Chiral Effective Field Theory and the High-Density Nuclear Equation of State
CChiral Effective Field
Theory and theHigh-Density Nuclear
Equation of State
C. Drischler, , , J. W. Holt, andC. Wellenhofer, , Department of Physics, University of California, Berkeley, California 94720,USA Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley,California 94720, USA Facility for Rare Isotope Beams, Michigan State University, Michigan 48824,USA; email: [email protected] Cyclotron Institute and Department of Physics and Astronomy, Texas A&MUniversity, College Station, Texas 77843, USA; email: [email protected] Institut f¨ur Kernphysik, Technische Universit¨at Darmstadt, 64289 Darmstadt,Germany; email: [email protected] ExtreMe Matter Institute EMMI, GSI Helmholtzzentrum f¨urSchwerionenforschung GmbH, 64291 Darmstadt, GermanyAnnu. Rev. Nucl. Part. Sci. 2021. 71:1–29This article’s doi:10.1146/annurev-nucl-102419-041903Copyright © Keywords chiral effective field theory, nuclear matter, neutron stars, many-bodyperturbation theory, bayesian uncertainty quantification
Abstract
Recent advances in neutron star observations have the potential to con-strain the properties of strongly interacting matter at extreme densi-ties and temperatures that are otherwise difficult to access through di-rect experimental investigation. At the same time, chiral effective fieldtheory has developed into a powerful theoretical framework to studynuclear interactions and nuclear matter properties with quantified un-certainties in the regime of astrophysical interest for modeling neutronstars. In this article, we review recent developments in the chiral effec-tive field theory approach to constructing microscopic nuclear forces andfocus on many-body perturbation theory as a computationally efficienttool for calculating the structure, phases, and linear-response proper-ties of hot and dense nuclear matter. We also demonstrate how effec-tive field theory combined with Bayesian methods enables statisticallymeaningful comparisons between nuclear theory predictions, nuclearexperiments, and observational constraints on the nuclear equation ofstate. a r X i v : . [ nu c l - t h ] J a n ontents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22. From microscopic interactions to the nuclear equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.1. Chiral effective field theory for nuclear forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2. Perturbative chiral nuclear interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.3. Many-body perturbation theory at zero and finite temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.4. Other many-body methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.5. Implementing three-body forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103. Nuclear equation of state at zero and finite temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1. Confronting nuclear forces with empirical constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2. Nuclear symmetry energy and the isospin-asymmetry expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143.3. Nuclear thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174. Applications to neutron star physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.1. Scales in hot and dense stellar matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.2. Neutron star structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3. Neutron star mergers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.4. Core-collapse supernovae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 245. Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1. Introduction
Neutron stars are one of Nature’s most intriguing astronomical objects and provide a uniquelaboratory for studying strongly interacting, neutron-rich matter under extreme conditions.With masses ≈ − ≈
10 km, neutron starscontain the densest form of matter in the observable Universe and lie just at the thresholdfor collapse to a black hole. Much has already been learned about neutron stars throughmass and radius measurements, pulsar timing, x-ray observations, and gravitational-wavemeasurements of binary mergers in the new era of multimessenger astronomy (see, e.g. ,Refs. (1, 2, 3) for reviews). But many interesting questions remain to be answered, especiallyregarding the nature of ultra-compressed matter located in the inner cores of heavy neutronstars where a variety of exotic new states of matter have been theorized to exist.
The mass of the Sunis M (cid:12) ≈ × kg. While neutron stars are bound together by gravity acting over macroscopic length scales,strong short-ranged nuclear interactions provide the essential pressure support to counteractgravitational collapse. The central densities in the heaviest neutron stars may reach up to5 − n , where n ≈ .
16 fm − is the nucleon number density typical of heavy atomic nuclei(the associated mass density is ρ ≈ . × g cm − ). Although the strong interaction isin principle described by quantum chromodynamics (QCD) over all relevant energy scales,at present no systematic computational method is available to calculate the properties ofthe high-density matter in the inner cores of heavy neutron stars. QCD: quantumchromodynamics
ChEFT: chiraleffective field theory
With chiral effective field theory (ChEFT), however, a powerful tool has emerged tocarry out microscopic calculations of nuclear matter properties at densities up to around2 n . Instead of QCD’s quarks and gluons, ChEFT is formulated in terms of nucleons andpions (and delta isobars), which are the effective strong interaction degrees of freedompresent throughout most of the neutron star interior. In its range of validity, ChEFTprovides a systematic expansion for two- and multi-nucleon interactions consistent with thesymmetries of low-energy QCD. The unresolved short-distance physics is parameterized in erms of contact interactions whose low-energy couplings are fitted to experimental data.An essential advantage over phenomenological approaches is that theoretical uncertaintiescan be quantified by analyzing the order-by-order convergence of the ChEFT expansion. Inthe last few years, the combination of systematic nuclear matter predictions from ChEFT,uncertainty quantification, and neutron star observations has developed into a new avenuefor constraining the high-density regime of the nuclear equation of state (EOS). EOS: equation ofstate
MBPT: many-bodyperturbation theory
In this review our aim is to describe recent advances in microscopic ChEFT calculationsof the nuclear EOS and their application to neutron stars. We highlight many-body per-turbation theory (MBPT) as an efficient framework for nuclear matter calculations at zeroand finite temperature based on chiral two- and multi-nucleon interactions. We also dis-cuss Bayesian methods for quantifying and propagating statistically meaningful theoreticaluncertainties. Together with nuclear experiments, astrophysical simulations, and neutronstar observations, next-generation ChEFT calculations will be crucial to infer the nature ofthe extreme matter hidden deep beneath the surface of neutron stars.
See, e.g. ,Refs. (4, 5, 6, 7) forrecent reviews ofnuclear structurecalculations withChEFT.
The review is organized as follows. In Section 2 we focus on recent progress in derivingnuclear forces from ChEFT and renormalization group (RG) methods to improve the many-body convergence in nuclear matter calculations. We then dedicate Section 3 to recenthigh-order MBPT calculations of the moderate-density nuclear EOS at zero temperatureand advances in the Bayesian quantification of EFT truncation errors. We also discussfinite-temperature calculations and nuclear thermodynamics. In Section 4 we review thepresent status of the high-density nuclear EOS constrained by nuclear theory, experiment,and observation in the era of multimessenger astronomy, emphasizing the importance ofChEFT. Section 5 ends the review with our summary and perspectives on future advancesin nuclear matter calculations and their applications to astrophysics.
RG: renormalizationgroup
2. From microscopic interactions to the nuclear equation of state
In this section, we briefly review delta-less ChEFT and the construction of chiral nuclearinteractions as microscopic input for many-body calculations. Applying RG methods allowsone to systematically generate (perturbative) low-momentum interactions, for which thenuclear EOS and related observables can be efficiently calculated using MBPT. We discussboth zero- and finite-temperature MBPT, complementary many-body approaches, and theimplementation of 3N interactions in nuclear matter calculations.
The interactions among nucleons arise as an effective low-energy phenomenon of QCD, thetheory of the strong interaction. At the momentum scales relevant for nuclear physics, p ∼ m π , QCD is strongly coupled and features nonperturbative effects such as sponta-neous chiral symmetry breaking and the confinement of quarks and gluons into hadrons.Direct applications of QCD to hadronic physics at finite density, where lattice QCD faces aformidable sign problem, are therefore extremely challenging and not feasible at present andin the near future. However, one can construct a systematic description of nuclear physicsin terms of the effective degrees of freedom at low energies: nucleons and pions (and deltaisobars). This effective description is given by ChEFT (8, 9, 10, 11, 12).The starting point of ChEFT is to write down the most general Lagrangian consistentwith the symmetries of low-energy QCD, in particular, the spontaneously broken chiral ••
The interactions among nucleons arise as an effective low-energy phenomenon of QCD, thetheory of the strong interaction. At the momentum scales relevant for nuclear physics, p ∼ m π , QCD is strongly coupled and features nonperturbative effects such as sponta-neous chiral symmetry breaking and the confinement of quarks and gluons into hadrons.Direct applications of QCD to hadronic physics at finite density, where lattice QCD faces aformidable sign problem, are therefore extremely challenging and not feasible at present andin the near future. However, one can construct a systematic description of nuclear physicsin terms of the effective degrees of freedom at low energies: nucleons and pions (and deltaisobars). This effective description is given by ChEFT (8, 9, 10, 11, 12).The starting point of ChEFT is to write down the most general Lagrangian consistentwith the symmetries of low-energy QCD, in particular, the spontaneously broken chiral •• Chiral EFT and the High-Density Nuclear EOS 3 ymmetry, for which pions are the (pseudo) Nambu-Goldstone bosons. This naturally sets alimit for the applicability of ChEFT, i.e. , the breakdown scale Λ b will be of order of the chiralsymmetry breaking scale Λ χ ∼ separation of scales inherent inChEFT, i.e. , the power counting is according to powers of a typical momentum p (or thepion mass) over the ChEFT breakdown scale, Q = max( p, m π ) / Λ b . A Bayesian analysisof free-space NNscattering withseveral (not-too-soft)chiral NN potentialsin Weinberg powercounting estimatedΛ b ≈
600 MeV (13).
In perturbative EFT, both power counting and ultraviolet renormalization are essen-tially unambiguous and straightforward. The situation is different for applications ofChEFT in nuclear physics, where the calculational framework must be able to accountfor nonperturbative effects such as bound states (atomic nuclei) and large S -wave scatter-ing lengths in NN scattering. While there has been some controversy in the literature asto how the ChEFT expansion should be set up precisely (see, e.g. , Refs. (10, 12) and refer-ences therein), the prevalent and most successful power counting for ChEFT (in particularregarding many-body applications) is the one first suggested by Weinberg. NN: nucleon-nucleon three-nucleon four-nucleon
Within Weinberg power counting, chiral nuclear interactions (and currents) are orga-nized according to naive ( i.e. , perturbative) dimensional analysis. The nuclear potentialsconstructed at a given truncation order in the ChEFT expansion are then used for com-puting observables. Renormalization in this approach is approximative, and carried out byequipping the potentials with regulator functions that suppress contributions above a cutoffscale Λ (cid:46) Λ b , typically chosen in the range 450 −
600 MeV. That is, the cutoff independenceof the observables will be achieved only approximatively through Λ-dependent low-energyconstants (LECs), which have to be fit to experimental data at a given scale. The residualcutoff dependence can then be attributed to higher-order terms in the expansion, so resultsare expected to become less cutoff dependent with increasing truncation order.
LECs: low-energyconstants are inpractice optimizedfor a given value ofΛ to reproducelow-energy NNscattering data andfew-nucleonobservables, see, e.g. , Refs. (12, 14).
