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 CFL6_EOS_H 00024 #define CFL6_EOS_H 00025 00026 #include <iostream> 00027 #include <o2scl/test_mgr.h> 00028 #include <o2scl/cfl_njl_eos.h> 00029 00030 #ifndef DOXYGENP 00031 namespace o2scl { 00032 #endif 00033 00034 /** \brief CFL NJL EOS with a color-superconducting 't Hooft interaction 00035 \nothing 00036 00037 Beginning with the Lagrangian: 00038 \f[ 00039 {\cal L} = {\cal L}_{Dirac} + {\cal L}_{NJL} + 00040 {\cal L}_{'t Hooft} + {\cal L}_{SC} + {\cal L}_{SC6} 00041 \f] 00042 \f[ 00043 {\cal L}_{Dirac} = {\bar q} \left( i \partial -m - 00044 \mu \gamma^0 \right) q 00045 \f] 00046 \f[ 00047 {\cal L}_{NJL} = G_S \sum_{a=0}^8 00048 \left[ \left( {\bar q} \lambda^a q \right)^2 00049 - \left( {\bar q} \lambda^a \gamma^5 q \right)^2 \right] 00050 \f] 00051 \f[ 00052 {\cal L}_{'t Hooft} = G_D \left[ 00053 \mathrm{det}_f {\bar q} \left(1-\gamma^5 \right) q 00054 +\mathrm{det}_f {\bar q} \left(1+\gamma^5 \right) q 00055 \right] 00056 \f] 00057 \f[ 00058 {\cal L}_{SC} = G_{DIQ} 00059 \left( {\bar q}_{i \alpha} i \gamma^5 00060 \varepsilon^{i j k} \varepsilon^{\alpha \beta \gamma} 00061 q^C_{j \beta} \right) 00062 \left( {\bar q}_{\ell \delta} i \gamma^5 00063 \epsilon^{\ell m k} 00064 \epsilon^{\delta \varepsilon \gamma} 00065 q^C_{m \varepsilon} \right) 00066 \f] 00067 \f[ 00068 {\cal L}_{SC6} = K_D 00069 \left( {\bar q}_{i \alpha} i \gamma^5 00070 \varepsilon^{i j k} \varepsilon^{\alpha \beta \gamma} 00071 q^C_{j \beta} \right) 00072 \left( {\bar q}_{\ell \delta} i \gamma^5 00073 \epsilon^{\ell m n} 00074 \epsilon^{\delta \varepsilon \eta} 00075 q^C_{m \varepsilon} \right) 00076 \left( {\bar q}_{k \gamma} q_{n \eta} \right) 00077 \f] 00078 00079 We can simplify the relevant terms in \f${\cal L}_{NJL}\f$: 00080 \f[ 00081 {\cal L}_{NJL} = G_S \left[ 00082 \left({\bar u} u\right)^2+ 00083 \left({\bar d} d\right)^2+ 00084 \left({\bar s} s\right)^2 00085 \right] 00086 \f] 00087 and in \f${\cal L}_{'t Hooft}\f$: 00088 \f[ 00089 {\cal L}_{NJL} = G_D \left( 00090 {\bar u} u {\bar d} d {\bar s} s 00091 \right) 00092 \f] 00093 00094 Using the definition: 00095 \f[ 00096 \Delta^{k \gamma} = \left< {\bar q} i \gamma^5 00097 \epsilon \epsilon q^C_{} \right> 00098 \f] 00099 and the ansatzes: 00100 \f[ 00101 ({\bar q}_1 q_2) ({\bar q}_3 q_4) \rightarrow 00102 {\bar q}_1 q_2 \left< {\bar q}_3 q_4 \right> 00103 +{\bar q}_3 q_4 \left< {\bar q}_1 q_2 \right> 00104 -\left< {\bar q}_1 q_2 \right> \left< {\bar q}_3 q_4 \right> 00105 \f] 00106 \f[ 00107 ({\bar q}_1 q_2) ({\bar q}_3 q_4) ({\bar q}_5 q_6) \rightarrow 00108 {\bar q}_1 q_2 \left< {\bar q}_3 q_4 \right> 00109 \left< {\bar q}_5 q_6 \right> 00110 +{\bar q}_3 q_4 \left< {\bar q}_1 q_2 \right> 00111 \left< {\bar q}_5 q_6 \right> 00112 +{\bar q}_5 q_6 \left< {\bar q}_1 q_2 \right> 00113 \left< {\bar q}_3 q_4 \right> 00114 -2\left< {\bar q}_1 q_2 \right> \left< {\bar q}_3 q_4 \right> 00115 \left< {\bar q}_5 q_6 \right> 00116 \f] 00117 for the mean field approximation, we can rewrite the Lagrangian 00118 \f[ 00119 {\cal L}_{NJL} = 2 G_S \left[ 00120 \left( {\bar u} u \right) \left< {\bar u} u \right> 