# Fock contributions to nuclear symmetry energy and its slope parameter based on Lorentz-covariant decomposition of nucleon self-energies

###### Abstract

Using relativistic Hartree-Fock (RHF) approximation, we study the effect of Fock terms on the nuclear properties not only around the saturation density, , but also at higher densities. In particular, we investigate how the momentum dependence due to the exchange contribution affects the nuclear symmetry energy and its slope parameter, using the Lorentz-covariant decomposition of nucleon self-energies in an extended version of the RHF model, in which the exchange terms are adjusted so as to reproduce the single-nucleon potential at . We find that the Fock contribution suppresses the kinetic term of nuclear symmetry energy at the densities around and beyond . It is noticeable that not only the isovector-vector () meson but also the isoscalar mesons () and pion make significant influence on the potential term of nuclear symmetry energy through the exchange diagrams. Furthermore, the exchange contribution prevents the slope parameter from increasing monotonically at high densities.

###### pacs:

21.65.-f, 21.65.Ef, 21.30.Fe, 24.10.Jv## I Introduction

The nuclear symmetry energy, , which is defined as the difference between the energies of pure neutron and symmetric nuclear matter, is recognized to be an important physical quantity in nuclear physics and astrophysics Lattimer:2014scr (); Li:2008gp (). It can account for many experimental facts at low nuclear densities, especially the existence of neutron skin and the different distributions of neutrons and protons in a nucleus Danielewicz:2008cm (); Danielewicz:2013upa (); Danielewicz:2016bgb (); Chen:2007ih (); Choi:2017afh (). It also plays an important role to explain the properties of isospin-asymmetric nuclear matter in the density region beyond the saturation density, , experimentally realized by the heavy-ion collisions Russotto:2011hq (); Russotto:2016ucm (); Danielewicz:2002pu (); Fuchs:2005zg (); Li:2005jy (); Klahn:2006ir (); Tsang:2008fd (). At the same time, some astrophysical observations, for instance the mass-radius relations of neutron stars and the cooling process of proto-neutron stars, strongly depend on Lattimer:2012xj (); Lattimer:2012nd (); Kolomeitsev:2016sjl (); Oertel:2016bki ().

So far, many theoretical discussions on the properties of symmetric and asymmetric nuclear matter have been performed. The phenomenological calculations based on the effective many-body interaction with the Skyrme or Gogny force have been adopted for a long time to examine the structure of finite nuclei and infinite nuclear matter RikovskaStone:2003bi (); Dutra:2012mb (); Decharge:1979fa (). Recently, the relativistic mean-field (RMF) models based on Quantum Hadrodynamics (QHD) have been widely applied to study astrophysical phenomena as well as the properties of nuclear matter at low densities Glendenning:2000 (); Glendenning:2001pe (); Shen:1998gq (); Shen:2011qu (); Ishizuka:2008gr (). Meanwhile, microscopic studies, for example the so-called Dirac-Brueckner-Hartree-Fock (DBHF) approach Brockmann:1990cn (); Katayama:2013zya (); Katayama:2015dga () and chiral perturbation theory Scherer:2002tk (); Entem:2003ft (); Machleidt:2011zz () based on realistic interactions, have also succeeded in the description of nuclear saturation features. However, depending on the models, different results of the high-dense behavior of have been reported so far, and, in particular, its density dependence beyond is undetermined yet Tsang:2012se ().

Since the observations of massive neutron-stars and the hyperon puzzle in astrophysics have provided valuable information on the equation of state (EoS) for neutron-star matter, it helps us study at high densities Lonardoni:2014bwa (); Bednarek:2011gd (); Bombaci:2016xzl (). In addition, the recent tidal deformation data induced by the gravitational wave from binary neutron star merger detected by LIGO scientific and Virgo collaborations TheLIGOScientific:2017qsa (); Abbott:2018wiz () may be useful to make a constraint on the EoS. Several models have been proposed recently, in which the astrophysical constraints as well as the terrestrial experimental data in nuclear physics are satisfied simultaneously. Within relativistic calculations, those models can be classified into two categories. One is calculated in relativistic Hartree (RH) approximation with additional repulsive force using SU(3) flavor symmetry, or nonlinear terms with multi-meson couplings Miyatsu:2013yta (); Miyatsu:2014wca (); Weissenborn:2011ut (); Lopes:2013cpa (); Zhao:2014rra (); Oertel:2014qza (); Gomes:2017zkc (); Tsubakihara:2012ic (); Maslov:2015msa (); Maslov:2015wba (). The other is based on relativistic Hartree-Fock (RHF) approximation Miyatsu:2011bc (); Katayama:2012ge (); Miyatsu:2015kwa (); Miyatsu:2013hea (); Whittenbury:2013wma (); Thomas:2013sea (); Whittenbury:2015ziz (); Li:2018jvz (); Li:2018qaw (). Since only the direct diagram is considered in the RH model, the exchanging mesons take zero momentum transfer. In contrast, the inclusion of Fock terms automatically allows us to explore the momentum dependence at meson-nucleon vertices. In addition, the pion contribution can be taken into account through the exchange diagram. However, it has not yet been investigated in detail how the exchange terms affect the properties of dense matter. Especially, we should discuss the effect of nucleon Fermi velocity on because the momentum dependence largely changes the Fermi velocity without varying the other properties of nuclear matter Maruyama:1999ye (); Maruyama:2005iw ().

