Grzegorz Biskowski | Searcharxiv

arXiv2026

A perturbative Liouville prescription for the celestial three-gluon amplitude

We study the celestial three-gluon amplitude in a dilaton background through the Mellin-Liouville formulation proposed by Stieberger, Taylor and Zhu (STZ). The original map contains an ambiguity in the identification of Liouville and Mellin variables; we resolve it by requiring global conformal covariance and compatibility with the semiclassical expansion of Liouville theory. This uniquely fixes the operator normalization and the parameter dictionary, and leads to a controlled expansion in the Liouville coupling $b$. Starting from the full Liouville DOZZ three-point function, we derive the leading and first subleading terms in the $b^2$ expansion. The leading term reproduces the tree-level Yang-Mills amplitude in the small total momentum limit, as anticipated in the STZ proposal. The one-loop correction can be written in closed form using modified Bessel functions, and its soft limit exhibits a clear separation into geometric and logarithmic contributions. The resulting framework extends the STZ proposal to finite-$b$ corrections in a consistent and computable way.

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arXiv2025

2D CFT and efficient Bethe ansatz for exactly solvable Richardson-Gaudin models

This work inaugurates a series of complementary studies on Richardson-Gaudin integrable models. We begin by reviewing the foundations of classical and quantum integrability, recalling the algebraic Bethe ansatz solution of the Richardson (reduced BCS) and Gaudin (central spin) models, and presenting a proof of their integrability based on the Knizhnik-Zamolodchikov equations and their generalizations to perturbed affine conformal blocks. Building on this foundation, we then describe an alternative CFT-based formulation. In this approach, the Bethe ansatz equations for these exactly solvable models are embedded within two-dimensional Virasoro CFT via irregular, degenerate conformal blocks. To probe new formulations within the Richardson-Gaudin class, we develop a high-performance numerical solver. The Bethe roots are encoded in the Baxter polynomial, with initial estimates obtained from a secular matrix eigenproblem and subsequently refined using a deflation-assisted hybrid Newton-Raphson/Laguerre algorithm. The solver proves effective in practical applications: when applied to picket-fence, harmonic oscillator, and hydrogen-like spectra, it accurately reproduces known rapidity trajectories and reveals consistent merging and branching patterns of arcs in the complex rapidity plane. We also explain how to generalize our computational approach to finite temperatures, allowing us to calculate temperature-dependent pairing energies and other thermodynamic observables directly within the discrete Richardson model. We propose an application of the solver to Gaudin-type Bethe equations, which emerge in the classical (large central charge) limit of Virasoro conformal blocks. We conclude by outlining future directions: direct minimization of the Yang-Yang function as an alternative root-finding strategy; revisiting time-dependent extensions; and ... .

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