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Pentagon functions for scattering of five massless particles. (English) Zbl 1457.81126
Summary: We complete the analytic calculation of the full set of two-loop Feynman integrals required for computation of massless five-particle scattering amplitudes. We employ the method of canonical differential equations to construct a minimal basis set of transcendental functions, pentagon functions, which is sufficient to express all planar and nonplanar massless five-point two-loop Feynman integrals in the whole physical phase space. We find analytic expressions for pentagon functions which are manifestly free of unphysical branch cuts. We present a public library for numerical evaluation of pentagon functions suitable for immediate phenomenological applications.
81V05 Strong interaction, including quantum chromodynamics
81T15 Perturbative methods of renormalization applied to problems in quantum field theory
81U20 \(S\)-matrix theory, etc. in quantum theory
81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
Full Text: DOI arXiv
[1] van Hameren, A., OneLOop: For the evaluation of one-loop scalar functions, Comput. Phys. Commun., 182, 2427 (2011) · Zbl 1262.81253
[2] 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
[3] Carrazza, S.; Ellis, RK; Zanderighi, G., QCDLoop: a comprehensive framework for one-loop scalar integrals, Comput. Phys. Commun., 209, 134 (2016) · Zbl 1375.81229
[4] S. Amoroso et al., Les Houches 2019: Physics at TeV Colliders: Standard Model Working Group Report, in 11th Les Houches Workshop on Physics at TeV Colliders: PhysTeV Les Houches, (2020) [arXiv:2003.01700] [INSPIRE].
[5] G. Heinrich, Collider Physics at the Precision Frontier, arXiv:2009.00516 [INSPIRE].
[6] Chetyrkin, KG; Tkachov, FV, Integration by Parts: The Algorithm to Calculate β-functions in 4 Loops, Nucl. Phys. B, 192, 159 (1981)
[7] Chawdhry, HA; Lim, MA; Mitov, A., Two-loop five-point massless QCD amplitudes within the integration-by-parts approach, Phys. Rev. D, 99 (2019)
[8] Peraro, T., Scattering amplitudes over finite fields and multivariate functional reconstruction, JHEP, 12, 030 (2016) · Zbl 1390.81631
[9] Peraro, T., FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs, JHEP, 07, 031 (2019)
[10] Bendle, D., Integration-by-parts reductions of Feynman integrals using Singular and GPI-Space, JHEP, 02, 079 (2020) · Zbl 1435.81076
[11] Wang, Y.; Li, Z.; Ul Basat, N., Direct reduction of multiloop multiscale scattering amplitudes, Phys. Rev. D, 101 (2020)
[12] Klappert, J.; Lange, F., Reconstructing rational functions with FireFly, Comput. Phys. Commun., 247, 106951 (2020)
[13] J. Klappert, S.Y. Klein and F. Lange, Interpolation of Dense and Sparse Rational Functions and other Improvements in FireFly, arXiv:2004.01463 [INSPIRE].
[14] J. Klappert, F. Lange, P. Maierhöfer and J. Usovitsch, Integral Reduction with Kira 2.0 and Finite Field Methods, arXiv:2008.06494 [INSPIRE].
[15] Böhm, J.; Georgoudis, A.; Larsen, KJ; Schönemann, H.; Zhang, Y., Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections, JHEP, 09, 024 (2018) · Zbl 1398.81264
[16] Ita, H., Two-loop Integrand Decomposition into Master Integrals and Surface Terms, Phys. Rev. D, 94, 116015 (2016)
[17] Abreu, S.; Dormans, J.; Febres Cordero, F.; Ita, H.; Page, B., Analytic Form of Planar Two-Loop Five-Gluon Scattering Amplitudes in QCD, Phys. Rev. Lett., 122 (2019) · Zbl 1416.81202
[18] Guan, X.; Liu, X.; Ma, Y-Q, Complete reduction of integrals in two-loop five-light-parton scattering amplitudes, Chin. Phys. C, 44 (2020)
[19] von Manteuffel, A.; Schabinger, RM, A novel approach to integration by parts reduction, Phys. Lett. B, 744, 101 (2015) · Zbl 1330.81151
[20] S. Badger et al., Applications of integrand reduction to two-loop five-point scattering amplitudes in QCD, PoSLL2018 (2018) 006 [arXiv:1807.09709] [INSPIRE].
