# Colour-kinematics duality The original statement of colour-kinematics duality says that it is possible to reorganise the perturbative expansion of tree-level amplitudes in $D$-dimensional pure YM theory with a general gauge group $G$ in terms of cubic diagrams where the kinematic numerators obey the same Jacobi relations and symmetry properties as their colour factors. More generally, in many gauge theories, extending well beyond YM theories with or without matter, it should be possible to reorganise the perturbative expansion so that there is a one-to-one map between the Lie-algebra identities of the colour factors carried by certain diagrams (with cubic or higher-point vertices) and the identities of the kinematic numerators of the same diagrams. In other words, think of CK duality as a constraint on fields, group representations, interactions and operators such that the above can be achieved. There are two main aspects to colour-kinematics duality: - *kinematic algebra*: the observation that amplitudes can be represented in such a way that the kinematic (colour-ordered) parts obey a Jacobi identity, akin to the colour structure parts. Colour and kinematics are “dual” in the sense of respecting the same algebraic relations. - *[[0067 Double copy|double copy]]*: the observation that amplitudes in the dual form can be squared to generate the amplitudes of other theories. Most famously this leads to gravity as the square of a gauge theory, but also the Galileon as the square of the chiral Lagrangian. To see an example, consider 4-point gluon amplitude: $A\left(1_{a} 2_{b} 3_{c} 4_{d}\right)=\frac{c_{s} n_{s}}{s}+\frac{c_{t} n_{t}}{t}+\frac{c_{u} n_{u}}{u},$ where $c_i$ are colour factors and $n_i$ are kinematic factors. E.g. $c_{s}=f_{a b e} f_{c d e}$. Now colour structures satisfy $c_{s}+c_{t}+c_{u}=0$, and Mandelstam variables satisfy $s+t+u=0$. The remarkable thing is that BCJ conjectures that there always *exists* a form of the amplitude where the kinematic numerators also satisfy a 'Jacobi identity': $n_{s}+n_{t}+n_{u}=0$. It is a result of having a huge redundancy in the theory due to gauge freedom. ## Refs - originals - [[BernCarrascoJohansson2008]] - [[BernCarrascoJohansson2010]] (conjecture to hold off-shell) - review - [[2019#Bern, Carrasco, Chiodaroli, Johansson, Roiban (Review)]] - proofs - [[Bjerrum-BohrDamgaardVanhove2009]] - [[Stieberger200907]] - [[FengHuangJia2010]] - squaring relation: [[BernDennenHuangKiermaier2010]] - explicit description of numerators - [[Kiermaier2011]] - [[Bjerrum-BohrDamgaardSondergaardVanhove2010]] - [[MafraSchlottererStieberger2011]] - general constraints from BCJ (BCJ boostrap) - [[2023#Brown, Kampf, Oktem, Paranjape, Trnka]] - related - important for [[0067 Double copy|double copy]] ## BCJ relations - $\sum_{i=2}^{m-1} p_{1} \cdot\left(p_{2}+\ldots+p_{i}\right) A_{m}^{\text {tree }}(2, \ldots, i, 1, i+1, \ldots, m)=0$ - reduces the number of independent colour-ordered tree [[0071 Yang-Mills|YM]] amplitudes to $(n-3)!$ - they arise whenever they duality between colour and kinematics and gauge invariance conspire to prevent the relation between partial amplitudes and numerator to be inverted - e.g. 4-point YM amplitude - $i \mathcal{A}_{4}^{\text {tree }}=g^{2}\left(\frac{n_{s} c_{s}}{s}+\frac{n_{t} c_{t}}{t}+\frac{n_{u} c_{u}}{u}\right)$\equiv i g^{2} A_{4}^{\text {tree}}(1,2,3,4) c_{s}-i g^{2} A_{4}^{\text {tree}}(1,3,2,4) c_{u}$ - $i\left(\begin{array}{l}A_{4}^{\text {tree }}(1,2,3,4) \\ A_{4}^{\text {tree }}(1,3,2,4)\end{array}\right)=\left(\begin{array}{cc}\frac{1}{s}+\frac{1}{t} & \frac{1}{t} \\ -\frac{1}{t} & -\frac{1}{u}-\frac{1}{t}\end{array}\right)\left(\begin{array}{l}n_{s} \\ n_{u}\end{array}\right)$ - reason: partial amplitudes are gauge invariant but numerators are not ## Comments - true in full [[0071 Yang-Mills|YM]], but also in [[0136 Self-dual Yang-Mills|SDYM]] ## Extensions - [[AlbayrakKharelMeltzer2020]] - covariant version - recent paper [[CheungMangan2021]][](https://arxiv.org/abs/2108.02276) - off-shell version (would establish a purely kinematic relation) - [[Ben-ShaharJohansson2021]][](https://arxiv.org/pdf/2112.11452.pdf) - **geometry-kinemetics** duality - [[2022#Cheung, Helset, Parra-Martinez]] - for form factors - [[2022#Lin, Yang]] and [[2023#Lin, Yang]] and refs therein - higher derivative - [[2023#Bonnefoy, Durieux, Nepveu]] \[*Acknowledgement: Most of the introductory sentences on this page are directly taken from notes of N. Craig's Gauge Theory class at UCSB.*\]