A new generating-function framework turns IBP relations into differential equations in a non-commutative algebra, yielding an iterative algorithm that derives symbolic reduction rules and checks completeness for topologies such as the sunset and double-box diagrams.
Feynman integral reduction with intersection theory made simple
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abstract
Feynman integral reduction based on intersection theory provides an alternative to the traditional integration-by-parts method, yet its practical application has been constrained by the large number of variables required in the computation. In this Letter, we demonstrate that by employing the recently introduced branch representation, the reduction of $L$-loop Feynman integrals with an arbitrary number of external legs can be achieved through the computation of at most $(3L-3)$-variable intersection numbers. This constitutes a significant simplification compared to existing approaches, particularly for multi-leg integrals where the number of variables in conventional methods scales with the total number of propagators. We validate the proposed method through explicit calculations of two-loop diagrams, demonstrating substantial improvements in computational efficiency relative to both traditional intersection-theory approaches and standard integration-by-parts reduction techniques.
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2026 2verdicts
UNVERDICTED 2representative citing papers
A parity-split IBP system for n-propagator families in de Sitter space is identified, along with a conjecture that dlog-form differential equations extend to dS integrands with Hankel functions, verified for the one-loop bubble.
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An Algorithm for the Symbolic Reduction of Multi-loop Feynman Integrals via Generating Functions
A new generating-function framework turns IBP relations into differential equations in a non-commutative algebra, yielding an iterative algorithm that derives symbolic reduction rules and checks completeness for topologies such as the sunset and double-box diagrams.
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Loop integrals in de Sitter spacetime: The parity-split IBP system and $\mathrm{d}\log$-form differential equations
A parity-split IBP system for n-propagator families in de Sitter space is identified, along with a conjecture that dlog-form differential equations extend to dS integrands with Hankel functions, verified for the one-loop bubble.