Neutron matter at next-to-next-to-next-to-leading order in chiral effective field theory
Neutron matter presents a unique system for chiral effective field theory (EFT), because all many-body forces among neutrons are predicted to next-to-next-to-next-to-leading order (NLO). We present the first complete NLO calculation of the neutron matter energy. This includes the subleading three-nucleon (3N) forces for the first time and all leading four-nucleon (4N) forces. We find relatively large contributions from NLO 3N forces. Our results provide constraints for neutron-rich matter in astrophysics with controlled theoretical uncertainties.
pacs:21.65.Cd, 21.30.-x, 26.60.Kp, 12.39.Fe
The physics of neutron matter ranges from universal properties at low densities to the structure of extreme neutron-rich nuclei and the densest matter we know to exist in neutron stars. For these extreme conditions, controlled calculations with theoretical error estimates are essential. Chiral EFT provides such a systematic expansion for nuclear forces RMP (). This is particularly exciting for neutron matter and neutron-rich systems, because all three- and four-neutron forces are predicted to NLO nm ().
Neutron matter based on chiral EFT has been studied using lattice simulations NLOlattice () at low densities, (with saturation density ), and following an in-medium chiral perturbation theory approach Wolfram1 (); Wolfram2 (), where low-energy couplings are adjusted to empirical nuclear matter properties. In addition, the renormalization group (RG) has been used to evolve chiral EFT interactions to low momenta PPNP (), which has enabled perturbative calculations for nucleonic matter nm (); nucmatt2 (). While these constrain the properties of neutron-rich matter to a much higher degree than is reflected in neutron star modeling nstar (), the dominant uncertainties are due to 3N forces, which were included only to NLO. A consistent inclusion of higher-order many-body forces is therefore key.
Here we present the first calculations at nuclear densities based directly on chiral EFT interactions without RG evolution. To this end, we have studied the perturbative convergence of chiral two-nucleon (NN) potentials for neutron matter in detail, and found that the available NLO and NLO potentials with lower cutoffs are perturbative. This is supported by small Weinberg eigenvalues at low energies indicating the perturbative convergence in the particle-particle channel PPNP (). In neutron matter, it comes as a result of effective range effects dEFT (), which weaken NN interactions at higher momenta, combined with weaker tensor forces among neutrons, and with limited phase space at finite density due to Pauli blocking nucmatt1 ().
At the NN level we use the NLO and NLO potentials developed by Epelbaum, Glöckle and Meißner (EGM) EGM () with and ( denotes the cutoff in the Lippmann-Schwinger equation and in the two-pion-exchange spectral-function regularization, respectively). We also use the NLO NN potential of Entem and Machleidt (EM) EM (), which is most commonly used in nuclear structure calculations. The larger NN potentials of EGM and EM have been found to be nonperturbative Note1 () and are therefore not included. Moreover, the LO NN contact couplings in the and EGM potentials break Wigner symmetry perturbatively (at the interaction level), with a repulsive spin-independent and an unnaturally large spin-dependent , leading to unexpectedly large -dependent 3N forces.
In this Letter, we include for the first time all NLO 3N and 4N forces, which have been derived only recently IR (); N3LOlong (); N3LOshort (); 4N (), in addition to the NLO 3N forces. Figure 1 shows our complete NLO calculation of the neutron matter energy as our main result, where the bands include estimates of the theoretical uncertainties due to the many-body calculation and in the many-body forces.
For neutrons, only the two-pion-exchange 3N forces contribute at NLO nm (). For the corresponding low-energy constants and , we take the range of values from a high-order analysis Krebs2012 (), at NLO: and (which includes the values in the EGM and EM NN potentials), and when the NLO 3N forces are included in an NLO calculation: and . It has been shown nm () that the NLO 3N force contributions in neutron matter can be to a good approximation calculated at the Hartree-Fock level. In this first calculation, we therefore evaluate the NLO 3N and 4N force contributions to the energy per particle at the Hartree-Fock level. The -body contributions are then given by
with short-hand notation . denotes the -body antisymmetrizer and the Fermi-Dirac distributions at zero temperature. We use a Jacobi-momenta regulator; in terms of given by with and 3N/4N cutoff . For the nucleon and pion mass, we use and , and for the axial coupling and the pion decay constant .
Chiral 3N forces at NLO can be grouped into
where we take the long-range parts, the subleading two-pion-exchange, the two-pion–one-pion-exchange and the pion-ring 3N forces, from Ref. N3LOlong (), and the short-range parts, the two-pion-exchange–contact and relativistic -corrections 3N forces from Ref. N3LOshort (). In Fig. 2, we give the individual Hartree-Fock contributions to the neutron matter energy. The evaluation is aided because parts of the different 3N force topologies vanish for neutrons, and the results have been checked by two independent calculations. The details of the calculation will be presented in a future paper. At the Hartree-Fock level, the 3N/4N contributions change by if the cutoff is taken to infinity (i.e., ), but we will also include NLO 3N forces beyond Hartree-Fock. This requires a consistently used regulator. Estimates of the theoretical uncertainty are provided by varying the 3N/4N cutoff.
