Abstract
We present the analytic calculation of the Master Integrals for the twoloop, non-planar topologies that enter the calculation of the amplitude for top-quark pair hadroproduction in the quark-annihilation channel. Using the method of differential equations, we expand the integrals in powers of the dimensional regulator ϵ and determine the expansion coefficients in terms of generalized harmonic polylogarithms of two dimensionless variables through to weight four.
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P. Bärnreuther, M. Czakon and A. Mitov, Percent level precision physics at the Tevatron: first genuine NNLO QCD corrections to q \( \overline{q} \)→ t \( \overline{t} \)+ X, Phys. Rev. Lett.109 (2012) 132001 [arXiv:1204.5201] [INSPIRE].
M. Czakon and A. Mitov, NNLO corrections to top-pair production at hadron colliders: the all-fermionic scattering channels, JHEP12 (2012) 054 [arXiv:1207.0236] [INSPIRE].
M. Czakon and A. Mitov, NNLO corrections to top pair production at hadron colliders: thequark-gluon reaction, JHEP01 (2013) 080 [arXiv:1210.6832] [INSPIRE].
M. Czakon, P. Fiedler and A. Mitov, Total top-quark pair-production cross section at hadroncolliders through O \( \left({\alpha}_S^4\right) \), Phys. Rev. Lett.110 (2013) 252004 [arXiv:1303.6254] [INSPIRE].
M. Czakon, P. Fiedler and A. Mitov, Resolving the Tevatron top quark forward-backwardasymmetry puzzle: fully differential next-to-next-to-leading-order calculation, Phys. Rev. Lett.115 (2015) 052001 [arXiv:1411.3007] [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].
M. Czakon, D. Heymes and A. Mitov, Dynamical scales for multi-TeV top-pair production at the LHC, JHEP04 (2017) 071 [arXiv:1606.03350] [INSPIRE].
M. Czakon, P. Fiedler, D. Heymes and A. Mitov, NNLO QCD predictions for fully-differential top-quark pair production at the Tevatron, JHEP05 (2016) 034 [arXiv:1601.05375] [INSPIRE].
S. Catani et al., Top-quark pair hadroproduction at next-to-next-to-leading order in QCD, Phys. Rev.D 99 (2019) 051501 [arXiv:1901.04005] [INSPIRE].
M. Czakon et al., Top-pair production at the LHC through NNLO QCD and NLO EW, JHEP10 (2017) 186 [arXiv:1705.04105] [INSPIRE].
S. Dittmaier, P. Uwer and S. Weinzierl, NLO QCD corrections to t \( \overline{t} \)+ jet production at hadron colliders, Phys. Rev. Lett.98 (2007) 262002 [hep-ph/0703120] [INSPIRE].
G. Bevilacqua, M. Czakon, C.G. Papadopoulos and M. Worek, Dominant QCD backgrounds in Higgs boson analyses at the LHC: a study of pp → t \( \overline{t} \)+ 2 jets at next-to-leading order, Phys. Rev. Lett.104 (2010) 162002 [arXiv:1002.4009] [INSPIRE].
G. Bevilacqua, M. Czakon, C.G. Papadopoulos and M. Worek, Hadronic top-quark pair production in association with two jets at Next-to-Leading Order QCD, Phys. Rev. D84 (2011) 114017 [arXiv:1108.2851] [INSPIRE].
K. Melnikov and M. Schulze, NLO QCD corrections to top quark pair production in association with one hard jet at hadron colliders, Nucl. Phys.B 840 (2010) 129 [arXiv:1004.3284] [INSPIRE].
G. Abelof, A. Gehrmann-De Ridder, P. Maierhofer and S. Pozzorini, NNLO QCD subtraction for top-antitop production in the q \( \overline{q} \)channel, JHEP08 (2014) 035 [arXiv:1404.6493] [INSPIRE].
G. Abelof and A. Gehrmann-De Ridder, Light fermionic NNLO QCD corrections to top-antitop production in the quark-antiquark channel, JHEP12 (2014) 076 [arXiv:1409.3148] [INSPIRE].
G. Abelof, A. Gehrmann-De Ridder and I. Majer, Top quark pair production at NNLO in the quark-antiquark channel, JHEP12 (2015) 074 [arXiv:1506.04037] [INSPIRE].
R. Bonciani et al., The q Tsubtraction method for top quark production at hadron colliders, Eur. Phys. J.C 75 (2015) 581 [arXiv:1508.03585] [INSPIRE].
R. Angeles-Martinez, M. Czakon and S. Sapeta, NNLO soft function for top quark pair production at small transverse momentum, JHEP10 (2018) 201 [arXiv:1809.01459] [INSPIRE].
