Asymptotic freedom and color confinement are undoubtedly the most remarkable aspects of quantum chromodynamics (QCD). Indeed, it is because of these features that QCD is universally accepted as the quantum field theory of strong interactions. On the one hand, asymptotic freedom, meaning that the theory approaches a noninteracting one in the high energy limit, allows for a perturbative treatment, so that the QCD Lagrangian can be used to derive analytical expressions describing high-energy processes. On the other hand, color confinement, implying that all observable states are color-neutral, is a purely nonperturbative phenomenon, which prevents an analytical calculation of the low-energy interactions such as the ones binding quarks into hadrons. Hence, a description of the hadron spectrum from QCD has been pursued by means of various approximate nonperturbative techniques. In particular, the study of heavy mesons provides an ideal benchmark for such techniques, since the high mass of the heavy quarks (i.e., bottom, charm) allows to simplify their treatment. The objective of this thesis is to build a unified framework based on QCD for the study of heavy mesons below and above the energy threshold for the production of open-flavor meson-meson pairs. This goal has been pursued through many intermediate objectives: i) the consistent study of the mass spectrum of quarkonium and quarkonium hybrid mesons; ii) the detailed calculation of the radiative transitions rates between quarkonium states, beyond the approximated expression usually found in
the literature; iii) the building of a phenomenological model for the calculation of strong decay widths of quarkonium and quarkonium hybrid states to openflavor meson-meson pairs; iv) the derivation of a phenomenological scheme for the treatment of heavy
mesons made of both quarkonium and open-flavor meson-meson components based on lattice QCD studies of string breaking, including the development of analytical and numerical techniques for solving the corresponding dynamical equation.
The quark binding interaction may be studied using the Bethe-Salpeter (BS) equation. The BS approach to the bound state problem has the virtue of being formally exact and completely relativistic, but its solution presents formidable difficulties. As an alternative, one may let himself be guided by the incredible success of potential quark models to try to reduce the BS formulation to a more transparent one. This reduction can be done systematically under the static approximation and the nonrelativistic limit. These two conditions, which may be met by states of a heavy quark-antiquark pair, allow to reduce the BS equation to a Schrödinger equation for the equal-time wave function. The quark binding interaction is then described by an effective potential, which can be determined in a gauge invariant way through the Wilson loop formalism. Concretely, this potential has been calculated ab initio in lattice QCD using the Born-Oppenheimer (BO) approximation for heavy-quark mesons. In this approximation, based on the mass of the heavy quarks being much larger than the QCD intrinsic energy scale, the quark-antiquark potential is calculated from the energy levels of stationary gluon and light quark fields in presence of static color sources (the heavy quarks).
The fundamental assumption of the BO approximation is that the components of a physical system may be classified distinctively as “heavy” and “light” on the basis of some energy scale, and that the dynamics of the light fields can be solved by neglecting the motion of the heavy degrees of freedom. The physical idea behind this approximation is that the time-scale for the evolution of the light fields is so short that, in comparison, the heavy degrees of freedom can be treated as being still. Then, once the light fields have been integrated in this static limit, the motion of the heavy degrees of freedom is determined from a nonrelativistic Schrödinger equation with effective potentials enclosing all the information on the light field dynamics. In atomic molecules, where the nuclei weigh several thousands times more than the electrons, the nuclei are treated as heavy degrees of freedom, while electrons and photons constitute the light fields. In heavy-quark systems, the distinction between “heavy” and “light” is provided by the QCD energy scale
associated to the gluon field. So, heavy quark flavors, charm and bottom, whose mass is much bigger than the QCD energy scale, can be considered as heavy degrees of freedom. On the other hand, gluons and light quark flavors, up, down, and strange, can be treated as
light fields.
