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.

Highlights

  • Gauge invariance has been shown to provide a strong constraint on scattering amplitudes in recent years

  • One interesting application is that a recursive expansion of single-trace Einstein-Yang-Mills (EYM) amplitudes can be determined by gauge invariance conditions of external gravitons in addition with a proper ansatz [5, 6] (the approach based on Cachazo-He-Yuan (CHY) formula [7,8,9,10] is provided in [11])

  • By expressing the coefficients in the gauge invariance induced identity according to the refined graphic rule and collecting those terms with the same structure of the lines corresponding to coefficients · and · k in particular examples, we find that the gauge invariance induced identity can always be expressed by a combination of BCJ relations

Read more

Summary

Introduction

Gauge invariance has been shown to provide a strong constraint on scattering amplitudes in recent years. Applying the recursive expansion repeatedly, one expresses an arbitrary tree level single-trace EYM amplitude in terms of pure Yang-Mills (YM) ones whose coefficients are conveniently constructed by a graphic rule [12]. This expansion can be regarded as a generalization of the earlier studies on EYM amplitudes with few gravitons [13,14,15,16]. We investigate the relationship between an arbitrary gauge invariance induced identity, which is derived from the expansion of tree level single-trace EYM amplitudes or pure GR amplitudes, and general BCJ relations. We review the recursive expansion of single-trace EYM amplitudes, the graphic rule, the gauge invariance induced identity from single-trace EYM amplitudes as well as the BCJ relations for Yang-Mills amplitudes

Recursive expansion of tree level single-trace EYM amplitudes
Gauge invariance induced identity from tree level single-trace EYM amplitudes
BCJ relation
Refined graphic rule for single trace tree-level EYM amplitudes
Examples for the refined graphic rule
The main idea
Direct evaluations
A typical example
Common features of the examples
Constructing all physical and spurious graphs for a given skeleton
The construction of the final upper and lower blocks for a given skeleton
The sum over all physical and spurious graphs
Graph-based BCJ relation as a combination of traditional BCJ relations
The general proof
Gauge invariance identities of tree-level YM and GR amplitudes
Conclusions and further discussions
A Conventions and definitions
B More examples for section 4

Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call

Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.