zbMATH — the first resource for mathematics

Analytic form of the planar two-loop five-parton scattering amplitudes in QCD. (English) Zbl 1416.81202
Summary: We present the analytic form of all leading-color two-loop five-parton helicity amplitudes in QCD. The results are analytically reconstructed from exact numerical evaluations over finite fields. Combining a judicious choice of variables with a new approach to the treatment of particle states in \(D\) dimensions for the numerical evaluation of amplitudes, we obtain the analytic expressions with a modest computational effort. Their systematic simplification using multivariate partial-fraction decomposition leads to a particularly compact form. Our results provide all two-loop amplitudes required for the calculation of next-to-next-to-leading order QCD corrections to the production of three jets at hadron colliders in the leading-color approximation.

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
Full Text: DOI arXiv
[1] Badger, S.; Frellesvig, H.; Zhang, Y., A Two-Loop Five-Gluon Helicity Amplitude in QCD, JHEP, 12, 045, (2013)
[2] 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].
[3] D.C. Dunbar and W.B. Perkins, Two-loop five-point all plus helicity Yang-Mills amplitude, Phys. Rev.D 93 (2016) 085029 [arXiv:1603.07514] [INSPIRE].
[4] D.C. Dunbar, G.R. Jehu and W.B. Perkins, Two-loop six gluon all plus helicity amplitude, Phys. Rev. Lett.117 (2016) 061602 [arXiv:1605.06351] [INSPIRE].
[5] D.C. Dunbar, J.H. Godwin, G.R. Jehu and W.B. Perkins, Analytic all-plus-helicity gluon amplitudes in QCD, Phys. Rev.D 96 (2017) 116013 [arXiv:1710.10071] [INSPIRE].
[6] 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].
[7] 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].
[8] 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].
[9] Abreu, S.; Febres Cordero, F.; Ita, H.; Page, B.; Sotnikov, V., Planar Two-Loop Five-Parton Amplitudes from Numerical Unitarity, JHEP, 11, 116, (2018)
[10] Peraro, T., Scattering amplitudes over finite fields and multivariate functional reconstruction, JHEP, 12, 030, (2016) · Zbl 1390.81631
[11] 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
[12] 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].
[13] Papadopoulos, CG; Tommasini, D.; Wever, C., The Pentabox Master Integrals with the Simplified Differential Equations approach, JHEP, 04, 078, (2016)
[14] Gehrmann, T.; Henn, JM; Lo Presti, NA, Pentagon functions for massless planar scattering amplitudes, JHEP, 10, 103, (2018) · Zbl 1402.81256
[15] Badger, S.; Mogull, G.; Ochirov, A.; O’Connell, D., A Complete Two-Loop, Five-Gluon Helicity Amplitude in Yang-Mills Theory, JHEP, 10, 064, (2015) · Zbl 1388.81274
[16] S. Abreu, L.J. Dixon, E. Herrmann, B. Page and M. Zeng, The two-loop five-point amplitude in\( \mathcal{N}=4 \)super-Yang-Mills theory, Phys. Rev. Lett.122 (2019) 121603 [arXiv:1812.08941] [INSPIRE].
[17] D. Chicherin, T. Gehrmann, J.M. Henn, P. Wasser, Y. Zhang and S. Zoia, All master integrals for three-jet production at NNLO, arXiv:1812.11160 [INSPIRE].
[18] R.H. Boels, Q. Jin and H. Lüo, Efficient integrand reduction for particles with spin, arXiv:1802.06761 [INSPIRE].
[19] 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].
[20] 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]. · Zbl 1116.81067
[21] Ellis, RK; Giele, WT; Kunszt, Z., A Numerical Unitarity Formalism for Evaluating One-Loop Amplitudes, JHEP, 03, 003, (2008)
[22] Giele, WT; Kunszt, Z.; Melnikov, K., Full one-loop amplitudes from tree amplitudes, JHEP, 04, 049, (2008) · Zbl 1246.81170
[23] 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].
[24] 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].
[25] 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].
[26] 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].
