Quark Confinement and the Hadron Spectrum IV

PREFACE.

COMMITTEES AND SPONSORS.

CONTENTS.

When chiral symmetry is spontaneously broken, the low-energy part of the Dirac operator spectrum can be computed analytically in the chiral limit. The tool is effective field theory or, equivalently in this case, Random Matrix Theory.

In this work, we investigate application of the hypervirial theorem for judging accuracy of the trial wavefucntions with single or a few variational parameters in the potential model. Comparing with other criteria we can be convinced that this method may be the most practical and powerful one because it does not need to know the exact solution of the Schrödinger equation with a complicated potential.

We illustrate the contemporary application of Dyson-Schwinger equations using two examples: the calculation of pseudoscalar meson masses, an associated model-independent mass formula and the approach to the heavy-quark limit; and the study of nucleon observables, including a calculation of its mass, M, via a covariant Fadde'ev equation and an estimate of pion-loop contributions to M.

We review aspects of confinement in the covariant and local description of QCD and discuss to what extend our present knowledge of the infrared behavior of QCD Green functions can support this description. In particular, we emphasize: the positivity violations of transverse gluon and quark states, the Kugo-Ojima confinement criterion, and the conditions necessary to avoid the decomposition property for colored clusters. We summarize how these issues relate to the infrared behavior of the propagators in Landau gauge QCD as extracted from solutions to truncated Dyson-Schwinger equations and lattice simulations.

We review the phenomenology of the dimension d = 2 vacuum condensate in pure gauge theories, which is the vacuum expectation of the minimal value of the gauge potential squared. Both Abelian and non-Abelian cases are discussed. In case of the compact U(1) the non-perturbative part of the condensate is saturated by monopoles. In the non-Abelian case, a two-component picture for the condensate is presented when finite values of order are associated both with large and short distances. Possible physical manifestations of the condensate are discussed.

The ¹H(e,e′K⁺)Λ reaction was studied as a function of squared four-momentum transfer, Q², between 0.52 and 2.00 (GeV/c)², and of the virtual photon polarization parameter, ∊, at Jefferson Lab. From this data, the first accurate separation of the longitudinal and transverse cross sections in this reaction was achieved. The results are discussed and compared to representative model calculations.

We start with an effective field theory containing classical vortex solutions and show that the fluctuations of these vortices are described by an effective string theory. Viewed as a model for long distance QCD, this theory provides a concrete picture of the QCD string as a fluctuating Abrikosov-Nielsen-Olesen vortex of a dual superconductor on the border between type I and type II. We present arguments which suggest that the action of the effective string theory is the Nambu-Goto action, i.e. the rigidity vanishes. We then use this theory to calculate the corrections to classical Regge trajectories due to string fluctuations.

We represent QCD at the hadronic scale by means of an effective Hamiltonian, H, formulated in the Coulomb gauge. As in the Nambu-Jona-Lasinio model, chiral symmetry is dynamically broken, however our approach is renormalizable and also includes confinement through a linear potential with slope specified by lattice gauge theory. We perform a comparative study of alternative many-body techniques for approximately diagonalizing H: BCS for the vacuum ground state; TDA and RPA for the excited hadron states. We adequately describe the experimental meson and lattice glueball spectra and perform the first relativistic, three quasiparticle calculation for hybrid mesons. In general agreement with alternative theoretical approaches, we predict the lightest hybrid states near but above 2 GeV, indicating the two recently observed J^PC = 1^-+ exotics at 1.4 and 1.6 GeV are of a different, perhaps four quark, structure. We also detail a new isospin dependent interaction from color octet annihilation (analogous to ortho positronium) which splits I = 0 and I = 1 states.

I briefly review the current status of chiral perturbation theory (CHPT) in the meson sector. Emphasis is given on the quest for higher precision. I discuss two examples: one where it is difficult to make a good prediction (K_L → π⁰γγ), and where CHPT, even when pushed to higher orders, cannot yield an increase in accuracy. The second one is ππ scattering where a very sharp prediction can be made by combining CHPT at the two–loop level with dispersion relations.

The main ideas behind the new ways to preserve chiral symmetries for lattice fermions are presented. The discussion is focused on vector-like fermions, the case relevant for lattice QCD.

The problem of nonperturbative derivation from QCD, of quark kernels is discussed. The physics of spontaneous breakdown of chiral symmetric is discussed in the framework of microscopic quark models. An effective-in the pion Hilbert space- Hamiltonian is presented together with its derivation from microscopic models. π - π scattering is also discussed and the Weinberg results for I = 0 and I = 2 are rederived. Finally the gluon-gluon correlator is discussed for the heavy-light quark system.