[21] Badger, S.; Brønnum-Hansen, C.; Hartanto, HB; Peraro, T., First look at two-loop five-gluon scattering in QCD, Phys. Rev. Lett., 120 (2018) · Zbl 1409.81155
[22] Abreu, S.; Febres Cordero, F.; Ita, H.; Page, B.; Zeng, M., Planar Two-Loop Five-Gluon Amplitudes from Numerical Unitarity, Phys. Rev. D, 97, 116014 (2018)
[23] Badger, S.; Brønnum-Hansen, C.; Hartanto, HB; Peraro, T., Analytic helicity amplitudes for two-loop five-gluon scattering: the single-minus case, JHEP, 01, 186 (2019) · Zbl 1409.81155
[24] Abreu, S.; Febres Cordero, F.; Ita, H.; Page, B.; Sotnikov, V., Planar Two-Loop Five-Parton Amplitudes from Numerical Unitarity, JHEP, 11, 116 (2018) · Zbl 1416.81202
[25] Abreu, S.; Dormans, J.; Febres Cordero, F.; Ita, H.; Page, B.; Sotnikov, V., Analytic Form of the Planar Two-Loop Five-Parton Scattering Amplitudes in QCD, JHEP, 05, 084 (2019) · Zbl 1416.81202
[26] Chicherin, D.; Gehrmann, T.; Henn, JM; Wasser, P.; Zhang, Y.; Zoia, S., Analytic result for a two-loop five-particle amplitude, Phys. Rev. Lett., 122, 121602 (2019) · Zbl 1414.83096
[27] Abreu, S.; Dixon, LJ; Herrmann, E.; Page, B.; Zeng, M., The two-loop five-point amplitude in \(\mathcal{N} = 4\) super-Yang-Mills theory, Phys. Rev. Lett., 122, 121603 (2019)
[28] Chicherin, D.; Gehrmann, T.; Henn, JM; Wasser, P.; Zhang, Y.; Zoia, S., The two-loop five-particle amplitude in \(\mathcal{N} = 8\) supergravity, JHEP, 03, 115 (2019) · Zbl 1414.83096
[29] Abreu, S.; Dixon, LJ; Herrmann, E.; Page, B.; Zeng, M., The two-loop five-point amplitude in \(\mathcal{N} = 8\) supergravity, JHEP, 03, 123 (2019) · Zbl 1414.83094
[30] Badger, S., Analytic form of the full two-loop five-gluon all-plus helicity amplitude, Phys. Rev. Lett., 123 (2019)
[31] Dalgleish, AR; Dunbar, DC; Perkins, WB; Strong, JMW, Full color two-loop six-gluon all-plus helicity amplitude, Phys. Rev. D, 101 (2020)
[32] Abreu, S.; Ita, H.; Moriello, F.; Page, B.; Tschernow, W.; Zeng, M., Two-Loop Integrals for Planar Five-Point One-Mass Processes, JHEP, 11, 117 (2020)
[33] Hartanto, HB; Badger, S.; Brønnum-Hansen, C.; Peraro, T., A numerical evaluation of planar two-loop helicity amplitudes for a W-boson plus four partons, JHEP, 09, 119 (2019) · Zbl 1409.81155
[34] Papadopoulos, CG; Wever, C., Internal Reduction method for computing Feynman Integrals, JHEP, 02, 112 (2020)
[35] Chawdhry, HA; Czakon, ML; Mitov, A.; Poncelet, R., NNLO QCD corrections to three-photon production at the LHC, JHEP, 02, 057 (2020)
[36] A.V. Kotikov, Differential equation method: The calculation of N point Feynman diagrams, Phys. Lett. B267 (1991) 123 [Erratum ibid.295 (1992) 409] [INSPIRE]. · Zbl 1020.81734
[37] Kotikov, AV, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B, 254, 158 (1991)
[38] Remiddi, E., Differential equations for Feynman graph amplitudes, Nuovo Cim. A, 110, 1435 (1997)
[39] Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys. B412 (1994) 751 [hep-ph/9306240] [INSPIRE]. · Zbl 1007.81512
[40] T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B580 (2000) 485 [hep-ph/9912329] [INSPIRE]. · Zbl 1071.81089
[41] Henn, JM, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett., 110, 251601 (2013)
[42] Meyer, C., Algorithmic transformation of multi-loop master integrals to a canonical basis with CANONICA, Comput. Phys. Commun., 222, 295 (2018)
[43] Dlapa, C.; Henn, J.; Yan, K., Deriving canonical differential equations for Feynman integrals from a single uniform weight integral, JHEP, 05, 025 (2020)
[44] Henn, J.; Mistlberger, B.; Smirnov, VA; Wasser, P., Constructing d-log integrands and computing master integrals for three-loop four-particle scattering, JHEP, 04, 167 (2020)
[45] J. Chen, X. Xu and L.L. Yang, Constructing Canonical Feynman Integrals with Intersection Theory, arXiv:2008.03045 [INSPIRE].
