# Effects of short-range correlations on nuclear symmetry energy within a modified Gogny-Hartree-Fock energy density functional approach

###### Abstract

Within a modified Gogny-Hartree-Fock (GHF) energy density functional
(EDF) encapsulating the nucleon-nucleon short-range correlations
(SRC)-induced high momentum tail (HMT) in the single-nucleon
momentum distribution, we investigate effects of the SRC-induced HMT
on the density dependence of nuclear symmetry energy . After
re-optimizing the modified GHF-EDF by reproducing the same empirical
properties of symmetric nuclear matter (SNM),
symmetry energy and its slope as well as major features of nucleon optical potential at saturation density , the is found to decrease at both sub-saturation and supra-saturation densities, leading to a reduced curvature of and subsequently a smaller coefficient for the isospin-dependence of nuclear incompressibility in better agreement with its experimental value. Astrophysical implications of the SRC-modified symmetry energy are also discussed. In particular, the SRC effects are found to decrease the proton fraction as well as the core-crust transition density and pressure in neutron stars at equilibrium. Moreover, the SRC-modified EOS and single-nucleon potentials can be used in future transport model simulations of heavy-ion collisions to investigate SRC effects in dense neutron-rich matter in terrestrial laboratories.

###### pacs:

21.65.Ef, 24.10.Ht, 21.65.CdIntroduction: The Gogny-Hartree-Fock (GHF) nuclear
energy density functional (EDF) has been playing an important role
in nuclear physics because of its special feature that the
finite-range part of the Gogny interaction Gogny () leads to a
momentum dependent single-particle potential through the exchange
term in the potential energy density within Hartree-Fock
calculations. In particular, the resulting single-nucleon potential
and equation of state (EOS) of nuclear matter have been widely used
in understanding many interesting phenomena in heavy-ion collisions
from low to relativistic energies, see, e.g.,
refs. Gal87 (); Pra88 (); Wel88 (); Gale90 (); Pan93 (); Zha94 () as well as in
studying various properties of stellar matter and neutron stars,
see, e.g., refs. Mis93 (); XuJ09 (); Beh11 (); Con15 (); XuJ15 (). Earlier
GHF-EDFs for symmetric nuclear matter (SNM) were extended in recent
years to isospin-asymmetric nuclear matter (ANM) by incorporating
the isospin dependence, see, e.g., refs. Bom01 (); Das03 (); Cxu10 ().
Applying them in transport model simulations of nuclear reactions
involving neutron-rich nuclei, see, e.g.,
refs. LIBA04 (); Che14 (); Che07a (); LCK08 (); Cozma (), have allowed us to
learn many interesting new physics regarding both the role of
isospin degree of freedom and the elusive density dependence of
nuclear symmetry energy . The latter is
critically important for both nuclear physics and astrophysics but
still poorly known especially at supra-saturation densities, see,
e.g., reviews in ref. EPJA (). In this work, using an
isospin-dependent single-nucleon momentum distribution with a high
momentum tail (HMT) in ANM at zero temperature constrained by recent
electron-nucleus scattering data and a new regulating function
within the GHF-EDF, we study effects of the HMT induced by
short-range nucleon-nucleon correlations (SRC) on the and
their astrophysical implications. It is found that the HMT reduces
the at both sub-saturation and supra-saturation densities,
leading to an isospin-dependent incompressibility in
better agreement with the experimental data. Implications of the
SRC-modified on the cooling mechanisms as well as the density
and pressure at the core-crust transition point in neutron stars are
also discussed.

