Abstract
The $s$-wave pion-pion ($\ensuremath{\pi}\ensuremath{\pi}$) scattering lengths are computed below the inelastic threshold by the L\"uscher technique with pion masses ranging from 240 to 463 MeV. In the Asqtad-improved staggered fermion formulation, we calculate the $\ensuremath{\pi}\ensuremath{\pi}$ four-point functions for the $I=0$ and 2 channels with ``moving'' wall sources without gauge fixing, and we 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 $\ensuremath{\pi}\ensuremath{\pi}$ scattering lengths as ${m}_{\ensuremath{\pi}}{a}_{\ensuremath{\pi}\ensuremath{\pi}}^{I=0}=0.214(4)(7)$ and ${m}_{\ensuremath{\pi}}{a}_{\ensuremath{\pi}\ensuremath{\pi}}^{I=2}=\ensuremath{-}0.04430(25)(40)$ for the $I=0$ and 2 channels, respectively, where the first uncertainties are statistical and the second ones are our estimates of several systematic effects. Our lattice results for the $s$-wave $\ensuremath{\pi}\ensuremath{\pi}$ scattering lengths are in good 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 the disconnected diagram. These lattice computations are carried out with the MILC $2+1$ flavor gauge configurations at two lattice spacings, $a\ensuremath{\approx}0.15$ and 0.12 fm.
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