Abstract
In the three-dimensional \(\mathcal{N}\) = 6 Chern-Simons matter (ABJM) theory, the integrand for the logarithm of the scattering amplitude admits a decomposition in terms of negative geometries, which implies that all the infrared divergences concentrate in the last loop integration. We compute the infrared-finite functions that arise from performing a three-loop integration over the four-loop integrand for the logarithm of the four-point amplitude, for which we use the method of differential equations. Our results provide a direct computation of the four-loop cusp anomalous dimension of the theory, in agreement with the current all-loop integrability-based proposal. We find an apparent simplicity in the leading singularities of the integrated results, provided one works in the frame in which the unintegrated loop variable goes to infinity. Finally, our results suggest an alternating sign pattern for the integrated negative geometries in the Euclidean region.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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].
S. Franco, D. Galloni, A. Mariotti and J. Trnka, Anatomy of the Amplituhedron, JHEP 03 (2015) 128 [arXiv:1408.3410] [INSPIRE].
N. Arkani-Hamed, H. Thomas and J. Trnka, Unwinding the Amplituhedron in Binary, JHEP 01 (2018) 016 [arXiv:1704.05069] [INSPIRE].
N. Arkani-Hamed, C. Langer, A. Yelleshpur Srikant and J. Trnka, Deep Into the Amplituhedron: Amplitude Singularities at All Loops and Legs, Phys. Rev. Lett. 122 (2019) 051601 [arXiv:1810.08208] [INSPIRE].
D. Damgaard, L. Ferro, T. Lukowski and M. Parisi, The Momentum Amplituhedron, JHEP 08 (2019) 042 [arXiv:1905.04216] [INSPIRE].
L. Ferro and T. Lukowski, The Loop Momentum Amplituhedron, JHEP 05 (2023) 183 [arXiv:2210.01127] [INSPIRE].
L. Ferro and T. Lukowski, Amplituhedra, and beyond, J. Phys. A 54 (2021) 033001 [arXiv:2007.04342] [INSPIRE].
E. Herrmann and J. Trnka, The SAGEX review on scattering amplitudes Chapter 7: Positive geometry of scattering amplitudes, J. Phys. A 55 (2022) 443008 [arXiv:2203.13018] [INSPIRE].
N. Arkani-Hamed, Y. Bai, S. He and G. Yan, Scattering Forms and the Positive Geometry of Kinematics, Color and the Worldsheet, JHEP 05 (2018) 096 [arXiv:1711.09102] [INSPIRE].
N. Arkani-Hamed, P. Benincasa and A. Postnikov, Cosmological Polytopes and the Wavefunction of the Universe, arXiv:1709.02813 [INSPIRE].
Y.-T. Huang, R. Kojima, C. Wen and S.-Q. Zhang, The orthogonal momentum amplituhedron and ABJM amplitudes, JHEP 01 (2022) 141 [arXiv:2111.03037] [INSPIRE].
S. He, C.-K. Kuo and Y.-Q. Zhang, The momentum amplituhedron of SYM and ABJM from twistor-string maps, JHEP 02 (2022) 148 [arXiv:2111.02576] [INSPIRE].
S. He, C.-K. Kuo, Z. Li and Y.-Q. Zhang, All-Loop Four-Point Aharony-Bergman-Jafferis-Maldacena Amplitudes from Dimensional Reduction of the Amplituhedron, Phys. Rev. Lett. 129 (2022) 221604 [arXiv:2204.08297] [INSPIRE].
S. He, Y.-T. Huang and C.-K. Kuo, The ABJM Amplituhedron, JHEP 09 (2023) 165 [Erratum ibid. 04 (2024) 064] [arXiv:2306.00951] [INSPIRE].
T. Lukowski and J. Stalknecht, Momentum Amplituhedron for N = 6 Chern-Simons-Matter Theory: Scattering Amplitudes from Configurations of Points in Minkowski Space, Phys. Rev. Lett. 131 (2023) 161601 [arXiv:2306.07312] [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
N. Arkani-Hamed et al., Coulomb Branch Amplitudes from a Deformed Amplituhedron Geometry, arXiv:2311.10814 [INSPIRE].
N. Arkani-Hamed, J. Henn and J. Trnka, Nonperturbative negative geometries: amplitudes at strong coupling and the amplituhedron, JHEP 03 (2022) 108 [arXiv:2112.06956] [INSPIRE].
L.F. Alday, E.I. Buchbinder and A.A. Tseytlin, Correlation function of null polygonal Wilson loops with local operators, JHEP 09 (2011) 034 [arXiv:1107.5702] [INSPIRE].
T. Adamo, Correlation functions, null polygonal Wilson loops, and local operators, JHEP 12 (2011) 006 [arXiv:1110.3925] [INSPIRE].
