Neutron matter at next-to-next-to-next-to-leading order in chiral effective field theory

Wavefunction matching for solving quantum many-body problems

Ab initio calculations have an essential role in our fundamental understanding of quantum many-body systems across many subfields, from strongly correlated fermions1-3 to quantum chemistry4-6 and from atomic and molecular systems7-9 to nuclear physics10-14. One of the primary challenges is to perform accurate calculations for systems where the interactions may be complicated and difficult for the chosen computational method to handle. Here we address the problem by introducing an approach called wavefunction matching. Wavefunction matching transforms the interaction between particles so that the wavefunctions up to some finite range match that of an easily computable interaction. This allows for calculations of systems that would otherwise be impossible owing to problems such as Monte Carlo sign cancellations. We apply the method to lattice Monte Carlo simulations15,16 of light nuclei, medium-mass nuclei, neutron matter and nuclear matter. We use high-fidelity chiral effective field theory interactions17,18 and find good agreement with empirical data. These results are accompanied by insights on the nuclear interactions that may help to resolve long-standing challenges in accurately reproducing nuclear binding energies, charge radii and nuclear-matter saturation in ab initio calculations19,20.

An updated nuclear-physics and multi-messenger astrophysics framework for binary neutron star mergers

The multi-messenger detection of the gravitational-wave signal GW170817, the corresponding kilonova AT2017gfo and the short gamma-ray burst GRB170817A, as well as the observed afterglow has delivered a scientific breakthrough. For an accurate interpretation of all these different messengers, one requires robust theoretical models that describe the emitted gravitational-wave, the electromagnetic emission, and dense matter reliably. In addition, one needs efficient and accurate computational tools to ensure a correct cross-correlation between the models and the observational data. For this purpose, we have developed the Nuclear-physics and Multi-Messenger Astrophysics framework NMMA. The code allows incorporation of nuclear-physics constraints at low densities as well as X-ray and radio observations of isolated neutron stars. In previous works, the NMMA code has allowed us to constrain the equation of state of supranuclear dense matter, to measure the Hubble constant, and to compare dense-matter physics probed in neutron-star mergers and in heavy-ion collisions, and to classify electromagnetic observations and perform model selection. Here, we show an extension of the NMMA code as a first attempt of analyzing the gravitational-wave signal, the kilonova, and the gamma-ray burst afterglow simultaneously. Incorporating all available information, we estimate the radius of a 1.4M⊙ neutron star to be [Formula: see text] km.

Strongly interacting matter exhibits deconfined behavior in massive neutron stars

Neutron-star cores contain matter at the highest densities in our Universe. This highly compressed matter may undergo a phase transition where nuclear matter melts into deconfined quark matter, liberating its constituent quarks and gluons. Quark matter exhibits an approximate conformal symmetry, predicting a specific form for its equation of state (EoS), but it is currently unknown whether the transition takes place inside at least some physical neutron stars. Here, we quantify this likelihood by combining information from astrophysical observations and theoretical calculations. Using Bayesian inference, we demonstrate that in the cores of maximally massive stars, the EoS is consistent with quark matter. We do this by establishing approximate conformal symmetry restoration with high credence at the highest densities probed and demonstrating that the number of active degrees of freedom is consistent with deconfined matter. The remaining likelihood is observed to correspond to EoSs exhibiting phase-transition-like behavior, treated as arbitrarily rapid crossovers in our framework.

QCD Equation of State of Dense Nuclear Matter from a Bayesian Analysis of Heavy-Ion Collision Data

Bayesian methods are used to constrain the density dependence of the QCD equation of state (EOS) for dense nuclear matter using the data of mean transverse kinetic energy and elliptic flow of protons from heavy ion collisions (HICs), in the beam energy range sqrt[s_{NN}]=2-10  GeV. The analysis yields tight constraints on the density dependent EOS up to 4 times the nuclear saturation density. The extracted EOS yields good agreement with other observables measured in HIC experiments and constraints from astrophysical observations both of which were not used in the inference. The sensitivity of inference to the choice of observables is also discussed.

