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The coupling of various charmonium mesons to a photon is studied using lattice QCD, giving access to radiative form factors and transitions, and probing the mesons' structure. Methods are developed which allow the robust determination of amplitudes, including those involving an excited state as well as multiple form factors, for a range of kinematics. These are applied in dynamical lattice QCD calculations using the distillation technique to compute the underlying three-point correlation functions. Form factors and amplitudes involving the low-lying charmonia, $\eta_c$, $J/\psi$, $\chi_{c0}$ and $\eta_c'$, are calculated, demonstrating the methods and providing the first dynamical lattice QCD results for some quantities.

Isospin-1/2 $D��$ scattering amplitudes are computed using lattice QCD, working in a single volume of approximately $(3.6\; \mathrm{fm})^3$ and with a light quark mass corresponding to $m_��\approx239$ MeV. The spectrum of the elastic $D��$ energy region is computed yielding 20 energy levels. Using the L��scher finite-volume quantisation condition, these energies are translated into constraints on the infinite-volume scattering amplitudes and hence enable us to map out the energy dependence of elastic $D��$ scattering. By analytically continuing a range of scattering amplitudes, a $D_0^\ast$ resonance pole is consistently found strongly coupled to the $S$-wave $D��$ channel, with a mass $m\approx 2200$ MeV and a width $��\approx400$ MeV. Combined with earlier work investigating the $D_{s0}^\ast$, and $D_0^\ast$ with heavier light quarks, similar couplings between each of these scalar states and their relevant meson-meson scattering channels are determined. The mass of the $D_0^\ast$ is consistently found well below that of the $D_{s0}^\ast$, in contrast to the currently reported experimental result., 40 pages, 15 figures. Various small changes to match the version accepted by JHEP

Focusing on three-pion states with maximal isospin ($��^+��^+��^+$), we present the first non-perturbative determination of an energy-dependent three-hadron scattering amplitude from first-principles QCD. The calculation combines finite-volume three-hadron energies, extracted using numerical lattice QCD, with a relativistic finite-volume formalism, required to interpret the results. To fully implement the latter, we also solve integral equations that relate an intermediate three-body K matrix to the physical three-hadron scattering amplitude. The resulting amplitude shows rich analytic structure and a complicated dependence on the two-pion invariant masses, represented here via Dalitz-like plots of the scattering rate., 22 pages, 7 figures, 11 tables, CERN-TH-2020-147, JLAB-THY-20-3242, v2: minor edits, add references, added a curve to Fig. 2; corresponds to version published in PRL

We present the first lattice QCD calculation of coupled $����$ and $����$ scattering, incorporating coupled $S$ and $D$-wave $����$ in $J^P=1^+$. Finite-volume spectra in three volumes are determined via a variational analysis of matrices of two-point correlation functions, computed using large bases of operators resembling single-meson, two-meson and three-meson structures, with the light-quark mass corresponding to a pion mass of $m_��\approx 391$ MeV. Utilizing the relationship between the discrete spectrum of finite-volume energies and infinite-volume scattering amplitudes, we find a narrow axial-vector resonance ($J^{PC}=1^{+-}$), the analogue of the $b_1$ meson, with mass $m_{R}\approx1380$ MeV and width $��_{R}\approx 91$ MeV. The resonance is found to couple dominantly to $S$-wave $����$, with a much-suppressed coupling to $D$-wave $����$, and a negligible coupling to $����$ consistent with the `OZI rule'. No resonant behavior is observed in $����$, indicating the absence of a putative low-mass $Z_s$ analogue of the $Z_c$ claimed in $��J/��$. In order to minimally present the contents of a unitary three-channel scattering matrix, we introduce an $n$-channel generalization of the traditional two-channel Stapp parameterization.

