Many of the XY Z mesons discovered in the last decade can be identified as bound states in BornOppenheimer (B-O) potentials for a heavy quark and antiquark. They include quarkonium hybrids, which are bound states in excited flavor-singlet B-O potentials, and quarkonium tetraquarks, which are bound states in flavor-nonsinglet B-O potentials. We present simple parameterizations of the deepest flavor-singlet B-O potentials. We infer the deepest flavor-nonsinglet B-O potentials from lattice QCD calculations of static adjoint mesons. Selection rules for hadronic transitions are used to identify XY Z mesons that are candidates for ground-state energy levels in the B-O potentials for charmonium hybrids and tetraquarks. The energies of the lowest-energy charmonium hybrids are predicted by using the results of lattice QCD calculations to calculate the energy splittings between the ground states of different B-O potentials and using the Schroedinger equation to determine the splittings between energy levels within a B-O potential.

We derive universal relations for the rf spectroscopy of a two-dimensional Fermi gas consisting of two spin states interacting through an S-wave scattering length. The rf transition rate has a high-frequency tail that is proportional to the contact and displays logarithmic scaling violations, decreasing asymptotically like 1/(ω2ln2ω). Its coefficient is proportional to ln2'(a'(2D)/a(2D)), where a(2D) and a'(2D) are the two-dimensional scattering lengths associated with initial-state and final-state interactions. The clock shift is proportional to the contact and to ln(a'(2D)/a(2D)). If |ln(a'(2D)/a(2D))| >> 1, the clock shift arises as a cancellation between much larger contributions proportional to ln2(a'(2D)/a(2D)) from bound-bound and bound-free rf transitions.

Many of the XY Z mesons discovered in the last decade can be identified as bound states of a heavy quark and antiquark in Born-Oppenheimer (B-O) potentials defined by the energy of gluon and light-quark fields in the presence of static color sources. The mesons include quarkonium hybrids, which are bound states in excited flavor-singlet B-O potentials, and quarkonium tetraquarks, which are bound states in flavor-nonsinglet B-O potentials. The deepest hybrid potentials are known from lattice QCD calculations. The deepest tetraquark potentials can be inferred from lattice QCD calculations of static adjoint mesons. Selection rules for hadronic transitions are derived and used to identify XY Z mesons that are candidates for ground-state energy levels in the B-O potentials for charmonium hybrids and tetraquarks. A full decade has elapsed since the discovery of the first XY Z meson, the X(3872) [7], but no compelling explanation for the pattern of XY Z mesons has emerged. In simple constituent models, an XY Z meson consists of a heavy quark (Q) and antiquark (Q) and possibly additional constituents that could be gluons (g) or light quarks (q) and light antiquarks (q). The models that have been proposed can be classified according to how the constituents are clustered within the me- The B-O approximation is used in atomic and molecular physics to understand the binding of atoms into molecules. It exploits the large ratio of the time scale for the motion of the atomic nuclei to that for the electrons, which is a consequence of the large ratio of the nuclear and electron masses. The electrons respond almost instantaneously to the motion of the nuclei, which can be described by the Schroedinger equation in a B-O potential defined by the energy of the electrons in the presence of static electric charges. The B-O approximation for QQ mesons in QCD was developed by Juge, Kuti, and Morningstar [13]. It exploits the large ratio of the time scale for the motion of the Q andQ to that for the evolution of gluon fields, which is a consequence of the large ratio of the heavy-quark mass to the nonperturbative momentum scale Λ QCD . The gluon field responds almost instantaneously to the motion of the QQ pair, which can be described by the Schroedinger equation in a B-O potential defined by the energy of the gluon field in the presence of static color sources. Conventional quarkonia are energy levels of a QQ pair in the ground-state B-O potential. The energy levels in the excited-state B-O potentials are quarkonium hybrids. Juge, Kuti, and Morningstar calculated many of the B-O potentials using quenched lattice QCD [13]. They calculated the spectra of charmonium hybrids and bottomonium hybrids by

We calculate the spectrum of three-body Efimov bound states near a Feshbach resonance within a model which accounts both for the finite range of interactions and the presence of background scattering. The latter may be due to direct interactions in an open channel or a second overlapping Feshbach resonance. It is found that background scattering gives rise to substantial changes in the trimer spectrum as a function of the detuning away from a Feshbach resonance, in particular in the regime where the background channel supports Efimov states on its own. Compared to the situation with negligible background scattering, the regime where van der Waals universality applies is shifted to larger values of the resonance strength if the background scattering length is positive. For negative background scattering lengths, in turn, van der Waals universality extends to even small values of the resonance strength parameter, consistent with experimental results on Efimov states in 39 K. Within a simple model, we show that short-range three-body forces do not affect van der Waals universality significantly. Repulsive three-body forces may, however, explain the observed variation between around −8 and −10 of the ratio between the scattering length where the first Efimov trimer appears and the van der Waals length.

In a system of ultracold atoms near a Feshbach resonance, pairs of atoms can be associated into universal dimers by an oscillating magnetic field with frequency near that determined by the dimer binding energy. We present a simple expression for the transition rate that takes into account many-body effects through a transition matrix element of the contact. In a thermal gas, the width of the peak in the transition rate as a function of the frequency is determined by the temperature. In a dilute Bose-Einstein condensate of atoms, the width is determined by the inelastic scattering rates of a dimer with zero-energy atoms. Near an atom-dimer resonance, there is a dramatic increase in the width from inelastic atom-dimer scattering and from atom-atom-dimer recombination. The recombination contribution provides a signature for universal tetramers that are Efimov states consisting of two atoms and a dimer.