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.

We present a quantization condition for the spectrum of a system composed of three identical bosons in a finite volume with periodic boundary conditions. This condition gives a relation between the finite volume spectrum and infinite volume scattering amplitudes. The quantization condition presented is an integral equation that in general must be solved numerically. However, for systems with an attractive two-body force that supports a two-body bound-state, a diboson, and for energies below the diboson breakup, the quantization condition reduces to the well-known Lüscher formula with exponential corrections in volume that scale with the diboson binding momentum. To accurately determine infinite volume phase shifts, it is necessary to extrapolate the phase shifts obtained from the Lüscher formula for the boson-diboson system to the infinite volume limit. For energies above the breakup threshold, or for systems with no two-body bound-state (with only scattering states and resonances) the Lüscher formula gets power-law volume corrections and consequently fails to describe the three-particle system. These corrections are nonperturbatively included in the quantization condition presented.

We perform a model-independent, non-perturbative investigation of two-point and three-point finite-volume correlation functions in the energy regime where two-particle states can go on-shell. We study three-point functions involving a single incoming particle and an outgoing two-particle state, relevant, for example, for studies of meson decays (e.g., B 0 → K * + − → πK + − ) or meson photo production (e.g., πγ * → ππ). We observe that, while the spectrum solely depends on the on-shell scattering amplitude, the correlation functions also depend on off-shell amplitudes. The main result of this work is a generalization of the Lellouch-Lüscher formula relating matrix elements of currents in finite and infinite spatial volumes. We extend that work by considering a theory with multiple, strongly-coupled channels and by accommodating external currents which inject arbitrary four-momentum as well as arbitrary angular momentum. The result is exact up to exponentially suppressed corrections governed by the pion mass times the box size. We also apply our master equation to various examples, including the two processes mentioned above as well as examples where the final state is an admixture of two open channels.

Working in relativistic quantum field theory, we derive the quantization condition satisfied by coupled two-and three-particle systems of identical scalar particles confined to a cubic spatial volume with periodicity L. This gives the relation between the finite-volume spectrum and the infinite-volume 2 → 2, 2 → 3 and 3 → 3 scattering amplitudes for such theories. The result holds for relativistic systems composed of scalar particles with nonzero mass m, whose center of mass energy lies below the four-particle threshold, and for which the two-particle K matrix has no singularities below the three-particle threshold. The quantization condition is exact up to corrections of the order O(e −mL ) and holds for any choice of total momenta satisfying the boundary conditions. arXiv:1701.07465v2 [hep-lat] 30 Apr 2017 , FIG. 1: An example of a Feynman diagram contributing to the finite-volume correlator. Above the three-particle threshold, this diagram has cuts due to both two-and three-particle states, as shown on the right-hand side. The three-particle cut runs through a self-energy bubble, meaning that such diagrams must be explicitly displayed (rather than subsumed into a dressed propagator) in order to properly identify all finite-volume effects.generalize those of Refs.[29] and [30], respectively. The first, Eq. (79), has a form reminiscent of the coupled twoparticle result [7,[10][11][12][13]. The finite-volume effects are contained in a diagonal two-by-two matrix with entries F 2 in the two-particle sector and F 3 in the three-particle sector. Aside from minor technical changes, these are the same finite-volume quantities that arise in the previously derived two-and three-particle quantization conditions [4, 5, 8, 12-14, 29, 30]. The coupling between channels is captured by the generalized divergence-free K matrix. This contains diagonal elements, mediating two-to-two and three-to-three transitions, as well as off-diagonal elements that encode the two-to-three transitions.To obtain both the quantization condition and the relation to the scattering amplitude from a single calculation, we use a matrix of finite-volume correlators, M L , chosen so that it goes over to the corresponding matrix of infinitevolume scattering amplitudes when the L → ∞ limit is taken appropriately. This differs from the type of correlator used in Ref. [29], but is the direct generalization of that considered in Ref. [30].The results of this work, like those given in Refs. [4, 5, 8, 12-14, 29, 30], are derived by analyzing an infinite set of finite-volume Feynman diagrams and identifying the power-law finite-volume effects. The central complication new to the present derivation comes from diagrams such as that of Fig. 1, in which a two-to-three transition is mediated by a one-to-two transition together with a spectator particle. The cuts on the right-hand side of the figure indicate that this diagram gives rise to finite-volume effects from both two-and three-particle states. As we describe in detail below, a consequence of such diagrams is...

The spectrum of a system with multiple channels composed of two hadrons with nonzero total momentum is determined in a finite cubic volume with periodic boundary conditions using effective field theory methods. The results presented are accurate up to exponentially suppressed corrections in the volume due to the finite range of hadronic interactions. The formalism allows one to determine the phase shifts and mixing parameters of ππ − KK isosinglet coupled channels directly from Lattice Quantum Chromodynamics. We show that the extension to more than two channels is straightforward and present the result for three channels. From the energy quantization condition, the volume dependence of electroweak matrix elements of two-hadron processes is extracted. In the non-relativistic case, we pay close attention to processes that mix the 1 S 0 − 3 S 1 two-nucleon states, e.g. proton-proton fusion (pp → d + e + + ν e ), and show how to determine the transition amplitude of such processes directly from lattice QCD.