Impact parameter representation for separable potentials

An exact impact parameter representation for nonlocal separable potential has been written down. This representation is valid at all angles and energies. Further, it is shown that the Blankenbecler-Goldberger representation follows naturally for such a potential and has also an $\text{N}\text{D}$ form.     

Effective range parameters of separable potentials

Formulas are obtained which express the effective range parameters of a rank-two separable potential as non-singular integrals of the form factors. These results are extended to allow for a Coulomb interaction.     

Integral equation theory for fluids ordered by an external field: separable interactions

The structural and thermodynamical properties of classical fluids orientationally ordered by an external field are investigated by means of integral equation theories. A general theoretical framework for handling these theories is developed and detailed for the particular case of separable interactions between fluid particles. This approach is then illustrated for the case of two (off lattice) models: the ferromagnetic Heisenberg model and a simple liquid crystal model, for which the numerical solution of integral equations such as the Percus-Yevick, the hypernetted chain, and the reference hypernetted chain closure equations are presented and compared with Monte Carlo simulation results and the analytical solution of the mean spherical approximation. The zero-field case is also examined, and the spontaneous ordering is analyzed in detail, mainly in what concerns the appearance of infinite wavelength singularity in the Ornstein-Zernike equation and the relation with the one-body closure equations and the long range orientational ordering that occurs. In particular, it is shown that the Wertheim one-body closure equation appears as a sum rule compatible with the Ornstein-Zernike equation. The relation between the elastic constant and the long range tail of the pair correlation function is made explicit. In particular, the long range behavior of the various terms in the expansion of the pair correlation function is depicted. The numerical investigation of the two models shows that it is not possible to discriminate between the four integral equations, as to which one would be the most accurate in all cases. The general trends in the thermodynamical and structural properties seem to indicate that the Percus-Yevick approximation is generally better in the strong ordering case, whereas the reference hypernetted chain approximation might be better suited to the study of the isotropic phase and the low ordering regimes.     

Energy-dependent Lorentz covariant representation of the NN interaction

A phenomenological Lorentz covariant representation of the NN t-matrix has been obtained with parameters that are explicitly energy dependent. The real and imaginary parts of the t-matrix are separately expanded in a complete set of on-shell Lorentz invariant amplitudes, each of which is represented by the exchange of isoscalar and isovector “mesons” modified by form factors. Direct and exchange contributions are handled explicitly in the model. The model parameters are fit to the summer 1994 Arndt amplitudes over the laboratory kinetic energy range from 200 to 500 MeV. In contrast to the earlier work of Horowitz, a single χ² minimization is carried out over the full energy range considered. Two separate fits are obtained, one with energy dependence confined to the coupling strengths, the other with energy dependence in the form factor masses as well. Although the second fit is somewhat better, both fits reproduce the empirical amplitudes reasonably well. Results are also given for various np and pp observables and compared with the data.     

Elastic unitarity and the K-matrix in a Lorentz covariant representation of the NN interaction

A Lorentz covariant representation of the NN t-matrix has been obtaained over the energy range from 150 to 500 MeV by solving the integral equation that connects the t-matrix with the K-matrix. The K-matrix is expanded in a complete set of on-shell Lorentz invariant amplitudes represented phenomenologically by isoscalar and isovector “meson” exchanges. The real part of the K-matrix is fit over the energy range from 150 to 500 MeV using coupling strengths that are allowed to vary quadratically with energy. Above the pion production threshold at T_lab = 280 MeV, the real K-matrix is supplemented by an imaginary part with linear energy dependence. The K-matrix parameters are fit to thesmost recent (January 1999) Arndt amplitudes [R.A. Arndt, D. Roper, VPI and SU Scattering Analysis Interactive Dial-in Program and Data Base]. Direct and exchange contributions to the K-matrix are handled explicitly in the formalism. The resulting t-matrix satisfies elastic unitary below the pion production threshold and contains non-local terms that are not present in direct Love-Franey parameterizations of the t-matrix. Results are given for the NN amplitudes and compared with both the Arndt amplitudes and amplitudes obtained from a direct fit of the t-matrix [O.V. Maxwell, Nucl. Phys. A 600 (1996) 509]. Results are also given for a selected set of np and pp observables.     

Convex-envelope formulation for separable many-particle interactions

The exact result for the free energy per particle in systems described by a hamiltonian of the type NP(V_N), where NV_N denotes a set of short-range operators, is reformulated in terms of a convex-envelope construction. A comparison is given with results obtained for classical systems with interactions of the so-called Kac-type.     

Local realistic NN interactions in the four-nucleon problem

The ⁴He problem with realistic NN interactions is solved and systematically studied. All such interactions are found to underbind ⁴He significantly. Possible sources of this underbinding are discussed. The method of hyperspherical harmonics is used.     

Systems with separable many-particle interactions. II

The treatment of a previous paper on systems with many-particle interactions is generalized to hamiltonians containing an analytic function of a number of short-range interaction operators V, which are normalized. An exact expression for the free energy per particle in the thermodynamic limit can be obtained from a trial hamiltonian which is linear in the operators V.     

Pauli-correct separable potentials for composite particle interactions

Experimental totalβ-decay energies for short-lived isotopes of gallium and germanium are presented. The sources were produced as mass-separated fission products at the OSIRIS on-line separator at Studsvik. By applyingβ-γ-coincidence methodsQ _α-values were determined for^80,81Ga and^79,81,82Ge. For these nuclei on or close to the shellN =50 and in the vicinity of ther-process “seed”, the atomic mass excess is derived and compared with predictions from mass formulae.     

