Singular Integral Equations. By N.I. Muskhelishvili. Translated fromthe 2nd Russian edition by J.R.M. Radok. Pp. 447. F1. 28.50. 1953. (Noordhoff, Groningen)

Nuclear forces can be systematically derived using effective chiral Lagrangians consistent with the symmetries of QCD. I review the status of the calculations for two- and three-nucleon forces and their applications in few-nucleon systems. I also address issues like the quark mass dependence of the nuclear forces and resonance saturation for four-nucleon operators.

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Modern theory of nuclear forces

Neutron Stars 1: Equation of State and Structure

Nowadays it has become customary in nuclear physics to denote by “tradition” the approach that considers nucleons and mesons as the relevant degrees of freedom. It is the purpose of this chapter to review this “traditional” approach in the area of nuclear forces and their applications to nuclear structure.

The Meson Theory of Nuclear Forces and Nuclear Structure

https://arxiv.org/pdf/cond-mat/0410417.pdf

Universality in few-body systems with large scattering length

The two pion exchange three-nucleon potential and nuclear matter

We introduce new values of the strength constants (i.e., a, b, c, and d coefficients) of the Tucson-Melbourne (TM) 2π-exchange three-nucleon potential. The new values come from contemporary dispersion-relation analyses of meson-factory πN-scattering data. We make variational Monte-Carlo calculations of the triton with the original and updated three-body forces to study the effects of this update. We remove a short-range–π-range part of the potential due to the c coefficient and discuss the effect on the triton binding energy.

Reworking the Tucson-Melbourne Three-Nucleon Potential

We present the complete momentum space three-nucleon potential of the two-pion-exchange type in the partial wave decomposition needed for the Faddeev equations of the three-nucleon bound state. The potential arises from an off-mass-shell model for $\ensuremath{\pi}N$ scattering based upon current algebra and a dispersion-theoretical axial vector amplitude dominated by the $\ensuremath{\Delta}(1230)$ isobar. The potential is manifestly Hermitian and defined for all three nucleon momenta. We display some matrix elements of the potential in the five three-body partial waves corresponding to the $^{1}S_{0}$ and $^{3}S_{1}\ensuremath{-}^{3}D_{1}$ states of the two-body subsystem. These matrix elements show a striking contrast to those of an older three-body potential mediated only by the $\ensuremath{\Delta}(1230)$ $p$-wave resonance.NUCLEAR STRUCTURE Three-body potential; few-nucleon system, Faddeev approach, partial wave decomposition in Jacobi variables.

Two-pion-exchange three-nucleon potential: Partial wave analysis in momentum space

We study recently proposed ultraviolet and infrared momentum regulators of the model spaces formed by construction of a variational trial wave function which uses a complete set of many-body basis states based upon three-dimensional harmonic oscillator (HO) functions. These model spaces are defined by a truncation of the expansion characterized by a counting number ($\mathcal{N}$) and by the intrinsic scale ($\ensuremath{\hbar}\ensuremath{\omega}$) of the HO basis---in short by the ordered pair ($\mathcal{N},\ensuremath{\hbar}\ensuremath{\omega}$). In this study we choose for $\mathcal{N}$ the truncation parameter ${N}_{\mathrm{max}}$ related to the maximum number of oscillator quanta, above the minimum configuration, kept in the model space. The uv momentum cutoff of the continuum is readily mapped onto a defined uv cutoff in this finite model space, but there are two proposed definitions of the ir momentum cutoff inherent in a finite-dimensional HO basis. One definition is based upon the lowest momentum difference given by $\ensuremath{\hbar}\ensuremath{\omega}$ itself and the other upon the infrared momentum which corresponds to the maximal radial extent used to encompass the many-body system in coordinate space. Extending both the uv cutoff to infinity and the ir cutoff to zero is prescribed for a converged calculation. We calculate the ground-state energy of light nuclei with ``bare'' and ``soft'' nucleon-nucleon ($NN$) interactions. By doing so, we investigate the behaviors of the uv and ir regulators of model spaces used to describe ${}^{2}$H, ${}^{3}$H, ${}^{4}$He, and ${}^{6}$He with $NN$ potentials Idaho N${}^{3}$LO and JISP16. We establish practical procedures which utilize these regulators to obtain the extrapolated result from sequences of calculations with model spaces characterized by ($\mathcal{N},\ensuremath{\hbar}\ensuremath{\omega}$).

Convergence properties of ab initio calculations of light nuclei in a harmonic oscillator basis

We study the radial Schr\"odinger equation for a particle of mass m in the field of a singular attractive $\ensuremath{\alpha}{/r}^{2}$ potential with $2m\ensuremath{\alpha}g1/4.$ This potential is relevant to the fabrication of nanoscale atom optical devices, is said to be the potential describing the dipole-bound anions of polar molecules, and is the effective potential underlying the universal behavior of three-body systems in nuclear physics and atomic physics, including aspects of Bose-Einstein condensates, first described by Efimov. New results in three-body physical systems motivate the present investigation. Using the regularization method of Beane et al., we show that the corresponding ``renormalization-group flow'' equation can be solved analytically. We find that it exhibits a limit cycle behavior and has infinitely many branches. We show that a physical meaning for self-adjoint extensions of the Hamiltonian arises naturally in this framework.

https://arxiv.org/pdf/quant-ph/0302199.pdf

Sidney A. Coon

Papers

The two pion exchange three-nucleon potential and nuclear matter

Reworking the Tucson-Melbourne Three-Nucleon Potential

Two-pion-exchange three-nucleon potential: Partial wave analysis in momentum space

Convergence properties of ab initio calculations of light nuclei in a harmonic oscillator basis

Singular inverse square potential, limit cycles, and self-adjoint extensions