Figure 1 depicts the hierarchy of nuclear interactions up to fifth order (or N LO) inthe chiral expansion without delta isobars. At each order the interactions are composedof short-range contact interactions as well as one- and multi-pion exchanges at long- andintermediate distances, respectively. The LECs associated with pion exchanges have re-cently been determined with high precision through an analysis of pion-nucleon scatteringwithin the framework of Roy-Steiner equations (15). The short-range LECs correspondingto NN couplings are generally fixed by matching to NN scattering data. Figure 1 showsthat ChEFT naturally predicts the observed hierarchy of two- and multi-nucleon interac-tions, i.e. , V NN > V > V , etc. The first nonvanishing 3N forces appear at N LO inthree topologies; from left to right: the long-range two-pion exchange (involving the pion-nucleon LECs c , c , and c ), intermediate-range one-pion exchange-contact ( ∝ c D ), andshort-range 3N contact interaction ( ∝ c E ). At N LO the 3N forces are significantly moreinvolved and operator-rich, and also 4N interactions start to contribute. Apart from thetwo N LO 3N LECs, c D and c E , chiral interactions up to N LO are completely determinedby the π N and NN system. While N LO NN forces have already been worked out, partlyeven at N LO, the derivation of N LO 3N interactions has not been finished yet. The3N LECs c D and c E can be fit to (uncorrelated) few-body observables; for instance, the H binding energy combined with, e.g. , the charge radius of He, the H β -decay half-life,or the nucleon-deuteron scattering cross section. Also heavier nuclei and even saturationproperties in infinite nuclear matter have been used to constrain 3N forces. ChEFT with explicitdelta isobars iscurrently lessdeveloped than thedelta-less version wefocus on here. Forrecent work ondelta-full ChEFT,see, e.g. ,Refs. (16, 17). N k LO: (next-to) k leading order Although the residual regulator and cutoff dependence of obervables at a given chiralorder is expected to decrease at higher orders, actual calculations show significant influence LO (Q )N LO (Q )N LO (Q )NLO (Q )LO (Q ) 3N forces 4N forcesNN forces Figure 1: Hierarchy of chiral nuclear interactions up to fifth order (or N LO) in the chiralexpansion without delta isobars (12). Nucleons (pions) are depicted by solid (dashed) lines.The annotation gives the number of short-range contact LECs. See the main text for details.of these so-called regulator artifacts on the ChEFT convergence depending on the specificregularization scheme and computational framework. These issues have resulted in thedevelopment of a flurry of chiral potentials with nonlocal, local as well as semilocal regulatorsfor a range of cutoff values; see, e.g. , Table I of Ref. (18). Moreover, as discussed inSection 2.2, RG methods allow one to modify a given set of two- and multi-nucleon potentialssuch that observables are left invariant (up to RG truncations) but the convergence of many-body calculations is optimized. These RG transformations are most suitably formulatedat the operator ( i.e. , Hamiltonian) level. The nuclear Hamiltonian constructed at a givenorder in the ChEFT expansion reads H = T kin + V NN (Λ , c i ) + V (Λ , c i ) + V (Λ , c i ) + . . . , where Λ stands for the (initial) cutoff or resolution scale, and c i for the set of LECs inferredfrom fits to experimental data. The nuclear Hamiltonian is not an observable, and the basicidea of the RG is to exploit this feature to generate more perturbative Hamiltonians. The strong short-range repulsion (“hard core”) and tensor force found in nuclear potentialsconstructed at cutoff scales Λ (cid:38)
500 MeV question the applicability of perturbation theoryfor many-body calculations. In fact, nuclear many-body calculations were historically con-sidered a nonperturbative problem (see also Section 2.3). Both features give rise to strongcouplings between high- and low-momentum states, i.e. , large off-diagonal matrix elements,which enhance the intermediate-state summations in perturbation theory. RG methodsallow one to amend this feature while preserving nonperturbative few-body results.The initial application (19) of RG methods to study the scale dependence of nuclearforces was based on T -matrix equivalence, but in recent years the similarity renormalization ••
500 MeV question the applicability of perturbation theoryfor many-body calculations. In fact, nuclear many-body calculations were historically con-sidered a nonperturbative problem (see also Section 2.3). Both features give rise to strongcouplings between high- and low-momentum states, i.e. , large off-diagonal matrix elements,which enhance the intermediate-state summations in perturbation theory. RG methodsallow one to amend this feature while preserving nonperturbative few-body results.The initial application (19) of RG methods to study the scale dependence of nuclearforces was based on T -matrix equivalence, but in recent years the similarity renormalization •• Chiral EFT and the High-Density Nuclear EOS 5 roup (SRG) has been the standard RG method for “softening” nuclear interactions. TheSRG decouples high- and low-momentum states through continuous infinitesimal unitarytransformations, H s = U s HU † s , described by a differential flow equation in the evolutionparameter s . As the SRG flow progresses, the matrix-elements of the NN potential aredriven toward a band-diagonal (or block-diagonal) form in momentum space, see Ref. (20)for illustrations. While NN observables are by construction invariant under any RG evo-lution of the NN potential, A -body observables will remain so only if one also consistentlyevolves the multi-nucleon part of the nuclear Hamiltonian. The SRG allows one to imple-ment this in principle exactly, in terms of so-called “induced” many-body forces. However,for practical applications, a truncation of the consistent evolution of multi-nucleon inter-actions is required, e.g. , at the 3N level. Neutron matter calculations with SRG-evolvedchiral interactions truncated at the 3N level indicate that induced 4N forces are (in thatcase) negligible within uncertainties for a wide range of SRG resolution scales (21, 22). SRG: similarityrenormalizationgroup
There have been many developments recently in the application of ChEFT and SRGtechnology for the construction of high-precision nuclear potentials. Hebeler et al. (23)explored a set of low-momentum N LO NN potentials combined with unevolved N LO 3Nforces where the two 3N LECs were fit to reproduce few-body data (assuming that the3N contact interactions capture dominant contributions from induced 3N forces). For thesoftest of these potentials (with λ = 1 . = 2 . − ), which was found to predictnuclear saturation properties (24) and ground-state energies of light- to medium-mass nu-clei in agreement with experiment (25), Stroberg et al. (26) have computed ground-stateand separation energies of nearly 700 isotopes up to iron. Moreover, H¨uther et al. (27)have constructed a family of SRG-evolved NN and 3N potentials up to N LO. Also,Reinert et al. (14) have developed the first chiral NN potentials up to N LO with semilocalregulators in momentum space. They showed that several N LO contact terms present inprevious generations of chiral NN potentials can be eliminated using unitary transforma-tions, leading to considerably softer potentials (even without SRG evolution).
The semilocalregulators used byReinert et al. leavethe long-rage part ofpion-exchangeinvariant, arguably abenefit compared tononlocal regulators.
The Weinberg eigenvalue analysis (28, 18) is a powerful tool for quantifying and mon-itoring the perturbativeness of nuclear forces at different resolution scales. Given an NNpotential, the Weinberg eigenvalues η ν ( W ) of the operator G ( W ) V NN determine the (rateof) convergence of the Born series for NN scattering. Here, G ( W ) is the (free-space orin-medium) propagator as a function of the complex energy W . The Born series convergesif and only if all eigenvalues satisfy | η ν ( W ) | <
1. Bound states of the potential (such asthe deuteron) correspond to η ν ( W ) = 1 at energies W <
0, so the free-space Born seriesdiverges even for soft potentials. In nuclear matter at sufficiently high densities, however,Pauli blocking suppresses the (in-medium) eigenvalues associated with bound (or nearly-bound) states. For potentials constructed at Λ (cid:46)
550 MeV, other sources of nonperturbativebehavior (such as the hard core) are suppressed as well, both in free-space and in-medium(see, e.g. , Ref. (20) for details). This implies that a nonperturbative treatment of in-mediumNN scattering in the particle-particle channel (see Section 2.3) is not mandatory for theseinteractions. Instead, order-by-order MBPT calculations can be used for systematicallystudying the many-body convergence of (low-momentum) chiral nuclear interactions.
The free propagatoris given by G ( W ) =( W − H ) − , with H the kineticenergy operator.The in-mediumpropagator involvesalso Pauli-blocking. MBPT starts with partitioning the nuclear Hamiltonian H into a reference one-body part H = T kin + U and a perturbation H = V − U , where T kin is the kinetic-energy operator nd U is an effective single-particle potential. We consider here NN-only potentials anddiscuss the implementation of 3N interactions in Section 2.5. The standard choice for U is the Hartree-Fock potential given by U (HF) i = (cid:80) j V ij,ij f j , with the antisymmetrized NNmatrix elements V ij,ab = (cid:104) k i k j | (1 − P ) V NN | k a k b (cid:105) , the Pauli exchange operator P , themomentum integral (cid:80) j = (cid:82) d k j / (2 π ) , and the zero-temperature distribution function f j = θ ( k F − k j ). (For simplicity, we assume here a single-species system and neglect spin-isospin degrees of freedom.) In zero-temperature MBPT, the ground-state energy density E is obtained by expanding H int about its reference value E . Truncating the many-bodyexpansion at a finite order L then leads to the approximation E ( k F ) (cid:39) E ( k F )+ (cid:80) Ll =1 E l ( k F ),where the Fermi momentum k F is in one-to-one correspondence with the particle numberdensity via n ( k F ) = (cid:80) i f i ( k F ). The MBPT series isin fact a divergentasymptotic series,but the divergentbehavior is expectedto appear only forhigh truncationorders L (cid:38)
20 (29).
The first-order correction is determined by the expectation value of U (HF) i (30). Athigher orders it is useful to represent the contributions diagrammatically, e.g. , as Hugenholtzdiagrams. The diagram and expression for the second-order contribution E are given by E ( k F ) = = − (cid:88) ijab V ij,ab V ab,ij f ij ¯ f ab D ab,ij , (1)with the distribution functions f ij = f i f j (holes) and ¯ f ab = (1 − f a )(1 − f b ) (particles), energydenominator D ab,ij = ε a + ε b − ε i − ε j , and single-particle energies ε i = k i / (2 M ) + U (HF) i .Writing down the expression associated with a diagram follows these simple rules: In Hartree-FockMBPT, the − U partof H = V − U cancels all diagramsinvolvingsingle-vertexloops (30, 24, 22). • each vertex gives a factor V ij,ab , with i and j ( a and b ) being the lines directedtowards (away from) the vertex, • downwards lines give factors of f i while upwards lines give (1 − f i ), corresponding tohole and particle excitations of the reference ground-state, respectively, and • for adjacent vertices there is an energy denominator given by subtracting the energyof the reference ground-state from the excited state corresponding to the particle andhole lines that are crossed by a virtual horizontal line between the two vertices.Each diagram’s overall factor can be inferred from the diagrammatic structure as well (30).For instance, the expression of the third-order particle-particle (pp) diagram reads E ,pp ( k F ) = = 18 (cid:88) ijabcd V ij,ab V ab,cd V cd,ij f ij ¯ f abcd D ab,ij D cd,ij . (2)Finding all valid diagrams (and associated expressions) at a given MBPT order has beenformalized using graph-theory methods (31). Together with automated code generation forthe efficient Monte Carlo integration of arbitrary MBPT diagrams developed in Ref. (24),a fully automated approach to MBPT calculations has become available. Using graph-theorymethods one findsthat in Hartree-FockMBPT there are(1 , , , , l = (2 , , , , In the traditional Brueckner (or G -matrix) approach (32), the pp ladder diagrams areresummed to all orders, motivated by the large high-momentum components of traditionalNN potentials to which the pp ladders are particularly sensitive. The pp bubbles in thesediagrams are even ultraviolet divergent if the potential is not sufficiently suppressed at highmomenta. For modern low-momentum potentials, however, the pp ladders no longer playa distinguished role in the many-body expansion, and explicit MBPT calculations at thirdand fourth order have shown that they are not enhanced compared to other diagrams atthe same order (24). Nevertheless, partial diagrammatic resummations are still pertinent ••
The first-order correction is determined by the expectation value of U (HF) i (30). Athigher orders it is useful to represent the contributions diagrammatically, e.g. , as Hugenholtzdiagrams. The diagram and expression for the second-order contribution E are given by E ( k F ) = = − (cid:88) ijab V ij,ab V ab,ij f ij ¯ f ab D ab,ij , (1)with the distribution functions f ij = f i f j (holes) and ¯ f ab = (1 − f a )(1 − f b ) (particles), energydenominator D ab,ij = ε a + ε b − ε i − ε j , and single-particle energies ε i = k i / (2 M ) + U (HF) i .Writing down the expression associated with a diagram follows these simple rules: In Hartree-FockMBPT, the − U partof H = V − U cancels all diagramsinvolvingsingle-vertexloops (30, 24, 22). • each vertex gives a factor V ij,ab , with i and j ( a and b ) being the lines directedtowards (away from) the vertex, • downwards lines give factors of f i while upwards lines give (1 − f i ), corresponding tohole and particle excitations of the reference ground-state, respectively, and • for adjacent vertices there is an energy denominator given by subtracting the energyof the reference ground-state from the excited state corresponding to the particle andhole lines that are crossed by a virtual horizontal line between the two vertices.Each diagram’s overall factor can be inferred from the diagrammatic structure as well (30).For instance, the expression of the third-order particle-particle (pp) diagram reads E ,pp ( k F ) = = 18 (cid:88) ijabcd V ij,ab V ab,cd V cd,ij f ij ¯ f abcd D ab,ij D cd,ij . (2)Finding all valid diagrams (and associated expressions) at a given MBPT order has beenformalized using graph-theory methods (31). Together with automated code generation forthe efficient Monte Carlo integration of arbitrary MBPT diagrams developed in Ref. (24),a fully automated approach to MBPT calculations has become available. Using graph-theorymethods one findsthat in Hartree-FockMBPT there are(1 , , , , l = (2 , , , , In the traditional Brueckner (or G -matrix) approach (32), the pp ladder diagrams areresummed to all orders, motivated by the large high-momentum components of traditionalNN potentials to which the pp ladders are particularly sensitive. The pp bubbles in thesediagrams are even ultraviolet divergent if the potential is not sufficiently suppressed at highmomenta. For modern low-momentum potentials, however, the pp ladders no longer playa distinguished role in the many-body expansion, and explicit MBPT calculations at thirdand fourth order have shown that they are not enhanced compared to other diagrams atthe same order (24). Nevertheless, partial diagrammatic resummations are still pertinent •• Chiral EFT and the High-Density Nuclear EOS 7 or consistent calculations of in-medium single-particle properties and response functions asperformed in the self-consistent Green’s functions method (for more details see Section 2.4).The consistent generalization of MBPT to finite temperatures (
T >
0) is a nontrivialissue. From the standard finite- T perturbation series for the grand-canonical potentialΩ( T, µ ) (cid:39) Ω ( T, µ ) + L (cid:88) l =1 Ω l ( T, µ ) , (3)the free energy density F ( T, µ ) is obtained via the thermodynamic relation F ( T, µ ) =Ω(
T, µ ) + µ n ( T, µ ). Here, the density is given by n ( T, µ ) = − ∂ Ω( T, µ ) /∂µ . The issue isnow that the relations between ( F , n ) and ( E , n ) obtained in finite- and zero- T MBPT,respectively, do not match in the limit T → The use of thegrand-canonicalensemble is requiredfor the evaluation ofquantum-statisticalaverages in thethermodynamiclimit.
