# Tensor Fermi liquid parameters in nuclear matter

from chiral effective field theory

###### Abstract

We compute from chiral two- and three-body forces the complete quasiparticle interaction in symmetric nuclear matter up to twice nuclear matter saturation density. Second-order perturbative contributions that account for Pauli-blocking and medium polarization are included, allowing for an exploration of the full set of central and noncentral operator structures permitted by symmetries and the long-wavelength limit. At the Hartree-Fock level, the next-to-next-to-leading order three-nucleon force contributes to all noncentral interactions, and their strengths grow approximately linearly with the nucleon density up that of saturated nuclear matter. Three-body forces are shown to enhance the already strong proton-neutron effective tensor interaction, while the corresponding like-particle tensor force remains small. We also find a large isovector cross-vector interaction but small center-of-mass tensor interactions in the isoscalar and isovector channels. The convergence of the expansion of the noncentral quasiparticle interaction in Landau parameters and Legendre polynomials is studied in detail.

## I Introduction

Fermi liquid theory Landau (1957a, b, 1959); Migdal and Larkin (1964) is widely used to describe the transport, response and dynamical properties of nuclear many-body systems Brown (1971); Iwamoto and Pethick (1982); Haensel and Jerzak (1982); Bäckman et al. (1985); Wambach et al. (1993); Friman and Rho (1996); Benhar and Valli (2007); Pastore et al. (2012); Benhar et al. (2013). The key quantity in this theory is the quasiparticle interaction, defined as the second functional derivative of the energy with respect to the quasiparticle distribution function. For many years the primary focus of investigation has been the central part of the quasiparticle interaction and its associated Fermi liquid parameters, which are directly related to static properties of the interacting ground state such as the incompressibility, isospin-asymmetry energy and magnetic susceptibility. The central terms include scalar operators in spin and isospin space, but more recently noncentral contributions Haensel and Dabrowski (1975); Schwenk and Friman (2004) that couple spin and momenta have been studied together with their impact on the density and spin-density response functions of neutron matter Olsson et al. (2004); Bacca et al. (2009); Pethick and Schwenk (2009); Davesne et al. (2015). Extending these results to nuclear matter with equal numbers of protons and neutrons and to systems with arbitrary isospin asymmetry will be needed to better understand neutrino transport and emissivity in neutron stars, proto-neutron star cooling Roberts et al. (2012), electron transport in neutron stars Bertoni et al. (2015), the evolution of shell structure and single-particle states in nuclei far from stability Otsuka et al. (2005, 2006), and nuclear collective excitations (spin and spin-isospin modes together with rotational modes of deformed nuclei) Friman and Haensel (1981); Cao et al. (2009); Co’ et al. (2012).

An important motivation of the present work is to provide microscopic guidance for the tensor forces employed in modern mean field effective interactions and nuclear energy density functionals. Including as well new estimates and uncertainties on the central Fermi liquid parameters, which are more directly related to nuclear observables, the present study will complement other recent efforts Pudliner et al. (1996); Brown and Schwenk (2014); Roggero et al. (2015); Buraczynski and Gezerlis (2016); Rrapaj et al. (2016); Zhang et al. (2018) to constrain energy density functionals from microscopic many-body theory. The importance of tensor forces in mean field modeling is a question of ongoing debate. While there is skepticism Lesinski et al. (2007); Lalazissis et al. (2009) that tensor forces can lead to a meaningful improvement in fits to nuclear ground state energies, there is strong evidence that the description of single-particle energies Otsuka et al. (2005, 2006); Colò et al. (2007); Zou et al. (2008), beta-decay half-lives Minato and Bai (2013), and spin-dependent collective excitations Cao et al. (2009); Co’ et al. (2012) are systematically improved with the inclusion of tensor forces (for a recent review, see Ref. Sagawa and Colò (2014)). One of the main driving questions is the extent to which the effective medium-dependent tensor force in mean field models resembles the fundamental tensor component of the free-space nucleon-nucleon interaction arising from + meson exchange. A main conclusion of the present work is that the proton-neutron effective tensor force is enhanced over the free-space tensor interaction due to three-body forces and second-order perturbative contributions. On the other hand, the proton-proton and neutron-neutron tensor forces are considerably smaller in magnitude. In addition, we find evidence for a large isovector cross-vector interaction that to our knowledge has not been previously studied in phenomenological mean field modeling of nuclei.

