# Self-consistent Many-Body Theory and Nuclear Matter in a Chiral Dirac-Hartree-Fock Approximation

## Main Article Content

## Abstract

**Abstract**

A self-consistent chiral Dirac-Hartree-Fock (CDHF) approximation generated by an effective model of the (σ, ω, π) quantum hadrodynamics is discussed and applied to nuclear matter and neutron stars. The CDHF approximation maintains conditions of thermodynamic consistency connected to the fundamental requirement of density functional theory (DFT). The self-consistent conditions to nuclear matter approximations generate functional equations for self-energies; accurate and rigorous solutions to self-energies are obtained and examined. The difference of solutions constructed by thermodynamic consistency (or DFT) and Feynman diagram approach is compared and discussed explicitly, which should be declared as an open question for many-body theory.

Exchange interactions are more important than direct interactions at nuclear matter saturation density, which suggests that an appropriate nuclear ground state approximation be the HF approximation rather than the mean-field (Hartree) approximation. The current CDHF approximation produces incompressibility and symmetry energy, K = 218 MeV and a4 = 21.3 MeV. The application to neutron stars yields Mmax star /M⊙ = 2.21 in the unit of solar mass and radius R = 11.6 km, which improves mean-field results.

## Article Details

**Quarterly Physics Review**, [S.l.], v. 3, n. 3, oct. 2017. ISSN 2572-701X. Available at: <https://esmed.org/MRA/qpr/article/view/1488>. Date accessed: 28 nov. 2022.

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## References

[1] B. D. Serot and J. D. Walecka, Advances in Nuclear Physics, edited by J. W. Negele and

E. Vogt (Plenum, New York, 1986), Vol. 16.

[2] Walecka, J.D. A Theory of Highly Condensed Matter. Annals of Physics, 83, (1974) 491.

[3] B. D. Serot, Quantum hadrodynamics, Reports on Progress in Physics, 55 (1992) 1855.

[4] Serot, B.D. and Uechi, H., Neutron Stars in Relativistic Hadron-Quark Models. Annals of

Physics, 179, (1987) 272.

[5] B. D. Serot and J. D. Walecka, Recent Progress in Many-Body Theories, vol. 3, T. L.

Ainsworth, C. E. Campbell, B. E. Clements, and E. Krotscheck, eds. (Plenum, New York,

1992), p. 49.

[6] B. D. Serot, Covariant effective field theory for nuclear structure and nuclear currents,

Lcct. Notes Phys. 641 (2004) 31.

[7] S. T. Uechi, H. Uechi, Hardon-Quark Hybrid Stars Constructed by the Nonlinear σ, ω, ρ

Mean-Field Model and MIT-Bag Model, Open Access Library Journal, 2, (2015).

[8] S. T. Uechi and H. Uechi, Density-Dependent Properties of Hadronic Matter in an Extended

Chiral (σ, π, ω) Mean-Field Model, Open Access Library Journal, 2, (2015).

[9] Uechi, S.T. and Uechi, H. (2016) Landau Theory of Fermi Liquid in a Relativistic Nonlinear

(σ, ω) Model at Finite Temperature. Open Access Library Journal, 3: e2757.

[10] J. D. Walecka, Theoretical Nuclear and Subnuclear Physics, Oxford University Press, 1995.

[11] Serot, B.D. and Walecka, J.D., Chiral QHD with Vector Mesons. Acta Physica Polonica

B, 23, (1992) 655.

[12] H. M¨uller and B. D. Serot, Relativistic mean-field theory and the high-density nuclear

equation of stateNucl. Phys. A606 (1996) 508.

[13] H. Uechi, S. T. Uechi, and B. S. Serot, B.D., Neutron Stars: The Aspect of High Density

Matter, Equations of State and Observables. Nova Science Publishers, New York (2012).

[14] H. Uechi, Properties of Nuclear and Neutron Matter in a Nonlinear σ-ω-ρ Mean-Field

Approximation with Self- and Mixed-Interactions. Nuclear Physics A, 780, (2006) 247.

[15] H. Uechi, Density-Dependent Correlations between Properties of Nuclear Matter and Neutron

Stars in a Nonlinear σ, π, ω Mean-Field Approximation. Nuclear Physics A, 799, (2008) 181.

[16] L. D. Landau, Theory of Fermi-liquids, Zh. Eksp. Teor. Fiz. 30, (1956) 1058, [Sov. Phys. JETP 3, 920 (1957)].

[17] L. D. Landau, Oscillations in a Fermi-liquid, Zh. Eksp. Teor. Fiz. 32, (1957) 59 [Sov. Phys. JETP 5, 101].

[18] W. Kohn and L. J. Sham, Self-Consistent Equations Including Exchange and Correlation

Effects. Physical Review, 140, (1965) A1133-A1138.

[19] W. Kohn, Nobel Lecture: Electronic Structure of Matter-Wave Functions and Density

Functional. Reviews of Modern Physics, 71, (1999) 1253.

[20] C. J. Horowitz and B. D. Serot, Properties of nuclear and neutron matter in a relativistic

Hartree-Fock theory, Nuclear Physics, A399 (1983) 529.

[21] H. Uechi, Fermi-liquid properties of nuclear matter in a Dirac-Hartree-fock approximation,

Nuclear Physics, A501 (1989) 813.

[22] H. Uechi, Self-consistent structure in a relativistic Dirac-Hartree-Fock approximation, Nuclear

Physics, A696 (2001) 511.

