Investigations of the mass and decays of the exotic hybrid are reviewed, including calculation of the pion1 to eta pion, eta-prime pion decay widths within the QCD sum rules technique. In this calculation, the recently-proposed eta, eta-prime quark mixing scheme is employed. The results indicate that the decay width Gamma(pion1 to eta pion) around 250 MeV is large compared with the decay width Gamma(pion1 to eta-prime pion) around 20 MeV. Inspired by these results, some phenomenological approaches are suggested to gain an understanding of the underlying mechanism of eta pion and eta-prime pion hybrid decays.

Starting with two coupled Bethe-Salpeter equations for the quark-antiquark, and for the quark-glue-antiquark component of the quarkonium, we solve the bound state equations perturbatively. The resulting admixture of glue can be partially understood in a semiclassical way, one has, however, to take care of the different use of time ordered versus retarded Green functions. Subtle questions concerning the precise definition of the equal time wave function arise, because the wave function for the Coulomb gluon is discontinuous with respect to the relative time of the gluon. A striking feature is that a one loop non abelian graph contributes to the same order as tree graphs, because the couplings of transverse gluons in the tree graphs are suppressed in the non relativistic bound state, while the higher order loop graph can couple to quarks via non suppressed Coulomb gluons. We also calculate the amplitude for quark and antiquark at zero distance in the quark-glue-antiquark component of the P-state. This quantity is of importance for annihilation decays of P-states. It shows a remarkable compensation between the tree graph and the non abelian loop graph contribution. An extension of our results to include non perturbative effects is possible.

Using the QCD string approach the adiabatic potentials and spectra of b\bar b-hybrid mesons are calculated. The results are compared to lattice studies.

QCD theory allows the existence of states which cannot be built by the naive quark model; both theoretical arguments and experimental data confirm the hypothesis that gluons may have freedom degrees at the constituent level, and should be confined. In this work, we use a phenomenological potential motivated by QCD (with some relativistic corrections) to determine the masses and the wavefunctions of several hybrid mesons, within the context of a constituent q-qbar-g model. We compare our estimates of the masses with the predictions of other theoretical models and with the observed masses of candidates.

We calculate the next-to-leading order (NLO) $\alpha_s$-corrections to the contributions of the condensates $<\alpha G^2>$ and $<\bar qq>^2$ in the current-current correlator of the hybrid current $g\barq(x)\gamma_{\nu}iF_{\mu\nu}^aT^aq(x)$ using the external field method in Feynman gauge. After incorporating these NLO contributions into the Laplace sum-rules, the mass of the $J^{PC}$=$1^{-+}$ light hybrid meson is recalculated using the QCD sum rule approach. We find that the sum rules exhibit enhanced stability when the NLO $\alpha_s$-corrections are included in the sum rule analysis, resulting in a $1^{-+}$ light hybrid meson mass of approximately 1.6 GeV.

The light-mass exotics with J^{PC}=1^{-+} observed at BNL and CERN may have a simple explanation as dynamically generated resonances in eta-prime pi rescattering in the final state interaction. This dynamics is mediated by the same anomalous glue which also generates the large mass of the eta-prime. OZI violating processes are also potentially important to eta-prime production in proton-proton collisions close to threshold.

We study the mixing of excited states of a Hydrogen atom in a cavity with de-excited states plus a confined photon as a model for the coupling of quark-antiquark and quark-antiquark-gluon hybrid states in QCD. For an interesting range of parameters, the results are analytic. We find a case for which wavefunctions (and hence decay patterns) may be at odds with mass with respect to identification of a state as hybrid or not.

Masses for vector 1--, axialvector 1+-, 1++, and exotic-vector 1-+ mesons are calculated within a quantum field theoretic framework based on the Dyson-Schwinger equations using a model form of the quark-antiquark interaction that is separable. The model provides an excellent description of pion and K-meson observables and the flavor-octet ground-state meson spectrum. With no adjustments to model parameters, a numerical solution of the Bethe-Salpeter equation yields two exotic 1-+ mesons with masses of 1439 and 1498 MeV.

