Public Member Functions | Public Attributes | Protected Member Functions | Protected Attributes | Private Member Functions | List of all members
o2scl::eos_quark_cfl Class Reference

Nambu Jona-Lasinio model with a schematic CFL di-quark interaction at finite temperature. More...

#include <eos_quark_cfl.h>

Inheritance diagram for o2scl::eos_quark_cfl:
o2scl::eos_quark_njl o2scl::eos_quark o2scl::eos_base o2scl::eos_quark_cfl6

Public Member Functions

virtual int set_parameters (double lambda=0.0, double fourferm=0.0, double sixferm=0.0, double fourgap=0.0)
 Set the parameters and the bag constant 'B0'. More...
 
virtual int calc_eq_temp_p (quark &u, quark &d, quark &s, double &qq1, double &qq2, double &qq3, double &gap1, double &gap2, double &gap3, double mu3, double mu8, double &n3, double &n8, thermo &qb, double temper)
 Calculate the EOS. More...
 
virtual int test_derivatives (double lmom, double mu3, double mu8, test_mgr &t)
 Check the derivatives specified by eigenvalues()
 
virtual int eigenvalues (double lmom, double mu3, double mu8, double egv[36], double dedmuu[36], double dedmud[36], double dedmus[36], double dedmu[36], double dedmd[36], double dedms[36], double dedu[36], double dedd[36], double deds[36], double dedmu3[36], double dedmu8[36])
 Calculate the energy eigenvalues as a function of the momentum. More...
 
int set_quartic (quartic_real_coeff &q)
 Set the routine for solving quartics.
 
int test_integration (test_mgr &t)
 Test the integration routines.
 
int test_normal_eigenvalues (test_mgr &t)
 Test the routine to compute the eigenvalues of non-superfluid fermions.
 
int test_gapped_eigenvalues (test_mgr &t)
 Test the routine to compute the eigenvalues of superfluid fermions.
 
virtual const char * type ()
 Return string denoting type ("eos_quark_cfl")
 
- Public Member Functions inherited from o2scl::eos_quark_njl
virtual int set_parameters (double lambda=0.0, double fourferm=0.0, double sixferm=0.0)
 Set the parameters and the bag constant B0. More...
 
virtual int calc_p (quark &u, quark &d, quark &s, thermo &lth)
 Equation of state as a function of chemical potentials. More...
 
virtual int calc_temp_p (quark &u, quark &d, quark &s, double T, thermo &th)
 Equation of state as a function of chemical potentials at finite temperature. More...
 
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, double temper)
 Equation of state and gap equations as a function of chemical potentials.
 
int gapfunms (size_t nv, const ubvector &x, ubvector &y)
 Calculates gap equations in y as a function of the constituent masses in x. More...
 
int gapfunqq (size_t nv, const ubvector &x, ubvector &y)
 Calculates gap equations in y as a function of the quark condensates in x. More...
 
int gapfunmsT (size_t nv, const ubvector &x, ubvector &y)
 Calculates gap equations in y as a function of the constituent masses in x. More...
 
int gapfunqqT (size_t nv, const ubvector &x, ubvector &y)
 Calculates gap equations in y as a function of the quark condensates in x. More...
 
int set_quarks (quark &u, quark &d, quark &s)
 Set the quark objects to use. More...
 
virtual int set_solver (mroot< mm_funct11, boost::numeric::ublas::vector< double >, jac_funct11 > &s)
 Set solver to use in set_parameters()
 
virtual int set_inte (inte<> &i)
 Set integration object.
 
- Public Member Functions inherited from o2scl::eos_quark
virtual int calc_e (quark &u, quark &d, quark &s, thermo &th)
 Calculate equation of state as a function of density.
 
virtual int calc_temp_e (quark &u, quark &d, quark &s, double temper, thermo &th)
 Calculate equation of state as a function of density.
 
- Public Member Functions inherited from o2scl::eos_base
virtual void set_thermo (thermo &th)
 Set class thermo object.
 
virtual const thermoget_thermo ()
 Get class thermo object.
 

