A generalization of the non-linear λ-model to four dimensions

The two-dimensional O(3)-nonlinear σ-model is generalized to four dimensions. The generalization, which includes three related models of the types O(5), CP² and HP¹, preserves conformal invariance and many of the geometrical properties of the non-linear σ-model. We examine the topological and geometrical properties of the O(5)-model in detail and, in particular, investigate its close connection with the Yang-Mills theory. The instantons of the model are examined and shown to correspond to conformal mappings from R⁴ into S⁴.     

A study of meson-baryon couplings in the constituent quark model

Within the framework of the constituent quark model we discuss the effects of different types of meson-baryon-baryon vertex operators on the form factors and the coupling strengths of the lowest-lying positive and negative parity non-strange baryons. We compare the quark pair creation model (³P₀-model) with the SU(6)-model in which mesons are treated as elementary fields that directly couple to the quarks. The latter model is employed both in the so-called static limit and in a modification motivated by Galilei invariance. It is demonstrated that the inclusion of non-static effects simulates some features of the³P₀ vertex. Especially the reaction πN→ππN is found to be very sensitive to the different assumptions on the dynamics of theq¯q pair creation process. More indirect hints for the internal structure of the mesons might be obtained from the predicted asymmetry for the two form factors ofN→Δ+π andΔ→N+π, which occurs in the³P₀-model, only.     

Classical solutions of CPn non linear σ -models; an algebraic geometrical description

The classical SO (3)-invariant σ-model and its suitably generalized versions are studied from the geometrical point of view. Known mathematical results concerning harmonic and holomorphic maps of a Riemann surface into the n-dimensional complex projective space are briefly reviewed. These are used to give a classification of classical solutions (both instantons and a certain subclass of unstable solutions) of $\text{C}$Pⁿ models and to study the properties of their energy spectrum.     

Generalized non-linear σ-models with gauge invariance

Generalizations of the non-linear σ-model and of the CP^N model are considered. They are invariant under a global group G and under a local gauge subgroup H. Whenever G/H defines a symmetric space, we can show that these models are asymptotically free. Furthermore, in two dimensions the local gauge invariance leads to confinement.     

On a factor associated with the unordered phase of λ-model on a cayley tree

In this paper we consider nearest-neighbours models, where the spin takes values in the set Φ = {η₁, η₁, …, η_q} and is assigned to the vertices of the Cayley tree Γ^k. The Hamiltonian is defined by some given λ-function. We find a condition for the function λ to determine a type of von Neumann algebra generated by the GNS construction associated with the unordered phase of the λ-model. We give also some physical applications of the obtained result.     

On the relationship between string low-energy effective actions and O(α′3) σ-model β-functions

We examine in detail the connection between low-energy string effective actions and the σ-model β-functions up to O(α^′3) in the torsion-free case. In particular we show that the relationship between, on the one hand, the equations of motion derived from the O(α^′3) string effective action, and, on the other hand, the O(α^′3) σ-model β-functions necessarily involves derivative operators acting on the β-functions.     

Renormalizations and Operator Expansion in σ-Model

The operator expansion (OPE) is studied for the Green function n(0) n(x)) at x² → 0 (n(x) is the dynamical field of σ-model) in the framework of the two-dimensional σ-model with the O(N) symmetry group at large N. As a preliminary step we formulate the ronormalization scheme which permits introduction of an arbitrary intermediate scale μ² in the framework of 1/N expansion and discuss factorization (separation) of small (p < μ ) and large (p > μ ) momentum region. It is shown that definition of composite local operators and coefficient functions figuring in OPE is unambiguous only in the leading order in 1/N expansion when dominant are the solutions with exrtemum of action. Corrections of order f(μ²)/N (here /(μ²) is the effective interaction constant at the point μ²) in composite operators and coefficient functions essentially depend on factorization method of high and low momentum regions. It is shown also that contributions to the power corrections of order m²x²/(μ²)/N in the Green function (here m is the dynamical mass-scale factor in σ-model) arise simultaneously from two sources: from the mean vacuum value of the composite operator n ∂²n and from the hard particle contributions in the coefficient function of unite operator. Due to the analogy between σ-model and QCD the obtained result indicates theoretical limitations to the sum rule method in QCD.     

Sigma-models having supermanifolds as target spaces

We study a topological sigma-model (A-model) in the case when the target space is an (m ₀|m ₁)-dimensional supermanifold. We prove under certain conditions that such a model is equivalent to an A-model having an (m ₀–m ₁)-dimensional manifold as a target space. We use this result to prove that in the case when the target space of A-model is a complete intersection in a toric manifold, this A-model is equivalent to an A-model having a toric supermanifold as a target space.Research supported in part by NSF grant No. DMS-9201366.     

Infrared convergence of Feynman integrals for the masslessA 4-model

For the masslessA ⁴-model it is proved that renormalization can be formulated such that each Feynman diagram yields an ultraviolet and infrared convergent contribution to the Green's functions.     

