Abstract
We present the first numerical results for the two-loop helicity amplitudes for the scattering of four partons and a W-boson in QCD. We use a finite field sampling method to reduce directly from Feynman diagrams to the coefficients of a set of master integrals after applying integration-by-parts identities. Since the basis of master integrals is not yet fully known analytically, we identify a set of master integrals with a simple divergence structure using local numerator insertions. This allows for accurate numerical evaluation of the amplitude using sector decomposition methods.
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J.R. Andersen et al., Les Houches 2015: Physics at TeV Colliders Standard Model Working Group Report, in 9th Les Houches Workshop on Physics at TeV Colliders (PhysTeV 2015) Les Houches, France, June 1–19, 2015, FERMILAB-CONF-16-175 (2016), [arXiv:1605.04692] [INSPIRE].
J.R. Andersen et al., Les Houches 2017: Physics at TeV Colliders Standard Model Working Group Report, FERMILAB-CONF-18-122 (2018), [arXiv:1803.07977] [INSPIRE].
A. von Manteuffel and R.M. Schabinger, A novel approach to integration by parts reduction, Phys. Lett. B 744 (2015) 101 [arXiv:1406.4513] [INSPIRE].
T. Peraro, Scattering amplitudes over finite fields and multivariate functional reconstruction, JHEP12 (2016) 030 [arXiv:1608.01902] [INSPIRE].
T. Peraro, FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs, JHEP07 (2019) 031 [arXiv:1905.08019] [INSPIRE].
C.G. Papadopoulos, D. Tommasini and C. Wever, The Pentabox Master Integrals with the Simplified Differential Equations approach, JHEP04 (2016) 078 [arXiv:1511.09404] [INSPIRE].
T. Gehrmann, J.M. Henn and N.A. Lo Presti, Pentagon functions for massless planar scattering amplitudes, JHEP10 (2018) 103 [arXiv:1807.09812] [INSPIRE].
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].
S. Badger, C. Brønnum-Hansen, H.B. Hartanto and T. Peraro, Analytic helicity amplitudes for two-loop five-gluon scattering: the single-minus case, JHEP01 (2019) 186 [arXiv:1811.11699] [INSPIRE].
S. Abreu, J. Dormans, F. Febres Cordero, H. Ita and B. Page, Analytic Form of Planar Two-Loop Five-Gluon Scattering Amplitudes in QCD, Phys. Rev. Lett. 122 (2019) 082002 [arXiv:1812.04586] [INSPIRE].
S. Abreu, J. Dormans, F. Febres Cordero, H. Ita, B. Page and V. Sotnikov, Analytic Form of the Planar Two-Loop Five-Parton Scattering Amplitudes in QCD, JHEP05 (2019) 084 [arXiv:1904.00945] [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].
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, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys. B 435 (1995) 59 [hep-ph/9409265] [INSPIRE].
Z. Bern, L.J. Dixon, D.A. Kosower and S. Weinzierl, One loop amplitudes for \( {e}^{+}{e}^{-}\to \overline{q}q\overline{Q}Q \), Nucl. Phys. B 489 (1997) 3 [hep-ph/9610370] [INSPIRE].
Z. Bern, L.J. Dixon and D.A. Kosower, One loop amplitudes for e +e −to four partons, Nucl. Phys. B 513 (1998) 3 [hep-ph/9708239] [INSPIRE].
J.M. Campbell and R.K. Ellis, Next-to-leading order corrections to W +2 jet and Z +2 jet production at hadron colliders, Phys. Rev. D 65 (2002) 113007 [hep-ph/0202176] [INSPIRE].
R.K. Ellis, W.T. Giele, Z. Kunszt, K. Melnikov and G. Zanderighi, One-loop amplitudes for W +3 jet production in hadron collisions, JHEP01 (2009) 012 [arXiv:0810.2762] [INSPIRE].
R.K. Ellis, K. Melnikov and G. Zanderighi, Generalized unitarity at work: first NLO QCD results for hadronic W +3jet production, JHEP04 (2009) 077 [arXiv:0901.4101] [INSPIRE].
C.F. Berger et al., An Automated Implementation of On-Shell Methods for One-Loop Amplitudes, Phys. Rev. D 78 (2008) 036003 [arXiv:0803.4180] [INSPIRE].
C.F. Berger et al., Next-to-Leading Order QCD Predictions for W+3-Jet Distributions at Hadron Colliders, Phys. Rev. D 80 (2009) 074036 [arXiv:0907.1984] [INSPIRE].
