Abstract
We obtain local numerators satisfying the BCJ color-kinematics duality at one loop for super-Yang-Mills theory in ten dimensions. This is done explicitly for six points via the field-theory limit of the genus-one open superstring correlators for different color orderings, in an analogous manner to an earlier derivation of local BCJ-satisfying numerators at tree level from disk correlators. These results solve an outstanding puzzle from a previous analysis where the six-point numerators did not satisfy the color-kinematics duality.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Z. Bern, J.J. Carrasco, M. Chiodaroli, H. Johansson and R. Roiban, The Duality Between Color and Kinematics and its Applications, arXiv:1909.01358 [INSPIRE].
Z. Bern, J.J. Carrasco, W.-M. Chen, H. Johansson and R. Roiban, Gravity Amplitudes as Generalized Double Copies of Gauge-Theory Amplitudes, Phys. Rev. Lett. 118 (2017) 181602 [arXiv:1701.02519] [INSPIRE].
Z. Bern, J.J.M. Carrasco, W.-M. Chen, H. Johansson, R. Roiban and M. Zeng, Five-loop four-point integrand of N = 8 supergravity as a generalized double copy, Phys. Rev. D 96 (2017) 126012 [arXiv:1708.06807] [INSPIRE].
N.E.J. Bjerrum-Bohr, P.H. Damgaard and P. Vanhove, Minimal Basis for Gauge Theory Amplitudes, Phys. Rev. Lett. 103 (2009) 161602 [arXiv:0907.1425] [INSPIRE].
S. Stieberger, Open & Closed vs. Pure Open String Disk Amplitudes, arXiv:0907.2211 [INSPIRE].
Z. Bern, J.J.M. Carrasco and H. Johansson, New Relations for Gauge-Theory Amplitudes, Phys. Rev. D 78 (2008) 085011 [arXiv:0805.3993] [INSPIRE].
N. Berkovits, Super Poincaré covariant quantization of the superstring, JHEP 04 (2000) 018 [hep-th/0001035] [INSPIRE].
N. Berkovits, ICTP lectures on covariant quantization of the superstring, ICTP Lect. Notes Ser. 13 (2003) 57 [hep-th/0209059] [INSPIRE].
J.A.M. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE].
M. Tentyukov and J.A.M. Vermaseren, The Multithreaded version of FORM, Comput. Phys. Commun. 181 (2010) 1419 [hep-ph/0702279] [INSPIRE].
C.R. Mafra and O. Schlotterer, Towards one-loop SYM amplitudes from the pure spinor BRST cohomology, Fortsch. Phys. 63 (2015) 105 [arXiv:1410.0668] [INSPIRE].
Z. Bern and D.A. Kosower, Efficient calculation of one loop QCD amplitudes, Phys. Rev. Lett. 66 (1991) 1669 [INSPIRE].
Z. Bern and D.A. Kosower, The Computation of loop amplitudes in gauge theories, Nucl. Phys. B 379 (1992) 451 [INSPIRE].
M.J. Strassler, Field theory without Feynman diagrams: One loop effective actions, Nucl. Phys. B 385 (1992) 145 [hep-ph/9205205] [INSPIRE].
Z. Bern, D.C. Dunbar and T. Shimada, String based methods in perturbative gravity, Phys. Lett. B 312 (1993) 277 [hep-th/9307001] [INSPIRE].
D.C. Dunbar and P.S. Norridge, Calculation of graviton scattering amplitudes using string based methods, Nucl. Phys. B 433 (1995) 181 [hep-th/9408014] [INSPIRE].
C. Schubert, Perturbative quantum field theory in the string inspired formalism, Phys. Rept. 355 (2001) 73 [hep-th/0101036] [INSPIRE].
N.E.J. Bjerrum-Bohr and P. Vanhove, Absence of Triangles in Maximal Supergravity Amplitudes, JHEP 10 (2008) 006 [arXiv:0805.3682] [INSPIRE].
C.R. Mafra and O. Schlotterer, Towards the n-point one-loop superstring amplitude. Part I. Pure spinors and superfield kinematics, JHEP 08 (2019) 090 [arXiv:1812.10969] [INSPIRE].
C.R. Mafra and O. Schlotterer, Towards the n-point one-loop superstring amplitude. Part II. Worldsheet functions and their duality to kinematics, JHEP 08 (2019) 091 [arXiv:1812.10970] [INSPIRE].
C.R. Mafra and O. Schlotterer, Towards the n-point one-loop superstring amplitude. Part III. One-loop correlators and their double-copy structure, JHEP 08 (2019) 092 [arXiv:1812.10971] [INSPIRE].
