Gravity as the square of gauge theory

We explore consequences of the recently discovered duality between color and kinematics, which states that kinematic numerators in a diagrammatic expansion of gauge-theory amplitudes can be arranged to satisfy Jacobi-like identities in one-to-one correspondence to the associated color factors. Using on-shell recursion relations, we give a field-theory proof showing that the duality implies that diagrammatic numerators in gravity are just the product of two corresponding gauge-theory numerators, as previously conjectured. These squaring relations express gravity amplitudes in terms of gauge-theory ingredients, and are a recasting of the Kawai, Lewellen, and Tye relations. Assuming that numerators of loop amplitudes can be arranged to satisfy the duality, our tree-level proof immediately carries over to loop level via the unitarity method. We then present a Yang-Mills Lagrangian whose diagrams through five points manifestly satisfy the duality between color and kinematics. The existence of such Lagrangians suggests that the duality also extends to loop amplitudes, as confirmed at two and three loops in a concurrent paper. By “squaring” the novel Yang-Mills Lagrangian we immediately obtain its gravity counterpart. We outline the general structure of these Lagrangians for higher points. We also write down various new representations of gauge-theory and gravity amplitudes that follow from the duality between color and kinematics.

Figure 1

A Jacobi relation between color factors of diagrams. According to the BCJ duality the diagrammatic numerators of amplitudes can be arranged so that they satisfy relations in one-to-one correspondence to the color Jacobi identities.

Figure 2

(a) In an on-shell recursion relation, a given residue ${\widehat{\mathcal{A}}}_n^{\alpha }$ is determined by diagrams sharing the same propagator labeled by $s_{\alpha }$. (b) We can obtain a diagrammatic expansion of the recursion relation either from the numerators ${\widehat{n}}_i$ of the shifted full amplitude ${\widehat{\mathcal{A}}}_n$, or from the numerators ${\widehat{n}}_{L,i}^{\alpha }$, ${\widehat{n}}_{R,i}^{\alpha }$ of the subamplitudes.

Figure 3

The product $i{\widehat{n}}_{L,i}{\widehat{n}}_{R,i}$ satisfies the duality relations satisfied by its factors ${\widehat{n}}_{L,i}$ and ${\widehat{n}}_{R,i}$.

Figure 4

A graphical representation of the color basis $c_{1,{\sigma }_2,\dots ,{\sigma }_{n-1},n}$ introduced in Ref. 18. Each vertex represents a structure constant ${\tilde{f}}^{abc}$, while each bond indicates contracted indices between the ${\tilde{f}}^{abc}$. This is also precisely the diagram associated with the kinematic numerator $n_{1,{\sigma }_2,\dots ,{\sigma }_{n-1},n}$.