C.F. Berger et al., Precise Predictions for W + 4-Jet Production at the Large Hadron Collider, Phys. Rev. Lett. 106 (2011) 092001 [arXiv:1009.2338] [INSPIRE].
Z. Bern et al., Next-to-Leading Order W + 5-Jet Production at the LHC, Phys. Rev. D 88 (2013) 014025 [arXiv:1304.1253] [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].
A.V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].
T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].
S. Abreu, B. Page and M. Zeng, Differential equations from unitarity cuts: nonplanar hexa-box integrals, JHEP01 (2019) 006 [arXiv:1807.11522] [INSPIRE].
J. Böhm, A. Georgoudis, K.J. Larsen, H. Schönemann and Y. Zhang, Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections, JHEP09 (2018) 024 [arXiv:1805.01873] [INSPIRE].
S. Abreu, L.J. Dixon, E. Herrmann, B. Page and M. Zeng, The two-loop five-point amplitude in 𝒩 = 4 super-Yang-Mills theory, Phys. Rev. Lett. 122 (2019) 121603 [arXiv:1812.08941] [INSPIRE].
D. Chicherin, T. Gehrmann, J.M. Henn, N.A. Lo Presti, V. Mitev and P. Wasser, Analytic result for the nonplanar hexa-box integrals, JHEP03 (2019) 042 [arXiv:1809.06240] [INSPIRE].
D. Chicherin, T. Gehrmann, J.M. Henn, P. Wasser, Y. Zhang and S. Zoia, All Master Integrals for Three-Jet Production at Next-to-Next-to-Leading Order, Phys. Rev. Lett. 123 (2019) 041603 [arXiv:1812.11160] [INSPIRE].
J. Gluza, K. Kajda and D.A. Kosower, Towards a Basis for Planar Two-Loop Integrals, Phys. Rev. D 83 (2011) 045012 [arXiv:1009.0472] [INSPIRE].
K.J. Larsen and Y. Zhang, Integration-by-parts reductions from unitarity cuts and algebraic geometry, Phys. Rev. D 93 (2016) 041701 [arXiv:1511.01071] [INSPIRE].
H. Ita, Two-loop Integrand Decomposition into Master Integrals and Surface Terms, Phys. Rev. D 94 (2016) 116015 [arXiv:1510.05626] [INSPIRE].
D.A. Kosower, Direct Solution of Integration-by-Parts Systems, Phys. Rev. D 98 (2018) 025008 [arXiv:1804.00131] [INSPIRE].
R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
A.V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun. 189 (2015) 182 [arXiv:1408.2372] [INSPIRE].
A.V. Smirnov and F.S. Chuharev, FIRE6: Feynman Integral REduction with Modular Arithmetic, arXiv:1901.07808 [INSPIRE].
A. von Manteuffel and C. Studerus, Reduze 2 — Distributed Feynman Integral Reduction, arXiv:1201.4330 [INSPIRE].
P. Maierhöfer, J. Usovitsch and P. Uwer, Kira — A Feynman integral reduction program, Comput. Phys. Commun. 230 (2018) 99 [arXiv:1705.05610] [INSPIRE].
R.H. Boels, Q. Jin and H. Lüo, Efficient integrand reduction for particles with spin, arXiv:1802.06761 [INSPIRE].
H.A. Chawdhry, M.A. Lim and A. Mitov, Two-loop five-point massless QCD amplitudes within the integration-by-parts approach, Phys. Rev. D 99 (2019) 076011 [arXiv:1805.09182] [INSPIRE].
G. Ossola, C.G. Papadopoulos and R. Pittau, Reducing full one-loop amplitudes to scalar integrals at the integrand level, Nucl. Phys. B 763 (2007) 147 [hep-ph/0609007] [INSPIRE].
P. Mastrolia and G. Ossola, On the Integrand-Reduction Method for Two-Loop Scattering Amplitudes, JHEP11 (2011) 014 [arXiv:1107.6041] [INSPIRE].
P. Mastrolia, E. Mirabella, G. Ossola and T. Peraro, Scattering Amplitudes from Multivariate Polynomial Division, Phys. Lett. B 718 (2012) 173 [arXiv:1205.7087] [INSPIRE].
Y. Zhang, Integrand-Level Reduction of Loop Amplitudes by Computational Algebraic Geometry Methods, JHEP09 (2012) 042 [arXiv:1205.5707] [INSPIRE].
