  We characterize Schatten $p$-class Hankel operators $H_f$ on the
Segal-Bargmann space when $0<p<\infty$ in terms of our recently introduced
notion of integral distance to analytic functions in $\mathbb{C}^n$. Our work
completes the study inspired by a theorem of Berger and Coburn on compactness
of Hankel operators and subsequently initiated twenty years ago by Xia and
Zheng, who obtained a characterization of the simultaneous membership of $H_f$
and $H_{\overline f}$ in Schatten classes $S_p$ when $1\le p<\infty$ in terms
of the standard deviation of $f$. As an application, we give a positive answer
to their question of whether $H_f\in S_p$ implies $H_{\overline{f}}\in S_p$
when $f\in L^\infty$ and $1<p<\infty$, which was previously solved for $p=2$
and $n=1$ by Xia and Zheng and for $p=2$ in any dimension by Bauer in 2004. In
addition, we prove our results in the context of weighted Segal-Bargmann
spaces, which include the standard and Fock-Sobolev weights.
