Calculation Parameters

The Chiral EFT Equation of State module has a variety of input parameters and output files which can be used to construct and study the Equation of State in low-energy nuclear matter.

As described in the Quick Start Guide, in order to run the Chiral EFT EoS module, the user must provide a config.yaml file. The purpose of this file is to provide all of the configuration data required to run the module, such as input data and program options. The rules for this input file are located in the OpenAPI 3.0.0 Specification file provided by the module.

Each time the module is executed, it reads the config.yaml file provided by the user and verifies that the input conforms with the OpenAPI specifications. If it does not, module execution will be unsuccessful. when this happens, make sure to check the logs to see what went wrong with your configuration.

It is not necessary to specify every single parameter in the configuration file. If any parameter is left out, it will be automatically filled in with the default value. The default values which are filled in for each parameter can be found in the OpenAPI specification. The only field which is required to be specified on each run is the run_name parameter to ensure that a non-empty configuration file has been provided.

Upon successful execution, there are a few possible output files that the module may produce. These are also described in the OpenAPI specification as part of the output schema. Note that not all of these files are necessarily created on every single run, and many of these require the user to specify a particular option in the config in order to be created. The only files guaranteed to be created on each run are the raw_output and output file containing the entire results of the Chiral EFT C++ module run.

Below are tables providing a brief overview of each module parameter and output file. These include parameter names, default values, and a brief description similar to what is found in the OpenAPI specification file in the module.

Input Parameters

Input parameters required to execute the Chiral EFT EoS module. All parameters have default values in case the user does not specify any (except run_name).

Category

Input Parameter

Default

Description

run_name

"default"

Name of the run (echoed in standard output logs)

chiraleft_parameters

fitted_parameter_set

none
Pre-fitted Chiral EFT potential parameter set
to use instead of manually specified parameter
values (overwrites all Chiral EFT parameter values)

Options are n3lo-450, n3lo-414, and none

cutoff_scale_MeV

450.0
Energy cutoff parameter of Gaussian regulator
cutoff function in \(\text{MeV}\)

cutoff_exponent

3.0
Cutoff exponent parameter of Gaussian regulator
cutoff function (default applied to all potential
contribution terms, 0 means no cutoff)

cutoff_exponent_LO

4.0
Cutoff exponent parameter of Gaussian regulator
cutoff function applied to Leading Order terms
(to prevent interference with higher order terms)

c_lecs
-0.81
3.28
-3.40
3.40
LEC’s of the dimension-two \(\pi\)-N Chiral EFT
Lagrangian, \(c_i\) in \(\text{GeV}^{-1}\)

This includes \(c_1\), \(c_2\), \(c_3\), \(c_4\)

d_lecs
3.06
-3.27
0.45
-5.65
LEC’s of the dimension-three \(\pi\)-N Chiral EFT
Lagrangian, \(\bar{d}_i\) in \(\text{GeV}^{-2}\)

This includes \(\bar{d}_{1}+\bar{d}_{2}\), \(\bar{d}_{3}\), \(\bar{d}_{5}\), \(\bar{d}_{14}-\bar{d}_{15}\)

contacts_LO
-0.154450
-0.155240
-0.1548085
0.000000
-0.142925
0.000000
Contact interaction LEC’s of the Leading Order (LO)
NN contact Lagrangian (2) in units of \(10^4\;\text{GeV}^{-2}\)
These LEC’s are charge-dependent and thus must be
specified for each interaction type. The order in
which they are specified here is:

\(C_{p\,p}\,,\;C_{n\,p}\,,\;C_{n\,n}\)

contacts_NLO
2.150000
1.240000
0.250000
-0.688000
0.610000
0.570000
-0.642500
Contact interaction LEC’s of the Next-to-Leading
Order (NLO) NN contact Lagrangian (7) in units
of \(10^4\;\text{GeV}^{-4}\)

contacts_N3LO
-4.250000
-16.400000
0.100000
2.100000
3.650000
12.000000
1.550000
-0.800000
2.650000
4.630000
-2.420000
-0.370000
1.892000
-0.610000
5.760000
Contact interaction LEC’s of the Next-to-Next-to-
Next-to-Leading Order (N3LO) NN
contact Lagrangian (15) in units of \(10^4\;\text{GeV}^{-6}\)

contact_lsj

true
Specifies whether the given contact LEC’s above are
in LSJ (partial-wave) formalism
If not, they are assumed to be in their standard
LEC form based on the NN contact Lagrangian

three_nucleon_forces
700.0
-0.240
-0.106
Parameters of the three-nucleon interaction used to
create the effective (in-medium) three-nucleon
potential. This includes the three-nucleon cutoff
energy scale in \(\text{MeV}\), and the two unitless LEC’s
appearing in the lowest-order (N2LO) 3N Lagrangian