O.T. Engelund and R. Roiban, On correlation functions of Wilson loops, local and non-local operators, JHEP 05 (2012) 158 [arXiv:1110.0758] [INSPIRE].
L.F. Alday, P. Heslop and J. Sikorowski, Perturbative correlation functions of null Wilson loops and local operators, JHEP 03 (2013) 074 [arXiv:1207.4316] [INSPIRE].
L.F. Alday, J.M. Henn and J. Sikorowski, Higher loop mixed correlators in N = 4 SYM, JHEP 03 (2013) 058 [arXiv:1301.0149] [INSPIRE].
J.M. Henn, G.P. Korchemsky and B. Mistlberger, The full four-loop cusp anomalous dimension in \(\mathcal{N}\) = 4 super Yang-Mills and QCD, JHEP 04 (2020) 018 [arXiv:1911.10174] [INSPIRE].
D. Chicherin and J.M. Henn, Symmetry properties of Wilson loops with a Lagrangian insertion, JHEP 07 (2022) 057 [arXiv:2202.05596] [INSPIRE].
J.M. Henn, M. Lagares and S.-Q. Zhang, Integrated negative geometries in ABJM, JHEP 05 (2023) 112 [arXiv:2303.02996] [INSPIRE].
S. He, C.-K. Kuo, Z. Li and Y.-Q. Zhang, Emergent unitarity, all-loop cuts and integrations from the ABJM amplituhedron, JHEP 07 (2023) 212 [arXiv:2303.03035] [INSPIRE].
L.F. Alday and J.M. Maldacena, Gluon scattering amplitudes at strong coupling, JHEP 06 (2007) 064 [arXiv:0705.0303] [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].
A. Brandhuber, P. Heslop and G. Travaglini, MHV amplitudes in N = 4 super Yang-Mills and Wilson loops, Nucl. Phys. B 794 (2008) 231 [arXiv:0707.1153] [INSPIRE].
J.M. Drummond, J. Henn, G.P. Korchemsky and E. Sokatchev, On planar gluon amplitudes/Wilson loops duality, Nucl. Phys. B 795 (2008) 52 [arXiv:0709.2368] [INSPIRE].
J.M. Henn, J. Plefka and K. Wiegandt, Light-like polygonal Wilson loops in 3d Chern-Simons and ABJM theory, JHEP 11 (2010) 053 [Erratum ibid. 11 (2011) 053] [arXiv:1004.0226] [INSPIRE].
M.S. Bianchi et al., Scattering Amplitudes/Wilson Loop Duality In ABJM Theory, JHEP 01 (2012) 056 [arXiv:1107.3139] [INSPIRE].
W.-M. Chen and Y.-T. Huang, Dualities for Loop Amplitudes of N = 6 Chern-Simons Matter Theory, JHEP 11 (2011) 057 [arXiv:1107.2710] [INSPIRE].
M. Leoni, A. Mauri and A. Santambrogio, On the amplitude/Wilson loop duality in N = 2 SCQCD, Phys. Lett. B 747 (2015) 325 [arXiv:1502.07614] [INSPIRE].
N. Gromov and P. Vieira, The all loop AdS4/CFT3 Bethe ansatz, JHEP 01 (2009) 016 [arXiv:0807.0777] [INSPIRE].
D. Chicherin and J. Henn, Pentagon Wilson loop with Lagrangian insertion at two loops in \(\mathcal{N}\) = 4 super Yang-Mills theory, JHEP 07 (2022) 038 [arXiv:2204.00329] [INSPIRE].
R. Hernandez and J.M. Nieto, Holographic correlation functions of hexagon Wilson loops with one local operator, Phys. Lett. B 726 (2013) 417 [arXiv:1301.7220] [INSPIRE].
D.H. Correa, V.I. Giraldo-Rivera and M. Lagares, Integrable Wilson loops in ABJM: a Y-system computation of the cusp anomalous dimension, JHEP 06 (2023) 179 [arXiv:2304.01924] [INSPIRE].
N. Beisert, The Analytic Bethe Ansatz for a Chain with Centrally Extended su(2|2) Symmetry, J. Stat. Mech. 0701 (2007) P01017 [nlin/0610017] [INSPIRE].
J.A. Minahan and K. Zarembo, The Bethe ansatz for superconformal Chern-Simons, JHEP 09 (2008) 040 [arXiv:0806.3951] [INSPIRE].
J.A. Minahan, W. Schulgin and K. Zarembo, Two loop integrability for Chern-Simons theories with N = 6 supersymmetry, JHEP 03 (2009) 057 [arXiv:0901.1142] [INSPIRE].
D. Bak, D. Gang and S.-J. Rey, Integrable Spin Chain of Superconformal U(M) × anti-U(N) Chern-Simons Theory, JHEP 10 (2008) 038 [arXiv:0808.0170] [INSPIRE].
