# Book Title

###### Abstract

The formalism of next-to-leading order Fermi Liquid Theory is employed to calculate the thermal properties of symmetric nuclear and pure neutron matter in a relativistic many-body theory beyond the mean field level which includes two-loop effects. For all thermal variables, the semi-analytical next-to-leading order corrections reproduce results of the exact numerical calculations for entropies per baryon up to 2. This corresponds to excellent agreement down to subnuclear densities for temperatures up to MeV. In addition to providing physical insights, a rapid evaluation of the equation of state in the homogeneous phase of hot and dense matter is achieved through the use of the zero-temperature Landau effective mass function and its derivatives.

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Chapter 1 Thermal Effects in Dense Matter

Beyond Mean Field Theory

Institute for Advanced Simulation, Institute für Kernphysik, and Jülich Center for Hadron Physics, Forschungszentrum
Jülich, D-52425 Jülich, Germany

c.constantinou@fz-juelich.de

Constantinos Constantinou, Sudhanva Lalit and Madappa Prakash]Sudhanva Lalit and Madappa Prakash

Department of Physics and Astronomy,

Ohio University, Athens, Ohio 45701, United States

sl897812@ohio.edu and
prakash@ohio.edu

### Contents

\@starttoctoc

### 1 Introduction

Core-collapse supernovae, neutron stars from their birth to old age, and binary mergers involving neutron stars all pass through stages in which there are considerable variations in the baryon density, temperature, and lepton content. Simulations of these astrophysical phenomena involve general relativistic hydrodynamics and neutrino transport with special relativistic effects. Convection, turbulence, magnetic fields, etc., also play crucial roles. The macroscopic evolution in each case is governed by microphysics involving strong, weak and electromagnetic interactions. Depending on the baryon density , temperature , and the lepton content of matter (characterized by when neutrinos are trapped or by the net electron concentration in neutrino-free matter), various phases of matter are encountered. For sub-nuclear densities ( and temperatures , different inhomogeneous phases are encountered. A homogeneous phase of nucleonic and leptonic matter prevails at near- and supra-nuclear densities ( at all temperatures. With progressively increasing density, homogeneous matter may contain hyperons, quark matter and Bose condensates.

Central to an understanding of the above astrophysical phenomena is the equation of state (EOS) of matter as a function of , and (or ) as it is as an integral part of hydrodynamical evolution, and controls electron capture and neutrino interactions in ambient matter. The EOS of dense matter has been investigated in the literature extensively, but for the most part those for use in the diverse physical conditions of relevance to astrophysical applications have been based on mean field theory in both non-relativistic potential or relativistic field-theoretical approaches. A recent article honoring Gerry Brown reviews the current status and advances made to date in the growing field of neutron star research [1].

The objective of this work is to assess the extent to which the model independent formalism of Fermi Liquid Theory (FLT) [2] is able to accurately describe thermal effects in dense homogeneous nucleonic matter under degenerate conditions for models beyond mean field theory (MFT). Recently, a next-to-leading order (NLO) extension of the leading-order FLT was developed in Ref. [3] incorporating its relativistic generalization in Ref. [4]. The FLT+NLO formalism was applied to non-relativistic potential models with contact and finite-range interactions as well as to relativistic models of dense matter at the mean field level in Ref. [3] . Excellent agreement with the results of exact numerical calculations for all thermal variables was found with the semi-analytical FLT+NLO results. In this contribution, we present similar excellent agreement with the exact numerical results of a relativistic field-theoretical model beyond the MFT level that includes two-loop (exchange) effects recently reported in Ref. [5]. The gratifying result is that the FLT+NLO formalism extends agreement with the exact numerical results for all and for which the entropy per baryon . This means that, for MeV, the method can adequately describe state variables down to a density of . For densities below , inhomogeneous phases occur for which a separate treatment is required. This development not only provides a check of time-consuming many-body calculations of dense matter at finite temperature, but also serves to accurately (and, to rapidly) calculate thermal effects from a knowledge of the zero-temperature single-particle spectra for up to 2 for which effects of interactions are relatively important.

The organization of this contribution is as follows. In Sec. 2, we summarize the NLO formalism of FLT recently developed in Ref. [3]. Section 3 contains a brief description of the relativistic field-theoretical model that extends mean-field theory (MFT) to include two-loop (TL) effects as implemented in Ref. [5]. Working formulas required for the evaluation of the degenerate-limit thermal effects (in particular, expressions for the single particle spectra) are given in this section which also includes our results and associated discussion. A summary of our work along with conclusions are in Sec. 4. Personal tributes to Gerry Brown from two of the authors (Constantinou and Prakash) form the content of Sec. 5.

### 2 Next-to-Leading Order Fermi Liquid Theory

The thermodynamics of fermion systems entails evaluation of integrals of the type

(0) |

where is the temperature, is the chemical potential, and is the single-particle spectrum of the underlying model. The functional form of is particular to the state property in question. Equivalently, we can write

(0) |

where

(0) | |||||

(0) |

Above, is inclusive of all those terms in the spectrum which depend only on the density . The Landau effective mass function, , and its derivatives with respect to momentum play crucial roles in determining the thermal effects.

