Microscopic Three-Body Force Effect on Nucleon-Nucleon Cross Sections in Symmetric Nuclear Matter

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1 Commun. Theor. Phys. (Beijing, China) 50 (2008) pp c Chinese Physical Society Vol. 50, No. 5, November 15, 2008 Microscopic Three-Body Force Effect on Nucleon-Nucleon Cross Sections in Symmetric Nuclear Matter ZHANG Hong-Fei, 1,2 ZUO Wei, 1,2,3 Lombardo Umberto, 4 LI Zeng-Hua, 4 and LI Jun-Qing 2,3 1 School of Nuclear Science and Technology, Lanzhou University, Lanzhou , China 2 Center of Theoretical Nuclear Physics, National Laboratory of Heavy Ion Accelerator, Lanzhou , China 3 Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou , China 4 Dipartimento di Fisica, Università di Catania, and INFN-LNS, Via S. Sofia 64, I Catania, Italy (Received January 30, 2008) Abstract We provide a microscopic calculation of neutron-proton and proton-proton cross sections in symmetric nuclear matter at various densities, using the Brueckner Hartree Fock approximation scheme with the Argonne V 14 potential including the contribution of microscopic three-body force. We investigate separately the effects of three-body force on the effective mass and on the scattering amplitude. In the present calculation, the rearrangement contribution of three-body force is considered, which will reduce the neutron and proton effective mass, and depress the amplitude of cross section. The effect of three body force is shown to be repulsive, especially in high densities and large momenta, which will suppress the cross section markedly. PACS numbers: f, Cn, z Key words: nucleon-nucleon cross section, nuclear matter, three-body force 1 Introduction During the impact phase between two heavy ions from high-energy central collisions, the nucleon density can reach values 2 3 times of the saturation density. Therefore one expects that, even if the mean field and the Pauli blocking still play a role in the medium modification of the nucleon-nucleon (N N) cross section, the effect of the modification of the interaction strength has the main role indeed. In the high-density domain the N N interaction is in fact deeply affected by the presence of the three-body force (3BF). [1,2] Heavy-ion collisions (HICs) are theoretically described by transport-model simulations, whose input data are the in-medium cross section and the nuclear mean field. [3,4] The latter, being intimately related to each other within the nuclear matter equation of state (EoS), must be consistently determined. The microscopic approaches to EoS, i.e. based on the NN bare two-body force (2BF), exhibit the unique advantage of treating both inputs on a unified basis and thus can adopt approximations which stay on the same footing. In the Brueckner theory the G-matrix plays the role of the in-medium scattering amplitude, the medium effects being the mean field and the Pauli blocking. In the zero density limit the G-matrix goes over into the T-matrix, and the Brueckner Bethe Goldstone (BBG) equation into the Lippman Schwinger equation. In the high-density limit the additional medium effects are vertex corrections due to N N excitations and nucleon resonances (isobar or Roper N (1440)). Their effect can be described in terms of a 3BF. [2] In turn, the latter is transformed by an average procedure into a density-dependent two-body force and added to the bare 2BF in the Brueckner scheme. The G-matrix is expected to be strongly affected by such vertex corrections in consideration that they play a crucial role in the saturation mechanism of nuclear matter. Beyond the scattering amplitude, the N N collisions in nuclear matter is also ruled by kinematic degrees of freedom, i.e. entrance flow and density of states in the exit channel; both are related to the nucleon effective mass, which is in turn related to the self-energy. Recently it has been proved that the rearrangement contribution of TBF to the self-energy is quite large, [5] and since it gives rise to a large reduction of the effective mass, one has to expect that it might have a strong influence on the in-medium cross section, being quadratic in the effective mass. 2 Theoretical Model 2.1 Brueckner Hartree Fock Theory with Microscopic TBF In the context of the Brueckner theory the role of the scattering amplitude between two nucleons embedded in a nuclear environment is played by the G matrix, The project supported by the Asia-Link project (CN/ASIA-LINK/008(94791)) of the European Commission, and in part by National Natural Science Foundation of China under Grant Nos , , , and , the Knowledge Innovative Project of CAS under Grant No. KJCX3-SYW-N2, and the Major Prophase Research Project of Fundamental Research of the Ministry of Science and Technology of China under Grant No. 2007CB815004