physical_parameters

hbarc

197.327
Reduced Planck constant times speed of light
in \(\text{MeV}\cdot\text{fm}\) used for unit conversions

proton_mass

938.272

Mass of the proton in \(\text{MeV}\)

neutron_mass

939.5653

Mass of the neutron in \(\text{MeV}\)

average_nucleon_mass

938.9182
Average mass of a nucleon in \(\text{MeV}\)
(used for neutron-proton interactions)

neutral_pion_mass

134.9766

Mass of the neutral pion (\(\pi^0\)) in \(\text{MeV}\)

charged_pion_mass

139.5702

Mass of the charged pions (\(\pi^{+}/\pi^{-}\)) in \(\text{MeV}\)

average_pion_mass

138.0390
Average mass of a pion in \(\text{MeV}\)
(used for most interaction terms)

gA

1.29
Nucleon axial-vector coupling constant
with Goldberger-Treiman correction

fpi

92.4

Pion decay constant

alpha

0.0072967957
Fine structure constant
(used for pion-photon interactions)

gamma

0.0
constant factor in pion-photon vertex
renormalization
(usually either 0 or Euler’s gamma constant)

computational_parameters
chiraleft_potential
->
minpts

1
Minimum number of Gauss-Legendre mesh points
to use for partial-wave angular integrations in
calculating the Chiral EFT nuclear potential
chiraleft_potential
->
maxpts

96
Maximum number of Gauss-Legendre mesh points
to use for partial-wave angular integrations in
calculating the potential
chiraleft_potential
->
cutoff_limit

80.0
Largest allowed value of Gaussian cutoff regulator
exponent in the potential
Note: Typically chosen to be 80.0 for agreement
of deuteron wavefunctions. For a potential which
vanishes at large momentum, use the value 1000.0
or larger
chiraleft_potential
->
momentumpts

80
Number of Gauss-Legendre mesh points to use for
momentum integrations in the effective (in-medium)
three-nucleon interaction terms of the potential
chiraleft_eos
->
epsilon

1.0e-12
Relative precision used for comparison of floating-
point values. Two floating point numbers \(a\)
and \(b\) are approximately equal when
\[| a - b | < \epsilon | a + b |\]
chiraleft_eos
->
interpolation
30
3
20
3
30
3
30
3
Computational parameters involved in evaluating and
interpolating tables of data used in place of
expensive computations. This includes:
  • potential_nn_points: Number of momentum
    grid values on which to evaluate the Chiral
    EFT Nucleon-Nucleon potential for storing in
    a matrix-elements data table
  • potential_nn_order: Polynomial order of
    local interpolation over the Chiral EFT
    Nucleon-Nucleon potential data table
  • potential_3n_points: Number of momentum
    grid values on which to evaluate the Chiral
    EFT Effective Three-Nucleon potential for
    storing in a matrix-elements data table
  • potential_3n_order: Polynomial order of
    local interpolation over the Chiral EFT
    Effective Three-Nucleon potential data table
  • angular_sum_points: Number of \(\cos{\theta}\)
    grid values on which to evaluate the angular
    sum function \(\sum\,\mathcal{C}(\theta_1,\theta_2)\)
    appearing in Second Order MBPT contributions
  • angular_sum_order: Polynomial order of
    local interpolation over the Angular sum
    function data tables
  • self_energy_points: Number of momentum
    values on which to evaluate the Nucleon
    Self-Energy for storing in a data table
  • self_energy_order: Polynomial order of
    local interpolation over the Nucleon Self-
    Energy data table for use in the Second
    Order MBPT contributions
chiraleft_eos
->
first_order_eos
6
10
10
Computational parameters involved in calculating
the First Order (Hartree-Fock) MBPT contribution.
This includes:
  • jmax: Maximum value of total angular
    momentum \(J\) in sum over partial waves
  • p_points_a: Number of Gauss-Legendre
    quadrature points (order of quadrature) to use in
    first subinterval of momentum integration
  • p_points_b: Number of Gauss-Legendre
    quadrature points (order of quadrature) to use in
    second subinterval of momentum integration
chiraleft_eos
->
first_order_self_energy
6
10
10
3.5
2.5
Computational parameters involved in calculating
the Nucleon Self-Energy MBPT contribution to
First Order. This includes:
  • jmax: Maximum value of total angular
    momentum \(J\) in sum over partial waves
  • p_points_a: Number of Gauss-Legendre
    quadrature points (order of quadrature) to use in
    first subinterval of momentum integration
  • p_points_b: Number of Gauss-Legendre
    quadrature points (order of quadrature) to use in
    second subinterval of momentum integration
  • k_max: Maximum momentum value at which
    to evaluate the Nucleon Self-Energy
    (momentum ranges from 0 to k_max)
  • k_fit_max: Maximum momentum value of
    fitting window when calculating effective
    mass approximation (0 to k_max)
chiraleft_eos
->
second_order_eos
4
10
10
10
8
8
Computational parameters involved in calculating
the Second Order MBPT contribution. This includes:
  • jmax: Maximum value of total angular
    momentum \(J\) in sum over partial waves
  • K_points: Number of Gauss-Legendre quadrature
    points (quadrature order) to use for \(K\)
    momentum integration
  • p1_points: Number of Gauss-Legendre quadrature
    points (quadrature order) to use for \(p_1\)
    momentum integration
  • p2_points: Number of Gauss-Legendre quadrature
    points (quadrature order) to use for \(p_2\)
    momentum integration
  • x1_points: Number of Gauss-Legendre quadrature
    points (quadrature order) to use for
    \(x_1 = \cos{\theta_1}\) integration
  • x2_points: Number of Gauss-Legendre quadrature
    points (quadrature order) to use for
    \(x_2 = \cos{\theta_2}\) integration

