  The quantum $XY$ spin chain with the interactions decaying as a power law
$1/r^{1+ \sigma}$ of the distance between spins $r$ was investigated with the
exact diagonalization method. Here, the constituent spin is set to $S=1$, which
enables us to incorporate the biquadratic interactions so as to realize the
order-disorder transition with the O$(2)$ symmetry maintained. Thereby, in the
ordered phase, we resolved the Higgs mass $m_H$ out of the Goldstone-excitation
continuum by specifying Higgs-particle's quantum numbers to adequate indices.
We then turn to the analysis of the critical amplitude ratio $m_H / \Delta$
($\Delta$: paramagnetic gap in the disordered phase). As the power of the
algebraic decay $\sigma$ increases, the amplitude ratio $m_H/\Delta$ gets
enhanced gradually in agreement with the
$\epsilon(=4-D)$-expansion-renormalization-group result; here, we resort to the
$\sigma \leftrightarrow D$ relation advocated recently in order to establish a
relationship between the renormalization-group result and ours.
