Abstract
We use generalized unitarity at the integrand-level to directly construct local, manifestly dual-conformally invariant formulae for all two-loop scattering amplitudes in planar, maximally supersymmetric Yang-Mills theory (SYM). This representation separates contributions into manifestly finite and manifestly divergent terms — in a way that renders all infrared-safe observables (including ratio functions) calculable without any need for regulation. These results perfectly match the all-loop BCFW recursion relations, to which we provide a closed-form solution valid through two-loop-order. Finally, we describe and document a Mathematica package which implements these results, available as part of this work’s source files on the arXiv.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
R. Britto, F. Cachazo and B. Feng, Computing one-loop amplitudes from the holomorphic anomaly of unitarity cuts, Phys. Rev. D 71 (2005) 025012 [hep-th/0410179] [INSPIRE].
R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in \( \mathcal{N}=4 \) super-Yang-Mills, Nucl. Phys. B 725 (2005) 275 [hep-th/0412103] [INSPIRE].
F. Cachazo, Sharpening The Leading Singularity, arXiv:0803.1988 [INSPIRE].
J.L. Bourjaily, S. Caron-Huot and J. Trnka, Dual-Conformal Regularization of Infrared Loop Divergences and the Chiral Box Expansion, JHEP 01 (2015) 001 [arXiv:1303.4734] [INSPIRE].
D.A. Kosower and K.J. Larsen, Maximal Unitarity at Two Loops, Phys. Rev. D 85 (2012) 045017 [arXiv:1108.1180] [INSPIRE].
H. Johansson, D.A. Kosower and K.J. Larsen, An Overview of Maximal Unitarity at Two Loops, PoS(LL2012)066 [arXiv:1212.2132] [INSPIRE].
H. Johansson, D.A. Kosower and K.J. Larsen, Two-Loop Maximal Unitarity with External Masses, Phys. Rev. D 87 (2013) 025030 [arXiv:1208.1754] [INSPIRE].
Y. Zhang, Integrand-Level Reduction of Loop Amplitudes by Computational Algebraic Geometry Methods, JHEP 09 (2012) 042 [arXiv:1205.5707] [INSPIRE].
J.M. Henn and V.A. Smirnov, Analytic results for two-loop master integrals for Bhabha scattering I, JHEP 11 (2013) 041 [arXiv:1307.4083] [INSPIRE].
M. Søgaard and Y. Zhang, Multivariate Residues and Maximal Unitarity, JHEP 12 (2013) 008 [arXiv:1310.6006] [INSPIRE].
H. Johansson, D.A. Kosower and K.J. Larsen, Maximal Unitarity for the Four-Mass Double Box, Phys. Rev. D 89 (2014) 125010 [arXiv:1308.4632] [INSPIRE].
S. Badger, H. Frellesvig and Y. Zhang, Multi-loop Integrand Reduction with Computational Algebraic Geometry, J. Phys. Conf. Ser. 523 (2014) 012061 [arXiv:1310.4445] [INSPIRE].
S. Badger, H. Frellesvig and Y. Zhang, Multi-loop integrand reduction techniques, PoS(LL2014)010 [arXiv:1407.3133] [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, JHEP 05 (2014) 090 [arXiv:1402.7078] [INSPIRE].
R. Britto, F. Cachazo and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499 [hep-th/0412308] [INSPIRE].
R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602 [hep-th/0501052] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, S. Caron-Huot and J. Trnka, The All-Loop Integrand For Scattering Amplitudes in Planar \( \mathcal{N}=4 \) SYM, JHEP 01 (2011) 041 [arXiv:1008.2958] [INSPIRE].
J.M. Drummond, J. Henn, V.A. Smirnov and E. Sokatchev, Magic identities for conformal four-point integrals, JHEP 01 (2007) 064 [hep-th/0607160] [INSPIRE].
J.M. Drummond, G.P. Korchemsky and E. Sokatchev, Conformal properties of four-gluon planar amplitudes and Wilson loops, Nucl. Phys. B 795 (2008) 385 [arXiv:0707.0243] [INSPIRE].
J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, Dual superconformal symmetry of scattering amplitudes in \( \mathcal{N}=4 \) super-Yang-Mills theory, Nucl. Phys. B 828 (2010) 317 [arXiv:0807.1095] [INSPIRE].
J.M. Drummond, J.M. Henn and J. Plefka, Yangian symmetry of scattering amplitudes in \( \mathcal{N}=4 \) super Yang-Mills theory, JHEP 05 (2009) 046 [arXiv:0902.2987] [INSPIRE].
J.M. Drummond and J.M. Henn, All tree-level amplitudes in \( \mathcal{N}=4 \) SYM, JHEP 04 (2009) 018 [arXiv:0808.2475] [INSPIRE].
A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].
N. Arkani-Hamed, F. Cachazo, C. Cheung and J. Kaplan, A Duality For The S Matrix, JHEP 03 (2010) 020 [arXiv:0907.5418] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo, A.B. Goncharov, A. Postnikov and J. Trnka, Scattering Amplitudes and the Positive Grassmannian, arXiv:1212.5605 [INSPIRE].
N. Arkani-Hamed and J. Trnka, The Amplituhedron, JHEP 10 (2014) 030 [arXiv:1312.2007] [INSPIRE].
N. Arkani-Hamed and J. Trnka, Into the Amplituhedron, JHEP 12 (2014) 182 [arXiv:1312.7878] [INSPIRE].
N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Local Integrals for Planar Scattering Amplitudes, JHEP 06 (2012) 125 [arXiv:1012.6032] [INSPIRE].