Nuclear Equation of State for Arbitrary Proton Fraction and Temperature Based on Chiral Effective Field Theory and a Gaussian Process Emulator

We calculate the equation of state of asymmetric nuclear matter at finite temperature based on chiral effective field theory interactions to next-to-next-to-next-to-leading order. Our results assess the theoretical uncertainties from the many-body calculation and the chiral expansion. Using a Gaussian process emulator for the free energy, we derive the thermodynamic properties of matter through consistent derivatives and use the Gaussian process to access arbitrary proton fraction and temperature. This enables a first nonparametric calculation of the equation of state in beta equilibrium, and of the speed of sound and the symmetry energy at finite temperature. Moreover, our results show that the thermal part of the pressure decreases with increasing densities.

Constraining neutron-star matter with microscopic and macroscopic collisions

Interpreting high-energy, astrophysical phenomena, such as supernova explosions or neutron-star collisions, requires a robust understanding of matter at supranuclear densities. However, our knowledge about dense matter explored in the cores of neutron stars remains limited. Fortunately, dense matter is not probed only in astrophysical observations, but also in terrestrial heavy-ion collision experiments. Here we use Bayesian inference to combine data from astrophysical multi-messenger observations of neutron stars1-9 and from heavy-ion collisions of gold nuclei at relativistic energies10,11 with microscopic nuclear theory calculations12-17 to improve our understanding of dense matter. We find that the inclusion of heavy-ion collision data indicates an increase in the pressure in dense matter relative to previous analyses, shifting neutron-star radii towards larger values, consistent with recent observations by the Neutron Star Interior Composition Explorer mission5-8,18. Our findings show that constraints from heavy-ion collision experiments show a remarkable consistency with multi-messenger observations and provide complementary information on nuclear matter at intermediate densities. This work combines nuclear theory, nuclear experiment and astrophysical observations, and shows how joint analyses can shed light on the properties of neutron-rich supranuclear matter over the density range probed in neutron stars.

How Perturbative QCD Constrains the Equation of State at Neutron-Star Densities

We demonstrate in a general and analytic way how high-density information about the equation of state (EOS) of strongly interacting matter obtained using perturbative quantum chromodynamics constrains the same EOS at densities reachable in physical neutron stars. Our approach is based on utilizing the full information of the thermodynamic potentials at the high-density limit together with thermodynamic stability and causality. This requires considering the pressure as a function of chemical potential p(μ) instead of the commonly used pressure as a function of energy density p(ε). The results can be used to propagate the perturbative quantum chromodynamics calculations reliable around 40n_{s} to lower densities in the most conservative way possible. We constrain the EOS starting from only a few times the nuclear saturation density n≳2.2n_{s}, and at n=5n_{s} we exclude at least 65% of otherwise allowed area in the ε-p plane. This provides information complementary to astrophysical observations that should be taken into account in any complete statistical inference study of the EOS. These purely theoretical results are independent of astrophysical neutron-star input, and hence, they can also be used to test theories of modified gravity and beyond the standard model physics in neutron stars.

The Equation of State of Neutron-Rich Matter at Fourth Order of Chiral Effective Field Theory and the Radius of a Medium-Mass Neutron Star

We report neutron star predictions based on our most recent equations of state. These are derived from chiral effective field theory, which allows for a systematic development of nuclear forces, order by order. We utilize high-quality two-nucleon interactions and include all three-nucleon forces up to fourth order in the chiral expansion. Our ab initio predictions are restricted to the domain of applicability of chiral effective field theory. However, stellar matter in the interior of neutron stars can be up to several times denser than normal nuclear matter at saturation, and its composition is essentially unknown. Following established practices, we extend our microscopic predictions to higher densities matching piecewise polytropes. The radius of the average-size neutron star, about 1.4 solar masses, is sensitive to the pressure at normal densities, and thus it is suitable to constrain ab initio theories of the equation of state. For this reason, we focus on the radius of medium-mass stars. We compare our results with other theoretical predictions and recent constraints.

