Chiral Mean Field (CMF) model

The Chiral Mean Field (CMF) model is based on a non-linear realization of the \(SU(3)\) sigma model, where hadrons interact through meson exchange, including \(\sigma\), \(\zeta\), \(\delta\), \(\omega\), \(\phi\), and \(\rho\). It is constructed in a chirally invariant manner, with most of the particle masses deriving from interactions with the medium. Consequently, these masses decrease at high densities and/or temperatures [1]. The commonly used sigma model is enhanced by the non-linear realization, which employs pseudoscalar mesons as angular parameters for the chiral transformation. This model also offers an ideal mechanism for generating equations of state (EoS’s) for astrophysical applications and has been fitted to match low- and high-energy physics data [2]. It is applicable at zero temperature, intermediate, and higher temperatures, accommodating degrees of freedom expected to manifest in various astrophysical scenarios, including leptons, baryons, and quarks, all within a single description. Furthermore, it reproduces QCD features such as chiral symmetry restoration and the transition to quark matter (utilizing a Polyakov-inspired loop (\(\Phi\)) as an order parameter for the deconfinement phase transition) [3]. The model is relativistic and thus respects causality, provided that repulsive vector interactions do not exceed certain limits [4]. Moreover, the model has been extended to account for the influence of magnetic fields and anomalous magnetic moments [5].

Quickstart

Using Docker

The quickest way to obtain the module involves pulling one of its docker images from the CMF++ container registry. Therefore, Docker must be locally installed.

To pull a Docker image, for instance v0.9.3, use

docker pull registry.gitlab.com/nsf-muses/module-cmf/cmf:v0.9.3

Alternatively, to build the latest version, just clone the repository clone and build the docker image with

git clone https://gitlab.com/nsf-muses/module-cmf/cmf
cd cmf
bash build_docker.sh

the container tagged as cmf:local will be created.

Without using Docker

Required libraries

The module can also be compiled and executed locally without using Docker. For this purpose, the following libraries are required:

• yaml-cpp, install CMAKE, and build yaml-cpp from source with:

  git clone https://github.com/jbeder/yaml-cpp.git
  cd yaml-cpp
  mkdir build
  cd build
  cmake -DYAML_BUILD_SHARED_LIBS=OFF ..
  make
  make DESTDIR=/desired/path/to/installation install
  

  the file libyaml-cpp.a will be created upon successful compilation in DESTDIR.

  for macOS, use Homebrew brew install yaml-cpp. For Linux, use your package manager, for instance, in Debian use sudo apt-get install libyaml-cpp-dev or install from source using the instructions detailed above.)

• doctest, install CMAKE, and build doctest from source with:

  git clone https://github.com/doctest/doctest.git
  cd doctest
  cmake .
  make DESTDIR=/desired/path/to/installation install
  

  the file doctest.h will be created upon successful compilation in DESTDIR/PREFIX, the default PREFIX is /usr/local/.

  for macOS, use Homebrew brew install doctest. For Linux, use your package manager, for instance, in Debian use sudo apt-get install doctest-dev or install from source using the instructions detailed above.)

• Windows using Windows Subsystem for Linux (WSL)

  It is recommended to use Windows WSL instead of cygwin. For this purpose, open a PowerShell terminal and install WSL with wsl --install -d Debian, then reboot your system. Once your system is up again, a Powershell terminal will open itself and request a new UNIX username. Note that your username must fulfill the UNIX username conventions (must start with a lowercase letter, may only contain lowercase letters, underscore (_), and dash (-), and may optionally end with a dollar sign ($)). After accepting the username, a password needs to be provided. Then the terminal will open the Debian Linux subsystem inside Windows. To update your subsystem and install the required libraries, use

  sudo apt update && sudo apt upgrade -y
  sudo apt-get -y install build-essential libyaml-cpp-dev doctest-dev
  yes | pip3 install openapi-core
  

  For basic usage please refer to WSL documentation.

How to build?

After successful installation of the required libraries, type

git clone https://gitlab.com/nsf-muses/module-cmf/cmf
bash build.sh

The executable cmf will be created upon successful compilation and linking. Note that if the libraries are not installed in the default path, the Makefile will need some modifications, specifically in the flags -I and -L inside CXXFLAGS and LDFLAGS, respectively.

