00001 /* 00002 ------------------------------------------------------------------- 00003 00004 Copyright (C) 2006, 2007, 2008, Andrew W. Steiner 00005 00006 This file is part of O2scl. 00007 00008 O2scl is free software; you can redistribute it and/or modify 00009 it under the terms of the GNU General Public License as published by 00010 the Free Software Foundation; either version 3 of the License, or 00011 (at your option) any later version. 00012 00013 O2scl is distributed in the hope that it will be useful, 00014 but WITHOUT ANY WARRANTY; without even the implied warranty of 00015 MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the 00016 GNU General Public License for more details. 00017 00018 You should have received a copy of the GNU General Public License 00019 along with O2scl. If not, see <http://www.gnu.org/licenses/>. 00020 00021 ------------------------------------------------------------------- 00022 */ 00023 #ifndef O2SCL_CFL6_EOS_OLD_H 00024 #define O2SCL_CFL6_EOS_OLD_H 00025 00026 #include <iostream> 00027 #include <o2scl/columnify.h> 00028 #include <o2scl/omatrix_cx_tlate.h> 00029 #include <o2scl/cfl_njl_eos_old.h> 00030 00031 #ifndef DOXYGENP 00032 namespace o2scl { 00033 #endif 00034 00035 /** 00036 \brief CFL NJL EOS with a color-superconducting 't Hooft interaction 00037 00038 Beginning with the Lagrangian: 00039 \f[ 00040 {\cal L} = {\cal L}_{Dirac} + {\cal L}_{NJL} + 00041 {\cal L}_{'t Hooft} + {\cal L}_{SC} + {\cal L}_{SC6} 00042 \f] 00043 \f[ 00044 {\cal L}_{Dirac} = {\bar q} \left( i \partial -m - 00045 \mu \gamma^0 \right) q 00046 \f] 00047 \f[ 00048 {\cal L}_{NJL} = G_S \sum_{a=0}^8 00049 \left[ \left( {\bar q} \lambda^a q \right)^2 00050 - \left( {\bar q} \lambda^a \gamma^5 q \right)^2 \right] 00051 \f] 00052 \f[ 00053 {\cal L}_{'t Hooft} = G_D \left[ 00054 \mathrm{det}_f {\bar q} \left(1-\gamma^5 \right) q 00055 +\mathrm{det}_f {\bar q} \left(1+\gamma^5 \right) q 00056 \right] 00057 \f] 00058 \f[ 00059 {\cal L}_{SC} = G_{DIQ} 00060 \left( {\bar q}_{i \alpha} i \gamma^5 00061 \varepsilon^{i j k} \varepsilon^{\alpha \beta \gamma} 00062 q^C_{j \beta} \right) 00063 \left( {\bar q}_{\ell \delta} i \gamma^5 00064 \epsilon^{\ell m k} 00065 \epsilon^{\delta \varepsilon \gamma} 00066 q^C_{m \varepsilon} \right) 00067 \f] 00068 \f[ 00069 {\cal L}_{SC6} = K_D 00070 \left( {\bar q}_{i \alpha} i \gamma^5 00071 \varepsilon^{i j k} \varepsilon^{\alpha \beta \gamma} 00072 q^C_{j \beta} \right) 00073 \left( {\bar q}_{\ell \delta} i \gamma^5 00074 \epsilon^{\ell m n} 00075 \epsilon^{\delta \varepsilon \eta} 00076 q^C_{m \varepsilon} \right) 00077 \left( {\bar q}_{k \gamma} q_{n \eta} \right) 00078 \f] 00079 00080 We can simplify the relevant terms in \f${\cal L}_{NJL}\f$: 00081 \f[ 00082 {\cal L}_{NJL} = G_S \left[ 00083 \left({\bar u} u\right)^2+ 00084 \left({\bar d} d\right)^2+ 00085 \left({\bar s} s\right)^2 00086 \right] 00087 \f] 00088 and in \f${\cal L}_{'t Hooft}\f$: 00089 \f[ 00090 {\cal L}_{NJL} = G_D \left( 00091 {\bar u} u {\bar d} d {\bar s} s 00092 \right) 00093 \f] 00094 00095 Using the definition: 00096 \f[ 00097 \Delta^{k \gamma} = \left< {\bar