The quasiparticle interaction can be computed microscopically from realistic two- and three-body forces starting from the perturbative expansion of the energy density and taking appropriate functional derivatives with respect to the Fermi distribution functions. For nuclear or astrophysical systems with densities near or above that of saturated nuclear matter, it is essential to consider the effects of three-body forces. To date three-nucleon forces have been included in calculations of the central and exchange-tensor quasiparticle interaction in nuclear matter Brown et al. (1977); Shen et al. (2003); Zuo et al. (2003); Kaiser (2006); Gambacurta et al. (2011); Holt et al. (2012) and the full quasiparticle interaction in neutron matter Holt et al. (2013). In the present work our aim is to extend the calculations in Ref. Holt et al. (2013) to the case of symmetric nuclear matter. This is a natural step before considering the more general case of isospin-asymmetric nuclear matter.

We take as a starting point a class Entem and Machleidt (2003); Coraggio et al. (2007); Marji et al. (2013); Coraggio et al. (2013, 2014); Sammarruca et al. (2015) of realistic two and three-body nuclear forces derived within the framework of chiral effective field theory Weinberg (1979); Epelbaum et al. (2009); Machleidt and Entem (2011). The two-body force is treated at both next-to-next-to-leading order (N2LO) and N3LO in the chiral power counting, while the three-body force is only considered at N2LO. Although the inclusion of consistent three-body forces at N3LO Bernard et al. (2008, 2011) in the chiral power counting will be needed for improved theoretical uncertainty estimates Tews et al. (2013); Drischler et al. (2016, 2017), the present set of nuclear force models has been shown to give a good description of nuclear matter saturation Coraggio et al. (2014); Holt and Kaiser (2017), the nuclear liquid-gas phase transition Wellenhofer et al. (2014), and the volume components of nucleon-nucleus optical potentials Holt et al. (2013, 2016) when used at second order in many-body perturbation theory. In addition to the order in the chiral expansion, the resolution scale (related to the momentum-space cutoff in the nuclear potential) is varied in order to assess the theoretical uncertainties.

The paper is organized as follows. In Section II we review the derivation of the quasiparticle interaction and associated Fermi liquid parameters from microscopic nuclear two- and three-body interactions. We present a general method to extract the central and noncentral components of the quasiparticle interaction from appropriate linear combinations of spin- and isospin-space matrix elements. We also benchmark the numerical calculations of the second-order contributions to the quasiparticle interaction against semi-analytical results for model interactions of one-boson exchange type. In Section III we present analytical expressions for the Landau parameters arising from the leading NLO chiral three-body force together with numerical results for the second-order contributions from two- and three-body forces. We end with a summary and conclusions.

## Ii Quasiparticle interaction in symmetric nuclear matter

### ii.1 General structure of the quasiparticle interaction

The quasiparticle interaction in symmetric nuclear matter has the general form Schwenk and Friman (2004)

(1) |

where

(2) | |||||

and analogously for except with the replacement . The relative momentum is defined by and the center of mass momentum is given by . The tensor operator has the usual form . The interaction in Eq. (2) is invariant under rotations, time-reversal, parity, and the exchange of particle labels. The presence of the medium breaks Galilean invariance, and two new structures (the “center-of-mass tensor” and “cross-vector” operators) arise Schwenk and Friman (2004) that depend explicitly on the center-of-mass momentum . Neither of these terms are found in the free-space nucleon-nucleon potential.

By assumption the two quasiparticle momenta and lie on the Fermi surface () and therefore the scalar functions admit a decomposition in Legendre polynomials:

(3) |

where and and . The expansion coefficients are referred to as the Fermi liquid parameters. In relating the Fermi liquid parameters to nuclear observables, it is often convenient to multiply by the density of states

(4) |

with the effective nucleon mass, to obtain dimensionless parameters .

The ten scalar functions in Eq. (2) can be extracted from linear combinations of the spin-space and isospin-space matrix elements, but the decomposition will depend on the orientation of the orthogonal vectors and . For instance, if and , then

(5) |

with the notation .

The quasiparticle interaction is defined as the second functional derivative of the energy with respect to the occupation probabilities :

(6) | |||||

where is a normalization volume. The quasiparticle interaction in momentum space has units fm, labels the spin quantum number of quasiparticle , and labels the isospin quantum number. In the present work we consider contributions to the quasiparticle interaction up to second order in many-body perturbation theory.