Usage

The basic execution involves one input yaml file called config.yaml, this file can be created with the python script create_config.py. Then, this file has to be validated against the openAPI CMF specifications yaml, with the python script yaml_preprocess.py, where all the missing values will be filled into their default value and the validated_config.yaml file will be created. If the config.yaml is valid then the cmf executable can be called. The cmf executable admits either no arguments (default case) or just one: the path for the validated_config.yaml file, which by default is ../input/validated_config.yaml. After this, clean_output.py, a python script layer is called to clean, interpolate, and filter the data into quarks, baryons and vacuum, and then split it into stable, metastable, and unstable solutions. Finally, the postprocess.py script is executed to create the output files for the other MUSES modules based on the cleaned ouput files from the previous step. In order to expedite the preprocess-execution-postprocess cycle, two execution scripts are provided: run_docker.sh, for docker execution with a config.yaml inside the default input folder; and run.sh for local execution with a config.yaml inside the default input folder

Using Docker

The most basic execution is called the default one, which will use this YAML configuration file . After successfully built of the container, it can be executed with

cd /path/to/git/clone/of/cmf/repo/
bash run_docker_default.sh

the output files will be located in /path/to/git/clone/of/cmf/repo/output/.

For a run with different parameters, just create a new YAML configuration file inside /path/to/git/clone/of/cmf/repo/input/ using the command line options for the python script create_config.py, then:

cd /path/to/git/clone/of/cmf/repo/
bash run_docker.sh

Without using Docker

After successful building via bash build.sh, and creation of a config.yaml file with the python script create_config.py, type

cd /path/to/git/clone/of/cmf/repo/
bash run.sh

For default execution without Docker, type

cd /path/to/git/clone/of/cmf/repo/
bash run_default.sh

the output files will be located in /path/to/git/clone/of/cmf/repo/output/.

Output

Successful execution for a production run will produce multiple output files in the /path/to/git/clone/of/cmf/repo/output/ folder depending on the value of the flags output_Lepton, output_flavor_equilibration, output_particle_properties, and output_debug:

• CMF_output_stable.csv/h5, tabulated CSV/HDF5 output for the stable EoS as defined in the schema inside the OpenAPI-Specifications file

• CMF_output_metastable.csv/h5, tabulated CSV/HDF5 output for the metastable EoS as defined in the schema inside the OpenAPI-Specifications file

• CMF_output_unstable.csv/h5, tabulated CSV/HDF5 output for the unstable EoS as defined in the schema inside the OpenAPI-Specifications file

• CMF_output_particle_properties_baryons.csv/h5, tabulated CSV/HDF5 output for the baryons stable+metastable EoS population details as defined in the schema inside the OpenAPI-Specifications file. Outputed if output_particle_properties is set to true.

• CMF_output_particle_properties_quarks.csv/h5, tabulated CSV/HDF5 output for the quark stable+metastable EoS population details as defined in the schema inside the OpenAPI-Specifications file. Outputed if output_particle_properties is set to true.

• CMF_output_for_Lepton_quarks.csv/h5, tabulated CSV/HDF5 output for the quarks stable+metastable EoS for Lepton module as defined in the schema inside the OpenAPI-Specifications file. Outputed if output_Lepton is set to true.

• CMF_output_for_Lepton_baryons.csv/h5, tabulated CSV/HDF5 output for the baryons stable+metastable EoS for Lepton module as defined in the schema inside the OpenAPI-Specifications file. Outputed if output_Lepton is set to true.

• CMF_output_for_Flavor_equilibration.csv/h5, tabulated CSV/HDF5 output for the stable EoS for Flavor equilibration module as defined in the schema inside the OpenAPI-Specifications file. Outputed if output_Flavor_equilibration is set to true.

Scientific motivation and algorithm

Introduction

Since the inception of the Big Bang theory, the realm of physics spanning from Planck’s time (\(10^{-43} \text{s}\)) to the present has piqued significant interest. According to this theory, scientists posit the existence of a singularity, characterized by quantum fluctuations, at the universe’s inception. Suddenly, a colossal explosion ensued, releasing an immense surge of energy. In this extraordinary environment of scorching temperatures, matter was conspicuously absent, leaving only pure radiation fields in its wake.

