Relativistic field-theoretical equations for the three-body multichannel πN and γN scattering reactions and propagator of the Δ-resonance

A new form of the relativistic three-body equations for the coupled πN and γN scattering reactions with three particle final states ππN and γπN is suggested. These equations are derived in the framework of the time-ordered three dimensional field theory. The solutions of the considered equations satisfy the unitarity condition and are exactly gauge invariant. The form of these three-body equations is does not depend on the choice of the model of Lagrangian and is also the same for the formulations with and without quark degrees of freedom.

The effective potentials of the suggested equations are defined by the vertex functions with two on-mass shell particles. It is emphasized that these input vertex functions can be constructed from experimental data. Special attention is given to the construction of the propagator of the Δ-resonance in the framework of the separable πN potential model. The strong dependence of the multichannel πN and γN cross sections on the form of the Δ-resonance propagator[2] is discussed.

The used formulation of the relativistic three-dimensional and three-body equations allows us to overcome a number of approximations which are usually used by practical calculations of the πN and γN scattering reactions.     

πN scattering and electromagnetic corrections in the perturbative chiral quark model

We apply the perturbative chiral quark model to give predictions for the electromagnetic O(p²) low-energy couplings of the ChPT effective Lagrangian that define the electromagnetic mass shifts of nucleons and first-order (e²) radiative corrections to the πN scattering amplitude. We estimate the leading isospin-breaking correction to the strong energy shift of the π⁻p atom in the 1s state, which is relevant for the experiment “Pionic Hydrogen” at PSI.     

The NN → NNη reaction close to threshold

The pp → ppη and np → npη reactions at energies near the η production threshold are studied in a non-relativistic one boson exchange model, where the N^* (1535 MeV) S11 resonance is excited through the exchange of π, η, and ω mesons and subsequently decays into an ηN pair. Energy integrated cross sections and energy spectra of the out going η's are reported. Providing NN and ηN final state interactions are taken into account coherently, the model reproduces both the scale and energy dependence of the cross section for the pp → ppη reactions up to 100 MeV. Final state interaction corrections due to the nucleon-nucleon and meson-nucleon forces influence strongly the scale and shape of the cross sections. The shape of the energy spectra of the outgoing η's provides a clear signature of the ηN force.     

Double pion photoproduction on the nucleon: study of the isospin channels

A model for the γp → π⁺π⁻p reaction developed earlier is extended to account for all isospin channels. The model includes N, Δ(1232), N^*(1440) and N^*(1520) as intermediate baryonic states and the meson as an intermediate 2π resonance. Although many terms contribute to the cross section, some channels exhibit particular sensitivity to certain mechanisms of resonance excitation or decay and the reactions provide novel information on such mechanisms. In particular the γN → N^*(1520) → Δπ process affects all the channels and is a key ingredient in the interpretation of the data. Comparison is made with all available data and the agreement is good in some channels. The remaining discrepancies in some other channels are discussed.     

N=1 formulation of general N=2 Yang-Mills supergravity couplings

We formulate general interactions of an N=2 Yang-Mills supermultiplet coupled to N=2 supergravity in terms of N=1 invariant functions. We show that the N=2 scalar potential of the complex scalar fields in the adjoint representation, superpartners of the Yang-Mills gauge fields, splits into a conventional gauge contribution as well as in a superpotential term contribution proportional to the SO(2) gauge coupling of N=2 de Sitter supergravity. The relation of these results to the σ model structure of N4 extended supergravities is also discussed, particularly in connection with the coset disintegration of scalar manifolds.     

Pion–nucleon amplitude near threshold: the sigma-term and scattering lengths beyond few loops

The pion–nucleon amplitude is considered in the vicinity of the elastic scattering threshold within a relativistic dynamical model dressing the πNN and πNΔ vertices self-consistently with an infinite number of meson loops. The dressing is formulated as solution of a system of coupled integral equations incorporating unitarity, crossing symmetry and analyticity constraints. The calculated scattering lengths and the sigma-term agree with recent data analyses. The dressing is important in this model both below and at threshold. The contribution of the Δ resonance is discussed, including effects of the consistent dressing of the πNΔ vertex. A comparison with the approaches of chiral perturbation theory and the Bethe–Salpeter equation is outlined.     

η in the nuclear medium within a chiral unitary approach

The s-wave η self-energy in the nuclear medium is calculated in a chiral unitary approach. A coupled channel Bethe–Salpeter equation is solved to obtain the effective ηN interaction in the medium. The base model reproduces well the free space πN elastic and inelastic scattering at the ηN threshold or N^*(1535) region. The Pauli blocking on the nucleons, binding potentials for the baryons and self-energies of the mesons are incorporated, including the η self-energy in a self-consistent way. Our calculation predicts about −54−i29 MeV for the optical potential at normal nuclear matter for an η at threshold but also shows a strong energy dependence of the potential.     

