We calculate the equation of state of isospin-symmetric nuclear matter in the threeloop approximation of chiral perturbation theory. The contributions to the energy per particleĒ(k f ) from one-and two-pion exchange diagrams are ordered in powers of the Fermi momentum k f (modulo functions of k f /m π ). It is demonstrated that, already at order O(k 4 f ), two-pion exchange produces realistic nuclear binding. The underlying saturation mechanism is surprisingly simple (in the chiral limit), namely the combination of an attractive k 3 f -term and a repulsive k 4 f -term. The empirical saturation point and the nuclear compressibility K ≃ 250 MeV are well reproduced at order O(k 5 f ) with a momentum cut-off of Λ ≃ 0.65 GeV which parametrizes short-range dynamics. No further short-distance terms are required in our calculation of nuclear matter. In the same framework we calculate the density-dependent asymmetry energy and find A 0 ≃ 34 MeV at the saturation point, in good agreement with the empirical value. The pure neutron matter equation of state is also in fair qualitative agreement with sophisticated many-body calculations and a resummation result of effective field theory, but only for low neutron densities ρ n < 0.25 fm −3 .
We extend a recent three-loop calculation of nuclear matter in chiral perturbation theory by including the effects from two-pion exchange with single and double virtual ∆(1232)-isobar excitation. Regularization dependent short-range contributions from pionloops are encoded in a few NN-contact coupling constants. The empirical saturation point of isospin-symmetric nuclear matter,Ē 0 = −16 MeV, ρ 0 = 0.16 fm −3 , can be well reproduced by adjusting the strength of a two-body term linear in density (and tuning an emerging three-body term quadratic in density). The nuclear matter compressibility comes out as K = 304 MeV. The real single-particle potential U (p, k f 0 ) is substantially improved by the inclusion of the chiral πN ∆-dynamics: it grows now monotonically with the nucleon momentum p. The effective nucleon mass at the Fermi surface takes on a realistic value of M * (k f 0 ) = 0.88M . As a consequence of these features, the critical temperature of the liquid-gas phase transition gets lowered to the value T c ≃ 15 MeV. In this work we continue the complex-valued single-particle potential U (p, k f ) + i W (p, k f ) into the region above the Fermi surface p > k f . The effects of 2π-exchange with virtual ∆-excitation on the nuclear energy density functional are also investigated. The effective nucleon mass associated with the kinetic energy density is M * (ρ 0 ) = 0.64M . Furthermore, we find that the isospin properties of nuclear matter get significantly improved by including the chiral πN ∆-dynamics. Instead of bending downward above ρ 0 as in previous calculations, the energy per particle of pure neutron matterĒ n (k n ) and the asymmetry energy A(k f ) now grow monotonically with density. In the density regime ρ = 2ρ n < 0.2 fm −3 relevant for conventional nuclear physics our results agree well with sophisticated many-body calculations and (semi)-empirical values.
Using the two-loop approximation of chiral perturbation theory, we calculate the momentum and density dependent single particle potential of nucleons in isospin-symmetric nuclear matter. The contributions from one-and two-pion exchange diagrams give rise to a potential depth for a nucleon at rest of U (0, k f 0 ) = −53.2 MeV at saturation density. The momentum dependence of the real part of the single particle potential U (p, k f 0 ) is non-monotonic and can be translated into a mean effective nucleon mass ofM * ≃ 0.8M . The imaginary part of the single particle potential W (p, k f ) is generated to that order entirely by iterated one-pion exchange. The resulting half width of a nucleon hole-state at the bottom of the Fermi sea comes out as W (0, k f 0 ) = 29.7 MeV. The basic theorems of Hugenholtz-Van-Hove and Luttinger are satisfied in our perturbative two-loop calculation of the nuclear mean field. PACS: 12.38.Bx, 21.65.+f Keywords: Effective field theory at finite density, Real and imaginary part of the single particle potential in nuclear matter, effective nucleon mass.
We calculate the nuclear energy density functional relevant for N=Z even-even nuclei in the systematic framework of chiral perturbation theory. The calculation includes the onepion exchange Fock diagram and the iterated one-pion exchange Hartree and Fock diagrams. From these few leading order contributions in the small momentum expansion one obtains already a very good equation of state of isospin symmetric nuclear matter. We find that in the region below nuclear matter saturation density the effective nucleon mass M * (ρ) deviates by at most 15% from its free space value M , with 0.89M < M * (ρ) < M for ρ < 0.11 fm −3 and M * (ρ) > M for higher densities. The parameterfree strength of the ( ∇ρ) 2 -term, F ∇ (k f ), is at saturation density comparable to that of phenomenological Skyrme forces. The magnitude of F J (k f ) accompanying the squared spin-orbit density J 2 comes out somewhat larger. The strength of the nuclear spin-orbit interaction, F so (k f ), as given by iterated one-pion exchange is about half as large as the corresponding empirical value, however, with the wrong negative sign. The novel density dependencies of M * (ρ) and F ∇,so,J (k f ) as predicted by our parameterfree calculation should be examined in nuclear structure calculations (after introducing an additional short range spin-orbit contribution constant in density).
We extend a recent three-loop calculation of nuclear matter in the systematic framework of chiral perturbation theory to finite temperatures T. The contributions from one- and two-pion exchange diagrams which cause nuclear binding and saturation at T=0 are included for T>0 in the density and temperature dependent free energy per particle, $\bar F(rho,T)$. The so-called anomalous 2pi-exchange contribution $\bar A(rho,T)$ (with no counterpart in the ground state energy density at T=0) is consistently included. The calculated pressure isotherms display the familiar first-order liquid-gas phase transition of isospin symmetric nuclear matter with a critical point at T_c = 25.5 MeV and rho_c = 0.09 fm^{-3}. The too high value of the critical temperature originates from the strong momentum dependence of the underlying single-particle potential U(p,k_{f0}) near the Fermi-surface. We also consider pure neutron matter at T>0 in the same framework and find fair agreement with sophisticated many-body calculations for neutron densities rho_n < 0.2 fm^{-3}.Comment: 10 pages, 4 figures, submitted to Physics Letters
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