arXiv subjects

Yuanche Liu

Publications and source records attributed to Yuanche Liu.

6 recordsLinked to original sources

arXiv2026

CHESS: CHEbyshev pSeudo-Spectral transport for Feynman integral differential equations

We present CHESS (CHEbyshev pSeudo Spectrum), a Wolfram Language package for high-precision one-dimensional transport of {\epsilon}-factorized differential equations for Feynman master integrals. The solver works with the matrix obtained by pulling a differential one-form to a chosen path. This matrix may be supplied directly, or assembled from constant matrices and precomputed scalar pullbacks of the one-forms. The program combines Chebyshev-Lobatto spectral collocation, sparse matrix assembly, sequential propagation in the {\epsilon}-expansion, and residue-based regularization of spurious regular singular endpoints. Benchmarks for large multi-scale integral families show rapid node convergence and agreement with independent reference data where such data are available. In the fixed local-series comparison used here, the Chebyshev transports also give shorter wall times; the reported process-tree memory usage is comparable for the smaller parallel runs and lower for the largest benchmark system in that comparison.

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arXiv2026

An Algorithm for the Symbolic Reduction of Multi-loop Feynman Integrals via Generating Functions

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.

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arXiv2026

Two-loop Six-point Planar Massless Feynman Integrals to Higher $\epsilon$ Orders

In this work, we calculate two-loop six-point planar massless Feynman integrals at higher orders in the dimensional regulator $\epsilon$, corresponding to higher transcendental weights. In previous works, these integrals were calculated up to weight four for the purpose of two-loop gauge theory amplitudes. Using modern rational reconstruction methods, we identify the complete alphabet with $269$ letters relevant to all weights, derive the analytic canonical differential equation and obtain the symbols up to weight six. As a proof of concept, using a new method with Chebyshev pseudospectral transport, we show that the corresponding pure basis can be efficiently evaluated up to weight six, i.e., to $ \mathcal{O}(\epsilon^2)$ in a physical scattering region. The results of this work can be applied to future three-loop amplitudes and provide new data for the formal study of symbols and cluster algebras.

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arXiv2025

Uncovering Singularities in Feynman Integrals via Machine Learning

We introduce a machine-learning framework based on symbolic regression to extract the full symbol alphabet of multi-loop Feynman integrals. By targeting the analytic structure rather than reduction, the method is broadly applicable and interpretable across different families of integrals. It successfully reconstructs complete symbol alphabets in nontrivial examples, demonstrating both robustness and generality. Beyond accelerating computations case by case, it uncovers the analytic structure universally. This framework opens new avenues for multi-loop amplitude analysis and provides a versatile tool for exploring scattering amplitudes.

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arXiv2025

Symbolic Reduction of Multi-loop Feynman Integrals via Generating Functions

We introduce a novel, systematic method for the complete symbolic reduction of multi-loop Feynman integrals, leveraging the power of generating functions. The differential equations governing these generating functions naturally yield symbolic recurrence relations. We develop an efficient algorithm that utilizes these recurrences to reduce integrals to a minimal set of master integrals. This approach circumvents the exponential growth of traditional integration-by-parts relations, enabling the reduction of high-rank, multi-loop integrals critical for state-of-the-art calculations in perturbative quantum field theory.

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arXiv2024

An Analytic Computation of Three-Loop Five-Point Feynman Integrals

We evaluate the three-loop five-point pentagon-box-box massless integral family in the dimensional regularization scheme, via canonical differential equation. We use tools from computational algebraic geometry to enable the necessary integral reductions. The boundary values of the differential equation are determined analytically in the Euclidean region. To express the final result, we introduce a new representation of weight six functions in terms of one-fold integrals over the product of weight-three functions with weight-two kernels that are derived from the differential equation. Our work paves the way to the analytic computation of three-loop multi-leg Feynman integrals.

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