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 184 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 double thd_potential (quark &u, quark &d, quark &s, double mu3, double mu8, const double ltemper)
 Direct calculation of the thermodynamic potential.
virtual int eigenvalues (double mom, ovector_view &egv, double mu3, double mu8)
 Calculate the energy eigenvalues as a function of the momentum.
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

virtual double tpot (double var, void *&pa)
 The function used to take derivatives of the thermodynamic potential (used by calc_eq_temp_p() ).

Protected Attributes

gsl_complex zero
 To clear all of the matrix entries.
omatrix_cx m6
 Storage for the inverse propagator.
ovector eval6
 Storage for the eigenvalues.
gsl_eigen_herm_workspace * w6
 GSL workspace for the eigenvalue computation.

Static Protected Attributes

static const int 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. The quarks must be set through quark_eos::quark_set() before use.

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 eigenvalues ( double  mom,
ovector_view egv,
double  mu3,
double  mu8 
) [virtual]

Calculate the energy eigenvalues as a function of the momentum.

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]. No space is allocated for the array by the function.

Todo:
This function may make some inappropriate assumptions on the vector egv.

Reimplemented from cfl_njl_eos.


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