Scrutinizing <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>π</mml:mi><mml:mi>π</mml:mi></mml:mrow></mml:math> 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 lattice QCD approach are analyzed using a novel dispersive solution of S-matrix, i.e. PKU representation, in which unitarity and analyticity amplitudes automatically satisfied conveniently decomposed into contributions cuts various poles, including bound states, virtual states resonances. The contribution left-hand cut is estimated $SU(2)$ chiral perturbation theory to $\mathcal{O}(p^4)$. Balanchandran-Nuyts-Roskies relations considered as constraints meet requirements crossing symmetry. It found that $IJ=00$ obtained at $m_\pi=391$ MeV Hadron Spectrum Collaboration (HSC) reveal presence both state pole below threshold rather than only one for $\sigma$. To reproduce MeV, virtual-state $IJ=20$ channel be necessary order balance effects from amplitudes. Similar discussions also carried out results with $m_\pi=236$ HSC. observed behaviors positions respect variation pion masses can provide deep insights our understanding dynamical origin $\sigma$ resonance.