Pion-pion scattering and the timelike pion form factor from $N_f=2+1$ lattice QCD simulations using the stochastic LapH method

Abstract

We report on progress applying the stochastic LapH method to estimate all-to-all propagators required in correlation functions of multi-hadron operators relevant for pion-pion scattering. Largevolume results for I = 2 and I = 1 pion-pion scattering phase shifts with good statistical precision are obtained from an N f = 2 + 1 anisotropic Wilson clover ensemble with m π = 240MeV. We also present a preliminary determination of the I = 1 pion-pion scattering phase shift and timelike pion form factor on an isotropic N f = 2 + 1 flavour ensemble generated by the Coordinated Lattice Simulation (CLS) community effort.Pion-pion scattering and the timelike pion form factor Lattice QCD simulations are inevitably carried out in finite volume and euclidean time, which complicates scattering calculations [1]. Since most excited hadrons are unstable resonances which appear experimentally as features in scattering cross sections, first-principles calculation of hadronhadron scattering amplitudes is desirable. The relation between finite-volume two-hadron spectra and infinite-volume elastic scattering amplitudes was formulated by Lüscher [2, 3] more than two decades ago and extended to moving frames in Ref. [4]. However, only recently have algorithmic advances in lattice QCD spectroscopy enabled finite-volume spectra to be calculated efficiently in large volume with light pion masses.These advances concern the treatment of all-to-all propagators, which are required to give definite momenta to all hadrons and to treat valence-quark-line-disconnected Wick contractions. Laplacian-Heaviside (LapH) quark smearing projects the quark propagator onto the subspace spanned by the lowest-lying N v eigenmodes of the three-dimensional covariant Laplace operator [5]. Stochastic noise introduced only in the LapH subspace results in more efficient estimators for allto-all quark propagators compared to noise on the entire lattice [6].The spatial profile of the LapH subspace projector is approximately Gaussian, as with other quark smearing procedures. In order to maintain a constant physical smearing radius, N v must increase proportionally to the spatial volume. However, in Ref. [6] it was demonstrated that with a moderate amount of dilution in the LapH subspace, the number of quark matrix inversions can be held constant as the spatial volume is increased without increasing the stochastic estimation error relative to the gauge noise. This enables all-to-all quark propagators to be estimated efficiently in large spatial volumes for a reasonable cost.The efficient treatment of all-to-all quark propagators in turn enables precision calculation of correlation functions containing multi-hadron interpolating operators with definite momenta and/or disconnected Wick contractions. From these correlation functions, finite-volume energies can be precisely extracted, which then yield elastic scattering amplitudes. The application of these techniques to extract elastic pion-pion scattering amplitudes from large volume ensembles is the subject of this work. Se...

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