Lattice QCD study of the s-wave $\pi\pi $ scattering lengths in the I=0 and 2 channels

The s-wave pion-pion ($\pi\pi$) scattering lengths are computed below the inelastic threshold by the L\"uscher technique with pion masses ranging from 240 MeV to 463 MeV. In the Asqtad-improved staggered fermion formulation, we calculate the $\pi\pi$ four-point functions for the I=0 and 2 channels with "moving" wall sources without gauge fixing, and analyze them at the next-to-leading order in the continuum three-flavor chiral perturbation theory. At the physical pion mass, we secure the s-wave $\pi\pi$ scattering lengths as $m_\pi a_{\pi\pi}^{I=0} = 0.214(4)(7)$ and $m_\pi a_{\pi\pi}^{I=2} = -0.04430(25)(40)$ for the I=0 and 2 channels, respectively, where the first uncertainties are statistical and second ones are our estimates of several systematic effects. Our lattice results for the s-wave $\pi\pi$ scattering lengths are in well accordance with available experimental reports and theoretical forecasts at low momentum. A basic ingredient in our study for the I=0 case is properly incorporating disconnected diagram. These lattice computations are carried out with the MILC 2+1 flavor gauge configurations at two lattice spacings $a \approx 0.15$ and 0.12 fm.