Single-Nucleon Momentum Distribution Function Encapsulating SRC-induced high momentum tail: It is well known that the SRC leads to a high (low) momentum tail (depletion) in the single-nucleon momentum distribution function denoted by above (below) the nucleon Fermi surface in cold nucleonic matter Bethe (); Ant88 (); Arr12 (); Cio15 (); Ry15 (); Hen16x (). Significant efforts have been made in recent years both theoretically and experimentally to constrain the isospin-dependent parameters characterizing the SRC-modified in neutron-rich nucleonic matter Wei15 (); Wei15a (); Hen14 (); Hen15 (); Col15 (); Egi06 (). In particular, it has been found via analyzing electron-nucleus scattering data that the percentage of nucleons in the HMT above the Fermi surface is as high as about 28%4% in SNM but decreases gradually to about only 1%2% in pure neutron matter (PNM) Hen14 (); Hen15 (). On the other hand, the predicted size of the HMT still depends on the model and interaction used. For instance, the self-consistent Green’s function (SCGF) theory using the AV18 interaction predicts a 11%13% HMT for SNM at saturation density and a 4%5% HMT in PNM Rio09 ().

For completeness and the ease of the following discussions, we first briefly describe the SRC-modified single-nucleon momentum distribution function encapsulating a HMT constrained by the available SRC data that we shall use in this work. The single-nucleon momentum distribution function in cold ANM has the following form Cai15 (); Cai16a (); Cai16b (); Cai16c ()

(1) |

Here, is the Fermi momentum where , and , respectively, and is the isospin asymmetry. The above form of was found consistent with the well-known predictions of microscopic nuclear many-body theories Bethe (); Ant88 (); Arr12 (); Cio15 () and the recent experimental findings Hen14 (); Hen15 (); Wei15 (); Wei15a (); Col15 (). This form of has been applied to address several issues regarding the HMT effects recently in both nuclear physics and astrophysics Cai15 (); Cai16a (); Cai16b (); Cai16c (); CXu11 (); CXu13 (); Hen15b (); Hen16 ().

The parameters , and are assumed to depend linearly on based on predictions of microscopic many-body theories Rio09 (); yin13 (); ZLi16 () Cai15 (). The amplitude and high-momentum cutoff coefficient determine the fraction of nucleons in the HMT via

(2) |

Moreover, the normalization condition between the density and the distribution , i.e., requires that only two of the three parameters, i.e., , and , are independent. Here we choose the first two as independent and determine the by

(3) |

Meanwhile, the adopted shape of the HMT both for
SNM and PNM is strongly supported by recent studies both
theoretically and experimentally Hen14 (); Tan08 (). It is
interesting to point out that the form of the HMT is
also found in Bose system theoretically Bra11 (); Wer12 (); Smi14 ()
and experimentally Mak14 (); Fle17 (), indicating a very general
feature of the HMT. For comparisons, we use two HMT parameter sets.
The adopting a 28% HMT in SNM and a 1.5% HMT in
PNM Hen14 (); Hen15 () is abbreviated as the HMT-exp set, and that
adopting a 12% HMT in SNM and a 4% HMT in PNM Rio09 () as the
HMT-SCGF set Cai16c (). Moreover, the model using a step
function for the is denoted as the free Fermi gas
(FFG) set as a reference. As discussed in more details in
ref. Cai16c (), the HMT parameters in the HMT-exp (HMT-SCGF)
parameter set are (),
(),
(), and
, respectively.

Incorporating single-nucleon momentum distributions with HMTs in Gogny Hartree-Fock energy density functionals: In most studies of heavy-ion collisions using transport models, one parameterizes the EDFs and determine their parameters by reproducing empirical properties of SNM at the saturation density , a selected value of symmetry energy and its slope as well as main features of nucleon optical potentials extracted from analyzing nucleon-nucleus scatterings, such as the isosclar and isovector nucleon effective masses and their asymptotic values at high momenta at , etc., see, e.g., ref. Che14 () for detailed discussions. For example, using a modified Gogny-type momentum-dependent interaction (MDI) first proposed by Das et al Das03 (); Che05 (), a modified GHF-EDF in terms of the average energy per nucleon in ANM at density and isospin asymmetry can be written as

(4) |

The first term is the kinetic energy while the three terms in the second line are the usual zero-range 2-body and effective 3-body contributions characterized by their strength parameters and as well as the density dependence of the 3-body force Das03 (); Che05 ()

(5) |

where controls the competition between the isosinglet and isotriplet 2-body interactions, and it affects only the slope but not the be design Das03 (). The last term in Eq. (Effects of short-range correlations on nuclear symmetry energy within a modified Gogny-Hartree-Fock energy density functional approach) is the contribution to the EOS from the finite-range 2-body interactions characterized by the strength parameter for like and for unlike nucleon paris, respectively, using the notations and . The and are the nucleon phase space distribution function and momentum distribution function, respectively. In equilibrated nuclear matter at zero temperature, they are related by

(6) |

For example, in the FFG, with the standard step function, then .

