G., Tree-loop duality relation beyond simple poles, JHEP, 03, 025 (2013) · Zbl 1371.81125 · doi:10.1007/JHEP03(2013)025 |
| [49] | Baumeister, R.; Mediger, D.; Pečovnik, J.; Weinzierl, S., Vanishing of certain cuts or residues of loop integrals with higher powers of the propagators, Phys. Rev. D, 99 (2019) · doi:10.1103/PhysRevD.99.096023 |
| [50] | N. Agarwal, L. Magnea, C. Signorile-Signorile and A. Tripathi, The infrared structure of perturbative gauge theories, arXiv:2112.07099 [INSPIRE]. |
| [51] | D. Kreimer, Outer space as a combinatorial backbone for Cutkosky rules and coactions, Springer (2021) [arXiv:2010.11781] [INSPIRE]. · Zbl 1484.81043 |
| [52] | D. Kreimer and K. Yeats, Algebraic interplay between renormalization and monodromy, arXiv:2105.05948 [INSPIRE]. · Zbl 1156.81033 |
| [53] | M. Berghoff and D. Kreimer, Graph complexes and Feynman rules, arXiv:2008.09540 [INSPIRE]. |
| [54] | Catani, S.; Gleisberg, T.; Krauss, F.; Rodrigo, G.; Winter, J-C, From loops to trees by-passing Feynman’s theorem, JHEP, 09, 065 (2008) · Zbl 1245.81117 · doi:10.1088/1126-6708/2008/09/065 |
| [55] | Bierenbaum, I.; Catani, S.; Draggiotis, P.; Rodrigo, G., A tree-loop duality relation at two loops and beyond, JHEP, 10, 073 (2010) · Zbl 1291.81381 · doi:10.1007/JHEP10(2010)073 |
| [56] | Capatti, Z.; Hirschi, V.; Kermanschah, D.; Ruijl, B., Loop-tree duality for multiloop numerical integration, Phys. Rev. Lett., 123 (2019) · Zbl 1436.81144 · doi:10.1103/PhysRevLett.123.151602 |
| [57] | R. Runkel, Z. Szőr, J.P. Vesga and S. Weinzierl, Causality and loop-tree duality at higher loops, Phys. Rev. Lett.122 (2019) 111603 [Erratum ibid.123 (2019) 059902] [arXiv:1902.02135] [INSPIRE]. |
| [58] | Jesús Aguilera-Verdugo, J.; Hernández-Pinto, RJ; Rodrigo, G.; Sborlini, GFR; Torres Bobadilla, WJ, Mathematical properties of nested residues and their application to multi-loop scattering amplitudes, JHEP, 02, 112 (2021) · doi:10.1007/JHEP02(2021)112 |
| [59] | Aguilera-Verdugo, JJ, Open loop amplitudes and causality to all orders and powers from the loop-tree duality, Phys. Rev. Lett., 124 (2020) · doi:10.1103/PhysRevLett.124.211602 |
| [60] | G. Sterman, An introduction to quantum field theory, Cambridge University Press (1993). |
| [61] | M.D. Schwartz, Quantum field theory and the Standard Model, Cambridge University Press (2014). |
| [62] | Mantovani, L.; Pasquini, B.; Xiong, X.; Bacchetta, A., Revisiting the equivalence of light-front and covariant QED in the light-cone gauge, Phys. Rev. D, 94 (2016) · doi:10.1103/PhysRevD.94.116005 |
| [63] | Bourjaily, JL; Hannesdottir, H.; McLeod, AJ; Schwartz, MD; Vergu, C., Sequential discontinuities of Feynman integrals and the monodromy group, JHEP, 01, 205 (2021) · Zbl 1459.81045 · doi:10.1007/JHEP01(2021)205 |
| [64] | R.P. Feynman, Quantum theory of gravitation, Acta Phys. Polon.24 (1963) 697 [INSPIRE]. |
| [65] | Catani, S.; Gleisberg, T.; Krauss, F.; Rodrigo, G.; Winter, J-C, From loops to trees by-passing Feynman’s theorem, JHEP, 09, 065 (2008) · Zbl 1245.81117 · doi:10.1088/1126-6708/2008/09/065 |