Regarding this, we first consider the finite- T expression for the second-order diagram,Ω ( T, µ ) = = − (cid:88) ijab V ij,ab V ab,ij f ij ¯ f ab G . (4)Equation (4) differs only slightly from E ( k F ) in Eq. (1). First, the energy denominator isreplaced by G = (1 − e − D ab,ij /T ) / (2 D ab,ij ). The numerator in this expression vanishes atany zero of the denominator, i.e. , there are no poles at finite T . In the T → G separate intotwo equivalent parts (with integrable poles at the integral boundaries), i.e. , G → /D ab,ij for T →
0. These features pertain for higher-order diagrams (33). The second difference atfinite T compared to T = 0 is that the f i = f i ( T, µ ) are Fermi-Dirac distributions insteadof step functions centered at the Fermi energy ε k F . The poles (at theintegral boundary)at T = 0 lead tononanalyticities inthe asymmetrydependence of thenuclear EOS, seeSection 3.2. Similar to the free Fermi gas ( i.e. , MBPT with U = 0 and L = 0), for Hartree-FockMBPT at L = 1, the chemical potential µ at T = 0 matches the reference Fermi energy ε k F ,with n ( T, µ ) = (cid:80) i f i ( T, µ ) (33). But these relations cease to be valid at higher orders dueto higher-order contributions in the expression for n ( T, µ ). Note that these contributionsinvolve factors ∂f i /∂µ = f i (1 − f i ) /T , which become δ ( ε i − µ ) at T = 0 (so there is anonvanishing contribution at T = 0). Contributions involving factors f i (1 − f i ) /T are alsopresent in certain perturbative contributions to Ω, starting at fourth order for Hartree-FockMBPT (33). [For U = 0, they appear already at second order.] These contributions can beassociated with the presence of additional so-called anomalous diagrams in finite- T MBPT,see Refs. (34, 33) for more details. As evident from the discussion above ( i.e. , below Eq. (4)),the T → T expressions for normal ( i.e. , not anomalous) contributions Ω l matches the corresponding zero- T contributions E l , except that the reference Fermi energyis replaced by the (true) chemical potential. Therefore, a consistent finite- T version ofHartree-Fock MBPT for L (cid:54) F ( T, ˜ µ ) (cid:39) F ( T, ˜ µ ) + L (cid:88) l =1 F l ( T, ˜ µ ) , (5)where F l = Ω l (for l = 1 , ,
3) and the auxiliary “chemical potential” ˜ µ is related to thedensity via n ( T, ˜ µ ) = (cid:80) i f i ( T, ˜ µ ), implying ˜ µ → ε k F in the T → The true chemicalpotential is obtainedfrom F ( T, ˜ µ ) via µ = ∂ F /∂n . In the U = 0 case, the method for constructing a finite- T perturbation series of the formof Eq. (5) for any L is well known (34): one expands each contribution to F ( T, µ ) about ˜ µ according to µ = ˜ µ + (cid:80) Ll =1 µ l ( T, ˜ µ ) while neglecting all terms beyond the truncation order . This process can also be applied to Hartree-Fock MBPT (22), with the caveat that thesingle-particle potential has to be evaluated at ˜ µ , i.e. , no derivatives of U (HF) i ( T, ˜ µ ) in ˜ µ appear. In both cases, U = 0 and Hartree-Fock MBPT, the resulting perturbation series forthe free energy reproduces zero- T MBPT at each truncation order L , even though the terms F l contain anomalous contributions for l (cid:62) l (cid:62)
2, for U = 0). The fact that Eq. (5) resultsfrom a truncated re-expansion shows explicitly that the original grand-canonical series is notconsistent with zero- T MBPT. For general arguments why the free-energy series is expectedto give improved results compared to grand-canonical MBPT, see Refs. (33, 35).
The terms µ l aredetermined by therequirement that thetruncated expansionof n ( T, µ ) about ˜ µ reproduces n ( T, ˜ µ ) = (cid:80) i f i ( T, ˜ µ ). Altogether, MBPT as formulated in the free-energy series [Eq. (5)] provides a consistentframework for nuclear matter calculations at zero- and finite-temperature, where many-bodyuncertainties can be systematically assessed by increasing the truncation order L . Althoughthe number of MBPT diagrams increases rapidly with L , the technologies recently developedfor automated diagram generation and evaluation (31, 24) enable calculations at high-enough orders to probe in detail the many-body convergence for chiral low-momentum NNand 3N interactions. Furthermore, exploring MBPT with single-particle potentials beyondthe Hartree-Fock level is an important task for future research. In particular, the single-particle potential U can be chosen at each truncation order such that the grand-canonicaland free-energy series are also equivalent for L > L = 1. First investigations of this order-by-order renormalizationof the single-particle potential have shown that higher-order contributions to U can have asignificant effect on low-order MBPT results and the many-body convergence (36). The advances in ChEFT and RG methods have established MBPT as a central approachfor studying the nuclear EOS at zero and finite temperature. While MBPT is the focus ofthis review, various other many-body methods have been applied in initial nuclear matterstudies with chiral NN and 3N interactions. In particular, nonperturbative frameworks areimportant to benchmark the MBPT convergence and probe aspects of many-body physicsbeyond the nuclear EOS. Below we will briefly discuss the self-consistent Green’s functions(SCGF) approach and
Quantum Monte Carlo (QMC) methods . Other methods not dis-cussed here for brevity are coupled-cluster (CC) theory (37), the in-medium SRG (4), andlattice EFT (38). Systematic comparisons between different many-body frameworks willprovide a coherent picture of microscopic interactions and nuclear many-body properties.The SCGF approach (39, 40) is based on the self-consistent computation of in-mediumpropagators (or Green’s functions) in Fourier (Matsubara) space, corresponding to the to-all-orders resummation of some perturbative contributions to the propagators. SCGF calcu-lations of the nuclear EOS at zero and finite temperature (41) have been implemented usingthe in-medium T -matrix approximation, where the ladder diagrams are resummed to allorders, providing a thermodynamically consistent generalization of Brueckner theory (40).Furthermore, SCGF calculations have been used to benchmark the order-by-order conver-gence of MBPT (up to third order) in neutron matter (42). The energy per particle obtainedin SCGF and MBPT was found to agree well for a range of unevolved chiral NN and 3Ninteractions up to N LO. The SCGF approach allows for fully consistent computationsof response functions and transport properties, which will be vital for comparisons withMBPT calculations of these quantities. ••
Quantum Monte Carlo (QMC) methods . Other methods not dis-cussed here for brevity are coupled-cluster (CC) theory (37), the in-medium SRG (4), andlattice EFT (38). Systematic comparisons between different many-body frameworks willprovide a coherent picture of microscopic interactions and nuclear many-body properties.The SCGF approach (39, 40) is based on the self-consistent computation of in-mediumpropagators (or Green’s functions) in Fourier (Matsubara) space, corresponding to the to-all-orders resummation of some perturbative contributions to the propagators. SCGF calcu-lations of the nuclear EOS at zero and finite temperature (41) have been implemented usingthe in-medium T -matrix approximation, where the ladder diagrams are resummed to allorders, providing a thermodynamically consistent generalization of Brueckner theory (40).Furthermore, SCGF calculations have been used to benchmark the order-by-order conver-gence of MBPT (up to third order) in neutron matter (42). The energy per particle obtainedin SCGF and MBPT was found to agree well for a range of unevolved chiral NN and 3Ninteractions up to N LO. The SCGF approach allows for fully consistent computationsof response functions and transport properties, which will be vital for comparisons withMBPT calculations of these quantities. •• Chiral EFT and the High-Density Nuclear EOS 9
MC refers to a family of stochastic methods that solve the many-body Schr¨odingerequation through random sampling (6). As such, QMC methods are truly nonperturba-tive and provide important benchmarks for many-body methods with basis expansions.However, apart from the fermion sign problem a caveat is that most QMC methods requirelocal nuclear potentials to obtain low-variance results, restricting both the regularizationscheme and the interaction operators that can be included in the ChEFT expansion. QMCcalculations with local chiral NN and 3N potentials up to N LO have been carried outin neutron matter (43, 44) and recently also symmetric nuclear matter (45). The regula-tor artifacts are (due to Fierz-invariance breaking) significantly larger compared to MBPTcalculations with nonlocal potentials. On the other hand, since QMC methods are notrestricted to soft interactions, a much wider range of momentum cutoffs can be studiedwith QMC. Hence, QMC methods can provide important insights into the residual cutoffdependence of observables and the breakdown scale of ChEFT at high densities.
