Few-Body Systems 0, 11 16 (2003) Few- Body Systems c by Springer-Verlag 2003 Printed in Austria Selected Topics in Correlated Hyperspherical Harmonics A. Kievsky INFN and Physics Department, Universita di Pisa, 56100 Pisa, Italy Abstract. Correlated hyperspherical harmonic basis functions are used to expand the three and four nucleon wave functions. Bound and scattering states are considered. Results for the binding energies of 3 H, 3 He and 4 He calculated using modern two and three nucleon forces are given and discussed. For scattering states, results for N d differential cross section and vector analyzing powers are shown. The importance of the Coulomb and magnetic moment effects are discussed. 1 Introduction In the description of few-body systems different aspects have to be considered. The interaction between particles can include not only pair interactions but also higher order terms as three body interactions. Many body forces appear naturally in those systems in which the interactive particles has internal structure. For example, three nucleon forces (3NF s) are needed in the description of fewnucleon systems to reproduce the corresponding binding energies. Furthermore, the pair interaction presents a strong repulsion at short distances due to the Pauli principle which prevents the nucleons for overlapping. These characteristics suggest some caution in the selection of the methods used to describe such systems. Standard expansion methods often show a slow convergence making the extraction of conclusive results difficult. This is an important point since few-body systems sometimes are used to test potentials models. Therefore the theoretical uncertainties should be reduced as much as possible at the moment of making comparisons to experimental data. In the case of nuclear systems, a new generation of NN potentials have been recently constructed including explicitly charge dependent terms. These modern interactions can be put in the general form E-mail address:kievsky@pi.infn.it v(nn) = v EM (NN) + v π (NN) + v R (NN). (1)
12 Selected Topycs in Correlated Hyperspherical Harmonics The long range part is represented by the one-pion-exchange potential v π (NN) and the electromagnetic potential v EM (NN) whereas the short range part v R (NN) includes a certain number of parameters (around 40), which are determined by fitting the N N scattering data and the deuteron binding energy (BE). The quality of the fit is such that modern potentials reproduce the available NN data with χ 2 per datum 1. In the present paper we will show some applications of the the correlated hyperspherical harmonic (CHH) and the pair correlated hyperspherical harmonic (PHH) expansions [1, 2] in the description of three and four nucleon systems. Bound and scattering states will be considered. The capability of the new NN interaction to describe the few nucleon dynamics will be discussed. 2 The three and four nucleon bound states The wave function corresponding to a three nucleon bound state can be written as a sum of three amplitudes Ψ = i ψ(x i, y i ) (2) Each amplitude corresponding to the i-th permutation of the Jacobi coordinates x i, y i has total angular momentum JJ z, total isospin T T z and definite parity. Using the LS coupling it is decomposed in partial wave amplitudes ψ(x i, y i ) = Y α (jk, i) = N c α φ α (x i, y i )Y α (jk, i) (3) { [Y lα (ˆx i )Y Lα (ŷ i )] Λα [s jk α s i α] Sα } JJ z [t jk α t i α] T Tz, (4) where x i, y i are the moduli of the Jacobi coordinates and Y α is the angularspin-isospin function for each channel. N c is the maximum number of channels considered. The two-dimensional amplitudes φ α are expanded in terms of the correlated basis ] φ α (x i, y i ) = ρ lα+lα F α [ K u α K(ρ) (2) P lα,lα K (φ i ), (5) where the hyperspherical variables are defined by the relations x i = ρ cos φ i and y i = ρ sin φ i and (2) P l,l K (φ) is a hyperspherical polynomial. The correlation factor F α is included in order to accelerate the convergence of the expansion. It helps to describe the system at short distances where, due the strong repulsion of the potential, Ψ 0. Among other possibilities, the correlation factor can be chosen as a pair correlation function, F α f α (x i ), (PHH expansion) or a Jastrow correlation factor, F α 3 i=1 f α(x i ), (CHH expansion). The description of the method to calculate the correlations functions is given in Ref. [1] whereas the extension of the CHH expansion to the four nucleon system is given in Ref. [3]. In Table I we show the results for the bound states of the three-nucleon system, 3 H and 3 He, using the Argonne v 18 (AV18) NN interaction plus the