### ii.2 Two-body force contributions

The first- and second-order terms in the perturbative expansion of the ground-state energy from two-body forces are given by

(7) |

(8) |

where , , and indicates an antisymmetrized interaction. In Eqs. (7) and (8) the sums run over momentum, spin and isospin.

Functionally differentiating Eq. (7) with respect to and yields for the first-order contribution to the quasiparticle interaction

(9) |

shown diagrammatically in Fig. 1(a). Since this is just the antisymmetrized free-space nucleon-nucleon potential, only the four central and two exchange-tensor terms in the quasiparticle interaction can be generated and the total spin (with associated quantum number ) is conserved. From the second-order energy in Eq. (8), three different contributions to the quasiparticle interaction arise which are distinguished by intermediate particle-particle, hole-hole, and particle-hole states shown diagrammatically in Figs. 1(b), 1(c), and 1(d), respectively. They have the form

(10) |

(11) |

(12) |

Eqs. (9)–(12) can be evaluated for realistic nuclear interactions by first decomposing the potential matrix elements into a partial-wave sum. The Fermi liquid parameters are then obtained by integrating over the angle between and with appropriate Legendre polynomials as weighting functions. For the first-order term, as well as the second-order particle-particle and hole-hole diagrams, the partial-wave decomposition is straightforward since the two quasiparticle states are both in the incoming or outgoing state. However, the evaluation of the second-order particle-hole diagram is more complicated due to the cross-coupling of the quasiparticle states in one incoming and one outgoing state. We state here the final expressions, and for additional details the reader is referred to Ref. Holt et al. (2013). As already mentioned, the first-order contribution to the quasiparticle interaction is just the antisymmetrized free-space potential:

(13) | |||||

where . The second-order terms are given by

(14) |

(15) |

(16) | |||||

where in the particle-particle and hole-hole diagrams , , , are the associated Legendre functions, and with . In the particle-hole diagram we have additionally , , , and . From the matrix elements of the second-order particle-hole contribution in the uncoupled spin and isospin basis, it is trivial through recoupling to generate the terms needed in Eq. (5) to extract the Fermi liquid parameters.

Given the numerical complexity of Eqs. (14)–(16), we have benchmarked our codes against semi-analytical results from model interactions. As a first case we consider a modified pseudoscalar interaction of the form

(17) |

where is the momentum transfer, is a dimensionless coupling constant, and is the mass parameter chosen to be large enough to achieve good convergence in momentum integrals and partial wave summations. We choose for concreteness and MeV. As a second interesting case, we consider the interference between an isoscalar central and spin-orbit interaction of the form

(18) |

(19) |

where is the incoming relative momentum. We choose the same values for and as in the modified pseudoscalar case above.

In Ref. Holt et al. (2013) it was shown that the second-order particle-particle and hole-hole diagrams can only generate the central, relative momentum tensor, and center-of-mass tensor components of the quasiparticle interaction. In fact, only the combination of a spin-orbit force with any non-spin-orbit force in the second-order particle-hole diagram can lead to a cross-vector term. These general conclusions are exemplified in the test interactions considered in Eqs. (17)–(19). In particular, the isoscalar relative-momentum tensor and center-of-mass tensor Fermi liquid parameters from modified pseudoscalar exchange at second order are shown in the left and middle plots of Fig. 2 employing both the partial-wave decomposition in Eqs. (14)–(16) as well as semi-analytical expressions similar to those in Ref. Holt et al. (2013). The isovector contributions only differ from the isoscalar contributions by integer factors and therefore are not shown explicitly. The modified pseudoscalar interaction at second order also gives rise to central components of the quasiparticle interaction, but these have been considered in previous work Holt et al. (2011). From Fig. 2 we see that the numerical agreement across the full range of densities considered, , is excellent. In the rightmost plot of Fig. 2 we show the Fermi liquid parameters associated with the cross-vector interaction from the interference term between a central and spin-orbit force. Again the numerical agreement between the two methods is very good.

