By assuming that the isospin- and momentum-dependent MDI interaction has a form similar to the Gogny-like effective two-body interaction with a Yukawa finite-range term and the momentum dependence only originates from the finite-range exchange interaction, we determine its parameters by comparing the predicted potential energy density functional in uniform nuclear matter with what has been usually given and used extensively in transport models for studying isospin effects in intermediate-energy heavy-ion collisions as well as in investigating the properties of hot asymmetric nuclear matter and neutron star matter. We then use the density matrix expansion to derive from the resulting finite-range exchange interaction an effective Skyrme-like zero-range interaction with density-dependent parameters. As an application, we study the transition density and pressure at the inner edge of neutron star crusts using the stability conditions derived from the linearized Vlasov equation for the neutron star matter.
For the coefficients of the amplitude a set of simultaneous equations is derived in momentum space. By the auxiliary conditions they are equivalent to nonrelativistic equations and suitable for the investigation of two-nucleon system.
We extend density-matrix expansions in nuclei to higher orders in derivatives of densities and test their convergence properties. The expansions allow for converting the interaction energies characteristic to finite- and short-range nuclear effective forces into quasi-local density functionals. We also propose a new type of expansion that has excellent convergence properties when benchmarked against the binding energies obtained for the Gogny interaction.
A finite-dimensional matrix model for the nucleon-nucleon cross section operator is used to calculate the dispersive correction to nucleon-nucleus total cross sections, and the leading terms in its expansion in the number of inelastic transitions in the high-energy limit where the longitudnal momentum transfers can be ignored.
In this follow-up article to [Shadow poles in the alternative parametrization of R-matrix theory, Ducru (2020)], we establish new results on scattering matrix pole expansions for complex wavenumbers in R-matrix theory. In the past, two branches of theoretical formalisms emerged to describe the scattering matrix in nuclear physics: R-matrix theory, and pole expansions. The two have been quite isolated from one another. Recently, our study of Brune's alternative parametrization of R-matrix theory has shown the need to extend the scattering matrix (and the underlying R-matrix operators) to comple…
Recently, a microscopically motivated nuclear energy density functional was derived by applying the density matrix expansion to the Hartree-Fock (HF) energy obtained from long-range chiral effective field theory two- and three-nucleon interactions. However, the HF approach cannot account for all many-body correlations. One class of correlations is included by Brueckner-Hartree-Fock (BHF) theory, which gives an improved definition of the one-body HF potential by replacing the interaction by a reaction matrix $G$. In this paper, we find that the difference between the $G$-matrix and the nucleon-…
A first step toward a universal nuclear energy density functional based on low-momentum interactions is taken using the density matrix expansion (DME) of Negele and Vautherin. The DME is adapted for non-local momentum-space potentials and generalized to include local three-body interactions. Different prescriptions for the three-body DME are compared. Exploratory results are given at the Hartree-Fock level, along with a roadmap for systematic improvements within an effective action framework for Kohn-Sham density functional theory.
We apply the density matrix expansion (DME) at Hartree-Fock level with long-range chiral effective field theory interactions defined in coordinate space up to next-to-next-to-leading order. We consider chiral potentials both with and without explicit Delta isobars. The challenging algebra associated with applying the DME to three-nucleon forces is tamed using a new organization scheme, which will also facilitate generalizations. We include local regulators on the interactions to mitigate the effects of singular potentials on the DME couplings and simplify the optimization of generalized Skyrme…
We explore the analytic structure of the three-channel $S$ matrix by generalizing uniformization and making a single-valued map for the three-channel $S$ matrix. First, by means of the inverse Jacobi's elliptic function we construct a transformation from eight Riemann sheets of the center-of-mass energy squared complex plane onto a torus, on which the three-channel $S$ matrix is represented single-valued. Secondly, we show that the Mittag-Leffler expansion, a pole expansion, of the three-channel scattering amplitude includes not only topologically trivial but also nontrivial contributions and …
Microscopic input to a universal nuclear energy density functional can be provided through the density matrix expansion (DME), which has recently been revived and improved. Several DME implementation strategies are tested for neutron drop systems in harmonic traps by comparing to Hartree-Fock (HF) and ab initio no-core full configuration (NCFC) calculations with a model interaction (Minnesota potential). The new DME with exact treatment of Hartree contributions is found to best reproduce HF results and supplementing the functional with fit Skyrme-like contact terms shows systematic improvement…
