I=2$ $\pi\pi$ $s$-wave scattering length from lattice QCD

The $I=2$ $\pi\pi$ elastic $s$-wave scattering phase shift is measured by lattice QCD with $N_f=3$ flavors of the Asqtad-improved staggered fermions. The lattice-calculated energy-eigenvalues of $\pi\pi$ systems at one center of mass frame and some moving frames using the moving wall source technique are utilized to secure phase shifts by L\"uscher's formula. Our computations are fine enough to obtain threshold parameters: scattering length $a$, effective range $r$, and shape parameter $P$, which can be extrapolated at the physical point by NLO in chiral perturbation theory, and our relevant NNLO predictions from expanding NPLQCD's works are novelly considered as the systematic uncertainties. Our outcomes are consistent with Roy equation determinations, newer experimental data, and lattice estimations. Numerical computations are performed with a coarse ($a\approx0.12$~fm, $L^3 T = 32^3 64$), two fine ($a\approx0.09$~fm, $L^3 T = 40^3 96$) and a superfine ($a\approx0.06$~fm, $L^3 T = 48^3 144$) lattice ensembles at four pion masses of $m_\pi\sim247~{\rm MeV}$, $249~{\rm MeV}$, $275~{\rm MeV}$, and $384~{\rm MeV}$, respectively.