J.A. Minahan, O. Ohlsson Sax and C. Sieg, Magnon dispersion to four loops in the ABJM and ABJ models, J. Phys. A 43 (2010) 275402 [arXiv:0908.2463] [INSPIRE].
J.A. Minahan, O. Ohlsson Sax and C. Sieg, Anomalous dimensions at four loops in N = 6 superconformal Chern-Simons theories, Nucl. Phys. B 846 (2011) 542 [arXiv:0912.3460] [INSPIRE].
M. Leoni et al., Superspace calculation of the four-loop spectrum in N = 6 supersymmetric Chern-Simons theories, JHEP 12 (2010) 074 [arXiv:1010.1756] [INSPIRE].
N. Gromov and G. Sizov, Exact Slope and Interpolating Functions in N = 6 Supersymmetric Chern-Simons Theory, Phys. Rev. Lett. 113 (2014) 121601 [arXiv:1403.1894] [INSPIRE].
A. Cavaglià, N. Gromov and F. Levkovich-Maslyuk, On the Exact Interpolating Function in ABJ Theory, JHEP 12 (2016) 086 [arXiv:1605.04888] [INSPIRE].
T.V. Brown, U. Oktem, S. Paranjape and J. Trnka, Loops of Loops Expansion in the Amplituhedron, arXiv:2312.17736 [INSPIRE].
J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].
A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].
Y.-T. Huang and A.E. Lipstein, Dual Superconformal Symmetry of N = 6 Chern-Simons Theory, JHEP 11 (2010) 076 [arXiv:1008.0041] [INSPIRE].
D. Gang et al., Tree-level Recursion Relation and Dual Superconformal Symmetry of the ABJM Theory, JHEP 03 (2011) 116 [arXiv:1012.5032] [INSPIRE].
T. Bargheer, F. Loebbert and C. Meneghelli, Symmetries of Tree-level Scattering Amplitudes in N = 6 Superconformal Chern-Simons Theory, Phys. Rev. D 82 (2010) 045016 [arXiv:1003.6120] [INSPIRE].
S. Lee, Yangian Invariant Scattering Amplitudes in Supersymmetric Chern-Simons Theory, Phys. Rev. Lett. 105 (2010) 151603 [arXiv:1007.4772] [INSPIRE].
A.V. Smirnov and M. Zeng, FIRE 6.5: Feynman Integral Reduction with New Simplification Library, arXiv:2311.02370 [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].
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].
A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical Polylogarithms for Amplitudes and Wilson Loops, Phys. Rev. Lett. 105 (2010) 151605 [arXiv:1006.5703] [INSPIRE].
A. Brandhuber et al., Two-loop Sudakov Form Factor in ABJM, JHEP 11 (2013) 022 [arXiv:1305.2421] [INSPIRE].
C. Duhr and F. Dulat, PolyLogTools — polylogs for the masses, JHEP 08 (2019) 135 [arXiv:1904.07279] [INSPIRE].
A.M. Polyakov, Gauge Fields as Rings of Glue, Nucl. Phys. B 164 (1980) 171 [INSPIRE].
V.S. Dotsenko and S.N. Vergeles, Renormalizability of Phase Factors in the Nonabelian Gauge Theory, Nucl. Phys. B 169 (1980) 527 [INSPIRE].
R.A. Brandt, F. Neri and M.-A. Sato, Renormalization of Loop Functions for All Loops, Phys. Rev. D 24 (1981) 879 [INSPIRE].
I.A. Korchemskaya and G.P. Korchemsky, On lightlike Wilson loops, Phys. Lett. B 287 (1992) 169 [INSPIRE].
G.P. Korchemsky and A.V. Radyushkin, Infrared factorization, Wilson lines and the heavy quark limit, Phys. Lett. B 279 (1992) 359 [hep-ph/9203222] [INSPIRE].
C. Ahn and R.I. Nepomechie, N = 6 super Chern-Simons theory S-matrix and all-loop Bethe ansatz equations, JHEP 09 (2008) 010 [arXiv:0807.1924] [INSPIRE].
D. Bak and S.-J. Rey, Integrable Spin Chain in Superconformal Chern-Simons Theory, JHEP 10 (2008) 053 [arXiv:0807.2063] [INSPIRE].
D. Gaiotto, S. Giombi and X. Yin, Spin Chains in N = 6 Superconformal Chern-Simons-Matter Theory, JHEP 04 (2009) 066 [arXiv:0806.4589] [INSPIRE].