[46] A.B. Goncharov, Multiple polylogarithms and mixed Tate motives, math/0103059 [INSPIRE]. · Zbl 0919.11080
[47] Goncharov, AB; Spradlin, M.; Vergu, C.; Volovich, A., Classical Polylogarithms for Amplitudes and Wilson Loops, Phys. Rev. Lett., 105, 151605 (2010)
[48] F. Brown, Iterated integrals in quantum field theory, in 6th Summer School on Geometric and Topological Methods for Quantum Field Theory, pp. 188-240, 2013, DOI [INSPIRE].
[49] Chen, K-T, Iterated path integrals, Bull. Am. Math. Soc., 83, 831 (1977) · Zbl 0389.58001
[50] Duhr, C.; Gangl, H.; Rhodes, JR, From polygons and symbols to polylogarithmic functions, JHEP, 10, 075 (2012) · Zbl 1397.81355
[51] T. Gehrmann, J.M. Henn and N.A. Lo Presti, Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD, Phys. Rev. Lett.116 (2016) 062001 [Erratum ibid.116 (2016) 189903] [arXiv:1511.05409] [INSPIRE]. · Zbl 1356.81169
[52] Papadopoulos, CG; Tommasini, D.; Wever, C., The Pentabox Master Integrals with the Simplified Differential Equations approach, JHEP, 04, 078 (2016)
[53] Gehrmann, T.; Henn, JM; Lo Presti, NA, Pentagon functions for massless planar scattering amplitudes, JHEP, 10, 103 (2018) · Zbl 1402.81256
[54] Chicherin, D.; Gehrmann, T.; Henn, JM; Lo Presti, NA; Mitev, V.; Wasser, P., Analytic result for the nonplanar hexa-box integrals, JHEP, 03, 042 (2019) · Zbl 1414.81255
[55] Abreu, S.; Page, B.; Zeng, M., Differential equations from unitarity cuts: nonplanar hexa-box integrals, JHEP, 01, 006 (2019) · Zbl 1409.81157
[56] Chicherin, D.; Gehrmann, T.; Henn, JM; Wasser, P.; Zhang, Y.; Zoia, S., All Master Integrals for Three-Jet Production at Next-to-Next-to-Leading Order, Phys. Rev. Lett., 123 (2019)
[57] T. Gehrmann and E. Remiddi, Two loop master integrals for γ∗ → 3 jets: The planar topologies, Nucl. Phys. B601 (2001) 248 [hep-ph/0008287] [INSPIRE].
[58] T. Gehrmann and E. Remiddi, Two loop master integrals for γ∗ → 3 jets: The nonplanar topologies, Nucl. Phys. B601 (2001) 287 [hep-ph/0101124] [INSPIRE].
[59] Caron-Huot, S.; Chicherin, D.; Henn, J.; Zhang, Y.; Zoia, S., Multi-Regge Limit of the Two-Loop Five-Point Amplitudes in \(\mathcal{N} = 4\) Super Yang-Mills and \(\mathcal{N} = 8\) Supergravity, JHEP, 10, 188 (2020)
[60] Caron-Huot, S.; Henn, JM, Iterative structure of finite loop integrals, JHEP, 06, 114 (2014) · Zbl 1333.81217
[61] Byers, N.; Yang, CN, Physical Regions in Invariant Variables for n Particles and the Phase-Space Volume Element, Rev. Mod. Phys., 36, 595 (1964)
[62] S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A15 (2000) 5087 [hep-ph/0102033] [INSPIRE]. · Zbl 0973.81082
[63] Smirnov, AV; Smirnov, VA, How to choose master integrals, Nucl. Phys. B, 960, 115213 (2020)
[64] J. Usovitsch, Factorization of denominators in integration-by-parts reductions, arXiv:2002.08173 [INSPIRE].
