A Surface Integrand for the Inverse KLT Kernel

TL;DR

This work introduces the inverse KLT integrand, a loop-level planar generalization of the inverse KLT kernel, defined on the Arkani-Hamed kinematic surface and computed by a novel cubic Berends-Giele recursion. It reveals a remarkable tree-level simplification: the infinite tower of odd-point interactions in the inverse KLT kernel collapses to a single stringy cubic root vertex, aligning the integrand with tr$\phi^3$ BAS amplitudes at all loops. The construction extends to loops via surface kinematics, producing all-loop integrands $m_{L,n}^{\alpha'}$ that include loop-specific contact terms, tadpoles, and loop-topologies, and connects to stringy pion (NLSM) amplitudes through α$'$-shifts. At leading order in $\alpha'$, the loop pion integrands reproduce the perfect NLSM structure, preserving Adler zeros and forward-limit behavior, thus unifying cubic scalars and pions across loop orders. The framework sets the stage for potential loop-level double-copy relations, loop-moduli-space interpretations, and extensions to other EFTs with special soft behavior.

Abstract

We propose a loop-level generalization of the inverse string theory Kawai-Lewellen-Tye (KLT) kernel: the planar inverse KLT integrand. The integrand is defined constructively via a novel Berends-Giele-like recursion that exposes the inverse KLT kernel as the simplest toy model of a ``stringy amplitude''. We show that, to all loop orders, the inverse KLT integrand is structurally equivalent to integrands in the cubic scalar tr$φ^3$ theory. This simplicity is obscured in the conventional Feynman diagram approach, where the inverse KLT integrand receives contributions from an infinity of infinite towers of contact interactions. The inverse KLT integrand is a rational function of stringified kinematic variables and is naturally defined on the kinematic surface proposed by Arkani-Hamed et al.. It provides an elementary analogue of the surfacehedron integrand for the tr$φ^3$ theory involving only scalar resonances and unifies the scattering of cubic scalars and pions in the non-linear sigma model (NLSM) to all loop orders via kinematic $α'$-shifts.

Figures (8)

• Figure 1: A generic Berends-Giele diagram is specified by a root vertex $V$ (valency $v$) and a compatible partition $P_V$ of external legs. The grey blobs can be identified with lower-point semi-on-shell amplitudes or Berends-Giele currents. The $n$-point amplitude is obtained recursively by summing over all Berends-Giele diagrams.
• Figure 2: Example for the canonical labeling of propagators in a planar Feynman diagram by the planar loop variables \ref{['defXloop1']}.
• Figure 3: Kinematic $X$-variables can be associated with curves on the kinematic surface. From left to right: examples for the tree-level $X$-variables \ref{['defX']} and the loop-level $X$-variables \ref{['defXloop1']}.
• Figure 4: Kinematic tadpole variables $X_{ii}$, $X_{z_iz_i}$ and off-shellness variables $X_{i,i+1}$ have a natural interpretation as curves on the kinematic surface involving one or more loop punctures $z_i$.
• Figure 5: Diagrammatic interpretation of kinematic tadpole variables $X_{ii}$, $X_{z_iz_i}$ and off-shellness variables $X_{i,i+1}$ as tadpole and bubble corrections on external and internal lines.