[27] R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in N = 4 super-Yang-Mills, Nucl. Phys.B 725 (2005) 275 [hep-th/0412103] [INSPIRE].
[28] H. Ita, Two-loop Integrand Decomposition into Master Integrals and Surface Terms, Phys. Rev.D 94 (2016) 116015 [arXiv:1510.05626] [INSPIRE].
[29] 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].
[30] Abreu, S.; Febres Cordero, F.; Ita, H.; Jaquier, M.; Page, B.; Zeng, M., Two-Loop Four-Gluon Amplitudes from Numerical Unitarity, Phys. Rev. Lett., 119, 142001, (2017)
[31] 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].
[32] F.R. Anger and V. Sotnikov, On the Dimensional Regularization of QCD Helicity Amplitudes With Quarks, arXiv:1803.11127 [INSPIRE].
[33] R.K. Ellis, W.T. Giele, Z. Kunszt and K. Melnikov, Masses, fermions and generalized D-dimensional unitarity, Nucl. Phys.B 822 (2009) 270 [arXiv:0806.3467] [INSPIRE].
[34] R. Boughezal, K. Melnikov and F. Petriello, The four-dimensional helicity scheme and dimensional reconstruction, Phys. Rev.D 84 (2011) 034044 [arXiv:1106.5520] [INSPIRE].
[35] Hodges, A., Eliminating spurious poles from gauge-theoretic amplitudes, JHEP, 05, 135, (2013) · Zbl 1342.81291
[36] E.W.N. Glover, Two loop QCD helicity amplitudes for massless quark quark scattering, JHEP04 (2004) 021 [hep-ph/0401119] [INSPIRE].
[37] A. De Freitas and Z. Bern, Two-loop helicity amplitudes for quark-quark scattering in QCD and gluino-gluino scattering in supersymmetric Yang-Mills theory, JHEP09 (2004) 039 [hep-ph/0409007] [INSPIRE].
[38] S. Catani, The Singular behavior of QCD amplitudes at two loop order, Phys. Lett.B 427 (1998) 161 [hep-ph/9802439] [INSPIRE].
[39] G.F. Sterman and M.E. Tejeda-Yeomans, Multiloop amplitudes and resummation, Phys. Lett.B 552 (2003) 48 [hep-ph/0210130] [INSPIRE].
[40] 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].
[41] Gardi, E.; Magnea, L., Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP, 03, 079, (2009)
[42] R. Baumeister, D. Mediger, J. Pečovnik and S. Weinzierl, On the vanishing of certain cuts or residues of loop integrals with higher powers of the propagators, arXiv:1903.02286 [INSPIRE].
[43] J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun.167 (2005) 177 [hep-ph/0410259] [INSPIRE]. · Zbl 1196.65045
[44] Goncharov, AB; Spradlin, M.; Vergu, C.; Volovich, A., Classical Polylogarithms for Amplitudes and Wilson Loops, Phys. Rev. Lett., 105, 151605, (2010)
[45] Duhr, C.; Gangl, H.; Rhodes, JR, From polygons and symbols to polylogarithmic functions, JHEP, 10, 075, (2012) · Zbl 1397.81355
[46] Duhr, C., Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP, 08, 043, (2012) · Zbl 1397.16028
[47] D. Cox, J. Little and D. O’Shea, Ideals, Varieties, and Algorithms, Undergraduate Texts in Mathematics, Springer, Cham (2015).
[48] Nogueira, P., Automatic Feynman graph generation, J. Comput. Phys., 105, 279, (1993) · Zbl 0782.68091
[49] Ochirov, A.; Page, B., Full Colour for Loop Amplitudes in Yang-Mills Theory, JHEP, 02, 100, (2017) · Zbl 1377.81108
[50] A. Ochirov and B. Page, Multi-quark colour decompositions from unitarity, in preparation (2019).
[51] W. Decker, G.-M. Greuel, G. Pfister and H. Schönemann, Singular 4-1-0A computer algebra system for polynomial computations, https://www.singular.uni-kl.de/ (2016).
[52] T. Gautier, J.-L. Roch and G. Villard, Givaro, https://casys.gricad-pages.univ-grenoble-alpes.fr/givaro/ (2017).