Three-nucleon forces are crucial for understanding properties of finite nuclei and nuclearmatter (46), such as drip lines along isotopic chains and nuclear saturation in SNM. Eventhough partial-wave decomposed matrix elements of chiral 3N forces have become availablerecently up to N LO (47), implementing 3N forces in many-body calculations remains com-putationally difficult and usually requires approximations (48). The large uncertainties dueto 3N forces, e.g. , in the nuclear EOS at densities n (cid:38) n , emphasize the need for improvingthese approximations as well as developing novel chiral NN and 3N potentials in general.Normal ordering allows one to include dominant 3N contributions in many-body frame-works using density-dependent effective two-body potentials (49). Through Wick’s theoremthe general three-body Hamiltonian can be exactly normal ordered with respect to a finite-density reference state ( e.g. , the Fermi sea of noninteracting nucleons or the Hartree-Fockground state) instead of the free-space vacuum (20). This shifts contributions from thethree-body Hamiltonian operator to effective zero-body, one-body, and two-body operatorsplus a residual (reduced) three-body operator. A many-body framework built for NN inter-actions can then incorporate a density-dependent effective interaction V medNN derived from V as V NN → V NN + ξ V medNN . The combinatorial factor ξ is determined by Wick’s theoremand depends on the many-body calculation of interest. The matrix elements of V medNN areobtained by summing one particle over the occupied states in the reference state: (cid:68) (cid:48) (cid:48) (cid:12)(cid:12)(cid:12) V medNN (cid:12)(cid:12)(cid:12) (cid:69) = ¯ V = (cid:88) σ τ (cid:90) d k (2 π ) f (cid:10) (cid:48) (cid:48) (cid:12)(cid:12) ¯ V (cid:12)(cid:12) (cid:11) , (6)with the shorthand notation | i (cid:105) = | k i σ i τ i (cid:105) , antisymmetrized 3N interactions ¯ V , andmomentum distribution function of the reference state f .In contrast to the (Galilean-invariant) NN potential, the effective two-body potential (6)depends on the center-of-mass momentum P of the two remaining particles. Hence, bothpotentials cannot be straightforwardly combined in a partial-wave basis and different ap-proximations for the P dependence have been used to enable applications to nuclear matter.Under the assumption that P = 0 first implementations evaluated Eq. (6) semi-analyticallyin symmetric nuclear matter and pure neutron matter starting from the N LO 3N interac-tions (50, 51, 23). Extensions to asymmetric nuclear matter and finite temperature havefollowed (35, 52, 42, 53), and a new method that allows for the construction of an effective
10 Drischler, Holt, and Wellenhofer wo-body potential from any partial-wave decomposed 3N interaction in an improved P angle-averaging approximation has been developed (53). The latter approach is especiallyadvantageous for studying 3N forces at N LO (53), bare and SRG-evolved, and in differentregularization schemes. Semi-analytic expressions along the lines of Ref. (50) have beenderived up to N LO and also partially to N LO (54).The three-body term in the normal-ordered Hamiltonian cannot be implemented usingeffective two-body potentials. In nuclear matter such residual 3N contributions have beenstudied in CC (55) and MBPT calculations (56, 57, 58, 24). Explicit calculations of theresidual 3N diagram in MBPT at second order (see the margin note) showed for a range ofchiral interactions that its contribution is typically much smaller than both the overall EFTtruncation error and the individual contributions from the other MBPT diagrams up to thisorder (24). While these findings give some justification for the commonly used approxima-tion where residual 3N contributions are neglected, the automated approach introduced inRef. (24) implements chiral NN, 3N, and 4N interactions exactly in nuclear matter calcula-tions using a single-particle spin-isospin basis. Combined with high-performance computing,this method sets the stage for systematic studies of ChEFT interactions in MBPT up tohigh orders and without the mentioned approximations.
Residual 3N diagramat second order:
3. Nuclear equation of state at zero and finite temperature
In this Section we survey recent nuclear matter calculations up to n ≈ n in MBPT withchiral NN and 3N interactions. We discuss advances in the quantification and propagationof EFT truncation errors, confront different microscopic constraints on the nuclear sym-metry energy with experiment, and examine contributions beyond the standard quadraticexpansion of the EOS in the isospin asymmetry. We conclude the Section with results forthe nuclear liquid-gas phase transition at finite temperature. Figure 2 (left) illustrates the nuclear EOS at zero temperature as a function of density n for a representative set of isospin asymmetries δ = ( n n − n p ) /n , where n n ( n p ) is theneutron (proton) number density. The uncertainty bands in the energy per particle E/A were obtained in Ref. (53) by second-order MBPT calculations based on the Hebeler et al. interactions (23). Several general observations can be gleaned. Nuclear interactions aremuch stronger in SNM compared to PNM, which is closer to the free Fermi gas (FFG,solid lines). Consequently, the uncertainties are larger in SNM, especially for densities n (cid:38) n . In PNM they are well controlled for n (cid:46) n , and a wide range of chiral NN and3N interactions leads to similar results for PNM (see, e.g. , Refs. (60, 46, 61)). Increasinguncertainties toward higher densities are predominantly due to 3N interactions. Althoughthe complexity of 3N interactions is much reduced in PNM (51), they provide at all valuesof δ important repulsive contributions that grow stronger with the density than those of NNinteractions. The 3N interactions are therefore crucial for understanding the high-densityEOS and the structure of neutron stars. In PNM all chiral interactions up to N LO arecompletely determined by the π N and NN system. The intermediate- and short-range 3Ninteractions at N LO that are proportional to the LECs c D and c E , respectively, vanish(for regulators symmetric in the particle labels) due to the coupling of pions to spin andthe Pauli principle, respectively. Also the long-range two-pion exchange 3N forces at N LO ••
In this Section we survey recent nuclear matter calculations up to n ≈ n in MBPT withchiral NN and 3N interactions. We discuss advances in the quantification and propagationof EFT truncation errors, confront different microscopic constraints on the nuclear sym-metry energy with experiment, and examine contributions beyond the standard quadraticexpansion of the EOS in the isospin asymmetry. We conclude the Section with results forthe nuclear liquid-gas phase transition at finite temperature. Figure 2 (left) illustrates the nuclear EOS at zero temperature as a function of density n for a representative set of isospin asymmetries δ = ( n n − n p ) /n , where n n ( n p ) is theneutron (proton) number density. The uncertainty bands in the energy per particle E/A were obtained in Ref. (53) by second-order MBPT calculations based on the Hebeler et al. interactions (23). Several general observations can be gleaned. Nuclear interactions aremuch stronger in SNM compared to PNM, which is closer to the free Fermi gas (FFG,solid lines). Consequently, the uncertainties are larger in SNM, especially for densities n (cid:38) n . In PNM they are well controlled for n (cid:46) n , and a wide range of chiral NN and3N interactions leads to similar results for PNM (see, e.g. , Refs. (60, 46, 61)). Increasinguncertainties toward higher densities are predominantly due to 3N interactions. Althoughthe complexity of 3N interactions is much reduced in PNM (51), they provide at all valuesof δ important repulsive contributions that grow stronger with the density than those of NNinteractions. The 3N interactions are therefore crucial for understanding the high-densityEOS and the structure of neutron stars. In PNM all chiral interactions up to N LO arecompletely determined by the π N and NN system. The intermediate- and short-range 3Ninteractions at N LO that are proportional to the LECs c D and c E , respectively, vanish(for regulators symmetric in the particle labels) due to the coupling of pions to spin andthe Pauli principle, respectively. Also the long-range two-pion exchange 3N forces at N LO •• Chiral EFT and the High-Density Nuclear EOS 11 .
00 0 .
05 0 .
10 0 .
15 0 . n [fm − ] − − − − E n e r g y p e r P a r t i c l e E / A [ M e V ] δ = . δ = . δ = . δ = . δ = . δ = . S v LK n E Nuclear Equation of State at T = .
14 0 .
16 0 . n [fm − ] − − − − − − − − S a t u r a t i o n E n e r g y E [ M e V ] Nuclear Saturation ( . . )( . . )( . . )( . . )( . . )( . . )( s i m ) ( s i m )( s i m ) ( s i m )( s i m ) ( H o l t )( H o l t )( H o l t ) (NNLOsat)(∆NNLOGO 394)(∆NNLOGO 450)(∆NLOGO 450) (GP-B 500 2 σ )(GP-B 450 2 σ ) Figure 2: (left) Nuclear EOS at T = 0 as a function of density n for a representativeset of isospin asymmetries δ as obtained in Ref. (53). Key observables that character-ize E ( n ≈ n , δ ) /A are illustrated. (right) Saturation points of numerous chiral inter-actions from fourth- (circles) and third-order (squares) MBPT calculations, as well asCC theory (triangles). The ellipses show the 2 σ regions of order-by-order calculationsup to N LO in MBPT with EFT truncation errors fully quantified (59). The whitebox in each panel depicts the empirical saturation point, E = − . ± .
57 MeV with n = 0 . ± .
007 fm − (53). The right panel has been modified from Ref. (24).are simplified since the LEC c does not contribute. This allows for tight low-densityconstraints on the neutron-rich matter EOS from PNM calculations and systematic high-density extrapolations (see Section 4). PNM: pure neutronmatter ( δ = 1) ANM: asymmetricnuclear matter(0 < δ < SNM: symmetricnuclear matter( δ = 0) FFG: free Fermi gas
Nuclear matter represents an ideal system for testing nuclear interactions at the densitiesaccessible to laboratory experiments and their implementation in many-body methods. Asillustrated in Fig. 2 (left), the nuclear EOS in the vicinity of n is (to good approximation)characterized by only a few experimentally accessible quantities. That is, the EOS ofSNM can be expanded about its minimum n as E ( n, δ = 0) /A ≈ E + ( K/ η , withthe saturation energy E = E ( n , /A , incompressibility K , and η = ( n − n ) / (3 n ).Further, explicit ANM calculations with chiral NN and 3N interactions (53, 62, 63) haveshown that the asymmetry dependence of the nuclear EOS is reasonably well reproducedby the standard quadratic approximation E ( n, δ ) /A = E ( n, /A + E sym ( n ) δ , where thesymmetry energy expanded in density reads E sym ( n ) ≈ S v + L η . In this approximationone finds E ( n, /A ≈ ( E + S v ) + L η for PNM. Microscopic predictions and empiricalconstraints for ( n , E , , K ) and ( S v , L ) can then be confronted with one another. CC: coupled cluster(see Sec. 2.4)
Nuclear saturation emerges from a delicate cancellation between kinetic and interactioncontributions to the EOS. Reproducing empirical constraints on ( n , E , K ) is therefore animportant benchmark of nuclear interactions, especially 3N forces (providing the necessaryrepulsion). Figure 2 (right) depicts the saturation points of numerous chiral potentials as
12 Drischler, Holt, and Wellenhofer redicted by fourth- (circles) and third-order (squares) MBPT calculations; specifically, theNN and 3N interactions by Hebeler et al. (as in Fig. 2 (left), “ λ/ Λ [ fm − ]”), Carlsson et al. (“sim Λ [ MeV]”), and Holt et al. (“Holt Λ [ MeV]”). The saturation points are alignedalong a Coester-like band (gray anticorrelation band), which overlaps with the empiricalsaturation point (white boxes, see legend), determined from a set of energy density func-tionals (64). Also shown are results from CC calculations with NNLO sat ( ▶ ) and the newdeltafull chiral potentials “∆N(N)LO GO Λ [ MeV]” ( ▶ , ▶ , ▶ ). Only the latter fall into theempirical range for ( n , E ). However, judging the extent a nuclear potential reproducesempirical (saturation) properties can be quite misleading without taking meaningful uncer-tainties into account; especially, the truncation of the EFT expansion at a finite order canresult in sizable EFT truncation errors (even at N LO) that need to be quantified.
See Ref. (65) for areview of the linkbetween the nuclearEOS and nuclearobservables; e.g. ,from measurementsof the isoscalar giantmonopole resonance K ≈ −
