Abstract
In this paper, we demonstrate that using differential operators one can construct the complete unified web for expansions of amplitudes for a wide range of theories. We first re-derive the expansion of multi-trace Einstein-Yang-Mills amplitudes to Kleiss-Kuijf basis of color-ordered Yang-Mills amplitudes, by applying proper differential operators which modify the coefficients in the recursive expansion of single-trace Einstein- Yang-Mills amplitudes. Next, through differential operators which act on amplitudes only, we obtain expansions of amplitudes of Yang-Mills theory, Yang-Mills-scalar theory, ϕ4 theory, non-linear sigma model, bi-adjoint scalar theory, Born-Infeld theory, Dirac-Born-Infeld theory and special Galileon theory. Then, together with other results in literatures, the complete unified web is achieved. This web for expansions is the dual version of the unified web for differential operators. Thus, connections among amplitudes of a variety of theories, which are reflected by Cachazo-He-Yuan integrands and differential operators previously, can also be represented by expansions. We also find that amplitudes of all theories in the web can be expanded to double color-ordered bi-adjoint scalar amplitudes in the double copy formula.
Highlights
CHY integrands for a variety of theories including Einstein-Yang-Mills (EYM), EinsteinMaxwell (EM), Born-Infeld (BI), YM, Yang-Mills-scalar (YMS), φ4, non-linear sigma model (NLSM), bi-adjoint scalar (BAS), Dirac-Born-Infeld (DBI), as well as special Galileon (SG), can be generated from integrands for GR.1 Recently, the same web for connections has been reproduced by introducing differential operators [11]
We first re-derive the expansion of multi-trace Einstein-Yang-Mills amplitudes to Kleiss-Kuijf basis of color-ordered Yang-Mills amplitudes, by applying proper differential operators which modify the coefficients in the recursive expansion of single-trace EinsteinYang-Mills amplitudes
We find that amplitudes of all theories in the web can be expanded to double color-ordered bi-adjoint scalar amplitudes in the double copy formula
Summary
We introduce differential operators proposed by Cheung, Shen and Wen, as well as unifying relations among a variety of amplitudes which are described by these operators. We discuss the choices of basis for expansions, and emphasize that basis can be determined only through differential operators. We rapidly review the recursive expansions for singletrace EYM and GR amplitudes, which emergence naturally from connections between amplitudes described by differential operators. Some notations which are used through out the paper will be provided
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