  In this paper we propose a dispersive method to describe two-body scattering
with unitarity imposed. This approach is applied to elastic $\pi\pi$
scattering. The amplitudes keep single-channel unitarity and describe the
experimental data well, and the low-energy amplitudes are consistent with that
of chiral perturbation theory. The pole locations of the $\sigma$, $f_0(980)$,
$\rho(770)$ and $f_2(1270)$ and their couplings to $\pi\pi$ are obtained. A
virtual state appearing in the isospin-two S-wave is confirmed. The
correlations between the left (and right) hand cut and the poles are discussed.
Our results show that the poles are more sensitive to the right hand cut rather
than the left hand cut. The proposed method could be used to study other
two-body scattering processes.