S. Badger, H. Frellesvig and Y. Zhang, Hepta-Cuts of Two-Loop Scattering Amplitudes, JHEP04 (2012) 055 [arXiv:1202.2019] [INSPIRE].
P. Mastrolia, E. Mirabella, G. Ossola and T. Peraro, Integrand-Reduction for Two-Loop Scattering Amplitudes through Multivariate Polynomial Division, Phys. Rev. D 87 (2013) 085026 [arXiv:1209.4319] [INSPIRE].
P. Mastrolia, E. Mirabella, G. Ossola and T. Peraro, Multiloop Integrand Reduction for Dimensionally Regulated Amplitudes, Phys. Lett. B 727 (2013) 532 [arXiv:1307.5832] [INSPIRE].
P. Mastrolia, T. Peraro and A. Primo, Adaptive Integrand Decomposition in parallel and orthogonal space, JHEP08 (2016) 164 [arXiv:1605.03157] [INSPIRE].
D.A. Kosower and K.J. Larsen, Maximal Unitarity at Two Loops, Phys. Rev.D 85 (2012) 045017 [arXiv:1108.1180] [INSPIRE].
S. Caron-Huot and K.J. Larsen, Uniqueness of two-loop master contours, JHEP10 (2012) 026 [arXiv:1205.0801] [INSPIRE].
S. Abreu, F. Febres Cordero, H. Ita, M. Jaquier, B. Page and M. Zeng, Two-Loop Four-Gluon Amplitudes from Numerical Unitarity, Phys. Rev. Lett. 119 (2017) 142001 [arXiv:1703.05273] [INSPIRE].
S. Abreu, F. Febres Cordero, H. Ita, M. Jaquier and B. Page, Subleading Poles in the Numerical Unitarity Method at Two Loops, Phys. Rev.D 95 (2017) 096011 [arXiv:1703.05255] [INSPIRE].
S. Badger, C. Brønnum-Hansen, H.B. Hartanto and T. Peraro, First look at two-loop five-gluon scattering in QCD, Phys. Rev. Lett.120 (2018) 092001 [arXiv:1712.02229] [INSPIRE].
S. Abreu, F. Febres Cordero, H. Ita, B. Page and M. Zeng, Planar Two-Loop Five-Gluon Amplitudes from Numerical Unitarity, Phys. Rev. D 97 (2018) 116014 [arXiv:1712.03946] [INSPIRE].
S. Badger et al., Applications of integrand reduction to two-loop five-point scattering amplitudes in QCD, PoS(LL2018) 006 (2018) [arXiv:1807.09709] [INSPIRE].
S. Abreu, F. Febres Cordero, H. Ita, B. Page and V. Sotnikov, Planar Two-Loop Five-Parton Amplitudes from Numerical Unitarity, JHEP11 (2018) 116 [arXiv:1809.09067] [INSPIRE].
D. Chicherin, T. Gehrmann, J.M. Henn, P. Wasser, Y. Zhang and S. Zoia, Analytic result for a two-loop five-particle amplitude, Phys. Rev. Lett.122 (2019) 121602 [arXiv:1812.11057] [INSPIRE].
S. Abreu, L.J. Dixon, E. Herrmann, B. Page and M. Zeng, The two-loop five-point amplitude in 𝒩 = 8 supergravity, JHEP03 (2019) 123 [arXiv:1901.08563] [INSPIRE].
D. Chicherin, T. Gehrmann, J.M. Henn, P. Wasser, Y. Zhang and S. Zoia, The two-loop five-particle amplitude in 𝒩 = 8 supergravity, JHEP03 (2019) 115 [arXiv:1901.05932] [INSPIRE].
S. Badger et al., Analytic form of the full two-loop five-gluon all-plus helicity amplitude, Phys. Rev. Lett. 123 (2019) 071601 [arXiv:1905.03733] [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].
A.V. Smirnov, FIESTA4: Optimized Feynman integral calculations with GPU support, Comput. Phys. Commun. 204 (2016) 189 [arXiv:1511.03614] [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].
T. Gehrmann and E. Remiddi, Two loop master integrals for γ* → 3 jets: The Planar topologies, Nucl. Phys. B 601 (2001) 248 [hep-ph/0008287] [INSPIRE].
T. Gehrmann, A. von Manteuffel and L. Tancredi, The two-loop helicity amplitudes for \( q\overline{q}^{\prime}\to {V}_1{V}_2\to 4 \)leptons, JHEP09 (2015) 128 [arXiv:1503.04812] [INSPIRE].