The two-pion-exchange 3N forces at NLO can be largely written as shifts of the low-energy constants, and N3LOlong () of the NLO 3N forces, plus a smaller contribution. The resulting energy of about per particle at saturation density in Fig. 2 is of the NLO 3N energy, as expected based on the chiral EFT power counting. In contrast, the two-pion–one-pion-exchange 3N force contributions, which include 14 diagrams, are relatively large with per particle at saturation density. Of similar, but opposite size are the pion-ring 3N force contributions, with per particle at . The shorter-range parts of NLO 3N forces depend on the momentum-independent NN contacts, and , which we take consistently from the NLO EM/EGM potential used. The contributions from the two-pion-exchange–contact 3N forces include 11 diagrams and depend only on . The resulting energy ranges from to at depending on the NN potential used. These larger 3N results at NLO are consistent with contributions from the large constants at NLO exactly in these three topologies Krebs2012 (). This shows that higher-order many-body forces still need to be investigated and that a chiral EFT with explicit excitations may be more efficient, since this would capture these effects already at NLO. Finally, the relativistic-corrections 3N forces depend also on and N3LOshort () and contribute at the few hundred keV level.
The 4N force contributions in Fig. 2 are an order of magnitude smaller than those from the NLO 3N forces and of similar size as the 3N relativistic corrections. We follow the 4N force notation through of Ref. 4N (), and include the direct and all 23 exchange terms. Due to the spin-isospin structure, only 3 topologies contribute to neutron matter: the three-pion-exchange 4N forces and , and the pion-pion-interaction 4N forces . The 4N forces and involving the contact vanish in neutron matter due to their spin structure. We find a total 4N force contribution of per particle at . The and energies largely cancel Riska (), and their sum agrees with the very small per particle at of Ref. Fiorilla2011 (), which considered these two parts.
Since diagrams beyond Hartree-Fock involving NN interactions and NLO 3N forces (in particular with the larger at NLO 3N and without RG evolution) provide non-negligible contributions nm (), we include all such diagrams to second order, as well as particle-particle diagrams to third order, which is technically possible based on Ref. nucmatt2 (). In addition to using NN potentials with different cutoffs and varying the 3N/4N cutoffs, we include estimates of the theoretical uncertainties of the constants and in the convergence of the many-body calculation. The latter is probed by studying the sensitivity of the energy to the single-particle spectrum used. We find that the energy changes from second to third order, employing a free or Hartree-Fock spectrum, by () per particle at () for the EGM 450/500, 450/700, EM 500 NLO potentials, respectively. The results, which include all these uncertainties, are displayed by the bands in Fig. 1. Understanding the cutoff dependence and developing improved power counting schemes remain important open problems in chiral EFT powercounting (). For the neutron matter energy at , our first complete NLO calculation yields per particle. If we were to omit the results based on the EM 500 NLO potential, as it converges slowest at , the range would be .
As we find relatively large contributions from NLO 3N forces, it is important to study the EFT convergence from NLO to NLO. This is shown in Fig. 3 for the EGM potentials (NLO is not available for EM), where the NLO results are found to overlap with the NLO band across a range around at saturation density. As expected from the net-attractive NLO 3N contributions in Fig. 2, the NLO band yields lower energies. For the NLO band, we have estimated the theoretical uncertainties in the same way, and the neutron matter energy ranges from per particle at . The theoretical uncertainty is reduced from NLO to NLO to , but not by a factor based on the power counting estimate. This reflects the large 3N contributions at NLO, and is similar to the convergence pattern observed in chiral NN potentials RMP ().
The neutron matter energy in Fig. 1 is in very good agreement with NLO lattice results NLOlattice () and Quantum Monte Carlo simulations GC () at very low densities (see also the inset) and approximately reproduces the scaling , which we attribute to effective-range effects combined with low cutoffs dEFT (). At nuclear densities, we compare our NLO results with variational calculations based on phenomenological potentials (APR) APR (), which are within the NLO band, but do not provide theoretical uncertainties. In addition, we compare the density dependence with results from Auxiliary Field Diffusion MC calculations (GCR) GCR () based on nuclear force models adjusted to an energy difference of between neutron matter and the empirical saturation point. The density dependence is similar to the NLO band, but the GCR results are higher below .