Systems with separable many-particle interactions. I

In this paper we give an exact evaluation of the free energy per particle for systems with separable many-particle interactions described by a hamiltonian of the type ? = ∑_kT(k) + NP (N^-1 ∑_kV(k)), where P is an arbitrary polynomial. In the proof use is made of a fundamental theorem due to Bogoliubov Jr. for ferromagnetic quadratic operators. The free energy can be obtained from a trial hamiltonian, which is linear in the operators T and V.     

The potential of effective field theory in NN scattering

We study an effective field theory of interacting nucleons at distances much greater than the pion's Compton wavelength. In this regime the NN potential is conjectured to be the sum of a delta function and its derivatives. The question we address is whether this sum can be consistently truncated at a given order in the derivative expansion, and systematically improved by going to higher orders. Regularizing the Lippmann-Schwinger equation using a cutoff we find that the cutoff can be taken to infinity only if the effective range is negative. A positive effective range — which occurs in nature — requires that the cutoff be kept finite and below the scale of the physics which has been integrated out, i.e. O(m_π). Comparison of cutoff schemes and dimensional regularization reveals that the physical scattering amplitude is sensitive to the choice of regulator. Moreover, we show that the presence of some regulator scale, a feature absent in dimensional regularization, is essential if the effective field theory of NN scattering is to be useful. We also show that one can define a procedure where finite cutoff dependence in the scattering amplitude is removed order by order in the effective potential. However, the characteristic momentum in the problem is given by the cutoff, and not by the external momentum. It follows that in the presence of a finite cutoff there is no small parameter in the effective potential, and consequently no systematic truncation of the derivative expansion can be made. We conclude that there is no effective field theory of NN scattering with nucleons alone.     

An algorithm for separable representations of realistic interactions

A general method for expanding in a separable form a local interaction is developed. The algorithm makes use of the Gamow vectors and momenta, calculated from an eigenvalue problem, and preserves the analytic properties of the original S-matrix. Applications are presented for realistic nucleon-nucleon interactions and phenomenological optical potentials. The agreement between the separable and exact scattering matrices is quite good on and off the energy shell, for a wide range of energies.     

On the search for a narrow penta-quark Z+-baryon in NN interactions

The possibility for an observation of a narrow penta-quark Z⁺-baryon in NN reactions is discussed. It is shown that the pp↦nΣ⁺ K ⁺ reaction at excess energies around 100 MeV above threshold provides optimal conditions for Z⁺-baryon detection by an analysis of the nK⁺ invariant mass spectrum, if the Z⁺ mass is located around 1.5 GeV involving a rather narrow width. Received: 24 May 2000 / Accepted: 3 August 2000     

Quantum interactions in a stochastic representation and two-level systems

The interaction between two quantum systems is formulated using a stochastic representation that allows one of them to be replaced by equivalent commutative random sources. The proposed method is applied to two-level systems in contact with a thermal bath. Strong-coupling effects and long-lived fluctuations of the total response of two systems in a common thermal bath are discussed.     

The effective NN interaction in the Blankenbecler-Sugar equation in a φ3 theory

The Bethe-Salpeter equation (BSE) and the Blankenbecler-Sugar equation (BBSE) are studied for scalar nucleons exchanging scalar bosons. Around the values of the coupling constant g, which allow a low-energy two-nucleon bound state, the truncation of the sum of irreducible diagrams in the BSE and the perturbative calculation of the effective interaction W in the BBSE with respect to the difference of the exact and the approximate two-nucleon propagator start to become unjustified. Generalizing a known result of the 4th order in g we show that all 6th order diagrams in W cancel in the limit of infinite nucleon mass. This opens a practical way to calculate W in the BBSE and justifies the dominance of the one-boson exchange in that model. This is supported by our numerical results for finite nucleon masses.     

Effective interactions are effective interactions

In recent years it has been realized that effective interactions—those valid over a limited range of energy-momentum—are often preferable even when more complete theories are available. We shall here present examples of such methods over a range of physical applications—from classical and quantum mechanics, to quantum gravity.     

Local potentials phase equivalent to separable two-nucleon interactions

Local energy-dependent potentials have been constructed phase equivalent to members of a family of phase-equivalent separable two-nucleon potentials in the¹ S ₀-state. It has been shown that these potentials obey the known off-shell constraints in the¹ S ₀-state and can therefore not be regarded as unrealistic in this sense. They have the same shape as the energy-independent local Kermode potentials. However we also find that the off-shell behaviour of a separable¹ S ₀ potential and its local equivalent can differ considerably.     

Effective theory of excitations in a Feshbach-resonant superfluid

A strongly interacting Fermi gas, such as that of cold atoms operative near a Feshbach resonance, is difficult to study by perturbative many-body theory to go beyond mean-field approximation. Here I develop an effective field theory for the resonant superfluid based on broken symmetry. The theory retains both fermionic quasiparticles and superfluid phonons, the interaction between them being derived nonperturbatively. The theory converges and can be improved order by order, in a manner governed by a low energy expansion rather than by a coupling constant. I apply the effective theory to calculate the specific heat and discuss the theory with a recent heat capacity experiment.     

NN scattering in higher-derivative formulation of baryon chiral perturbation theory

We consider a new approach to the nucleon-nucleon scattering problem in the framework of the higher-derivative formulation of baryon chiral perturbation theory. Starting with a Lorentz-invariant form of the effective Lagrangian we work out a new symmetry-preserving framework where the leading-order amplitude is calculated by solving renormalizable equations and corrections are taken into account perturbatively. Analogously to the KSW approach, the (leading) renormalization scale dependence to any finite order is absorbed in the redefinition of a finite number of parameters of the effective potential at given order. On the other hand, analogously to Weinberg’s power counting, the one-pion-exchange potential is of leading order and is treated non-perturbatively.