cfl6_eos Class Reference

#include <cfl6_eos.h>

Inheritance diagram for cfl6_eos:

cfl_njl_eos nambujl_eos quark_eos eos

Detailed Description

CFL NJL EOS with a color-superconducting 't Hooft interaction.

Beginning with the Lagrangian:

\[ {\cal L} = {\cal L}_{Dirac} + {\cal L}_{NJL} + {\cal L}_{'t Hooft} + {\cal L}_{SC} + {\cal L}_{SC6} \]

\[ {\cal L}_{Dirac} = {\bar q} \left( i \partial -m - \mu \gamma^0 \right) q \]

\[ {\cal L}_{NJL} = G_S \sum_{a=0}^8 \left[ \left( {\bar q} \lambda^a q \right)^2 - \left( {\bar q} \lambda^a \gamma^5 q \right)^2 \right] \]

\[ {\cal L}_{'t Hooft} = G_D \left[ \mathrm{det}_f {\bar q} \left(1-\gamma^5 \right) q +\mathrm{det}_f {\bar q} \left(1+\gamma^5 \right) q \right] \]

\[ {\cal L}_{SC} = G_{DIQ} \left( {\bar q}_{i \alpha} i \gamma^5 \varepsilon^{i j k} \varepsilon^{\alpha \beta \gamma} q^C_{j \beta} \right) \left( {\bar q}_{\ell \delta} i \gamma^5 \epsilon^{\ell m k} \epsilon^{\delta \varepsilon \gamma} q^C_{m \varepsilon} \right) \]

\[ {\cal L}_{SC6} = K_D \left( {\bar q}_{i \alpha} i \gamma^5 \varepsilon^{i j k} \varepsilon^{\alpha \beta \gamma} q^C_{j \beta} \right) \left( {\bar q}_{\ell \delta} i \gamma^5 \epsilon^{\ell m n} \epsilon^{\delta \varepsilon \eta} q^C_{m \varepsilon} \right) \left( {\bar q}_{k \gamma} q_{n \eta} \right) \]

We can simplify the relevant terms in ${\cal L}_{NJL}$:

\[ {\cal L}_{NJL} = G_S \left[ \left({\bar u} u\right)^2+ \left({\bar d} d\right)^2+ \left({\bar s} s\right)^2 \right] \]

and in ${\cal L}_{'t Hooft}$:

\[ {\cal L}_{NJL} = G_D \left( {\bar u} u {\bar d} d {\bar s} s \right) \]

Using the definition:

\[ \Delta^{k \gamma} = \left< {\bar q} i \gamma^5 \epsilon \epsilon q^C_{} \right> \]

and the ansatzes:

\[ ({\bar q}_1 q_2) ({\bar q}_3 q_4) \rightarrow {\bar q}_1 q_2 \left< {\bar q}_3 q_4 \right> +{\bar q}_3 q_4 \left< {\bar q}_1 q_2 \right> -\left< {\bar q}_1 q_2 \right> \left< {\bar q}_3 q_4 \right> \]

\[ ({\bar q}_1 q_2) ({\bar q}_3 q_4) ({\bar q}_5 q_6) \rightarrow {\bar q}_1 q_2 \left< {\bar q}_3 q_4 \right> \left< {\bar q}_5 q_6 \right> +{\bar q}_3 q_4 \left< {\bar q}_1 q_2 \right> \left< {\bar q}_5 q_6 \right> +{\bar q}_5 q_6 \left< {\bar q}_1 q_2 \right> \left< {\bar q}_3 q_4 \right> -2\left< {\bar q}_1 q_2 \right> \left< {\bar q}_3 q_4 \right> \left< {\bar q}_5 q_6 \right> \]

for the mean field approximation, we can rewrite the Lagrangian

\[ {\cal L}_{NJL} = 2 G_S \left[ \left( {\bar u} u \right) \left< {\bar u} u \right> +\left( {\bar d} d \right) \left< {\bar d} d \right> +\left( {\bar s} s \right) \left< {\bar s} s \right> - \left< {\bar u} u \right>^2 - \left< {\bar d} d \right>^2 - \left< {\bar s} s \right>^2 \right] \]

\[ {\cal L}_{'t Hooft} = - 2 G_D \left[ \left( {\bar u} u \right) \left< {\bar u} u \right> \left< {\bar s} s \right> + \left( {\bar d} d \right) \left< {\bar u} u \right> \left< {\bar s} s \right> + \left( {\bar s} s \right) \left< {\bar u} u \right> \left< {\bar d} d \right> - 2 \left< {\bar u} u \right>\left< {\bar d} d \right> \left< {\bar s} s \right> \right] \]

