Reconstructing the Nucleon-Nucleon Potential by a New Coupled-Channel Inversion Method

Abstract

A second-order supersymmetric transformation is presented, for the two-channel Schrödinger equation with equal thresholds. It adds a Breit-Wigner term to the mixing parameter, without modifying the eigenphase shifts, and modifies the potential matrix analytically. The iteration of a few such transformations allows a precise fit of realistic mixing parameters in terms of a Padé expansion of both the scattering matrix and the effective-range function. The method is applied to build an exactly-solvable potential for the neutron-proton 3 S1-3 D1 case.PACS numbers: 03.65. Nk, 03.65.Ge, 13.75.Cs In quantum scattering theory, the fundamental inverse problem consists in deducing the interaction potential between two colliding particles from their experimental elastic-scattering cross sections [1]. These cross sections have first to be parametrized in terms of energy-dependent partial-wave phase shifts or scattering matrices. For a central interaction V (r), the partial waves decouple and a sequence of single-channel inverse problems have to be solved. For more complicated interactions, like the tensor interaction in nuclear physics, partial waves may be coupled and matrix potentials have to be constructed from coupled-channel scattering matrices.A formal solution to these inverse problems can be written in terms of integral equations [1]. In practical applications, experimental data turn out to be precisely parametrizable in terms of separable kernels for these equations. These correspond to scattering matrices that are rational functions of the wave number. The integral equations can then be solved analytically and the corresponding potentials are also expressed in a separable form. This procedure applies to both the singleand coupled-channel cases [2,3]. In the single-channel case, the same potentials can be constructed with the help of supersymmetric quantum mechanics (SUSYQM) [4,5]. This method is much simpler than the integral-equation one, as the potential is directly related to its scattering-matrix poles. Moreover, a small number of poles is in general sufficient to fit experimental data on the whole elastic-scattering energy range; this is in particular the case for the neutron-proton singlet (spin 0) channels [6,7], that decouple because of the vanishing tensor interaction.The present Letter aims at extending this very efficient single-channel method to the two-channel case without threshold difference, and at applying it to the neutron-proton triplet (spin 1) coupled channels. This system was studied in the framework of the integral-equation method by Newton and Fulton [2], at low energy, and by Kohlhoff and von Geramb [3], on the whole elastic-scattering range. The present paper subsumes these works, by separating the effect of the coupling between channels in the inversion procedure, by parametrizing the data with a minimal number of poles, and by deriving simple expressions for the corresponding matrix potential.We consider two channels with equal thresholds and angular momenta l and l + 2. The scatter...

Supersymmetric inversion of effective-range expansions

“…The poles are then deduced from the above equations. The advantage is that they automatically satisfy conditions (7). The drawback of this approach is that these poles can be either imaginary or complex, while complex poles sometimes lead to oscillations in the potentials deduced from supersymmetric quantum mechanics.…”

“…Typically, for the nucleon-nucleon S wave, this expansion is usually believed to be useful to fit experimental phase shifts up to 5 MeV only [3]. It was however realized by several authors [4][5][6][7] that a Padé approximant or rational expansion of the effective-range function was much more appropriate than a Taylor expansion to parametrize data efficiently. For instance, as shown below, for the nucleonnucleon S wave, a Padé expansion of order [3/2], which * bikash.midya@gmail.com † jmspar@ulb.ac.be depends on six parameters, fits the effective-range function on the whole elastic-scattering region.…”

Single- and coupled-channel radial inverse scattering with supersymmetric transformations

“…Iterations are necessary in realistic inversions. This generalization for pairs of complex conjugate transformations was employed in [83]. In the two-channel case, according to (2.41), a physical potential has the diagonal long-range behaviour (2.43),…”

“…Such pairs have a strong practical interest and were applied to the neutron-proton system in [83]. They were recently generalized to an arbitrary number of channels [90], where they were shown to be applicable to an even number of channels only.…”

“…Eigenphase-preserving transformations Equation (6.70) is general but not easy to use. An important particular case does not share that drawback [42,83] when U ∞ is proportional to an orthogonal matrix. This is realized when the diagonal elements of the 2×2 matrix are real and equal and the off-diagonal elements are real and opposite, hence if and only if the coefficients of σ x and σ z vanish, i.e.…”