CHY-construction of Planar Loop Integrands of Cubic Scalar Theory

TL;DR

The paper extends the CHY formulation to all loops by interpreting loop momenta as massless in a higher dimension, recasting loop amplitudes as forward limits of tree-level diagrams. It develops a concrete two-loop planar CHY integrand for color-ordered bi-adjoint φ^3 theory by classifying planar diagrams, carefully subtracting forward singularities, and using a systematic mapping rule to express loop contributions as sums over tree amplitudes with inserted loop pairs. The authors provide explicit CHY-integrands for two-loop orderings and verify consistency by counting the number of terms against Feynman-diagram calculations, finding perfect agreement. This framework suggests a path to generalize CHY to higher loops and non-planar sectors, with potential implications for extending to Yang-Mills and gravity theories.

Abstract

In this paper, by treating massive loop momenta to massless momenta in higher dimension, we are able to treat all-loop scattering equations as tree ones. As an application of the new aspect, we consider the CHY-construction of bi-adjoint phi_3 theory. We present the explicit formula for two-loop planar integrands. We discuss carefully how to subtract various forward singularities in the construction. We count the number of terms obtained by our formula and by direct Feynman diagram calculation and find the perfect match, thus provide a strong support for our results.

Figures (9)

• Figure 1: The CHY-graph for Feynman diagrams with pole $s_{n1}$. (a) The CHY-graph for full $n$-point tree-level amplitude; (b) The "pinching" picture where the new vertex $A$ represents the combination of vertexes $1,n$. (c) The CHY-graph for Feynman diagrams containing pole $s_{n1}$ obtained from (b) after lifting the $(n-1)$-point graph to the $n$-point graph. (d) The CHY-graph after subtracting (c) from (a), where we have used arrows to indicate the direction.
• Figure 2: (a) The CHY-graph for full $n$-point tree-level amplitude; (b) The "pinching" picture where the new vertex $A$ represents the combination of vertexes $n,1,2$. (c) The CHY-graph for all Feynman diagrams with pole $s_{n12}$ obtained from (b) after lifting the $(n-2)$-point graph to the $n$-point graph. (d) The CHY-graph having the fixed poles $s_{n12}$ and $s_{n1}$; (e) The CHY-graph having the fixed poles $s_{n12}$ and $s_{12}$;
• Figure 3: The excluded one-loop Feynman diagrams of $\phi^3$ theory and their corresponding trees after the cut
• Figure 4: General planar two-loop Feynman diagrams Type (A) and Type (B) of $\phi^3$ theory. There are some special two-loop diagrams: (A-1) one-loop tadpole; (A-2) one-loop massless bubble; (B-1) two-loop tadpole; (B-2) two-loop massless bubble; and (B-3) Reducible two-loop diagrams.
• Figure 5: Singular contributions from the one-loop tadpoles and one-loop massless bubbles. At the four corners, we have four general cases. For example, the corner "L1-tadpole" means that the left one-loop is tadpole while the right one-loop can be general. Each pair of nearby corners has an intersection. For example, between the corner "L1-tadpole" and the corner "L2-bubble" we will have the diagram where the left one-loop is tadpole and the right one-loop is massless bubble. For each loop diagram, we have also drawn the corresponding tree diagrams after the cut. These pictures will be very useful when we discuss how to write down the CHY-integrand.