In the BO approximation, heavy mesons are obtained as solutions of a Schrödinger-like equation where the static energy levels calculated in lattice QCD play the role of effective potentials. In quenched lattice QCD, this is, without dynamical light quarks, the ground state potential is associated to the quarkonium configuration extensively studied in quark models, while excited state potentials correspond to quarkonium hybrids. In general, these configurations may mix with each other in a multichannel Schrödinger-like equation, however, the non-adiabatic couplings between different channels are usually neglected. This results in the single channel BO approximation, where each channel obeys a separate Schrödinger equation whose solution yield the spectrum of heavy meson states in the corresponding configuration.
The wave functions calculated within the BO approximation can be used to calculate decay widths of quarkonium and quarkonium hybrids states. This study focused on decays by a single photon emission and strong decays to open-flavor meson-meson pairs.
Radiative decays are important in that quantum electrodynamics (QED), the quantum field theory of electromagnetic interactions, is well understood, so that the calculated widths may be expected to be affected exclusively by the uncertainty in the underlying potential model, and not in the theory used for their calculation. From the QED interaction Hamiltonian one can derive a transition operator between quarkonium states which, sandwiched between the initial and final wave functions, gives the radiative decay width. The transition operator is usually reduced to a simplified form given by the dipole approximation, which consists of two separate approximations: the long wavelength approximation for the emitted photon and the nonrelativistic approximation for the heavy quark currents. However, these approximations may be lifted using some angular momentum algebra and by introducing several sums over intermediate quarkonium states.
Strong decays into an open-flavor meson-meson pair, on the other hand, are exceptionally difficult to calculate from the underlying theory of QCD. However, calculating the width for such decays is extremely important since they are expected to be the main decay channels of quarkonium and quarkonium hybrid mesons, when kinematically allowed. In fact, these are usually the preferred channels for the discovery of most heavy quark mesons. Because the quenched lattice potentials lack information on the dynamics of the light quarks, decays to open-flavor meson-meson pair have to be calculated using some decay model. More precisely, the decay of an initial
quarkonium or quarkonium hybrid state to a final open-flavor heavy-light meson pair is expected to occur from the creation of a light quark pair from the hadronic medium and their recombination with the heavy quarks. In a BO framework, one may assume the creation of the light quark pair to occur much instantaneously with respect to the timescale for the evolution of the heavy quarks. Therefore, the quantum numbers of the created light quark pair can be assumed to be the same as those of the hadronic medium in which the decay takes place, which can be identified with BO configuration of the initial state. This fact, along with the reasonable assumption of the decay being dominated by the lowest possible values of the total angular momentum, fixes the quantum numbers of the quark pair creation model. Thanks to this, quark pair creation models for decays of quarkonium states and the lowest bottomonium hybrid can be constructed in a consistent framework, and the two of them can be compared in a sensible way.
The validity of the single channel BO approximation should not be taken for granted just because the heavy quark mass is much bigger than the QCD energy scale, for the non-adiabatic couplings between different channels may not be negligible. In that case, the single channel approximation may be deemed reasonable only as long as the wave function has no significant overlap with the non-adiabatic couplings. This condition may be met by quarkonium and quarkonium hybrids with quenched lattice potentials, but this is not anymore the case when using unquenched lattice QCD potentials, this is, including dynamical light quarks. In fact, the unquenched lattice potentials show a mixing between a quarkonium and open-flavor meson-meson configurations due to string breaking. As a consequence of string breaking, the non-adiabatic couplings in the Schrödinger-like equation cannot be neglected, and one should solve the complete system of coupled equations. However, this is unpractical for at least two reasons: first, the connection between the non-adiabatic couplings and lattice QCD is not straightforward, second, the adiabatic wave function components would not be associated to a well-defined configuration, but rather to a mixing of quarkonium and meson-meson.
The shortcomings of the single channel BO approximation may be easily averted by using the diabatic framework, where the quarkonium and meson-meson component obey a multichannel Schrödinger equation where the potential matrix, including the potential of each channel as well as the mixing potentials between them, enjoys a direct connection with lattice QCD. In fact, the form of the diabatic potential matrix elements can be inferred from the static energy levels calculated in unquenched lattice QCD.