calculation_options

use_multithreading

false
Whether to use multithreading, implemented with
OpenMP library in C++

n_threads

8
Number of parallel threads on which to run the EoS
calculation when multithreading is enabled

use_three_nucleon_forces

true
Whether to use three-nucleon forces in the Chiral
EFT nuclear potential calculation
(implemented using the in-medium effective 3N force)

use_first_order_eos

true
Whether to calculate the First Order (Hartree-Fock)
MBPT contribution to the Equation of State

use_second_order_eos

true
Whether to calculate the Second Order MBPT
contribution to the Equation of State

use_first_order_self_energy

true
Whether to calculate the First Order Nucleon
Self-Energy MBPT contribution (used to correct the
Second Order EoS contribution)

use_quadratic_asymmetry_expansion

false
Whether to use the quadratic expansion in the
isospin asymmetry to determine the asymmetric matter
EoS instead of the exact result (performance boost)

use_free_energy_ansatz_fit

true
Whether to use the fit the Many-Body Free Energy
results to an ansatz function, resulting in smoother
derivatives requiring less mesh points (performance
boost)
Note: The compuational parameters (number of
mesh points, etc.) have been optimized to work with
the ansatz fit. If this option is turned off, the
number of mesh points must be increased to obtain
a smooth Equation of State.

output_options

output_format

"csv"

File format for all output files

output_precision

12
Number of digits of numerical precision at which to
write EoS data to all output files

include_output_stable

true
Whether to create an output file of the Equation of
State results in the stable regime (removing
unstable and metastable/spinodal regions)

include_output_lepton

true
Whether to create an output file of the Equation of
State results formatted for use by the Lepton module

include_output_flavor

true
Whether to create an output file of the Equation of
State results formatted for use by the Flavor
Equilibration module

include_output_self_energy

false
Whether to create an output file of the Nucleon
Self-Energy calculation data (single-particle
dispersion relation)

Note: Mainly used for debugging

include_output_saturation_properties

false
Whether to create an output file of the nuclear
matter saturation and symmetry properties

verbose

false
Whether to display various extra elements to
standard output such as logos, loading bars, and
some calculated values
Note: In the Calculation Engine, standard output
is forwarded to the logs. Thus this option is mainly
useful for local debugging

eos_grid

density_start

0.032

Initial value of nucleon density in \(\text{fm}^{-3}\)

density_end

0.32

Final value of nucleon density in \(\text{fm}^{-3}\)

density_step

0.032

Step in nucleon density in \(\text{fm}^{-3}\)

isospin_asymmetry_start

0.0

Initial value of isospin asymmetry parameter

\[\left(\delta = \frac{n_n - n_p}{n_n + n_p}\right)\]

isospin_asymmetry_end

1.0

Final value of isospin asymmetry parameter

isospin_asymmetry_step

1.0

Step in isospin asymmetry parameter

temperature_start

0.0

Initial value of temperature in \(\text{MeV}\)

Note: As of MUSES 1.0, the Chiral EFT module
only calculates the EoS at zero temperature.
Finite temperature will be implemented in a later
release

temperature_end

0.0

Final value of temperature in \(\text{MeV}\)

Note: As of MUSES 1.0, the Chiral EFT module
only calculates the EoS at zero temperature

temperature_step

0.0

Step in temperature in \(\text{MeV}\)

Note: As of MUSES 1.0, the Chiral EFT module
only calculates the EoS at zero temperature

Output Files

The Chiral EFT EoS module has many possible output files which can be generated for the user. Below is a brief description of each possible output file and what it contains. For more information on the specific columns, see their description in the module OpenAPI Specification.

Note that most output files require an option to be set to true in order to generate them. These options are also included in the table.

