arXiv:1007.3475v2 [hep-th] 5 Nov 2010Constraints and Generalized Gauge Transformations on Tree-Level Gluon and Graviton Amplitudes Diana Vaman*, York-Peng Yao†1 *Department of Physics, The University of Virginia Charlottesville, VA 22904, USA †Department of Physics, The University of Michigan Ann Arbor, MI 48109, USA Abstract Writing the fully color dressed and graviton amplitudes, respectively, asA= 〈C|A〉=〈C|M|N〉andAgr=〈˜N|M|N〉, where|A〉is a set of Kleiss-Kuijf color ordered basis,|N〉,|˜N〉and|C〉are the similarly ordered numerators and color co- efficients, we show that the propagator matrixMhas (n−3)(n−3)! independent eigenvectors|λ0 j〉with zero eigenvalue, forn-particle processes. The resulting equa- tions〈λ0 j|A〉= 0 are relations among the color ordered amplitudes. The freedom to shift|N〉 → |N〉+∑ jfj|λ0 j〉and similarly for|˜N〉, wherefjare (n−3)(n−3)! ar- bitrary functions, encodes generalized gauge transformations. They yield both BCJ amplitude and KLT relations, when such freedom is accounted for.Furthermore, fjcan be promoted to the role of effective Lagrangian vertices in the field operator space. 1E-mail addresses: dv3h@virginia.edu, yyao@umich.edu
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