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Register a new userDensity-dependent NN-interaction from subsubleading chiral 3N-force: Intermediate-range contributions
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N. Kaiser
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From the subsubleading chiral three-nucleon force [intermediate-range contributions, published in Phys. Rev. C,87, 054007 (2013)] a density-dependent NN-interaction $V_text{med}$ is derived in isospin-symmetric nuclear matter. Following the division of the pertinent 3N-diagrams into two-pion-one-pion exchange topology and ring topology, one evaluates for these all selfclosings and concatenations of nucleon-lines to an in-medium loop. In the case of the $2pi 1pi$-exchange topology, the momentum- and $k_f$-dependent potentials associated with the isospin-operators ($1$ and $vectau_1 !cdot! vectau_2$) and five independent spin-structures require at most one numerical integration. For the more challenging (concatenations of the) ring diagrams proportional to $c_{1,2,3,4}$, one ends up with regularized double-integrals $int_0^lambda dr,r int_0^{pi/2} dpsi$ from which the $lambda^2$-divergence has been subtracted and the logarithmic piece $sim ln (m_pi/lambda)$ is isolated. The derived semi-analytical results are most helpful to implement the subsubleading chiral 3N-forces into nuclear many-body calculations.
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We derive from the subleading contributions to the chiral three-nucleon force (long-range terms, published in Phys.,Rev.,C,77, 064004 (2008)) a density-dependent two-nucleon interaction $V_text{med}$ in isospin-symmetric, spin-saturated nuclear matter. Following the division of the pertinent 3N-diagrams into two-pion exchange topology, two-pion-one-pion exchange topology and ring topology, we evaluate for these all self-closings and concatenations of nucleon-lines to an in-medium loop. The momentum and $k_f$-dependent potentials associated with the isospin operators ($1$ and $vectau_1!cdot!vectau_2$) and five independent spin-structures are expressed in terms of functions, which are either given in closed analytical form or require at most one numerical integration. In the same way we treat the $2pi$-exchange 3N-force up to fourth order. Our results for $V_text{med}$ are most helpful to implement the long-range subleading chiral 3N-forces into nuclear many-body calculations.
The long-range terms of the subleading chiral three-nucleon force [published in Phys.,Rev.,C77, 064004 (2008)] are specified to the case of three neutrons. From these $3n$-interactions an effective density-dependent neutron-neutron potential $V_text{med}$ in pure neutron matter is derived. Following the division of the pertinent 3n-diagrams into two-pion exchange, two-pion-one-pion exchange and ring topology, all self-closings and concatenations of two neutron-lines to an in-medium loop are evaluated. The momentum and $k_n$-dependent potentials associated with the spin-operators $1,, vecsigma_1!cdot!vecsigma_2,, vecsigma_1!cdot!vec q, vecsigma_2!cdot!vec q,, i( vecsigma_1!+!vecsigma_2)!cdot ! (vec q!times ! vec p,),, (vecsigma_1!cdot!vec p,vecsigma_2!cdot!vec p+vecsigma_1!cdot!vec p,, vecsigma_2!cdot!vec p,)$ and $ vecsigma_1!cdot ! (vec q!times ! vec p,)vecsigma_2!cdot ! (vec q!times ! vec p,)$ are expressed in terms of functions, which are either given in closed analytical form or require at most one numerical integration. The subsubleading chiral 3N-force is treated in the same way. The obtained results for $V_text{med}$ are helpful to implement the long-range chiral three-body forces into advanced neutron matter calculations.
We discuss the building blocks for a consistent inclusion of chiral three-nucleon (3N) interactions into ab initio nuclear structure calculations beyond the lower p-shell. We highlight important technical developments, such as the similarity renormalization group (SRG) evolution in the 3N sector, a JT-coupled storage scheme for 3N matrix elements with efficient on-the-fly decoupling, and the importance truncated no-core shell model with 3N interactions. Together, these developments make converged ab initio calculations with explicit 3N interactions possible also beyond the lower p-shell. We analyze in detail the impact of various truncations of the SRG-evolved Hamiltonian, in particular the truncation of the harmonic-oscillator model space used for solving the SRG flow equations and the omission of the induced beyond-3N contributions of the evolved Hamiltonian. Both truncations lead to sizable effects in the upper p-shell and beyond and we present options to remedy these truncation effects. The analysis of the different truncations is a first step towards a systematic uncertainty quantification of all stages of the calculation.
The nuclear symmetry energy is a key quantity in nuclear (astro)physics. It describes the isospin dependence of the nuclear equation of state (EOS), which is commonly assumed to be almost quadratic. In this work, we confront this standard quadratic expansion of the EOS with explicit asymmetric nuclear-matter calculations based on a set of commonly used Hamiltonians including two- and three-nucleon forces derived from chiral effective field theory. We study, in particular, the importance of non-quadratic contributions to the symmetry energy, including the non-analytic logarithmic term introduced by Kaiser [Phys.~Rev.~C textbf{91}, 065201 (2015)]. Our results suggest that the quartic contribution to the symmetry energy can be robustly determined from the various Hamiltonians employed, and we obtain 1.00(8) MeV (or 0.55(8) MeV for the potential part) at saturation density, while the logarithmic contribution to the symmetry energy is relatively small and model-dependent. We finally employ the meta-model approach to study the impact of the higher-order contributions on the neutron-star crust-core transition density, and find a small 5% correction.
The effective field theory of NN interactions in nuclear matter is considered. Due to the Pauli principle the effective NN amplitude is not affected by the shallow bound states. We show that the next-to-leading order terms in the chiral expansion of the effective NN potential can be interpreted as corrections so the expansion is systematic. The value of potential energy per particle is calculated and some issues concerning the chiral effective theory of nuclear matter are outlined.
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