  The aim of this paper is to study the Lelong number, the integrability index
and the Monge-Amp\`ere mass at the origin of an $S^1$-invariant
plurisubharmonic function on a balanced domain in $\mathbb{C}^n$ under the
Schwarz symmetrization. We prove that $n$ times the integrability index is
exactly the Lelong number of the symmetrization, and if the function is further
toric with a single pole at the origin, then the Monge-Amp\`ere mass is always
decreasing under the symmetrization
