An Algorithm for the Symbolic Reduction of Multi-loop Feynman Integrals via Generating Functions
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866917479166509056 |
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| author | Feng, Bo Li, Xiang Liu, Yuanche Ma, Yanqing Zhang, Yang |
| author_facet | Feng, Bo Li, Xiang Liu, Yuanche Ma, Yanqing Zhang, Yang |
| contents | We develop a generating-function formulation for the symbolic reduction of multi-loop Feynman integrals. In this framework, integration-by-parts identities are rewritten as differential equations for sector-wise generating functions, so the reduction problem can be studied in a non-commutative algebra of differential operators rather than only through relations among individual integrals. This viewpoint leads to an iterative algorithm that generates candidate equations, extracts symbolic reduction rules, updates the active rule set, and tests completeness on the lattice of integral indices. We illustrate the method with the sunset topology, planar and non-planar massless double-box topologies, representative subsectors, and a degenerate example in which the top sector contains no master integral. Together, these examples show how symbolic reduction rules, descendant equations, and completeness criteria can be organized within a single algebraic framework. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_09541 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Algorithm for the Symbolic Reduction of Multi-loop Feynman Integrals via Generating Functions Feng, Bo Li, Xiang Liu, Yuanche Ma, Yanqing Zhang, Yang High Energy Physics - Phenomenology High Energy Physics - Theory We develop a generating-function formulation for the symbolic reduction of multi-loop Feynman integrals. In this framework, integration-by-parts identities are rewritten as differential equations for sector-wise generating functions, so the reduction problem can be studied in a non-commutative algebra of differential operators rather than only through relations among individual integrals. This viewpoint leads to an iterative algorithm that generates candidate equations, extracts symbolic reduction rules, updates the active rule set, and tests completeness on the lattice of integral indices. We illustrate the method with the sunset topology, planar and non-planar massless double-box topologies, representative subsectors, and a degenerate example in which the top sector contains no master integral. Together, these examples show how symbolic reduction rules, descendant equations, and completeness criteria can be organized within a single algebraic framework. |
| title | An Algorithm for the Symbolic Reduction of Multi-loop Feynman Integrals via Generating Functions |
| topic | High Energy Physics - Phenomenology High Energy Physics - Theory |
| url | https://arxiv.org/abs/2605.09541 |