We review lattice evidence showing that center-vortex condensation is a serious candidate for the mechanism of colour confinement in quantum chromodynamics.

The properties of center vortices are discussed within continuum Yang-Mills theory. By starting from the lattice theory and carefully performing the continuum limit the gauge potential of center vortices is obtained and the continuum analog of the maximal center gauge fixing is extracted. It is shown, that the Pontryagin index of center vortices is given by their self-intersection number, which vanishes unless the center vortices host magnetic monpoles, which make the vortex sheets non-oriented.

The ladder-rainbow truncation of the set of Dyson–Schwinger equations is used to study light mesons. The parameters in the effective interaction are constrained by the chiral condensate and f_π; the current quark masses are fitted to m_π and m_K. The dressed quark propagators are in qualitative agreement with recent lattice-QCD results at low q² while having the correct perturbative behavior at large q². The resulting vector meson masses are within 5% of the experimental values. The obtained electromagnetic form factors and strong and electroweak coupling constants are also in good agreement with the data. At finite temperature, this truncation leads to a mean-field chiral phase transition. The spatial pion mass is almost constant below this transition, but rises with T close to and above T_c. The mass of its chiral partner, an idealized σ meson, decreases with T until T_c, where it becomes degenerate with the pion.

We describe the observation of the B_c meson through its semileptonic decays, B_c → J/ψlν, and the measurements of the B_c mass, lifetime and production rate in the CDF detector at Fermilab. We also present estimates for B_c production and decay into other final states in the forthcoming run of the upgraded CDF and Tevatron.

The study of heavy quarks on the lattice are faced with two difficulties, the appropriate formulation for heavy quarks when m_Qa > 1, where m_Q is the mass of the heavy quark and a the lattice spacing, and the effect of dynamical "sea" quarks, which until recently have been too expensive computationally to include. Here I review progress in both these areas and suggest that much more work will be necessary. I focus in particular on the states.

We review lattice QCD results for glueballs (including a discussion of mixing with scalar mesons), hybrid mesons and exotic mesons (such as B_sB_s molecules). We also discuss string breaking as a mixing between colour flux states and states.

We review the connection between QCD and potential models of quarkonia in the framework of effective field theories, with an emphasis on non-perturbative methods such as lattice simulations. The static and heavy quark potentials are introduced and the applicability of the non-relativistic and adiabatic approximations are discussed. The rôles of so-called hybrid potentials are explored and we comment on the QCD analogue of the QED Lamb shift.

I review and discuss a selected sample of recent results in pNRQCD.

Some brief thoughts on Confinement IV, suggestions for future research in confinement, and speculations about an alternative picture.

We study the configuration space of the Tomboulis SO(3) × Z(2) formulation with periodic boundary conditions. The dynamical variables are constrained by the required coincidence of Z(2) and SO(3) monopoles. Furthermore, there is an additional constraint coming from the boundary conditions. We propose an update algorithm that satisfies the constraints and is straightforward to implement.

A model for the infrared sector of SU(2) Yang-Mills theory, based on magnetic vortices represented by (closed) random surfaces, is presented. The model quantitatively describes both confinement and the topological aspects of Yang-Mills theory. Details (including an adequate list of references) can be found in the e-prints hep-lat/9912003 and hep-lat/0004013, both to appear in Nucl. Phys. B.

It is shown that the instanton in the singular gauge is located at the Gribov horizon of the maximally Abelian gauge. Monopoles arising as gauge copies of the instanton are found to be kinematically suppressed because they increase the gauge fixing functional.

In SU(2) lattice gauge theory, a new self-restricted cooling procedure is developed to uncover the gauge invariant vortex vacuum texture. The emerging vortex vacuum structure amounts to the full string tension and gives rise to a mass dimension four condensate which is of pure vortex origin.

No abstract received.

The local action of an SU(2) gauge theory in general covariant Abelian gauges and the associated equivariant BRST symmetry that guarantees the perturbative renormalizability of the model are given. A global SL(2,R) symmetry of the model is spontaneously broken by ghost-antighost condensation at arbitrarily small coupling. This leads to propagators that are finite at Euclidean momenta for all elementary fields except the Abelian "photon". Ward Identities show that the symmetry breaking gives rise to massless BRST-quartets with ghost numbers (1, 2, -2, -1) and (0, 1, -1, 0). The latter quartet is interpreted as due to an Abelian Higgs mechanism in the dual description of the model.

A dual formulation of three dimensional SU(2) lattice gauge model is constructed. We calculate duals for the partition function and Wilson loop. Using the auxiliary fields we derive a local form of the plaquette representation of the Wilson action and discuss its applicability for the investigation of the confinement properties of the model at weak coupling.