S. Caron-Huot and K.J. Larsen, Uniqueness of two-loop master contours, JHEP 10 (2012) 026 [arXiv:1205.0801] [INSPIRE].
J.L. Bourjaily, Positroids, Plabic Graphs and Scattering Amplitudes in Mathematica, arXiv:1212.6974 [INSPIRE].
P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley Classics Library, John Wiley & Sons Inc., New York U.S.A. (1978).
M. Søgaard and Y. Zhang, Elliptic Functions and Maximal Unitarity, Phys. Rev. D 91 (2015) 081701 [arXiv:1412.5577] [INSPIRE].
L.J. Dixon, J.M. Drummond and J.M. Henn, Bootstrapping the three-loop hexagon, JHEP 11 (2011) 023 [arXiv:1108.4461] [INSPIRE].
L.J. Dixon, J.M. Drummond, M. von Hippel and J. Pennington, Hexagon functions and the three-loop remainder function, JHEP 12 (2013) 049 [arXiv:1308.2276] [INSPIRE].
L.J. Dixon, J.M. Drummond, C. Duhr and J. Pennington, The four-loop remainder function and multi-Regge behavior at NNLLA in planar \( \mathcal{N}=4 \) super-Yang-Mills theory, JHEP 06 (2014) 116 [arXiv:1402.3300] [INSPIRE].
L.J. Dixon, J.M. Drummond, C. Duhr, M. von Hippel and J. Pennington, Bootstrapping six-gluon scattering in planar \( \mathcal{N}=4 \) super-Yang-Mills theory, PoS(LL2014)077 [arXiv:1407.4724] [INSPIRE].
L.J. Dixon and M. von Hippel, Bootstrapping an NMHV amplitude through three loops, JHEP 10 (2014) 065 [arXiv:1408.1505] [INSPIRE].
B. Basso, A. Sever and P. Vieira, Spacetime and Flux Tube S-Matrices at Finite Coupling for \( \mathcal{N}=4 \) Supersymmetric Yang-Mills Theory, Phys. Rev. Lett. 111 (2013) 091602 [arXiv:1303.1396] [INSPIRE].
B. Basso, A. Sever and P. Vieira, Space-time S-matrix and Flux tube S-matrix II. Extracting and Matching Data, JHEP 01 (2014) 008 [arXiv:1306.2058] [INSPIRE].
B. Basso, A. Sever and P. Vieira, Space-time S-matrix and Flux-tube S-matrix III. The two-particle contributions, JHEP 08 (2014) 085 [arXiv:1402.3307] [INSPIRE].
B. Basso, A. Sever and P. Vieira, Space-time S-matrix and Flux-tube S-matrix IV. Gluons and Fusion, JHEP 09 (2014) 149 [arXiv:1407.1736] [INSPIRE].
B. Basso, J. Caetano, L. Córdova, A. Sever and P. Vieira, OPE for all Helicity Amplitudes, JHEP 08 (2015) 018 [arXiv:1412.1132] [INSPIRE].
J. Golden and M. Spradlin, The differential of all two-loop MHV amplitudes in \( \mathcal{N}=4 \) Yang-Mills theory, JHEP 09 (2013) 111 [arXiv:1306.1833] [INSPIRE].
J. Golden, A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Motivic Amplitudes and Cluster Coordinates, JHEP 01 (2014) 091 [arXiv:1305.1617] [INSPIRE].
J. Golden and M. Spradlin, A Cluster Bootstrap for Two-Loop MHV Amplitudes, JHEP 02 (2015) 002 [arXiv:1411.3289] [INSPIRE].
J. Golden, M.F. Paulos, M. Spradlin and A. Volovich, Cluster Polylogarithms for Scattering Amplitudes, J. Phys. A 47 (2014) 474005 [arXiv:1401.6446] [INSPIRE].
L.J. Dixon, J.M. Drummond and J.M. Henn, Analytic result for the two-loop six-point NMHV amplitude in \( \mathcal{N}=4 \) super Yang-Mills theory, JHEP 01 (2012) 024 [arXiv:1111.1704] [INSPIRE].
J.M. Henn, Analytic results for multiloop scattering amplitudes, PoS(LL2012)012 [arXiv:1209.5879] [INSPIRE].
J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].
J.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys. A 48 (2015) 153001 [arXiv:1412.2296] [INSPIRE].
B.L. van der Waerden, Spinoranalyse, Nach. Ges. Wiss. Göttingen Math.-Phys. 1 (1929) 100.
L.J. Mason and D. Skinner, Dual Superconformal Invariance, Momentum Twistors and Grassmannians, JHEP 11 (2009) 045 [arXiv:0909.0250] [INSPIRE].
N. Arkani-Hamed, F. Cachazo and C. Cheung, The Grassmannian Origin Of Dual Superconformal Invariance, JHEP 03 (2010) 036 [arXiv:0909.0483] [INSPIRE].
N. Arkani-Hamed, F. Cachazo and J. Kaplan, What is the Simplest Quantum Field Theory?, JHEP 09 (2010) 016 [arXiv:0808.1446] [INSPIRE].
J.L. Bourjaily, Efficient Tree-Amplitudes in \( \mathcal{N}=4 \) : Automatic BCFW Recursion in Mathematica, arXiv:1011.2447 [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1505.05886
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Bourjaily, J.L., Trnka, J. Local integrand representations of all two-loop amplitudes in planar SYM. J. High Energ. Phys. 2015, 119 (2015). https://doi.org/10.1007/JHEP08(2015)119
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2015)119