Translating Neutron Star Observations to Nuclear Symmetry Energy via Deep Neural Networks

One of the most significant challenges involved in efforts to understand the equation of state of dense neutron-rich matter is the uncertain density dependence of the nuclear symmetry energy. In particular, the nuclear symmetry energy is still rather poorly constrained, especially at high densities. On the other hand, detailed knowledge of the equation of state is critical for our understanding of many important phenomena in the nuclear terrestrial laboratories and the cosmos. Because of its broad impact, pinning down the density dependence of the nuclear symmetry energy has been a long-standing goal of both nuclear physics and astrophysics. Recent observations of neutron stars, in both electromagnetic and gravitational-wave spectra, have already constrained significantly the nuclear symmetry energy at high densities. The next generation of telescopes and gravitational-wave observatories will provide an unprecedented wealth of detailed observations of neutron stars, which will improve further our knowledge of the density dependence of nuclear symmetry energy, and the underlying equation of state of dense neutron-rich matter. Training deep neural networks to learn a computationally efficient representation of the mapping between astrophysical observables of neutron stars, such as masses, radii, and tidal deformabilities, and the nuclear symmetry energy allows its density dependence to be determined reliably and accurately. In this work, we use a deep learning approach to determine the nuclear symmetry energy as a function of density directly from observational neutron star data. We show, for the first time, that artificial neural networks can precisely reconstruct the nuclear symmetry energy from a set of available neutron star observables, such as masses and radii as measured by, e.g., the NICER mission, or masses and tidal deformabilities as measured by the LIGO/VIRGO/KAGRA gravitational-wave detectors. These results demonstrate the potential of artificial neural networks to reconstruct the symmetry energy and the equation of state directly from neutron star observational data, and emphasize the importance of the deep learning approach in the era of multi-messenger astrophysics.

Astrophysical Constraints on the Symmetry Energy and the Neutron Skin of ^{208}Pb with Minimal Modeling Assumptions

The symmetry energy and its density dependence are crucial inputs for many nuclear physics and astrophysics applications, as they determine properties ranging from the neutron-skin thickness of nuclei to the crust thickness and the radius of neutron stars. Recently, PREX-II reported a value of 0.283±0.071  fm for the neutron-skin thickness of ^{208}Pb, implying a slope parameter L=106±37  MeV, larger than most ranges obtained from microscopic calculations and other nuclear experiments. We use a nonparametric equation of state representation based on Gaussian processes to constrain the symmetry energy S_{0}, L, and R_{skin}^{^{208}Pb} directly from observations of neutron stars with minimal modeling assumptions. The resulting astrophysical constraints from heavy pulsar masses, LIGO/Virgo, and NICER clearly favor smaller values of the neutron skin and L, as well as negative symmetry incompressibilities. Combining astrophysical data with PREX-II and chiral effective field theory constraints yields S_{0}=33.0_{-1.8}^{+2.0}  MeV, L=53_{-15}^{+14}  MeV, and R_{skin}^{^{208}Pb}=0.17_{-0.04}^{+0.04}  fm.

Global Microscopic Description of Nucleon-Nucleus Scattering with Quantified Uncertainties

We develop for the first time a microscopic global nucleon-nucleus optical potential with quantified uncertainties suitable for analyzing nuclear reaction experiments at next-generation rare-isotope beam facilities. Within the improved local density approximation and without any adjustable parameters, we begin by computing proton-nucleus and neutron-nucleus optical potentials from a set of five nuclear forces from chiral effective field theory for 1800 target nuclei in the mass range 12≤A≤242 for energies between 0  MeV<E≲150  MeV. We then parameterize a global optical potential for each chiral force that depends smoothly on the projectile energy as well as the target nucleus mass number and isospin asymmetry. Uncertainty bands for elastic scattering observables are generated from a full covariance analysis of the parameters entering in the description of our global optical potential and benchmarked against existing experimental data for stable target nuclei. Since our approach is purely microscopic, we anticipate a similar quality of the model for nucleon scattering on unstable isotopes.