Public Attributes

double eq_limit
 The equal mass threshold.
 
bool integ_test
 Set to true to test the integration (default false)
 
quartic_real_coeff_cern def_quartic
 The default quartic routine. More...
 
double gap_limit
 Smallest allowable gap (default 0.0) More...
 
bool zerot
 If this is true, then finite temperature corrections are ignored (default false) More...
 
bool fixed_mass
 Use a fixed quark mass and ignore the quark condensates.
 
bool color_neut
 If true, then ensure color neutrality.
 
double GD
 Diquark coupling constant (default 3 G/4) More...
 
double inte_epsabs
 The absolute precision for the integration (default $ 10^{-4} $ ) More...
 
double inte_epsrel
 The relative precision for the integration (default $ 10^{-4} $ ) More...
 
size_t inte_npoints
 The number of points used in the last integration (default 0) More...
 
- Public Attributes inherited from o2scl::eos_quark_njl
double limit
 Accuracy limit for Fermi integrals for finite temperature. More...
 
bool fromqq
 Calculate from quark condensates if true (default true) More...
 
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.
 
mroot_hybrids< mm_funct11, boost::numeric::ublas::vector< double >, boost::numeric::ublas::matrix< double >, jac_funct11def_solver
 The default solver.
 
inte_qag_gsl def_it
 The default integrator.
 
double up_default_mass
 
double down_default_mass
 
double strange_default_mass
 
quark def_up
 
quark def_down
 
quark def_strange
 
- Public Attributes inherited from o2scl::eos_quark
fermion_eval_thermofet
 Object for computing fermion thermodynamics.
 
fermion_eff def_fet
 Default fermion thermodynamics.
 
- Public Attributes inherited from o2scl::eos_base
thermo def_thermo
 The default thermo object.
 

Protected Member Functions

virtual int integrands (double p, double res[])
 The integrands. More...
 
int normal_eigenvalues (double m, double lmom, double mu, double lam[2], double dldmu[2], double dldm[2])
 Compute ungapped eigenvalues and the appropriate derivatives.
 
int gapped_eigenvalues (double m1, double m2, double lmom, double mu1, double mu2, double tdelta, double lam[4], double dldmu1[4], double dldmu2[4], double dldm1[4], double dldm2[4], double dldg[4])
 Treat the simply gapped quarks in all cases gracefully. More...
 
For the integration
double rescale_error (double err, double result_abs, double result_asc)
 The error scaling function for integ_err.
 
int integ_err (double a, double b, const size_t nr, ubvector &res, double &err2)
 A new version of inte_qng_gsl to integrate several functions at the same time.
 
- Protected Member Functions inherited from o2scl::eos_quark_njl
int B0fun (size_t nv, const ubvector &x, ubvector &y)
 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, const njtp &pa)
 The integrand for the quark condensate.
 
double ide (double x, const njtp &pa)
 The integrand for the density.
 
double ied (double x, const njtp &pa)
 The integrand for the energy density.
 
double ipr (double x, const njtp &pa)
 The integrand for the pressure.
 

Protected Attributes

double temper
 Temperature.
 
double smu3
 3rd gluon chemical potential
 
double smu8
 8th gluon chemical potential
 
For computing eigenvalues
gsl_matrix_complex * iprop
 Inverse propagator matrix.
 
gsl_matrix_complex * eivec
 The eigenvectors.
 
ubmatrix_complex dipdgapu
 The derivative of the inverse propagator wrt the ds gap.
 
ubmatrix_complex dipdgapd
 The derivative of the inverse propagator wrt the us gap.
 
ubmatrix_complex dipdgaps
 The derivative of the inverse propagator wrt the ud gap.
 
gsl_vector * eval
 The eigenvalues.
 
gsl_eigen_hermv_workspace * w
 Workspace for eigenvalue computation.
 
- Protected Attributes inherited from o2scl::eos_quark_njl
inteit
 The integrator for finite temperature integrals.
 
mroot< mm_funct11, boost::numeric::ublas::vector< double >, jac_funct11 > * 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()
 
- Protected Attributes inherited from o2scl::eos_base
thermoeos_thermo
 A pointer to the thermo object.
 

Private Member Functions

 eos_quark_cfl (const eos_quark_cfl &)
 
eos_quark_cfloperator= (const eos_quark_cfl &)
 

Numerical methods

typedef boost::numeric::ublas::vector< double > ubvector
 
typedef boost::numeric::ublas::vector< std::complex< double > > ubvector_complex
 
typedef boost::numeric::ublas::matrix< std::complex< double > > ubmatrix_complex
 
quartic_real_coeffquartic
 The routine to solve quartics.
 

