Critical fourth-order logarithmic asymptotics

Determine whether a term proportional to ε⁴ log(1/|ε|) occurs in the full-collision asymptotics of Sineβ correlation functions at the critical boundary mβ=3, and compute its coefficient, including exceptional profiles and parameter values for which the residue vanishes.

Background

At mβ=3, the fourth inverse moments reach their integrability boundary and the regular fourth-order coefficient develops an apparent pole. The paper notes that this pole alone does not establish the form of the critical asymptotics, because collision-scale effects may produce a logarithmic correction and cancellations may occur for special profiles or parameters.

References

At $m\beta=3$, identifying the coefficient of a possible $\varepsilon4\log(1/|\varepsilon|)$ term requires a collision-scale argument.

— Fourth-Order Fusion Asymptotics for $\mathrm{Sine}_β$ Correlation Functions  (2609.11543 - Fang, 10 Sep 2026) in Section 9, “Further questions”