00121 +\left( {\bar d} d \right) \left< {\bar d} d \right> 00122 +\left( {\bar s} s \right) \left< {\bar s} s \right> 00123 - \left< {\bar u} u \right>^2 00124 - \left< {\bar d} d \right>^2 00125 - \left< {\bar s} s \right>^2 00126 \right] 00127 \f] 00128 \f[ 00129 {\cal L}_{'t Hooft} = - 2 G_D \left[ 00130 \left( {\bar u} u \right) \left< {\bar u} u \right> 00131 \left< {\bar s} s \right> 00132 + \left( {\bar d} d \right) \left< {\bar u} u \right> 00133 \left< {\bar s} s \right> 00134 + \left( {\bar s} s \right) \left< {\bar u} u \right> 00135 \left< {\bar d} d \right> 00136 - 2 \left< {\bar u} u \right>\left< {\bar d} d \right> 00137 \left< {\bar s} s \right> 00138 \right] 00139 \f] 00140 \f[ 00141 {\cal L}_{SC} = G_{DIQ} \left[ 00142 \Delta^{k \gamma} 00143 \left( {\bar q}_{\ell \delta} i \gamma^5 00144 \epsilon^{\ell m k} 00145 \epsilon^{\delta \varepsilon \gamma} 00146 q^C_{m \varepsilon} \right) 00147 + \left( {\bar q}_{i \alpha} i \gamma^5 00148 \varepsilon^{i j k} \varepsilon^{\alpha \beta \gamma} 00149 q^C_{j \beta} \right) 00150 \Delta^{k \gamma \dagger} 00151 - \Delta^{k \gamma} 00152 \Delta^{k \gamma \dagger} 00153 \right] 00154 \f] 00155 \f[ 00156 {\cal L}_{SC6} = K_D \left[ 00157 \left( {\bar q}_{m \varepsilon} q_{n \eta} \right) 00158 \Delta^{k \gamma} \Delta^{m \varepsilon \dagger} 00159 + \left( {\bar q}_{i \alpha} i \gamma^5 00160 \varepsilon^{i j k} \varepsilon^{\alpha \beta \gamma} 00161 q^C_{j \beta} \right) 00162 \Delta^{m \varepsilon \dagger} 00163 \left< {\bar q}_{m \varepsilon} q_{n \eta} \right> 00164 \right] 00165 \f] 00166 \f[ 00167 + K_D \left[\Delta^{k \gamma} 00168 \left( {\bar q}_{\ell \delta} i \gamma^5 00169 \epsilon^{\ell m n} 00170 \epsilon^{\delta \varepsilon \eta} 00171 q^C_{m \varepsilon} \right) 00172 \left< {\bar q}_{m \varepsilon} q_{n \eta} \right> 00173 -2 00174 \Delta^{k \gamma} \Delta^{m \varepsilon \dagger} 00175 \left< {\bar q}_{m \varepsilon} q_{n \eta} \right> 00176 \right] 00177 \f] 00178 00179 If we make the definition \f$ {\tilde \Delta} = 00180 2 G_{DIQ} \Delta \f$ 00181 00182 */ 00183 class cfl6_eos : public cfl_njl_eos { 00184 public: 00185 00186 cfl6_eos(); 00187 00188 virtual ~cfl6_eos(); 00189 00190 /** \brief Calculate the EOS 00191 \nothing 00192 00193 Calculate the EOS from the quark condensates. Return the mass 00194 gap equations in \c qq1, \c qq2, \c qq3, and the normal gap 00195 equations in \c gap1, \c gap2, and \c gap3. 00196 00197 Using \c fromqq=true as in nambujl_eos and nambujl_temp_eos 00198 does not work here and will return an error. 00199 00200 If all of the gaps are less than gap_limit, then the 00201 nambujl_temp_eos::calc_temp_p() is used, and \c gap1, \c gap2, 00202 and \c gap3 are set to equal \c u.del, \c d.del, and \c s.del, 00203 respectively. 