Quark-exchange Kernel Research Articles

G-Matrix Equation in the Quark-Model Resonating-Group Method for Baryon-Baryon Interaction

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.

NN tensor and LS potentials in a quark model with quark-antiquark excitations

A more detailed study is made of the NN interaction in a quark model in which the (qq̄) excitations inherent in the quark-gluon interaction are explicitly incorporated into the model space. A unified treatment for all types of exchange terms of the (3q)-(3q) to (3q)-(3q)(qq̄) coupling kernels is used to calculate the space parts of the full coupling kernels and their Wigner transforms in complete analytic form. The present investigation focuses on the noncentral parts of the NN interaction. The tensor force gains almost its full strength from coupling kernels of Nπ and Nρ type. If the Nπ contribution is adjusted to fit the experimental pion-nucleon coupling constant the predicted strength of the full tensor force is in reasonable agreement with that of conventional OBEP's over the range in which the tensor force can act. The LS force gains contributions from both the pure (3q)-(3q) and the coupling kernels, but the dominant contributions (about 60–65% of the triplet-odd LS potential in the 0.7–1.0 fm range), come from the coupling kernels and particularly from the Nω and Nρ components. The triplet-odd LS potential derived from the full quark-exchange kernel is in remarkably good agreement with the OBEP LS potential over the significant range. Both the tensor and LS potentials are approximated surprisingly well over their full range by the simple (qq̄) exchange terms of our model. The 3P RGM phase shifts are calculated to show that both tensor and LS forces of our quark model are in good agreement with the experimental facts.

Meson exchange potentials in an NN quark model with quark-antiquark excitations

In a recent quark-model study of the NN interaction, we explicitly incorporated the qq̄ excitations inherent in the quark-gluon interaction lagrangian into the model space, and examined the contributions to the NN potential of particular (3 q)( q q ) components of the single-nucleon wave function, with specific baryon and meson quantum numbers, through the Wigner transforms of the resultant coupling terms of the quark exchange kernels. The present investigation focuses on the long-range parts of the potentials. It is shown that these arise almost entirely through two of the 25 possible types of exchange terms, and that these two correspond to the simple exchange of a q q pair between the two nucleons and contribute only through coupling terms for which the (3q) components have the quantum numbers of a nucleon and the ( q q ) component the color-singlet character of a real pseudoscalar or vector meson. The potentials arising from these simple color-singlet q q exchanges have been evaluated in explicit analytic form. In their dependence on nucleon ( σ 1, · σ 2) and ( τ 1 · τ 2) factors and the relative importance and signs of spin-spin, spin-independent central, LS, and tensor terms they have all the characteristics of conventional OBEP's. They are in remarkably good agreement with conventional OBEP's in the r ⩾ 1.2 fm range and have the same qualitative radial features over an even wider range, the one exception being the steep short-range repulsive rise of the spin-independent central term of the ω-exchange potential. The corresponding q q exchange potential instead changes sign to become attractive at r < 0.8 fm.

The NN interaction in a quark model with quark-antiquark excitations

Quark-antiquark excitations inherent in the quark-gluon interaction have been incorporated into a quark model of the nucleon to study the effects of such excitations on the NN interaction within the framework of the resonating group method. The three-quark (3q) components of a single nucleon are augmented by ( 3 q)( q q ) excitations with the 24 possible spin, isospin, color combinations for the energetically lowest p-wave relative motion function. Quark exchange kernels are then calculated for the two-nucleon system described by these improved nucleon internal functions; and these exchange kernels are converted to phase-shift-equivalent effective NN potentials by the Wigner-transform WKB technique. The off-shell qq̄ pair-creation terms are derived from the one-gluon exchange diagrams in the Breit approximation in analogy with the derivation of the qq and qq̄ potentials. The parameters of the interaction are chosen to be consistent with the experimental Δ-N mass difference, the nucleon-vector-meson coupling constants, and the nucleon magnetic moments. Within these constraints, the predicted amplitudes of the ( 3 q)( q q ) components of the nucleon internal functions have been shown to be insensitive to the precise values of the model parameters. In particular, they pass the crucial test of being insensitive to very large changes in the magnitude of the confinement potential constant which is a necessary ingredient of the model. The qq̄ excitations lead to the following effects in the S-wave NN potentials: (i) The repulsive core heights of the simple 3q-3q model are greatly reduced but retain their strong essentially linear energy dependence, with numerical values very similar to those of the short-range phenomenological terms of the Paris potential. (ii) The effective potentials have acquired an attractive part in the 0.8-1.5 fm range. However, this attraction is too weak to bind the deuteron or fit the extreme low-energy S-wave phase shifts.