2 1220 ZHANG Hong-Fei, ZUO Wei, Lombardo Umberto, LI Zeng-Hua, and LI Jun-Qing Vol. 50 which satisfies the Brueckner Bethe Goldstone (BBG) equation, [6 8] k 1 k 2 Q(k 1, k 2 ) k 1 k 2 G(ρ, β, ω) = v NN + v NN ω ɛ(k 1 ) ɛ(k 2 ) + iη k 1 k 2 G(ρ, β, ω), (1) where ω is the starting energy, and Q is the Pauli operator, which prevents the two intermediate nucleons from being scattered into the states below the Fermi sea, and ɛ(k) is the single particle energy given by ɛ(k) ɛ(k; ρ) = 2 k 2 /(2m) + U(k; ρ). The single particle potential U(k) is calculated from the real part of the on-shell G-matrix and we adopt for it the so-called continuous choice, [8] i.e. for any momentum k below and above the Fermi surface, U BHF (k; ρ) = k n(k )Re kk G[e(k)+e(k )] kk a, (2) where the subscript a indicates antisymmetrization of the matrix element. Due to the occurrence of U(k) in Eqs. (1) and (2), the latter constitutes a coupled system of equations that have to be solved in a self-consistent way. In addition to the two-body bare interaction, a 3BF must be considered to take into account virtual excitations of nucleon resonances and N N excitations. [1] The 3BF adopted in the present calculation is based on the mesonexchange current model and it is described in full detail in Refs. [2] and [5]. In order to be applied in the Brueckner approach, 3BF is converted into an effective two-body force by averaging out the third particle weighted on the two-body correlation function, i.e. r 1 r 2 V τ 1τ 2 3 r 1 r 2 = 1 d r 3 d r 4 3φ n(τ 3 r 3) τ 3 σ 3 n (1 η τ1,τ 3 (r 13))(1 η τ2,τ 3 (r 23)) W 3 ( r 1 r 2 r 3 r 1 r 2 r 3 )φ n (τ 3 r 3 ) (1 η τ1,τ 3 (r 13 ))(1 η τ2,τ 3 (r 23 )).(3) The function η τ1 τ 2 (r) is the average over spin and momenta in the Fermi sea of the defect function, of which only the most important partial wave components have been included, i.e. the 1 S 0 and 3 S 1 3D 1 partial waves. The transformation of the 3BF to an effective 2BF entails a self-consistent coupling between 3BF and Brueckner procedure of solving the Brueckner Bethe Goldstone equations. One first calculates the correlation function with only the 2BF and then builds up the effective 3BF, which in turn is added to the 2BF, and again calculates the correlation function and so on up to the convergence is reached. The partial wave expansion of full interaction has been truncated at l max = 6. Fig. 1 Real and imaginary parts of the 1 S 0 components of the G matrix with (solid lines) and without (dotted lines) the contribution of TBF to the mean field for different nuclear density. In Fig. 1 the G-matrix in the 1 S 0 -channel is plotted. While 3BF is negligible at low density, it starts to be operating at the saturation density and becomes more effective at high density. As a consequence the real part of G- matrix gets reduced, since the BHF mean field becomes less attractive. The imaginary part, which is related to the particle-hole excitations, gets larger by the 3BF enhancement of the ground state correlations. As it is well

3 No. 5 Microscopic Three-Body Force Effect on Nucleon-Nucleon Cross Sections in Symmetric Nuclear Matter 1221 known, the G-matrix shows a spurious singular behavior around the Fermi energy due to the missing pairing correlations from BHF approximation. In fact, when the latter are included in the calculations the singularity disappears. Since the singularity in the real part of G-matrix is very narrow it was easily washed out by hand. [9] The same has been done for the high density imaginary part. In addition to the in-medium scattering amplitude, the NN cross section is also depending on the density of states, which is given by the inverse derivative of the single particle energy with respect to momentum. In the vacuum it is determined by the kinetic energy and it turns out to be proportional to the bare mass m, in the medium it receives contribution from the self-energy too. With reasonable approximation the self-energy effect results in replacing the bare mass with the effective mass. The latter is given by m (k) m = k m ( dɛ(k) ) 1. (4) dk In the BHF approximation with TBF the self-energy contains three terms ɛ(k) = ɛ BHF (k) + ɛ corr (k) + ɛ TBF (k), (5) where the first term is the mean field with inert core, the second term accounts for the core polarization and the third term corresponds to the rearrangement contribution associated to the density dependence of the effective 3BF. [5] In Fig. 2 the effective mass is plotted. The repulsive nature of TBF leads to an enhancement of the effective mass so to competitively interplay with G-matrix in the cross section. Fig. 2 Momentum dependence of nucleon effective mass m /m in symmetric nuclear matter for three densities. Solid and dotted lines correspond to the presence and absent of the TBF effect, respectively. 2.2 In-Medium Cross Section In the c.m. frame, the elastic differential cross section for the neutron-proton (np) scattering from unpolarized beams is given by [9] σ np (θ) = m 2 4π 2 4 SS z S z G S S z S z (θ) 2. (6) In the case of collision between identical particles, i.e. neutron-neutron (nn) or proton-proton (pp) collisions, the antisymmetrization requires to sum up the two scattering amplitudes G(θ) and G(π θ) before taking the modulus square: σ nn (θ) = σ pp (θ) = m 2 16π 2 4 SS z S z G S S z S z (θ)+( 1)S G S S z S z (π θ) 2.(7) The Coulomb force is neglected in this investigation. The G-matrix is the on-shell in-medium scattering amplitude, including the Pauli blocking and the dispersive effect of the mean field, as shown in the BBG equation. The prefactor is the density of states of the nuclear matter in the entrance and exit channels. The effective mass, which comes from of the momentum dependence of the

