Large N_C QCD 2004

Prologue

List of Participants

Contents

A discussion is given of the confinement mechanism in terms of the Abelian projection scheme, for a general number N_c of colors. There is a difficulty in the N_c → ∞ limit that requires a careful treatment, as the charges of the condensing magnetic monopoles tend to infinity. We suggest that Bose condensation of electric or magnetic charges is indicative for the kind of confinement that takes place, but the actual mechanism of confinement depends on other features as well.

I review some developments in the large-N gauge theory since 1974. The main attention is payed to: multicolor QCD, matrix models, loop equations, reduced models, 2D quantum gravity, free random variables, noncommutative theories, AdS/CFT correspondence.

We review our current understanding of instanton effects in the large N_c limit of QCD.

This talk is about the planar equivalence between gluodynamics (super-Yang-Mills theory) and a non-supersymmetric “orientifold field theory.” We outline an “orientifold” large N expansion, analyze its possible phenomenological consequences in one-flavor massless QCD, and make a first attempt at extending the correspondence to three massless flavors. An analytic calculation of the quark condensate in one-flavor QCD starting from the gluino condensate in gluodynamics is thoroughly discussed.

The main leitmotiv of the present talk is an analysis of different topological objects such as strings and domain walls in QCD. As is well known, the standard model in general (and QCD in particular) does not support any such objects due to some simple topological arguments. However, the situation cardinally changes when one considers QCD in the large N_c limit or in the limit of large baryon density when the unique light η′ field emerges. I discuss η′-domain walls and η′ global strings which occur in both systems: in high density QCD and in QCD at large N_c. I also discuss the effects of quantum anomalies in the presence of topological objects in high-density QCD. The anomaly-induced interactions lead to a number of interesting phenomena (such as the induced currents along the strings), which may have phenomenological consequences observable in neutron stars.

We review the construction of superstring models which are dual to non-supersymmetric confining gauge theories. We discuss the calculation of the mass spectrum, Wilson loops, topological susceptibility and other physical properties.

A holographic dual of N_f-flavour large-N_c QCD is constructed by placing N_f D6-brane probes in the background of N_c D4-branes. For N_f = 1 it exhibits spontaneous chiral symmetry breaking and the corresponding massless pseudoscalar in the spectrum. A holographic version of the Vafa-Witten theorem is presented. The U(1)_A chiral anomaly and the associated η′ physics are discussed.

I summarise what recent lattice calculations tell us about the large-N limit of SU(N) gauge theories in 3+1 dimensions. The focus is on confinement, how close SU(∞) is to SU(3), new stable strings at larger N, deconfinement, topology and θ-vacua. I discuss the effective string theory description, as well as master fields, space-time reduction and non-analyticity.

It is possible to numerically solve QCD in the planar limit using standard numerical techniques on existing computer clusters. The basic ideas behind the computational strategy and recent numerical results in the fermionic sector of large N QCD are reviewed.

In the first part of the talk, I discussed results on the determination of the ratios of the light quark masses from large N_c chiral perturbation theory, to be described elsewhere. The following notes contain material from the second part of the talk, which concerns the implications of large N_c for resonance dominance estimates of the low energy coupling constants in chiral perturbation theory.

I review a recent analytic method for computing matrix elements of electroweak operators based on the large-N_c expansion. In particular, I give a rather detailed description of the matching of the quark Lagrangian onto the meson chiral Lagrangian governing mixing. In the ΔS = 1 sector, I explain how to obtain an estimate for both ε′/ε and the ΔI = 1/2 rule. Finally, I give an example of how the method can also be useful for lattice calculations.

We present work going on calculating the kaon B_K parameter in the 1/N_c expansion. The goal of this work is to analyze analytically in the presence of chiral corrections this phenomenologically very important parameter. We present the method used and preliminary results for the chiral limit value for which we get . We also give some analytical indications of why the large N_c prediction of may have small 1/N_c corrections in the real case.

The properties of baryons can be computed in a systematic expansion in 1/N_c, making use of an SU(2N_F) spin-flavor symmetry that is exact at N_c = ∞, and the structure of the symmetry-breaking 1/N_c corrections. This talk reviews the applications of the 1/N_c expansion to baryons. Recent results on the 1/N_c expansion to exotic baryons are also presented.

Based on a valence-quark picture of large N_c baryons, I describe in some detail the 1/N_c power counting for decays and spin-flavor configuration mixings in baryons.