[53] P. Barrett, Implementing the Rivest Shamir and Adleman Public Key Encryption Algorithm on a Standard Digital Signal Processor, Springer, Berlin, Heidelberg (1987).
[54] J. van der Hoeven, G. Lecerf and G. Quintin, Modular SIMD arithmetic in mathemagix, CoRRabs/1407.3383 (2014) [arXiv:1407.3383].
[55] F.A. Berends and W.T. Giele, Recursive Calculations for Processes with n Gluons, Nucl. Phys.B 306 (1988) 759 [INSPIRE].
[56] F.R. Anger, F. Febres Cordero, H. Ita and V. Sotnikov, NLO QCD predictions for\( Wb\overline{b} \)production in association with up to three light jets at the LHC, Phys. Rev.D 97 (2018) 036018 [arXiv:1712.05721] [INSPIRE].
[57] J.C. Collins, Renormalization, Cambridge Monographs on Mathematical Physics, vol. 26, Cambridge University Press, Cambridge (1986).
[58] M. Kreuzer, Lecture notes: Supersymetry, http://hep.itp.tuwien.ac.at/ kreuzer/inc/susy.pdf (2010).
[59] 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].
[60] 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)
[61] S. Abreu, L.J. Dixon, E. Herrmann, B. Page and M. Zeng, The two-loop five-point amplitude in\( \mathcal{N}= 8 \)supergravity, JHEP03 (2019) 123 [arXiv:1901.08563] [INSPIRE].
[62] D. Chicherin, T. Gehrmann, J.M. Henn, P. Wasser, Y. Zhang and S. Zoia, The two-loop five-particle amplitude in\( \mathcal{N}=8 \)supergravity, JHEP03 (2019) 115 [arXiv:1901.05932] [INSPIRE]. · Zbl 1414.83096
[63] P.S. Wang, A p-adic algorithm for univariate partial fractions, in Proceedings of the Fourth ACM Symposium on Symbolic and Algebraic Computation, SYMSAC ’81, New York, NY, U.S.A., pp. 212-217, ACM (1981) [https://doi.org/10.1145/800206.806398].
[64] E.K. Leinartas, Factorization of rational functions of several variables into partial fractions, Izv. Vyssh. Uchebn. Zaved. Mat. (1978) 47.
[65] A. Raichev, Leinartass partial fraction decomposition, arXiv:1206.4740.
[66] Meyer, C., Transforming differential equations of multi-loop Feynman integrals into canonical form, JHEP, 04, 006, (2017) · Zbl 1378.81064
[67] Zhang, Y., Integrand-Level Reduction of Loop Amplitudes by Computational Algebraic Geometry Methods, JHEP, 09, 042, (2012) · Zbl 1397.81183
[68] 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].
[69] A.V. Smirnov and V.A. Smirnov, Applying Grobner bases to solve reduction problems for Feynman integrals, JHEP01 (2006) 001 [hep-lat/0509187] [INSPIRE].
[70] Z. Bern, L.J. Dixon and D.A. Kosower, One loop corrections to five gluon amplitudes, Phys. Rev. Lett.70 (1993) 2677 [hep-ph/9302280] [INSPIRE].
[71] Z. Kunszt, A. Signer and Z. Trócsányi, One loop radiative corrections to the helicity amplitudes of QCD processes involving four quarks and one gluon, Phys. Lett.B 336 (1994) 529 [hep-ph/9405386] [INSPIRE].
[72] Z. Bern, L.J. Dixon and D.A. Kosower, One loop corrections to two quark three gluon amplitudes, Nucl. Phys.B 437 (1995) 259 [hep-ph/9409393] [INSPIRE].
[73] 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].
[74] B. Ruijl, A. Plaat, J. Vermaseren and J. van den Herik, Why Local Search Excels in Expression Simplification, arXiv:1409.5223 [INSPIRE].
[75] J.L. Bourjaily, Efficient Tree-Amplitudes in N = 4: Automatic BCFW Recursion in Mathematica, arXiv:1011.2447 [INSPIRE].
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.