The regulating function Gal87 (); Das03 () originating from the meson exchange theory of nuclear force normally has the form of

(7) |

where k and are the momenta of two interacting nucleons and is a parameter regulating the momentum dependence of the single-particle potential. For applications to SNM, it is usually determined by fixing the nucleon isoscalar effective mass at the Fermi surface to an empirical value Gal87 (); Das03 (). The single-nucleon potential corresponding to Eq. (Effects of short-range correlations on nuclear symmetry energy within a modified Gogny-Hartree-Fock energy density functional approach) is given by

(8) |

In applying the above formalisms to transport model simulations of nuclear reactions, the and are calculated self-consistently from solving dynamically the coupled Boltzmann-Uehling-Uhlenbeck (BUU) transport or molecular dynamics equations for quasi-nucleons Bert88 (); Aich91 (). While in studying thermal properties of hot nuclei or stellar matter in thermal equilibrium, the Fermi-Dirac distributions at finite temperatures are used.

Traditionally, one writes the EDF as a sum of kinetic EOS of FFG
plus several potential terms. Before making any applications, the
model parameters of the EDFs are normally fixed by using step
functions for the and
as in a FFG at zero temperature in reproducing properties of nuclei
or nuclear matter in their ground states. In reality, however, since
all nucleons interact with each other in nuclear medium, they
naturally become quasi-nucleons. The normal practice of optimizing
the EDFs puts all effects of interactions into the potential part of
the EDF thus ignores interaction effects on the kinetic energy of
quasi-nucleons. The momentum distribution of these quasi-nucleons in
the ground state of the system considered is not simply a step
function if SRC effects are considered as we discussed in the
previous section. Here, we separate the total EDF into a kinetic
energy and several potential parts of quasi-nucleons. The
and with HMTs
constrained by the SRC experiments are used in evaluating both the
kinetic and the momentum-dependent potential parts of the EDF in ANM
at zero temperature. At least for simulating heavy-ion collisions
using transport models, how the total EDFs are separated into their
kinetic and potential parts are important and have practical
consequences in predicting experimental
observables Hen15b (); LI-shi (). Interestingly, how the SRC may
affect the symmetry energy, heavy-ion reactions and properties of
neutron stars are among the central issues in our pursuit of
understanding the nature of neutron-rich nucleonic matter. Previous
attempts to incorporate the experimentally constrained
and with HMT in the
non-relativistic EDF and examine their effects on heavy-ion
collisions and neutron stars were found very
difficult LiGuo (). This is mainly because of the nontrivial
momentum dependence of the and the EDF
when the SRC-modified and
are used. Since one needs to solve 8-coupled
equations simultaneously to obtain self-consistently all model
parameters from inverting empirical properties of ANM and nucleon
optical potentials at , numerical problems associated with
the momentum integrals in
Eqs. (Effects of short-range correlations on nuclear symmetry energy within a modified Gogny-Hartree-Fock energy density functional approach) and (Effects of short-range correlations on nuclear symmetry energy within a modified Gogny-Hartree-Fock energy density functional approach) using the original are very difficult to solve.

A new high-momentum regulating function: To avoid the problem mention above, we propose a new high-momentum regulating function that approximates very well the original one while enables all integrals in the EDF and to be analytically expressed. Perturbatively, if is large compared to the momenta scale in the problems under investigation, the in Eq. (7) can be expanded as . Using this as a hint, we parameterize the as

(9) |

where and are two new parameters. It is interesting to note that this is invariant under the transformation , and , indicating that we have the freedom to first fix one of them without affecting the physical results. Here we set and then determine the and using known constraints as we shall discuss in the following.