How Well Do We Know the Neutron-Matter Equation of State at the Densities Inside Neutron Stars? A Bayesian Approach with Correlated Uncertainties

We introduce a new framework for quantifying correlated uncertainties of the infinite-matter equation of state derived from chiral effective field theory (χEFT). Bayesian machine learning via Gaussian processes with physics-based hyperparameters allows us to efficiently quantify and propagate theoretical uncertainties of the equation of state, such as χEFT truncation errors, to derived quantities. We apply this framework to state-of-the-art many-body perturbation theory calculations with nucleon-nucleon and three-nucleon interactions up to fourth order in the χEFT expansion. This produces the first statistically robust uncertainty estimates for key quantities of neutron stars. We give results up to twice nuclear saturation density for the energy per particle, pressure, and speed of sound of neutron matter, as well as for the nuclear symmetry energy and its derivative. At nuclear saturation density, the predicted symmetry energy and its slope are consistent with experimental constraints.

Symmetric Nuclear Matter from the Strong Interaction

We study the equation of state of symmetric nuclear matter at zero temperature over a wide range of densities using two complementary theoretical approaches. At low densities, up to twice nuclear saturation density, we compute the energy per particle based on modern nucleon-nucleon and three-nucleon interactions derived within chiral effective field theory. For higher densities, we derive for the first time constraints in a Fierz-complete setting directly based on quantum chromodynamics using functional renormalization group techniques. We find remarkable consistency of the results obtained from both approaches as they come together in density and the natural emergence of a maximum in the speed of sound c_{S} at supranuclear densities. The presence of this maximum appears tightly connected to the formation of a diquark gap. Notably, this maximum is observed to exceed the asymptotic value c_{S}^{2}=1/3 while its exact position in terms of the density cannot yet be determined conclusively.

Chiral Interactions up to Next-to-Next-to-Next-to-Leading Order and Nuclear Saturation

We present an efficient Monte Carlo framework for perturbative calculations of infinite nuclear matter based on chiral two-, three-, and four-nucleon interactions. The method enables the incorporation of all many-body contributions in a straightforward and transparent way, and makes it possible to extract systematic uncertainty estimates by performing order-by-order calculations in the chiral expansion as well as the many-body expansion. The versatility of this new framework is demonstrated by applying it to chiral low-momentum interactions, exhibiting a very good many-body convergence up to fourth order. Following these benchmarks, we explore new chiral interactions up to next-to-next-to-next-to-leading order (N^{3}LO). Remarkably, simultaneous fits to the triton and to saturation properties can be achieved, while all three-nucleon low-energy couplings remain natural. The theoretical uncertainties of nuclear matter are significantly reduced when going from next-to-next-to-leading order to N^{3}LO.

Tidal Deformabilities and Radii of Neutron Stars from the Observation of GW170817

We use gravitational-wave observations of the binary neutron star merger GW170817 to explore the tidal deformabilities and radii of neutron stars. We perform a Bayesian parameter estimation with the source location and distance informed by electromagnetic observations. We also assume that the two stars have the same equation of state; we demonstrate that, for stars with masses comparable to the component masses of GW170817, this is effectively implemented by assuming that the stars' dimensionless tidal deformabilities are determined by the binary's mass ratio q by Λ_{1}/Λ_{2}=q^{6}. We investigate different choices of prior on the component masses of the neutron stars. We find that the tidal deformability and 90% credible interval is Λ[over ˜]=222_{-138}^{+420} for a uniform component mass prior, Λ[over ˜]=245_{-151}^{+453} for a component mass prior informed by radio observations of Galactic double neutron stars, and Λ[over ˜]=233_{-144}^{+448} for a component mass prior informed by radio pulsars. We find a robust measurement of the common areal radius of the neutron stars across all mass priors of 8.9≤R[over ^]≤13.2  km, with a mean value of ⟨R[over ^]⟩=10.8  km. Our results are the first measurement of tidal deformability with a physical constraint on the star's equation of state and place the first lower bounds on the deformability and areal radii of neutron stars using gravitational waves.