J.G. Korner, Z. Merebashvili and M. Rogal, NNLO O \( \left({\alpha}_s^4\right) \)results for heavy quark pair production in quark-antiquark collisions: The One-loop squared contributions, Phys. Rev.D 77 (2008) 094011 [Erratum ibid.D 85 (2012) 119904] [arXiv:0802.0106] [INSPIRE].
B. Kniehl, Z. Merebashvili, J.G. Korner and M. Rogal, Heavy quark pair production in gluon fusion at next-to-next-to-leading O \( \Big({\alpha_s^4}^{\Big)} \)order: One-loop squared contributions, Phys. Rev.D 78 (2008) 094013 [arXiv:0809.3980] [INSPIRE].
C. Anastasiou and S.M. Aybat, The one-loop gluon amplitude for heavy-quark production at NNLO, Phys. Rev. D78 (2008) 114006 [arXiv:0809.1355] [INSPIRE].
M. Czakon, Tops from light quarks: full mass dependence at two-loops in QCD, Phys. Lett.B 664 (2008) 307 [arXiv:0803.1400] [INSPIRE].
A. Ferroglia, M. Neubert, B.D. Pecjak and L.L. Yang, Two-loop divergences of scattering amplitudes with massive partons, Phys. Rev. Lett.103 (2009) 201601 [arXiv:0907.4791] [INSPIRE].
A. Ferroglia, M. Neubert, B.D. Pecjak and L.L. Yang, Two-loop divergences of massive scattering amplitudes in non-abelian gauge theories, JHEP11 (2009) 062 [arXiv:0908.3676] [INSPIRE]
A. Goncharov, Polylogarithms in arithmetic and geometry, Proc. Int. Congr. Math.1,2 (1995) 374.
A.B. Goncharov, Multiple polylogarithms and mixed Tate motives, math/0103059.
E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys.A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun. 167 (2005) 177 [hep-ph/0410259] [INSPIRE].
R. Bonciani et al., Two-loop fermionic corrections to heavy-quark pair production: the quark-antiquark channel, JHEP07 (2008) 129 [arXiv:0806.2301] [INSPIRE].
R. Bonciani, A. Ferroglia, T. Gehrmann and C. Studerus, Two-loop planar corrections to heavy-quark pair production in the quark-antiquark channel, JHEP08 (2009) 067 [arXiv:0906.3671] [INSPIRE].
P. Bärnreuther, M. Czakon and P. Fiedler, Virtual amplitudes and threshold behaviour of hadronic top-quark pair-production cross sections, JHEP02 (2014) 078 [arXiv:1312.6279] [INSPIRE].
R. Bonciani et al., Two-loop leading color corrections to heavy-quark pair production in the gluon fusion channel, JHEP01 (2011) 102 [arXiv:1011.6661] [INSPIRE].
A. von Manteuffel and C. Studerus, Massive planar and non-planar double box integrals for light N fcontributions to gg → tt, JHEP10 (2013) 037 [arXiv:1306.3504] [INSPIRE].
R. Bonciani et al., Light-quark two-loop corrections to heavy-quark pair production in the gluon fusion channel, JHEP12 (2013) 038 [arXiv:1309.4450] [INSPIRE].
L.-B. Chen and J. Wang, Master integrals of a planar double-box family for top-quark pair production, Phys. Lett.B 792 (2019) 50 [arXiv:1903.04320] [INSPIRE].
A. von Manteuffel and L. Tancredi, A non-planar two-loop three-point function beyond multiple polylogarithms, JHEP06 (2017) 127 [arXiv:1701.05905] [INSPIRE].
L. Adams, E. Chaubey and S. Weinzierl, Planar double box integral for top pair production with a closed top loop to all orders in the dimensional regularization parameter, Phys. Rev. Lett.121 (2018) 142001 [arXiv:1804.11144] [INSPIRE].
L. Adams, E. Chaubey and S. Weinzierl, Analytic results for the planar double box integral relevant to top-pair production with a closed top loop, JHEP10 (2018) 206 [arXiv:1806.04981] [INSPIRE].
R. Bonciani, P. Mastrolia and E. Remiddi, Vertex diagrams for the QED form-factors at the two loop level, Nucl. Phys.B 661 (2003) 289 [Erratum ibid.B 702 (2004) 359] [hep-ph/0301170] [INSPIRE].
R. Bonciani, P. Mastrolia and E. Remiddi, Master integrals for the two loop QCD virtual corrections to the forward backward asymmetry, Nucl. Phys.B 690 (2004) 138 [hep-ph/0311145] [INSPIRE].
