A note on kinematic flow and differential equations for two-site one-loop graph in FRW spacetime

In this work, we systematically study the differential systems governing loop-level wavefunction coefficients of conformally-coupled scalar field theory within a general power-law FRW cosmology. By utilizing the twisted cohomology, hyperplane arrangements, and IBP techniques, we derive the canonical differential equations for two-site one-loop bubble and tadpole systems, revealing distinct structural differences. We present new insights into the one-loop tadpole system, uncovering that its integral family can include multiple parent functions due to distinct pairs of relative hyperplane associated with each function, unlike the single parent function appearing in the one-loop bubble case. Moreover, we demonstrate that the tadpole correlator selectively probes only a subset of the cohomology space, despite the hyperplane arrangement suggesting a higher-dimensional structure. Another novel contribution of this work is the extension of kinematic flow framework to the loop-level scenarios for the first time. Using a graphical approach based on family trees generated by marked tubing graphs, which encode singularity structures, we efficiently construct the differential equations and uncover the hierarchical relationships among the associated master integrals. Additionally, we provide a preliminary discussion on generalization to two-site higher-loop configurations. We propose a general decomposition formula for the canonical form of a two-site diagram with arbitrary loops, breaking it into unshifted and shifted components associated with the fundamental tree-level and bubble-like structures, and establish a block-wise decomposition rule for the matrix \( \overset{\sim }{A} \) in the corresponding differential system. These advancements provide a unified framework for two-site loop-level correlators and lay the groundwork for future study of more complex multi-site loop systems.