Approximately \(10^{-36} \text{s}\) later, the universe underwent a remarkable expansion known as the inflation era, during which the four fundamental forces segregated. Following this era of inflation, the universe continued to expand, and temperatures gradually subsided, allowing matter to take form. At this juncture, a scalding concoction of quark-gluon plasma (QGP), comprising liberated quarks, gluons, photons, and leptons, emerged. From this primordial plasma, hadrons emerged.

The dynamics in this energy regime are governed by quantum chromodynamics (QCD), which can be categorized into two sub-regimes: the high-energy realm (\(10^2 \text{GeV}\)) and the low-energy realm (\(10^{-2} \text{GeV}\)). In the high-energy realm, quarks and gluons are the dominant degrees of freedom, with quarks exhibiting asymptotic freedom, characterized by a small coupling constant and the applicability of perturbation theory. Conversely, in the low-energy realm, quarks and gluons are confined within hadrons, resulting in a large coupling constant and the inapplicability of perturbation theory. This low-energy realm is also referred to as the non-perturbative regime of QCD. The interplay of asymptotic freedom and quark confinement confers distinctive properties upon the strong force, setting it apart from other forces.

As the universe expanded and cooled, it gave rise to galaxies, stars, and ultimately life. To this day, 13.7 billion years after the Big Bang, the universe continues its expansion.

To unravel the mysteries of the early universe, physicists delve into high-energy interactions. This quest is pursued through relativistic heavy-ion collisions (HICs). Relativistic HICs recreate the extreme conditions of QCD matter by colliding two beams of heavy ions, each hurtling at approximately 99% of the speed of light, head-on. Under specific conditions, this collision melts heavy ions, releasing their constituent gluons and quarks. As the collision zone expands and temperatures plummet, hadronization occurs, causing quarks and gluons to coalesce into hadrons. Throughout this process, the properties of these particles undergo profound alterations, offering insights into the events within the collision zone. Empirical data is meticulously analyzed using appropriate techniques to decipher the intricacies of QCD physics.

The property of asymptotic freedom in strong interactions endows the QGP with chiral symmetry. Within chiral symmetry, the mass of light quarks becomes negligible in comparison to that of heavy quarks, a scenario known as the chiral limit. At low temperatures and moderate densities, hadrons materialize due to QCD confinement. The transition from QGP to hadrons, marking the breaking of chiral symmetry, has been estimated through recent lattice calculations to occur at around \(T_c \sim 155 \text{MeV}\). Moreover, at low baryonic densities and high temperatures, a crossover region materializes where quarks and hadrons coexist, constituting a second-order phase transition. Within the cores of neutron stars, where densities can reach several times the nuclear saturation density, this environment serves as a natural laboratory for probing QCD physics.

Neutron star interiors encompass a vast range of densities, spanning from approximately \(10^{4} \text{g cm}^{-3}\) in the crust to about \(10^{15} \text{g cm}^{-3}\) in the stellar core, representing several times the nuclear saturation density. While our understanding of matter holds reasonably well up to slightly beyond saturation density, knowledge about higher densities remains limited. An exception occurs at extremely high densities, surpassing those within neutron star cores, where perturbative QCD (PQCD) yields reliable results. Given that lattice QCD has not yet been extrapolated to high densities, effective models, such as the CMF model, calibrated for different energy regimes, become a natural choice for describing such environments. These models can generate dependable results pertaining to the matter’s equation of state (EoS) and the associated particle population. In the context of neutron stars, this data can be employed in dynamic simulations encompassing core-collapse supernova explosions, star cooling, and compact star mergers, necessitating tabulated information on microphysics as input.