On the pp → ppη(η′) reactions near threshold

The production rate for η′ in pp → ppη′ at rest is calculated in a covariant one boson exchange model, previously applied to study π⁰ and η production in NN collisions. The transition amplitudes for the elementary BN → η′N processes with B being the meson exchanged (B = π, σ, η, , ω and a₀) are taken to be the sum of s- and u-channels with a nucleon in the intermediate states, and an a₀ meson pole in a t-channel. The couplings of the η′ to hadrons are a factor 0.4 weaker than the respective η-hadron couplings, as suggested by a quark model and a singlet-octet mixing angle θ = −23°. The model reproduces near threshold cross sections for the quasielastic processes π⁻p → nη(η′) and pp → ppη(η′) reactions.     

Four-body calculation of ΛH and ΛHe with realistic YN and NN interactions

We carried out four-body calculations of _Λ⁴H and _Λ⁴He taking both the 3N + Λ and 3N + Σ channels explicity with the use of realistic NN and YN interactions. The Σ-channel component plays an important role in binding energies of the A = 4 hypernuclei though the admixture is approximately 1%. The ΛN - ΣN coupling is found to be of central-force type in the Nijmegen model D and of tensor-force type in the model F.     

The large N limit of the O(N) Heisenberg spin system in two dimensions

The large N limit of the O(N) Heisenberg spin system in two dimensions (a lattice version of the non-linear σ-model) is analysed by the collective field technique. A compact expression for the mass gap and the β-function is obtained for arbitrary but fixed gN. The strong and weak coupling limits of this expression correctly reproduce the known results.     

The O(N) vector model in the large-N limit revisited: multicritical points and double scaling limit

The multicritical points of the O(N)-invariant N vector model in the large-N limit are re-examined. Of particular interest are the subtleties involved in the stability of the phase structure at critical dimensions. In the limit N → ∞ while the coupling g → g_c in a correlated manner (the double scaling limit) a massless bound state O(N) singlet is formed and powers of 1/N are compensated by IR singularities. The persistence of the N → ∞ results beyond the leading order is then studied with particular interest in the possible existence of a phase with propagating small mass vector fields and a massless singlet bound state. We point out that under certain conditions the double scaled theory of the singlet field is non-interacting in critical dimensions.     

Final-state hyperon-nucleon interaction in the inclusive K photoproduction from the deuteron

We calculate d(γ, K⁺) inclusive cross sections with the full inclusion of the final ΛN - ΣN interaction. Modern hyperon-nucleon forces and a recently updated production operator for the γ + N → K⁺ + Y process are used. Significant effects of the hyperon-nucleon final-state interaction have been found especially around the K⁺ΣN threshold.     

Exact renormalization group study of fermionic theories

The exact renormalization group approach (ERG) is developed for the case of pure fermionic theories by deriving a Grassmann version of the ERG equation and applying it to the study of fixed point solutions and critical exponents of the two-dimensional chiral Gross-Neveu model. An approximation based on the derivative expansion and a further truncation in the number of fields is used. Two solutions are obtained analytically in the limit N → ∞, with N being the number of fermionic species. For finite N some fixed point solutions, with their anomalous dimensions and critical exponents, are computed numerically. The issue of separation of physical results from the numerous spurious ones is discussed. We argue that one of the solutions we find can be identified with that of Dashen and Frishman, whereas the others seem to be new ones.     

Renormalization of N = 2, 4 Yang-Mills theories with soft breaking of an extended supersymmetry by N = 1 mass term

N = 2, 4 Yang-Mills theories with soft breaking of an extended supersymmetry by mass terms are considered. It is proved that for N = 4 there are no ultraviolet divergences in the mass renormalization constants to all orders of perturbation theory. For N = 2 our two-loop calculations show that the charge and mass renormalization constants contain only one-loop divergences and are the same in this order. It is shown by direct calculation that mass terms can acquire finite quantum corrections starting from the two-loop approximation. The renormalization scheme dependence of N = 4 renormalization group functions is investigated. We have found that unlike renormalization schemes with minimal subtractions of divergences other renormalization schemes give a nonzero β-function. At nonzero masses the β-function in MOM schemes is not zero even at the one-loop level. In the massless case β≠0 beginning from the two-loop approximation.     

