Self-decomposability of $\alpha$-Cauchy distributions

In 2009, Yano, Yano and Yor asked whether the $\alpha$-Cauchy distributions are self-decomposable or at least infinitely divisible in their study of hitting times for symmetric stable processes. Recently, the first named author proved that $\alpha$-Cauchy distribution is infinitely divisible if and only if $1<\alpha \le 2$. In this paper, we prove that $\alpha$-Cauchy distribution is self-decomposable if and only if $\alpha =2$. The proof is based on a new criterion for self-decomposability in the symmetric case. As an application, we show that the first hitting time of a nonzero point by a symmetric stable process of index $1<\alpha <2$ starting from zero is not self-decomposable.