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

We analyze 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 renormalized. 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 renormalized correlators in four dimensions where the flat space limit of stress tensor correlators is controlled by the coefficients in the trace anomaly.

Figure 1

The momenta in the 3-point function form a triangle by momentum conservation, with angle ${\varphi }_i$ appearing opposite the side of length $|p_i|$.

Figure 2

(a) As we increase $\theta$ from 0 to $\pi$, $z$ and $\overline{z}$ 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 $\overline{z}$ crosses the branch cut between $(-\infty ,0)$. (b) The trajectory of $z$ and $\overline{z}$ as we continue from a collinear initial configuration to one with $E=0$. The flat space limit thus corresponds to bringing $z$ and $\overline{z}$ 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 $⟨JJJ⟩$, where the numbered operations refer to Table 1, and ${\mathbf{5}}^{n-1}$ means applying operation $\mathbf{5}$ a total of $n-1$ times.

Figure 4

Reduction scheme for the regulated triple-$K$ integrals appearing in $⟨TTT⟩$, where the numbered operations refer to those in Table 1.

Figure 5

Reduction scheme for the regulated triple-$K$ integrals appearing in $⟨JJ\mathcal{O}⟩$.

Figure 6

Reduction scheme for the regulated triple-$K$ integrals appearing in $⟨TT\mathcal{O}⟩$. All numbered operations refer to those in Table 1.

Figure 7

Region momenta relating the massless box to the 3-point master integral $I_{1\{000\}}$. All external momenta are taken as ingoing.