Algorithm

In computational terms, the CMF model is represented as a collection of seven coupled non-linear algebraic equations for the variables \(\sigma\), \(\zeta\), \(\delta\), \(\omega\), \(\rho\), \(\phi\), and \(\Phi\):

\[\begin{equation} \sigma: \quad \sum_i g_{i\sigma} \rho_{s,i} = -k_0\chi_0^2\sigma+4k_1(\sigma^2+\zeta^2+\delta^2)\sigma+2k_2(\sigma^2+3\delta^2)\sigma+2k_3\chi_0\sigma\zeta+\frac{2\epsilon}{3}\chi_0^4\frac{\sigma}{\sigma^2-\delta^2}-m_\pi^2f_\pi, \end{equation}\]

\[\begin{equation} \delta: \quad \sum_i g_{i\delta}\rho_{s,i} = -k_0\chi_0^2\delta+4k_1(\sigma^2+\zeta^2+\delta^2)\delta+2k_2(3\sigma^2+\delta^2)\delta-2k_3\chi_0\delta\zeta-\frac{2\epsilon}{3}\chi_0^4\frac{\delta}{\sigma^2-\delta^2}, \end{equation}\]

\[\begin{equation} \zeta: \quad \sum_i g_{i\zeta}\rho_{s,i} = -k_0\chi_0^2\zeta+4k_1(\sigma^2+\zeta^2+\delta^2)\zeta+4k_2\zeta^3+k_3\chi_0(\sigma^2-\delta^2)+\frac{\epsilon}{3\zeta}\chi_0^4-\bigg(\sqrt2m_K^2f_K-\frac{1}{\sqrt2}m_\pi^2f_\pi\bigg), \end{equation}\]

\[\begin{split}\begin{equation} \omega: \quad \sum_i g_{i\omega}n_{B,i} = m_\omega^2\omega+2 g_4 \begin{cases} \text { C1: } 2 \omega \left(\omega^2 + 3 \rho^2 \right),\\ \text { C2: } \omega \left(2 \omega^2 + 3 \phi^2 \right),\\ \text { C3: } 2 \omega \left(\omega^2 + \rho^2 + \phi^2 \right),\\ \text { C4: } \left(2\omega^3+3\phi^2\omega+3\sqrt2\phi\omega^2+\frac{\phi^3}{\sqrt{2}} \right), \end{cases} \\ \end{equation}\end{split}\]

\[\begin{split}\begin{equation} \phi: \quad \sum_i g_{i\phi}n_{B,i} = m_\phi^2\phi+2 g_4\begin{cases} \text { C1: } 4 \phi^3,\\ \text { C2: } \phi \left( \phi^2 + 3 \left(\omega^2+\rho^2\right)\right),\\ \text { C3: } 2 \phi \left( \omega^2 + \phi^2 + \rho^2 \right),\\ \text { C4: } \frac{\phi^3}{2} + 3 \omega^2 \phi + \sqrt{2} \omega^3 + \frac{3}{\sqrt{2}} \omega \phi^2, \end{cases} \\ \end{equation}\end{split}\]

\[\begin{split}\begin{equation} \rho: \quad \sum_i g_{i\rho}n_{B,i} = m_\rho^2\rho+2 g_4\begin{cases} \text { C1: } 2 \rho ( 3\omega^2 + \rho^2),\\ \text { C2: } \rho ( 3 \phi^2 + 2\rho^2 ),\\ \text { C3: } 2 \rho ( \omega^2 + \phi^2 + \rho^2),\\ \text { C4: } 0, \end{cases} \end{equation}\end{split}\]

\[\begin{equation} \Phi: \quad \sum_i g_{i\Phi}\rho_{s,i} = 2 a_{1} \mu_{B}^{4} \Phi + a_3T_0^4\frac{12\Phi}{3\Phi^2-2\Phi-1} \end{equation}\]

where the index \(i\) runs for quarks (up, down, strange), Baryons (Octet and Decuplet), Mesons (scalar, pseudoscalar, and vector), and their correspondent antiparticles. The coupling factors \(g_{i\sigma}\), \(g_{i\zeta}\), \(g_{i\delta}\), \(g_{i\omega}\), \(g_{i\phi}\), \(g_{i\Phi}\) correlate particle \(i\)-th with the mean fields \(\sigma\), \(\zeta\), \(\delta\), \(\omega\), \(\phi\), and \(\rho\), respectively. \(\rho_{s}\) is the scalar density and \(n_B\) is the vector density.

The following flowchart represents the basic algorithm of the code:

Therefore, the main solving routine used is the multidimensional root solver fsolve based on the MINPACK library.

References

1. Papazoglou:1998vr

2. Dexheimer:2008ax

3. Dexheimer:2009hi

4. Dexheimer:2017nse

5. Dexheimer:2011pz