The advantages of using this new regulating function is twofold: firstly, the basically 1/2 and 1/3 power in the second and third term in (9) is relevant for describing properly the energy dependence of nucleon optical potential Ham90 (); secondly, it enables analytical expressions for the EOS and in ANM. We notice that the function is only perturbatively effective at momenta smaller than the momentum scale , indicating that the EDF constructed can only be used to a restricted range of momentum/density. It turns out that the cut-off of the HMT in ANM up to about is significantly smaller than the parameter we use here. The above non-relativistic GHF-EDF is denoted as abMDI in the following.

Quantity | Value | Coupling | FFG | HMT-SCGF | HMT-exp |
---|---|---|---|---|---|

(fm) | |||||

(MeV) | |||||

(MeV) | |||||

(MeV) | |||||

(MeV) | |||||

(MeV) |

We fix all parameters in the model EDF using empirical properties of
SNM, ANM and main features of nucleon optical potentials at
. More specifically, for SNM we adopt
at the saturation density
with the EOS of
SNM, its incompressibility You99 (); Shl06 (); Pie10 (); Che12 (); Col14 (),
the isoscalar nucleon k-mass, i.e.,
LiBA15 (), is
selected as , and
. For the isospin-dependent part in
ANM, we adopt for the symmetry
energy, LiBA13 () for the
slope of the symmetry energy and
XuJ15 () for
the symmetry potential, respectively. Moreover, the value of
is constrained to fall within a reasonable range to
guarantee the effect of the high order terms in in the EOS
of ANM mainly characterized by the fourth order symmetry energy,
i.e.,
,
is smaller than 3 MeV at , to be consistent with
predictions of microscopic many-body theories. Consequently,
is obtained
and the study based on is used as the
default one. It is worth noting that the single-nucleon potential in
SNM thus constructed is consistent with the global relativistic
nucleon optical potential extracted from analyzing nucleon-nucleus
scattering data Ham90 (). Thus, totally five isoscalar
parameters, i.e., and
for SNM, and three isovector parameters, i.e.,
and are
all fixed. Details values of these parameters for the three cases using the same set of input physical properties are shown in Tab. 1 .

Short-range correlation effects on the density dependence of nuclear symmetry energy: Now we turn to effects of the SRC on nuclear symmetry energy. Shown in Fig. 1 are the results obtained using the FFG, HMT-SCGF and HMT-exp parameter sets. By construction, they all have the same and at . Also shown are the constraints on the around from analyzing intermediate energy heavy-ion collisions (HIC) Tsa12 () and the isobaric analog states (IAS) Dan14 (). Although the predicted using the three parameter sets can all pass through these constraints, they behave very differently especially at supra-saturation densities. The uncertainty of the due to that of the parameter is also shown in Fig. 1 for the HMT-exp set with the gray dash-dot lines. It is seen that the uncertainty is much smaller than the SRC effect. For example, the variation of the symmetry energy at owing to the uncertainty of is about 2.3 MeV while the SRC effect is about 14.5 MeV. Since the parameter mainly affects the high density/momentum behavior of the EOS, its effects become smaller at lower densities. The reduction of the at both sub-saturation and supra-saturation densities leads to a reduction of the curvature coefficient of the symmetry energy. More quantitatively, we find that the changes from MeV in the FFG set to about and MeV in the HMT-SCGF and HMT-exp set, respectively. It is interesting to stress that this SRC reduction of help reproduce the experimentally measured isospin-dependence of incompressibility in ANM where . The skewness of SNM is approximately , and in the FFG, HMT-SCGF and HMT-exp set, respectively. The resulting is found to change from in the FFG set to about and in the HMT-SCGF and HMT-exp set, respectively. The latter is in good agreement with the best estimate of MeV from analyzing several different kinds of experimental data currently available Col14 ().

