We determine elastic and coupled-channel amplitudes for isospin-1 meson-meson scattering in P -wave, by calculating correlation functions using lattice QCD with light quark masses such that mπ = 236 MeV in a cubic volume of ∼ (4 fm)3 . Variational analyses of large matrices of correlation functions computed using operator constructions resembling ππ, KK and qq, in several moving frames and several lattice irreducible representations, leads to discrete energy spectra from which scattering amplitudes are extracted. In the elastic ππ scattering region we obtain a detailed energydependence for the phase-shift, corresponding to a ρ resonance, and we extend the analysis into the coupled-channel KK region for the first time, finding a small coupling between the channels.The study of hadron spectroscopy from first principles QCD is entering a new stage of development where the relationship between the discrete spectrum of the theory in a finite-volume and the infinite-volume scattering amplitudes is being practically utilized to study resonances. The tool which allows us access to the spectrum is lattice QCD in which the quark and gluon fields are considered on a finite-grid of points with only systematically improvable approximations being made.Predictably it is the simplest resonant scattering channel which has attracted the greatest initial interest [1][2][3][4][5], that of ππ with isospin=1, in which a low-lying elastic vector resonance called the ρ appears. These works have made use of the formalism relating the discrete spectrum at rest and in moving frames to elastic scattering amplitudes, which has been in place for many years [6][7][8][9][10]. Recently the extension to coupled-channels has been presented [11][12][13][14], and the first lattice QCD study of a coupled-channel system, that of πK, ηK in S, P and Dwaves, has appeared [15,16], showing that the energy dependence and resonant content of the scattering matrix for such a system can be extracted from finite volume spectra.To date virtually all determinations of hadron scattering amplitudes in lattice QCD calculations have worked with artificially heavy u, d quark mass values, a choice which leads to heavier than physical pseudoscalar mesons -this reduces the computational cost, allowing calculations in smaller volumes (where m π L remains large), and pushes up in energy the thresholds for multihadron scattering such as ππππ, for which a finite-volume formalism is not yet in place (but see Refs. [17][18][19][20] The Hadron Spectrum Collaboration previously computed ππ scattering using 391 MeV pions [21,22], extracting detailed spectra of QCD eigenstates from variational analysis of two-point correlation functions computed in several moving frames in three different volumes. By obtaining a significant number of energy levels in the elastic scattering region they were able to map out the energy dependence of the scattering amplitude and show that there is a narrow ρ resonance barely above ππ threshold.In this paper we deliver an extension of the work presen...

The vast majority of hadrons observed in nature are not stable under the strong interaction, rather they are resonances whose existence is deduced from enhancements in the energy dependence of scattering amplitudes. The study of hadron resonances offers a window into the workings of quantum chromodynamics (QCD) in the low-energy non-perturbative region, and in addition, many probes of the limits of the electroweak sector of the Standard Model consider processes which feature hadron resonances. From a theoretical standpoint, this is a challenging field: the same dynamics that binds quarks and gluons into hadron resonances also controls their decay into lighter hadrons, so a complete approach to QCD is required. Presently, lattice QCD is the only available tool that provides the required non-perturbative evaluation of hadron observables. In this article, we review progress in the study of few-hadron reactions in which resonances and bound-states appear using lattice QCD techniques. We describe the leading approach which takes advantage of the periodic finite spatial volume used in lattice QCD calculations to extract scattering amplitudes from the discrete spectrum of QCD eigenstates in a box. We explain how from explicit lattice QCD calculations, one can rigorously garner information about a variety of resonance properties, including their masses, widths, decay couplings, and form factors. The challenges which currently limit the field are discussed along with the steps being taken to resolve them.

The quantization condition for interacting energy eigenvalues of the two-nucleon system in a finite cubic volume is derived in connection to the nucleon-nucleon scattering amplitudes. This condition is derived using an auxiliary (dimer) field formalism that is generalized to arbitrary partial waves in the context of non-relativistic effective field theory. The quantization condition presented gives access to the scattering parameters of the two-nucleon systems with arbitrary parity, spin, isospin, angular momentum and center of mass motion, from a lattice QCD calculation of the energy eigenvalues. In particular, as it includes all non-central interactions, such as the two-nucleon tensor force, it makes explicit the dependence of the mixing parameters of nucleon-nucleon systems calculated from lattice QCD when there is a physical mixing among different partial-waves, e. g. S-D mixing in the deuteron channel. We provide explicit relations among scattering parameters and their corresponding point group symmetry class eigenenergies with orbital angular momentum l ≤ 3, and for center of mass boost vectors of the form. L denotes the special extent of the cubic volume and n 1 , n 2 , n 3 are integers. Our results are valid below inelastic thresholds up to exponential volume corrections that are governed by the pion mass. a briceno@uw.edu b davoudi@uw.edu c luu5@llnl.gov 1 arXiv:1305.4903v2 [hep-lat]

We present for the first time a determination of the energy dependence of the isoscalar ππ elastic scattering phase-shift within a first-principles numerical lattice approach to QCD. Hadronic correlation functions are computed including all required quark propagation diagrams, and from these the discrete spectrum of states in the finite volume defined by the lattice boundary is extracted. From the volume dependence of the spectrum we obtain the S-wave phase-shift up to the KK threshold. Calculations are performed at two values of the u, d quark mass corresponding to mπ = 236, 391 MeV and the resulting amplitudes are described in terms of a σ meson which evolves from a bound-state below ππ threshold at the heavier quark mass, to a broad resonance at the lighter quark mass.