| [66] | Plätzer, S.; Ruffa, I., Towards colour flow evolution at two loops, JHEP, 06, 007 (2021) · Zbl 1466.81137 · doi:10.1007/JHEP06(2021)007 |
| [67] | Z. Capatti, V. Hirschi and B. Ruijl, Local unitarity: a KLN-based approach to hadronic cross-sections, to appear. |
| [68] | R.E. Cutkosky, Singularities and discontinuities of Feynman amplitudes, J. Math. Phys.1 (1960) 429 [INSPIRE]. · Zbl 0122.22605 |
| [69] | A.G. Baydin, B.A. Pearlmutter, A.A. Radul and J.M. Siskind, Automatic differentiation in machine learning: a survey, J. Mach. Learn. Res.18 (2018) 1. · Zbl 06982909 |
| [70] | N.N. Bogoliubov and O.S. Parasiuk, On the multiplication of the causal function in the quantum theory of fields, Acta Math.97 (1957) 227 [INSPIRE]. · Zbl 0081.43302 |
| [71] | W.E. Caswell and A.D. Kennedy, A simple approach to renormalization theory, Phys. Rev. D25 (1982) 392 [INSPIRE]. · Zbl 1267.81251 |
| [72] | W. Zimmermann, Convergence of Bogolyubov’s method of renormalization in momentum space, Commun. Math. Phys.15 (1969) 208 [Lect. Notes Phys.558 (2000) 217] [INSPIRE]. · Zbl 0192.61203 |
| [73] | K. Hepp, Proof of the Bogolyubov-Parasiuk theorem on renormalization, Commun. Math. Phys.2 (1966) 301 [INSPIRE]. · Zbl 1222.81219 |
| [74] | Herzog, F., Zimmermann’s forest formula, infrared divergences and the QCD β-function, Nucl. Phys. B, 926, 370 (2018) · Zbl 1380.81229 · doi:10.1016/j.nuclphysb.2017.11.011 |
| [75] | K.G. Chetyrkin and F.V. Tkachov, Infrared R operation and ultraviolet counterterms in the MS scheme, Phys. Lett. B114 (1982) 340 [INSPIRE]. |
| [76] | K.G. Chetyrkin and V.A. Smirnov, R∗ operation corrected, Phys. Lett. B144 (1984) 419 [INSPIRE]. |
| [77] | V.A. Smirnov and K.G. Chetyrkin, R∗ operation in the minimal subtraction scheme, Theor. Math. Phys.63 (1985) 462 [Teor. Mat. Fiz.63 (1985) 208] [INSPIRE]. |
| [78] | Herzog, F.; Ruijl, B., The R∗-operation for Feynman graphs with generic numerators, JHEP, 05, 037 (2017) · Zbl 1380.81133 · doi:10.1007/JHEP05(2017)037 |
| [79] | K.G. Chetyrkin, Combinatorics of R-, R^−1-, and R∗-operations and asymptotic expansions of Feynman integrals in the limit of large momenta and masses, arXiv:1701.08627 [INSPIRE]. |
| [80] | J.H. Lowenstein and W. Zimmermann, On the formulation of theories with zero mass propagators, Nucl. Phys. B86 (1975) 77 [INSPIRE]. |
| [81] | A.A. Vladimirov, Method for computing renormalization group functions in dimensional renormalization scheme, Theor. Math. Phys.43 (1980) 417 [Teor. Mat. Fiz.43 (1980) 210] [INSPIRE]. |
| [82] | M. Gomes, J.H. Lowenstein and W. Zimmermann, Generalization of the momentum-space subtraction procedure for renormalized perturbation theory, Commun. Math. Phys.39 (1974) 81 [INSPIRE]. |
| [83] | J.H. Lowenstein, Convergence theorems for renormalized Feynman integrals with zero-mass propagators, Commun. Math. Phys.47 (1976) 53 [INSPIRE]. · Zbl 0323.46070 |
| [84] | Pikelner, A., FMFT: Fully Massive Four-loop Tadpoles, Comput. Phys. Commun., 224, 282 (2018) · Zbl 07694311 · doi:10.1016/j.cpc.2017.11.017 |
