Off-energy-shell continuation of the two-nucleon transition matrix in the presence of tensor forces

For a coupled partial wave with a bound state, the entire T-matrix is determined by the bound-state energy and wave function, the on-shell mixing parameter and a real symmetric matrix function σ(k₂,₂ k₁₁) of the continuous momentum variables k and the discrete eigenchannel labels . The on-shell part of σ is given by the eigenphases, the arbitrariness of the T-matrix is the off-shell continuation of this phase-shift matrix σ(k₂, k₁) into two dimensions. Three possible techniques are suggested to go from to the entire T-matrix. This construction procedure for T avoids the usual complications of a potential fit.     

Off-energy-shell continuation of the πN transition matrix element

General relationships of the fully off-shell T-matrix element for the central forces are derived. The problem of finding the fully off-shell T-matrix element can be reduced to finding a real two-parameter function φ(p, k). It can be shown in the case of no bound state that if φ(p, k) is separable, the whole problem becomes equivalent to the inverse scattering problem. An approximate method to include inelastic effects is proposed and the πN off-shell T-matrix element is calculated.     

Approximately linear relations between two-nucleon and three-nucleon parameters

The correlation between the triton binding energyE _t and then-d scattering length² a and the zero energy singlet and triplet wound integral I_os and I_ot is investigated, with phaseequivalent separableNN interactions obeying all the important off-shell constraints. It is found that the correlation betweenE _t and² a is stronger than betweenE _t (² a) andI _os ,I _ot separately. A strong correlation is also found between the percentageD-state in the deuteronP _d andI _ot . Fixing all the other off-shell parameters but allowing variations ofI _os andI _ot leads to surprisingly accurate linear relations betweenE _t(² a) andI _os ,I _ot (P _d). These relations depend strongly on the other off-shell parameters.     

Off-energy-shell continuation of the two-body transition matrix in the impact parameter representation

In the impact parameter representation, the symmetric part σ(k′, b, k) of the half-shell T-matrix, H, can be given arbitrarily. This in turn determines the entire T-matrix. Since σ(k, b, k) is essentially the impact parameter phase shift, the arbitrary part of the T-matrix appears to be a continuation into two dimensions of the phase shift function with symmetry being the only restriction. Since we can work with the T-matrix instead of the potential V, our formalism stays closer to experiments. Only two assumptions are necessary, namely, high energy and small intermediate angles.     

Quality of the three-nucleon bound-state wave function from a two-nucleon separable expansion method

The numerical quality of the³H wave function obtained by the separable expansion method of Ernst, Shakin, and Thaler is examined. Separable approximations to the Paris potential with increasing accuracy are used in the¹ S ₀ and³ S ₁-³ D ₁ partial waves to calculate the binding energy, wave function, wave-function component percentages, and theS- andD-wave asymptotic normalization constants of³H. The results are compared with existing five-channel calculations obtained directly (without expansion) from the Paris potential to determine convergence. It is found that the results converge rapidly to the right values, indicating that the³H wave function thus obtained is of high quality and essentially indistinguishable from that obtained directly from the Paris interaction.Dedicated to Profs. Erich Schmid and Ivo laus on the occasion of their 60th birthdays     

Off-shell effects in the three-nucleon problem

Using a previously reported solution of the inverse scattering problem for rank two separable potentials, off-shell effects on the binding energy of the model triton are investigated. Much larger effects are found than previously reported in the literature.     

Properties of the transition amplitude for two-nucleon transfer reactions

A general discussion is given of the angular momentum structure of the transition amplitude for two-nucleon transfer, particularly in light-ion-induced reactions. For example, it is shown that, within the model which is usually assumed, the relative phase between simultaneous and sequential transfer parts of the amplitude, for a given two-nucleon configuration, cannot depend upon the value of J, the total angular momentum transfer. A dependence on the oddness or evenness of J can emerge when there are additional sources of angular momentum transfer. Some of these are indicated.     

Faddeev equations and the method of hyperspherical harmonics in the problem of three-nucleon continuum

A method for solving the Faddeev equations in configuration space is developed for a three-nucleon system in the continuum by using the decomposition over a hyperspherical basis. The wave functions of Nd-system, phase shifts, and cross sections of Nd-scattering at subthreshold energies are calculated. Also, within the framework of this method, one-dimensional integral equations are formulated for the problem of infinite motion of all three strongly interacting particles, and the Faddeev equations for a system of three hadrons with Coulomb interaction in the continuum are modified. Similar methods of investigation of three-particle systems are reviewed.     