The first ten coefficients in the perturbative expansion of the plaquette in Lattice SU(3) are computed both on a 8⁴ and on a 24⁴ lattice. They are shown to be fully consistent with the growth dictated by the first IR Renormalon and with the expected finite size effects on top of that. As already pointed out a few years ago, this leads to a puzzling result on the smaller lattice: when the contribution associated with the Renormalon is subtracted from Monte Carlo measurements of the plaquette, what is left over does not scale (as expected) as a⁴, but as a². While the analysis is not yet complete on the larger lattice, the implications of such a finding is discussed.

The colour confinement criterion proposed by Kugo and Ojima is tested in the lattice Landau gauge QCD simulation. The renormalization effects are studied by measuring the gluon propagator, ghost propagator, three gluon vertex and the ghost-antighost-gluon vertex. The running coupling α_s from ghost-antighost-gluon vertex in the infrared region yields , consistent to that from three gluon vertex in high momentum region.

I discuss the different aspects of the quark-anti-quark wave functional in Hamiltonian lattice QCD.

We compute the ratio between the scale Λ_L associated with a lattice formulation of QCD using the overlap-Dirac operator, and . To this end, the one-loop relation between the lattice coupling g₀ and the coupling renormalized in the scheme is calculated, using the lattice background field technique. We also compute the one-loop renormalization Z_Γ of the two-quark operators , where Γ denotes a generic Dirac matrix. Furthermore, we study the renormalization of quark bilinears which are more extended and have better chiral properties. Finally, we present improved estimates of Z_Γ, coming from cactus resummation and from mean field perturbation theory.

In this talk, I report on a work done in collaboration with R. Markazi and E.H. Saidi [1].

We discuss the model of confinement, in which the decay of hadron into the free quarks and gluons is not strictly forbidden, but the lifetime of hadron with respect to quark-gluon channel of decay is greater or at least of the order of age of the Universe. Our model provides the confinement of a massive quark within the limited region of space by means of constant component of the effective potential which arises as a result of reorganization of vacuum of the scalar field, which effectively describes gluon degrees of freedom, with condensation of corresponding Higgs field. On a concrete example it is shown that the lifetime of hadron equal to age of the Universe leads to the Higgs boson with the mass m_H > 40 GeV for realistic coupling and astrophysical constants.

The standard QCD action is improved by the addition of irrelevant operators built with chiral composites. An effective Lagrangian is derived in terms of auxiliary fields, which has the form of the phenomenological chiral Lagrangians. Our improved QCD action appears promising for numerical simulations as the pion physics is explicitely accounted for by the auxiliary fields.

Regge trajectories of quark-antiquark mesons can be well approximated for phenomenology purposes by a specific nonlinear form, reflecting that the flux tubes cannot be arbitrarily large, but break due to the effect of pair-production. If confirmed, this would imply that there is only a finite number of bound states on each trajectory, and consequently, an existence of "spectroscopy windows" for each flavor. Here we present our results for these windows.

We study the topological susceptibility, χ in two flavour lattice QCD.¹ We find clear evidence for the expected suppression of χ at small quark mass. The estimate of the pion decay constant, , is consistent with the experimental value of approximately 93 MeV. We compare χ to the large-N_c prediction and find consistency over a large range of quark masses.

Properties of light-light mesons are described by the effective Hamiltonian with spinless quarks derived from QCD. The spectrum is computed by the WKB method and shown to reproduce the celebrated linear Regge trajectories even for the lowest levels. The correct string slope of the trajectories naturally appears in the present approach as the string dynamics is taken into account properly.

Similar method is applied to heavy-light mesons and a set of corrections to the Hamiltonian is taken into account including spin-spin and Tomas spin-orbit interactions. The numerical results for the spectrum are compared with the experimental data and with the results of recent lattice calculations.

We present results on electromagnetic excitations of nucleon resonances within an extended version of the Goldstone-boson-exchange constituent quark model. The agreement with the data turns out to be not very satisfactory. Relativistic boost effects may change the results drastically, however.

The pionium lifetime is calculated in the framework of the quasipotential-constraint theory approach, including the sizable electromagnetic corrections. The framework of generalized chiral perturbation theory allows then an analysis of the lifetime value as a function of the ππ S-wave scattering lengths with isospin I = 0, 2, the latter being dependent on the quark condensate value.

We present lattice results for the gluon propagator for SU(2) and SU(3) in the Laplacian gauge which avoids lattice Gribov copies. In SU(3) we compare with the most recent lattice calculation in Landau gauge and with various approximate solutions of the Dyson Schwinger equations (DSE).