It is also interesting to notice that the SRC-induced reduction of within the non-relativistic EDF approach here is qualitatively consistent with the earlier finding within the nonlinear Relativistic Mean-Field (RMF) theory Cai16b (). Nevertheless, since there is no explicit momentum dependence in the RMF EDF, the corresponding reduction of is smaller. Obviously, the momentum-dependent interaction makes the softening of the symmetry energy at supra-saturation densities more evident. This naturally leads us to the question why the SRC reduces the at both sub-saturation and supra-saturation densities. The SRC affects the through several terms. First of all, because of the momentum-squared weighting in calculating the average nucleon kinetic energy, the isospin dependence of the HMT makes the kinetic symmetry energy different from the FFG prediction as already pointed out in several earlier studies CXu11 (); CXu13 (); Hen15b (); Vid11 (); Lov11 (); Car12 (); Rio09 (); Car14 (); Cai15 (). More specifically, within the parabolic approximation of ANM’s EOS the is approximately the energy difference between PNM and SNM. Thus, the larger HMT due to the stronger SRC dominated by the neutron-proton isosinglet interaction increases significantly the average energy per nucleon in SNM but has little effect on that in PNM, leading to a reduction of the kinetic symmetry energy. More specifically, the factor defined via is a measure of the SRC effect on nucleon kinetic energy, and it is determined by the characteristics of via Cai15 ()

(10) |

Numerically, the is with the HMT-SCGF
parameters while it is with the HMT-exp parameters
consistent with earlier estimates Hen15b (). Of course, the HMT
also affects the momentum dependent part of the potential symmetry
energy. However, the strength parameters and
are not independently determined but through the global optimization
of the EDF. The reduction of the kinetic symmetry energy is
naturally compensated by the increased potential contribution in
optimizing the EDF to keep the and
consistent with empirical constraints at LiBA13 ().
However, because of the dependence of the kinetic
symmetry energy, after readjusting the model parameters in the
global optimization process using empirical properties of ANM and
SNM at , the reduction of kinetic still dominates
over the increase of potential symmetry energy at abnormal
densities, leading to the overall reduction of at both sub-saturation and supra-saturation densities.

Implications of the SRC-modified on properties of neutron stars: The SRC-modified EOS and single-particle potentials are expected to affect properties of both heavy-ion collisions and neutron stars. While investigations of explicit SRC effects on experimental observables are currently underway, we present here SRC effects on two internal properties of neutron stars that may have consequences on observables. Firstly, the pressure of neutron-rich nucleonic matter around can be written as

(11) |

where . Near , it is well known that the pressure is dominated by the term. Keeping the and the same by design in our study here, effects of the SRC on the pressure is through the decrease of and increase of as we discussed earlier. The overall effect, however, depends on the density and isospin asymmetry . For example, comparing the HMT-exp with the FFG model calculations, the increase in is MeV and the decrease in is MeV. For very neutron-rich matter in neutron stars with , the decrease in dominates, leading to an appreciable softening of the EOS. While for much smaller values of , such as the average isospin asymmetry in Pb or those normally reached in heavy-ion reactions, the SRC effect on pressure is small when the and are kept constants. The latter two quantities are known to affect the maximum masses and radii of neutron stars. We thus expect the SRC to have some effects on the mass-radius correlation of neutron stars. However, the results will depend on our choice for the and as we shall report elsewhere Cai16e ().

Many interesting microphysics in neutron stars depend critically on the proton fraction at -equilibrium. The latter is uniquely determined by the nuclear symmetry energy. For neutrino free -stable matter, the chemical equilibrium for the matter requires with ’s the respective chemical potentials of particles involved. Moreover, the denotes the high-order effect of the ANM’s EOS, where is the EOS of PNM. Then because of charge neutrality. Shown in Fig. 2 are the proton fractions in the matter as a function of density obtained using the three sets of model parameters in both the parabolic approximation and the one including the high-oder term , respectively. It is seen that the high-oder term plays a significant role at supra-saturation densities especially in the HMT-exp case. Although we have restricted the quartic symmetry energy at to be less than 3 MeV in the optimization of the model EDF, it is clear that the SRC induces a significant at supra-saturation densities. This is understandable as we have shown earlier analytically that the SRC-induced isospin dependence of the HMT leads to a significant quartic term in the kinetic symmetry energy Cai15 ().