Short-range part of the NN interaction: Lowest-order relativistic corrections to equivalent local potentials from quark exchange kernels

The full Breit hamiltonian, including the Darwin term and relativistic kinetic-energy correction, has been used to construct resonating group kernels in analytic form for the six-quark NN system. The nonlocal kernels have been converted to equivalent local potentials using the WKB approximation based on their Wigner transforms. The equivalent local potentials have been confirmed to be rather insensitive to a choice of parameters and to be fairly accurate by comparing with the phase shifts and the wave functions obtained with the RGM equation. The results indicate the importance of the Darwin term. The term leads to a large reduction of the strong repulsive cores given by the color magnetic interaction and makes the resulting potentials less repulsive than those which neglect this term. The relative importance of the various components in the Breit interaction and of the quark-gluon exchange diagrams is made clear.

Baryon-baryon spin-orbit and tensor interactions from quark-exchange kernels

Dibaryon quark-exchange kernels are constructed in explicit analytic form for the tensor and spin-orbit terms of the one-gluon-exchange quark-quark Breit interaction and for spin-orbit terms generated by quark-confinement mechanisms. The spin operators needed are defined through their spin-reduced matrix elements including those needed for interactions coupling NN, NΔ, and ΔΔ channels. Effective baryon-baryon spin-orbit potentials, generated through the Wigner transforms of the quark-exchange kernels with the use of a local momentum approximation, show that the NN spin-orbit interaction derived from the symmetric spin-orbit term of the one-gluon-exchange quark-quark interaction is in general agreement with the short-range part of phenomenological potentials derived from NN scattering. With the inclusion of the antisymmetric spin-orbit one-gluon-exchange terms and spin-orbit terms generated by confining potentials the full triplet-odd NN spin-orbit potential is greatly reduced in the 0.5–1 fm range. The uncertainties associated with spin-orbit terms generated by quark-confinement mechanisms are emphasized. The relative importance of various possible quark-gluon exchange terms is studied and shows that models which neglect some types of exchange terms are open to question. An SU(3)-flavor symmetric model for N-hyperon spin-orbit potentials leads to an NΛ spin-orbit potential only slightly weaker than the NN spin-orbit potential.

Short-range part of the NN interaction: Effects of changes in six-quark spatial symmetry

A study of nonrelativistic resonating group method quark exchange kernels shows that even large changes in the relative (6) and (42) space symmetry of the six-quark system do not make significant changes in the extreme short range parts of the NN interaction. The repulsive core heights of equivalent local potentials derived by the Wentzel-Kramers-Brillouin approximation from the Wigner transform of the quark exchange kernels are changed by only approx.20% under a large change in the relative (6) and (42) space symmetry of the antisymmetrized internal six-quark resonating group method wave function.

Short range part of theNNinteraction: Equivalent local potentials from quark exchange kernels

To focus on the nature of the short range part of the $\mathrm{NN}$ interaction, the intrinsically nonlocal interaction among the quark constituents of colorless nucleons is converted to an equivalent local potential using resonating group kernels which can be evaluated in analytic form. The WKB approximation based on the Wigner transform of the nonlocal kernels has been used to construct the equivalent potentials without recourse to the long range part of the $\mathrm{NN}$ interaction. The relative importance of the various components of the exchange kernels can be examined: The results indicate the importance of the color magnetic part of the exchange kernel for the repulsive part in the $(\mathrm{ST})=(10), (01)$ channels, in particular since the energy dependence of the effective local potentials seems to be set by this term. Large cancellations of color Coulombic and quark confining contributions, together with the kinetic energy and norm exchange terms, indicate that the exact nature of the equivalent local potential may be sensitive to the details of the parametrization of the underlying quark-quark interaction. The equivalent local potentials show some of the characteristics of the phenomenological short range terms of the Paris potential.NUCLEAR REACTIONS Short range part of $\mathrm{NN}$ interaction, six-quark model, RGM quark exchange kernels, equivalent local potentials via WKB and Wigner transforms.