QCD sum-rules are used to calculate the $\hat\rho(1^{-+})\to\pi\eta, \pi\eta'$ decay widths of the exotic hybrid in two different $\eta-\eta'$ mixing schemes. In the conventional flavour octet-singlet mixing scheme, the decay widths are both found to be small, while in the recently-proposed quark mixing scheme, the decay width $\Gamma_{\hat\rho\to\eta\pi}\approx 250 MeV$ is large compared with the decay width $\Gamma_{\hat\rho\to\eta^\prime\pi}\approx 20 MeV$. These results provide some insight into $\eta$-$\eta'$ mixing and hybrid decay features.

The short- and intermediate-distance behaviour of the hybrid adiabatic potentials is calculated in the framework of the QCD string model. The calculations are performed with the inclusion of Coulomb force. Spin-dependent force and the so-called string correction term are treated as perturbation at the leading potential-type regime. Reasonably good agreement with lattice measurements takes place for adiabatic curves excited with magnetic components of field strength correlators.

Hybrid adiabatic potentials are considered in the framework of the QCD string model. The einbein field formalism is applied to obtain the large-distance behaviour of adiabatic potentials. The calculated excitation curves are shown to be the result of interplay between potential-type longitudinal and string-type transverse vibrations. The results are compared with recent lattice data.

We investigate implications of quark-hadron duality for hybrid mesons in the large-Nc limit. A simple formalism is developed which implements duality for QCD two-point functions of currents of quark bilinears, with any number of gluons. We argue that the large-Nc meson masses share a common parameter, which is related to the QCD string tension. This parameter is fixed from correlators of conserved vector and axial-vector currents, and using lattice QCD determinations of the string tension. Our results predict towers of hybrid mesons which, within expected 1/Nc corrections, naturally accommodate the 1^(-+) experimental hybrid candidates.

We show how a field theory containing classical vortex solutions can be expressed as an effective string theory of long distance QCD describing the two transverse oscillations of the string. We use the semiclassical expansion of this effective string theory about a classical rotating string solution to obtain Regge trajectories for mesons with zero mass quarks. The first semiclassical correction adds the constant 1/12 to the classical Regge formula for the angular momentum of mesons on the leading Regge trajectory. In D spacetime dimensions, this additive constant is (D-2)/24. The excited states of the rotating string give rise to daughter Regge trajectories determining the spectrum of hybrid mesons.

We study the possible interpretation of the two exotic resonances $J^{PC}= 1^{-+}$ at 1400 and 1600 MeV, claimed to be observed by BNL, decaying respectively into $\eta\pi$, $\eta'\pi$, $f_{1}\pi$ and $ \rho\pi$. These objects are interpreted as hybrid mesons, in the quark-gluon constituent model using a chromoharmonic confining potentiel. The quantum numbers $J^{PC}I^{G} = 1^{-+} 1^{-}$ can be considered in a constituent model as an hybrid meson ($ q \bar q g$). The lowest $J^{PC}= 1^{-+}$states may be built in two ways : $l_{g}$=1 (gluon-excited) corresponding to an angular momentum between the gluon and ($ q \bar q$) system, while $l_{q \bar q}=1$ (quarks-excited) corresponds to an angular momentum between $q$ and $\bar q$. For the gluon-excited mode $1^{-+}$ hybrids, we find the decay dominated by the $b_{1}\pi$ channel, and by the $\rho \pi $ channel for the quark-excited mode. In our model, neither the quark-excited nor the gluon-excited $1^{-+}$ (1400 MeV) hybrids can decay into $\eta\pi$ and $\eta'\pi$, in contradiction with experiment. Hence, the 1400 MeV resonance seems unlikely to be an hybrid state. The $1^{-+}$ (1600 MeV) gluon-excited hybrid is predicted with too large a total decay width, to be considered as an hybrid candidate. On the contrary the quark-excited mode has a total decay width around 165 MeV, with a $\rho \pi $ preferred decay channel, in agreement with BNL. Our conclusion is that {\it{this resonance may be considered as a hybrid meson in the quark-excited mode}}

We calculate the radiative corrections to the current-current correlator of the hybrid current $g\bar q(x)\gamma_{\nu}iG_{\mu\nu}^aT^aq(x)$. Based on this new result we use the QCD sum rule approach to estimate lower bounds on the masses of the $J^{PC}$=$1^{-+}$ and $0^{++}$ light hybrids.