A. Kievsky 13 Hamiltonian 3 H 3 He B(MeV) B(MeV) AV18 (T = 1/2) 7.618 6.917 AV18 (T = 1/2, 3/2) 7.624 6.925 AV18+UR (T = 1/2) 8.474 7.742 AV18+UR (T = 1/2, 3/2) 8.479 7.750 Expt. 8.482 7.718 Table 1. Three-nucleon binding energies including a two body and a three body potential. The isospin T = 3/2 has been considered. 3NF of Urbana (UR) [4, 5]. The calculations include the case of isopin T = 3/2 which in general is omitted due to its very small contribution. The strength of the UR potential has been fixed in oder to reproduce the 3 H binding energy. On the other hand the 3 He binding energy is slightly overestimate. Therefore, it is of interest to analyze the different contributions to the 3 H 3 He mass difference. In Table II the contributions of the charge symmetry breaking terms of the potential are shown. All these terms with the exception of the last two, that have been calculated perturbatively, are included in the v EM (NN) interaction of the AV18 potential. Interaction term Nuclear CSB Point Coulomb Full Coulomb Magnetic moment Orbit-orbit force n-p mass difference Total (theory) Expt. B( 3 H)-B( 3 He) 65 kev 677 kev 648 kev 17 kev 7 kev 14 kev 751 kev 764 kev Table 2. Different contribution to the 3 H 3 He mass difference. As we see from the table, the mass difference is slightly underestimate by around 10 kev, giving some room for a charge dependence in the three nucleon interaction. The extension of the calculations to the A = 4 system is not simple. Recently, a benchmark for the α particle bound state has been produced [6] using a simplified version of the AV18. Very few calculations of the A = 4 bound state can be found in the literature which include all the complexities of the modern NN interactions plus 3NF s. In table III the binding and kinetic energies as well as the P and D states probabilities calculated using the HH expansion are given and compare to different calculations. The AV18 and the AV18+UR potential models have been considered. The agreement between the HH expansion and the solution of the Faddeev Yakubovsky (FY) equations is remarkable. The 50 kev difference gives the actual accuracy in the description of such a system. The Green Function Monte Carlo (GFMC) result underestimates the binding by
14 Selected Topycs in Correlated Hyperspherical Harmonics AV18 B(MeV) T(MeV) P P (%) P D (%) CHH [3] 24.18 97.79 0.34 13.69 HH 24.21 97.85 0.35 13.74 FY [7] 24.23 97.80 0.35 13.78 AV18+UR B(MeV) T(MeV) P P (%) P D (%) CHH [3] 28.00 111.72 0.65 15.78 HH 28.45 113.24 0.73 16.03 FY [7] 28.50 113.21 0.75 16.03 GFMC [8] 28.30 Expt. 28.3 Table 3. Binding energy (B) and kinetic energy (T) as well as P and D state probabilities for the four nucleon system. around 200 kev which, however, corresponds to an accuracy of 1%. 3 The three nucleon scattering states The extension of the PHH expansion to describe scattering states in the A = 3 system is given in Ref. [9]. Here we would like to show the quality of the description of some polarization observables taking into account different terms in the electromagnetic potential v EM (NN). In fact, in the description of the 3N continuum the magnetic moment (MM) interaction and corrections to the Coulomb potential has been systematically disregarded. This omission has been justified in the past by the intrinsic difficulties in solving the nuclear problem. At present days the 3N continuum is routinely solved by different techniques making possible the treatment of those electromagnetic terms beyond the Coulomb interaction. The starting point to describe N d scattering is the transition matrix