BCJ duality and double copy in the closed string sector
Alexander Ochirov, Piotr Tourkine
TL;DR
The paper investigates loop-level BCJ color-kinematics duality and the gravity double copy using closed-string worldsheet methods, with a focus on four-point one-loop amplitudes. It shows that tree-level BCJ numerators can be obtained from Mafra–Schlotterer–Stieberger chiral blocks, and extends the analysis to one-loop for N=2 SYM and symmetric N=4 supergravity, highlighting left-right string contractions as the origin of gravity’s squared structure. A central finding is that gauge-theory BCJ representations require total-derivative corrections (δW3, δW2) whose string-theoretic interpretation traces to left-right mixing and dimension shifts, rather than straightforward IBP manipulations. The work clarifies how left-right contractions in string theory produce gravity’s square-structures and how these connect to BCJ representations, offering a string-theoretic lens on generalized gauge invariance and set of open questions for extending MSS concepts to loop level. Formally, the gravity integrand can be expressed as a product of gauge-theory numerators with left-right contributions, i.e. $\mathcal{M}_n^L = i^{L+1} (\kappa/2)^{n+2L-2} \sum_i \int \cdots \frac{n_i(\ell)\tilde{n}_i(\ell)}{D_i(\ell)}$, with additional string-derived terms $W_2$ and total-derivative corrections encoding left-right mixing and dimension-shifting effects.$
Abstract
This paper is focused on the loop-level understanding of the Bern-Carrasco-Johansson double copy procedure that relates the integrands of gauge theory and gravity scattering amplitudes. At four points, the first non-trivial example of that construction is one-loop amplitudes in N=2 super-Yang-Mills theory and the symmetric realization of N=4 matter-coupled supergravity. Our approach is to use both field and string theory in parallel to analyze these amplitudes. The closed string provides a natural framework to analyze the BCJ construction, in which the left- and right-moving sectors separately create the color and kinematics at the integrand level. At tree level, in a five-point example, we show that the Mafra-Schlotterer-Stieberger procedure gives a new direct proof of the color-kinematics double copy. We outline the extension of that argument to n points. At loop level, the field-theoretic BCJ construction of N=2 SYM amplitudes introduces new terms, unexpected from the string theory perspective. We discuss to what extent we can relate them to the terms coming from the interactions between left- and right-movers in the string-theoretic gravity construction.