Lattice QCD study of the scalar mesons<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:msub><mml:mi>a</mml:mi><mml:mn>0</mml:mn></mml:msub><mml:mo stretchy="false">(</mml:mo><mml:mn>1450</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>and<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>σ</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mn>600</mml:mn><mml:mo stretchy="false">)</mml:mo></mml:math>

We study the ${a}_{0}$ and $\ensuremath{\sigma}$ mesons with the overlap fermion in the chiral regime with the pion mass as low as 182 MeV in the quenched approximation. After the ${\ensuremath{\eta}}^{\ensuremath{'}}\ensuremath{\pi}$ ghost states are separated, we find the ${a}_{0}$ mass with the $q\overline{q}$ interpolation field to be almost independent of the quark mass in the region below the strange quark mass. The chirally extrapolated results are consistent with ${a}_{0}(1450)$ being the $u\overline{d}$ meson and ${K}_{0}^{*}(1430)$ being the $u\overline{s}$ meson with calculated masses at $1.42\ifmmode\pm\else\textpm\fi{}0.13\text{ }\text{ }\mathrm{GeV}$ and $1.41\ifmmode\pm\else\textpm\fi{}0.12\text{ }\text{ }\mathrm{GeV}$, respectively. We also calculate the scalar mesonium with a tetraquark interpolation field. In addition to the two-pion scattering states, we find a state at $\ensuremath{\sim}550\text{ }\text{ }\mathrm{MeV}$. Through the study of volume dependence, we confirm that this state is a one-particle state, in contrast to the two-pion scattering states. This suggests that the observed state is a tetraquark mesonium which is quite possibly the $\ensuremath{\sigma}(600)$ meson.