### ii.3 Three-body force contributions

We next consider contributions to the quasiparticle interaction from three-body forces. The Hartree-Fock energy is given by

(20) |

where the totally antisymmetrized three-body potential is given by . In the present work we employ the N2LO chiral three-body force, which includes a long-range two-pion exchange component , a one-pion exchange contribution , and a pure contact force . The two-pion exchange three-nucleon interaction has the form

(21) |

where , MeV, MeV is the average pion mass, denotes the difference between the final and initial momenta of nucleon , and the isospin tensor is given by

(22) |

The one-pion exchange component of the three-nucleon interaction is defined by

(23) |

where MeV. Finally, the three-nucleon contact interaction has the form

(24) |

In pure neutron matter only the terms proportional to and contribute to the ground state energy and quasiparticle interaction, but for symmetric nuclear matter in general all terms are needed. Taking two functional derivatives of Eq. (20) with respect to and yields

(25) |

Since the three-body force is symmetric under the interchange of particle labels, we can rewrite Eq. (25) without loss of generality as

(26) |

In general there are nine distinct direct (and exchange) contributions to the quasiparticle interaction from a three-body force. In Fig. 3 we show the direct terms from the N2LO chiral three-nucleon interaction (exchange terms can be obtained by swapping the two outgoing lines). As seen in Fig. 3 there are three topologically distinct contributions from the two-pion exchange three-body force . Contribution ‘(2)’ represents the sum of four reflected diagrams, while contribution ‘(3)’ represents the sum of two reflected diagrams. The one-pion exchange contribution gives rise to two topologically distinct diagrams, shown as ‘(4)’ and ‘(5)’ in Fig. 3. Finally, there is a single diagram ‘(6)’ coming from the three-body contact force . As shown in the Appendix, this diagram contributes only to the central components of the quasiparticle interaction.

At second order in perturbation theory we include the effects of three-body forces by first constructing a density-dependent two-body force , as described in detail in Refs. Holt et al. (2009, 2010). In Eqs. (10)–(12) we then replace with . This approximation accounts for only a subset of the full second-order contributions from three-body forces.

## Iii Results

In the present section we focus on the noncentral components of the quasiparticle interaction from the five different chiral nuclear forces {n2lo450, n2lo500, n3lo414, n3lo450, n3lo500} Entem and Machleidt (2003); Coraggio et al. (2007); Marji et al. (2013); Coraggio et al. (2013, 2014); Sammarruca et al. (2015). We focus primarily on the role of three-body forces and second-order perturbative contributions. The quality of the nuclear force models and perturbative many-body method is benchmarked by comparing the nuclear incompressibility, isospin asymmetry energy, and effective mass (which are related to specific central Fermi liquid parameters) with empirical values. We also study the convergence of the Legendre polynomial decomposition for both central and noncentral forces.

### iii.1 First-order perturbative contributions to Fermi liquid parameters

At first order in perturbation theory, two-nucleon forces contribute only to the relative momentum tensor noncentral Fermi liquid parameters () due to the underlying Galilean invariance of the free-space interaction. In Fig. 4 we show as solid-circle and solid-square symbols the magnitude of the dimensionless Fermi liquid parameters () and () from two-body forces as a function of density. The error bars are calculated as the standard deviation of the results from the five nucleon-nucleon potentials considered in the present work and represent an estimate of systematic uncertainties in the construction of realistic two- and three-body force models. Note that in this section, we employ the Landau effective mass that enters into the density of states, Eq. (4), computed from the full quasiparticle interaction including second-order perturbative contributions (see Eq. (42) below). Comparing the results from two-body forces in Fig. 4, we find the approximate relationship and , which in fact exactly holds for all in the case of a pure one-pion exchange (OPE) nucleon-nucleon potential Brown et al. (1977).

Next we present analytical expressions for the noncentral Fermi-liquid parameters in nuclear matter from the N2LO chiral three-nucleon interaction. The associated low-energy constants have been fitted separately for each nuclear potential and are compiled in Table 1.

n2lo450 | |||||
---|---|---|---|---|---|

n2lo500 | |||||

n3lo414 | |||||

n3lo450 | |||||

n3lo500 |

We present individually the Fermi liquid parameters arising from the five diagrammatic contributions in Fig. 3. The pion self-energy correction leads to a relative-momentum tensor interaction

(27) |

where and is the Legendre polynomial of degree . The pion-exchange vertex correction, , likewise gives rise to relative-tensor force of the form

(28) |

Since and renormalize the one-pion exchange interaction through self-energy and vertex corrections, the associated Fermi liquid parameters obey the generic relationship as seen explicitly above.

The Pauli-blocked two-pion exchange contribution, , gives rise to a richer set of spin and isospin structures that lead to contributions to all of the noncentral Fermi liquid parameters (). For the relative tensor interaction we find

(29) | |||

with . The center-of-mass tensor Fermi liquid parameters in the isoscalar and isovector channel are given by