| [85] | B. Ruijl, F. Herzog, T. Ueda, J.A.M. Vermaseren and A. Vogt, The R∗-operation and five-loop calculations, PoSRADCOR2017 (2018) 011 [J. Phys. Conf. Ser.1085 (2018) 052006] [arXiv:1801.06084] [INSPIRE]. |
| [86] | K.G. Chetyrkin and F.V. Tkachov, Integration by parts: the algorithm to calculate β-functions in 4 loops, Nucl. Phys. B192 (1981) 159 [INSPIRE]. |
| [87] | R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE]. |
| [88] | R.N. Lee, LiteRed 1.4: a powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser.523 (2014) 012059 [arXiv:1310.1145] [INSPIRE]. |
| [89] | Kniehl, BA; Pikelner, AF; Veretin, OL, Three-loop massive tadpoles and polylogarithms through weight six, JHEP, 08, 024 (2017) · doi:10.1007/JHEP08(2017)024 |
| [90] | Martin, SP; Robertson, DG, Evaluation of the general 3-loop vacuum Feynman integral, Phys. Rev. D, 95 (2017) · doi:10.1103/PhysRevD.95.016008 |
| [91] | Borowka, S., pySecDec: a toolbox for the numerical evaluation of multi-scale integrals, Comput. Phys. Commun., 222, 313 (2018) · Zbl 07693053 · doi:10.1016/j.cpc.2017.09.015 |
| [92] | Denner, A., Techniques for calculation of electroweak radiative corrections at the one loop level and results for W physics at LEP-200, Fortsch. Phys., 41, 307 (1993) |
| [93] | Lang, J-N; Pozzorini, S.; Zhang, H.; Zoller, MF, Two-loop rational terms in Yang-Mills theories, JHEP, 10, 016 (2020) · doi:10.1007/JHEP10(2020)016 |
| [94] | Pozzorini, S.; Zhang, H.; Zoller, MF, Rational terms of UV origin at two loops, JHEP, 05, 077 (2020) · doi:10.1007/JHEP05(2020)077 |
| [95] | Ossola, G.; Papadopoulos, CG; Pittau, R., On the rational terms of the one-loop amplitudes, JHEP, 05, 004 (2008) · doi:10.1088/1126-6708/2008/05/004 |
| [96] | Blaschke, DN; Gieres, F.; Heindl, F.; Schweda, M.; Wohlgenannt, M., BPHZ renormalization and its application to non-commutative field theory, Eur. Phys. J. C, 73, 2566 (2013) · doi:10.1140/epjc/s10052-013-2566-8 |
| [97] | Weinzierl, S., Review on loop integrals which need regularization but yield finite results, Mod. Phys. Lett. A, 29, 1430015 (2014) · Zbl 1291.81178 · doi:10.1142/S0217732314300158 |
| [98] | Signer, A.; Stöckinger, D., Using dimensional reduction for hadronic collisions, Nucl. Phys. B, 808, 88 (2009) · Zbl 1192.81404 · doi:10.1016/j.nuclphysb.2008.09.016 |
| [99] | Alwall, J., The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations, JHEP, 07, 079 (2014) · Zbl 1402.81011 · doi:10.1007/JHEP07(2014)079 |
| [100] | J.C. Collins, F. Wilczek and A. Zee, Low-energy manifestations of heavy particles: application to the neutral current, Phys. Rev. D18 (1978) 242 [INSPIRE]. |
| [101] | W. Bernreuther and W. Wetzel, Decoupling of heavy quarks in the minimal subtraction scheme, Nucl. Phys. B197 (1982) 228 [Erratum ibid.513 (1998) 758] [INSPIRE]. |
| [102] | G.M. Prosperi, M. Raciti and C. Simolo, On the running coupling constant in QCD, Prog. Part. Nucl. Phys.58 (2007) 387 [hep-ph/0607209] [INSPIRE]. |
| [103] | I.V. Tyutin, Gauge invariance in field theory and statistical physics in operator formalism, arXiv:0812.0580 [INSPIRE]. |