\[ {\cal L}_{SC} = G_{DIQ} \left[ \Delta^{k \gamma} \left( {\bar q}_{\ell \delta} i \gamma^5 \epsilon^{\ell m k} \epsilon^{\delta \varepsilon \gamma} q^C_{m \varepsilon} \right) + \left( {\bar q}_{i \alpha} i \gamma^5 \varepsilon^{i j k} \varepsilon^{\alpha \beta \gamma} q^C_{j \beta} \right) \Delta^{k \gamma \dagger} - \Delta^{k \gamma} \Delta^{k \gamma \dagger} \right] \]

\[ {\cal L}_{SC6} = K_D \left[ \left( {\bar q}_{m \varepsilon} q_{n \eta} \right) \Delta^{k \gamma} \Delta^{m \varepsilon \dagger} + \left( {\bar q}_{i \alpha} i \gamma^5 \varepsilon^{i j k} \varepsilon^{\alpha \beta \gamma} q^C_{j \beta} \right) \Delta^{m \varepsilon \dagger} \left< {\bar q}_{m \varepsilon} q_{n \eta} \right> \right] \]

\[ + K_D \left[\Delta^{k \gamma} \left( {\bar q}_{\ell \delta} i \gamma^5 \epsilon^{\ell m n} \epsilon^{\delta \varepsilon \eta} q^C_{m \varepsilon} \right) \left< {\bar q}_{m \varepsilon} q_{n \eta} \right> -2 \Delta^{k \gamma} \Delta^{m \varepsilon \dagger} \left< {\bar q}_{m \varepsilon} q_{n \eta} \right> \right] \]

If we make the definition $ {\tilde \Delta} = 2 G_{DIQ} \Delta $

Definition at line 183 of file cfl6_eos.h.


Public Member Functions

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, const double temper)
 Calculate the EOS.
virtual int integrands (double p, double res[])
 The momentum integrands.
virtual int test_derivatives (double lmom, double mu3, double mu8, test_mgr &t)
 Check the derivatives specified by eigenvalues().
virtual int eigenvalues6 (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 and their derivatives.
virtual int make_matrices (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])
 Construct the matrices, but don't solve the eigenvalue problem.
virtual const char * type ()
 Return string denoting type ("cfl6_eos").

Data Fields

double KD
 The color superconducting 't Hooft coupling (default 0).
double kdlimit
 The absolute value below which the CSC 't Hooft coupling is ignored(default $ 10^{-6} $).

Protected Member Functions

int set_masses ()
 Set the quark effective masses from the gaps and the condensates.

Protected Attributes

omatrix_cx iprop6
 Storage for the inverse propagator.
omatrix_cx eivec6
 The eigenvectors.
omatrix_cx dipdgapu
 The derivative wrt the ds gap.
omatrix_cx dipdgapd
 The derivative wrt the us gap.
omatrix_cx dipdgaps
 The derivative wrt the ud gap.
omatrix_cx dipdqqu
 The derivative wrt the up quark condensate.
omatrix_cx dipdqqd
 The derivative wrt the down quark condensate.
omatrix_cx dipdqqs
 The derivative wrt the strange quark condensate.
ovector eval6
 Storage for the eigenvalues.
gsl_eigen_hermv_workspace * w6
 GSL workspace for the eigenvalue computation.

Static Protected Attributes

static const int mat_size = 36
 The size of the matrix to be diagonalized.

Private Member Functions

 cfl6_eos (const cfl6_eos &)
cfl6_eosoperator= (const cfl6_eos &)

Member Function Documentation

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,
const double  temper 
) [virtual]

Calculate the EOS.

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

Using fromqq=true as in nambujl_eos and nambujl_temp_eos does not work here and will return an error.

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

Reimplemented from cfl_njl_eos.

virtual int eigenvalues6 ( 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]

Calculate the energy eigenvalues and their derivatives.

Given the momentum mom, and the chemical potentials associated with the third and eighth gluons (mu3 and mu8), this computes the eigenvalues of the inverse propagator and the assocated derivatives.

Note that this is not the same as cfl_njl_eos::eigenvalues() which returns dedmu rather dedqqu.

virtual int make_matrices ( 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]

Construct the matrices, but don't solve the eigenvalue problem.

This is used by check_derivatives() to make sure that the derivative entries are right.


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

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