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arXiv:1209.2145 · Physics of Atomic Nuclei

Does the multiple-scattering series in the pion-deuteron scattering actually converge?

A. E. Kudryavtsev, A. I. Romanov, Vakhid A. Gani

Published Jul 1, 2013• 6 citations

Abstract

It is demonstrated that the well-known answer for the multiple-scattering series (MSS) for a light particle interacting to a pair of static nucleons, calculated in the Fixed Centers Approximation (FCA), works well for a wide region of the two-body complex scattering length a . However, this approach is not applicable in a narrow region surrounding the real positive a half-axis, where the MSS does not converge. Simultaneously, for real positive a ’s the 3-body system forms an infinite set of bound states.

DOI: 10.1134/s1063778813070090

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It is demonstrated that the well-known answer for the multiple-scattering series (MSS) for a light particle interacting to a pair of static nucleons, calculated in the Fixed Centers Approximation (FCA), works well for a wide region of the two-body complex scattering length a . However, this approach is not applicable in a narrow region surrounding the real positive a half-axis, where the MSS does not converge. Simultaneously, for real positive a ’s the 3-body system forms an infinite set of bound states.

The asymptotics of total scattering cross sections for a three-body quantum system with two heavy particles are studied when the masses tend to infinity. It is assumed that the initial channel is a two-cluster channel such that a light particle and a heavy one form a bound state and the relative velocity of another heavy particle to the pair is fixed. It is shown that the limits exist and they can be expressed in terms of transition probabilities which appear in the scattering theory for the charge transfer models.

We construct an eigenvalue problem by confining many-body system to a bounded domain with the boundary condition that the wave function vanishes. By changing the boundary, however, the eigenvalues of the energy can be varied continuously. The D-matrix is defined for a series of bounded problems with the same value for the ground state energy. The D-matrix is related to the S-matrix, enabling us to calculate the S-matrix at a given energy. The Schrodinger equation for the system is transformed to a diffusion equation by regarding time as imaginary. Initial ensemble, representing an approximate wave function, is evolved, through Monte Carlo simulation of random walks and branching, to the ground state ensemble. The limitations of investigation are: 1. Ingoing and outgoing channels have two fragments. 2. The interaction between the fragments is negligible outside the boundary mentioned above. 3. The particles are bosons or we know the zeros of the wave function. First we consider the scattering of a particle by a potential, which is equivalent to the two-body problem, in one dimension. Here we use the Poschl-Teller potential for which the exact solution is known. We use this case to investigate a new sampling method and study of various parameters. Next we consider three particles in one dimension. Here we take interaction to be a potential well, where at least one of the interactions is attractive so that a two-body bound state is possible.

It is demonstrated that the well-known answer for the multiple-scattering series (MSS) for a light particle interacting to a pair of static nucleons, calculated in the Fixed Centers Approximation (FCA), works well for a wide region of the two-body complex scattering length a.However, this approach is not applicable in a narrow region surrounding the real positive a half-axis, where the MSS does not converge.Simultaneously, for real positive a's the 3-body system forms an infinite set of bound states.

We develop an approach to scattering theory for generalized

N

-body systems. In particular we consider a general class of three quasi-particle systems, for which we prove Asymptotic Completeness.

We apply the functional renormalization group to few-nucleon systems. Our starting point is a local effective action that includes three- and four-nucleon interactions, expressed in terms of nucleon and two-nucleon boson fields. The evolution of the coupling constants in this action is described by a renormalization group flow. We derive these flow equations both in the limit of exact Wigner SU(4) symmetry and in the realistic case of broken symmetry. In the symmetric limit we find that the renormalization flow equations decouple and can be combined into two sets, one of which matches the known results for bosons, and the other result matches the one for a single flavor of spin

1/2

fermions. The equations show universal features in the unitary limit, which is obtained when the two-body scattering length tends to infinity. We calculate the spin-quartet neutron-deuteron scattering length and the deuteron-deuteron scattering lengths in the spin-singlet and spin-quintet channels.