Loop integrals in de Sitter spacetime: The parity-split IBP system and $\mathrm{d}\log$-form differential equations

TL;DR

This paper introduces a novel IBP reduction and differential equations framework for massive loop integrals in de Sitter spacetime, revealing a parity-based structural property and extending $ ext{d} ext{log}$-form techniques.

Contribution

It develops a parity-split IBP system, formulates a Baikov representation for dS integrals, and conjectures $ ext{d} ext{log}$-form differential equations extend to de Sitter space.

Findings

The IBP system splits into $2^n$ parity-based subsystems.

A Baikov representation and dimensional recurrence relations are formulated for dS integrals.

The conjecture that $ ext{d} ext{log}$-form integrands extend to dS is verified in the one-loop bubble case.

Abstract

We develop integration-by-parts (IBP) reduction and differential equations for massive loop integrals of cosmological correlators in de Sitter (dS) spacetime, demonstrating the feasibility of this approach. We identify a structural property of the dS IBP system: for an $n$-propagator family, it splits into $2^n$ closed subsystems classified by the parity of the propagator indices. We further formulate a Baikov representation for loop integrals in dS space and derive the corresponding dimensional recurrence relations. In flat spacetime, intersection theory shows that $\mathrm{d}\log$-form master integrands lead to $\mathrm{d}\log$-form differential equations. Motivated by fibration intersection theory, we conjecture that this construction extends to dS integrands involving Hankel functions. We verify this conjecture in the one-loop bubble family and determine the associated alphabet.