We present a determination of the isospin-1/2 elastic $��K$ scattering amplitudes in S and P partial waves using lattice Quantum Chromodynamics. The amplitudes, constrained for a large number of real-valued energy points, are obtained as a function of light-quark mass, corresponding to four pion masses between 200 and 400 MeV. Below the first inelastic threshold, the P-wave scattering amplitude is dominated by a single pole singularity that evolves from being a stable bound-state at the highest quark mass into a narrow resonance that broadens as the pion and kaon masses are reduced. As in experiment, the S-wave amplitude does not exhibit an obviously resonant behavior, but instead shows a slow rise from threshold, which is not inconsistent with the presence of a $��$/$K_0^\star(700)$-like resonance at the considered quark masses. As has been found in analyses of experimental scattering data, simple analytic continuations into the complex energy plane of precisely-determined lattice QCD amplitudes on the real energy axis are not sufficient to model-independently determine the existence and properties of this state. The spectra and amplitudes we present will serve as an input for increasingly elaborate amplitude analysis techniques that implement more of the analytic structure expected at complex energies., 11 pages, 5 figures

We present the first determination of $����$ scattering, incorporating dynamically-coupled partial-waves, using lattice QCD, a first-principles numerical approach to QCD. Considering the case of isospin-2 $����$, we calculate partial-wave amplitudes with $J \le 3$ and determine the degree of dynamical mixing between the coupled $S$ and $D$-wave channels with $J^P=1^+$. The analysis makes use of the relationship between scattering amplitudes and the discrete spectrum of states in the finite volume lattice. Constraints on the scattering amplitudes are provided by over one hundred energy levels computed on two lattice volumes at various overall momenta and in several irreducible representations of the relevant symmetry groups. The spectra follow from variational analyses of matrices of correlations functions computed with large bases of meson-meson operators. Calculations are performed with degenerate light and strange quarks tuned to the physical strange quark mass so that $m_��\sim 700$ MeV, ensuring that the $��$ is stable against strong decay. This work demonstrates the successful application of techniques, opening the door to calculations of scattering processes that incorporate the effects of dynamically-coupled partial-waves, including those involving resonances or bound states., Minor changes to match the published version

We present preliminary calculations towards radiative transitions in charmonium using anisotropic $\textit{Nf}$=2+1 dynamical ensembles generated by the Hadron Spectrum Collaboration. With the use of newer technologies we aim to investigate transitions between states, including potential exotic charmonium states, lying higher in the spectrum than in previous studies. A crucial ingredient in this work is the use of variationally optimised interpolating operators which allow for a reliable determination of the three-point correlation functions needed. Using these operators, we perform first calculations of relevant three-point correlation functions before discussing future directions., We thank our colleagues within the Hadron Spectrum Collaboration. COH acknowledges support from the School of Mathematics at Trinity College Dublin. GM acknowledges support from the Herchel Smith Fund at the University of Cambridge. SMR acknowledges support from Science Foundation Ireland [RFP-PHY-3201]. CET acknowledges support from the U.K. Science and Technology Facilities Council (STFC) [grant ST/L000385/1].

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 $m_��= 236$ MeV in a cubic volume of $\sim (4 \,\mathrm{fm})^3$. Variational analyses of large matrices of correlation functions computed using operator constructions resembling $����$, $K\overline{K}$ and $q\bar{q}$, in several moving frames and several lattice irreducible representations, leads to discrete energy spectra from which scattering amplitudes are extracted. In the elastic $����$ scattering region we obtain a detailed energy-dependence for the phase-shift, corresponding to a $��$ resonance, and we extend the analysis into the coupled-channel $K\overline{K}$ region for the first time, finding a small coupling between the channels., matches published version, minor typos corrected

We present the first ab initio calculation of a radiative transition of a hadronic resonance within quantum chromodynamics (QCD). We compute the amplitude for $����\to ����^\star$, as a function of the energy of the $����$ pair and the virtuality of the photon, in the kinematic regime where $����$ couples strongly to the unstable $��$ resonance. This exploratory calculation is performed using a lattice discretization of QCD with quark masses corresponding to $m_��\approx 400$~MeV. We obtain a description of the energy dependence of the transition amplitude, constrained at 48 kinematic points, that we can analytically continue to the $��$ pole and identify from its residue the $��\to ����^\star$ form factor., V2, version accepted for publication in PRL. More references added

