Let $u=(u_1,...,u_n), v=(v_1,...,v_n)$ be two random vectors independently and uniformly distributed on the unit sphere in $\mathbb{R}^n$. Define two other random variables $X=\sqrt{\sum_{i=1}^nu_i^2v_i^2}$, $Y=|u_1v_1|$. Consider the following ratio of expectation: $$r_n(\alpha)=\frac{\mathbb{E}\{\exp[-\frac{\alpha^2-\alpha^2X^2+\alpha X}{2}]\}}{\mathbb{E}\{\exp[-(\alpha^2-\alpha^2Y^2+\alpha Y)]\}}$$ Does there exist a finite upper bound for $r_n(\alpha)$, independent of $\alpha$, for all $\alpha\geq0$?