The NLO band provides key constraints for the nuclear equation of state and for astrophysics. Figure 4 shows, following Ref. LL (), the allowed range for the symmetry energy and its density dependence (for details on the determination of and see Ref. nstar ()). Compared to the results from RG-evolved chiral interactions with 3N forces at NLO only nstar (), we find the same correlation (with the same slope), but not as tight due to the additional density dependences at NLO. The NLO ranges for and are and . The two neutron-matter bands in Fig. 4 are complementary, because the RG evolution in Hebeler et al. nstar () improves the many-body convergence, while the band presented in this work is the first consistent NLO calculation. The predicted NLO range, as well as that of Hebeler et al. nstar (), are in agreement with constraints obtained from energy-density functionals for nuclear masses Kortelainen2010 () and from the Pb dipole polarizability Tamii2011 (). In the future, the NLO band can be narrowed further by a higher-order many-body calculation with NLO 3N forces and by taking into account excitations (explicitly or through large contributions at NLO Krebs2012 ()). Combined with the heaviest neutron star Demorest () and a general extension to high densities nstar (), our NLO energy band leads to a radius range of km for a typical neutron star, in remarkable agreement with Ref. nstar (). For an alternative determination using in-medium chiral perturbation theory for all densities see Ref. Wolfram2 ().
We have presented the first complete NLO calculation of the neutron matter energy, including NN, 3N and 4N forces, with the first application of NLO 3N forces to many-body systems. The significant contributions from NLO 3N forces show that their inclusion will also be very important for nuclear structure and reactions. Our results provide constraints for the nuclear equation of state and for neutron-rich matter in astrophysics, and highlight the exciting role neutron matter and neutron-rich systems play in chiral EFT, where all many-neutron forces are predicted. The large contributions from NLO 3N forces signal the importance of contributions at nuclear densities.
We thank E. Epelbaum, R. J. Furnstahl, N. Kaiser and H. Krebs for discussions. This work was supported by the Helmholtz Alliance Program of the Helmholtz Association, contract HA216/EMMI “Extremes of Density and Temperature: Cosmic Matter in the Laboratory”, the DFG through Grant SFB 634, the ERC Grant No. 307986 STRONGINT, and the NSF Grant No. PHY–1002478.
- (1) E. Epelbaum, H.-W. Hammer and U.-G. Meißner, Rev. Mod. Phys. 81, 1773 (2009).
- (2) K. Hebeler and A. Schwenk, Phys. Rev. C 82, 014314 (2010).
- (3) E. Epelbaum et al., Eur. Phys. J. A 40, 199 (2009).
- (4) N. Kaiser, S. Fritsch and W. Weise, Nucl. Phys. A 697, 255 (2002).
- (5) W. Weise, Prog. Part. Nucl. Phys. 67, 299 (2012).
- (6) S. K. Bogner, R. J. Furnstahl and A. Schwenk, Prog. Part. Nucl. Phys. 65, 94 (2010).
- (7) K. Hebeler et al., Phys. Rev. C 83, 031301(R) (2011).
- (8) K. Hebeler et al., Phys. Rev. Lett. 105, 161102 (2010) and in preparation.
- (9) A. Schwenk and C. J. Pethick, Phys. Rev. Lett. 95, 160401 (2005).
- (10) S. K. Bogner et al., Nucl. Phys. A 763, 59 (2005).
- (11) E. Epelbaum, W. Glöckle and U.-G. Meißner, Eur. Phys. J. A 19, 401 (2004); Nucl. Phys. A 747, 362 (2005).
- (12) D. R. Entem and R. Machleidt, Phys. Rev. C 68, 041001(R) (2003); Phys. Rept. 503, 1 (2011).
- (13) For the EGM potential, this is the case only with the larger in 3N forces at NLO.
- (14) S. Ishikawa and M. R. Robilotta, Phys. Rev. C 76, 014006 (2007).
- (15) V. Bernard et al., Phys. Rev. C 77, 064004 (2008).
- (16) V. Bernard et al., Phys. Rev. C 84, 054001 (2011).
- (17) E. Epelbaum, Phys. Lett. B 639, 456 (2006).
- (18) H. Krebs, A. Gasparyan and E. Epelbaum, Phys. Rev. C 85, 054006 (2012).
- (19) H. McManus and D. O. Riska, Phys. Lett. 92B, 29 (1980).
- (20) S. Fiorilla, N. Kaiser and W. Weise, Nucl. Phys. A 880, 65 (2012).
- (21) A. Nogga, R. G. E. Timmermans and U. van Kolck, Phys. Rev. C 72, 054006 (2005).
- (22) A. Gezerlis and J. Carlson, Phys. Rev. C 81, 025803 (2010).
- (23) A. Akmal, V. R. Pandharipande and D. G. Ravenhall, Phys. Rev. C 58, 1804 (1998).
- (24) S. Gandolfi, J. Carlson and S. Reddy, Phys. Rev. C 85, 032801(R) (2012).
- (25) J. M. Lattimer and Y. Lim, arXiv:1203.4286.
- (26) M. Kortelainen et al., Phys. Rev. C 82, 024313 (2010).
- (27) A. Tamii et al., Phys. Rev. Lett. 107, 062502 (2011).
- (28) P. B. Demorest et al., Nature 467, 1081 (2010).