Double copy structure and the flat space limit of conformal correlators in even dimensions

TL;DR

The paper shows that the flat space limit of momentum-space 3-point CFT correlators in even dimensions exhibits a double copy structure, mirroring results in odd dimensions. A central tool is the master 1-loop triangle integral $I_{1\{000\}}$, whose leading singularity after an analytic continuation governs the flat-space limit of all relevant correlators; in even dimensions the analysis must handle divergences and branch cuts via renormalisation. In four dimensions, the flat space limit of stress-tensor correlators is controlled by conformal anomaly coefficients $(a,c)$, linking holographic expectations to the amplitude structure. The results extend the double copy picture to general even dimensions and provide a systematic procedure to extract flat-space data from renormalised correlators, with potential implications for cosmology and higher-point functions.

Abstract

We analyse the flat space limit of 3-point correlators in momentum space for general conformal field theories in even spacetime dimensions, and show they exhibit a double copy structure similar to that found in odd dimensions. In even dimensions, the situation is more complicated because correlators contain branch cuts and divergences which need to be renormalised. We describe the analytic continuation of momenta required to extract the flat space limit, and show that the flat space limit is encoded in the leading singularity of a 1-loop triangle integral which serves as a master integral for 3-point correlators in even dimensions. We then give a detailed analysis of the renormalised correlators in four dimensions where the flat space limit of stress tensor correlators are controlled by the coefficients in the trace anomaly.

Figures (7)

• Figure 1: The momenta in the 3-point function form a triangle by momentum conservation, with angle $\phi_i$ appearing opposite the side of length $|p_i|$.
• Figure 2: (a) As we increase $\theta$ from $0$ to $\pi$, $z$ and $\bar{z}$ move clockwise in the complex plane following the solid blue and orange dashed paths respectively. Starting from generic complex conjugate initial values (corresponding to physical momentum configurations), they ultimately end up exchanging positions. In the process, $z$ crosses the branch cut between $(1,\infty)$ while $\bar{z}$ crosses the branch cut between $(-\infty,0)$. (b) The trajectory of $z$ and $\bar{z}$ as we continue from an collinear initial configuration to one with $E=0$. The flat space limit thus corresponds to bringing $z$ and $\bar{z}$ to a point on the real axis between $0$ and $1$ after crossing the cuts in the direction shown.
• Figure 3: Reduction scheme for the regulated triple-$K$ integrals appearing in $\langle JJJ\rangle$, where the numbered operations refer to Table \ref{['operations']}, and ${\bf 5}^{n-1}$ means applying operation ${\bf 5}$ a total of $n-1$ times.
• Figure 4: Reduction scheme for the regulated triple-$K$ integrals appearing in $\langle TTT\rangle$, where the numbered operations refer to those in Table \ref{['operations']}.
• Figure 5: Reduction scheme for the regulated triple-$K$ integrals appearing in $\langle JJ\mathcal{O}\rangle$.