Maciej A. Nowak

SU(3) extension of the skyrme model

Abstract The lagrangian and hamiltonian of the SU(3) L × SU(3) R non-linear sigma model are given in terms of collective variables. It is argued that the model describes the octet and decuplet of baryons, and the interpretation of the unbroken symmetries of the system is given. The model is quantized and the numerical results are compared with the data.

The Hyperons as Skyrmions With Vector Mesons

We show that when “strong” vector mesons are suitably incorporated, the SU(3) skyrmion model of Callan and Klebanov works very well to O(1Nc). A simplified model with an ω and φ is constructed and shown to provide the resolution of the strangeness problem encountered in the collective coordinate treatment of the hyperons.

Chiral effective action with heavy quark symmetry

We derive an effective action combining chiral and heavy-quark symmetry, using approximate bosonization techniques to QCD-inspired models of spontaneous symmetry breaking. We explicitly show that the heavy-quark limit is compatible with the large [ital N][sub [ital c]] (number of color) limit in the meson sector, and estimate the couplings between the light and heavy mesons ([ital D],[ital D][sup *], . . .) and their chiral partners to order one in the heavy-quark mass. The relevance of this effective action to solitons with heavy quarks is briefly discussed.

Eigenvalues and singular values of products of rectangular Gaussian random matrices

We derive exact analytic expressions for the distributions of eigenvalues and singular values for the product of an arbitrary number of independent rectangular gaussian random matrices in the limit of large matrix dimensions. We show that they both have power-law behavior at zero and determine the corresponding powers. We also propose a heuristic form of finite size corrections to these expressions which very well approximates the distributions for matrices of finite dimensions.

Chiral Disorder in QCD

Using the Gell{endash}Mann{endash}Oakes{endash}Renner relation and semiclassical arguments, we show that the bulk quark spectrum in QCD exhibits a variety of regimes including the ergodic one described by random matrix theory. We analyze the quark spectral form factor in the diffusive and ballistic regime. We suggest that a class of chiral transitions in QCD is possibly of the metal-insulator type, with a universal spectral statistics at the mobility edge. {copyright} {ital 1998} {ital The American Physical Society}

Wishart and anti-Wishart random matrices

We provide a compact exact representation for the distribution of the matrix elements of the Wishart-type random matrices †, for any finite number of rows and columns of , without any large N approximations. In particular, we treat the case when the Wishart-type random matrix contains redundant, non-random information, which is a new result. This representation is of interest for a procedure for reconstructing the redundant information hidden in Wishart matrices, with potential applications to numerous models based on biological, social and artificial intelligence networks.

NON-HERMITIAN RANDOM MATRIX MODELS

We introduce an extension of the diagrammatic rules in random matrix theory and apply it to non-hermitian random matrix models using the 1/N approximation. A number of one-and two-point functions are evaluated on their holomorphic and non-holomorphic supports to leading order in 1/N. The one-point functions describe the distribution of eigenvalues, while the two-point functions characterize their macroscopic cotrelations. The generic form for the two-point functions is obtained, generalizing the concept of macroscopic universality to non-hermitian random matrices. We show that the holomorphic and non-holomorphic one- and two-point functions condition the behavior of pertinent partition functions to order O(1/N). We derive explicit conditions for the location and distribution of their singularities. Most of our analytical results are found to be in good agreement with numerical calculations using large ensembles of complex matrices.

Dirac spectrum in QCD and quark masses

We use a chiral random matrix model to investigate the effects of massive quarks on the distribution of eigenvalues of QCD inspired Dirac operators. Kalkreuters lattice analysis of the spectrum of the massive (hermitian) Dirac operator for two colors and Wilson fermions is shown to follow from a cubic equation in the quenched approximation. The quenched spectrum shows a Mott transition from a (delocalized) Goldstone phase softly broken by the current mass, to a (localized) heavy quark phase, with quarks localized over their Compton wavelength. Both phases are distinguishable by the quark density of states at zero virtuality, with a critical quark mass of the order of 100–200 MeV.At the critical point, the quark density of states is given by νQ (λ) ∼ |λ|13. Using Grassmaniian techniques, we derive an integral representation for the resolvent of the massive Dirac operator with one flavor in the unquenched approximation, and show that near zero virtuality the distribution of eigenvalues is quantitatively changed by a non-zero quark mass. The generalization of our construction to arbitrary flavors is also discussed. Some recommendations for lattice simulations are suggested.

Infinite products of large random matrices and matrix-valued diffusion

Abstract We use an extension of the diagrammatic rules in random matrix theory to evaluate spectral properties of finite and infinite products of large complex matrices and large Hermitian matrices. The infinite product case allows us to define a natural matrix-valued multiplicative diffusion process. In both cases of Hermitian and complex matrices, we observe the emergence of a “topological phase transition”, when a hole develops in the eigenvalue spectrum, after some critical diffusion time τ crit is reached. In the case of a particular product of two Hermitian ensembles, we observe also an unusual localization–delocalization phase transition in the spectrum of the considered ensemble. We verify the analytical formulas obtained in this work by numerical simulation.

Large Nc confinement and turbulence

We suggest that the transition that occurs at large N_{c} in the eigenvalue distribution of a Wilson loop may have a turbulent origin. We arrived at this conclusion by studying the complex-valued inviscid Burgers-Hopf equation that corresponds to the Makeenko-Migdal loop equation, and we demonstrate the appearance of a shock in the spectral flow of the Wilson loop eigenvalues. This picture supplements that of the Durhuus-Olesen transition with a particular realization of disorder. The critical behavior at the formation of the shock allows us to infer exponents that have been measured recently in lattice simulations by Narayanan and Neuberger in d=2 and d=3. Our analysis leads us to speculate that the universal behavior observed in these lattice simulations might be a generic feature of confinement, also in d=4 Yang-Mills theory.