R. Bonciani and A. Ferroglia, Two-loop QCD corrections to the heavy-to-light quark decay, JHEP11 (2008) 065 [arXiv:0809.4687] [INSPIRE].
C. Bogner et al., Loopedia, a database for loop integrals, Comput. Phys. Commun.225 (2018) 1 [arXiv:1709.01266] [INSPIRE].
C.M. Carloni Calame, M. Passera, L. Trentadue and G. Venanzoni, A new approach to evaluate the leading hadronic corrections to the muon g-2, Phys. Lett.B 746 (2015) 325 [arXiv:1504.02228] [INSPIRE].
G. Abbiendi et al., Measuring the leading hadronic contribution to the muon g − 2 via μe scattering, Eur. Phys. J.C 77 (2017) 139 [arXiv:1609.08987] [INSPIRE].
P. Mastrolia, M. Passera, A. Primo and U. Schubert, Master integrals for the NNLO virtual corrections to μe scattering in QED: the planar graphs, JHEP11 (2017) 198 [arXiv:1709.07435] [INSPIRE].
S. Di Vita et al., Master integrals for the NNLO virtual corrections to μe scattering in QED: the non-planar graphs, JHEP09 (2018) 016 [arXiv:1806.08241] [INSPIRE].
R.N. Lee and K.T. Mingulov, Master integrals for two-loop C-odd contribution to e +e − → ℓ +ℓ −process, arXiv:1901.04441 [INSPIRE].
C. Anastasiou and A. Lazopoulos, Automatic integral reduction for higher order perturbative calculations, JHEP07 (2004) 046 [hep-ph/0404258] [INSPIRE].
R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
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].
P. Maierhöfer, J. Usovitsch and P. Uwer, Kira — A Feynman integral reduction program, Comput. Phys. Commun.230 (2018) 99 [arXiv:1705.05610] [INSPIRE].
A.V. Smirnov, Algorithm FIRE — Feynman Integral REduction, JHEP10 (2008) 107 [arXiv:0807.3243] [INSPIRE].
A.V. Smirnov and V.A. Smirnov, FIRE4, LiteRed and accompanying tools to solve integration by parts relations, Comput. Phys. Commun.184 (2013) 2820 [arXiv:1302.5885] [INSPIRE].
A.V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun.189 (2015) 182 [arXiv:1408.2372] [INSPIRE].
C. Studerus, Reduze-Feynman Integral Reduction in C++, Comput. Phys. Commun.181 (2010) 1293 [arXiv:0912.2546] [INSPIRE].
A. von Manteuffel and C. Studerus, Reduze 2 — Distributed Feynman integral reduction, arXiv:1201.4330 [INSPIRE].
F.V. Tkachov, A theorem on analytical calculability of four loop renormalization group functions, Phys. Lett.100B (1981) 65 [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].
T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys.B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
A.V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett.B 254 (1991) 158 [INSPIRE].
E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim.A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].
M. Argeri and P. Mastrolia, Feynman diagrams and differential equations, Int. J. Mod. Phys.A 22 (2007) 4375 [arXiv:0707.4037] [INSPIRE].
J.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys. A48 (2015) 153001 [arXiv:1412.2296] [INSPIRE].
J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett.110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].
M. Argeri et al., Magnus and Dyson series for master integrals, JHEP03 (2014) 082 [arXiv:1401.2979] [INSPIRE].
S. Di Vita, P. Mastrolia, U. Schubert and V. Yundin, Three-loop master integrals for ladder-box diagrams with one massive leg, JHEP09 (2014) 148 [arXiv:1408.3107] [INSPIRE].
J.M. Henn, A.V. Smirnov and V.A. Smirnov, Evaluating single-scale and/or non-planar diagrams by differential equations, JHEP03 (2014) 088 [arXiv:1312.2588] [INSPIRE].
T. Gehrmann, A. von Manteuffel, L. Tancredi and E. Weihs, The two-loop master integrals for q \( \overline{q} \)→ V V, JHEP06 (2014) 032 [arXiv:1404.4853] [INSPIRE].
A. von Manteuffel, R.M. Schabinger and H.X. Zhu, The two-loop soft function for heavy quark pair production at future linear colliders, Phys. Rev.D 92 (2015) 045034 [arXiv:1408.5134] [INSPIRE].
R.N. Lee, Reducing differential equations for multiloop master integrals, JHEP04 (2015) 108 [arXiv:1411.0911] [INSPIRE].