The critical proton fraction for the direct URCA process to occur in npe (npe) matter assuming nucleons have the FFG momentum distribution is also shown in Fig. 2. It was predicted a long time ago that the threshold value for the direct URCA is reduced by the SRC Fra08 (). However, how the neutrino emissivity near the threshold may be affected by the SRC has not been worked out consistently yet Isac (). We notice by passing that so far there is no confirmed observation of direct URCA in any neutron star although it has been predicted to occur by many models Cooling (). Obviously, more work on this issue remains to be done. Nevertheless, it is interesting to note that the obtained in the HMT-exp starts to decrease at about with a maximum proton fraction . On the other hand, with both the FFG and HMT-SCGF parameters, the continues to increase to high densities. If the conventional estimates for the direct URCA thresholds remain valid, the FFG and HMT-SCGF would allow while the HMT-exp would forbid the direct URCA process in neutron stars.

Another interesting quantity depends critically on the is the transition density , which separates the liquid core from the inner crust in neutron stars. It depends sensitively on the low-density behavior of and plays an important role in determining many properties of neutron stars, such as the fractional moment of inertia of the crust relevant for understanding the glitch phenomenon XuJ09 (); Hor01 (); Pro06 (); Duc08a (); Duc08b (); Lat07 (). One simple and widely used approach to determine the is to examine when the following stability condition is violated XuJ09 (); Kub07 ()

(12) |

Here, the is the energy per nucleon in -equilibrium npe matter (the core-crust transition occurs at low densities where muons generally do not appear), see, e.g., (Effects of short-range correlations on nuclear symmetry energy within a modified Gogny-Hartree-Fock energy density functional approach), and the pressure has contributions from both nucleons () and electrons (). The is found to be about , and for the FFG, HMT-SCGF and HMT-exp case, respectively. The corresponding transition pressure is found to be about , and , respectively. The parameter is found to have very small effects on the transition density and pressure . It is interesting to see that the SRC brings significant changes to the transition density and pressure. For example, compared to the FFG case, the in the HMT-exp calculation is reduced by about 57%.

The above discussions about effects of the SRC-modified on microphysics happening in neutron stars may help
establish a connection between the SRC and astrophysical observations. In addition, the SRC-modified single-nucleon potential and EOS of ANM
can be used directly in transport model simulations of nuclear reactions, providing a link between the SRC and observables in terrestrial
experiments. Explorations of these links and comparisons with data will help us better understand the underlying physics determining the elusive nature of neutron-rich nucleonic matter.

Summary: Within a modified non-relativistic GHF-EDF approach and using a new momentum regulating function, we studied effects of SRC-induced HMT in the single-nucleon momentum distribution on the density dependence of nuclear symmetry energy. After re-optimizing the modified GHF-EDF by reproducing the same empirical properties of ANM, SNM and major features of nucleon optical potential at saturation density, the was found to decrease at both sub-saturation and supra-saturation densities, leading to a reduced curvature of and subsequently a smaller for the isospin-dependence of nuclear incompressibility in better agreement with its experimental value. Astrophysical implications of this finding were also discussed. In particular, the proton fraction as well as the core-crust transition density and pressure in neutron stars at equilibrium are found to decrease as the fraction of HMT nucleons in SNM increases, linking clearly properties of neutrons stars with the microphysics of SRC. Moreover, the SRC-modified EOS and the single-nucleon potentials in ANM can be used in future transport model simulations of heavy-ion collisions to investigate SRC effects in dense neutron-rich matter in terrestrial laboratories.

Acknowledgement: We would like to thank L.W. Chen, O. Hen and E. Piasetzky for helpful discussions. This work was supported in part by the U.S. Department of Energy, Office of Science, under Award Number DE-SC0013702, the CUSTIPEN (China-U.S. Theory Institute for Physics with Exotic Nuclei) under the US Department of Energy Grant No. DE-SC0009971 and the National Natural Science Foundation of China under Grant No. 11320101004.

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