230 MeVwas inferred.
Until a few years ago, the prevalent way of estimating theoretical uncertainties in nuclearmatter calculations was parameter variation within some (arbitrary) range; that is, probingthe observable’s sensitivity to, e.g. , the 3N LECs or momentum cutoff. Recently, the focushas been more on the systematic quantification of EFT truncation errors (66), which can beestimated by assuming that an observable’s EFT convergence pattern at order k takes theform y k ( n ) = y ref ( n ) (cid:80) km =0 c m ( n ) Q m ( n ) (67). Here, y ref ( n ) sets a dimensionful referencescale, Q ( n ) is the dimensionless expansion parameter, and the c n ( n ) are the dimensionlesscoefficients not to be confused with the LECs of the interaction ( e.g. , y = E/A at N LO).Note that c = 0 in Weinberg power counting. For given choices of y ref ( n ) and Q ( n ), the c m (cid:54) k ( n ) are obtained from order-by-order calculations { y , y , . . . , y k } of the observable.Since y ref ( n ) and Q ( n ) factor in all physical scales, the c m ( n ) are expected to be of orderone ( i.e. , natural ), unless the coefficients are fine-tuned. The standard EFT uncertainty,which assumes that the truncation error is dominated by the first omitted term, has beenimplemented by Epelbaum et al. (68) and applied to a wide range of observables in fi-nite nuclei and infinite matter. This “EKM uncertainty” can be summarized at N j LO as δy ( n ) = y ref Q j +2 max( | c | , | c | , . . . , | c j +1 | ), whose point-wise estimates can be interpretedas Bayesian credibility regions under a particular choice of priors for c m (13). Both y ref ( n ) and Q ( n ) need to bechosen; e.g. , y ref (cid:39) y and Q ∼ p/ Λ b withtypical momentum p ∝ k F ( n ) and someestimate of the EFTbreakdown scale Λ b have been applied tothe nuclear EOS. The
Bayesian Uncertainty Quantification: Errors in Your EFT (BUQEYE) collabora-tion has recently introduced a Bayesian framework for quantifying correlated EFT uncer-tainties in the nuclear EOS (69, 59). In contrast to the standard EFT uncertainty, the newframework allows for the quantification and propagation of statistically meaningful uncer-tainties to derived quantities ( e.g. , the pressure) while accounting for correlations acrossdensities and between observables. Without considering these correlations, uncertaintiescan be overestimated. The framework also includes Bayesian model checking tools (70)for diagnosing and testing whether the in-medium ChEFT expansion works as assumed( e.g. , inference for Λ b ). Gaussian Processes (GPs) with physics-based hyperparametersare trained on the order-by-order calculations of the energy per particle under the as-sumption that all c m ( n ) are random curves drawn from a single GP (67). The Gaussianposterior for the c m ( n ) is then used to estimate the to-all-orders EFT truncation error δy k ( n ) = y ref ( n ) (cid:80) ∞ m = k +1 c m ( n ) Q m ( n ) and combined with additional ( e.g. , many-body)uncertainties. From the posterior also arbitrary derivatives in n can be obtained. BUQEYE:
BayesianUncertaintyQuantification:Errors in Your EFT;the collaboration’sframework forquantifyingcorrelated EFTtruncation errors ispublicly available at buqeye.github.io/software/ . GP(–B):
GaussianProcess(–BUQEYEcollaboration)
With this new framework Refs. (69, 59) studied the EFT convergence of the first order-by-order calculations with NN and 3N interactions up to N LO in PNM and SNM, con-ducted in Refs. (24, 71) using a novel Monte Carlo integration framework for MBPT. Theassociated N LO 4N Hartree-Fock energies have been found negligible compared to theoverall uncertainties (see also Ref. (72)). To construct a set of order-by-order NN and 3N ••
With this new framework Refs. (69, 59) studied the EFT convergence of the first order-by-order calculations with NN and 3N interactions up to N LO in PNM and SNM, con-ducted in Refs. (24, 71) using a novel Monte Carlo integration framework for MBPT. Theassociated N LO 4N Hartree-Fock energies have been found negligible compared to theoverall uncertainties (see also Ref. (72)). To construct a set of order-by-order NN and 3N •• Chiral EFT and the High-Density Nuclear EOS 13 . . . n [fm − ]1020304050 E / N ± σ [MeV] 0 . . . n [fm − ]0510152025 LONLON LON LO P ± σ [MeV fm − ] 0 . . . n [fm − ]0 . . . . . c s ± σ [ − ] Figure 3: Order-by-order predictions for the energy per particle
E/N (left panel), pressure P = n d( E/N ) / d n (middle panel), and speed of sound squared c s = ∂P/∂ε (right panel)in PNM as a function of the density (69) based on the MBPT calculations up to N LO inRef. (24). The energy density ε includes the rest mass contribution. Correlated uncertaintybands are given at the 1 σ confidence level. See the main text for details.interactions up to N LO, Ref. (24) adjusted the two 3N LECs to the triton and ( n , E )for two cutoffs. Several potentials with reasonable saturation properties were obtained,although generally underbound at N LO. This holds also at the 2 σ credibility level withEFT truncation errors quantified (59), as depicted by the ellipses “GP–B Λ [MeV]” inFig. 2 (right). Hoppe et al. (25) found that the corresponding binding energies (chargeradii) of medium-mass nuclei are predicted too small (too large) compared to experiment,in (dis)agreement with the expectations from SNM. Since both observables were also muchmore sensitive to the 3N LEC c D , SNM and medium-mass nuclei seem more intricatelyconnected than one might naively expect (25).Figure 3 shows the order-by-order predictions for the energy per particle, pressure, andspeed of sound squared in PNM at the 1 σ confidence level based on “GP–B 500” (69).The observables show an order-by-order convergence pattern at n (cid:46) . − , whereasN LO and N LO have a markedly different density dependence at n (cid:38) n due to repulsive3N contributions. This is also manifested in the Bayesian diagnostics (67). Assuming Q = k F / Λ b , the inferred breakdown scale Λ b ≈
600 MeV is consistent with free-spaceNN scattering and could be associated with n > n . The EFT truncation errors arestrongly correlated in density and to those in SNM. A correlated approach is thereforenecessary to propagate uncertainties reliably to derived quantities, although the standardEFT uncertainty for the energy per particle is broadly similar to the 1 σ confidence level (69). The nuclear symmetry energy is a key quantity to understand the structure of neutron-rich nuclei and neutron stars. Although masses of heavy nuclei constrain the value of thesymmetry energy well at nuclear densities, its density dependence is much less known (73).Studying the density-dependent symmetry energy from theory, experiment, and observationis therefore an important task in the era of multimessenger astronomy.
14 Drischler, Holt, and Wellenhofer S v [MeV]020406080100 S l o p e P a r a m e t e r L [ M e V ] H e a v y - I o n C o lli s i o n s Sn N e u tr o nS k i n G i a n t D i p o l e R e s o n a n ce s P b D i p o l e P o l a r i z a b ili t y M a ss e s I A S + ∆ R H GUGUG Analytic
GP–B 450GP–B 500 HK . . . . n [fm − ]0102030405060 S y mm e t r y E n e r g y E s y m [ M e V ] Lim ’18 (1 σ ∣ σ )Carbone ’18Lonardoni ’20 ( E ; E τ )GP–B 500 ’20 (1 σ ∣ σ )Akmal ’98Baldo ’97Muether ’87 Figure 4: (left) Constraints on E sym ( n ) based on chiral interactions (bands) and phe-nomenological potentials (symbols). The vertical band depicts the empirical saturationdensity. (right) Theoretical and experimental constraints for ( S v , L ) as well as the con-jectured UG bounds in comparison (see annotations). Gray ellipses (59) show the allowedregions from PNM and SNM calculations at N LO with truncation errors quantified (light:1 σ , dark: 2 σ ). The white area shows the joint experimental constraint without “IAS+∆ R ”.Figure 4 (left) summarizes theoretical constraints for E sym ( n (cid:54) n ) from a wide rangeof chiral NN and 3N forces as well as different many-body methods. Specifically, we showthe results for E sym ( n ) = E ( n, δ = 1) /A − E ( n, /A as obtained in the calculations byLim et al. (74) and Drischler et al. (59, 24) [“GP–B 500”] in MBPT, Carbone et al. (75)in the SCGF method, and Lonardoni et al. (76) using QMC methods. The latter wereconducted with two different parameterizations of the N LO 3N contact interaction ( i.e. ,distinct bands for E and E τ ) to demonstrate the significant regulator artifacts present inlocal chiral 3N potentials. Different uncertainty estimates were used in these calculations.The uncertainty bands by Carbone et al. probe parameter variations in the nuclear interac-tions, while those by Lonardoni et al. and Drischler et al. quantify truncation errors usingthe standard EFT uncertainty (up to N LO) and BUQEYE’s new Bayesian framework (upto N LO), respectively. Also many-body (or statistical Monte Carlo) uncertainties are in-cluded in the bands. Lim et al. performed a statistical analysis of MBPT calculations basedon a range of chiral potentials at different orders and two single-particle spectra to probethe chiral and many-body convergence. Only the results by Lim et al. and Drischler et al. (both MBPT) have a clear statistical interpretation, each at the 1 σ and 2 σ confidence level(different shadings). Overall, the constraints from ChEFT are consistent with each other,even at the highest densities shown, but the uncertainties in E sym ( n ) are generally sizable, e.g. , 20 . ± .
1, 31 . ± .
0, and 49 . ± . n / n , and 2 n , respectively, forLim et al. at the 1 σ confidence level. Drawing general conclusions from comparing the sizesof these bands can be misleading since the underlying methods for estimating uncertainties ••
0, and 49 . ± . n / n , and 2 n , respectively, forLim et al. at the 1 σ confidence level. Drawing general conclusions from comparing the sizesof these bands can be misleading since the underlying methods for estimating uncertainties •• Chiral EFT and the High-Density Nuclear EOS 15 re quite different. Order-by-order comparisons for a wider range of chiral NN and 3Ninteractions with EFT truncation errors quantified are called for to provide more insightsin and stringent constraints on E sym ( n ). The Bayesian statistical tools introduced by theBUQEYE collaboration allow for such systematic studies.Despite the large uncertainties in the SNM EOS (see Sec. 3.1), predictions for E sym ( n )[as an energy difference] can be made with significantly smaller uncertainties than those in E ( n, /A and E ( n, /A individually, if correlations are properly considered. Reference (59)found that the EFT truncation errors associated with the PNM and SNM calculations inRef. (24) are highly correlated , meaning that the uncertainty in E sym ( n ) is less than theusual in-quadrature sum of errors. Combined with order-by-order calculations up to N LOthis led to the narrow bands “GP–B 500” in Fig. 4 (left) based on the interactions usedwith Λ = 500 MeV ( e.g. , E sym (2 n ) = 45 . ± . σ confidence level. The bands agree with the constraints by Lim et al. at the 1 σ level (or even better) as well as the calculations by Akmal et al. , Baldo et al. ,and Muether et al. with phenomenological nuclear potentials. The latter, however, do notprovide uncertainties that could be used to judge the level of agreement. These correlationsneed to be investigated further using different many-body frameworks and interactions. The statisticalmeaning of the(uncorrelated)standard EFTuncertainty appliedto E sym ( n ) isunclear and requiresa choice for thetypical momentum p that depends on twodifferent k F at fixeddensity n . Figure 4 (right) compares various theoretical and experimental constraints in the S v – L plane (see annotations). The regions obtained by Hebeler et al. (“H”), Gandolfi et al. (“G”),and Holt and Kaiser (“HK”), which were derived from microscopic PNM calculations andthe empirical saturation point, agree well with each other and are consistent with therange in S v of the joint experimental constraint (white area), although L is predictedwith somewhat lower values. Constraints extracted from measurements of isobaric analogstates and isovector skins (“IAS + ∆ R ”) are not included in the white area because theybarely overlap. The 1 σ and 2 σ ellipses of “GP–B 500” (as in Fig. 4 (left)) are in excellentagreement with the joint experimental constraint [“GP–B 450” is slightly shifted to higher( S v , L )], indicating a stiffer neutron-rich matter EOS at n compared to the other theoreticalconstraints. This is, however, consistent with joint theory-agnostic posteriors from pulsar,gravitational-wave, and NICER observations ( e.g. , compare with Figure 1 in Ref. (77)). Animportant feature of the correlated GP approach is that the theoretical uncertainties in n (including truncation errors) are accounted for through marginalization over the Gaussianposterior for the saturation density predicted from the SNM calculations, pr( n ) ≈ . ± .
01 fm − . Apart from “HK” allowing slightly lower ( S v , L ), all shown theory calculationssatisfy the constraint (solid black line) derived from the conjecture (78) that the unitarygas (UG) sets a lower bound for the PNM EOS. Overall, Fig. 4 (right) shows that currentconstraints from nuclear theory and experiment predict the symmetry energy parametersin the range S v ≈ −
35 MeV and L ≈ −
72 MeV.
UG: unitary gas;Ref. (78) also madeadditionalassumptions toderive an analyticbound (“UGAnalytic”, dashedblack line inFig. 4 (right)).
While the standard quadratic approximation [ E ( n, δ ) /A = E ( n, /A + E sym ( n ) δ ] isin general sufficient to characterize the isospin-asymmetry dependence of the nuclear EOS,certain neutron-star properties, such as the crust-core transition (79) and the thresholdfor the direct URCA cooling process (80), are sensitive to nonquadratic contributions.Neglecting charge-symmetry breaking effects, the energy per particle may be assumed tohave an expansion in the asymmetry δ of the form E ( n, δ ) /A ≈ E ( n, /A + (cid:80) Ll =1 S l ( n ) δ l ,where the standard quadratic approximation corresponds to S l> ( n ) = 0. Note, however,that already the FFG contributes to the nonquadratic terms, e.g. , S FFG4 ( n ) (cid:39) .