The form factor for γπ⁺ → π⁺π⁰ was calculated in a simple–minded constituent model with a constant quark mass parameter, as well as in the Schwinger-Dyson approach. The comparative discussion of these and various other theoretical results on this anomalous process, as well as the scarce already available data (hopefully to be supplemented by more accurate CEBAF data soon), seem to favor Schwinger–Dyson modeling which would yield relatively small low–momentum values of the constituent (dynamically dressed) quark mass function.

In this work we extend the shiftd l-expansion method developped in [1] to the spectra of heavy-light mesons .

We present results for the mass of the b quark in the scheme obtained by calculating the binding energy of the B meson in the static limit. The self energy of a static quark, needed for this purpose, is now known to O(α³) in the quenched approximation. We find a preliminary value of at n_f = 0. The error is dominated by the remaining uncertainty in . In addition, using at O(α²), we estimate that the quark mass is reduced by approximately 70 MeV when two flavours of dynamical quarks are introduced.

The recent lattice calculations of Bali, Schilling and Wachter (BSW) have made possible an unprecedented level of contact with phenomenology since these authors were able to completely determine all of the leading relativistic corrections to the heavy quark potentials. Since this work is based on the effective Lagrangian of NRQCD, it shares with other lattice calculations a need for reliable inputs for the matching coefficients c_i(μ, m). We compare calculations based on the BSW determination of the matching coefficients from one-loop expressions with those from the tree-level approximation, namely, c_i(μ, m) = 1. Using the central potential generated from the BSW lattice calculation, we find that the upsilon energies are not sensitive to this difference, but charmonium energies give a preference for the one-loop values.

A brief review is given of attempts to understand the energies of four-quark systems calculated on a lattice in terms of nuclear-physics-inspired many-body techniques involving interquark potentials. Results are given for the next stage of this study where the wavefunctions of heavy-light mesons are also calculated on a lattice.

We calculate the form factors for weak decays of B_(s) and D_(s) mesons to light pseudoscalar and vector mesons within a relativistic dispersion approach based on the constituent quark picture. This approach gives the form factors as relativistic double spectral representations in terms of the wave functions of the initial and final mesons. The form factors have the correct analytic properties and satisfy general requirements of nonperturbative QCD in the heavy quark limit. The effective quark masses and meson wave functions are determined by fitting the quark model parameters to lattice QCD results for the B → ρ transition form factors at large momentum transfers and to the measured D → (K, K*)lν decay rates. This allows us to predict numerous form factors for all kinematically accessible q² values.

A dozen energy levels of two scalar particles bound by a harmonic-oscillator potential are calculated exactly using four different equations: the Bethe-Salpeter equation, the "spinless Salpeter" equation, a new relativistic two-body wave equation local in configuration space, and the non-relativistic Schrödinger equation. The four equations all disagree with each other, but the Bethe-Salpeter equation and the relativistic two-body wave equation agree most closely.

The Bethe–Salpeter equation for bound states of a fermion–antifermion pair in the instantaneous approximation for the involved interaction kernel is converted into an equivalent matrix eigenvalue problem with explicitly (algebraically) given matrices.

The recently established generalization of the Gell-Mann–Low theorem is applied to Wick–Cutkosky–like models, where the exchanged particle can have arbitrary mass. We derive the effective Schrödinger equation to lowest non–trivial order, identify the diagrams corresponding to the Bloch–Wilson Hamiltonian, and consider the non–relativistic and one–body limits.

We discuss the characteristic features of QCD sum rules in the medium, which distinguish them from their vacuum counterparts. Both mesonic and nucleonic correlation functions are considered.

We investigate the eigenvalue spectrum of the staggered Dirac matrix in two-color QCD at nonzero temperature and at baryon density when the eigenvalues become complex. The quasi-zero modes and their role for chiral symmetry breaking and the deconfinement transition are examined. The bulk of the spectrum and its relation to quantum chaos is considered. Comparison with predictions from random matrix theory is presented.

The NA50 experiment at CERN has found evidence for deconfinement in central Pb-Pb collisions at 158 A GeV/c at the CERN SPS. The anomalous (with respect to ordinary nuclear absorption) suppression of J/ψ has been observed in 1995 and 1996 data and has been confirmed with the 1998 data analysis, where a detailed study of the suppression pattern vs. centrality has been performed. NA50 results exclude present conventional hadronic models, while they find a natural explanation in the formation of a deconfined state of quarks and gluons.

We give a very short account of our recent investigations¹ on Abelian monopole condensation in SU(2) and SU(3) lattice gauge theories.

No abstract received.

No abstract received.

The strong decay amplitudes are studied in the 't Hooft model for the two-dimensional QCD. Special attention is payed to the pions in the final state, and it is demonstrated that any amplitude of the form A → π + C vanishes identically for any pion momentum.

LIST OF PARTICIPANTS.