# The properties of nuclear matter with lattice potential in relativistic Brueckner-Hartree-Fock theory

## Abstract

We study the properties of nuclear matter with lattice nucleon-nucleon () potential in the relativistic Brueckner-Hartree-Fock (RBHF) theory. To use this potential in such a microscopic many-body theory, we firstly have to construct a one-boson-exchange potential (OBEP) based on the latest lattice potential. Three mesons, pion, meson, and meson, are considered. Their coupling constants and cut-off momenta are determined by fitting the on-shell behaviors and phase shifts of the lattice force, respectively. Therefore, we obtain two parameter sets of the OBEP potential (named as LOBEP1 and LOBEP2) with these two fitting ways. We calculate the properties of symmetric and pure neutron matter with LOBEP1 and LOBEP2. In non-relativistic Brueckner-Hartree-Fock case, the binding energies of symmetric nuclear matter are around and MeV at saturation densities, while it becomes and MeV in relativistic framework with and channels using our two parameter sets. For the pure neutron matter, the equations of state in non-relativistic and relativistic cases are very similar due to only consideration channel with isospin case.

## I Introduction

It is an extremely challenging problem to solve the nuclear many-body problem directly from the quantum chromodynamics (QCD) theory in modern nuclear physics. Since the beginning of this century, we anticipate the dawn of treating such a problem from lattice QCD calculation. Hatsuda et al. (HAL collaboration) extracted the nucleon-nucleon () potential based on the imaginary-time Nambu-Bethe-Salpeter (NBS) wave functions in lattice QCD (1); (2); (3). The nuclear force is taken from the zero-strangeness sector of the octet-baryon potentials in the flavor- limit calculated on the lattice, where the renormalization group improved Iwasaki gauge action and the nonperturbatively improved Wilson quark action were employed on a lattice with the lattice spacing fm. The pion masses are ranging at five values between MeV to MeV(4); (5). These five potentials were able to describe the basic characters of potential, such as a strong repulsive force at short distance for the central interaction and an attractive tensor force at intermediate distance.

Recently, Inoue et al. applied such lattice potentials on the study of nuclear many-body system from nuclear matter to finite nuclei (6); (7). They obtained a saturation point ( fm, MeV) for symmetric nuclear matter with the lightest quark mass ( MeV, MeV) using a powerful microscopic nuclear many-body theory, Brueckner-Hartree-Fock (BHF) theory (8), which can deal with the short-range central and intermediate tensor forces properly. In this case, the maximum mass of the neutron star is found to be 0.53 times the solar mass. The calculated result is far from the empirical saturation property of symmetric nuclear matter, fm, MeV, but it still shows a possibility of ab initio calculation of nuclear many-body problem from QCD theory. Furthermore, the light doubly magic nuclei, O and Ca, were also investigated with the BHF theory in harmonic-oscillator basis (7). The binding energies per particle for O and Ca are MeV and MeV, respectively. These works help us greatly understand the connection between lattice QCD and the ground states of nuclear many-body system.

The relativistic Brueckner-Hartree-Fock (RBHF) theory, as a relativistic version of the BHF theory, is able to describe the saturation property of symmetric nuclear matter successfully, through taking the medium effect into the potential (9). It also can explain the spin-orbit force naturally by the scalar and vector potentials using Dirac equation. The relativistic effect can provide a repulsive contribution to the binding energy through a Z-graph process of nucleon-antinucleon excitation (10), instead of a phenomenological three-body force used in the BHF theory. Therefore, in this work, we would like to study the properties of nuclear matter using the lattice potential with the RBHF theory. We provide the necessary formula of the RBHF theory in Sect. II and discussions of our results in Sect. III, before drawing conclusions in Sect. IV.

## Ii Relativistic Brueckner-Hartree-Fock theory

In this section, we will show the basic framework of RBHF theory. The effective interaction, -matrix in RBHF theory can be written as the Bethe-Goldstone equation (9)

(1) |

where is the c.m. momentum, and are the initial, intermediate, and final relative momenta, respectively, for two particles in nuclear medium. is the Pauli operator projecting onto unoccupied states. We can solve this integral equation by the matrix inverse method. Furthermore, is the single-particle energy of nucleon, in nuclear matter, which can be defined as

(2) |

where is kinetic energy and is a single-particle potential related with -matrix,

(3) |

where and are nucleon states including single-particle momenta, spin, and isospin index. The propagator in Eq.(1) depends on the single-particle potential, through the single-particle energy, Eq.(2). Consequently, the determination of -matrix depends on the choice of . In many-body problem, a nucleon in the nuclear medium can be viewed as a ’bare’ nucleon that is ’dressed’ as a consequence of its effective two-body interactions with the other nucleons. Such a ’dressed’ nucleon state should satisfy the Dirac equation in nuclear matter,

(4) |

where, is the relativistic self-energy of nucleon and for neutron/proton. As symmetry required, the self-energy must have the general Lorentz structure