00204 00205 */ 00206 virtual int calc_eq_temp_p(quark &u, quark &d, quark &s, 00207 double &qq1, double &qq2, double &qq3, 00208 double &gap1, double &gap2, double &gap3, 00209 double mu3, double mu8, 00210 double &n3, double &n8, thermo &qb, 00211 const double temper); 00212 00213 /// The momentum integrands 00214 virtual int integrands(double p, double res[]); 00215 00216 /// Check the derivatives specified by eigenvalues() 00217 virtual int test_derivatives(double lmom, double mu3, double mu8, 00218 test_mgr &t); 00219 00220 /** \brief Calculate the energy eigenvalues and their derivatives 00221 \nothing 00222 00223 Given the momentum \c mom, and the chemical potentials 00224 associated with the third and eighth gluons (\c mu3 and \c mu8), 00225 this computes the eigenvalues of the inverse propagator and 00226 the assocated derivatives. 00227 00228 Note that this is not the same as cfl_njl_eos::eigenvalues() 00229 which returns \c dedmu rather \c dedqqu. 00230 */ 00231 virtual int eigenvalues6(double lmom, double mu3, double mu8, 00232 double egv[36], double dedmuu[36], 00233 double dedmud[36], double dedmus[36], 00234 double dedmu[36], double dedmd[36], 00235 double dedms[36], double dedu[36], 00236 double dedd[36], double deds[36], 00237 double dedmu3[36], double dedmu8[36]); 00238 00239 /** \brief Construct the matrices, but don't solve the eigenvalue 00240 problem 00241 \nothing 00242 00243 This is used by check_derivatives() to make sure that the derivative 00244 entries are right. 00245 */ 00246 virtual int make_matrices(double lmom, double mu3, double mu8, 00247 double egv[36], double dedmuu[36], 00248 double dedmud[36], double dedmus[36], 00249 double dedmu[36], double dedmd[36], 00250 double dedms[36], double dedu[36], 00251 double dedd[36], double deds[36], 00252 double dedmu3[36], double dedmu8[36]); 00253 00254 /// The color superconducting 't Hooft coupling (default 0) 00255 double KD; 00256 00257 /// Return string denoting type ("cfl6_eos") 00258 virtual const char *type() { return "cfl6_eos"; }; 00259 00260 /** \brief The absolute value below which the CSC 't Hooft coupling 00261 is ignored(default \f$ 10^{-6} \f$) 00262 */ 00263 double kdlimit; 00264 00265 protected: 00266 00267 #ifndef DOXYGEN_INTERNAL 00268 00269 /// Set the quark effective masses from the gaps and the condensates 00270 int set_masses(); 00271 00272 /// The size of the matrix to be diagonalized 00273 static const int mat_size=36; 00274 00275 /// Storage for the inverse propagator 00276 omatrix_cx iprop6; 00277 00278 /// The eigenvectors 00279 omatrix_cx eivec6; 00280 00281 /// The derivative wrt the ds gap 00282 omatrix_cx dipdgapu; 00283 00284 /// The derivative wrt the us gap 00285 omatrix_cx dipdgapd; 00286 00287 /// The derivative wrt the ud gap 00288 omatrix_cx dipdgaps; 00289 00290 /// The derivative wrt the up quark condensate 00291 omatrix_cx dipdqqu; 00292 00293 /// The derivative wrt the down quark condensate 00294 omatrix_cx dipdqqd; 00295 00296 /// The derivative wrt the strange quark condensate 00297 omatrix_cx dipdqqs; 00298 00299 /// Storage for the eigenvalues 00300 ovector eval6; 00301 00302 /// GSL workspace for the eigenvalue computation 00303 gsl_eigen_hermv_workspace *w6; 00304 00305 private: 00306 00307 cfl6_eos(const cfl6_eos &); 00308 cfl6_eos& operator=(const cfl6_eos&); 00309 00310 #endif 00311 00312 }; 00313 00314 #ifndef DOXYGENP 00315 } 00316 #endif 00317 00318 #endif
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