In the degenerate limit, characterized by large values of the parameter , Sommerfeld’s Lemma

(0) |

can be used for the approximate evaluation of such integrals. Truncation of the series at the first term recovers results for cold matter; the second term produces the familiar Fermi Liquid Theory (FLT) corrections and the third term represents the next-to-leading order (NLO) extension to FLT. Owing to the asymptotic nature of the Sommerfeld formula, the expansion will, in general, diverge at higher orders unless all terms are retained.

The number density of single-species fermions with internal degrees of freedom in 3 dimensions is (throughout we use units in which )

(0) |

In the present context, we take to be an independent variable as is appropriate for a system that does not exchange particles with an external reservoir but whose total volume is allowed to change. Thus

(0) |

where is the Fermi momentum and is the result of Eq. (2) evaluated according to Eq. (2). Perturbative inversion of Eq. (2) leads to

(0) |

where

(0) |

is the Landau effective mass. The combination of Eq. (2) with the Sommerfeld expansion of the entropy density, formally given by

(0) | |||||

(0) |

yields an expression for in terms of quantities defined on the Fermi surface:

(0) | |||||

(0) |

where is the level density parameter with denoting the Fermi temperature, and

(0) |

Then the entropy per particle is the simple ratio whereas the thermal energy, pressure and chemical potential are obtained via Maxwell’s relations (the integrals below are performed at constant density):

(0) | |||||

(0) | |||||

(0) | |||||

where

(0) |

Other quantities of interest such as the specific heats at constant volume and pressure, and the thermal index are given by standard thermodynamics:

(0) | |||||

(0) | |||||

(0) |

Note that while the NLO terms in the thermal quantities above have the same temperature dependences as those of a free Fermi gas, the accompanying density-dependent factors differ reflecting the effects of interactions.

### 3 Application to Models Beyond Mean Field Theory

In this work, we investigate the degenerate-limit thermodynamics of a relativistic field-theoretical model in which the nucleon-nucleon (NN) interaction is mediated by the exchange of , , and mesons (scalar, vector, iso-vector and pseudo-vector, respectively). Nonlinear self-couplings of the scalar field are also included. The model is described by the Lagrangian density [5, 6, 7]

(0) | |||||

(0) | |||||

(0) | |||||

where

(0) | |||||

(0) |

are the field-strength tensors and are the isospin matrices. We use the masses MeV, MeV, MeV, MeV and MeV, the couplings , , , and , the pion decay constant MeV and the nucleon axial current constant as in Ref. [5].

All thermodynamic quantities of interest can be derived from the grand potential density which is related to the pressure by . For an isotropic system in its rest-frame, the pressure is obtained from the diagonal elements of the spatial part of the energy-momentum tensor . In mean-field theory (MFT), the result is

(0) | |||||

where

(0) |

In the mean-field approximation, the spectrum that enters the Fermi distribution function is given by

(0) |

The +(-) sign corresponds to particles (antiparticles) and the subscripts to the two nucleon species. The Dirac effective mass results from the minimization of with respect to the expectation value of the scalar field.

The leading corrections to the mean-field arise from two-loop (TL) exchanges of the mesons involved in the model. These corrections are given by (see, e.g., [8, 5])

(0) | |||||

(0) | |||||

(0) | |||||

(0) | |||||

where

(0) | |||||

(0) | |||||

(0) |

The corresponding TL contributions to the single-particle spectrum [via with ; =nucleon species] are [8, 5] :

(0) | |||||

(0) | |||||

(0) | |||||

Note that at the TL level, the self-interactions of the scalar field bestow upon it an effective scalar-meson mass

(0) |

which is used in all exchange terms involving the - meson.

#### 3.1 Two-loop calculations of dense nucleonic matter

The degenerate limit formalism delineated in Sec. 2 requires for its implementation, in principle, only the parts of the spectrum [for ] and the pressure (for and ). Note, however, that for cold matter the statements and are equivalent (being that at , ) and that the energy density is somewhat easier to minimize with respect to in order to obtain . We therefore opt to work with the latter. Confining ourselves to symmetric nuclear matter (SNM) and pure neutron matter (PNM) in the interest of simplicity, we have for the energy density (in the notation of Ref. [8])

(0) | |||||

(0) | |||||

(0) | |||||

(0) | |||||

(0) | |||||

and for the spectrum [via ; =meson]

(0) | |||||

(0) | |||||

(0) | |||||

(0) | |||||

(0) | |||||

where

(0) | |||||

(0) | |||||

(0) | |||||

(0) | |||||

(0) | |||||

(0) |

With the inclusion of the TL contributions and noting that , the Dirac effective mass is determined by solving

(0) | |||||

(0) | |||||

(0) | |||||

(0) | |||||

(0) | |||||

(0) | |||||

where

(0) | |||||

(0) | |||||