(5) |

where and are an attractive scalar field and a repulsive vector field, respectively, and is the time component of the vector field. It has been shown that is much smaller than and in Ref. (9). Thus we can write

(6) |

Therefore, we can obtain the positive energy solution in above Dirac equation,

(7) |

where is a Pauli spinor, the effective nucleon mass and effective single-particle energy in relativistic framework,

(8) |

and

(9) |

It is formally identical to a free-space spinor, but with replaced by . Now, from Eq.(6), the single-particle potential becomes

(10) |

where is a Dirac spinor as shown in Eq.(7) and is the conjugate spinor. and are momentum dependent, which can be taken as the constants in a fixed density as an approximation (9). Thus, the single-particle potential becomes,

(11) |

One can easily see that this equation could be parameterized in terms of two constants, and . Therefore, in the actually numerical procedure, starting from some initial values of and , the -matrix equation is solved and a first approximation for is then obtained. This solution is again parameterized in terms of a new set of constants, and the calculation are repeated until the convergence is reached. The energy per neutron or proton in nuclear matter can be calculated through evaluating the expectation value of -matrix with relativistic Hartree-Fock wave functions.

## Iii Results and Discussion

The potential used in the RBHF theory should be the one described by quantum field theory with spinor structure in order to take into account the nuclear medium effect. Therefore, at the beginning, a one-boson-exchange potential (OBEP) should be constructed based on the present lattice potential which is represented in a relative coordinate space. In this work, we only consider the lattice potential with lightest quark mass ( MeV, MeV) worked out by Inoue et al., which is closest to the pion physical value and generates the most attractive binding energy for nuclear matter and finite nuclear system in the BHF theory (6).

In Fig. 1, we plot the potential from the lattice QCD calculation for various channels in coordinate space, using the potential of the pion mass MeV (La469). There are only four channels in lattice force, and until now with partial waves . The channel corresponds to the central force, while the channel to the tensor force. We also compare these potentials with a high precision charge-dependent realistic potential obtained by fitting the phase shifts of scattering data, AV18 potential (11). We can find that the behaviors of lattice potential are similar with the AV18 potential. At short distance in the and channels, there are strong repulsive cores, while in the channel, an attractive region appears in intermediate distance. The largest difference between these two potentials is that the attractive magnitude of the lattice QCD is much smaller than the one of AV18 potential. This should arise from the fact that the present pion mass is still far from its physical value, MeV.

Usually, the potential written in the momentum space is more convenient for the calculation of nuclear matter. Therefore, we would like to transform the present La469 potential into the momentum space by Fourier transformation:

(12) | |||||

where is the -order spherical bessel function.

We show the on-shell () matrix elements of potentials of La469 potential together with the AV18 potential for and channels in momentum space in Fig.2. Here, we include one more realistic potentials, Bonn A potential (10), which is the potential constructed by one-boson exchange potential (OBEP) with six mesons, and . The behavior of on-shell matrix elements in La469 potential is similar to those of AV18 potential and Bonn potential for and channels. However, there is an attractive force at low momentum in the channel of La469 potential, while it is repulsive in AV18 potential and Bonn potential.

It is necessary to use a OBEP in the RBHF theory to discuss the relativistic effect in nuclear matter. So, we try to construct a OBEP by using the La469 potential. Three mesons, pion, , and mesons are considered in the present OBEP. Pion can provide the tensor force and long range part of potential, while the and mesons provide middle-range and short-range contributions to potential, respectively. In this case, the Lagrangian of mesons coupling with nucleon can be written as,

(13) | |||||

where, the pseudovector coupling between pion and nucleon is adopted and .

To discuss the influences of on-shell and off-shell behaviours of lattice potential, two parameter sets of OBEP will be fitted. In the first parameter set, we fit the parameters in OBEP, the coupling constants between mesons and nucleon and the cut-off momenta, through the on-shell matrix elements of La469 potential, and obtain LOBEP1. The second parameter set, LOBEP2, is obtained by fitting the phase shift of La469 potential. In above two fittings, the parameters are determined by the data of La469 potential in and channels to avoid anomalous behaviors in channel of La469 potential seen in comparison with AV18 and Bonn potentials.

In Fig. 3, we plot the on-shell matrix elements of OBEP by fitting the La469 potential (LOBEP1) for , and channels in momentum space. We find that the LOBEP1 can describe the on-shell behaviors of La469 potential very well.

The La469 potential is extracted from the lattice results through Schroedinger equation in the non-relativistic framework. Therefore, the phase shifts of La469 potential is calculated within the non-relativistic propagator, while the OBEP in the RBHF model should be obtained in the relativistic form following the idea in Ref.(9). We use the Thompson equation to work out the phase shifts to fix LOBEP2 potential (12). Since the phase shifts are the observable quantities, they should be independent of frameworks. The phase shifts of La469 potential and the fitted LOBEP2 potentials, are given in Fig. 4 for , and channels. In the third panel, is the mixing parameter of coupled states. These phase shifts of LOBEP2 potential can describe La469 potential very well, where up to the laboratory energy of MeV.