M which can be decomposed as the sum of the Coulomb amplitude plus a MM amplitude plus a nuclear term J M SS νν (θ) = f c(θ)δ SS δνν + M SS νν (so) + 4π k L max LL 2L + 1 (L0Sν Jν)(L M S ν Jν) exp[i(σ L + σ L 2σ 0 )] J TLL SS Y L M (θ, 0). (6) This is a 6 6 matrix corresponding to the projections of the two possible couplings of the spin of the deuteron S d = 1 and the spin 1/2 of the third particle to S = 1/2 or 3/2. The quantum numbers L, L represent the relative orbital angular momentum between the deuteron and the third particle and J is the total angular momentum of the three-nucleon scattering state. J TT SS T are the T -matrix elements corresponding to a Hamiltonian containing nuclear plus Coulomb plus MM interactions. The n d case is recovered putting f c = σ l = 0. As a first example let us show the n d analyzing power A y at the neutron energy E n = 1.2 MeV that has been recently measured [10]. At this very low
A. Kievsky 15 energy the nuclear part of the transition matrix converges already for L max = 3. In Fig.1 three different theoretical curves are shown corresponding to calculations using the AV18 potential and neglecting the MM interaction (solid line), including the MM interaction but neglecting the MM amplitude in the transition matrix (dashed line) and including the MM amplitude (dotted dashed line). It is interesting to notice the forward-angle dip structure which already appears in n p scattering [11]. Only after including the MM amplitude it is possible to reproduce this particular behavior. We can conclude that in n d scattering the MM interaction produces a pronounced modification of A y at forward angles but has a very small effect around the peak. 0.04 0.03 0.02 0.01 A y 0-0.01-0.02-0.03 0 45 90 135 180 θ c.m. [deg] Figure 1. The n d vector analyzing power A y at E n = 1.2 MeV. Finally we want to discuss the importance of the Coulomb effects in N d scattering as the energy increases. In fact, up to E lab = 30 MeV we can observe appreciable differences in the description of n d and p d elastic scattering that however tend to diminish [9]. Experimental data are not always conclusive since experiments with neutrons have larger uncertainties than those performed with protons. On the other hand the description of p d scattering has been often performed using n d calculations, in particular at high energies [12]. To make clear the importance of Coulomb effects as the energy of the collision increases, in Fig.2 we show a comparison of n d and p d calculations at E lab = 65 MeV. The differential cross section and A y are shown. Three curves are displayed corresponding to p d AV18 (solid line), n d AV18 (dashed line) and p d AV18+UR (dotted line) and compared to experimental data. At this energy Coulomb effects are very reduced. On the other hand 3NF effects are appreciable at the minimum of the differential cross section.
16 Selected Topycs in Correlated Hyperspherical Harmonics 100 0.4 dσ/dω [mb/sr] 10 A y 0.0-0.4 1 0 45 90 135 180 θ c.m. [deg] -0.8 0 45 90 135 180 θ c.m. [deg] Figure 2. The differential cross section and A y at E lab = 65 MeV. Acknowledgement. I would like to thank M. Viviani and S. Rosati since many of the results given in this work has been done with their collaboration and P. Barletta for useful discussions. References 1. Kievsky A., Viviani M., and Rosati S.: Nucl. Phys. A551, 241 (1993) 2. Kievsky A., Viviani M., and Rosati S.: Nucl. Phys. A577, 511 (1994) 3. Viviani M., Kievsky A., and Rosati S.: Few-Body Syst. 18, 25 (1995) 4. Wiringa R.B., Stoks V.G.J., and Schiavilla R.: Phys. Rev. C51, 38 (1995) 5. Pudliner B.S. et al.: Phys. Rev. Lett. 74, 4396 (1995) 6. Kamada H. et al.: Phys. Rev. C64, 044001 (2001) 7. Nogga A., Kamada H., Glöckle W., and Barrett B.R.: Phys. Rev. C65, 054003 (2002) 8. Wiringa R.B., Pieper S.C., Carlson J., and Pandharipande V.R.: Phys. Rev. C62, 014001 (2000) 9. Kievsky A., Viviani M., and Rosati S.: Phys. Rev. C64, 024002 (2001) 10. Neidel E.M. et al.: Phys. Lett. B552, 29 (2003) 11. Stoks V.G.J., and de Swart J.J.: Phys. Rev. C42, 1235 (1990) 12. Wita la H. et al.: Phys. Rev. C63, 024007 (2001)