Session L13: Nuclear Theory I: Few-Body

Scattering using Euclidean Green functions

We show that it is possible to compute differential cross sections using matrix elements of polynomials in $e^{-\beta H}$ in normalizable states. These matrix elements can be calculated by quadrature using reflection-positive Euclidean Green functions. The the proposed method is based on an explicit ``time-dependent'' computation of the M{\o}ller wave operators using the Kato-Birman invariance principle to replace $H$ by $-e^{-\beta H}$ in the expression for the wave operators. The compact spectrum of $-e^{-\beta H}$ allows uniform polynomial approximations of continuous functions of $-e^{-\beta H}$. We tested the method using a solvable model with the range and strength of a typical nucleon-nucleon interaction and found convergence to the transition matrix elements for energies up to 1.5 GeV.   

Chiral potentials, perturbation theory, and the $^{1}$S$_{0}$ channel of NN scattering

Nucleon-nucleon phase shifts obtained from experimental data and the chiral expansion for the long-distance part of the NN interaction are used to obtain information about the short-distance piece of the NN potential that is at work in the $^{1}$S$_{0}$ channel. The energy dependence produced by short-distance dynamics is well approximated by a two-term polynomial for T$_{ lab}\le $ 200 MeV if the scale of separation between ``long-'' and ``short-'' distance physics is chosen to be less than 1.8 fm. A quantitative description of NN dynamics is possible, at least in this channel, by treating the long-distance parts of the chiral NN potential in perturbation theory.   

Study of alpha particle scattering by nuclei using the Klein-Gordon equation.

The calculations of scattering cross sections of alpha particles on target nuclei using the Klein-Gordon equation will be presented. Analytical problems for including Coulomb potential in the treatment will be discussed and has been resolved in the new code. Calculations done by this code are compared to those performed in using a relativistic version of the Schrodinger equation for the scattering of 1.37 GeV alpha particles by calcium. Differences in the calculations require the use of a new potential in the Klein-Gordon equation. Since data points exist only in the forward angles, more experiments in the larger angles are required to realize a better understanding of the nuclear potential.   

Pionless effective field theory and the 4-nucleon scattering system

The effective field theory without pions at next-to-leading-order is used to analyse universal bound-state and scattering properties of the 4-nucleon system. Results of five phase shift equivalent nucleon-nucleon potentials for the singlet S-wave $^3$He neutron scattering length, $a_0(^3\textrm{He-n})$, the phase shifts of the $^4$He system, and bound state properties for $^3$H, $^3$He, and $^4$He, are presented. The calculations are performed within the refined resonating group model and include a full treatment of the coulomb interaction. All results are compared with experimental data and values from AV18/UIX model calculations. A correlation between $a_0(^3\textrm{He-n})$ and the $^4$He binding energy is found. Furthermore, we confirm the linear correlations, already investigated at leading-order, between the $^3$H binding energy and the $^3$H charge radius, and the Tjon line. Our results demonstrate the usefulness of the pionless theory at next-to-leading-order in the $^4$He system, and confirm that no 4-nucleon force is needed to renormalize the theory at this order.   

Separable Expansions of $V_{low}$ for 2- and 3-Nucleon Systems

We present an alternative organizational scheme for developing effective theories of 2- and 3-body systems that is systematic, accurate, and efficient with controlled errors. To illustrate our approach we consider the bound state and scattering properties of the 3 S1 and 4 S3/2 2- and 3-nucleon systems. Our approach combines the computational benefits of using separable potentials with the improved convergence properties of potentials evolved with a renormalization group procedure. Long ago Harms showed that any potential can be expanded in a series of separable terms, but this fact is only useful if the expansion can be truncated at low order. The separable expansion provides an attractive organizational scheme that incorporates finite range effects at the outset in contrast to the familiar effective range theory starting with contact interactions. We show that when applied to a renormalization group-evolved potential, the separable expansion converges rapidly, with accurate results for both 2- and 3-body scattering processes using only two separable terms.   

Charge radius of $^{6}$He and Halo nuclei in the Gamow Shell Model

We calculated the $^{6}$He charge radius in the framework of the Gamow Shell Model (GSM). The charge radius reflects both the size of the halo, due to the motion of the $\alpha $-core around the nuclear center of mass, and also provides us with information on how the several subsystems interact with each other. The motivation for this work was given by the recent very precise measurements of $^{6,8}$He, $^{11}$Li and $^{11}$Be charge radii. The proper treatment of the continuum turns out to be very important for their correct description, and the GSM is a very advanced theory in this direction. For first time in GSM calculations we used a Hamiltonian that is free from spurious center of mass motion, by adopting an intrinsic set of coordinates. Our calculations show that, if we aim to calculate the $^{6}$He charge radius right, the total two-body wavefunction should contain an $\sim $ 91{\%} of a p3/2 partial wave. We are convinced that for $^{6}$He, the charge radius is very sensitive to i) the halo extent, namely the binding of the system and ii) the p3/2 occupation. This observation will help us to constraint our Hamiltonian and construct the effective interaction in the p and p-sd shells, which will facilitate the description of weakly bound systems.   

Checker Board Model

The Checker Board Model (CBM) is a 2D model of the nucleus that proposes that the synchronization of two outer rotating quarks in the nucleons accounts for magnetic moment of the nucleons and that the resulting magnetic flux couples (weaves) into the 2D checker board array structures and this 2D magnetic coupling in addition to electrostatic forces of the two rotating and one stationary quark accounts for the apparent strong nuclear force. The symmetry of the He nucleus helps explain why this 2D structure is stable. This model explain the mass of the proton and neutron, along with their magnetic moments and their absolute and relative sizes and predict the masses of two newly proposed quarks $^{(1)}$: the ``up'' and the ``dn'' quarks. Since the masses of the ``up'' and ``dn'' quark determined by the CBM (237.31 MeV and 42.392 MeV respectively) did not fit within the standard model as candidates for u and d, a new model (New Physics) had to be invented. The details of this new nuclear physics model can be found at: http://checkerboard.dnsalias.net/ (1). T.M. Lach, Checkerboard Structure of the Nucleus, Infinite Energy, Vol. 5, issue 30, (2000). (2). T.M. Lach, Masses of the Sub-Nuclear Particles, nucl-th/0008026, @http://xxx.lanl.gov/