4 1222 ZHANG Hong-Fei, ZUO Wei, Lombardo Umberto, LI Zeng-Hua, and LI Jun-Qing Vol. 50 mean field (Schrödinger mass). The spin components of the G-matrix are obtained by summing up the angular momentum partial waves G S S z S z (θ) = LL J Y L S z S z (θ, 0) L0, SS z JS z L S z S z, SS z JS z k; L SJ G k; LSJ, (8) where the brackets are the Clebsch Gordan coefficients. Employing the latter expression, the differential cross section can be explicitly integrated over the solid angle to give the total cross section valid for both like and unlike particles section σ tot (E) = m 2 16π 2 4 [1 ( 1) S+L+T ] 2 SJ L L 2J + 1 4π GSJ L L 2. (9) The total cross section for like particles turns out to be smaller than that for the neutron-proton scattering since the isospin T = 0 channels do not contribute in the former case. The prefactor in Eqs. (7) and (8) is the density of states of the nuclear matter in the entrance and exit channels. The effective mass, which comes from the momentum dependence of the mean field (Schrödinger mass), should not be confused with the effective mass related to the renormalization of the self-energy due to the scalar field (Dirac mass). [10,11] 3 Results and Discussions The in-medium N N cross section within Brueckner theory has been already discussed in several papers. [9,10,12 19] But, since the range of applicability of the Brueckner theory is restricted to low density, the corresponding in-medium cross section cannot describe the entire evolution of an HIC and other phenomena, where high baryon density is involved. The recent extensions of the Brueckner with 3BF permit to extend the framework of Brueckner theory to very high density. Therefore, the in-medium cross section has been revisited with new calculations. Therefore, the main concern of this note is to emphasize the role of 3BF, specially above the saturation density. Apart from the sizeable enhancement of the repulsive component of the interaction operating into the scattering amplitude (G-matrix in Brueckner theory), the level density in the entrance and exit channels is deeply affected via the effective mass, as shown in Ref. [2]. The potential used in our calculations is the Argonne V 14, [20] which is molded on the experimental phase shifts. Therefore the theoretical NN cross sections in free space will nicely fit the experimental one, as shown in Fig. 3. It has to be noticed that σ pp is smaller than σ pn for the symmetrization requirement. Both calculated σ pp and σ pn converge to the corresponding experimental values [21 24] rapidly with the increasing of partial wave numbers. Fig. 3 The free space cross section for different partial waves, and compared with experimental data. 3.1 Identical Particles We examine four typical situations, i.e. low density (ρ = fm 3 ), saturation density (ρ = 0.17 fm 3 ) and high density (ρ = 0.34 fm 3 ), (ρ = 0.51 fm 3 ). In Fig. 4 the in-medium cross section from a BHF calculation is displayed with and without three-body force. The free cross section is also plotted for comparison. Up to the saturation density the effect of 3BF is small and the medium suppression is mainly controlled by the reduction of space of states due to the Pauli blocking. The asymptotic value with and without 3BF is the same. At the higher density the 3BF produces a more pronounced reduction of the cross section, which persists up to high energy. The latter is mainly due to the strong 3BF renormalization of the effective mass shown in Fig. 2. In all calculations the momentum dependence of the effective mass is taken into account. The scattering amplitude is also affected by 3BF, as shown in Fig. 1 for the

5 No. 5 Microscopic Three-Body Force Effect on Nucleon-Nucleon Cross Sections in Symmetric Nuclear Matter 1223 channel 1 S 0, but it is difficult to disentangle the reduction of attractive two-body channels from the increase of the repulsive ones. Fig. 4 Total proton-proton cross section σ pp with (solid lines) and without (dotted lines) the effect of TBF. The free space cross section is also shown for comparison. 3.2 Non-identical Particles In the scattering between distinguishable nucleons also the component T = 0 of the interaction takes place in scattering amplitude. As a consequence, the free cross section between unlike particles is larger than that between like particles and the medium suppression does not affect such a property. The 3BF has no special effects in this case, as shown in Fig. 5. It is worth while to be noticed that the low energy uprise of the in-medium σ pn is a remnant of the pairing anomaly, which cannot be removed by hand at low density. Fig. 5 Total proton-neutron cross section σ pn with (solid lines) and without (dotted lines) the effect of TBF. The free space section (dot-dashed line) is shown for comparison.

