  The criticality of the low-frequency conductivity for the bilayer quantum
Heisenberg model was investigated numerically. The dynamical conductivity
(associated with the O$(3)$ symmetry) displays the inductor $\sigma (\omega)
=(i\omega L)^{-1}$ and capacitor $i \omega C$ behaviors for the ordered and
disordered phases, respectively. Both constants, $C$ and $L$, have the same
scaling dimension as that of the reciprocal paramagnetic gap $\Delta^{-1}$.
Then, there arose a question to fix the set of critical amplitude ratios among
them. So far, the O$(2)$ case has been investigated in the context of the
boson-vortex duality. In this paper, we employ the exact diagonalization
method, which enables us to calculate the paramagnetic gap $\Delta$ directly.
Thereby, the set of critical amplitude ratios as to $C$, $L$ and $\Delta$ are
estimated with the finite-size-scaling analysis for the cluster with $N \le 34$
spins.
