Neutron Matter at Next-to-Next-to-Next-to-Leading Order in Chiral Effective Field Theory

Neutron matter presents a unique system for chiral effective field theory because all many-body forces among neutrons are predicted to next-to-next-to-next-to-leading order (${\mathrm{N}}^3\mathrm{LO}$). We present the first complete ${\mathrm{N}}^3\mathrm{LO}$ calculation of the neutron matter energy. This includes the subleading three-nucleon forces for the first time and all leading four-nucleon forces. We find relatively large contributions from ${\mathrm{N}}^3\mathrm{LO}$ three-nucleon forces. Our results provide constraints for neutron-rich matter in astrophysics with controlled theoretical uncertainties.

Figure 1

Neutron matter energy per particle as a function of density including NN, 3N, and 4N forces at ${\mathrm{N}}^3\mathrm{LO}$. The three overlapping bands are labeled by the different NN potentials and include uncertainty estimates due to the many-body calculation, the low-energy $c_i$ constants, and by varying the $3\mathrm{N}/4\mathrm{N}$ cutoffs (see text for details). For comparison, results are shown at low densities (see also the inset) from NLO lattice [3] and quantum Monte Carlo (QMC) simulations [22], and at nuclear densities from variational (APR, the different points are with or without boost corrections) [23] and auxiliary field diffusion MC calculations (GCR) [24] based on adjusted nuclear force models.

Figure 2

Energy per particle versus density for all individual ${\mathrm{N}}^3\mathrm{LO}$ 3N and 4N force contributions to neutron matter at the Hartree-Fock level. The bands are obtained by varying the $3\mathrm{N}/4\mathrm{N}$ cutoff $\Lambda =2–2.5{\mathrm{fm}}^{-1}$. For the two-pion-exchange–contact and the relativistic-corrections 3N forces, the different bands correspond to the different NN contacts, $C_T$ and $C_S$, determined consistently for the ${\mathrm{N}}^3\mathrm{LO}$ EM or EGM potentials. The inset diagram illustrates the $3\mathrm{N}/4\mathrm{N}$ force topology.

Figure 3

Neutron matter energy per particle as a function of density at ${\mathrm{N}}^2\mathrm{LO}$ (upper blue band that extends to the dashed line) and ${\mathrm{N}}^3\mathrm{LO}$ (lower red band). The bands are based on the EGM NN potentials and include uncertainty estimates as in Fig. 1.

Figure 4

Range for the symmetry energy $S_v$ and its density dependence $L$ obtained at ${\mathrm{N}}^3\mathrm{LO}$ (this work) versus including 3N forces at ${\mathrm{N}}^2\mathrm{LO}$ (Hebeler et al. [8]). For comparison [25], we show constraints obtained from energy-density functionals for nuclear masses (Kortelainen et al. [26]) and from the ${}^{208}\mathrm{Pb}$ dipole polarizability (Tamii et al. [27]).