Including the radiative perturbative corrections and the short distance tachyonic gluon mass effects which mimic the ones of UV renormalons, we re-estimate the decay amplitudes, masses and widths of light hybrid mesons from QCD spectral sum rules. We show that the effects are tiny and confirm the previous lowest order results. We discuss the phenomenological impacts of our results for the vector hybrids.

QCD and QED exhibit an infinite set of three-point Green's functions that contain only OZI rule violating contributions, and (for QCD) vanish exactly in the large N_c limit. This is due to symmetrization, do not follow from any Lagrangian symmetry, and constitutes the discovery of a new symmetry principle in QCD and QED. We prove that the amplitude for a neutral hybrid {1,3,5...}^{-+} exotic current to create eta pi0 vanishes exactly for QCD in the large N_c limit. Crystal Barrel's experimental claim of a resonant 1^{-+} decaying to eta pi0 suggests that the 1^{-+} is not a hybrid meson.

Nonperturbative QCD approach is systematically derived starting from the QCD Lagrangian. Treating spin effects as a perturbation, one obtains the universal effective Hamiltonian describing mesons, hybrids and glueballs. Constituent mass of quark and gluon is calculated via string tension. The resulting spectrum of mesons, hybrids and glueballs obtained is in good overall agreement with lattice data and experiment.

For some experimental guidelines of the next millenium, I review the determinations of the masses, decays and mixings of the gluonia, scalar and hybrid mesons from QCD spectral sum rules and low-energy theorems, and compare them with the lattice.

We present model independent predictions on the short range behavior of the energies of the gluonic excitations between static quarks (hybrid potentials) in an effective field theory framework (pNRQCD).

Within an effective field theory framework we study heavy-quark--antiquark systems with a typical distance between the heavy quark and the antiquark smaller than $1/\Lambda_{\rm QCD}$. A suitable definition of the potential is given within this framework, while non-potential (retardation) effects are taken into account in a systematic way. We explore different physical systems. Model-independent results on the short distance behavior of the energies of the gluonic excitations between static quarks are obtained. Finally, we show how infrared renormalons affecting the static potential get cancelled in the effective theory.

To include the explicit valence gluon degrees of freedom into spectroscopy of the lowest states of heavy quarkonia, we consider, within an adiabatic model, the properties of the lowest hybrid $\bar{Q}Qg$-mesons and estimate the effects of their mixing with low-lying vector $\bar{c}c$-charmonia. The perspectives of compatibility of the resulting picture with data are discussed.

The decay widths of the $0^{++}$ and $1^{-+}$ heavy-light hybrids to B(D) and pion are calculated by using the QCD sum rules. The interpolated current of the hybrid is chosen as $g\bar q\gamma_{\alpha}G_{\alpha\mu}^aT^ah_{\it v}(x)$. In order to avoid the infrared problem, a two point correlation function between the pion and vacuum instead of the three point correlation function between the vacuum is used. We also keep the leading order of $1/M_Q$ expansion in our calculation for convenience. The decay width of the first type is $1\sim 2$ MeV while the second type is several hundred KeV.

We have calculated the binding energy of the hybrid quarkonium up to the order of ${\cal O}(1/m_Q)$ and found their decay constants scale like $m_Q^{5/4}$ as $m_Q\to \infty$. The $0^{--}$ and $0^{++}$ hybrid quarkonium is exactly degenerate in the limit of $m_Q\to\infty$ while the ${\cal O}(1/m_Q)$ correction renders the $0^{--}$ mass lower than that for $0^{++}$. The $1^{-+}$ and $1^{+-}$ hybrid is nearly degenerate and lies 0.7 GeV lower than $0^{--}$. With $m_b=4.8$GeV the mass of $1^{-+}$ $b\bar b g$ state is $(10.75\pm 0.10)$, which is the lightest exotic hybrid quarkonium.