| [104] | C. Becchi, A. Rouet and R. Stora, Renormalization of gauge theories, Annals Phys.98 (1976) 287 [INSPIRE]. |
| [105] | T. Kugo and I. Ojima, Local covariant operator formalism of non-Abelian gauge theories and quark confinement problem, Prog. Theor. Phys. Suppl.66 (1979) 1 [INSPIRE]. · Zbl 1098.81592 |
| [106] | Herzog, F.; Ruijl, B.; Ueda, T.; Vermaseren, JAM; Vogt, A., On Higgs decays to hadrons and the R-ratio at N^4LO, JHEP, 08, 113 (2017) · doi:10.1007/JHEP08(2017)113 |
| [107] | K.G. Chetyrkin, J.H. Kühn and M. Steinhauser, Three loop polarization function and \(O\left({\alpha}_S^2\right)\) corrections to the production of heavy quarks, Nucl. Phys. B482 (1996) 213 [hep-ph/9606230] [INSPIRE]. |
| [108] | K.G. Chetyrkin, A.H. Hoang, J.H. Kühn, M. Steinhauser and T. Teubner, Double bubble corrections to heavy quark production, Phys. Lett. B384 (1996) 233 [hep-ph/9603313] [INSPIRE]. |
| [109] | Y. Kiyo, A. Maier, P. Maierhofer and P. Marquard, Reconstruction of heavy quark current correlators at \(O\left({\alpha}_s^3\right) \), Nucl. Phys. B823 (2009) 269 [arXiv:0907.2120] [INSPIRE]. · Zbl 1196.81240 |
| [110] | Chen, L.; Dekkers, O.; Heisler, D.; Bernreuther, W.; Si, Z-G, Top-quark pair production at next-to-next-to-leading order QCD in electron positron collisions, JHEP, 12, 098 (2016) |
| [111] | Bernreuther, W.; Chen, L.; Dekkers, O.; Gehrmann, T.; Heisler, D., The forward-backward asymmetry for massive bottom quarks at the Z peak at next-to-next-to-leading order QCD, JHEP, 01, 053 (2017) · Zbl 1373.81373 · doi:10.1007/JHEP01(2017)053 |
| [112] | Wang, S-Q; Meng, R-Q; Wu, X-G; Chen, L.; Shen, J-M, Revisiting the bottom quark forward-backward asymmetry A_FBin electron-positron collisions, Eur. Phys. J. C, 80, 649 (2020) · doi:10.1140/epjc/s10052-020-8234-x |
| [113] | L. Chen, Forward-backward asymmetries of the heavy quark pair production in e^+ e^− collisions at \(O\left({\alpha}_s^2\right) \), in International workshop on future linear colliders, (2021) [arXiv:2105.06213] [INSPIRE]. |
| [114] | D.E. Soper, QCD calculations by numerical integration, Phys. Rev. Lett.81 (1998) 2638 [hep-ph/9804454] [INSPIRE]. |
| [115] | D.E. Soper, Techniques for QCD calculations by numerical integration, Phys. Rev. D62 (2000) 014009 [hep-ph/9910292] [INSPIRE]. |
| [116] | S.D. Ellis and D.E. Soper, Successive combination jet algorithm for hadron collisions, Phys. Rev. D48 (1993) 3160 [hep-ph/9305266] [INSPIRE]. |
| [117] | M. Cacciari, FastJet: a code for fast k_tclustering, and more, in 41^strencontres de Moriond: QCD and hadronic interactions, (2006), p. 487 [hep-ph/0607071] [INSPIRE]. |
| [118] | Kermanschah, D., Numerical integration of loop integrals through local cancellation of threshold singularities, JHEP, 01, 151 (2022) · Zbl 1521.65017 · doi:10.1007/JHEP01(2022)151 |
| [119] | Del Duca, V.; Deutschmann, N.; Lionetti, S., Momentum mappings for subtractions at higher orders in QCD, JHEP, 12, 129 (2019) · doi:10.1007/JHEP12(2019)129 |
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.