Coupled-channel $��K$ and $��K$ scattering amplitudes are determined by studying the finite-volume energy spectra obtained from dynamical lattice QCD calculations. Using a large basis of interpolating operators, including both those resembling a $q\bar{q}$ construction and those resembling a pair of mesons with relative momentum, a reliable excited-state spectrum can be obtained. Working at ${m_��=391\,\mathrm{MeV}}$, we find a gradual increase in the $J^P=0^+$ $��K$ phase-shift which may be identified with a broad scalar resonance that couples strongly to $��K$ and weakly to $��K$. The low-energy behavior of this amplitude suggests a virtual bound-state that may be related to the $��$ resonance. A bound state with $J^P=1^-$ is found very close to the $��K$ threshold energy, whose coupling to the $��K$ channel is compatible with that of the experimental $K^\star(892)$. Evidence is found for a narrow resonance in $J^P=2^+$. Isospin--3/2 $��K$ scattering is also studied and non-resonant phase-shifts spanning the whole elastic scattering region are obtained., 32 pages, 27 figures - replaced to better match version appearing in PRD. Previously incorrect JLab preprint number also corrected

Using first-principles calculation within Quantum Chromodynamics, we are able to determine a pattern of strangeness=1 resonances which appear as complex singularities within coupled $��K$, $��K$ scattering amplitudes. We make use of numerical computation in the lattice discretized approach to the quantum field theory with light quark masses corresponding to $m_��\sim 400\, \textrm{MeV}$ and at a single lattice spacing. The energy dependence of scattering amplitudes is extracted through their relationship to the discrete spectrum in a finite-volume, which we map out in unprecedented detail., V3: version accepted for publication in PRL. A code bug which led to incorrect normalization of certain correlation functions is corrected - qualitative features of the results remain unchanged

We present highly excited spectra of charm-light and charm-strange mesons from dynamical lattice QCD. Our calculations are performed on anisotropic $N_{f} = 2+1$ dynamical ensembles generated by the Hadron Spectrum Collaboration. The use of novel techniques and a large basis of interpolating operators have allowed us to extract these spectra to a high degree of statistical precision, extract states of high spin and observe candidate hybrid mesons. We interpret and discuss our results in light of the current experimental situation., Comment: 8 pages, 3 figures, to appear in the proceedings of the Confinement X conference (8-12 October 2012, TUM Campus Garching, Munich.)

The isospin-2 pi pi system provides a useful testing ground for determining elastic hadron scattering parameters from finite-volume spectra obtained using lattice QCD computations. A reliable determination of the excited state spectrum of two pions in a cubic box follows from variational analysis of correlator matrices constructed using a large basis of operators. A general operator construction is presented which respects the symmetries of a multi-hadron system in flight. This is applied to the case of pi pi and allows for the determination of the scattering phase-shifts at a large number of kinematic points, in both S-wave and D-wave, within the elastic region. The technique is demonstrated with a calculation at a pion mass of 396 MeV, where the elastic scattering is found to be well described by a scattering length parameterisation., Comment: Tables of little-group CGCs in ancillary file; v2: minor changes to reflect published version

Finite-volume lattice QCD calculations offer the possibility of extracting resonance parameters from the energy-dependent elastic phase-shift computed using the Luescher technique. In this letter, as a trial of the method, we report on the extraction of the non-resonant phase-shift for S and D-wave pi-pi isospin-2 scattering from dynamical lattice QCD computations. We define a variational basis of operators resembling pairs of pions of definite relative momentum and extract a spectrum of excited states that maps to phase-shifts at a set of discrete scattering momenta. Computations are performed with pion masses between 400 and 520 MeV on multiple spatial volumes. We observe no significant quark mass dependence in the phase-shifts extracted which are in reasonable agreement with the available experimental data at low momentum., 5 pages, updated to reflect published version

The status and prospects for investigations of exotic and conventional hadrons with lattice QCD are discussed. The majority of hadrons decay strongly via one or multiple decay-channels, including most of the experimentally discovered exotic hadrons. Despite this difficult challenge, the properties of several hadronic resonances have been determined within lattice QCD. To further discern the spectroscopic properties of various hadrons and to help resolve their nature we present our suggestions for future analytic and lattice studies., Comment: 23 pages + references, Submitted to the Proceedings of the US Community Study on the Future of Particle Physics (Snowmass 2021); v2: few references added