The s-wave nucleon-nucleon (NN) scattering matrix ($S$-matrix) exhibits UV/IR symmetries which are hidden in the effective field theory (EFT) action and scattering amplitudes, and which explain some generic features of the phase shifts. These symmetries offer clarifying interpretations of existing pionless EFT expansions, and suggest starting points for novel expansions. The leading-order (LO) $S$-matrix obtained in the pionless EFT with scattering lengths treated exactly is shown to have a UV/IR symmetry which leaves the sum of s-wave phase shifts invariant. A new scheme, which treats effecti…
We use the Negele-Vautherin density matrix expansion to derive a quasi-local density functional for the description of systems of fermions interacting with short-ranged interactions composed of arbitrary finite-range central, spin-orbit, and tensor components. Terms that are absent in the original Negele-Vautherin approach owing to the angle averaging of the density matrix are fixed by employing a gauge invariance condition. We obtain the Kohn-Sham interaction energies in all spin-isospin channels, including the exchange terms, expressed as functions of the local densities and their derivative…
We employ Chiral Perturbation Theory (ChPT) to evaluate the complete set of pseudoscalar matrix elements, <0|P_f|pi^0,eta>, with P_f any of the flavor-diagonal pseudoscalar currents (f=u,d,s), to order (m_d-m_u), and to next-to-leading order in the chiral expansion. These matrix elements represent the basic input to a QCD sum rule analysis of isospin breaking in the pi NN couplings using the three-point-function method. We discuss also how one could use the results to construct a one parameter family of interpolating fields for the $π^0$, all of whose members have zero vacuum-to-$η$ matrix ele…
Nuclear data libraries (ENDF, JEFF, JENDL, CENDL, etc.) document our phenomenological knowledge of nuclear cross sections as interpreted by R-matrix theory. The R-matrix scattering model can parameterize the energy dependence of the scattering matrix in different ways. This article establishes new results on three such sets of parameters: the Wigner-Eisenbud, the Brune, and the Siegert-Humblet parameters. We show how the latter two arise from invariance to the arbitrary boundary condition, and how they can trade-off more complex parameters for a simpler energy dependence parameterization: the …
The density matrix expansion is used to derive a local energy density functional for finite range interactions with a realistic meson exchange structure. Exchange contributions are treated in a local momentum approximation. A generalized Slater approximation is used for the density matrix where an effective local Fermi momentum is chosen such that the next to leading order off-diagonal term is canceled. Hartree-Fock equations are derived incorporating the momentum structure of the underlying finite range interaction. For applications a density dependent effective interaction is determined from…
Similarly to the standard effective range expansion that is done near the threshold energy, we obtain a generalized power-series expansion of the multi-channel Jost-matrix that can be done near an arbitrary point on the Riemann surface of the energy within the domain of its analyticity. In order to do this, we analytically factorize its momentum dependencies at all the branching points on the Riemann surface. The remaining single-valued matrix functions of the energy are then expanded in the power-series near an arbitrary point in the domain of the complex energy plane where it is analytic. A …
Empirical energy density functionals (EDFs) are generally successful in describing nuclear properties across the table of nuclides. But their limitations motivate using the density-matrix expansion (DME) to embed long-range pion interactions into a Skyrme functional. Recent results on the impact of the pion were both encouraging and puzzling, necessitating a careful re-examination of the DME implementation. Here we take the first steps, focusing on two-body scalar terms in the DME. Exchange energies with long-range one-pion contributions are well approximated by all DME implementations conside…
By considering a flavour expansion about the SU(3)-flavour symmetric point, we investigate how flavour-blindness constrains octet baryon matrix elements after SU(3) is broken by the mass difference between quarks. Similarly to hadron masses we find the expansions to be constrained along a mass trajectory where the singlet quark mass is held constant, which provides invaluable insight into the mechanism of flavour symmetry breaking and proves beneficial for extrapolations to the physical point. Expansions are given up to third order in the expansion parameters. Considering higher orders would g…
We present a lattice QCD determination of the distribution amplitude (DA) of the pion and the first few Mellin moments from an analysis of the quasi-DA matrix element within the leading-twist framework. We perform our study on a HISQ ensemble with $a=0.076$ fm lattice spacing with the Wilson-Clover valence quark mass tuned to the physical point. We analyze the ratios of pion quasi-DA matrix elements at short distances using the leading-twist Mellin operator product expansion (OPE) at the next-to-leading order and the conformal OPE at the leading-logarithmic order. We find a robust result for t…
