Many-body perturbation theory for the nuclear equation of state up to fifth order

Abstract

We present an automated, GPU-accelerated framework for many-body perturbation theory (MBPT) calculations of the zero-temperature nuclear equation of state (EOS) based on chiral nucleon-nucleon (NN) and three-nucleon (3N) interactions. Automated diagram generation and evaluation enable the computation of all diagrams up to fifth order in the MBPT expansion at the normal-ordered two-body level in infinite matter, with residual three-body contributions explicitly included up to third order. Multi-GPU acceleration of 3N normal ordering, a novel Monte Carlo integrator (called PVegas), and further advances in high-performance computing enable us to evaluate all 840 fifth-order diagrams with controlled numerical uncertainties. We investigate the MBPT convergence up to fifth order in pure neutron matter (PNM) and symmetric nuclear matter (SNM) for two sets of chiral interactions, study neutron star matter, and present fourth-order results for asymmetric matter including normal-ordered 3N forces. The framework enables systematic MBPT studies with harder interactions and benchmarks against nonperturbative methods. It can be further extended to finite-temperature EOS calculations and to improved uncertainty quantification using emulation and resummation techniques.

Figures (14)

• Figure 1: Energy per particle in PNM (a) and SNM (b), and the symmetry energy \ref{['eq:esym']} (c) based on MBPT(5) and the six Hebeler et al. interactions in Table \ref{['tab:hebeler_interactions']}. Each line corresponds to one of these Hamiltonians, and their spread at a given density, indicated by the bands, serves as a simple uncertainty estimate. The diagrammatic content of these calculations is summarized in Table \ref{['tab:mbpt']}. All NN and 3N contributions up to fifth order at the normal-ordered two-body level and residual 3N contributions up to third order are included. The empirical saturation point, a bivariate student-$t$ distribution inferred in Ref. Drischler:2024ebw from a wide range of density functionals, is depicted by the blue ellipse, which encompasses the 95% confidence region. To guide the eye, the corresponding marginalized empirical saturation density at the 95% confidence level is shown by the gray vertical bands. The extracted saturation points in SNM are depicted in Fig. \ref{['fig:coester_plot']}.
• Figure 2: Correlation between the predicted saturation density $n_0$ and saturation energy $E_0/A$ for the six Hebeler et al. NN and 3N interactions (grouped by colors) obtained at second, third, fourth, and fifth order in the MBPT expansion (different markers). The values for interaction's SRG resolution scale and 3N momentum cutoff are annotated as "$(\lambda_\mathrm{SRG}/\Lambda_\mathrm{3N})$" in units of $\, \text{fm}^{-1}$ (see also Table \ref{['tab:hebeler_interactions']}). These results correspond to the SNM calculations depicted in the bottom panel of Fig. \ref{['fig:eos_pnm_snm_5th']}. The gray band depicts the Coester band obtained by a linear least-squares fit to all results shown. Its width is determined by the minimum width that covers all results. The colored ellipses represent the empirical saturation point estimated in Refs. Drischler:2024ebw based on Bayesian model mixing across a wide range of energy-density functionals at four confidence levels (see the legend). The associated bivariate student-$t$ distribution is given by Eq. \ref{['eq:emp_sat_point']}.
• Figure 3: Residual 3N contributions to the energy per particle as a function of the density at third (top panel) and second order (bottom panel) in SNM based on the six Hebeler et al. potentials (see Table \ref{['tab:hebeler_interactions']}). The uncertainty bands represent the GPs discussed in the main text and correspond to the $1\sigma$ credibility region. The vertical bands depict the empirical range for the saturation density, corresponding to Eq. \ref{['eq:emp_sat_point']}.
• Figure 4: Posterior (dark blue bands) and prior distributions (light blue bands) of the PNM EOS as predicted by the EOS model \ref{['eq:eos_model']}. Panel (a) shows the energy per particle, panel (b) the pressure, and panel (c) the sound speed squared at the 95% confidence level (see the legend). The corresponding nuclear matter parameters and model parameters are depicted in Fig. \ref{['fig:corner']}. The $2\sigma$-error bars in panel (a) show the four data points used to calibrate the EOS model \ref{['eq:eos_model']}. Note that the EOS prior is chiral EFT-agnostic and weakly informed, e.g., as is evident at low densities in panel (a). The gray vertical bands represent the 95% confidence interval for the empirical saturation density.
• Figure 5: Posterior distribution of the inferred nuclear matter parameters, obtained from our Bayesian inference based on the EOS model \ref{['eq:eos_model']}. The parameters are the nuclear saturation point $(n_0, E_0/A)$, the incompressibility $K$, and the symmetry energy $S_v$ and its slope parameter $L$ evaluated at $n_0$. All parameters are in $\, \text{MeV}$, except for $n_0$, which is in $\, \text{fm}^{-3}$. The titles on the diagonal panels state the 95% confidence regions centered on the medians of the corresponding marginal distributions (dashed vertical lines), all of which fall within canonical ranges. By construction, the joint distribution of $(n_0,E_0/A)$ approximates the empirical saturation point \ref{['eq:emp_sat_point']} to which it was calibrated. The contour lines, moving outward, enclose the $0.5\sigma$, $1\sigma$, $1.5\sigma$, and $2\sigma$ confidence regions.