QCD sum rules are derived for the heavy hybrid mesons with one heavy quark and $J^{PC}=1^{-+}, 0^{++}, 1^{+-}, 0^{--}$ in the leading order of heavy quark effective theory. We obatin the binding energy $\Lambda_H =(1.1\pm 0.1), (1.51\pm 0.10), (1.28\pm 0.10), (1.5\pm 0.1)$ GeV and decay constant $f_H =(0.205\pm 0.02), (0.415\pm 0.04), (0.217\pm 0.02), (0.317\pm 0.03) $GeV$^{{7\over 2}}$ for $1^{-+}, 0^{++}, 1^{+-}, 0^{--}$ respectively when the light quark is up or down quark. In all the four channels the binding energy is 0.13 GeV larger when the light quark is strange quark.

The relativistic four-quark equations are found in the framework of the dispersion relation technique. The solutions of these equations using the method based on the extraction of leading singularities of the amplitudes are obtained. The mass spectrum values of lowest hybrid mesons are calculated.

We calculate masses of the light-quark hybrid mesons with the quantum number $0^{++}$ and $0^{-+}$ by using the QCD sum rules. Two kinds of the interpolated currents with the same quantum number are employed. We find that the approximately equal mass is predicted for the $0^{-+}$ hybrid state from the different current and the different mass is obtained for the $0^{++}$ hybrid state from the different current. The prediction depends on the interaction between the gluon and quarks in the low-lying hybrid mesons. The mixing effect on the mass of the light-quark hybrid meson through Low-energy theorem has been examined too, and it is found that this mixing shifts the mass of hybrid meson and glueball a little.

Hybrid meson energies are calculated in the static quark limit with the Dynamical Quark Model (DQM). In the DQM, transverse gluons are represented as effective constituents with a dynamically generated mass. Hybrid masses are determined within the Tamm-Dancoff approximation for the resulting relativistic Salpeter equation. The model calculation agrees quite well with recent lattice data. The successes and shortcomings of the model provide clear indications about the nature of glue at low energy.

We construct the B.S. equation for the hybrid mesons under instantaneous approximation. The kernel is chosen as the sum of an one-gluon exchange potential and a linear confining potential. The equations are solved by numerical method, and the spectrum of hybrid mesons $b\bar{b}g$ and $c\bar{c}g$ are obtained.

Exotic mesons with quantum numbers of a hybrid ($\bar{q} q g$) such as $J^{PC} = 1^{-+}$are shown to exist as narrow resonant states in the large $N_c$ limit of QCD. The width of these states is proportional to $1/N_c$. Possible phenomenological implications for the world of $N_c=3$ are discussed

We calculate the strong decays of hybrid mesons to conventional mesons for all the lowest lying J^PC hybrids of flavour uu, dd, ss, cc and bb. A decay operator developed from the heavy quark expansion of quantum chromodynamics is employed. We show that the selection rule that hybrid mesons do not decay to identical S-wave mesons, found in other models, is preserved. We predict decays of charmonium hybrids, discuss decays of J^PC=1^-+ exotic isovector hybrids of various masses, and interpret the \pi(1800) as a hybrid meson.

The heavy quark expansion of Quantum Chromodynamics and the strong coupling flux tube picture of nonperturbative glue are employed to develop the phenomenology of hybrid meson decays. The decay mechanism explicitly couples gluonic degrees of freedom to the pair produced quarks and hence does not obey the well known, but model-dependent, selection rule which states that hybrids do not decay to pairs of L=0 mesons. However, the nonperturbative nature of gluonic excitations in the flux tube picture leads to a new selection rule: light hybrids do not decay to pairs of identical mesons. New features of the model are highlighted and partial widths are presented for several low lying hybrid states.

The model for hybrid excitations of the QCD string with quarks is presented starting from the perturbation theory in the nonperturbative background. The propagation of a system containing $q \bar q$-pair and gluon is considered. The simplified version of the Hamiltonian, including both long-range nonperturbative interaction and Coulomb force, is derived. The masses of the lowest $q \bar q g$ hybrids are evaluated, and numerical results for the spectra are listed.

Model-independent features of hybrid states with heavy quarks and adiabatic approximation for such systems are discussed.These general arguments are applied to the QCD-based constituent gluon model.The leading- order Hamiltonian is derived from the general formalism. The masses of heavy quarks hybrids, including charmonium hybrid states,are calculated. We predict $c\bar c g$ hybrids at 4.0$\pm$0.1 GeV and $b \bar b g$ ones at 10.5$\pm$0.1 GeV.Comparison with some other models are made.