C.R. Mafra, Berends-Giele recursion for double-color-ordered amplitudes, JHEP 07 (2016) 080 [arXiv:1603.09731] [INSPIRE].
C.R. Mafra and O. Schlotterer, One-loop open-string integrals from differential equations: all-order α’-expansions at n points, JHEP 03 (2020) 007 [arXiv:1908.10830] [INSPIRE].
C.R. Mafra and O. Schlotterer, Multiparticle SYM equations of motion and pure spinor BRST blocks, JHEP 07 (2014) 153 [arXiv:1404.4986] [INSPIRE].
N. Berkovits, Multiloop amplitudes and vanishing theorems using the pure spinor formalism for the superstring, JHEP 09 (2004) 047 [hep-th/0406055] [INSPIRE].
N. Berkovits and C.R. Mafra, Some Superstring Amplitude Computations with the Non-Minimal Pure Spinor Formalism, JHEP 11 (2006) 079 [hep-th/0607187] [INSPIRE].
C.R. Mafra and O. Schlotterer, One-loop superstring six-point amplitudes and anomalies in pure spinor superspace, JHEP 04 (2016) 148 [arXiv:1603.04790] [INSPIRE].
C.R. Mafra, O. Schlotterer and S. Stieberger, Explicit BCJ Numerators from Pure Spinors, JHEP 07 (2011) 092 [arXiv:1104.5224] [INSPIRE].
M. Bianchi, A.L. Guerrieri, Y.-t. Huang, C.-J. Lee and C. Wen, Exploring soft constraints on effective actions, JHEP 10 (2016) 036 [arXiv:1605.08697] [INSPIRE].
E. Casali, S. Mizera and P. Tourkine, Loop amplitudes monodromy relations and color-kinematics duality, JHEP 03 (2021) 048 [arXiv:2005.05329] [INSPIRE].
C.R. Mafra, O. Schlotterer, S. Stieberger and D. Tsimpis, A recursive method for SYM n-point tree amplitudes, Phys. Rev. D 83 (2011) 126012 [arXiv:1012.3981] [INSPIRE].
C.R. Mafra, O. Schlotterer and S. Stieberger, Complete N-Point Superstring Disk Amplitude I. Pure Spinor Computation, Nucl. Phys. B 873 (2013) 419 [arXiv:1106.2645] [INSPIRE].
C.R. Mafra, O. Schlotterer and S. Stieberger, Complete N-Point Superstring Disk Amplitude II. Amplitude and Hypergeometric Function Structure, Nucl. Phys. B 873 (2013) 461 [arXiv:1106.2646] [INSPIRE].
C.R. Mafra and O. Schlotterer, Cohomology foundations of one-loop amplitudes in pure spinor superspace, arXiv:1408.3605 [INSPIRE].
E. Bridges and C.R. Mafra, Algorithmic construction of SYM multiparticle superfields in the BCJ gauge, JHEP 10 (2019) 022 [arXiv:1906.12252] [INSPIRE].
C.R. Mafra, Towards Field Theory Amplitudes From the Cohomology of Pure Spinor Superspace, JHEP 11 (2010) 096 [arXiv:1007.3639] [INSPIRE].
J. Broedel, O. Schlotterer and S. Stieberger, Polylogarithms, Multiple Zeta Values and Superstring Amplitudes, Fortsch. Phys. 61 (2013) 812 [arXiv:1304.7267] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles: Scalars, Gluons and Gravitons, JHEP 07 (2014) 033 [arXiv:1309.0885] [INSPIRE].
C.R. Mafra, Planar binary trees in scattering amplitudes, (2020), DOI [arXiv:2011.14413] [INSPIRE].
C. Reutenauer, Free Lie Algebras, London Mathematical Society Monographs (1993), pp 1–286.
Z. Bern, J.J.M. Carrasco and H. Johansson, Perturbative Quantum Gravity as a Double Copy of Gauge Theory, Phys. Rev. Lett. 105 (2010) 061602 [arXiv:1004.0476] [INSPIRE].
Z. Bern, J.J.M. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, Simplifying Multiloop Integrands and Ultraviolet Divergences of Gauge Theory and Gravity Amplitudes, Phys. Rev. D 85 (2012) 105014 [arXiv:1201.5366] [INSPIRE].
Z. Bern, S. Davies, T. Dennen and Y.-t. Huang, Absence of Three-Loop Four-Point Divergences in N = 4 Supergravity, Phys. Rev. Lett. 108 (2012) 201301 [arXiv:1202.3423] [INSPIRE].
Z. Bern, S. Davies, T. Dennen and Y.-t. Huang, Ultraviolet Cancellations in Half-Maximal Supergravity as a Consequence of the Double-Copy Structure, Phys. Rev. D 86 (2012) 105014 [arXiv:1209.2472] [INSPIRE].