N. Gromov and A. Sever, Analytic Solution of Bremsstrahlung TBA, JHEP 11 (2012) 075 [arXiv:1207.5489] [INSPIRE].
L. Griguolo, D. Marmiroli, G. Martelloni and D. Seminara, The generalized cusp in ABJ(M) N = 6 Super Chern-Simons theories, JHEP 05 (2013) 113 [arXiv:1208.5766] [INSPIRE].
L.J. Dixon, J.M. Drummond and J.M. Henn, Bootstrapping the three-loop hexagon, JHEP 1 (2011) 023 [arXiv:1108.4461] [INSPIRE].
L.J. Dixon, J.M. Drummond and J.M. Henn, Analytic result for the two-loop six-point NMHV amplitude in N = 4 super Yang-Mills theory, JHEP 01 (2012) 024 [arXiv:1111.1704] [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 N = 4 super-Yang-Mills theory, JHEP 06 (2014) 116 [arXiv:1402.3300] [INSPIRE].
L.J. Dixon, M. von Hippel and A.J. McLeod, The four-loop six-gluon NMHV ratio function, JHEP 01 (2016) 053 [arXiv:1509.08127] [INSPIRE].
S. Caron-Huot, L.J. Dixon, A. McLeod and M. von Hippel, Bootstrapping a Five-Loop Amplitude Using Steinmann Relations, Phys. Rev. Lett. 117 (2016) 241601 [arXiv:1609.00669] [INSPIRE].
J.M. Drummond, G. Papathanasiou and M. Spradlin, A Symbol of Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon, JHEP 03 (2015) 072 [arXiv:1412.3763] [INSPIRE].
L.J. Dixon et al., Heptagons from the Steinmann Cluster Bootstrap, JHEP 02 (2017) 137 [arXiv:1612.08976] [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 et al., Grassmannian Geometry of Scattering Amplitudes, Cambridge University Press (2016) [https://doi.org/10.1017/CBO9781316091548] [INSPIRE].
Y.-T. Huang and C.K. Wen, ABJM amplitudes and the positive orthogonal grassmannian, JHEP 02 (2014) 104 [arXiv:1309.3252] [INSPIRE].
Y.-T. Huang, C. Wen and D. Xie, The positive orthogonal Grassmannian and loop amplitudes of ABJM, J. Phys. A 47 (2014) 474008 [arXiv:1402.1479] [INSPIRE].
T. Bargheer, N. Beisert, F. Loebbert and T. McLoughlin, Conformal Anomaly for Amplitudes in \(\mathcal{N}\) = 6 Superconformal Chern-Simons Theory, J. Phys. A 45 (2012) 475402 [arXiv:1204.4406] [INSPIRE].
M.S. Bianchi et al., One Loop Amplitudes In ABJM, JHEP 07 (2012) 029 [arXiv:1204.4407] [INSPIRE].
A. Brandhuber, G. Travaglini and C. Wen, All one-loop amplitudes in N = 6 superconformal Chern-Simons theory, JHEP 10 (2012) 145 [arXiv:1207.6908] [INSPIRE].
S. Caron-Huot and Y.-T. Huang, The two-loop six-point amplitude in ABJM theory, JHEP 03 (2013) 075 [arXiv:1210.4226] [INSPIRE].
L. Bianchi and M.S. Bianchi, Nonplanarity through unitarity in the ABJM theory, Phys. Rev. D 89 (2014) 125002 [arXiv:1311.6464] [INSPIRE].
M.S. Bianchi and M. Leoni, On the ABJM four-point amplitude at three loops and BDS exponentiation, JHEP 11 (2014) 077 [arXiv:1403.3398] [INSPIRE].
S. He, Y.-T. Huang, C.-K. Kuo and Z. Li, The two-loop eight-point amplitude in ABJM theory, JHEP 02 (2023) 065 [arXiv:2211.01792] [INSPIRE].
M.S. Bianchi et al., ABJM amplitudes and WL at finite N , JHEP 09 (2013) 114 [arXiv:1306.3243] [INSPIRE].
Z. Li, Integrating the full four-loop negative geometries and all-loop ladder-type negative geometries in ABJM theory, arXiv:2402.17023 [INSPIRE].
Acknowledgments
We would like to especially thank Johannes Henn for insightful discussions and guidance. Moreover, we thank Song He, Chia-Kai Kuo, Jungwon Lim, Antonela Matijašić, Julian Miczajka and Giulio Salvatori for useful discussions. We would also like to thank Diego Correa for discussions and useful comments on the manuscript. ML is supported by fellowships from CONICET (Argentina) and DAAD (Germany). This research received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 725110), Novel structures in scattering amplitudes.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2402.17432
Rights and permissions
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.
About this article
Cite this article
Lagares, M., Zhang, SQ. Higher-loop integrated negative geometries in ABJM. J. High Energ. Phys. 2024, 142 (2024). https://doi.org/10.1007/JHEP05(2024)142
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP05(2024)142