The solutions of the diabatic Schrödinger equation with energy below the lowest open-flavor meson-meson threshold represent bound states. For energies far below the meson-meson threshold mass, the open-flavor meson-meson components are too much kinematically forbidden to play any role, and the corresponding solutions are simply quarkonium states. However, for energies closer below threshold, open-flavor meson-meson components may appear as molecular components thanks to the diabatic mixing induced by string breaking. For energy above one or more open-flavor meson-meson thresholds, the solutions of the diabatic Schrödinger equation possess some components that oscillate indefinitely. These components are not normalizable, therefore their interpretation as wave functions of a heavy meson state is not straightforward. To deal with this issue, one may use a bound state approximation in which, at a first stage, the mixing potentials with open thresholds are neglected. Then the coupling between the approximated bound states and the continuum of open-flavor meson-meson states may be taken into account. As a result, the approximated bound states acquire a mass correction and a decay width, thus becoming resonances. These resonances can then associated with heavy mesons above threshold, decaying to open-flavor meson-meson pairs through string breaking.
As an alternative to the bound state approximation, a more complete treatment of heavy mesons above threshold can be given within a mesonmeson scattering framework. In fact, the oscillating behavior of the wave function components associated to open channel is, though troublesome for a bound-state analysis, well known analytically. These wave functions, in fact, represent the free stream of open-flavor meson-meson pairs scattering from each other through their mixing with quark-antiquark mediated by string breaking. These asymptotic solutions can be transformed in the usual representation of the stationary scattering states in terms of the scattering amplitudes. This allows to calculate the S matrix directly from the diabatic potential matrix, which is related to the static energy levels in lattice QCD, without any additional approximation. Then, heavy mesons above threshold are naturally identified with the peaks in the calculated open-flavor meson-meson cross-sections.
We have pursued a comprehensive study of hidden-flavor heavy mesons for energies below and above open-flavor meson-meson thresholds by means of various theoretical schemes. In the first part of this study, we have used the BO approximation with potentials derived from quenched lattice QCD to analyze quarkonium and quarkonium hybrids. In the second part, we have adapted the diabatic framework from molecular physics to strong interactions in order to have a description of heavy mesons, made of quark-antiquark and meson-meson components, based on unquenched lattice QCD studies of string breaking. Specifically, the conclusions of each chapter are given below.
Chapter 1: We have reviewed the BO approximation and parametrized the quarkonium and quarkonium hybrid potentials obtained in quenched lattice QCD. We have calculated the spectrum of quarkonium states with phenomenological values of the parameters, then included some spindependent corrections, then showed that the experimental low-lying spectrum of heavy mesons is well described in terms of quarkonium states. We have also calculated the masses of the lowest bottomonium and charmonium hybrid states using the same values of the parameters as in the quarkonium potential, without adding any new free parameter.
Chapter 2: We have reviewed the derivation of the usual dipole transition operator for radiative quarkonium transitions from the QED interaction Hamiltonian. We have briefly reviewed the long wavelength and nonrelativistic approximations and their conditions of validity. Since these conditions may not be met by some radiative decays between quarkonium states, we have lifted the long wavelength and ronrelativistic approximations and derived more general formulae for the electromagnetic transition operator. We have finally shown, through a short review of a radiative decays between charmonium states, that the results from these general formulae are very sensitive to the details of the quarkonium wave functions, and hence may serve as a stringent test of different phenomenological models when comparing with data.
Chapter 3: We have developed quark pair creation (QPC) models for strong decays to open-flavor mesonmeson pairs respecting the symmetries of the BO approximation. We have shown that the quantum numbers of the BO potential of the decaying state, associated with the hadronic medium, may be used to determine the quantum numbers of the created light quark pair. Specifically, in the quarkonium case, the hadronic medium is in its ground state with vacuum-like quantum numbers, which corresponds to the customary 3P0 QPC model extensively used in phenomenological studies. As for the lowest quarkonium hybrid, whose hadronic medium has the quantum numbers of the ground state gluelump, the corresponding QPC model has quantum numbers 1P1. Thus, we have reviewed the 3P0 model for quarkonium and developed the 1P1 model for the lowest quarkonium hybrid.