45 MeV × ( n/n ) / . Parametric fits to microscopic ANM calculations have been used to investi-gate the nonquadratic contributions and found them to be relatively small (53, 62, 81, 63).
16 Drischler, Holt, and Wellenhofer .
00 0 .
05 0 .
10 0 . n [fm − ]05101520 T e m p e r a t u r e T [ M e V ] gas gas-liquid liquid n3lo414n3lo450n3lo500 0 . . . . . . δT c ( δ ) T κ T ( δ ) gas-liquid n3lo414n3lo450 Figure 5: (left) Liquid-gas coexistence boundary (binodal) of SNM from second-orderMBPT calculations based on three sets of N LO NN potentials and N LO 3N interactionswith Λ = 414, 450, and 500 MeV (35, 52). The zero-temperature limit of the coexistenceboundary corresponds to the nuclear saturation point. The white box shows the empiricalrange for the critical point from Ref. (84). (right) Asymmetry dependence of the criticaltemperature T c ( δ ) and the temperature T κ T ( δ ) where the region with negative κ − T vanishes.Recently, however, Kaiser (82) has shown that MBPT at second order gives rise to addi-tional logarithmic contributions ∼ δ l ln | δ | with l (cid:62)
2. Furthermore, Wellenhofer et al. (63)found that the analogous expansion of the free energy exhibits convergent behavior for δ (cid:54) T → S l> ), as implied by the logarithmic terms at T = 0. Nevertheless, Wen and Holt (83) demonstrated that the coefficients of the normaland logarithmic terms at T = 0 can be extracted up to O ( δ ) from high-precision MBPTcalculations with chiral interactions. Such calculations allow for the improvement of ex-isting parametrizations in δ at T = 0 and help motivate the investigation of alternativeschemes, such as an expansion in terms of the proton fraction x = n p /n = (1 − δ ) / While thermal effects are negligible in isolated neutron stars, they become important inneutron star mergers and core-collapse supernovae, where T (cid:46)
100 MeV can be reached.Dense matter at such high temperatures not only consists of nucleons and leptons but also ofadditional particles such as pions and hyperons. The consistent inclusion of these particlesin medium is work in progress (85, 86). In the nascent field of multimessenger astronomy,one of the immediate theoretical needs is consistent modeling of (i) cold neutron stars,(ii) hot hypermassive neutron stars formed in the aftermath of compact object mergers,and (iii) core-collapse supernovae so that observations and simulations in any one of theseastrophysical regimes can be propagated to constrain the others. Finite-temperature MBPTwith ChEFT interactions is a suitable framework for this purpose, and here we describe someof the results on nuclear thermodynamics in recent years (for reviews, see Refs. (87, 88)).The salient thermodynamic feature of homogeneous nuclear matter at sub-saturationdensities is the presence of a liquid-gas type instability toward the formation of clustered ••
100 MeV can be reached.Dense matter at such high temperatures not only consists of nucleons and leptons but also ofadditional particles such as pions and hyperons. The consistent inclusion of these particlesin medium is work in progress (85, 86). In the nascent field of multimessenger astronomy,one of the immediate theoretical needs is consistent modeling of (i) cold neutron stars,(ii) hot hypermassive neutron stars formed in the aftermath of compact object mergers,and (iii) core-collapse supernovae so that observations and simulations in any one of theseastrophysical regimes can be propagated to constrain the others. Finite-temperature MBPTwith ChEFT interactions is a suitable framework for this purpose, and here we describe someof the results on nuclear thermodynamics in recent years (for reviews, see Refs. (87, 88)).The salient thermodynamic feature of homogeneous nuclear matter at sub-saturationdensities is the presence of a liquid-gas type instability toward the formation of clustered •• Chiral EFT and the High-Density Nuclear EOS 17 tructures. In neutron stars, this instability corresponds to the crust-core transition, involv-ing such intricate features as a variety of “pasta” shapes (89). The nuclear liquid-gas insta-bility is also connected to the observed multifragmentation events in intermediate-energyheavy-ion collisions. In the idealized case of (infinite) nuclear matter, there is a liquid-gasphase transition of van-der-Waals type. Nuclear matter calculations at finite temperaturewith chiral interactions have provided predictions for the properties of this phase transition,in particular the location of the critical point. Figure 5 (left) shows the second-order MBPTresults for the boundary of the liquid-gas coexistence region (so-called binodal) of SNM ob-tained in Ref. (35). The predicted critical point, especially the associated temperature T c ≈ −
19 MeV, is consistent with estimates ( e.g. , T c ≈ −
20 MeV (90)) extracted frommultifragmentation, nuclear fission, and compound nuclear decay experiments (90, 84).
The binodal of SNMwas recentlycomputed using theSCGF method (91)and with latticeEFT (38); theresults are similar tothose of Fig. 5.
In the interior of the binodal a region where the homogeneous system is unstable withrespect to infinitesimal density fluctuations can be found. The boundary of this region iscalled spinodal. Between the binodal and spinodal the uniform system is metastable. [Thetwo boundaries coincide at the critical point.] For SNM, the unstable region is identified bya negative inverse isothermal compressibility, κ − T = n ( ∂P/∂n ) <
0. An equivalent stabilitycriterion is ∂µ/∂n >
0, corresponding to a strictly convex free energy density F ( T, n ) as afunction of n . If charge-symmetry breaking effects are neglected, SNM can be treated asa pure substance with one particle species (nucleons), whereas ANM is a binary mixturewith two thermodynamically distinct particles (neutrons and protons). This implies thatthe stability criteria are different in the two cases, and for ANM the region with κ − T < F ( T, n n , n p )is a strictly convex function of n n and n p (see Ref. (52) for details). The MBPT resultsfor the asymmetry dependence of the critical temperature T c ( δ ) from Ref. (52) are shownin Fig. 5 (right). For comparison, we also show the trajectory of the temperature T κ T ( δ )where the subregion with negative κ − T vanishes. The trajectory of T c ( δ ) reaches its T = 0endpoint at a small proton fraction x ; i.e. , while PNM is stable at all densities, alreadysmall x lead to a region where the system undergoes a phase separation (52, 93). While for SNM thebinodal correspondsto the Maxwellconstruction, forANM the moreinvolved Gibbsconstruction isrequired (93).
A useful characteristic for the temperature dependence of the nuclear EOS is the ther-mal index Γ th ( T, n, δ ) = 1 + P th ( T, n, δ ) / E th ( T, n, δ ), where P th is the thermal part of thepressure, and E th is the thermal energy density. For a free gas of nucleons with effectivemasses m ∗ n,p ( n, δ ) one obtains for Γ th the temperature-independent expression The thermal indexof a free Fermi gas isΓ th,free = 5 / Γ (cid:63) th ( n, δ ) = 53 − (cid:88) t =n , p n t ( n, δ ) m ∗ t ( n, δ ) ∂m ∗ t ( n, δ ) ∂n . (7)[To be precise, for δ / ∈ { , } the above expression is valid only in the classical limit, but itprovides a good approximation to Γ (cid:63) th ( n, x ) for intermediate values of δ as Ref. (61) showed.]Recently, Refs. (22, 94) showed that Γ (cid:63) th ( n, δ ) reproduces the exact Γ th with high accuracy.This implies that the temperature-dependence of the EOS can be characterized in terms ofa temperature-independent effective mass (see Ref. (61) for a recent implementation), whichis in particular useful for monitoring thermal effects in astrophysical applications (95, 96).
4. Applications to neutron star physics
In this Section our goal is to emphasize the prominent role of nuclear theory in modelingneutron stars, core-collapse supernovae, and neutron star mergers. We begin by placing
18 Drischler, Holt, and Wellenhofer igh-energy nuclear astrophysics in the more general context of the QCD phase diagramand discuss under what ambient conditions ChEFT can serve as a tool to constrain theproperties of hot and dense matter. Specific applications include the neutron-star mass-radius relation, moment of inertia, and tidal deformability, as well as the nuclear EOS andneutrino opacity for astrophysical simulations.
The extreme astrophysical environments found in core-collapse supernovae, neutron starinteriors, and neutron-star mergers span baryon number densities n B ∼ − − n ,temperatures T ∼ −
100 MeV, and isospin asymmetries δ ∼ − Y e ∼ − .
5) (97). In Sections 2 and 3 we have shown that ChEFTprovides a suitable framework to constrain the EOS, transport, and response properties ofhadronic matter when the physical energy scale is well below the chiral symmetry breakingscale of Λ χ ∼ n ≈ − n and temperatures T (cid:46)
30 MeV. Therefore, additional modeling is needed at high densities and temperatures tocover all regions of astrophysical interest. For this purpose, high-energy heavy-ion collisionsat RHIC, LHC, and especially FAIR aim to probe states of matter similar to those thatexist naturally in neutron stars, but reaching sufficiently large proton-neutron asymmetriesremains a significant challenge that may be addressed with next-generation radioactiveion beam facilities, such as FRIB. The interplay of microscopic ChEFT, whose convergencepattern is not especially sensitive to the isospin asymmetry, together with upcoming nuclearexperiments that create and study hot, dense, and neutron-rich matter, will provide a directline of inquiry probing neutron-star physics from low to high densities.From the observational side, measurements of neutron star masses, radii, tidal deforma-bilities, and moments of inertia are expected to place constraints on the pressure of beta-equilibrium matter at n (cid:38) n (98, 74, 99). In Fig. 6, we present a qualitative overviewof the QCD phase diagram and highlight regions probed by nuclear experiments (RHIC,LHC, FAIR, and FRIB), theory (lattice QCD and ChEFT), and astrophysical simulationsof neutron stars, supernovae, and neutron star mergers. We see that ChEFT intersectsstrongly with the region of FRIB experiments and nuclear astrophysics, providing a bridgebetween new discoveries in the laboratory and their implications for neutron stars. Thenext decade is expected to witness a strong interplay among all of these different fields,with nuclear theory predictions getting confronted with stringent empirical tests. RHIC:
RelativisticHeavy Ion Collider
LHC:
Large HadronCollider
FAIR:
Facility forAntiproton and IonResearch
FRIB:
Facility forRare Isotope Beams
The mass-radius relation of non-rotating neutron stars is determined from the EOS bythe general relativistic equations for hydrostatic equilibrium, the Tolmann-Oppenheimer-Volkoff (TOV) equations:d p d r = − G ( M ( r ) + 4 πr p )( ε + p ) r ( r − GM ( r )) , d M d r = 4 πr ε, (8)where r is the radial distance from the center of the star, M ( r ) is the mass enclosed within r , ε is the energy density, and p is the pressure. Analysis of spectral data from neutron starsin quiescent low-mass x-ray binaries and x-ray bursters (100, 101) have resulted in radiusmeasurements R . = 10 −
13 km for typical 1 . (cid:12) neutron stars. More recently, the NICER ••
13 km for typical 1 . (cid:12) neutron stars. More recently, the NICER •• Chiral EFT and the High-Density Nuclear EOS 19
RIB
Figure 6: Schematic view of the QCD phase diagram. We highlight regions probed by ex-periments (RHIC, LHC, FAIR, and FRIB), regions of validity for lattice QCD and ChEFT,and environments reached in neutron stars, supernovae, and neutron star mergers.x-ray telescope has observed hot spot emissions from the accretion-powered x-ray pulsarPSR J0030+045. Pulse profile modeling of the x-ray spectrum from two independent groupshave yielded consistent results for the neutron star’s mass M = 1 . +0 . − . M (cid:12) (102) and M =1 . +0 . − . M (cid:12) (103) and radius R = 13 . +1 . − . km (102) and R = 12 . +1 . − . km (103) at the68% credibility level. Future large area x-ray timing instruments, such as STROBE-X andeXTP, have the potential to reduce uncertainties in the neutron-star mass-radius relationto ∼
2% at a given value of the mass. This would significantly constrain the neutron-richmatter EOS at n ≈ n and when combined with mass and radius measurements of theheaviest neutron stars could give hints about the composition of the inner core (104). TOV:
Tolmann-Oppenheimer-Volkoff
NICER:
Neutron starInterior CompositionExploreR
STROBE-X:
SpectroscopicTime-ResolvingObservatory forBroadband EnergyX-rays eXTP: enhancedX-ray Timing andPolarimetry
In recent years numerous works have studied constraints on the neutron star EOS fromChEFT. In Ref. (105) the EOS of neutron-rich matter was calculated up to saturationdensity with MBPT using chiral NN and 3N interactions. To extrapolate to higher densities,a series of piecewise polytropes was used to parameterize the EOS. It was found that ChEFTgenerically gives rise to soft EOSs that lead to 1 . (cid:12) neutron stars with radii in the range R . = 10 −
14 km. Subsequent studies ( e.g. , Refs. (106, 107, 61)) have employed a widerrange of chiral forces, increased the assumed range of validity for ChEFT calculations to2 n , and explored other high-density EOSs, including smooth extrapolations and speed ofsound parameterizations. The choice of transition density at which ChEFT predictionsare replaced by model-dependent high-density parameterizations has a particularly largeinfluence on neutron-star radius constraints. For instance, when the transition densitywas raised to n t = 2 n , Ref. (108) obtained R . = 9 . − . R . = 10 . − .
