nd Scattering Observables Predicted by the Quark-Model Baryon-Baryon Interaction

Abstract. We solve thend scattering in the Faddeev formalism, employing the NN sector of the quarkmodel baryon-baryon interaction fss2. The energy-dependence of the NN interaction, inherent to the (3 q)-(3q) resonating-group formulation, is eliminated by the standard o ff-shell transformation utilizing the 1 / √ N factor, whereN is the normalization kernel for the (3 q)-(3q) system. This procedure yields an extra nonlocality, whose effect is very important to reproduce all the scattering observables below En ≤ 65 MeV. The di fferent off-shell properties from the standard meson-exchange potentials, related to the non-locality of the quark-model baryonbaryon interaction, yields appreciable e ffects to the di fferential cross sections and polarization observables of the nd elastic scattering, which are usually attributed to the specific properties of three-body forces.


Introduction
The QCD-inspired spin-flavor S U 6 quark model (QM) for the baryon-baryon interaction, developed by the Kyoto-Niigata group, has achieved accurate description of available nucleon-nucleon (NN) and hyperon-nucleon experimental data [1].These QM baryon-baryon interactions are characterized by the nonlocality and the energy dependence inherent to the framework of the resonatinggroup method (RGM) for two three-quark systems.In the strangeness sector, the Pauli forbidden state sometimes exists as the result of the exact antisymmetrization of six quarks.The short-range repulsion of the baryon-baryon interaction is mainly described by the quark-exchange kernel, which gives quite different off-shell properties from the standard meson-exchange potentials.The energy dependence of the interaction is eliminated [2] by the standard off-shell transformation, utilizing the 1/ √ N factor for the interaction Hamiltonian and the renormalized relative wave function between two clusters.This procedure yields an extra nonlocality, whose effect was examined in detail for the three-nucleon bound state and for the hypertriton [3].The advantage of the larger triton binding energy by the QM NN interaction; namely, the deficiency of 350 keV, predicted by the model fss2, is still much smaller than the standard values 0.5 MeV -1 MeV, given by the modern meson-exchange NN potentials.It is therefore interesting to examine the QM predictions to the three-nucleon scattering observables, especially in this renormalized framework without the explicit energy dependence of the RGM kernel.
In this contribution, we apply our QM NN interaction fss2 to the neutron-deuteron (nd) scattering in the Faddeev formalism for composite systems [4].The Alt-Glassbergera e-mail: yfujiwar@scphys.Kyoto-u.ac.jpSandhas (AGS) equations are solved in the momentum representation, using the off-shell RGM T -matrix obtained from the energy-independent renormalized RGM kernel [2].The Gaussian nonlocal potentials constructed from the fss2 is essentially used in the isospin basis.The singularity of the NN T -matrix from the deuteron pole is handled in the Noyes-Kowalski method.Another notorious moving singularity of the free three-body Green function is treated by the the standard spline interpolation technique developed by the Böchum group [5].We use the channel-spin formalism which is convenient to discuss the nd scattering.The NN interaction up to I max = 4 is included in the present calculations up to about E n = 65 MeV.
In order to confirm the importance of the present offshell transformation, we have also calculated the differential cross sections and various types of analyzing powers, the polarization transfer, and spin correlation parameters, using a simple prescription of assuming a constant energy of two-nucleon subsystem; namely, the deuteron energy, ε = −ε d = −2.223MeV.In this prescription, we come across some disagreements with experimental data.Only when the off-shell transformation is properly taken into account, we can achieve an overall agreement with experiment.We therefore show in this paper only the results obtained with the renormalized RGM kernel of the QM BB interaction.

NN phase shifts predicted by fss2
We show in Fig. 1 the NN phase shifts predicted by the model fss12.More detailed information on the QM BB interaction is obtained from the QMPACK homepage [6].
The energy-independent renormalized RGM kernel V RGM for a two-cluster system reads [2] EPJ Web of Conferences where V D is the direct potential, and G is the sum of the exchange kinetic-energy and interaction kernels.The nonlocal kernel W appears through the elimination of the energy-dependence, and is given by Here K is the exchange normalization kernel, h denotes h 0 + V D + G with h 0 being the kinetic energy for the twocluster relative motion, and Λ = 1 − |u u| is a two-cluster Pauli projection operator, where |u is a Pauli-forbidden state satisfying K|u = |u .In the NN sector, there appears no Pauli forbidden state at the quark level, so that we can simply set Λ = 1 in the following formulations.An advantage of using the V RGM is that the two-cluster RGM equation takes the form of the usual Schrödinger equation in the Pauli-allowed model space, and the relative wave function is properly normalized.This Schrödingertype equation for the relative wave function gives the same asymptotic behavior as the original RGM equation, thus preserving the phase shifts and physical observables for the two-cluster system.The difference between the previous energy-dependent RGM kernel, , and V RGM in Eq. ( 1) is essentially a replacement of Λ(εK)Λ with W.Here ε is the two-cluster relative energy measured from its threshold.The value is however not defined in the three-cluster system, in particular, for the scattering systems.In the following, we will consistently use the energy-independent renormalized RGM kernel V RGM in Eq. ( 1), both for the bound-state solution and the scattering problems.