6 1224 ZHANG Hong-Fei, ZUO Wei, Lombardo Umberto, LI Zeng-Hua, and LI Jun-Qing Vol Conclusions and Outlooks In summary, the medium modifications of the NN cross section have been revisited within the BHF approximation with 3BF. The inclusion of 3BF effect allows to extend the range of application of the new results up to 2 3 times the saturation density. In this approximation the scattering amplitude (G-matrix) embodies not only the mean-field dispersive effect and Pauli blocking but also a number of important medium modifications due to the virtual excitation of N N couples and nucleon resonances. The 3BF induces a further suppression to the total cross section of identical nucleons as well as non-identical nucleons. But here the main effect is the strong reduction of the density of states in the entrance and exit channels due to the rearrangement term of the self-energy, which however is also traced to the 3BF. The effect of three-body force is shown to be repulsive, especially in high densities and large momenta, which suppresses the cross section markedly. In the next work, the study on nucleon-nucleon will be extended for asymmetric nuclear matter, which will give more important information on HICs. Acknowledgments We are thankful to Profs. Zhong-Yu Ma, En-Guang Zhao, C. Fuchs, F. Sammarruca, and R. Machleidt for valuable discussions. H.F. Zhang and W. Zuo acknowledge the warm hospitality they received at LNS-INFN, Catania where this work was started. References [1] P. Grange, A. Lejeune, M. Martzolff, and J.F. Mathiot, Phys. Rev. C 40 (1989) [2] W. Zuo, A. Lejeune, U. lombardo, J.F. Mathiot, Eur. Phys. J. A 14 (2002) 469; W. Zuo, A. Lejeune, U. Lombardo, and J.F. Mathiot, Nucl. Phys. A 706 (2002) 418. [3] P. Danielewicz, R. Lacey, and W.G. Lynch, Science 298 (2002) [4] B.A. Li, Phys. Rev. Lett. 88 (1998) [5] W. Zuo, U. Lombardo, H.-J. Schulze, and Z.H. Li, Phys. Rev. C 74 (2006) [6] J.P. Jeukenne, A. Lejeune, and C. Mahaux, Phys. Rep. 25 (1976) 83. [7] I. Bombaci and U. Lombardo, Phys. Rev. C 44 (1991) [8] W. Zuo, I. Bombaci, and U. Lombardo, Phys. Rev. C 60 (1999) [9] G. Giansiracusa1, U. Lombardo1, and N. Sandulescu, Phys. Rev. C 53 (1996) R1478. [10] C. Fuchs, A. Faessler, and M. El-Shabshiry, Phys. Rev. C 64 (2001) [11] E.N.E. van Dalen, C. Fucks, and Ammand Faessler, Phys. Rev. C 72 (2005) [12] A. Bohnet, N. Ohtsuka, J. Aichelin, R. Linden, and Amand Faessler, Nucl. Phys. A 494 (1989) 349. [13] G.Q. Li and R. Machleidt, Phys. Rev. C 48 (1993) 1702; ibid. 49 (1994) 566. [14] H.-J. Schulze, A. Schnell, G. Röpke, and U. Lombardo, Phys. Rev. C 55 (1997) [15] Joaquin Diaz-Alonso, and Lysiane Morns, Nucl. Phys. A 629 (1998) 679. [16] Bao-An Li and Lie-Wen Chen, Phys. Rev. C 72 (2005) ; Bao-An Li, Pawel Danielewicz, and William G. Lynch, Phys. Rev. C 72 (2005) [17] F. Sammarruca and P. Krastev, Phys. Rev. C 73 (2006) [18] Qing-Feng Li, Zhu-Xia Li, and Guang-Jun Mao, Phys. Rev. C 62 (2000) [19] Ying-Xun Zhang, Zhu-Xia Li, and Pawel Danielewicz, Phys. Rev. C 75 (2007) [20] R.B. Wiringa, R.A. Smith, and T.L. Ainsworth, Phys. Rev. C 29 (1984) [21] A. Bol, P. Devescovi, P. Leleux, P. Lipnik, P. Macq, and J.P. Meulders, Phys. Rev. C 32 (1985) 623. [22] V. Grundies, et al., Phys. Lett. B 158 (1985) 15. [23] K. Chen, et al., Phys. Rev. 166 (1968) 949. [24] J.W. Wilson and C.M. Costner, NASA TN D-8107 (1975).