A graphic approach to gauge invariance induced identity
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Springer
Received: December 11, Revised: April 5, Accepted: April 25, Published: May 3,
2018 2019 2019 2019
Linghui Houa,1 and Yi-Jian Dua,b,2 a
Center for Theoretical Physics, School of Physics and Technology, Wuhan University, No. 299 Bayi Road, Wuhan 430072, China b Suzhou Institute of Wuhan University, No. 377 Linquan Street, Suzhou, 215123, China
E-mail: [email protected], [email protected] Abstract: All tree-level amplitudes in Einstein-Yang-Mills (EYM) theory and gravity (GR) can be expanded in terms of color ordered Yang-Mills (YM) ones whose coefficients are polynomial functions of Lorentz inner products and are constructed by a graphic rule. Once the gauge invariance condition of any graviton is imposed, the expansion of a tree level EYM or gravity amplitude induces a nontrivial identity between color ordered YM amplitudes. Being different from traditional Kleiss-Kuijf (KK) and Bern-Carrasco-Johansson (BCJ) relations, the gauge invariance induced identity involves polarizations in the coefficients. In this paper, we investigate the relationship between the gauge invariance induced identity and traditional BCJ relations. By proposing a refined graphic rule, we prove that all the gauge invariance induced identities for single trace tree-level EYM amplitudes can be precisely expanded in terms of traditional BCJ relations, without referring any property of polarizations. When further considering the transversality of polarizations and momentum conservation, we prove that the gauge invariance induced identity for tree-level GR (or pure YM) amplitudes can also be expanded in terms of traditional BCJ relations for YM (or bi-scalar) amplitudes. As a byproduct, a graph-based BCJ relation is proposed and proved. Keywords: Scattering Amplitudes, Gauge Symmetry ArXiv ePrint: 1811.12653 1
The unusual ordering of authors is just to let authors get proper recognition of contributions under outdated practice in China. 2 Corresponding author
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2019)012
JHEP05(2019)012
A graphic approach to gauge invariance induced identity
Contents 1 Introduction
1
3 Refined graphic rule and the main idea 3.1 Refined graphic rule for single trace tree-level EYM amplitudes 3.1.1 Examples for the refined graphic rule 3.2 The main idea
6 6 8 9
4 Direct evaluations 4.1 A typical example 4.2 Common features of the examples
10 10 14
5 General structure of skeletons
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6 Constructing all physical and spurious graphs for a given skeleton 6.1 The construction of the final upper and lower blocks for a given skeleton 6.2 Physical and spurious graphs for a given configuration of the final upper and lower blocks 6.3 The sum over all physical and spurious graphs
19 19 21 24
7 Graph-based BCJ relation as a combination of traditional BCJ relations 29 7.1 Examples 30 7.2 The general proof 34 8 Gauge invariance identities of tree-level YM and GR amplitudes 8.1 The old-version graphic rule for the BCJ numerators n1|ζζ |n 8.2 Understanding the gauge inva
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