Gessel-Type Expansion for the Circular $\beta$-Ensemble and Central Limit Theorem for the Sine-$\beta$ Process for $\beta\le 2$

We obtain a Gessel-type expansion in Jack polynomials for the expectations of multiplicative functionals in the circular $\beta$-ensemble. As a consequence, we establish a Szeg\H{o}-type limit theorem for all $H^{1/2}(\mathbb{T})$ functions when $\beta \le 2$, together with an explicit rate of convergence for functions from $H^1(\mathbb{T})$. The estimate is stable under the scaling limit to the sine-$\beta$ process and yields a Soshnikov-type central limit theorem for the sine-$\beta$ process in the full $H^{1/2}(\mathbb{R})$ class.