Z. Bern, S. Davies, T. Dennen, A.V. Smirnov and V.A. Smirnov, Ultraviolet Properties of N = 4 Supergravity at Four Loops, Phys. Rev. Lett. 111 (2013) 231302 [arXiv:1309.2498] [INSPIRE].
Z. Bern, S. Davies and T. Dennen, Enhanced ultraviolet cancellations in \( \mathcal{N} \) = 5 supergravity at four loops, Phys. Rev. D 90 (2014) 105011 [arXiv:1409.3089] [INSPIRE].
S. Lee, C.R. Mafra and O. Schlotterer, Non-linear gauge transformations in D = 10 SYM theory and the BCJ duality, JHEP 03 (2016) 090 [arXiv:1510.08843] [INSPIRE].
J.J.M. Carrasco and H. Johansson, Five-Point Amplitudes in N = 4 Super-Yang-Mills Theory and N = 8 Supergravity, Phys. Rev. D 85 (2012) 025006 [arXiv:1106.4711] [INSPIRE].
E.Y. Yuan, Virtual Color-Kinematics Duality: 6-pt 1-Loop MHV Amplitudes, JHEP 05 (2013) 070 [arXiv:1210.1816] [INSPIRE].
S. He and O. Schlotterer, New Relations for Gauge-Theory and Gravity Amplitudes at Loop Level, Phys. Rev. Lett. 118 (2017) 161601 [arXiv:1612.00417] [INSPIRE].
K. Dilcher, Sums of products of bernoulli numbers, J. Number Theory 60 (1996) 23.
M.B. Green, J.H. Schwarz and E. Witten, Superstring Theory. Vol. 2: Loop Amplitudes, Anomalies And Phenomenology, University Press, Cambridge U.K. (1987).
M.B. Green, J.H. Schwarz and L. Brink, N = 4 Yang-Mills and N = 8 Supergravity as Limits of String Theories, Nucl. Phys. B 198 (1982) 474 [INSPIRE].
P. Tourkine, Tropical Amplitudes, Annales Henri Poincaré 18 (2017) 2199 [arXiv:1309.3551] [INSPIRE].
P.H. Frampton and T.W. Kephart, Explicit Evaluation of Anomalies in Higher Dimensions, Phys. Rev. Lett. 50 (1983) 1343 [Erratum ibid. 51 (1983) 232] [INSPIRE].
P.H. Frampton and T.W. Kephart, The Analysis of Anomalies in Higher Space-time Dimensions, Phys. Rev. D 28 (1983) 1010 [INSPIRE].
M.B. Green and J.H. Schwarz, Anomaly Cancellation in Supersymmetric D = 10 Gauge Theory and Superstring Theory, Phys. Lett. B 149 (1984) 117 [INSPIRE].
M.B. Green and J.H. Schwarz, The Hexagon Gauge Anomaly in Type I Superstring Theory, Nucl. Phys. B 255 (1985) 93 [INSPIRE].
W.-M. Chen, Y.-t. Huang and D.A. McGady, Anomalies without an action, arXiv:1402.7062 [INSPIRE].
Z. Bern, C. Boucher-Veronneau and H. Johansson, N ≥ 4 Supergravity Amplitudes from Gauge Theory at One Loop, Phys. Rev. D 84 (2011) 105035 [arXiv:1107.1935] [INSPIRE].
V. Del Duca, L.J. Dixon and F. Maltoni, New color decompositions for gauge amplitudes at tree and loop level, Nucl. Phys. B 571 (2000) 51 [hep-ph/9910563] [INSPIRE].
S. He, R. Monteiro and O. Schlotterer, String-inspired BCJ numerators for one-loop MHV amplitudes, JHEP 01 (2016) 171 [arXiv:1507.06288] [INSPIRE].
N.E.J. Bjerrum-Bohr, T. Dennen, R. Monteiro and D. O’Connell, Integrand Oxidation and One-Loop Colour-Dual Numerators in N = 4 Gauge Theory, JHEP 07 (2013) 092 [arXiv:1303.2913] [INSPIRE].
I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Product, Academic press (2014).
N.E.J. Bjerrum-Bohr, P.H. Damgaard, T. Sondergaard and P. Vanhove, Monodromy and Jacobi-like Relations for Color-Ordered Amplitudes, JHEP 06 (2010) 003 [arXiv:1003.2403] [INSPIRE].
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: 2102.12943
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
Bridges, E., Mafra, C.R. Local BCJ numerators for ten-dimensional SYM at one loop. J. High Energ. Phys. 2021, 31 (2021). https://doi.org/10.1007/JHEP07(2021)031
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP07(2021)031