Additional Inherited Members

- Public Types inherited from o2scl::eos_quark_njl
typedef boost::numeric::ublas::vector< double > ubvector
 
typedef struct o2scl::eos_quark_njl::njtp_s njtp
 A structure for passing parameters to the integrands.
 

Detailed Description

The variable B0 must be set before use.

The original Lagrangian is

\[ {\cal L} = {\cal L}_{\mathrm{Dirac}} + {\cal L}_{\mathrm{4-fermion}} + {\cal L}_{\mathrm{6-fermion}} + {\cal L}_{CSC1} + {\cal L}_{CSC2} \]

\[ {\cal L}_{\mathrm{Dirac}} = \bar{q}_{i \alpha} \left( i {\partial} \delta_{i j} \delta_{\alpha \beta} - m_{i j} \delta_{\alpha \beta} - \mu_{i j,~\alpha \beta} \gamma^0 \right) q_{j \beta} \]

\[ {\cal L}_{\mathrm{4-fermion}} = G_S \sum_{a=0}^8 \left[ \left( \bar{q} \lambda^a_f q \right)^2 + \left( \bar{q} i \gamma_5 \lambda^a_f q \right)^2 \right] \]

\[ {\cal L}_{\mathrm{6-fermion}} = - G_{D} \left[ {\mathrm det}_{i j} \, \bar{q}_{i \alpha} \left( 1 + i \gamma_5 \right) q_{j \beta} + {\mathrm det}_{i j} \, \bar{q}_{i \alpha} \left( 1 - i \gamma_5 \right) q_{j \beta} \right] \delta_{\alpha \beta} \]

\[ {\cal L}_{CSC1} = G_{DIQ} \sum_k \sum_{\gamma} \left[ \left(\bar{q}_{i \alpha} \epsilon_{i j k} \epsilon_{\alpha \beta \gamma} q^C_{j \beta}\right) \left(\bar{q}_{i^{\prime} \alpha^{\prime}}^C \epsilon_{i^{\prime} j^{\prime} k} \epsilon_{\alpha^{\prime} \beta^{\prime} \gamma} q_{j^{\prime} \beta^{\prime}}\right)\right] \]

\[ {\cal L}_{CSC2} = G_{DIQ} \sum_k \sum_{\gamma} \left[ \left(\bar{q}_{i \alpha} i \gamma_5 \epsilon_{i j k} \epsilon_{\alpha \beta \gamma} q^C_{j \beta}\right) \left(\bar{q}_{i^{\prime} \alpha^{\prime}}^C i \gamma_5 \epsilon_{i^{\prime} j^{\prime} k} \epsilon_{\alpha^{\prime} \beta^{\prime} \gamma} q_{j^{\prime} \beta^{\prime}}\right) \right] \,, \]

where $ \mu $ is the quark number chemical potential. couplings $ G_S $, $ G_D $, and $ G_{DIQ} $ ultra-violet three-momentum cutoff, $ \Lambda $

The thermodynamic potential is

\[ \Omega(\mu_i,\left<\bar{q} q\right>_i,\left< q q\right>_i,T) = \Omega_{\mathrm{vac}}+\Omega_{\mathrm{stat}} + \Omega_0 \]

where $ i $ runs over all nine (three colors times three flavors) quarks. We assume that the condensates are independent of color and that the quark chemical potentials are of the form $ \mu_Q=\mu_{\mathrm{Flavor(Q)}}+\mu_{\mathrm{Color(Q)}} $ with

\[ \mu_{\mathrm{red}} = \mu_3 + \mu_8/\sqrt{3} \quad \mu_{\mathrm{green}} = -\mu_3 + \mu_8/\sqrt{3} \quad \mu_{\mathrm{blue}} = -2 \mu_8 /\sqrt{3} \]

With these assumptions, the thermodynamic potential as given by the function thd_potential(), is a function of 12 variables

\[ \Omega(\mu_u, \mu_d, \mu_s, \mu_3, \mu_8, \left<\bar{u} u\right>, \left<\bar{d} d\right>, \left<\bar{s} s\right>, \left< u d\right>, \left< u s\right>, \left< d s\right>, T) \]