The derivation of the 1/N_c expansions for the baryon vector and axial-vector currents, in the SU(3) symmetry limit and including perturbative SU(3) breaking, is reviewed. Symmetry breaking effects in the hyperon semileptonic decay form factors are reanalyzed. Fits to experimental data yield corrections to the leading vector and axial-vector form factors consistent with expectations. The values of the Cabibbo-Kobayashi-Maskawa matrix elements V_ud and V_us also extracted from the analysis are comparable to the ones recommended by the Particle Data Group.

The 1/N_c expansion of QCD provides a valuable semiquantitative tool to study baryon scattering amplitudes and the short-lived baryon resonances embedded within them. A generalization of methods originally applied in chiral soliton models in the 1980’s provides the key to deriving a rigorous 1/N_c expansion. One obtains model-independent relations among amplitudes that impose mass and width degeneracies among resonances of various quantum numbers. Phenomenological evidence confirms that patterns of resonant decay predicted by 1/N_c agree with data. One may extend the analysis to subleading orders in 1/N_c, where again agreement with data is evident, in both meson-baryon scattering and pion photoproduction.

The hadronic decays of non-strange negative parity baryons are analyzed in the framework of the 1/N_c expansion. A complete basis of spin-flavor operators for the pseudoscalar meson S and D partial wave amplitudes is established to order 1/N_c and the unknown effective coefficients are determined by fitting to the empirically known widths. A set of relations between widths that result at the leading order, i.e. order , is given and tested with the available data.

We use the 1/N_c expansion of QCD to analyze the spectrum of positive parity resonances with strangeness S = 0, −1, −2 and −3 in the 2–3 GeV mass region, supposed to belong to the [56, 4⁺] multiplet. The mass operator is similar to that of [56, 2⁺], previously studied in the literature. The analysis of the latter is revisited. In the [56, 4⁺] multiplet we find that the spin-spin term brings the dominant contribution and that the spin-orbit term is entirely negligible in the hyperfine interaction, in agreement with constituent quark model results. More data are strongly desirable, especially in the strange sector in order to fully exploit the power of this approach.

This talk describes progress at understanding the properties of the nucleon and its excitations from lattice QCD. I begin with a review of recent lattice results for the lowest-lying states of the excited baryon spectrum. The need to approach physical values of the light quark masses is emphasized, enabling the effects of the pion cloud to be revealed, I then outline the development of techniques that will enable the extraction of the masses of the higher resonances. I will describe how such calculations provide insight into the structure of the hadrons, and enable comparison both with experiment, and with QCD-inspired pictures of hadron structure, such as calculations in the limit of large N_c.

Instantons generate attraction in the scalar anti-triplet diquark channel. Direct instanton-induced diquark correlations are maximally enhanced for N_c = 2 and are suppressed in the large N_c limit. In this work we study the strength of such correlations, for N_c = 3. We identify some correlation functions, which allow to probe directly the diquark content of the nucleon. We use the Random Instanton Liquid Model to compute these matrix elements to all orders in the ’t Hooft interaction and we compare these results with the predictions of the mean-field Quark Chiral Soliton Model, which provides the solution to leading order in 1/N_c. We find strong discrepancies between the “exact” numerical calculations and the analytic mean-field calculations. Such discrepancies signal that, for N_c = 3, the instanton-induced diquark correlations are very strong.

We have computed the baryon spectrum in the context of super-conformal Yang-Mills theory using AdS/CFT duality. Baryons are included in the theory by adding an open string sector, corresponding to quarks in the fundamental and higher representations. The hadron mass scale is introduced by imposing boundary conditions at the wall at the end of AdS space. The quantum numbers of each baryon are identified by matching the fall-off of the string wavefunction Ψ(x, r) at the asymptotic 3 + 1 boundary to the operator dimension of the lowest three-quark Fock state, subject to appropriate boundary conditions. Higher Fock states are matched quanta to quanta with quantum fluctuations of the bulk geometry about the fixed AdS background, maintaining conformal invariance. The resulting four-dimensional spectrum displays a remarkable resemblance to the physical baryon spectrum of QCD, including the suppression of spin-orbit interactions.

The 1/N_c expansion for transverse-momentum dependent parton distributions is compatible with the existence of nonzero T-odd distributions. The isospin structure of the leading order is described for all leading-twist distributions.