$I=0$ $\pi\pi$ $s$-wave scattering length from lattice QCD

Understand

We deliver lattice results for the $I=0$ $\pi\pi$ elastic $s$-wave scattering length calculated with the MILC $N_f=3$ flavors of the Asqtad-improved staggered fermions.

• The scattering phase shifts are determined by L\"uscher's formula from the energy-eigenvalues of $\pi\pi$ systems at one center of mass frame and four moving frames using the moving wall source technique.
• Our measurements are good enough to resolve the scattering length $a$ and effective range $r$, moreover, it allows us to roughly estimate the shape parameter $P$.
• Using our lattice results, the scattering length $a$ and effective range $r$ at the physical point are extrapolated by chiral perturbation theory.