Pair numbers and two-nucleon transfer

Counting numbers of different kinds of nucleon pairs is considered within the context of the isovector-pairing SO(5) model. Connection of the pair numbers with the two-nucleon-transfer-reaction strength is discussed. Presented at Workshop on calculation of double-beta-decay matrix elements (MEDEX’97), Prague, May 27–31, 1997. This work has been supported by the Grant Agency of the Academy of Sciences of the Czech Republic under grant No. A1048504.     

The role of effective two-nucleon potential in the electric monopole transition in C12 and O16

In the present paper the Shiff's model is used to explain the O⁺-transition mechanism in nuclei. In contrary to the Shiff's study more realistic effective potentials suggested bySiemens andSprung andBanerjee are applied. They give good results.The author would like to thank Prof. I.Úlehla and Z.Pluha for encouraging interest in this work.     

Properties of simultaneous and sequential two-nucleon transfer

Approximate forms of the first- and second-order distorted-wave Born amplitudes are used to study the overall structure, particularly the selection rules, of the amplitudes for simultaneous and sequential transfer of two nucleons. The rôle of the spin-state assumed for the intermediate deuterons in sequential (t, p) reactions is stressed. The similarity of one-step and two-step amplitudes for (α, d) reactions is exhibited, and the consequent absence of any obvious J-dependence in their interference is noted.     

Adiabatic calculation of the three-nucleon force

A preliminary estimate of the three-nucleon force within the adiabatic framework has already been done. In this work we try to improve the first calculation by using relativistic kinematics for the pion and including theP 11 absorption channel, as well as theP 33 and the less importantS 13,S 11,P 13,P 31 channels, in theN interaction.Presented at the symposium Mesons and Light Nuclei, Bechyn, Czechoslovakia. May 27–June 1, 1985.     

Pion-mass dependence of three-nucleon observables

We use an effective field theory (EFT) which contains only short-range interactions to study the dependence of a variety of three-nucleon observables on the pion mass. The pion-mass dependence of input quantities in our “pionless” EFT is obtained from a recent chiral EFT calculation. To the order we work at, these quantities are the ¹ S ₀ scattering length and effective range, the deuteron binding energy, the ³ S ₁ effective range, and the binding energy of one three-nucleon bound state. The chiral EFT input we use has the inverse ³ S ₁ and ¹ S ₀ scattering lengths vanishing at m _π ^crit = 197.8577 MeV. At this “critical” pion mass, the triton has infinitely many excited states with an accumulation point at the three-nucleon threshold. We compute the binding energies of these states up to next-to-next-to-leading order in the pionless EFT and study the convergence pattern of the EFT in the vicinity of the critical pion mass. Furthermore, we use the pionless EFT to predict how doublet and quartet nd scattering lengths depend on m _π in the region between the physical pion mass and m _π = m _π ^crit .     

Three-nucleon forces and the three-nucleon systems

The basic principles and philosophy which have guided the area of few-nucleon physics are motivated and discussed. Recent advances have made it possible to solve accurately the Schrödinger (or Faddeev) equation for many of the configurations of the few-nucleon systems, A brief review is given of some of the results of these calculations, which also Indicate that one-pion exchange plays roughly the same role in binding these systems that it plays in electromagnetic meson-exchange currents. Finally, a qualitative discussion of chiral perturbation theory is presented, which highlights the role that chiral symmetry plays in the nuclear force. Three-body forces from different areas of physics are discussed, as well as the significance of three-nucleon forces in few-nucleon systems.Lectures held at the Indian Summer School on Electromagnetic and Weak Interactions of Particles with Nuclei, Sázava, Czechoslovakia, 6–11 September 1992The work of J. L. F. was performed under the auspices of the U. S. Department of Energy. The support of the Alexander von Humboldt-Stiftung is gratefully acknowledged. The author gratefully acknowledges discussions with B. Holstein, whose superb lectures on this topic introduced him to chiral perturbation theory.