0067 Double copy - otherworldlines

# Double copy The double copy is a relation between graviton amplitudes and gluons amplitudes, or more generally between the scattering amplitudes of two different theories. Since perturbation theory is far simpler in [[0071 Yang-Mills|YM]] theory than in standard approaches to gravity, the double copy is a very powerful relation. The idea of double copy is not restricted to amplitude, though, as some gravity *solutions* are related to [[0071 Yang-Mills|YM]] solutions in a double-copy fashion. ## Origin - long ago, it was found that the Veneziano scattering amplitude [[Veneziano1968]], A(s, t), (later identified as an open-string scattering amplitude) and the Virasoro-Shapiro amplitude ([[Virasoro1969]] and [[Shapiro1969]]), M(s, t, u), (later identified as a closed-string amplitude) are related: $M(s, t, u)=\frac{\sin \left(\pi \alpha^{\prime} s\right)}{\pi \alpha^{\prime}} A(s, t) A(s, u)$ - in the low energy limit, this reduces to the [[0554 Einstein gravity|GR]]-[[0071 Yang-Mills|YM]] double copy: $\mathcal{M}_{4}^{\text {tree }}(1,2,3,4)=\left(\frac{\kappa}{2}\right)^{2} s A_{4}^{\text {tree }}(1,2,3,4) A_{4}^{\text {tree }}(1,2,4,3)$ ## Classification - early days [[0398 KLT relations]] - then [[0152 Colour-kinematics duality|BCJ relations]] - even better (in [[0234 Self-dual gravity]] and [[0136 Self-dual Yang-Mills]]): [[2011#Monteiro, O'Connell]] ## Using [[0152 Colour-kinematics duality]] - starting from [[0071 Yang-Mills|YM]] amplitude $A\left(1_{a} 2_{b} 3_{c} 4_{d}\right)=\frac{c_{s} n_{s}}{s}+\frac{c_{t} n_{t}}{t}+\frac{c_{u} n_{u}}{u}$ - replace colour factors by kinematic factors - get graviton amplitude: $A_{4}\left(1^{--} 2^{--} 3^{++} 4^{++}\right)=\frac{n_{s}^{2}}{s}+\frac{n_{t}^{2}}{t}+\frac{n_{u}^{2}}{u}=-\frac{\langle 12\rangle^{4}[34]^{4}}{s t u}$ - more generally: $\mathcal{M}_{m}^{\text {tree }}=\left.\mathcal{A}_{m}^{\text {tree }}\right|_{c_{i} \rightarrow n_{i}}$=\sum_{\tau \in S_{m-2}} A_{m}^{\text {tree }}(1, \tau(2, \ldots, m-1), m) n(1, \tau(2, \ldots, m-1), m)$=-i \sum_{\sigma, \rho \in S_{m-3}} A_{m}^{\text {tree }}(1, \sigma, m-1, m) S[\sigma|\rho] \tilde{A}_{m}^{\text {tree }}(1, \rho, m, m-1)$ ## Related topics - [[0398 KLT relations]] - [[0152 Colour-kinematics duality]] ## Refs - recent review - [[2019#Bern, Carrasco, Chiodaroli, Johansson, Roiban (Review)]] - for EFTs - [[2012#Broedel, Dixon]]: $F^3$, $R^3$, $F^4$, $\phi R^2$, etc. - [[BonnefoyDurieuxGrojeanMachadoNepveu2021]][](https://arxiv.org/pdf/2112.11453.pdf) - in AdS - [[Zhou2021]][](https://arxiv.org/pdf/2106.07651.pdf) - [[HerderscheeRoibanTeng2022]][](https://arxiv.org/pdf/2201.05067.pdf) - [[CheungParra-MartinezSivaramakrishnan2022]][](https://arxiv.org/pdf/2201.05147.pdf) - [[BissiFardelliManentiZhou2022]][](https://arxiv.org/pdf/2209.01204.pdf) - [[Li2022]][](https://arxiv.org/pdf/2212.13195.pdf) - using twisted forms - [[2022#Mazloumi, Stieberger]] - recent - [[DelaCruzMaybeeOConnellRoss2020]][](https://arxiv.org/pdf/2009.03842.pdf) - [[MonteiroOConnellVeigaSergola2020]][](https://arxiv.org/pdf/2012.11190.pdf): in split signature ## Comments - in the presence of adjoint-rep. fermions, the duality implies SUSY [[ChiodaroliJinRoiban2013]] ## GR from collinear limit of [[0071 Yang-Mills|YM]] - [[2015#Stieberger, Taylor (Feb)]] ## Double copy of asymptotic symmetry - [[2024#Ferrero, Francia, Heissenberg, Romoli]]