Three-nucleon bound state
The three-cluster equation for the 3N bound state reads where α, β, and γ denote three independent pairs of twocluster subsystems, H 0 is the free three-body kinetic-energy operator, and V RGM α stands for the RGM kernel in Eq. ( 1) for the α-pair, etc.The correlation between the triton binding energy, B t = −E( 3 H), and the D-state probability of the deuteron is plotted in Fig. 2 for fss2 and FSS, as well as many other Faddeev calculations using modern realistic meson-exchange potentials.In the traditional meson-exchange potentials, the calculated points are located on a straight line, which is similar to the Coester line for the saturation point of symmetric nuclear matter.We find that fss2 gives a larger binding energy than the modern realistic meson-exchange potentials like Bonn-C and AV18, while the result of FSS is not very far from that of Bonn-C.It is interesting to note that our QM points are apparently off the line on which the data points of the modern meson-exchange potentials fall.By taking into account the charge-dependence correction of 190 keV, we conclude that the quark-model potential underbinds the triton by approximately 350 keV [3].Thus the energy to be accounted for by three-nucleon forces may not be as large as 0.5 -1 MeV, which most of standard mesonexchange potentials [8] predict.The different predictions for the contributions of the three-body force prompt us to examine what results the present quark-model NN interaction produces in other three-nucleon observables, especially in the nd and pd scatterings.

Effective range parameters
The accurate calculation of the effective range parameters of the nd scattering is very difficult especially for the spindoublet (S c = 1/2) channel, because of the pole structure of k cot δ near the zero-energy threshold.We apply an extended effective range expansion in this particular case.We have obtained 2 a = 0.60 fm for I max = 4 with a pole located at Here, I max is the maximum value of the NN total angular momentum included in the calculation.This value of the doublet scattering length is comparable with the experimental value 2 a exp = 0.65 ± 0.04 fm.On the other hand, the quartet scattering length is almost independent of the model space truncation, and is well reproduced as 4 a = 6.27 fm vs. 4 a exp = 6.35 ± 0.02 fm.We compare our results with some predictions by meson exchange potentials in Table 1.Note that the charge dependence of the NN interaction is neglected in our calculation, but the effect is rather small.We find that the fss2 results almost reproduce the experimental values of E B ( 3 H), 2 a, and 4 a, without reinforcing the two-body NN force with the threebody forces.

Total cross sections
The elastic and breakup total cross sections up to 40 MeV, predicted by fss2, are plotted in Fig. 3, together with the experimental data.Although the experimental values of total breakup cross sections have large error bars especially at the low-energy region, the elastic and total cross sections are nicely reproduced based on the constraint of the optical theorem.
The salient features of the obtained results are as follows: 1.The minimum values of the differential cross sections have an opposite energy dependence to the one given by the standard meson-exchange potentials.For the QM 03029-p.3NN interaction, the cross section minima are higher for higher energies.This behavior of the differential cross sections is consistent with the bound-state calculation of 3 H, in which fss2 predicts a large binding energy close to the experiment without introducing any threebody forces.

A slight overestimation of the differential cross sec-
tions is developing at forward angles when the energy becomes higher than E n ∼ 35 MeV.(See Fig. 5.) This overestimation is not improved by the renormalized RGM prescription of the QM interaction.Although the correct treatment of the W term in Eq. ( 1) is important, the effect of an extra nonlocality from this prescription may not be so large.

The maximum height of the neutron analyzing power
A y (θ) in E n ≤ 20 MeV (the so-called A y puzzle) is improved and is almost similar to the old results by the separable approximation of the realistic NN potentials.However, there still remains the deficiency of the order of 10%.(See Fig. 6.) On the other hand, the vector-type deuteron analyzing power iT 11 (θ) is generally well reproduced.(See Fig. 7.) The tensor-type deuteron analyzing power T 2m (θ) in Fig. 8 seems to imply the importance of the Coulomb effect even at the energy E n ∼ 9 MeV [10,11].

Summary
We have applied our quark-model NN interaction fss2 to the neutron-deuteron (nd) scattering in the Faddeev formalism for composite systems.The energy-dependence of the (3q)-(3q) RGM kernel is eliminated by the standard off-shell transformation utilizing the 1/ √ N factor, where N is the normalization kernel for the (3q)-(3q) system.This procedure yields an extra nonlocality, whose effect is very important to reproduce all the scattering observables below E n ≤ 65 MeV.We have found many new features which seem to be related to the different off-shell properties possessed by the quark-model baryon-baryon interaction.These include: 1) a large triton binding energy, 2) reproduction of the doublet scattering length 2 a, 3) the energy dependence of the differential cross sections at the minimum points, 4) the maximum hight of the nucleon analyzing power A y (θ) in the low-energy region E n ≤ 20 MeV.We are now investigating the observables related to the deuteron breakup processes.

Fig. 1 .
Fig.1.Calculated np phase shifts by fss2 in the isospin basis, compared with the phase-shift analysis SP99 by Arndt et al.[7]

Fig. 2 .
Fig. 2. Calculated 3 H binding energies B t as a function of the deuteron D-state probability P D .Calculations are made in the isospin basis, using the np interaction, for fss2, FSS, Bonn-A, B, C and Chiral (denoted by black circles).The group, including CD-Bonn, Nijmegen I and AV18 (denoted by black diamonds), takes into account the effect of charge-dependence of the interaction.In the Paris and RSC results (denoted by the open diamonds), the 1 S 0 interaction is determined from the pp scattering data.Those energies denoted by black circles go down by about 190 keV when the charge dependence of the NN force is taken into account.The experimental value, B t = 8.482 MeV, is shown by the dashed line.

Fig. 3 .
Fig. 3.The fss2 predictions to the nd total cross sections up to E n = 40 MeV, compared with the experiment.

Table 1 .
[9] nd scattering length predicted by fss2 (I max = 4 with n=8-8-8).For fss2 results, the charge dependence of the NN force is neglected.The heading NN implies the calculation only by the two-nucleon force, and NN+TM including the three-nucleon force.The results by CD-Bonn 2000, AV18, and Nijm I, are taken from[9].