Recent experimental analyses on the density dependence of have been performed by using the free Fermi Gas model, in which can be separated into the kinetic and potential terms as , where is the total baryon density Tsang:2008fd (); Zhang:2007qv (); Sammarruca:2017edz (); Hen:2014yfa (); Dutra:2017ysk (). In this decomposition, is expressed in terms of only the degrees of freedom of nucleons, and it is typically parametrized as , where is the symmetry-energy value at . Meanwhile, is simply given by the power-low function as with a parameter .

On the other hand, some theoretical calculations on have been carried out in the RHF model Sun:2008zzk (); Zhao:2014bga (); Sun:2016ley (); Liu:2018far (), and is again often divided into and , which are expressed in terms of the derivatives of the expectation valuables, and , where the nuclear matter Hamiltonian is given as Vidana:2011ap (). Since this decomposition does not correspond to that in the experimental analyses, it may be misleading to compare the components, and , in the experimental results with the theoretical results calculated in Refs. Sun:2008zzk (); Zhao:2014bga (); Sun:2016ley (); Liu:2018far ().

In contrast, an alternative decomposition based on the Lorentz-covariant forms of nucleon self-energies has been proposed in Refs. Czerski:2002pz (); Cai:2012en (); Liu:2018kfv (), and it is more consistent with the experimental analyses. In the present paper, we adopt this decomposition to clarify the density dependence of and its slope parameter, , and study the properties of dense matter in detail within RHF approximation Serot:1984ey (); Bouyssy:1987sh (). The effect of Fock contributions, namely the momentum dependence of nucleon self-energies, is very important to understand the properties of asymmetric nuclear matter, including and .

This paper is organized as follows. In Section II, a brief review for the RHF formalism based on QHD is presented. The analytical derivation of and is then demonstrated. Numerical results and discussions are addressed in Section III. We here study the nuclear matter properties from point of view of the Fock terms. In particular, we investigate the momentum dependence of and at high densities, and compare our results with the recent experimental data, focusing on the parameter in . Finally, we give a summary in Section IV.

## Ii Theoretical formalism

### ii.1 Relativistic Hartree-Fock approach

For the description of uniform nuclear matter, we present the relativistic formulation based on the QHD model in Hartree-Fock approximation. The total Lagrangian density is written as Serot:1984ey (); Bouyssy:1987sh ()

(1) |

where is the nucleon field with the mass in vacuum, MeV.

The meson term reads

(2) |

with () being the field strength tensor for () meson. The meson masses are respectively chosen as MeV, MeV, MeV, and MeV.

The interaction Lagrangian density is given by

(3) |

where is the isospin matrix for nucleon, and . The -, -, -, and - coupling constants are respectively denoted by , , and , while is the tensor coupling constant for meson. In the present calculation, the tensor coupling for meson is neglected, since the - tensor coupling constant is small Bouyssy:1987sh (); Brockmann:1990cn ().

In order to obtain a quantitative description of the nuclear ground-state properties, the following nonlinear potential for meson is, at least, required to consider Boguta:1977xi ():

(4) |

Since, in RHF approximation, the precise treatment of nonlinear terms involves tremendous difficulties Massot:2008pf (); Massot:2009kk (), we simply replace the field in Eq. (4) by its ground-state expectation value, , in the present calculation Bernardos:1993re (); Weber:1999qn ().