Neutron Star Tidal Deformabilities Constrained by Nuclear Theory and Experiment

We confront observational data from gravitational wave event GW170817 with microscopic modeling of the cold neutron star equation of state. We develop and employ a Bayesian statistical framework that enables us to implement constraints on the equation of state from laboratory measurements of nuclei and state-of-the-art chiral effective field theory methods. The energy density functionals constructed from the posterior probability distributions are then used to compute consistently the neutron star equation of state from the outer crust to the inner core, assuming a composition consisting of protons, neutrons, electrons, and muons. In contrast to previous studies, we find that the 95% credibility range of predicted neutron star tidal deformabilities (136<Λ<519) for a 1.4 solar-mass neutron star is already consistent with the upper bound deduced from observations of the GW170817 event. However, we find that lower bounds on the neutron star tidal deformability will very strongly constrain microscopic models of the dense matter equation of state. We also demonstrate a strong correlation between the neutron star tidal deformability and the pressure of beta-equilibrated matter at twice saturation density.

Gravitational-Wave Constraints on the Neutron-Star-Matter Equation of State

The detection of gravitational waves originating from a neutron-star merger, GW170817, by the LIGO and Virgo Collaborations has recently provided new stringent limits on the tidal deformabilities of the stars involved in the collision. Combining this measurement with the existence of two-solar-mass stars, we generate a generic family of neutron-star-matter equations of state (EOSs) that interpolate between state-of-the-art theoretical results at low and high baryon density. Comparing the results to ones obtained without the tidal-deformability constraint, we witness a dramatic reduction in the family of allowed EOSs. Based on our analysis, we conclude that the maximal radius of a 1.4-solar-mass neutron star is 13.6 km, and that the smallest allowed tidal deformability of a similar-mass star is Λ(1.4  M_{⊙})=120.

Cool Quark Matter

We generalize the state-of-the-art perturbative equation of state of cold quark matter to nonzero temperatures, needed in the description of neutron star mergers and core collapse processes. The new result is accurate to O(g^{5}) in the gauge coupling, and is based on a novel framework for dealing with the infrared sensitive soft field modes of the theory. The zero Matsubara mode sector is treated via a dimensionally reduced effective theory, while the soft nonzero modes are resummed using the hard thermal loop approximation. This combination of known effective descriptions offers unprecedented access to small but nonzero temperatures, both in and out of beta equilibrium.

Holographic Quark Matter and Neutron Stars

We use a top-down holographic model for strongly interacting quark matter to study the properties of neutron stars. When the corresponding equation of state (EOS) is matched with state-of-the-art results for dense nuclear matter, we consistently observe a first-order phase transition at densities between 2 and 7 times the nuclear saturation density. Solving the Tolman-Oppenheimer-Volkov equations with the resulting hybrid EOSs, we find maximal stellar masses in excess of two solar masses, albeit somewhat smaller than those obtained with simple extrapolations of the nuclear matter EOSs. Our calculation predicts that no quark matter exists inside neutron stars.

Static Response of Neutron Matter

We generalize the problem of strongly interacting neutron matter by adding a periodic external modulation. This allows us to study from first principles a neutron system that is extended and inhomogeneous, with connections to the physics of both neutron-star crusts and neutron-rich nuclei. We carry out fully nonperturbative microscopic quantum Monte Carlo calculations of the energy of neutron matter at different densities, as well as different strengths and periodicities of the external potential. In order to remove systematic errors, we examine finite-size effects and the impact of the wave function ansatz. We also make contact with energy-density functional theories of nuclei and disentangle isovector gradient contributions from bulk properties. Finally, we calculate the static density-density linear response function of neutron matter and compare it with the response of other physical systems.

Chiral Three-Nucleon Interactions in Light Nuclei, Neutron-α Scattering, and Neutron Matter

We present quantum Monte Carlo calculations of light nuclei, neutron-α scattering, and neutron matter using local two- and three-nucleon (3N) interactions derived from chiral effective field theory up to next-to-next-to-leading order (N(2)LO). The two undetermined 3N low-energy couplings are fit to the (4)He binding energy and, for the first time, to the spin-orbit splitting in the neutron-α P-wave phase shifts. Furthermore, we investigate different choices of local 3N-operator structures and find that chiral interactions at N(2)LO are able to simultaneously reproduce the properties of A=3,4,5 systems and of neutron matter, in contrast to commonly used phenomenological 3N interactions.

