The $s$-wave pion-pion ($\pi \pi$) scattering lengths are computed below the inelastic threshold by the Lüscher technique with pion masses ranging from 240 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 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 $\pi \pi$ 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 the second ones are our estimates of several systematic effects. Our lattice results for the $s$-wave $\pi \pi$ 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, $a\approx 0.15$ and 0.12 fm.

• Received 28 January 2013

DOI:https://doi.org/10.1103/PhysRevD.87.074501