Characteristic polynomials of random Hermitian matrices and Duistermaat–Heckman localisation on non-compact Kähler manifolds

We reconsider the problem of calculating a general spectral correlation function containing an arbitrary number of products and ratios of characteristic polynomials for a N×N random matrix taken from the Gaussian Unitary Ensemble (GUE). Deviating from the standard “supersymmetry” approach, we integrate out Grassmann variables at the early stage and circumvent the use of the Hubbard–Stratonovich transformation in the “bosonic” sector. The method, suggested recently by J.V. Fyodorov [Nucl. Phys. B 621 [PM] (2002) 643], is shown to be capable of calculation when reinforced with a generalisation of the Itzykson–Zuber integral to a non-compact integration manifold. We arrive to such a generalisation by discussing the Duistermaat–Heckman localisation principle for integrals over non-compact homogeneous Kähler manifolds. In the limit of large-N the asymptotic expression for the correlation function reproduces the result outlined earlier by A.V. Andreev and B.D. Simons [Phys. Rev. Lett. 75 (1995) 2304].     

Interpolating between Ising, XY-, and non-linear σ-models

The β-functions of O(N)-symmetric non-linear σ-models on the lattice were recently discovered to be non-monotonic for N 3. We explain the non-monotonic behaviour as a non-perturbative lattice effect by relating it to the Kosterlitz-Thouless transition of the XY-model. We also relate the latter transition to the phase transition of the Ising model. These relationships are established by interpolating between the O(N)- and the O(N − 1)-symmetric non-linear σ-models by suppression of the Nth component of the N-vector field with a mass term. A critical line in the coupling-mass plane connects the critical point of the Ising model (N = 1) with the critical point of the XY-model (N = 2). This line extends towards the region of non-monotonic behaviour of the β-function of the O(3)-symmetric model. The nature of the transition lines is also investigated.     

Nucleon structure in the perturbative chiral quark model

We give an overview of recent applications of the perturbative chiral quark model (PCQM) in the analysis of the structure of the nucleon. The PCQM is based on an effective Lagrangian, where baryons are described by relativistic valence quarks and a perturbative cloud of Goldstone bosons as required by chiral symmetry. We discuss applications to the electromagnetic properties of the nucleon, to σ-term physics, to πN scattering including radiative corrections and to the strange form factors of the nucleon.     

Critical currents and the critical state in superconductors with magnetic ions

The critical state model is used to derive equations that relate the additional magnetic moment (ΔM) produced by the flux pinning to the critical current density (J_c) measured in transport measurements. The equations derived for conventional superconductors can be used for superconductors that contain magnetic ions, if ΔM is replaced by ΔM/(1 + χ′(H)) where χ′(H) is the differential susceptibility. In the critical state, the field gradient has contributions from both the macroscopic supercurrents and the Amperian currents from the magnetic ions. Magnetic measurements are sensitive to both contributions. Transport measurements only characterise the macroscopic supercurrents. For superconductors which contain rare-earth elements, the J_c values calculated using hysteretic magnetisation measurements without including the term χ′(H), can be in error by factors of 7.     

1/N expansion and the correlation energy of nuclear matter in the relativistic σω model

We propose the 1/N expansion method, where N can be identified with 2 due to the isospin SU(2), as a systematic expansion scheme for the relativistic meson-nucleon many-body theory. We derive a general formula for the computation of the effective action in the σω model based on the 1/N expansion, utilizing the functional integral method. In this scheme the leading contribution coincides with the familiar Hartree approximation. The elimination of the Landau ghost from the meson propagators in the subgraphs, which appears to be essential for our formulation, is also established.

As an application we compute the energy density of nuclear matter beyond the Hartree approximation including the next-to-leading order contributions in the 1/N expansion. In this order, one should include the ring energy contribution as a correlation energy addition to the exchange energy contribution. We find that we can describe the nuclear matter saturation properties without drastic changes of the overall physical picture which emerged in the Hartree approximation. The important role of the repulsive contribution due to the vacuum polarization effects in the ring energy is emphasized.     

Rigorous pion-pion scattering lengths from threshold πN → ππN data

A new evaluation of the universal ππ scattering length relation is used to extract the ππ s-wave scattering lengths from threshold pion production data. Previous work has shown that the chiral perturbation series relating threshold pion production to ππ scattering lengths appears to converge well only for the isospin-2 case, giving a₂ = −0.031 ± 0.007m_π⁻¹. A model-independent and data-insensitive universal curve then implies a₀ = 0.235 ± 0.03m_π⁻¹ for the isospin-0 scattering length.