Chapter 4: We have used the diabatic representation of BO, first introduced in molecular and atomic physics, to develop a phenomenological framework for the description of heavy mesons made of quark-antiquark and openflavor meson-meson components. We have shown that the dynamics is governed by a multichannel Schrödinger equation that, particularizing to a specific set of JPC quantum numbers, can be reduced to a radial form. Then, we have used the static energies calculated in unquenched lattice QCD to infer the form of the radial potential associated to the mixing between quark-antiquark and meson-meson. It turns out that this potential is significant only near the crossing between the quarkonium potential and the threshold mass.
Chapter 5: We have examined the spectrum of bound states obtained in the diabatic framework, with masses below the lowest open-flavor meson-meson threshold. Then we have briefly discussed the difficulties in solving the diabatic Schrödinger equation for energies above threhsold, and introduced a bound state approximation in which coupling to the open threshold is initially neglected. Then, we have showed that the reintroduction of the coupling with the meson-meson continua transforms the approximated bound states in Fano resonances, associated to heavy mesons decaying to open-flavor meson-meson pairs. We have then calculated a phenomenological spectrum of bottomoniumlike and charmoniumlike states and discussed the limitations of the bound state approximation.
Chapter 6: We have overcome the limitations of the bound-state approximation by developing a nonperturbative scattering formalism for open-flavor meson-meson pairs. We have shown that the solutions of the diabatic Schrödinger equation for energies above threshold are naturally interpreted as stationary scattering states. We have then connected these solutions with the usual representation of scattering states in terms of an incoming plane wave plus a spherical wave multiplied by the scattering amplitude. In this way, we have derived a nonperturbative scheme for calculating the on-shell S matrix directly from the diabatic potential matrix. Finally, we have carried out a phenomenological analysis of elastic open-bottom and open-charm cross sections in comparison with the calculated Fano resonances and with data. The resulting physical picture of heavy mesons below and above threshold corresponds to a spectrum of (quasi-) conventional quarkonium states, plus unconventional states lying close to some open-flavor meson-meson thresholds. We have also argued that an unconventional state with mass close below some threshold causes an enhancement of the low-momentum cross section in the corresponding channel, which may overshadow quasiconventional resonances lying close by.
From a phenomenological point of view, there are two well-established experimental unconventional states that provide ideal case studies for, respectively, the BO approximation and the diabatic framework developed here. On the one hand the Upsilon(10860), whose mass and decay properties are compatible with those of a bottomonium state mixing with the lowest bottomonium hybrid, as calculated within a BO framework equipped with consistent QPC models. On the other hand, the X(3872), which can be interpreted in the diabatic framework as a loosely bound open-charm meson-meson state with a compact charmonium core, as shown by a phenomenological calculation with an effective value of the energy gap.
It should be pointed out that the diabatic framework is perfectly general in the sense that it can also be applied for a description of heavy-meson systems made of quarkonium and quarkonium hybrid components, such as the Upsilon(10860). The current limitation preventing this is the lack of lattice QCD input to derive the form of the mixing potential in such systems. Furthermore, the diabatic treatment is also suited for systems containing quarkonium, quarkonium hybrid, and meson-meson components as well. Hence, we conclude that a completely unified study of hidden-flavor mesons below and above open-flavor thresholds is possible by means of the diabatic approach. It is important to notice that the diabatic potential matrix treats on equal grounds the potentials of different channels (e.g.: quarkonium, meson-meson, hybrid,...) as well as the mixing potentials between them, so that the descriptions below and above threshold are connected seamlessly