To sum up all orders of the tadpole (Hartree) and exchange (Fock) diagrams in the nucleon Green’s function, , we consider the Dyson’s equation

(5) |

where is the four momentum of nucleon, is the nucleon self-energy, and is the nucleon Green’s function in free space. The nucleon self-energy in matter can be written as (Serot:1984ey, )

(6) |

with being the unit vector along the (three) nucleon momentum, . It can be divided into the scalar (), time (), and space () components, which provide the effective nucleon mass, momentum, and energy in matter Miyatsu:2011bc (); Katayama:2012ge (); Miyatsu:2015kwa ():

(7) | |||

(8) | |||

(9) |

In the mean-field approximation, the meson fields are replaced by their constant expectation values: , , and (the field). The pion field vanishes in RH approximation, but it can be taken into account as the exchange contribution in RHF approximation. In the present calculation, the retardation effect in Fock terms is ignored, since it gives at most a few percent contribution to the nucleon self-energies (Serot:1984ey, ; Katayama:2012ge, ; Whittenbury:2013wma, ). All the components of nucleon self-energies, , in Eq. (6) are then calculated by Miyatsu:2011bc (); Katayama:2012ge (); Miyatsu:2015kwa ()

(10) | ||||

(11) | ||||

(12) |

with and . The is the Fermi momentum for nucleon , and the factor, , is the isospin weight at meson- vertex in the exchange diagrams. In addition, the functions , , , and are explicitly specified in Table 1, in which the following functions are used (Katayama:2012ge, ; Miyatsu:2015kwa, ):

(13) | ||||

(14) | ||||

(15) | ||||

(16) | ||||

(17) |

where

(18) | ||||

(19) | ||||

(20) | ||||

(21) |

– | ||||

– | ||||

– | ||||

– | ||||

– | – | – | ||

– |

with being a cutoff parameter at interaction vertex specified by , as shown in the 1st column of Table 1. In the present calculation, a dipole-type form factor is introduced at each interaction vertex Miyatsu:2011bc (); Katayama:2012ge (); Miyatsu:2015kwa ():

(22) |

We here employ GeV, GeV, GeV, and GeV Brockmann:1990cn (), and the effect of the form factor vanishes in the limit, .

As in the case of the RH model, the mean-field values of , , and in Eqs. (10) and (11) are given by

(23) | ||||

(24) | ||||

(25) |

where the scalar and nucleon densities are, respectively, written as

(26) | ||||

(27) |

Once the nucleon self-energies shown in Eqs. (10)–(12) are calculated, the total energy density for uniform nuclear matter is determined by the energy-momentum tensor. It can be given by a sum of the kinetic and potential terms of nucleon and the nonlinear term,

(28) |

with

(29) | ||||

(30) | ||||

(31) |

The pressure for uniform matter is then obtained from thermodynamics relation,

(32) |

### ii.2 Symmetry energy and its slope parameter

Using the Hugenholtz–Van Hove theorem, the nuclear symmetry energy is generally derived by Czerski:2002pz (); Cai:2012en ()

(33) |

with being the kinetic (potential) term for . Using the nucleon self-energies, the and in RHF approximation are respectively given by

(34) | ||||

(35) |

with , , and

(36) |

We here note that the definitions of the kinetic and potential terms shown in Eqs. (34) and (35) are different from those given in Refs. Zhao:2014bga (); Sun:2016ley (); Liu:2018far (). Since the nucleon self-energies, , can be separated into the direct and exchange contributions, the direct one in is exactly the same as in RH approximation Chen:2007ih (); Dutra:2014qga ():

(37) |

The slope parameter of nuclear symmetry energy, , is also given by the kinetic and potential terms,

(38) |

with

(39) | ||||

(40) |

If we ignore the exchange contribution, that is, we take , , , and , and are then equivalent to those in RH approximation. Thus, they are, respectively, written as Dutra:2014qga ()

(41) | ||||

(42) |

## Iii Numerical results and discussions

### iii.1 Determination of coupling constants

In the RMF model, the coupling constants are phenomenologically determined so as to reproduce the properties of infinite nuclear matter and finite nuclei Lalazissis:2009zz (); Sugahara:1993wz (). In the present calculation, for simplicity, the coupling constants, , , , and in Eqs. (3) and (4), are adjusted so as to fit the properties of symmetric nuclear matter at the saturation density, fm, namely the saturation energy ( MeV), the incompressibility ( MeV), and the effective nucleon mass (). These coupling constants are given in Table 2, in which all the cases are calculated using RH or RHF approximation.

