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Register a new userG-Matrix Equation in the Resonating-Group Method
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The G-matrix equation is most straightforwardly formulated in the resonating-group method if the quark-exchange kernel is directly used as the driving term for the infinite sum of all the ladder diagrams. The inherent energy-dependence involved in the exchange term of the normalization kernel plays the essential role to define the off-shell T-matrix uniquely when the complete Pauli-forbidden state exists. We analyze this using a simple solvable model with no quark-quark interaction, and calculating the most general T-matrix in the formulation developed by Noyes and Kowalski. This formulation gives a certain condition for the existence of the solution in the Lippmann-Schwinger resonating-group method. A new procedure to deal with the corrections for the reduced masses and the internal-energy terms in the Lambda N - Sigma N coupled-channel resonating-group equation is proposed.
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We calculate n alpha phase-shifts and scattering observables in the resonating-group method, using the nuclear-matter G-matrix of an SU_6 quark-model NN interaction. The G-matrix is generated in the recent energy-independent procedure of the quark-model NN interaction with the continuous prescription for intermediate spectra, by assuming an appropriate Fermi momentum k_F=1.2 fm^-1. The n alpha RGM interaction kernels are evaluated with explicit treatments of the nonlocality and momentum dependence of partial-wave G-matrix components. The momentum dependence of the G-matrix components is different for each of the nucleon-exchange and interaction types. Without introducing any artificial parameters except for k_F, the central and spin-orbit components of the n alpha Born kernel are found to have reasonable strengths under the assumption of a rigid translationally invariant shell-model wave function of the alpha-cluster. The characteristic behaviors of three different exchange terms, corresponding to knockout, heavy-particle pickup and nucleon-rearrangement processes, are essentially the same between the case of previous local effective NN forces and the case of nonlocal G-matrix NN interactions.
A two-cluster microscopic model is applied to study elastic alpha-alpha scattering and resonance structure of $^{8}$Be. The model is an algebraic version of the Resonating Group Method, which makes use complete set of oscillator functions to expand wave function of two-cluster system. Interaction between clusters is determined by well-known semi-realistic nucleon-nucleon potentials of Hasegawa-Nagata, Minnesota and Volkov. Detail analysis of resonance wave functions is carried out in oscillator, coordinate and momentum spaces. Effects of the Pauli principle on wave functions of the $^{8}$Be continuous spectrum states are thoroughly studied.
We carry out Faddeev calculations of three-alpha (3 alpha) and two-alpha plus Lambda (alpha alpha Lambda) systems, using two-cluster resonating-group method kernels. The input includes an effective two-nucleon force for the alpha alpha resonating-group method and a new effective Lambda N force for the Lambda alpha interaction. The latter force is a simple two-range Gaussian potential for each spin-singlet and triplet state, generated from the phase-shift behavior of the quark-model hyperon-nucleon interaction, fss2, by using an inversion method based on supersymmetric quantum mechanics. Owing to the exact treatment of the Pauli-forbidden states between the clusters, the present three-cluster Faddeev formalism can describe the mutually related, alpha alpha, 3 alpha and alpha alpha Lambda systems, in terms of a unique set of the baryon-baryon interactions. For the three-range Minnesota force which describes the alpha alpha phase shifts quite accurately, the ground-state and excitation energies of 9Be Lambda are reproduced within 100 - 200 keV accuracy.
The Effective Field Theory without pions at next-to-leading order is used to analyze universal bound state and scattering properties of the 3- and 4-nucleon system. Results of a variety of phase shift equivalent nuclear potentials are presented for bound state properties of 3H and 4He, and for the singlet S-wave 3He-neutron scattering length a_0(3He-n). The calculations are performed with the Refined Resonating Group Method and include a full treatment of the Coulomb interaction and the leading-order 3-nucleon interaction. The results compare favorably with data and values from AV18(+UIX) model calculations. A new correlation between a_0(3He-n) and the 3H binding energy is found. Furthermore, we confirm at next-to-leading order the correlations, already found at leading-order, between the 3H binding energy and the 3H charge radius, and the Tjon line. With the 3H binding energy as input, we get predictions of the Effective Field Theory without pions at next-to-leading order for the root mean square charge radius of 3H of (1.6pm 0.2) fm, for the 4He binding energy of (28pm 2.5) MeV, and for Re(a_0(3He-n)) of (7.5pm 0.6)fm. Including the Coulomb interaction, the splitting in binding energy between 3H and 3He is found to be (0.66pm 0.03) MeV. The discrepancy to data of (0.10mp 0.03) MeV is model independently attributed to higher order charge independence breaking interactions. We also demonstrate that different results for the same observable stem from higher order effects, and carefully assess that numerical uncertainties are negligible. Our results demonstrate the convergence and usefulness of the pion-less theory at next-to-leading order in the 4He channel. We conclude that no 4-nucleon interaction is needed to renormalize the theory at next-to-leading order in the 4-nucleon sector.
A systematic connection between QCD and nuclear few- and many-body properties in the form of the Effective Field Theory without pions is applied to $Ale 6$ nuclei to determine its range of applicability. We present results at next-to-leading order for the Tjon correlation and for a correlation between the singlet S-wave $^3$He-neutron scattering length and the triton binding energy. In the A=6 sector we performed leading order calculations for the binding energy and the charge and matter radii of the halo nucleus $^6$He. Also at leading order, the doublet S-wave 4-He-neutron phase shifts are compared with R-matrix data. These analysis provide evidence for a sufficiently fast convergence of the effective field theory, in particular, our results in $Ale 4$ predict an expansion parameter of about 1/3, and they converge to data within the predicted uncertainty band at this order. A properly adjusted three-body contact force which we include together with the Coulomb interaction in all calculations is found to correctly renormalize the pion-less theory at leading- and next-to-leading order, i.e. the power counting does not require four-body forces at the respective order.
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