The model for constituent glue is presented starting from the perturbation theory in nonperturbative QCD vacuum. Green function is constructed for a system containing qqbar pair and gluon propagating in the vacuum background field, and Hamiltonian approach to the problem is formulated. The masses of lowest qqbarg hybrids with light quarks are evaluated, and implication of results for the spectroscopy of exotic states is discussed.

We present results from analytical and numerical studies of a flux tube model of hybrid mesons. Our numerical results use a Hamiltonian Monte Carlo algorithm and so improve on previous analytical treatments, which assumed small flux tube oscillations and an adiabatic separation of quark and flux tube motion. We find that the small oscillation approximation is inappropriate for typical hadrons and that the hybrid mass is underestimated by the adiabatic approximation. For physical parameters in the ``one-bead" flux tube model we estimate the lightest hybrid masses (${}_\Lambda L = {}_1 P$ states) to be 1.8-1.9~GeV for $u\bar u$ hybrids, 2.1-2.2~GeV for $s\bar s$ and 4.1-4.2~GeV for $c\bar c$. We also determine masses of conventional $q\bar q$ mesons with $L=0$ to $L=3$ in this model, and confirm good agreement with experimental $J$-averaged multiplet masses. Mass estimates are also given for hybrids with higher orbital and flux-tube excitations. The gap from the lightest hybrid level (${}_1P$) to the first hybrid orbital excitation (${}_1D$) is predicted to be $\approx 0.4$~GeV for light quarks $(q=u,d)$ and $\approx 0.3$~GeV for $q=c$. Both ${}_1P$ and ${}_1D$ hybrid multiplets contain the exotics $1^{-+}$ and $2^{+-}$; in addition the ${}_1P$ has a $0^{+-}$ and the ${}_1D$ contains a $3^{-+}$. Hybrid mesons with doubly-excited flux tubes are also considered. The implications of our results for spectroscopy are discussed, with emphasis on charmonium hybrids, which may be accessible at facilities such as BEPC, KEK, a Tau-Charm Factory, and in $\psi$ production at hadron colliders.

An analytic calculation of the breaking of excited chromoelectric flux-tubes is performed in an harmonic oscillator approximation and applied to predict the dynamics of all $J^{PC}$ low-lying gluonic excitations of mesons (hybrids). Widths, branching ratios and production dynamics of some recently discovered $J^{PC} = 1^{-+}, 0^{-+}$ and $1^{--}$ mesons are found to be in remarkable agreement with these results. We introduce the selection rules that can be used to understand the systematics of numerical decay calculations and we find possible significant breaking of these rules for specific channels that may enable enhanced production and detection of hybrids.

We show that the dynamics of the Quark-Gluon Plasma is such that during hadronization the creation of hybrids will predominate over the creation of mesons, giving a novel signature of the existence of QGP. At T = 0 the $(q\bar{q}g)$ hybrids are known to decay strongly into a pair of mesons. We find that at temperatures relevant to the QGP, this channel is forbidden. This would lead to significant modifications of the photonic signals of the QGP.

A new formalism is developed for the two photon production of hybrid mesons via intermediate hadronic decays. In an adiabatic and non-relativistic context with spin 1 pair creation we obtain the first absolute estimates of unmixed hybrid production strengths to be small (0.03 - 3 eV) in relation to experimental meson widths (0.1 - 5 keV). Within this context, two photon experiments at Babar, Cleo II, LEP2 and LHC therefore strongly discriminate between hybrid and conventional meson wave function components, filtering out conventional meson components. Decay widths of unmixed hybrids vanish. Conventional meson two photon widths are roughly in agreement with experiment.

We consider hybrid states with heavy quarks in the frame of recently proposed constituent gluon model. Limiting regimes for model Hamiltonian are discussed. It is shown that these regimes match smoothly, hence predictions of the model are definite enough. We predict lowest $c \bar c g$ hybrids at $4.1 \pm 0.1$ GeV and $b \bar b g$ hybrids at $10.5 \pm 0.1$ GeV. Regge-trajectories for hybrids are derived from the formalism. Experimental aspects of our predictions are discussed

Published in Z.Phys.C33:265-273,1986

Published in Phys.Lett.B112:71-75,1982