Abstract
Tree-level color-ordered Yang-Mills (YM) amplitudes can be decomposed in terms of (n − 2)! bi-scalar (BS) amplitudes, whose expansion coefficients form a basis of Bern-Carrasco-Johansson (BCJ) numerators. By the help of the recursive expansion of Einstein-Yang-Mills (EYM) amplitudes, the BCJ numerators are given by polynomial functions of Lorentz contractions which are conveniently described by graphic rule. In this work, we extend the expansion of YM amplitudes to off-shell level. We define different types of off-shell extended numerators that can be generated by graphs. By the use of these extended numerators, we propose a general decomposition formula of off-shell Berends-Giele currents in YM. This formula consists of three terms: (i). an effective current which is expanded as a combination of the Berends-Giele currents in BS theory (The expansion coefficients are one type of off-shell extended numerators) (ii). a term proportional to the total momentum of on-shell lines and (iii). a term expressed by the sum of lower point Berends-Giele currents in which some polarizations and momenta are replaced by vectors proportional to off-shell momenta appropriately. In the on-shell limit, the last two terms vanish while the decomposition of effective current precisely reproduces the decomposition of on-shell YM amplitudes with the expected coefficients (BCJ numerators in DDM basis). We further symmetrize these coefficients such that the Lie symmetries are satisfied. These symmetric BCJ numerators simultaneously satisfy the relabeling property of external lines and the algebraic properties (antisymmetry and Jacobi identity).
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References
C.-H. Fu, Y.-J. Du, R. Huang and B. Feng, Expansion of Einstein-Yang-Mills Amplitude, JHEP 09 (2017) 021 [arXiv:1702.08158] [INSPIRE].
M. Chiodaroli, M. Günaydin, H. Johansson and R. Roiban, Explicit Formulae for Yang-Mills-Einstein Amplitudes from the Double Copy, JHEP 07 (2017) 002 [arXiv:1703.00421] [INSPIRE].
F. Teng and B. Feng, Expanding Einstein-Yang-Mills by Yang-Mills in CHY frame, JHEP 05 (2017) 075 [arXiv:1703.01269] [INSPIRE].
Y.-J. Du and F. Teng, BCJ numerators from reduced Pfaffian, JHEP 04 (2017) 033 [arXiv:1703.05717] [INSPIRE].
Y.-J. Du, B. Feng and F. Teng, Expansion of All Multitrace Tree Level EYM Amplitudes, JHEP 12 (2017) 038 [arXiv:1708.04514] [INSPIRE].
S. Stieberger and T.R. Taylor, New relations for Einstein-Yang-Mills amplitudes, Nucl. Phys. B 913 (2016) 151 [arXiv:1606.09616] [INSPIRE].
D. Nandan, J. Plefka, O. Schlotterer and C. Wen, Einstein-Yang-Mills from pure Yang-Mills amplitudes, JHEP 10 (2016) 070 [arXiv:1607.05701] [INSPIRE].
L. de la Cruz, A. Kniss and S. Weinzierl, Relations for Einstein-Yang-Mills amplitudes from the CHY representation, Phys. Lett. B 767 (2017) 86 [arXiv:1607.06036] [INSPIRE].
O. Schlotterer, Amplitude relations in heterotic string theory and Einstein-Yang-Mills, JHEP 11 (2016) 074 [arXiv:1608.00130] [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].
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].
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].
X. Gao, S. He and Y. Zhang, Labelled tree graphs, Feynman diagrams and disk integrals, JHEP 11 (2017) 144 [arXiv:1708.08701] [INSPIRE].
S. He, L. Hou, J. Tian and Y. Zhang, Kinematic numerators from the worldsheet: cubic trees from labelled trees, JHEP 08 (2021) 118 [arXiv:2103.15810] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles in Arbitrary Dimensions, Phys. Rev. Lett. 113 (2014) 171601 [arXiv:1307.2199] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering equations and Kawai-Lewellen-Tye orthogonality, Phys. Rev. D 90 (2014) 065001 [arXiv:1306.6575] [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].
F. Cachazo, S. He and E.Y. Yuan, Einstein-Yang-Mills Scattering Amplitudes From Scattering Equations, JHEP 01 (2015) 121 [arXiv:1409.8256] [INSPIRE].
