Colour-kinematics duality and the Drinfeld double of the Lie algebra of diffeomorphisms
Colour-kinematics duality suggests that Yang-Mills (YM) theory possesses some hidden Lie algebraic structure. So far this structure has resisted understanding, apart from some progress in the self-dual sector. We show that there is indeed a Lie algebra behind the YM Feynman rules. The Lie algebra we...
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Springer
2017
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| Online Access: | https://eprints.nottingham.ac.uk/44795/ |
| _version_ | 1848796998995542016 |
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| author | Fu, Chih-Hao Krasnov, Kirill |
| author_facet | Fu, Chih-Hao Krasnov, Kirill |
| author_sort | Fu, Chih-Hao |
| building | Nottingham Research Data Repository |
| collection | Online Access |
| description | Colour-kinematics duality suggests that Yang-Mills (YM) theory possesses some hidden Lie algebraic structure. So far this structure has resisted understanding, apart from some progress in the self-dual sector. We show that there is indeed a Lie algebra behind the YM Feynman rules. The Lie algebra we uncover is the Drinfeld double of the Lie algebra of vector fields. More specifically, we show that the kinematic numerators following from the YM Feynman rules satisfy a version of the Jacobi identity, in that the Jacobiator of the bracket defined by the YM cubic vertex is cancelled by the contribution of the YM quartic vertex. We then show that this Jacobi-like identity is in fact the Jacobi identity of the Drinfeld double. All our considerations are off-shell. Our construction explains why numerators computed using the Feynman rules satisfy the colour-kinematics at four but not at higher numbers of points. It also suggests a way of modifying the Feynman rules so that the duality can continue to hold for an arbitrary number of gluons. Our construction stops short of producing explicit higher point numerators because of an absence of a certain property at four points. We comment on possible ways of correcting this, but leave the next word in the story to future work. |
| first_indexed | 2025-11-14T19:56:54Z |
| format | Article |
| id | nottingham-44795 |
| institution | University of Nottingham Malaysia Campus |
| institution_category | Local University |
| last_indexed | 2025-11-14T19:56:54Z |
| publishDate | 2017 |
| publisher | Springer |
| recordtype | eprints |
| repository_type | Digital Repository |
| spelling | nottingham-447952020-05-04T18:30:25Z https://eprints.nottingham.ac.uk/44795/ Colour-kinematics duality and the Drinfeld double of the Lie algebra of diffeomorphisms Fu, Chih-Hao Krasnov, Kirill Colour-kinematics duality suggests that Yang-Mills (YM) theory possesses some hidden Lie algebraic structure. So far this structure has resisted understanding, apart from some progress in the self-dual sector. We show that there is indeed a Lie algebra behind the YM Feynman rules. The Lie algebra we uncover is the Drinfeld double of the Lie algebra of vector fields. More specifically, we show that the kinematic numerators following from the YM Feynman rules satisfy a version of the Jacobi identity, in that the Jacobiator of the bracket defined by the YM cubic vertex is cancelled by the contribution of the YM quartic vertex. We then show that this Jacobi-like identity is in fact the Jacobi identity of the Drinfeld double. All our considerations are off-shell. Our construction explains why numerators computed using the Feynman rules satisfy the colour-kinematics at four but not at higher numbers of points. It also suggests a way of modifying the Feynman rules so that the duality can continue to hold for an arbitrary number of gluons. Our construction stops short of producing explicit higher point numerators because of an absence of a certain property at four points. We comment on possible ways of correcting this, but leave the next word in the story to future work. Springer 2017-01-17 Article PeerReviewed Fu, Chih-Hao and Krasnov, Kirill (2017) Colour-kinematics duality and the Drinfeld double of the Lie algebra of diffeomorphisms. Journal of High Energy Physics, 2017 . 75/1-75/40. ISSN 1029-8479 Scattering Amplitudes Differential and Algebraic Geometry https://link.springer.com/article/10.1007%2FJHEP01%282017%29075 doi:10.1007/JHEP01(2017)075 doi:10.1007/JHEP01(2017)075 |
| spellingShingle | Scattering Amplitudes Differential and Algebraic Geometry Fu, Chih-Hao Krasnov, Kirill Colour-kinematics duality and the Drinfeld double of the Lie algebra of diffeomorphisms |
| title | Colour-kinematics duality and the Drinfeld double of the Lie algebra of diffeomorphisms |
| title_full | Colour-kinematics duality and the Drinfeld double of the Lie algebra of diffeomorphisms |
| title_fullStr | Colour-kinematics duality and the Drinfeld double of the Lie algebra of diffeomorphisms |
| title_full_unstemmed | Colour-kinematics duality and the Drinfeld double of the Lie algebra of diffeomorphisms |
| title_short | Colour-kinematics duality and the Drinfeld double of the Lie algebra of diffeomorphisms |
| title_sort | colour-kinematics duality and the drinfeld double of the lie algebra of diffeomorphisms |
| topic | Scattering Amplitudes Differential and Algebraic Geometry |
| url | https://eprints.nottingham.ac.uk/44795/ https://eprints.nottingham.ac.uk/44795/ https://eprints.nottingham.ac.uk/44795/ |