q} i \gamma^5 00098 \epsilon \epsilon q^C_{} \right> 00099 \f] 00100 and the ansatzes: 00101 \f[ 00102 ({\bar q}_1 q_2) ({\bar q}_3 q_4) \rightarrow 00103 {\bar q}_1 q_2 \left< {\bar q}_3 q_4 \right> 00104 +{\bar q}_3 q_4 \left< {\bar q}_1 q_2 \right> 00105 -\left< {\bar q}_1 q_2 \right> \left< {\bar q}_3 q_4 \right> 00106 \f] 00107 \f[ 00108 ({\bar q}_1 q_2) ({\bar q}_3 q_4) ({\bar q}_5 q_6) \rightarrow 00109 {\bar q}_1 q_2 \left< {\bar q}_3 q_4 \right> 00110 \left< {\bar q}_5 q_6 \right> 00111 +{\bar q}_3 q_4 \left< {\bar q}_1 q_2 \right> 00112 \left< {\bar q}_5 q_6 \right> 00113 +{\bar q}_5 q_6 \left< {\bar q}_1 q_2 \right> 00114 \left< {\bar q}_3 q_4 \right> 00115 -2\left< {\bar q}_1 q_2 \right> \left< {\bar q}_3 q_4 \right> 00116 \left< {\bar q}_5 q_6 \right> 00117 \f] 00118 for the mean field approximation, we can rewrite the Lagrangian 00119 \f[ 00120 {\cal L}_{NJL} = 2 G_S \left[ 00121 \left( {\bar u} u \right) \left< {\bar u} u \right> 00122 +\left( {\bar d} d \right) \left< {\bar d} d \right> 00123 +\left( {\bar s} s \right) \left< {\bar s} s \right> 00124 - \left< {\bar u} u \right>^2 00125 - \left< {\bar d} d \right>^2 00126 - \left< {\bar s} s \right>^2 00127 \right] 00128 \f] 00129 \f[ 00130 {\cal L}_{'t Hooft} = - 2 G_D \left[ 00131 \left( {\bar u} u \right) \left< {\bar u} u \right> 00132 \left< {\bar s} s \right> 00133 + \left( {\bar d} d \right) \left< {\bar u} u \right> 00134 \left< {\bar s} s \right> 00135 + \left( {\bar s} s \right) \left< {\bar u} u \right> 00136 \left< {\bar d} d \right> 00137 - 2 \left< {\bar u} u \right>\left< {\bar d} d \right> 00138 \left< {\bar s} s \right> 00139 \right] 00140 \f] 00141 \f[ 00142 {\cal L}_{SC} = G_{DIQ} \left[ 00143 \Delta^{k \gamma} 00144 \left( {\bar q}_{\ell \delta} i \gamma^5 00145 \epsilon^{\ell m k} 00146 \epsilon^{\delta \varepsilon \gamma} 00147 q^C_{m \varepsilon} \right) 00148 + \left( {\bar q}_{i \alpha} i \gamma^5 00149 \varepsilon^{i j k} \varepsilon^{\alpha \beta \gamma} 00150 q^C_{j \beta} \right) 00151 \Delta^{k \gamma \dagger} 00152 - \Delta^{k \gamma} 00153 \Delta^{k \gamma \dagger} 00154 \right] 00155 \f] 00156 \f[ 00157 {\cal L}_{SC6} = K_D \left[ 00158 \left( {\bar q}_{m \varepsilon} q_{n \eta} \right) 00159 \Delta^{k \gamma} \Delta^{m \varepsilon \dagger} 00160 + \left( {\bar q}_{i \alpha} i \gamma^5 00161 \varepsilon^{i j k} \varepsilon^{\alpha \beta \gamma} 00162 q^C_{j \beta} \right) 00163 \Delta^{m \varepsilon \dagger} 00164 \left< {\bar q}_{m \varepsilon} q_{n \eta} \right> 00165 \right] 00166 \f] 00167 \f[ 00168 + K_D \left[\Delta^{k \gamma} 00169 \left( {\bar q}_{\ell \delta} i \gamma^5 00170 \epsilon^{\ell m n} 00171 \epsilon^{\delta \varepsilon \eta} 00172 q^C_{m \varepsilon} \right) 00173 \left< {\bar q}_{m \varepsilon} q_{n \eta} \right> 00174 -2 00175 \Delta^{k \gamma} \Delta^{m \varepsilon \dagger} 00176 \left< {\bar q}_{m \varepsilon} q_{n \eta} \right> 00177 \right] 00178 \f] 00179 00180 If we make the definition \f$ {\tilde \Delta} = 00181 2 G_{DIQ} \Delta \f$ 00182 00183 */ 00184 class cfl6_eos_old : public cfl_njl_eos_old { 00185 public: 00186 00187 cfl6_eos_old(); 00188 00189 virtual ~cfl6_eos_old(); 00190 00191 /** 00192 \brief Calculate the EOS 00193 00194 Calculate the EOS from the quark condensates. Return the mass 00195 gap equations in \c qq1, \c qq2, \c qq3, and the normal gap 00196 equations in \c gap1, \c gap2, and \c gap3. 