Output File

Option

Columns

Description

raw_output

always created
nucleon_density,
isospin_asymmetry,
temperature,
charge_fraction,
proton_density,
neutron_density,
proton_fermi_momentum,
neutron_fermi_momentum,
proton_mu0,
neutron_mu0,
proton_effective_mass,
neutron_effective_mass,
proton_energy_shift,
neutron_energy_shift,
f_0,
f_1,
f_2,
free_energy
Contains the raw output of the Chiral EFT EoS C++
module. This includes microscopic quantities used in
the calculation such as nucleon fermi momenta and
non-interacting chemical potentials, as well as each
term in the Free Energy expansion from MBPT.
Note: This file is always written with the .csv
extension, even if the output format is set to a
different extension. It is mainly used for
debugging the C++ module.
If use_quadratic_asymmetry_expansion and/or
use_free_energy_ansatz_fit are set to true,
the data in the free_energy column is not
guaranteed to match the data found in output file.
These optimizations are handled during
post-processing.

output

always created
nucleon_density,
isospin_asymmetry,
temperature,
charge_fraction,
proton_density,
neutron_density,
proton_chemical_potential,
neutron_chemical_potential,
free_energy,
energy,
pressure,
entropy,
speed_of_sound,
proton_effective_mass,
neutron_effective_mass,
proton_energy_shift,
neutron_energy_shift
Contains the full output of the Chiral EFT Equation
of State for low-energy nuclear matter. This
includes all thermodynamic quantities derived from
the free energy. It also includes some microscopic
properties of the nucleons, such as individual
chemical potentials and effective masses/energy
shifts.
Depending on the EoS grid used, this Equation of
State may contain unstable or metastable (spinodal)
regions, typically where the liquid-gas phase
transition occurs.

output_stable

include_output_stable
nucleon_density,
isospin_asymmetry,
temperature,
charge_fraction,
proton_density,
neutron_density,
proton_chemical_potential,
neutron_chemical_potential,
free_energy,
energy,
pressure,
entropy,
speed_of_sound,
proton_effective_mass,
neutron_effective_mass,
proton_energy_shift,
neutron_energy_shift
Contains the full output of the Chiral EFT Equation
of State for low-energy nuclear matter like the
file above.
However, this data cuts out unstable and metastable
points, leaving only a stable Equation of State
with less points than the previous file.
The stability conditions for an Equation of State
at zero temperature are given by
\[\frac{\partial^2 P}{\partial \mu_B^2} \geq 0,\quad \frac{\partial^2 P}{\partial \mu_Q^2} \geq 0\]
In the Chiral EFT EoS module, this is equivalent to
\[\frac{\partial P}{\partial n_B} \geq 0,\quad \frac{\partial P}{\partial n_p} \geq 0\]

output_lepton

include_output_lepton
temperature,
muB,
muS,
muQ,
vector_density,
total_S_density,
total_Q_density,
energy_density,
pressure,
entropy_density
Contains the Chiral EFT Equation of State results
formatted as input to the MUSES Lepton module.
The Lepton module requires the EoS on a
\(\left(\mu_B,\,\mu_Q,\,\mu_S\right)\) grid in
order to introduce leptons into the nuclear EoS.

output_flavor_equilibration

include_output_flavor
temperature,
muB,
muS,
muQ,
vector_density,
total_S_density,
total_Q_density,
energy,
pressure,
entropy,
proton_effective_mass,
neutron_effective_mass,
proton_chemical_potential,
neutron_chemical_potential,
proton_vector_density,
neutron_vector_density,
proton_potential,
neutron_potential
Contains the Chiral EFT Equation of State results
formatted as input to the MUSES
Flavor Equilibration module.
The Flavor Equilibration module requires the
EoS on a \(\left(\mu_B,\,\mu_Q,\,\mu_S\right)\) grid as well
as microscopic properties of nucleons in order to
calculate quantities related to beta equilibration
in neutron star matter.

saturation_properties

include_output_saturation_properties
saturation_density,
saturation_energy,
saturation_compressibility,
symmetry_saturation_energy,
symmetry_slope_parameter,
isobaric_incompressibility
Contains Saturation and Symmetry Energy properties
of the calculated nuclear Equation of State.
For definitions of saturation quantities, see here.
For definitions of symmetry energy quantities, see
here.

self_energy

include_output_self_energy
nucleon_density,
isospin_asymmetry,
temperature,
momentum,
proton_kinetic_energy,
neutron_kinetic_energy,
proton_self_energy,
neutron_self_energy,
proton_sp_energy,
neutron_sp_energy
Contains the Nucleon Self-Energy contribution from
MBPT for both protons and neutrons. When
use_first_order_self_energy is enabled, this
function is calculated and used as the single-
particle dispersion relation for nucleons in the
Equation of State calculation.
Note: This file mainly used for debugging Self-
Energy results, however could also potentially be
used as input to future modules describing the full
dispersion relation of nucleons in low-energy
nuclear matter.