Transport properties of nuclear matter within Brueckner-Hartree-Fock Hongfei Zhang ( 张鸿飞） Lanzhou University Aug. 3, 2015 PKU-CUSTIPEN Workshop on " Advances in computations of nuclear structure and nuclear forces “ (August 1-6, 2015, Peking University, Beijing) 1

Collaborators: Umberto Lombardo, Wei Zuo, Xiaojun Bao, Zenghua Li, Jianming Dong, Francesca Sammarruca, 2

Contents 1 Motivation 2 Sketch of BHF with microscopic three-body force 3 In-medium nucleon-nucleon cross sections 4 Mean free path in asymmetric nuclear matter 5 Shear viscosity and thermal conductivity 6 Conclusions 3

1 Motivations Heavy-ion collisions are theoretically described by transport-model simulations whose input data are the in-medium cross sections and the nuclear mean field. In Brueckner theory, the G matrix plays the role of the in-medium scattering amplitude. [Eur. Phys. J. A 30 III (2006)]. The mean free path is a crucial quantity in the transport-model simulations of heavy-ion collisions and in the prediction of the nuclear transparency measured in (e, e′p) reactions. [Phys. Rev. C 45, 791 (1992)] Gravitational radiation emitted by r-modes in rapidly rotating neutron stars drives the instability of the system. Which dissipation mechanism tends to suppress this instability? A good candidate is the shear viscosity. Which constituent of NS mainly contribute to the viscosity? Neutrons, leptons, hyperons, quarks,… 4

2 Sketch of BHF with microscopic three- body forces The starting point is the reaction G matrix, which satisfies the Brueckner-Bethe Goldstone (BBG) equation, the single-particle energy is given by: the single-particle potential: 5

In present BHF calculation, the Three-Body-Force (TBF) has been reduced to an equivalently effective two-body interaction via a suitable average with respect to the third-nucleon degrees of freedom. A. Lejeune, U. Lombardo, W. Zuo, PLB 477 (2000) 45. where the trace is taken with respect to spin and isospin of the third nucleon. 6

3 In-medium NN cross sections In the c.m. frame, the nonrelativistic elastic differential cross section for neutron-proton (np) scattering from unpolarized beams is given by the differential cross section can be explicitly integrated over the solid angle to give the total cross section, 7

Zhang, Li, Lombardo, Luo, Sammarruca, Zuo, PRC 76, (2007) 8

Momentum dependence of the nucleon effective mass m ∗ /m in symmetric nuclear matter at three densities, in the presence and absence of the 3BF effect. Zhang, Li, Lombardo, Luo, Sammarruca, Zuo, Phys. Rev. C 76, (2007) 9

Free-space pn and NN cross sections for increasing values of the maximum angular momentum and as a function of the incident laboratory energy, E. The squares represent the experimental data. Zhang, Li, Lombardo, Luo, Sammarruca, Zuo, Phys. Rev. C 76, (2007) 10

Total cross sections for scattering of identical nucleons with and without the effect of 3BFs. The free-space cross section is shown for comparison. Zhang, Li, Lombardo, Luo, Sammarruca, Zuo, Phys. Rev. C 76, (2007) 11

Total cross sections for scattering of nonidentical nucleons Zhang, Li, Lombardo, Luo, Sammarruca, Zuo, Phys. Rev. C 76, (2007) 12

Center-of-mass differential cross section for scattering of identical nucleons with and without the effect of 3BFs at E =100 and 240 MeV. The free-space cross section is also shown for comparison Zhang, Li, Lombardo, Luo, Sammarruca, Zuo, Phys. Rev. C 76, (2007) 13

Center-of-mass differential cross section for scattering of nonidentical nucleons with and without the effect of 3BFs at E =100 and 240 MeV. The free-space cross section is also shown for comparison Zhang, Li, Lombardo, Luo, Sammarruca, Zuo, Phys. Rev. C 76, (2007) 14

Comparison with the NN cross sections of DBHF model. Identical nucleonsNonidentical nucleons Z diagram (left) and its contribution to the nuclear matter energy (right). Upward/downward lines represent positive/negative energy states. Zhang, Li, Lombardo, Luo, Sammarruca, Zuo, Phys. Rev. C 76, (2007) 15

In DBHF, relativistic corrections effectively take into account virtual antinucleon excitation in the nucleon self-energy. In BHF, the virtual antinucleon excitation is included by the Z-diagram. [20] F. Sammarruca, arXiv: v1 [nucl-th]. [9] W. Zuo, A. Lejeune, U. Lombardo, J. F. Mathiot, EPJA 14, 469 (2002); NPA706, 418 (2002). 16

4 Mean free path The nucleon mean free path can be calculated from imaginary part of the mass operator. BHF EBHF 17

Mean free path versus energy with and without 3BF for symmetric nuclear matter. Three densities and zero temperature are considered Bao, Zhang, Lombardo, Dong, Zuo, J. Phys. G, 41 (2014)

Mean free path for symmetric nuclear matter in the BHF (left) and EBHF (right) approximation: three densities at zero temperature (upper panels) and three temperatures （ lower panels ） at the saturation density are considered. Bao, Zhang, Lombardo, Dong, Zuo, J. Phys. G, 41 (2014)

Bao, Zhang, Lombardo, Dong, Zuo, J. Phys. G, 41 (2014)

Bao, Zhang, Lombardo, Dong, Zuo, J. Phys. G, 41 (2014)

5 Shear viscosity and thermal conductivity Shear viscosity Thermal conductivity PRL 21,(1968) The shear viscosity and thermal conductivity of Fermi liquid were derived by Abrikosov and Khalatnikov from Landan kinetic equation.

Shear viscosity Zhang, Lombardo, Zuo, Phys. Rev. C 82, (2010) 23

Thermal conductivity Zhang, Lombardo, Zuo, Phys. Rev. C 82, (2010) 24

6 Conclusions The in-medium NN cross sections were calculated within the BHF+3BF theory, showing that Pauli principle mainly suppress the forward and backward angles whereas the mass renormalization plays the most important role. The nucleon mean free path in isospin symmetric and asymmetric nuclear matter at finite temperature was calculated with BHF. The density, isospin and temperature dependence of the mean free path has been investigated. Transport parameters, shear viscosity and thermal conductivity were calculated in different configurations of nuclear matter, including beta-stable nuclear matter and neutron stars. 25

Thank you for your attention! 26