20 Drischler, Holt, and Wellenhofer E sym (2 n ) (MeV) R . ( k m ) − − −
30 40 50 60 700 . . . PDF R = 12 .
38 km
Figure 7: Correlations between the radius of a 1 . (cid:12) neutron star and the isospin asym-metry energy E sym at twice saturation density n = 2 n . Inset: probability distribution of E sym (2 n ) for the specific value R . = 12 .
38 km.to R . >
13 km. To demonstrate how a precise neutron-star mass and radius measurementcan constrain the EOS of beta-equilibrium matter at n = 2 n , in Fig. 7 we show thecorrelated probability distribution (74) for the radius of a 1 . (cid:12) neutron star and thenuclear symmetry energy at twice saturation density E sym (2 n ). In the inset we showthe conditional probability distribution for E sym (2 n ) assuming a precise measurement of R . = 12 .
38 km. For the specific EOS modeling used in Ref. (74), such a precise radiusconstraint determines E sym (2 n ) with an uncertainty of approximately 10%. In addition to ahigh-densityextrapolation, auniform-matter EOSfrom ChEFT needsto be supplementedwith a neutron-starcrust model, e.g. ,the BPS crustmodel (109).
In addition to radius measurements, there has long been the possibility (110, 111) toobtain a neutron-star moment of inertia measurement based on long-term radio timing ofPSR J0737-3039, a binary pulsar system in which the periastron advance receives a smallcorrection from relativistic spin-orbit coupling. A recent analysis (112) has shown that by2030 a moment of inertia measurement of PSR J0737-3039A to 11% precision at the 68%confidence level is achievable. The moment of inertia for a uniformly rotating neutron starof radius R and angular velocity Ω can be calculated in the slow-rotation approximation,valid for most millisecond pulsars, by solving the TOV equations together with I = 8 π (cid:90) R r ( ε + p ) e ( λ − ν ) / ¯ ω Ω d r , e − λ = (cid:18) − mr (cid:19) − , d ν d r = − ε + p d p d r , (9)where λ and ν are metric functions and ˜ ω is the rotational drag. In Refs. (113, 114) themoment of inertia of PSR J0737-3039A, which has a very well measured mass of M =1 .
338 M (cid:12) , was calculated from EOSs based on ChEFT. In Ref. (113) it was found that atthe 95% credibility level, the moment of inertia of J0737-3039A lies in the range 0 . × g cm < I < . × g cm , while Ref. (114) found a consistent but somewhat largerrange of 1 . × g cm < I < . × g cm . The moment of inertia is stronglycorrelated with the neutron star radius, and it has been shown (115) that measurements ofthe PSR J0737-3039A moment of inertia can constrain its radius to within ± The radius andmoment of inertia ofa typical 1 . (cid:12) neutron star isstrongly correlatedwith the value of E sym ( n ≈ n ) (98). In the past ten years, several neutron stars (116, 117, 118) with well measured massesof M (cid:38) (cid:12) have been observed. The maximum mass ( M TOVmax ) of a non-rotating neutron ••
338 M (cid:12) , was calculated from EOSs based on ChEFT. In Ref. (113) it was found that atthe 95% credibility level, the moment of inertia of J0737-3039A lies in the range 0 . × g cm < I < . × g cm , while Ref. (114) found a consistent but somewhat largerrange of 1 . × g cm < I < . × g cm . The moment of inertia is stronglycorrelated with the neutron star radius, and it has been shown (115) that measurements ofthe PSR J0737-3039A moment of inertia can constrain its radius to within ± The radius andmoment of inertia ofa typical 1 . (cid:12) neutron star isstrongly correlatedwith the value of E sym ( n ≈ n ) (98). In the past ten years, several neutron stars (116, 117, 118) with well measured massesof M (cid:38) (cid:12) have been observed. The maximum mass ( M TOVmax ) of a non-rotating neutron •• Chiral EFT and the High-Density Nuclear EOS 21 tar is a key quantity to probe the composition of the inner core, which must have asufficiently stiff EOS to support the enormous pressure due to the outer layers. To datethe strongest candidate for the heaviest measured neutron star is PSR J0740+6620, witha mass of M = 2 . +0 . − . M (cid:12) at the 95% credibility level (118). As mentioned previously,ChEFT generically gives rise to relatively soft EOSs just above nuclear saturation density.The existence of a very massive neutron star with M = 2 .
14 M (cid:12) would require a stiff EOSat high densities, revealing a slight tension with ChEFT (119). However, even smoothextrapolations (74, 61) of EOSs from ChEFT can produce maximum neutron star massesin the range 2 . (cid:12) (cid:46) M TOVmax (cid:46) . (cid:12) , and therefore more precise radius measurements(or the observation of heavier neutron stars) are needed to make strong inferences aboutthe EOS in the ChEFT validity region n (cid:46) n . Beyond M TOVmax ,additional stablebranches (120), suchas hybridquark-hadron starsor pure quark stars,may appear beforethe ultimate collapseto a black hole.
The advent of gravitational wave astronomy has opened a new window into the visibleUniverse. Current gravitational wave detectors (LIGO and Virgo) are sensitive to fre-quencies 10 Hz < f <
10 kHz, which is the prime range for compact object mergers andsupernovae. Gravitational wave astronomy therefore has major implications for the fieldof nuclear astrophysics (3). In particular, during the late-inspiral phase of binary neutronstar coalescence, a pre-merger neutron star will deform with induced quadrupole moment Q under the large tidal gravitational field E : Q ij = − λ E ij , where λ is the dimensionful tidaldeformability parameter. Tidal deformations enhance gravitational radiation and increasethe rate of inspiral. Gravitational wave detectors are sensitive to such phase differencesand hence the dense matter EOS, but such corrections enter formally at fifth order (121)in a post-Newtonian expansion of the waveform phase and are therefore difficult to extract.The tidal deformability is an important observable in its own right, but this quantity is alsostrongly correlated with both the neutron star radius (122), since more compact stars expe-rience a smaller deformation under a given tidal field, and especially the moment of inertiathrough the celebrated I-Love-Q relations (123). The post-merger gravitational wave signalfrom binary neutron star coalescence can also carry important information on the nuclearEOS. It has been shown (124) that the peak oscillation frequency f peak of a neutron-starmerger remnant is strongly correlated with neutron star radii. Moreover, a strong first-orderphase transition can show up as a deviation in the empirical correlation band between f peak and Λ (125). LIGO:
LaserInterferometerGravitational-WaveObservatory
The first observation (126), GW170817, of a neutron star merger through its gravita-tional wave emissions was accompanied by a short gamma-ray burst and optical counter-part (127). The combined multi-messenger observations of this single event have resultedin a wealth of new insights about the origin of the elements and the properties of neutronstars. Analysis of the gravitational waveform resulted in a prediction (128) Λ . = 190 +390 − for the dimensionless tidal deformability Λ = λ/M of a 1 . (cid:12) neutron star. Theoreticalpredictions from ChEFT (74, 129) predating the analysis in Ref. (128) yielded similarlysmall tidal deformabilities 140 < Λ <
520 (74). Analogous constraints on the binary tidaldeformability parameter˜Λ = 1613 ( m + 12 m ) m Λ + ( m + 12 m ) m Λ ( m + m ) (10)from gravitational-wave data [˜Λ = 300 +420 − (130)] and ChEFT [80 < ˜Λ <
580 (129)] weresimilarly consistent. From the strong correlation between neutron star radii and tidal de-
22 Drischler, Holt, and Wellenhofer
10 12 14 R . (km) p n ( M e V f m − ) n t = 2 n
200 400 600 Λ . . . . . . I . (10 g cm ) − − − Figure 8: Joint probability distribution for the pressure of beta-equilibrium matter p (2 n )with the radius of a 1 . (cid:12) neutron star (left), the tidal deformability of a 1 . (cid:12) neutronstar (middle), and the moment of inertia of PSR J0737-3039A with a mass of 1 .
338 M (cid:12) (right). Results obtained from the Bayesian modeling of the nuclear EOS in Ref. (74).formabilities, the LIGO/Virgo Scientific Collaboration reported (128) an inferred constraintof R = 11 . +1 . − . km for both of the neutron stars involved in the merger under the assump-tion that the shared EOS could support 2 M (cid:12) neutron stars. Only the combined mass M tot = 2 . +0 . − . M (cid:12) of the binary was very well measured from the gravitational wave-form, but neither of the individual component masses 1 .
17 M (cid:12) < M , < .
60 M (cid:12) wereexpected (126) to deviate more than 20% from the canonical value of M (cid:39) . (cid:12) , assum-ing low neutron star spins. In summary, GW170817 data were found to strongly favor thesoft EOSs predicted from ChEFT, though many other models (131) with generically stifferEOSs were consistent with the upper bounds on Λ and R . from GW170817.Current gravitational wave interferometers do not have large signal-to-noise ratios at thehigh frequencies expected during the post-merger ringdown phase and therefore GW170817provided no clues about the fate of the merger remnant. Nevertheless, an analysis of thespectral and temporal properties of the kilonova (132) optical counterpart to GW170817have been used (133, 134, 135, 136, 137) to infer the lifetime of the merger remnant.Depending on the component neutron star masses prior to merger (primarily the totalmass M tot ) as well as the maximum mass for a nonrotating neutron star M TOVmax , the mergerremnant can (i) undergo immediate collapse to a black hole, (ii) exist as a short-lived hy-permassive neutron star supported against collapse by differential rotation, (iii) persist asa longer-lived supermassive neutron star supported against collapse by rigid-body rotation,or (iv) form a stable massive neutron star. While there is still some uncertainty about whatranges of M tot will lead to each of the above four scenarios, it has been suggested (134)that prompt collapse will occur when M tot (cid:38) (1 . − . M TOVmax , hypermassive neutron starswill be created when 1 . M TOVmax (cid:38) M tot (cid:38) (1 . − . M TOVmax , and supermassive neutron starswill result when M tot (cid:46) . M TOVmax . Each merger outcome is expected to have a qualita-tively different optical counterpart and total mass ejection, since longer remnant lifetimesgenerically give rise to more and faster moving disk wind ejecta.Observations of the GW170817 kilonova suggest that the most likely outcome of theneutron star merger was the formation of a hypermassive neutron star, which would imply ••
60 M (cid:12) wereexpected (126) to deviate more than 20% from the canonical value of M (cid:39) . (cid:12) , assum-ing low neutron star spins. In summary, GW170817 data were found to strongly favor thesoft EOSs predicted from ChEFT, though many other models (131) with generically stifferEOSs were consistent with the upper bounds on Λ and R . from GW170817.Current gravitational wave interferometers do not have large signal-to-noise ratios at thehigh frequencies expected during the post-merger ringdown phase and therefore GW170817provided no clues about the fate of the merger remnant. Nevertheless, an analysis of thespectral and temporal properties of the kilonova (132) optical counterpart to GW170817have been used (133, 134, 135, 136, 137) to infer the lifetime of the merger remnant.Depending on the component neutron star masses prior to merger (primarily the totalmass M tot ) as well as the maximum mass for a nonrotating neutron star M TOVmax , the mergerremnant can (i) undergo immediate collapse to a black hole, (ii) exist as a short-lived hy-permassive neutron star supported against collapse by differential rotation, (iii) persist asa longer-lived supermassive neutron star supported against collapse by rigid-body rotation,or (iv) form a stable massive neutron star. While there is still some uncertainty about whatranges of M tot will lead to each of the above four scenarios, it has been suggested (134)that prompt collapse will occur when M tot (cid:38) (1 . − . M TOVmax , hypermassive neutron starswill be created when 1 . M TOVmax (cid:38) M tot (cid:38) (1 . − . M TOVmax , and supermassive neutron starswill result when M tot (cid:46) . M TOVmax . Each merger outcome is expected to have a qualita-tively different optical counterpart and total mass ejection, since longer remnant lifetimesgenerically give rise to more and faster moving disk wind ejecta.Observations of the GW170817 kilonova suggest that the most likely outcome of theneutron star merger was the formation of a hypermassive neutron star, which would imply •• Chiral EFT and the High-Density Nuclear EOS 23 value of M TOVmax = 2 . − .