Model | (fm) | |||||||
---|---|---|---|---|---|---|---|---|

RH | 9.52 | 10.36 | 2.63 | – | – | – | 18.11 | |

RH | 9.52 | 10.36 | 3.95 | – | – | – | 18.11 | |

RHF | 6.57 | 9.55 | 2.63 | 1.00 | 1.00 | 1.00 | 8.92 | |

RHF | 8.82 | 9.08 | 0.82 | 1.00 | 1.00 | 1.00 | 18.26 | |

ERHF(low) | 8.07 | 8.79 | 2.63 | 1.00 | 1.27 | 0.40 | 19.90 | |

ERHF(high) | 5.56 | 6.06 | 2.63 | 2.00 | 2.88 | 0.60 | 39.09 |

As for the - coupling constants, and , which are directly related to , we consider the following two ways: (1) as shown in the RH and RHF models of Table 2, we adopt the empirical values, and , which are suggested through the vector-meson-dominance model based on current algebra Sakurai:1969 (); Hohler:1974ht (); Hohler:1976ax (), (2) the coupling is chosen so as to satisfy the currently estimated value of at , namely MeV, and the relation is used in RHF approximation. The models with this choice are denoted by RH and RHF (without asterisk) in Table 2.

In addition, the pseudovector - coupling constant is fixed as , derived from the low-energy scattering data deSwart:1990 ().

Furthermore, we here consider an extended version of the RHF model (denoted by ERHF in Table 2). In the ERHF model, we replace the coupling constants, , and , in the exchange terms with new ones, , and , and the couplings in the direct terms remain unchanged (see Table 2). The purpose of this extension is to examine how the momentum dependence of nucleon self-energies contributes to various physical quantities through varying the strength of Fock terms Weber:1992qc (); Weber:1993et (); Maruyama:1993jb (). In this version, as in the RHF model, the coupling constants, , , , and , are determined so as to fit the nuclear saturation properties, adopting the empirical values, , , and . The new coupling constants in the exchange terms, , , and , are then determined by simulating the empirical value of at . In Table 2, we provide two parameter sets for the ERHF model, namely ERHF low (high), which can well reproduce the experimental data of single-nucleon potential at relatively low (high) kinetic energies. This issue will be discussed in detail later.

### iii.2 Nucleon self-energy and Schrödinger-equivalent potential

In order to clarify the effect of Fock terms on the nuclear-matter properties, it is of importance to study the momentum dependence of nucleon self-energies, and the ERHF model may be one way to see it.

The Dirac optical model Li:2013ck (); Hama:1990vr () is useful to investigate the momentum dependence. Although there are a lot of nonrelativistic single-nucleon potentials, in the present calculation, we consider the so-called Schrödinger-equivalent potential (SEP) based on the Dirac equation with Lorentz-covariant scalar and vector self-energies for nucleon Jaminon:1981xg ():

(43) |

where the nucleon kinetic energy, , reads with being the single-particle energy. With the nucleon self-energies shown in Eqs. (10)–(12), the single-particle energy is given by a solution of the transcendental equation,

(44) |

The momentum dependence of nucleon self-energies, , in symmetric nuclear matter at is shown in Figs. 1. We present the results of the RH, RHF, and ERHF models, which satisfy the saturation conditions and at . Because the direct contribution of directly couples to the mean-field values of the , , and mesons, all the components retain the constant values at any momentum in the RH model. It is found that the does not show any impact in symmetric nuclear matter within Hartree approximation, and that, even in RHF approximation, it is very small. In contrast, the momentum dependence due to the exchange contribution is clearly demonstrated in the RHF and ERHF models. Thus, it is expected that the self-energies, and , contribute dominantly to the single-nucleon potential at .

The energy dependence of single-nucleon potential (or nucleon optical potential), , is depicted in Fig. 2. We also show the results of the RH, RHF and ERHF models. In the ERHF low (high) model, the coupling constants, , , and , are adjusted so as to cover the scattering data for MeV. As and are constant and is proportional to in the RH model, it is difficult to reproduce the scattering data widely in RH approximation. Meanwhile, due to the momentum dependence which is intrinsically possessed in through Fock terms, depends on non-linearly in the RHF and ERHF models. Moreover, it is found that, in the ERHF(high) model, the enhanced exchange contribution makes it possible to well reproduce the scattering data at high . We note that strongly depends on the effective nucleon mass Danielewicz:1999zn (), which is fixed as at in the present calculation.

