Scrutinizing $\pi\pi$ scattering in light of recent lattice phase shifts

In this paper, the $IJ=00, 11, 20$ partial wave $\pi\pi$ scattering phase shifts determined by the lattice QCD approach are analyzed by using a novel dispersive solution of the S-matrix, i.e. the PKU representation, in which the unitarity and analyticity of scattering amplitudes are automatically satisfied and the phase shifts are conveniently decomposed into the contributions of the cuts and various poles, including bound states, virtual states and resonances. The contribution of the left-hand cut is estimated by the $SU(2)$ chiral perturbation theory to $\mathcal{O}(p^4)$. The Balanchandran-Nuyts-Roskies relations are considered as constraints to meet the requirements of the crossing symmetry. It is found that the $IJ=00$ $\pi\pi$ scattering phase shifts obtained at $m_\pi=391$ MeV by Hadron Spectrum Collaboration (HSC) reveal the presence of both a bound state pole and a virtual state pole below the $\pi\pi$ threshold rather than only one bound state pole for the $\sigma$. To reproduce the lattice phase shifts at $m_\pi=391$ MeV, a virtual-state pole in the $IJ=20$ channel is found to be necessary in order to balance the left-hand cut effects from the chiral amplitudes. Similar discussions are also carried out for the lattice results with $m_\pi=236$ MeV from HSC. The observed behaviors of the pole positions with respect to the variation of the pion masses can provide deep insights into our understanding of the dynamical origin of $\sigma$ resonance.