35 M (cid:12) (135, 134, 137). Eliminating the possibility of promptblack hole formation in GW170817 also rules out compact neutron stars with small radiiand tidal deformabilities. In Ref. (133) such arguments were used to infer that the radius ofa 1 . (cid:12) neutron star must be larger than R . (cid:38) . (cid:38) >
400 can ruleout a significant set of soft EOSs (138), roughly half of those allowed in the analysis ofRef. (74). Combined gravitational wave and electromagnetic observations of binary neutronstar mergers together with more precise radius measurements therefore have the possibilityto strongly constrain the dense matter EOS and related neutron star properties in theregime of validity of ChEFT (138, 139). As a demonstration, in Fig. 8 we show the jointprobability distributions (74) for the pressure of nuclear matter at n = 2 n with (i) theradius of a 1 . (cid:12) neutron star, (ii) the tidal deformability of a 1 . (cid:12) neutron star, and(iii) the moment of inertia of PSR J0737-3039A with a mass of 1 .
338 M (cid:12) . Neutron stars are born following the gravitational collapse and ensuing supernova explosionof massive stars ( M (cid:38) (cid:12) ). The core bounce probes densities only slightly above normalnuclear saturation density (97) and leaves behind a hot ( T ∼ −
50 MeV) nascent proto-neutron star. During the subsequent Kelvin-Helmholtz phase that lasts tens of seconds, theproto-neutron star emits neutrinos, cools to temperatures
T < M ∗ , re-duces thermal pressure and leads to enhanced contraction of the initial proto-neutron star.This results in the emission of higher-energy neutrinos that support the explosion throughthe neutrino reheating mechanism (140). Since microscopic calculations based on ChEFTtend to predict larger values of the effective mass than many mean-field models (145), theseobservations help motivate recent efforts (146) to include thermal constraints from ChEFTdirectly into supernova EOS tables.Neutrino reactions also affect the nucleosynthesis outcome in neutrino-driven wind out-flows and the late-time neutrino signal that will be measured with unprecedented de-tail during the next galactic supernova. Charged-current neutrino-absorption reactions ν e + n → p + e − and ¯ ν e + p → n + e + , which can be calculated from the imaginarypart of vector and axial vector response functions, are especially sensitive (147, 148) tonuclear interactions and in particular the difference ∆ U = U n − U p between proton andneutron mean fields. The isovector mean field is especially important in the neutrinosphere,the region of warm and dense matter where neutrinos decouple from the exploding star.
24 Drischler, Holt, and Wellenhofer ecently, the calculation (149, 145) of nuclear response functions that include mean-fieldeffects from ChEFT interactions have shown that terms beyond the Hartree-Fock approxi-mation are needed for accurate modeling. In particular, resummed particle-particle ladderdiagrams were shown (149) to produce larger isovector mean fields due to resonant, non-perturbative effects in the NN interaction. Moreover, for neutral-current neutrino reactions,such as neutrino pair bremsstrahlung and absorption, resonant NN interactions were shownto significantly enhance reaction rates at low densities compared to the traditional one-pionexchange approximation (150).
5. Summary and outlook
In this article, we have reviewed recent progress in ChEFT calculations of nuclear matterproperties (with quantified uncertainties) and their implications in the field of nuclear as-trophysics. Combined with observational and experimental constraints, these microscopiccalculations provide the basis for improved modeling of supernovae, neutron stars, and neu-tron star mergers. In particular, we have highlighted MBPT as an efficient framework forstudying the nuclear EOS and transport properties across a wide range of densities, isospinasymmetries, and temperatures. We have also shown how advances in high-performancecomputing have enabled the implementation of two- and multi-nucleon forces in MBPTup to high orders in the chiral and many-body expansions. Finally, we have describednew tools for quantifying theoretical uncertainties (especially EFT truncation errors) toconfront microscopic calculations of the nuclear EOS with empirical constraints. Such sys-tematic studies are particularly important in view of EOS constraints anticipated in thenew era of multimessenger astronomy; e.g. , from gravitational wave detection, mass-radiusmeasurements of neutron stars, and experiments with neutron-rich nuclei.In the following we briefly summarize several open research directions at the interfaceof nuclear EFT and high-energy nuclear astrophysics. (i) EFT truncation errors and theircorrelations in density and across observables need to be studied with different many-bodyframeworks and nuclear interactions at arbitrary isospin asymmetry and finite temperature.(ii) Together with EFT truncation errors, the uncertainties in the LECs parametrizing theinteractions need to be quantified and propagated to nuclear matter properties using acomprehensive Bayesian statistical analysis. (iii) The full uncertainty quantification of thenuclear EOS will be aided by the development of improved order-by-order chiral NN and3N potentials and the study of different regularization schemes as well as delta-full chiralinteractions. (iv) Many-body calculations of nuclear matter properties beyond the nuclearEOS ( e.g. , linear response and transport coefficients) with chiral NN and 3N interactions arerequired for more accurate numerical simulations of supernovae and neutron star mergers.(v) Future neutron star observations will provide stringent tests of nuclear forces and nuclearmany-body methods in a regime that is presently largely unconstrained. The interplaybetween observation, experiment, and theory in the next decade can be expected to resultin many further advances in our understanding of strongly interacting matter.
DISCLOSURE STATEMENT
The authors are not aware of any affiliations, memberships, funding, or financial holdingsthat might be perceived as affecting the objectivity of this review. ••
The authors are not aware of any affiliations, memberships, funding, or financial holdingsthat might be perceived as affecting the objectivity of this review. •• Chiral EFT and the High-Density Nuclear EOS 25
CKNOWLEDGMENTS
We are very grateful to our collaborators and colleagues for fruitful discussions over the yearsthat contributed in one way or another to the completion of this review article. We specifi-cally thank Evgeny Epelbaum for providing us with the diagrams in Figure 1 and YeunhwanLim for sharing the results of Figures 7 and 8. C.D. acknowledges support by the Alexandervon Humboldt Foundation through a Feodor-Lynen Fellowship and thanks the N3AS andBUQEYE collaboration for creating warm research environments. This material is basedupon work supported by the U.S. Department of Energy, Office of Science, Office of NuclearPhysics, under the FRIB Theory Alliance award DE-SC0013617. The work of J.W.H. wassupported by the National Science Foundation under Grant No. PHY1652199 and by theU.S. Department of Energy National Nuclear Security Administration under Grant No.DE-NA0003841. C.W. has been supported by the Deutsche Forschungsgemeinschaft (DFG,German Research Foundation) – Project-ID 279384907 – SFB 1245.
LITERATURE CITED
1. ¨Ozel F, Freire P.
Annu. Rev. Astron. Astrophys.
Rev. Mod. Phys.
Prog. Part. Nucl. Phys.
Front. in Phys.
Ann. Rev. Nucl. Part. Sci.
Ann. Rev. Nucl. Part. Sci.
Astrophys. J.
Rev. Mod. Phys.
Phys. Rep.
Rev. Mod. Phys.
Rev. Mod. Phys.
Front. in Phys.
Phys. Rev. C
Eur. Phys. J. A
Phys. Rev. Lett.
Phys. Rev. C
Front. in Phys.
Phys. Rev. C
Nucl. Phys. A
Prog. Part. Nucl. Phys.
Phys. Rev. C
Phys. Rev. C
Phys. Rev. Lett.
Phys. Rev. C
Phys. Lett. B
Phys. Rev.
J. Stat. Mech.
Rev. Mod. Phys.
Phys. Rev. C
Phys. Rev.
26 Drischler, Holt, and Wellenhofer
5. Wellenhofer C, Holt JW, Kaiser N, Weise W.
Phys. Rev. C
Phys. Rev. C
Rep. Prog. Phys
Phys. Rev. Lett.
Prog. Part. Nucl. Phys.
Front. in Phys.
Phys. Rev. C
Phys. Rev. C
Phys. Rev. C
Phys. Rev. Lett.
Phys. Rev. Res.
Ann. Rev. Nucl. Part. Sci.
Phys. Rev. C
Front. Phys.
Phys. Rev. C
Phys. Rev. C
Phys. Rev. C
Phys. Rev. C
Phys. Rev. C
Eur. Phys. J. A
Phys. Rev. C
Phys. Rev. C
Phys. Rev. C
Phys. Rev. C
Phys. Rev. C
Phys. Rev. C
Phys. Rev. C
Prog. Part. Nucl. Phys.
J. Phys. G
Phys. Rev. C
Eur. Phys. J. A
Phys. Rev. Lett.
Phys. Rev. C
Phys. Rev. Lett.
Phys. Rev. Lett.
J. Phys. G
Phys. Rev. Lett.
Phys. Rev. C
Phys. Rev. Res.
Phys. Rev. C
Astrophys. J.
Phys. Rev. C
Phys. Rev. C
Phys. Rev. C
Phys. Rev. C
Phys. Rev. C • Chiral EFT and the High-Density Nuclear EOS 27
6. Petschauer S, et al.
Front. in Phys.
Prog. Part. Nucl. Phys.
Phys. Rept.
Phys. Rev. C
Phys. Atom. Nucl.
Phys. Rev. C
Amer. Inst. Chem. Eng. J.
Nucl. Phys. A
Phys. Rev. C
Phys. Rev. Lett.
Phys. Rev. D
Rev. Mod. Phys.
Astrophys. J.
Phys. Lett. B
Astrophys. J.
Astron. Astrophys.
Astrophys. J.
Astrophys. J.
Nature Phys. (2020)105. Hebeler K, Lattimer J, Pethick C, Schwenk A.
Phys. Rev. Lett.
Eur. Phys. J. A
Eur. Phys. J. A
Astrophys. J.
Astrophys. J.
Science
Astrophys. J.
Mon. Not. Roy. Astr. Soc.
Phys. Rev. C
Astrophys. J.
Phys. Rev. C
Nature
Science
Nature Astronomy
Phys. Rev. D
Phys. Rev. D
Phys. Rev. Lett.
Phys. Rev. D
Phys. Rev. D
Phys. Rev. Lett.
Phys. Rev. Lett.
Astrophys. J. Lett.
Phys. Rev. Lett.
Phys. Rev. C
Phys. Rev. X
Phys. Rev. Lett.
Mon. Not. Roy. Astr. Soc.
Astrophys. J. Lett.
Astrophys. J. Lett.
Phys. Rev. D
Astrophys. J. Lett.
28 Drischler, Holt, and Wellenhofer
37. Rezzolla L, Most ER, Weih LR.
Astrophys. J. Lett.
Nature Astron.
Phys. Rept.
Astrophys. J.
Astrophys. J.
Publ. Astron. Soc. Aust.
Phys. Rev. C
Phys. Rev. C
Phys. Rev. C
Phys. Rev. C
Phys. Rev. Lett.
Phys. Rev. C
Phys. Rev. Lett. ••