A. von Manteuffel and L. Tancredi, The two-loop helicity amplitudes for gg → V 1V 2 → 4 leptons, JHEP06 (2015) 197 [arXiv:1503.08835] [INSPIRE].
J.M. Henn, K. Melnikov and V.A. Smirnov, Two-loop planar master integrals for the production of off-shell vector bosons in hadron collisions, JHEP05 (2014) 090 [arXiv:1402.7078] [INSPIRE].
F. Caola, J.M. Henn, K. Melnikov and V.A. Smirnov, Non-planar master integrals for the production of two off-shell vector bosons in collisions of massless partons, JHEP09 (2014) 043 [arXiv:1404.5590] [INSPIRE].
C.G. Papadopoulos, D. Tommasini and C. Wever, Two-loop Master Integrals with the Simplified Differential Equations approach, JHEP01 (2015) 072 [arXiv:1409.6114] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, The All-Loop Integrand For Scattering Amplitudes in Planar N = 4 SYM, JHEP01 (2011) 041 [arXiv:1008.2958] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Local Integrals for Planar Scattering Amplitudes, JHEP06 (2012) 125 [arXiv:1012.6032] [INSPIRE].
P. Nogueira, Automatic Feynman graph generation, J. Comput. Phys.105 (1993) 279 [INSPIRE].
G.’t Hooft and M.J.G. Veltman, Regularization and Renormalization of Gauge Fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].
Z. Bern, A. De Freitas, L.J. Dixon and H.L. Wong, Supersymmetric regularization, two loop QCD amplitudes and coupling shifts, Phys. Rev. D 66 (2002) 085002 [hep-ph/0202271] [INSPIRE].
J. Kuipers, T. Ueda, J.A.M. Vermaseren and J. Vollinga, FORM version 4.0, Comput. Phys. Commun. 184 (2013) 1453 [arXiv:1203.6543] [INSPIRE].
B. Ruijl, T. Ueda and J. Vermaseren, FORM version 4.2, arXiv:1707.06453 [INSPIRE].
G. Cullen, M. Koch-Janusz and T. Reiter, Spinney: A Form Library for Helicity Spinors, Comput. Phys. Commun. 182 (2011) 2368 [arXiv:1008.0803] [INSPIRE].
A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP05 (2013) 135 [arXiv:0905.1473] [INSPIRE].
P.A. Baikov, Explicit solutions of the multiloop integral recurrence relations and its application, Nucl. Instrum. Meth. A 389 (1997) 347 [hep-ph/9611449] [INSPIRE].
J.L. Bourjaily, E. Herrmann and J. Trnka, Prescriptive Unitarity, JHEP06 (2017) 059 [arXiv:1704.05460] [INSPIRE].
S. Catani, The singular behavior of QCD amplitudes at two loop order, Phys. Lett. B 427 (1998) 161 [hep-ph/9802439] [INSPIRE].
T. Becher and M. Neubert, On the Structure of Infrared Singularities of Gauge-Theory Amplitudes, JHEP06 (2009) 081 [Erratum ibid. 11 (2013) 024] [arXiv:0903.1126] [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].
E. Gardi and L. Magnea, Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP03 (2009) 079 [arXiv:0901.1091] [INSPIRE].
E. Panzer, On hyperlogarithms and Feynman integrals with divergences and many scales, JHEP03 (2014) 071 [arXiv:1401.4361] [INSPIRE].
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].
S. Badger, G. Mogull and T. Peraro, Local integrands for two-loop all-plus Yang-Mills amplitudes, JHEP08 (2016) 063 [arXiv:1606.02244] [INSPIRE].
R.K. Ellis and G. Zanderighi, Scalar one-loop integrals for QCD, JHEP02 (2008) 002 [arXiv:0712.1851] [INSPIRE].
C.G. Papadopoulos, Simplified differential equations approach for Master Integrals, JHEP07 (2014) 088 [arXiv:1401.6057] [INSPIRE].
D. Chicherin, J. Henn and V. Mitev, Bootstrapping pentagon functions, JHEP05 (2018) 164 [arXiv:1712.09610] [INSPIRE].
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Hartanto, H.B., Badger, S., Brønnum-Hansen, C. et al. A numerical evaluation of planar two-loop helicity amplitudes for a W-boson plus four partons. J. High Energ. Phys. 2019, 119 (2019). https://doi.org/10.1007/JHEP09(2019)119
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DOI: https://doi.org/10.1007/JHEP09(2019)119