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).
Highlights
The recursive expansion relation of tree level Einstein-Yang-Mills (EYM) amplitudes [1,2,3,4,5] serves as a bridge between Einstein gravity (GR) and Yang-Mills theory (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
From the three-point example (4.18), we learn that the Berends-Giele current in YM can be decomposed into three terms: (i). an effective current Jρ which is written in terms of BS currents whose coefficients are the type-A numerators in DDM form, (ii). a term Kρ which is proportional to the total momentum and (iii). an Lρ term that is a sum of currents where the polarization vector and momenta of some external lines are replaced by lower-point Kρ terms and the corresponding momenta, respectively
Summary
The recursive expansion relation of tree level Einstein-Yang-Mills (EYM) amplitudes [1,2,3,4,5] serves as a bridge between Einstein gravity (GR) and Yang-Mills theory (YM). The polynomial coefficients in this expansion can be considered as the Bern-Carrasco-Johansson (BCJ) [10, 11] numerators (which are characterized by cubic diagrams and satisfy antisymmetry and Jacobi identity) in Del Duca-Dixon-Maltoni (DDM) basis [12] It was shown in [1, 3,4,5, 13, 14] that the EYM recursive expansion and the resulted pure-YM expansion of GR amplitudes could be understood from the framework of CachazoHe-Yuan (CHY) [15,16,17,18,19] formula. When the Lorentz contractions between external polarizations and/or momenta are expressed by graphs [3,4,5, 24,25,26,27], the coefficients (BCJ numerators) for the pure-YM (BS) expansion of GR (YM) amplitudes can be given by a sum over proper graphs This method provides a convenient approach to the study of related properties of GR, EYM and YM amplitudes.
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