Integration Rules for Loop Scattering Equations

TL;DR

This work extends the CHY scattering-equation formalism from tree level to one loop in scalar $\varphi^3$-theory by formulating integration rules that avoid solving the loop-level equations directly. The authors build a forward-limit representation with two off-shell legs and derive a loop measure that respects GL(1) scaling, introducing a modified integrand that uses off-shell subsets via $[P]$ and excludes tadpole-containing regions. They demonstrate, through explicit 2-, 3-, and 4-point examples, that the loop CHY construction reproduces the corresponding Feynman one-loop amplitudes after regularization, with CHY graphs providing a compact bookkeeping tool. The results illustrate how to generalize tree-level integration rules to loop level and set the stage for higher-loop extensions, while noting the essential role of regularization to permit loop-momentum shifts and to regulate singular forward-limit contributions.

Abstract

We formulate new integration rules for one-loop scattering equations analogous to those at tree-level, and test them in a number of non-trivial cases for amplitudes in scalar $φ^3$-theory. This formalism greatly facilitates the evaluation of amplitudes in the CHY representation at one-loop order, without the need to explicitly sum over the solutions to the loop-level scattering equations.

Figures (9)

• Figure 1:
• Figure 2: The Feynman diagram at two points.
• Figure 3: The CHY graphs for two-point (left), three-point (middle) and four-point(right).
• Figure 4: The triangle contribution at three points.
• Figure 5: The three bubble contributions at three points.