Energy Correlators from Star Integrals via Mellin Space

Abstract

We explore the Mellin space representation for the collinear limit of $N$-point energy correlators in ${\cal N}=4$ super-Yang-Mills theory. We show that these correlators can be written as integro-differential operators acting on star integrals: one-loop $n$-gons in $n$ dimensions. For the three-point energy correlator, we obtain the Mellin representation, use it to relate the correlator to the massive box integral, and show how to solve this relation to match with the expected result. For the four-point energy correlator, we obtain the Mellin representation and use it to write the correlator to a sum of various box and hexagon integrals in special kinematics. Our results provide a systematic method to relate higher-point energy correlators in the collinear limit to star integrals, which are known exactly.

Figures (1)

• Figure 1: Star integrals corresponding to the five types of $\Gamma$ function structures in $\text{E}^\text{4}\text{C}$. (1) 4-mass box. (2) 4-mass hexagon: $u_2=u_9=1$ and $P_{45}=P_{56}=0$. (3) 3-mass hexagon: same as (2), additionally with $P_{12}=0$. (4) 4-mass hexagon: given by the integral shown in (\ref{['eq:deformedhexagon']}), with kinematics satisfying (\ref{['eq:type4constraints']}). Points connected with dashed lines are null-separated: $P_{14} = 0$. (5) 3-mass hexagon: the same integral (\ref{['eq:deformedhexagon']}), additionally with $P_{16}=0$.