L. Adams, E. Chaubey and S. Weinzierl, Simplifying differential equations for multiscale Feynman integrals beyond multiple polylogarithms, Phys. Rev. Lett.118 (2017) 141602 [arXiv:1702.04279] [INSPIRE].
J. Ablinger et al., Algorithms to solve coupled systems of differential equations in terms of power series, PoSLL2016 (2016) 005 [arXiv:1608.05376] [INSPIRE].
C. Meyer, Transforming differential equations of multi-loop Feynman integrals into canonical form, JHEP04 (2017) 006 [arXiv:1611.01087] [INSPIRE].
A. Georgoudis, K.J. Larsen and Y. Zhang, Azurite: an algebraic geometry based package for finding bases of loop integrals, Comput. Phys. Commun.221 (2017) 203 [arXiv:1612.04252] [INSPIRE].
O. Gituliar and V. Magerya, Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form, Comput. Phys. Commun.219 (2017) 329 [arXiv:1701.04269] [INSPIRE].
S. Di Vita et al., Master integrals for the NNLO virtual corrections to q \( \overline{q} \)→ t \( \overline{t} \)scattering in QCD: the non-planar graphs, JHEP06 (2019) 117 [arXiv:1904.10964] [INSPIRE].
T. Gehrmann and E. Remiddi, Analytic continuation of massless two loop four point functions, Nucl. Phys.B 640 (2002) 379 [hep-ph/0207020] [INSPIRE].
C. Meyer, Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA, Comput. Phys. Commun.222 (2018) 295 [arXiv:1705.06252] [INSPIRE].
M. Becchetti and R. Bonciani, Two-loop master integrals for the planar QCD massive corrections to di-photon and di-jet hadro-production, JHEP01 (2018) 048 [arXiv:1712.02537] [INSPIRE].
K.T. Chen, Iterated path integrals, Bull. Am. Math. Soc.83 (1977) 831.
L. Tancredi, private communication (2017).
T. Binoth and G. Heinrich, An automatized algorithm to compute infrared divergent multiloop integrals, Nucl. Phys.B 585 (2000) 741 [hep-ph/0004013] [INSPIRE].
J. Carter and G. Heinrich, SecDec: a general program for sector decomposition, Comput. Phys. Commun.182 (2011) 1566 [arXiv:1011.5493] [INSPIRE].
S. Borowka, J. Carter and G. Heinrich, Numerical evaluation of multi-loop integrals for arbitrary kinematics with SecDec 2.0, Comput. Phys. Commun.184 (2013) 396 [arXiv:1204.4152] [INSPIRE].
S. Borowka et al., SecDec-3.0: numerical evaluation of multi-scale integrals beyond one loop, Comput. Phys. Commun.196 (2015) 470 [arXiv:1502.06595] [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].
A.V. Smirnov and M.N. Tentyukov, Feynman Integral Evaluation by a Sector decomposiTion Approach (FIESTA), Comput. Phys. Commun.180 (2009) 735 [arXiv:0807.4129] [INSPIRE].
A.V. Smirnov, FIESTA 3: cluster-parallelizable multiloop numerical calculations in physical regions, Comput. Phys. Commun.185 (2014) 2090 [arXiv:1312.3186] [INSPIRE].
A.V. Smirnov, FIESTA4: Optimized Feynman integral calculations with GPU support, Comput. Phys. Commun.204 (2016) 189 [arXiv:1511.03614] [INSPIRE].
C. W. Bauer, A. Frink and R. Kreckel, Introduction to the GiNaC framework for symbolic computation within the C++ programming language, math.CS/0004015.
A. von Manteuffel, E. Panzer and R.M. Schabinger, A quasi-finite basis for multi-loop Feynman integrals, JHEP02 (2015) 120 [arXiv:1411.7392] [INSPIRE].
E. Panzer, On hyperlogarithms and Feynman integrals with divergences and many scales, JHEP03 (2014) 071 [arXiv:1401.4361] [INSPIRE].
A. von Manteuffel and R.M. Schabinger, Numerical multi-loop calculations via finite integrals and one-mass EW-QCD Drell-Yan master integrals, JHEP04 (2017) 129 [arXiv:1701.06583] [INSPIRE].
J.A.M. Vermaseren, Axodraw, Comput. Phys. Commun.83 (1994) 45 [INSPIRE].
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Becchetti, M., Bonciani, R., Casconi, V. et al. Master Integrals for the two-loop, non-planar QCD corrections to top-quark pair production in the quark-annihilation channel. J. High Energ. Phys. 2019, 71 (2019). https://doi.org/10.1007/JHEP08(2019)071
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DOI: https://doi.org/10.1007/JHEP08(2019)071