00197 00198 Using \c fromqq=true as in nambujl_eos and nambujl_temp_eos does 00199 not work here and will return an error. The quarks must be set 00200 through quark_eos::quark_set() before use. 00201 00202 If all of the gaps are less than gap_limit, then the 00203 nambujl_temp_eos::calc_temp_p() is used, and \c gap1, \c gap2, 00204 and \c gap3 are set to equal \c u.del, \c d.del, and \c s.del, 00205 respectively. 00206 00207 */ 00208 virtual int calc_eq_temp_p(quark &u, quark &d, quark &s, 00209 double &qq1, double &qq2, double &qq3, 00210 double &gap1, double &gap2, double &gap3, 00211 double mu3, double mu8, 00212 double &n3, double &n8, thermo &qb, 00213 const double temper); 00214 00215 /** \brief Direct calculation of the thermodynamic potential 00216 */ 00217 virtual double thd_potential(quark &u, quark &d, quark &s, 00218 double mu3, double mu8, 00219 const double ltemper); 00220 00221 00222 /** 00223 \brief Calculate the energy eigenvalues as a function of the momentum 00224 00225 Given the momentum 'mom', and the chemical potentials 00226 associated with the third and eighth gluons ('mu3' and 'mu8'), 00227 the energy eigenvalues are computed in egv[0] ... egv[35]. No 00228 space is allocated for the array by the function. 00229 00230 \todo This function may make some inappropriate assumptions 00231 on the vector egv. 00232 */ 00233 virtual int eigenvalues(double mom, ovector_view &egv, double mu3, 00234 double mu8); 00235 00236 /// The color superconducting 't Hooft coupling (default 0) 00237 double KD; 00238 00239 /// Return string denoting type ("cfl6_eos_old") 00240 virtual const char *type() { return "cfl6_eos_old"; }; 00241 00242 /** \brief The absolute value below which the CSC 't Hooft coupling 00243 is ignored(default \f$ 10^{-6} \f$) 00244 */ 00245 double kdlimit; 00246 00247 protected: 00248 00249 #ifndef DOXYGEN_INTERNAL 00250 00251 /// To clear all of the matrix entries 00252 gsl_complex zero; 00253 00254 /// The size of the matrix to be diagonalized 00255 static const int size=36; 00256 00257 /// Storage for the inverse propagator 00258 omatrix_cx m6; 00259 00260 /// Storage for the eigenvalues 00261 ovector eval6; 00262 00263 /// GSL workspace for the eigenvalue computation 00264 gsl_eigen_herm_workspace *w6; 00265 00266 /** \brief The function used to take derivatives of the thermodynamic 00267 potential (used by calc_eq_temp_p() ) 00268 */ 00269 virtual double tpot(double var, void *&pa); 00270 00271 private: 00272 00273 cfl6_eos_old(const cfl6_eos_old &); 00274 cfl6_eos_old& operator=(const cfl6_eos_old&); 00275 00276 #endif 00277 00278 }; 00279 00280 #ifndef DOXYGENP 00281 } 00282 #endif 00283 00284 #endif
Documentation generated with Doxygen and provided under the GNU Free Documentation License. See License Information for details.
Project hosting provided by
,
O2scl Sourceforge Project Page