nambujl_eos Class Reference

#include <nambujl_eos.h>

Inheritance diagram for nambujl_eos:

quark_eos eos cfl_njl_eos cfl6_eos

Detailed Description

Nambu Jona-Lasinio EOS at zero temperature.

Calculates everything from the quark condensates ([uds].qq) and the chemical potentials ([uds].mu). If "fromqq" is set to false, then instead it calculates everything from the dynamical masses ([uds].ms) and the chemical potentials. L, G, K, and B0 are fixed constants. [uds].pr returns the pressure due to the Fermi-gas contribution plus the bag pressure contribution. [uds.ed] is the energy density for each quark so that e.g. u.ed+u.pr=u.mu*u.n. B0 should be fixed using calc_B0() beforehand to ensure that the energy density and pressure of the vacuum is zero.

The functions set_parameters() should be called first.

The code is based on Buballa99.

The Lagrangian is

\[ {\cal L} = \bar q ( i \partial{\hskip-2.0mm}/ - {\hat m_0}) q \;+\; G \sum_{k=0}^8 [\,({\bar q}\lambda_k q)^2 + ({\bar q} i\gamma_5\lambda_k q)^2\,] + {\cal L}_6 \]

\[ {\cal L}_6 = - K \,[ \,{\rm det}_f ({\bar q}(1+\gamma_5) q) + {\rm det}_f ({\bar q}(1-\gamma_5) q) \,] \, . \]

And the corresponding thermodynamic potential is

\[ \Omega = \Omega_{FG} + \Omega_{Int} \]

where $\Omega_{FG}$ is the Fermi gas contribution and

\[ \frac{\Omega_{\mathrm{Int}}}{V} = - 2 N_c \sum_{i=u,d,s} \int \frac {d^3p}{(2\pi)^3} \sqrt{m_i^2 + p^2} + \frac{\Omega_{V}}{V} \]

\[ \frac{\Omega_{V}}{V} = \sum_{i=u,d,s} 2 G \langle\bar{q}_i q_i \rangle^2 - 4 K \langle \bar{q}_u q_u \rangle \langle \bar{q}_d q_d \rangle \langle \bar{q}_s q_s \rangle + B_0\,. \]

where $B_0$ is a constant defined to ensure that the energy density and the pressure of the vacuum is zero.

Unlike Buballa99, the bag constant, $\Omega_{Int}/V$ is defined without the term

\[ \sum_{i=u,d,s} 2 N_C \int_0^{\Lambda} \frac{d^3 p}{(2 \pi)^3} \sqrt{ m_{0,i}^2+p^2 } ~dp \]

since this allows an easier comparison to the finite temperature EOS. The constant $B_0$ in this case is therefore significantly larger, but the energy density and pressure are still zero in the vacuum.

The Feynman-Hellman theorem (Bernard88 ), gives

\[ \left< \bar{q} q \right> = \frac{\partial m^{*}}{\partial m} \]

The functions calc_e() and calc_p() never return a value other than zero, but will give nonsensical results for nonsensical inputs.

Finite T documentation

Calculates everything from the quark condensates ([uds].qq) and the chemical potentials ([uds].mu). If "fromqq" is set to false, then instead it calculates everything from the dynamical masses ([uds].ms) and the chemical potentials. L, G, K, and B0 are fixed constants. [uds].pr returns the pressure due to the Fermi-gas contribution plus the bag pressure contribution. [uds.ed] is the energy density for each quark so that e.g. u.ed+u.pr=u.mu*u.n. B0 is fixed to ensure that the energy density and pressure of the vacuum is zero.

This implementation includes contributions from antiquarks.

References: Buballa99, Hatsuda94.

Definition at line 125 of file nambujl_eos.h.


Public Member Functions

virtual int set_parameters (double lambda=0.0, double fourferm=0.0, double sixferm=0.0)
 Set the parameters and the bag constant B0.
virtual int calc_p (quark &u, quark &d, quark &s, thermo &lth)
 Equation of state as a function of chemical potentials.
virtual int calc_temp_p (quark &u, quark &d, quark &s, const double T, thermo &th)
 Equation of state as a function of chemical potentials at finite temperature.
virtual int calc_eq_p (quark &u, quark &d, quark &s, double &gap1, double &gap2, double &gap3, thermo &lth)
 Equation of state and gap equations as a function of chemical potential.
virtual int calc_eq_e (quark &u, quark &d, quark &s, double &gap1, double &gap2, double &gap3, thermo &lth)
 Equation of state and gap equations as a function of the densities.
int calc_eq_temp_p (quark &tu, quark &td, quark &ts, double &gap1, double &gap2, double &gap3, thermo &qb, const double temper)
 Equation of state and gap equations as a function of chemical potentials.
int gapfunms (size_t nv, const ovector_view &x, ovector_view &y, void *&pa)
 Calculates gap equations in y as a function of the constituent masses in x.
int gapfunqq (size_t nv, const ovector_view &x, ovector_view &y, void *&pa)
 Calculates gap equations in y as a function of the quark condensates in x.
int gapfunmsT (size_t nv, const ovector_view &x, ovector_view &y, void *&pa)
 Calculates gap equations in y as a function of the constituent masses in x.
int gapfunqqT (size_t nv, const ovector_view &x, ovector_view &y, void *&pa)
 Calculates gap equations in y as a function of the quark condensates in x.
int set_quarks (quark &u, quark &d, quark &s)
 Set the quark objects to use.
virtual const char * type ()
 Return string denoting type ("nambujl_eos").
virtual int set_solver (mroot< void *, mm_funct< void * > > &s)
 Set solver to use in set_parameters().
virtual int set_inte (inte< void *, funct< void * > > &i)
 Set integration object.