F. Cachazo, S. He and E.Y. Yuan, Scattering Equations and Matrices: From Einstein To Yang-Mills, DBI and NLSM, JHEP 07 (2015) 149 [arXiv:1412.3479] [INSPIRE].
C.R. Mafra, Berends-Giele recursion for double-color-ordered amplitudes, JHEP 07 (2016) 080 [arXiv:1603.09731] [INSPIRE].
J. Broedel, O. Schlotterer and S. Stieberger, Polylogarithms, Multiple Zeta Values and Superstring Amplitudes, Fortsch. Phys. 61 (2013) 812 [arXiv:1304.7267] [INSPIRE].
B. Feng, X. Li and K. Zhou, Expansion of Einstein-Yang-Mills theory by differential operators, Phys. Rev. D 100 (2019) 125012 [arXiv:1904.05997] [INSPIRE].
K. Zhou and G.-J. Zhou, Note on scalar-graviton and scalar-photon-graviton amplitudes, Eur. Phys. J. C 80 (2020) 930 [arXiv:2007.05910] [INSPIRE].
R. Huang, Y.-J. Du and B. Feng, Understanding the Cancelation of Double Poles in the Pfaffian of CHY-formulism, JHEP 06 (2017) 133 [arXiv:1702.05840] [INSPIRE].
C.S. Lam, Pfaffian Diagrams for Gluon Tree Amplitudes, Phys. Rev. D 98 (2018) 076002 [arXiv:1808.07575] [INSPIRE].
L. Hou and Y.-J. Du, A graphic approach to gauge invariance induced identity, JHEP 05 (2019) 012 [arXiv:1811.12653] [INSPIRE].
Y.-J. Du and L. Hou, A graphic approach to identities induced from multi-trace Einstein-Yang-Mills amplitudes, JHEP 05 (2020) 008 [arXiv:1910.04014] [INSPIRE].
H. Tian, E. Gong, C. Xie and Y.-J. Du, Evaluating EYM amplitudes in four dimensions by refined graphic expansion, JHEP 04 (2021) 150 [arXiv:2101.02962] [INSPIRE].
F.A. Berends and W.T. Giele, Recursive Calculations for Processes with n Gluons, Nucl. Phys. B 306 (1988) 759 [INSPIRE].
C.R. Mafra and O. Schlotterer, Berends-Giele recursions and the BCJ duality in superspace and components, JHEP 03 (2016) 097 [arXiv:1510.08846] [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].
E. Bridges and C.R. Mafra, Algorithmic construction of SYM multiparticle superfields in the BCJ gauge, JHEP 10 (2019) 022 [arXiv:1906.12252] [INSPIRE].
R. Kleiss and H. Kuijf, Multi-Gluon Cross-sections and Five Jet Production at Hadron Colliders, Nucl. Phys. B 312 (1989) 616 [INSPIRE].
Y.-J. Du, B. Feng and C.-H. Fu, BCJ Relation of Color Scalar Theory and KLT Relation of Gauge Theory, JHEP 08 (2011) 129 [arXiv:1105.3503] [INSPIRE].
C.-H. Fu, Y.-J. Du and B. Feng, Note on symmetric BCJ numerator, JHEP 08 (2014) 098 [arXiv:1403.6262] [INSPIRE].
Z. Bern and T. Dennen, A Color Dual Form for Gauge-Theory Amplitudes, Phys. Rev. Lett. 107 (2011) 081601 [arXiv:1103.0312] [INSPIRE].
C.R. Mafra and O. Schlotterer, Solution to the nonlinear field equations of ten dimensional supersymmetric Yang-Mills theory, Phys. Rev. D 92 (2015) 066001 [arXiv:1501.05562] [INSPIRE].
H. Frost, C.R. Mafra and L. Mason, A Lie bracket for the momentum kernel, arXiv:2012.00519 [INSPIRE].
C.R. Mafra, Planar binary trees in scattering amplitudes, DOI [arXiv:2011.14413] [INSPIRE].
S. Mizera and B. Skrzypek, Perturbiner Methods for Effective Field Theories and the Double Copy, JHEP 10 (2018) 018 [arXiv:1809.02096] [INSPIRE].
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Wu, K., Du, YJ. Off-shell extended graphic rule and the expansion of Berends-Giele currents in Yang-Mills theory. J. High Energ. Phys. 2022, 162 (2022). https://doi.org/10.1007/JHEP01(2022)162
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DOI: https://doi.org/10.1007/JHEP01(2022)162