RH | RHF | ERHF(low) | ERHF(high) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Direct | |||||||||||||

Exchange | – | – | – | ||||||||||

– | – | – | |||||||||||

– | – | – | |||||||||||

– | – | – | |||||||||||

– | – | – | |||||||||||

– | – | – | |||||||||||

Total |

The contents of nucleon self-energies, , in symmetric nuclear matter at are presented in Table 3. In the RH model, an attractive (repulsive) force comes from only the direct contribution due to the () meson exchange through (). On the other hand, in RHF approximation, all the components of are affected by the exchange contribution. As for the exchange contribution in the RHF and ERHF models, the () meson gives a repulsive (attractive) force in the scalar component, while both and mesons work as a repulsive force in the time component. Although the pion also influences all the components through the exchange diagram, its contribution is small. Moreover, even in symmetric nuclear matter, the meson contributes to through Fock terms, where the contribution due to tensor-tensor () mixing is relatively large comparing with those due to vector-vector () and vector-tensor () mixing. We note that is very small at .

In Figs. 3 and 4, the nucleon self-energies, , and the single-nucleon potential, , in symmetric nuclear matter are respectively presented as a function of . In the present calculation, the model dependence of and is very weak at densities below , while the Fock terms play important roles in both and at high densities. It is also interesting to compare the density dependence of with the Dirac-Brueckner-Hartree-Fock (DBHF) calculation. The self-energies, and , in RH approximation, are very similar to those in the DBHF result by the TUS group Katayama:2013zya (), whereas, with increasing , and in the ERHF(high) model become close to the results calculated by Brockmann and Machleidt Brockmann:1990cn (). In Fig. 4, although any Fock effect on is little seen up to , the exchange terms give a large contribution to at high densities.

### iii.3 Effective nucleon mass and nuclear equation of state

The density dependence of the effective nucleon mass in symmetric nuclear matter is displayed in Fig. 5. We show two types of the effective nucleon mass: one is the relativistic mass in matter, , defined in Eq. (7), and the other is the effective mass in a nonrelativistic framework, , which is the so-called Landau mass Maruyama:1999ye (); Typel:2005ba (). Compared with the Landau mass, the relativistic one decreases rapidly as the density increases. We can see that both masses in the RH model are smaller than those in the other models at the densities above . It is also found that the exchange contribution suppresses their sharp reduction at high densities.

In Fig. 6, the nuclear binding energy per nucleon, , for symmetric nuclear or pure neutron matter is presented. For symmetric nuclear matter, our results are close to the recent calculation based on chiral effective field theory (EFT) in the density region below 0.4 fm Sammarruca:2014zia (). We see that the Fock contribution diminishes at high densities. In contrast, for pure neutron matter, the effect of Fock terms is not small even at low densities. Moreover, the present results tend to be larger than those calculated by EFT and Quantum Monte Carlo (QuMoCa) method at high densities Gandolfi:2011xu (). It is found that, as seen in the ERHF(high) model, the Fock terms enhances the difference between for symmetric nuclear and pure neutron matter, which implies that a large exchange contribution enlarges at the densities above .

In Fig. 7, we illustrate pressure for symmetric nuclear or pure neutron matter in comparison with the experimental constraints from heavy-ion flow data Danielewicz:2002pu (); Fuchs:2005zg (). In both cases, pressure in the RH model exceeds the constraints at high densities, while those in the RHF and ERHF models are consistent with the analysis of heavy-ion collision data. It is found that the exchange contribution softens pressure at high densities.

### iii.4 Symmetry energy and its slope parameter

Model | (MeV) | (MeV) | (MeV) | (MeV) | (MeV) | (MeV) | |
---|---|---|---|---|---|---|---|

RH | 0.754 | 23.5 | 67.0 | 32.0 | |||

RH | 0.754 | 32.5 | 94.1 | 32.0 | |||

RHF | 0.733 | 46.3 | 123.6 | ||||

RHF | 0.763 | 32.5 | 81.1 | ||||

ERHF(low) | 0.762 | 32.5 | 94.6 | 42.2 | |||

ERHF(high) | 0.762 | 32.5 | 113.5 | 116.6 |

The properties of symmetric nuclear matter at is presented in Table 4. The Landau mass of nucleon and the third-order incompressibility are denoted by and , respectively. The nuclear symmetry energy, , around is approximately expressed as a power series of the isospin-asymmetry parameter, Chen:2007ih (); Chen:2009wv ():

(45) |

with

(46) | ||||