Data Fields

double limit
 Accuracy limit for Fermi integrals for finite temperature.
bool fromqq
 Calculate from quark condensates if true (default true).
double L
 The momentum cutoff.
double G
 The four-fermion coupling.
double K
 The 't Hooft six-fermion interaction coupling.
double B0
 The bag constant.
gsl_mroot_hybrids< void *,
mm_funct< void * > > 
def_solver
 The default solver.
The default quark masses
These are the values from Buballa99 which were used to fix the pion and kaon decay constants, and the pion, kaon, and eta prime masses. They are set in the constructor and are in units of $ \mathrm{fm}^{-1} $ .

double up_default_mass
double down_default_mass
double strange_default_mass
The default quark objects
The masses are automatically set in the constructor to up_default_mass, down_default_mass, and strange_default_mass.c

quark def_up
quark def_down
quark def_strange

Protected Types

typedef struct nambujl_eos::njtp_s njtp

Protected Member Functions

int B0fun (size_t nv, const ovector_view &x, ovector_view &y, void *&pa)
 Used by calc_B0() to compute the bag constant.
void njbag (quark &q)
 Calculates the contribution to the bag constant from quark q.
double iqq (double x, void *&pa)
 The integrand for the quark condensate.
double ide (double x, void *&pa)
 The integrand for the density.
double ied (double x, void *&pa)
 The integrand for the energy density.
double ipr (double x, void *&pa)
 The integrand for the pressure.

Protected Attributes

gsl_inte_qag< void *, funct
< void * > > 
def_it
 The default integrator.
inte< void *, funct< void * > > * it
 The integrator for finite temperature integrals.
mroot< void *, mm_funct< void * > > * solver
 The solver to use for set_parameters().
quarkup
 The up quark.
quarkdown
 The down quark.
quarkstrange
 The strange quark.
double cp_temp
 The temperature for calc_temp_p().

Data Structures

struct  njtp_s
 A structure for passing parameters to the integrands [protected]. More...

Member Function Documentation

virtual int set_parameters ( double  lambda = 0.0,
double  fourferm = 0.0,
double  sixferm = 0.0 
) [virtual]

Set the parameters and the bag constant B0.

This function allows the user to specify the momentum cutoff, lambda, the four-fermion coupling fourferm and the six-fermion coupling from the 't Hooft interaction sixferm . If 0.0 is given for any of the values, then the default is used ($ \Lambda=602.3/(\hbar c), G=1.835/\Lambda^2, K=12.36/\Lambda^5 $).

The value of the shift in the bag constant B0 is automatically calculated to ensure that the energy density and the pressure of the vacuum are zero. The functions set_quarks() and set_thermo() can be used before hand to specify the quark and thermo objects.

virtual int calc_p ( quark u,
quark d,
quark s,
thermo lth 
) [virtual]

Equation of state as a function of chemical potentials.

This function automatically solves the gap equations

Reimplemented from quark_eos.

virtual int calc_temp_p ( quark u,
quark d,
quark s,
const double  T,
thermo th 
) [virtual]

Equation of state as a function of chemical potentials at finite temperature.

This function automatically solves the gap equations

Reimplemented from quark_eos.

Reimplemented in cfl_njl_eos.

int gapfunms ( size_t  nv,
const ovector_view x,
ovector_view y,
void *&  pa 
)

Calculates gap equations in y as a function of the constituent masses in x.

The function utilizes the quark objects which can be specified in set_quarks() and the thermo object which can be specified in eos::set_thermo().

int gapfunqq ( size_t  nv,
const ovector_view x,
ovector_view y,
void *&  pa 
)

Calculates gap equations in y as a function of the quark condensates in x.

The function utilizes the quark objects which can be specified in set_quarks() and the thermo object which can be specified in eos::set_thermo().

int gapfunmsT ( size_t  nv,
const ovector_view x,
ovector_view y,
void *&  pa 
)

Calculates gap equations in y as a function of the constituent masses in x.

The function utilizes the quark objects which can be specified in set_quarks() and the thermo object which can be specified in eos::set_thermo().

int gapfunqqT ( size_t  nv,
const ovector_view x,
ovector_view y,
void *&  pa 
)

Calculates gap equations in y as a function of the quark condensates in x.

The function utilizes the quark objects which can be specified in set_quarks() and the thermo object which can be specified in eos::set_thermo().

int set_quarks ( quark u,
quark d,
quark s 
)

Set the quark objects to use.

The quark objects are used in gapfunms(), gapfunqq(), gapfunmsT(), gapfunqqT(), and B0fun().


Field Documentation

double limit

Accuracy limit for Fermi integrals for finite temperature.

limit is used for the finite temperature integrals to ensure that no numbers larger than exp(limit) or smaller than exp(-limit) are avoided. (Default: 20)

Definition at line 153 of file nambujl_eos.h.

bool fromqq

Calculate from quark condensates if true (default true).

If this is false, then computations are performed using the effective masses as inputs

Definition at line 161 of file nambujl_eos.h.


The documentation for this class was generated from the following file:

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