[65] Chicherin, D.; Henn, J.; Mitev, V., Bootstrapping pentagon functions, JHEP, 05, 164 (2018)
[66] Henn, JM; Smirnov, AV; Smirnov, VA, Evaluating single-scale and/or non-planar diagrams by differential equations, JHEP, 03, 088 (2014)
[67] Henn, JM; Mistlberger, B., Four-graviton scattering to three loops in \(\mathcal{N} = 8\) supergravity, JHEP, 05, 023 (2019) · Zbl 1416.83142
[68] Goncharov, AB, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett., 5, 497 (1998) · Zbl 0961.11040
[69] C.W. Bauer, A. Frink and R. Kreckel, Introduction to the GiNaC framework for symbolic computation within the C++ programming language, J. Symb. Comput.33 (2002) 1 [cs/0004015]. · Zbl 1017.68163
[70] Caron-Huot, S.; Dixon, LJ; McLeod, A.; von Hippel, M., Bootstrapping a Five-Loop Amplitude Using Steinmann Relations, Phys. Rev. Lett., 117, 241601 (2016)
[71] Dixon, LJ; Drummond, J.; Harrington, T.; McLeod, AJ; Papathanasiou, G.; Spradlin, M., Heptagons from the Steinmann Cluster Bootstrap, JHEP, 02, 137 (2017) · Zbl 1377.81197
[72] Dixon, LJ; Drummond, JM; Henn, JM, Bootstrapping the three-loop hexagon, JHEP, 11, 023 (2011) · Zbl 1306.81092
[73] Dixon, LJ; Drummond, JM; von Hippel, M.; Pennington, J., Hexagon functions and the three-loop remainder function, JHEP, 12, 049 (2013) · Zbl 1342.81159
[74] Dixon, LJ; Liu, Y-T, Lifting Heptagon Symbols to Functions, JHEP, 10, 031 (2020)
[75] https://gitlab.com/pentagon-functions/PentagonMI.
[76] https://gitlab.com/VasilySotnikov/Li2pp.
[77] A. van Hameren, J. Vollinga and S. Weinzierl, Automated computation of one-loop integrals in massless theories, Eur. Phys. J. C41 (2005) 361 [hep-ph/0502165] [INSPIRE].
[78] Kuipers, J.; Ueda, T.; Vermaseren, JAM, Code Optimization in FORM, Comput. Phys. Commun., 189, 1 (2015) · Zbl 1344.65050
[79] B. Ruijl, T. Ueda and J. Vermaseren, FORM version 4.2, arXiv:1707.06453 [INSPIRE].
[80] Takahasi, H.; Mori, M., Double exponential formulas for numerical integration, Publ. Res. Inst. Math. Sci., 9, 721 (1973) · Zbl 0293.65011
[81] Bailey, DH; Jeyabalan, K.; Li, XS, A comparison of three high-precision quadrature schemes, Exper. Math., 14, 317 (2005) · Zbl 1082.65028
[82] N. Thompson and J. Maddock, Double-exponential quadrature, https://www.boost.org/doc/libs/1_73_0/libs/math/doc/html/math_toolkit/double_exponential.html, (2017).
[83] Y. Hida, S. Li and D. Bailey, Quad-double arithmetic: Algorithms, implementation, and application, http://crd-legacy.lbl.gov/ dhbailey/mpdist/, (2001).
[84] https://gitlab.com/pentagon-functions/PentagonFunctions-cpp.
[85] ATLAS collaboration, Measurement of the production cross section of three isolated photons in pp collisions at \(\sqrt{s} = 8\) TeV using the ATLAS detector, Phys. Lett. B781 (2018) 55 [arXiv:1712.07291] [INSPIRE].
[86] Grazzini, M.; Kallweit, S.; Wiesemann, M., Fully differential NNLO computations with MATRIX, Eur. Phys. J. C, 78, 537 (2018)
[87] S. Abreu et al., Caravel: A C++ Framework for the Computation of Multi-Loop Amplitudes with Numerical Unitarity, arXiv:2009.11957 [INSPIRE].
[88] 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].
[89] Borowka, S.; Heinrich, G.; Jahn, S.; Jones, SP; Kerner, M.; Schlenk, J., A GPU compatible quasi-Monte Carlo integrator interfaced to pySecDec, Comput. Phys. Commun., 240, 120 (2019)
[90] Moriello, F., Generalised power series expansions for the elliptic planar families of Higgs + jet production at two loops, JHEP, 01, 150 (2020)
[91] M. Hidding, DiffExp, a Mathematica package for computing Feynman integrals in terms of one-dimensional series expansions, arXiv:2006.05510 [INSPIRE].
[92] J. Broedel, C. Duhr, F. Dulat and L. Tancredi, Elliptic polylogarithms and iterated integrals on elliptic curves. Part I: general formalism, JHEP05 (2018) 093 [arXiv:1712.07089] [INSPIRE].
[93] Broedel, J.; Duhr, C.; Dulat, F.; Tancredi, L., Elliptic polylogarithms and iterated integrals on elliptic curves II: an application to the sunrise integral, Phys. Rev. D, 97, 116009 (2018)
[94] Adams, L.; Weinzierl, S., Feynman integrals and iterated integrals of modular forms, Commun. Num. Theor. Phys., 12, 193 (2018) · Zbl 1393.81015
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