Nuclear matter from effective quark-quark interaction

We study neutron matter and symmetric nuclear matter with the quark-meson model for the two-nucleon interaction. The Bethe-Bruckner-Goldstone many-body theory is used to describe the correlations up to the three hole-line approximation with no extra parameters. At variance with other nonrelativistic realistic interactions, the three hole-line contribution turns out to be non-negligible and to have a substantial saturation effect. The saturation point of nuclear matter, the compressibility, the symmetry energy, and its slope are within the phenomenological constraints. Since the interaction also reproduces fairly well the properties of the three-nucleon system, these results indicate that the explicit introduction of the quark degrees of freedom within the considered constituent quark model is expected to reduce the role of three-body forces.

Auxiliary-field quantum Monte Carlo simulations of neutron matter in chiral effective field theory

We present variational Monte Carlo calculations of the neutron matter equation of state using chiral nuclear forces. The ground-state wave function of neutron matter, containing nonperturbative many-body correlations, is obtained from auxiliary-field quantum Monte Carlo simulations of up to about 340 neutrons interacting on a 10(3) discretized lattice. The evolution Hamiltonian is chosen to be attractive and spin independent in order to avoid the fermion sign problem and is constructed to best reproduce broad features of the chiral nuclear force. This is facilitated by choosing a lattice spacing of 1.5 fm, corresponding to a momentum-space cutoff of Λ=414  MeV/c, a resolution scale at which strongly repulsive features of nuclear two-body forces are suppressed. Differences between the evolution potential and the full chiral nuclear interaction (Entem and Machleidt Λ=414  MeV [L. Coraggio et al., Phys. Rev. C 87, 014322 (2013).

Quantum Monte Carlo calculations of neutron matter with nonlocal chiral interactions

We present fully nonperturbative quantum Monte Carlo calculations with nonlocal chiral effective field theory (EFT) interactions for the ground-state properties of neutron matter. The equation of state, the nucleon chemical potentials, and the momentum distribution in pure neutron matter up to one and a half times the nuclear saturation density are computed with a newly optimized chiral EFT interaction at next-to-next-to-leading order. This work opens the way to systematic order by order benchmarking of chiral EFT interactions and ab initio prediction of nuclear properties while respecting the symmetries of quantum chromodynamics.

Quantum Monte Carlo calculations with chiral effective field theory interactions

We present the first quantum Monte Carlo (QMC) calculations with chiral effective field theory (EFT) interactions. To achieve this, we remove all sources of nonlocality, which hamper the inclusion in QMC calculations, in nuclear forces to next-to-next-to-leading order. We perform auxiliary-field diffusion Monte Carlo (AFDMC) calculations for the neutron matter energy up to saturation density based on local leading-order, next-to-leading order, and next-to-next-to-leading order nucleon-nucleon interactions. Our results exhibit a systematic order-by-order convergence in chiral EFT and provide nonperturbative benchmarks with theoretical uncertainties. For the softer interactions, perturbative calculations are in excellent agreement with the AFDMC results. This work paves the way for QMC calculations with systematic chiral EFT interactions for nuclei and nuclear matter, for testing the perturbativeness of different orders, and allows for matching to lattice QCD results by varying the pion mass.

Optimized chiral nucleon-nucleon interaction at next-to-next-to-leading order

We optimize the nucleon-nucleon interaction from chiral effective field theory at next-to-next-to-leading order (NNLO). The resulting new chiral force NNLO(opt) yields χ(2)≈1 per degree of freedom for laboratory energies below approximately 125 MeV. In the A=3, 4 nucleon systems, the contributions of three-nucleon forces are smaller than for previous parametrizations of chiral interactions. We use NNLO(opt) to study properties of key nuclei and neutron matter, and we demonstrate that many aspects of nuclear structure can be understood in terms of this nucleon-nucleon interaction, without explicitly invoking three-nucleon forces.