[23] H. Uechi, The Effective Chiral (σ, π, ω) Model of Quantum Hadrodynamics Applied to

Nuclear Matter and Neutron Stars, Journal of Applied Mathematics and Physics, 3, (2015)

114. Published Online February 2015 in SciRes.

[24] H. Uechi, The Theory of conserving approximations and the density functional theory in

approximations for nuclear matter, Progress of Theoretical Physics, 111 (2004) 525.

[25] L. D. Miller, Exchange potentials in relativistic Hartree-Fock theory of closed-shell nuclei,

Phys. Rev. C 9, (1974) 537.

[26] L. D. Miller, Relativistic single-particle potentials for nuclei, Annals of Physics, V 91,

Issue 1, (1975) 40.

[27] S. Weinberg, Nonlinear Realizations of Chiral Symmetry, Phys. Rev. 166 (1968) 1568.

[28] S. Weinberg, Phenomenological Lagrangians, Physica, A96, (1979) 327.

[29] J. Schwinger, Chiral dynamics, Phys. Lett. 24B (1967) 473.

[30] R. D. Puff, Groud-state properties of nuclear matter, Annals of Physics, 13, (1961) 317.

[31] L. Wilets, R. D. Puff, D. Chiang and W. T. Nutt, Meson dynamics and the nuclear manybody

problem. II. Finite density Hartree-Fock, Physical Review C 14, (1976) 2269.

[32] A. F. Bielajew, and B. D. Serot, Renormalized Hartree-Fock equations in a nuclear relativistic

quantum field theory, Annals of Physics, V156, Issue 2, Elsevier, (1984) 215.

[33] N. M. Hugenholtz and L. Van Hove, A theorem on the single particle energy in a Fermi

gas with interaction, Physica 24 (1958) 363.

[34] B. D. Day, Current state of nuclear matter calculations, Rev. Mod. Phys. 50, (1978) 495.

[35] G. Baym and L. P. Kadanoff, Conservation Laws and Correlation Functions, Phys. Rev.

124 (1961) 287.

[36] G. Baym, Self-Consistent Approximations in Many-Body Systems, Phys. Rev. 127 (1962)

1391.

[37] M. Bonitz, R. Nareyka and D. Semkat, ed. Progress in Nonequilibrium Green's Functions,

Proceedings of the conference, World Scientific, 2000.

[38] M. Bonitz, R. Nareyka and D. Semkat, ed. Progress in Nonequilibrium Green's Functions II,

Proceedings of the conference, World Scientific, 2003.

[39] Y. Takada, Exact self-energy of the many-body problem from conserving approximations,

Physical Review B52, (1995) 12708.

[40] G. Stefanucci and R. Van Leeuwen, Nonequilibrium Many-Body Theory of Quantum Systems,

Cambridge University Press, (2013).

[41] N. P. Landsman and Ch. G. Van Weert, Real- and imaginary-time field theory at finite

temperature and density, Phys. Reports 145 (1987) 141.

[42] H. Uechi, Constraints on the self-consistent relativistic Fermi-sea particle formalism in the

quantum hadrodynamical model, Physical Review, C41 (1990) 744.

[43] K. A. Brueckner and C. A. Levinson, Interacting Particles to a Problem of Self-Consistent

Fields, Phys. Rev. 97 (1955) 1344

[44] K. A. Brueckner, Single Particle Energies in the Theory of Nuclear Matter, Phys. Rev.

110 (1958) 597.

[45] K. A. Brueckner and D. T. Goldman, Single Particle Energies in the Theory of Nuclear

Matter, Physical Review, 117 (1960) 207.

[46] K. A. Brueckner, J. L. Gammel and J. T. Kubis, Calculation of Single-Particle Energies in

the Theory of Nuclear Matter, Physical Review, 118 (1960) 1438.

[47] G. Baym and C. Pethick, in The physics of liquid and solid helium, Part II, ed. K. H. Bennemann

and J. B. Ketterson (John Wiley, New York,1978).

[48] A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover Pub.

New York (2003).

[49] M. Bonitz, Quantum Kinetic Theory, B.G. Tehbner Stuttgart, Leipzig, (1998).

[50] A. M. Zagoskin, Quantum Theory of Many-Body Systems, Springer, New York, Inc. (1998).

[51] B. D. Serot, A relativistic nuclear field theory with π and ρ mesons, Phys. Lett. 86 B

(1979) 146 [Erratum: Phys. Lett. 87B (1979) 403].

[52] H. Heiselberg and M. Hjorth-Jensen, Phases of dense matter in neutron stars, Phys. Rep.,

V328, (2000) 237.

[53] J. M. Lattimer and M. Prakash, Neutron star observations: Prognosis for equation of state

constraints, Phys. Rep., 442, (2007) 109.

[54] R. P. Drake, High-Energy-Density Physics, Springer-verlag, (2006).

[55] S. Weinberg, Gravitation and Cosmology, John Wiley and Sons, New York, 1972.

[56] C. W. Misner, K. S. Thorne and J. W. Wheeler, Gravitation, W. H. Freeman and Company,

New York, 1973.

[57] C. Constantinou, et.al., Thermal properties of hot and dense matter with finite range

interactions, Physical Review, C 92, 025801 (2015).

[58] J. M. Lattimer and A. W. Steiner, Neutron Star Masses and Radii from Quiescent Low-

Mass X-ray Binaries, The Astrophysical Journal, 784:123 (2014).

[59] P. B. Demorest, T. Pennucci, S. M. Ransom, M. S. E. Roberts, and J. W. T. Hessels, A

two-solar-mass neutron star measured using Shapiro delay, Nature (London) 467 (2010)

1081.

[60] J. Antoniadis et al., A Massive Pulsar in a Compact Relativistic Binary, Science 340, (2013)

6131.