The individual terms are

\[ \Omega_{\mathrm{stat}} = - \frac{1}{2} \int \frac{d^3 p}{\left(2 \pi\right)^3} \, \sum_{i=1}^{72} \left[ \frac{\lambda_i}{2} + T \ln{\left(1 + e^{-\lambda_i/T} \right)} \right] \]

\[ \Omega_{\mathrm{vac}} = - 2 G_S \sum_{i=u,d,s} \langle {\bar q_i} q_i \rangle^2 +4 G_D \left<{\bar u} u\right> \left<{\bar d} d \right> \left<{\bar s} s\right> + \sum_k \sum_{\gamma} \frac{\left|\Delta^{k \gamma}\right|^2}{4 G_{DIQ}} \]

where $ \lambda_i $ are the eigenvalues of the (72 by 72) matrix (calculated by the function eigenvalues())

\[ D = \left[ \begin{array}{cc} - \gamma^0 \vec{\gamma} \cdot \vec{p} - M_{i} \gamma^0 + \mu_{i \alpha} & \Delta i \gamma^0 \gamma_5 C \\ i \Delta \gamma^0 C \gamma_5 & - \gamma^0 \vec{\gamma}^T \cdot \vec{p} + M_{i} \gamma^0 - \mu_{i \alpha}\end{array} \right] \]

and $ C $ is the charge conjugation matrix (in the Dirac representation).

The values of the various condensates are usually determined by the condition

\[ \frac{\partial \Omega}{\left<\bar{q} q\right>_i} = 0 \quad \frac{\partial \Omega}{\left<q q\right>_i} = 0 \]

Note that setting fixed_mass to true and setting all of the gaps to zero when gap_limit is less than zero will reproduce an analog of the bag model with a momentum cutoff.

The variable eos_quark_njl::fromqq is automatically set to true in the constructor, as computations with fromqq=false are not implemented.

Idea for Future:

This class internally mixes ubvector, ubmatrix, gsl_vector and gsl_matrix objects in a confusing and non-optimal way. Fix this.

Allow user to change derivative object? This isn't possible right now because the stepsize parameter of the derivative object is used.


References:

Created for Steiner02.

Definition at line 208 of file eos_quark_cfl.h.

Member Function Documentation

◆ calc_eq_temp_p()

virtual int o2scl::eos_quark_cfl::calc_eq_temp_p ( quark u,
quark d,
quark s,
double &  qq1,
double &  qq2,
double &  qq3,
double &  gap1,
double &  gap2,
double &  gap3,
double  mu3,
double  mu8,
double &  n3,
double &  n8,
thermo qb,
double  temper 
)
virtual

Calculate the EOS from the quark condensates in u.qq, d.qq and s.qq. Return the mass gap equations in qq1, qq2, qq3, and the normal gap equations in gap1, gap2, and gap3.

Using fromqq=false as in eos_quark_njl and eos_quark_njl does not work here and will return an error. Also, the quarks must be set through eos_quark::quark_set() before use.

If all of the gaps are less than gap_limit, then the eos_quark_njl::calc_temp_p() is used, and gap1, gap2, and gap3 are set to equal u.del, d.del, and s.del, respectively.

Todo:
It surprises me that n3 is not -res[11]. Is there a sign error in the color densities?

Reimplemented in o2scl::eos_quark_cfl6.

◆ eigenvalues()

virtual int o2scl::eos_quark_cfl::eigenvalues ( double  lmom,
double  mu3,
double  mu8,
double  egv[36],
double  dedmuu[36],
double  dedmud[36],
double  dedmus[36],
double  dedmu[36],
double  dedmd[36],
double  dedms[36],
double  dedu[36],
double  dedd[36],
double  deds[36],
double  dedmu3[36],
double  dedmu8[36] 
)
virtual

Given the momentum mom, and the chemical potentials associated with the third and eighth gluons (mu3 and mu8), the energy eigenvalues are computed in egv[0] ... egv[35].