We tabulate the meson-nucleon coupling constants and cut-off momenta of LOBEP1 and LOBEP2 potentials in Table 1 and compare them with the corresponding values in Bonn A potential. The masses of pion, meson, and nucleon have been calculated in lattice QCD. The meson mass should be fitted in this work. These fitting parameters look reasonable comparing with Bonn A potential obtained by the scattering data.

Force | LOBEP1 | LOBEP2 | Bonn A | ||||||
---|---|---|---|---|---|---|---|---|---|

(MeV) | (MeV) | (MeV) | (MeV) | (MeV) | (MeV) | ||||

Now, with the LOBEP1 and LOBEP2 potentials, the equation of state (EOS) of nuclear matter can be calculated with the BHF theory and RBHF theory described in the Methods section as following (9), taking only and channels. The EOSs of symmetric nuclear matter () in the BHF and RBHF theories are given in the upper panel of Fig. 5. In the BHF theory, the binding energy is MeV at saturation density fm for the LOBEP1 potential. The saturation properties will be changed as MeV at saturation density fm for LOBEP2 potential, which are consistent with the results by Inoue et al. (6). The saturation properties of symmetric nuclear matter are MeV at fm in the RBHF theory with LOBEP1 potential, while they are MeV at fm for LOBEP2 potential. It looks that the non-relativistic case provides more repulsive effect with the present lattice potential, while with Bonn potential, the RBHF theory is more repulsive (9). But we should remember that there are only channels available in the present lattice potential. We do not have the data of lattice potential at channels.

Furthermore, we also calculate the EOSs of pure neutron matter with LOBEP1 and LOBEP2 potentials in BHF and RBHF theories shown in the lower panel of Fig. 5. They are almost identical, since in pure neutron matter there is only one contribution from channel from lattice potential with isospin channel.

Now, we can obtain the wave components of the LOBEP1 potential, which can be regarded as our prediction on the lattice potential. Then, this wave contribution is taken into account in the calculation of nuclear matter. With its contribution, the saturation energy in BHF becomes larger than the one in the RBHF theory now as shown in Fig. 6 and consistent with the RBHF theory to Bonn potential. It is demonstrated that the waves are very important for the relativistic effect to provide the repulsive effect. For the LOBEP2 potential, there is a similar result.

We also compare the EOSs of pure neutron matter with LOBEP1 and Bonn A potentials in the RBHF theory in Fig. 7. The EOS of Bonn A potential has more repulsive effect in the high density region, which includes the contribution not only from channel, but also from other channels, such as channel and so on. Therefore, we need more data of lattice potential for higher partial waves to describe the properties of neutron star correctly.

## Iv Conclusion

In conclusion, two kinds of one-boson-exchange potential (OBEP) were constructed based on the latest lattice potential (La469) to study the properties of nuclear matter with the relativistic Brueckner-Hartree-Fock (RBHF) theory. We fitted the OBEP with the on-shell matrix elements and phase shifts of La469 potential respectively, and obtained LOBEP1 and LOBEP2, which could completely reproduce the fitting data. The saturation properties of these two OBEPs in the BHF theory were consistent with the existing calculation for La469 potential directly including and channels by Inoue et al.. The non-relativistic EOS had more attractive contribution to the saturation energy than the RBHF theory. This result is opposite in comparison with the previous calculation in RBHF theory with Bonn potential, which is obtained by the nucleon-nucleon scattering data. This is because in our calculation, there are only channels provided by the La469 potential. Once the -waves with included in LOBEP1 and LOBEP2 potentials, the EOSs of BHF theory became more attractive than the one of RBHF theory, which is consistent with the previous calculation with Bonn potential. It demonstrated that the -waves were very important for the repulsive contributions in relativistic effect. The pure neutron matter is also calculated in BHF and RBHF theory. These EOSs are almost identical from both LOBEP1 potential and LOBEP2 potential. In this case, there are only channel contributing to the isospin system.

Although we can obtain the binding state of symmetric nuclear matter with present lattice potential, the saturation properties were still far from the empirical data. The pure neutron matter with lattice potential still need more repulsive contribution at high density to obtain the reasonable maximum mass of neutron star. We hope that the lattice QCD can provide more data on higher partial waves in the potential and less quark mass to approach the physical pion mass so that we can realize the ab initio calculation of nuclear many-body system from the QCD level.

## V Acknowledgments

J. Hu would like to thank Dr. T. Inoue for providing the subroutine of lattice potential and useful discussion and Dr. Y. Zhang for the corrections on language. This work was supported in part by the National Natural Science Foundation of China (Grants No. 11375089, and No. 11405090).

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