The ${\pi }^{+}{\pi }^{+}$ $s$-wave scattering phase shift is determined below the inelastic threshold using lattice QCD. Calculations were performed at a pion mass of $m_{\pi }\sim 390\mathrm{MeV}$ with an anisotropic $n_f=2+1$ clover fermion discretization in four lattice volumes, with spatial extent $L\sim 2.0$, 2.5, 3.0 and 3.9 fm, and with a lattice spacing of $b_s\sim 0.123\mathrm{fm}$ in the spatial direction and $b_t\sim b_s/3.5$ in the time direction. The phase shift is determined from the energy eigenvalues of ${\pi }^{+}{\pi }^{+}$ systems with both zero and nonzero total momentum in the lattice volume using Lüscher’s method. Our calculations are precise enough to allow for a determination of the threshold scattering parameters, the scattering length $a$, the effective range $r$, and the shape parameter $P$, in this channel and to examine the prediction of two-flavor chiral perturbation theory: $m_{\pi }^{2}ar=3+\mathcal{O}(m_{\pi }^2/{\Lambda }_{\chi }^2)$. Chiral perturbation theory is used, with the lattice QCD results as input, to predict the scattering phase shift (and threshold parameters) at the physical pion mass. Our results are consistent with determinations from the Roy equations and with the existing experimental phase shift data.

• Received 20 September 2011

DOI:https://doi.org/10.1103/PhysRevD.85.034505