◆ gapped_eigenvalues()

int o2scl::eos_quark_cfl::gapped_eigenvalues ( double  m1,
double  m2,
double  lmom,
double  mu1,
double  mu2,
double  tdelta,
double  lam[4],
double  dldmu1[4],
double  dldmu2[4],
double  dldm1[4],
double  dldm2[4],
double  dldg[4] 
)
protected

This function uses the quarks q1 and q2 to construct the eigenvalues of the inverse propagator, properly handling the either zero or finite quark mass and either zero or finite quark gaps. In the case of finite quark mass and finite quark gaps, the quartic solver is used.

The chemical potentials are separated so we can add the color chemical potentials to the quark chemical potentials if necessary.

This function is used by eigenvalues(). It does not work for the "ur-dg-sb" set of quarks which are paired in a non-trivial way.

Todo:
In the code, the equal mass case seems to be commented out. Why?

◆ integrands()

virtual int o2scl::eos_quark_cfl::integrands ( double  p,
double  res[] 
)
protectedvirtual
  • res[0] is the thermodynamic potential, $ \Omega $
  • res[1] is $ d -\Omega / d T $
  • res[2] is $ d \Omega / d \mu_u $
  • res[3] is $ d \Omega / d \mu_d $
  • res[4] is $ d \Omega / d \mu_s $
  • res[5] is $ d \Omega / d m_u $
  • res[6] is $ d \Omega / d m_d $
  • res[7] is $ d \Omega / d m_s $
  • res[8] is $ d \Omega / d \Delta_{ds} $
  • res[9] is $ d \Omega / d \Delta_{us} $
  • res[10] is $ d \Omega / d \Delta_{ud} $
  • res[11] is $ d \Omega / d \mu_3 $
  • res[12] is $ d \Omega / d \mu_8 $

Reimplemented in o2scl::eos_quark_cfl6.

◆ set_parameters()

virtual int o2scl::eos_quark_cfl::set_parameters ( double  lambda = 0.0,
double  fourferm = 0.0,
double  sixferm = 0.0,
double  fourgap = 0.0 
)
virtual

This function allows the user to specify the momentum cutoff, lambda, the four-fermion coupling fourferm, the six-fermion coupling from the 't Hooft interaction sixferm, and the color-superconducting coupling, fourgap. 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 $).

If the four-fermion coupling that produces a gap is not specified, it is automatically set to 3/4 G, which is the value obtained from the Fierz transformation.

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

Member Data Documentation

◆ def_quartic

quartic_real_coeff_cern o2scl::eos_quark_cfl::def_quartic

Slightly better accuracy (with slower execution times) can be achieved using poly_real_coeff_gsl which polishes the roots of the quartics. For example

eos_quark_cfl cfl;
poly_real_coeff_gsl gp;
cfl.set_quartic(gp);

Definition at line 314 of file eos_quark_cfl.h.

◆ gap_limit

double o2scl::eos_quark_cfl::gap_limit

If any of the gaps are below this value, then it is assumed that they are zero and the equation of state is simplified accordingly. If all of the gaps are less than gap_limit, then the results from eos_quark_njl are used in calc_eq_temp_p(), calc_temp_p() and thd_potential().

Definition at line 335 of file eos_quark_cfl.h.

◆ GD

double o2scl::eos_quark_cfl::GD

The default value is the one derived from a Fierz transformation. (Buballa04)

Definition at line 358 of file eos_quark_cfl.h.

◆ inte_epsabs

double o2scl::eos_quark_cfl::inte_epsabs

This is analogous to gsl_inte::epsabs

Definition at line 365 of file eos_quark_cfl.h.

◆ inte_epsrel

double o2scl::eos_quark_cfl::inte_epsrel

This is analogous to gsl_inte::epsrel

Definition at line 372 of file eos_quark_cfl.h.

◆ inte_npoints

size_t o2scl::eos_quark_cfl::inte_npoints

This returns 21, 43, or 87 depending on the number of function evaluations needed to obtain the desired precision. If it the routine failes to obtain the desired precision, then this variable is set to 88.

Definition at line 382 of file eos_quark_cfl.h.

◆ zerot

bool o2scl::eos_quark_cfl::zerot

This implements some simplifications in the momentum integration that are not possible at finite temperature.

Definition at line 343 of file eos_quark_cfl.h.


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

Documentation generated with Doxygen. Provided under the GNU Free Documentation License (see License Information).