The O(5) nonlinear σ-model as a SU(2) gauge theory

We show how the O(5) nonlinear σ-model, coupled with “gravity” in a generally covariant way, can be translated into the quaternionic model HP¹, invariant under a local SU (2) gauge, and we discuss the classical solutions.     

Conformally invariant σ models in 2n dimensions

The conformally invariant σ-model of Felsager and Leinaas in four dimensions (n=2) is generalised to 2n dimensions, and the unit-charge instantons are presented. This hierarchy of σ-models starts with the O(3) σ-model (n=1). The relationship of these models with the generalised YM systems for even n=2p is pointed out.     

The XY-model and the self-avoiding walk approximation

The SAW-approximation is calculated for the quantum-mechanical XY-model for spin $\text{1}\text{2}$ in a parallel magnetic field. For this purpose we derive an exact expression for the internal weight factors of the polygon graphs.In this approximation we obtain an analytic expression for the phase-separation line in the (H, T)-phase. The critical exponents are the same as for classical models.     

Dispersion relation for the Gell-Mann-Low function,the spectrality condition and the asymptotic behaviour of the invariant charge

It is pointed out that dispersion relation of the Gell-Mann-Low function together with a hypothesis about the spectrality define the asymptotic behaviour of the Gell-Mann-Low function ψ(g). For the g?⁴-model we get ψ(g) ≈ agatg ? 1.     

Renormalisation of Noncommutative ϕ 4-Theory by Multi-Scale Analysis

In this paper we give a much more efficient proof that the real Euclidean ? ⁴-model on the four-dimensional Moyal plane is renormalisable to all orders. We prove rigorous bounds on the propagator which complete the previous renormalisation proof based on renormalisation group equations for non-local matrix models. On the other hand, our bounds permit a powerful multi-scale analysis of the resulting ribbon graphs. Here, the dual graphs play a particular rôle because the angular momentum conservation is conveniently represented in the dual picture. Choosing a spanning tree in the dual graph according to the scale attribution, we prove that the summation over the loop angular momenta can be performed at no cost so that the power-counting is reduced to the balance of the number of propagators versus the number of completely inner vertices in subgraphs of the dual graph.     

On renormalizable two-dimensional σ-models with dynamical charges

(a) The general conditions are stated under which a non-linear σ-model in two dimensions has dynamical charges of the type found by Lüscher and Pohlmeyer for the O (N)-model. (b) he renormalization of a large class of σ-models including CPⁿ-models and chiral fields on Grassmann manifolds is discussed.     

Non-supersymmetric extremal RN-AdS black holes in $$ \mathcal{N} = 2 $$ gauged supergravity

We investigate extremal Reissner-Nordström-AdS black holes in fourdimensional \( \mathcal{N} = 2 \) abelian gauged supergravity. We find a new attractor equation which is not reduced to the one in the asymptotically flat spacetime. We also argue a formula which is available even in the presence of the scalar potential. We apply them to the T³-model and the STU-model in generic black hole charge distributions. In addition, focusing on the so-called T³-model with a single neutral vector multiplet, we obtain non-supersymmetric extremal Reissner-Nordström-AdS black hole solutions with regular event horizons in the D0-D4 and the D2-D6 black hole charge configurations. The negative cosmological constant emerges even without the Fayet-Iliopoulos parameters.     

Vacuum alignment by broken gauge interactions

We study the problem of vacuum alignment by the extended technicolour (ETC)forces which have been invoked to give mass to ordinary fermions (quarks and leptons) in technicolour theories. We find that the ETC interactions prefer to align the vacuum in such a way as to give the maximum possible mass to the ordinary fermions. We show that the P⁰, P³, P^+-, P₈⁰, P₈³, and P₈^+- pseudo-Goldstone bosons in one technifamily ETC models cannot acquire any mass from ETC interactions alone. Finally, we point out that the Pati-Salam unification which has been invoked to give mass to the P⁰ and P³ implies a remaining discrete degeneracy of the vacuum.     

On finite action solutions of the nonlinear σ-model

The instanton and anti-instanton solutions of the two-dimensional O(3) σ-model are special examples of harmonic maps, which have been studied extensively in the mathematical literature. We give an elementary and self-contained proof that these solutions are the only continuous maps for which the action is finite and stationary under variations, without assuming any additional boundary conditions at infinity. An element of the proof is the vanishing of the stress tensor for a finite action solution, which actually holds true for the general O(N) σ-model. For the two-dimensional O(2l + 1) σ-model we exhibit explicit finite action solutions that do not lie in any lower dimensional sphere; the existence of such solutions has been pointed out in the mathematical literature. We also present a rigorous proof, based on Derrick's scaling argument, that there are no nonconstant finite action solutions in more than two dimensions.     

A real-time approach to nonequilibrium quantum field theory of relativistic fields

Summary The basic ingredients of a real-time nonequilibrium quantumfield theoretical approach, based on a generalization of thermofield dynamics, are briefly reviewed. In particular, the possible advantages of its application to relativistic models, as the φ⁴-model for the Higgs field are pointed out. Also the differences between the above approach and others using the Feynman's path integral method are underlined. ?Angelo della Riccia? Fellow.