00001 /* 00002 ------------------------------------------------------------------- 00003 00004 Copyright (C) 2006, 2007, 2008, 2009, 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 An EOS like \ref cfl_njl_eos but 00035 with a color-superconducting 't Hooft interaction 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 \hline 00183 <b>References:</b> 00184 00185 Created for \ref Steiner05. 00186 */ 00187 class cfl6_eos : public cfl_njl_eos { 00188 public: 00189 00190 cfl6_eos(); 00191 00192 virtual ~cfl6_eos(); 00193 00194 /** \brief Calculate the EOS 00195 \nothing 00196 00197 Calculate the EOS from the quark condensates. Return the mass 00198 gap equations in \c qq1, \c qq2, \c qq3, and the normal gap 00199 equations in \c gap1, \c gap2, and \c gap3. 00200 00201 Using \c fromqq=true as in nambujl_eos and nambujl_temp_eos 00202 does not work here and will return an error. 00203 00204 If all of the gaps are less than gap_limit, then the 00205 nambujl_temp_eos::calc_temp_p() is used, and \c gap1, \c gap2, 00206 and \c gap3 are set to equal \c u.del, \c d.del, and \c s.del, 00207 respectively. 00208 00209 */ 00210 virtual int calc_eq_temp_p(quark &u, quark &d, quark &s, 00211 double &qq1, double &qq2, double &qq3, 00212 double &gap1, double &gap2, double &gap3, 00213 double mu3, double mu8, 00214 double &n3, double &n8, thermo &qb, 00215 double temper); 00216 00217 /// The momentum integrands 00218 virtual int integrands(double p, double res[]); 00219 00220 /// Check the derivatives specified by eigenvalues() 00221 virtual int test_derivatives(double lmom, double mu3, double mu8, 00222 test_mgr &t); 00223 00224 /** \brief Calculate the energy eigenvalues and their derivatives 00225 \nothing 00226 00227 Given the momentum \c mom, and the chemical potentials 00228 associated with the third and eighth gluons (\c mu3 and \c mu8), 00229 this computes the eigenvalues of the inverse propagator and 00230 the assocated derivatives. 00231 00232 Note that this is not the same as cfl_njl_eos::eigenvalues() 00233 which returns \c dedmu rather \c dedqqu. 00234 */ 00235 virtual int eigenvalues6(double lmom, double mu3, double mu8, 00236 double egv[36], double dedmuu[36], 00237 double dedmud[36], double dedmus[36], 00238 double dedmu[36], double dedmd[36], 00239 double dedms[36], double dedu[36], 00240 double dedd[36], double deds[36], 00241 double dedmu3[36], double dedmu8[36]); 00242 00243 /** \brief Construct the matrices, but don't solve the eigenvalue 00244 problem 00245 \nothing 00246 00247 This is used by check_derivatives() to make sure that the derivative 00248 entries are right. 00249 */ 00250 virtual int make_matrices(double lmom, double mu3, double mu8, 00251 double egv[36], double dedmuu[36], 00252 double dedmud[36], double dedmus[36], 00253 double dedmu[36], double dedmd[36], 00254 double dedms[36], double dedu[36], 00255 double dedd[36], double deds[36], 00256 double dedmu3[36], double dedmu8[36]); 00257 00258 /// The color superconducting 't Hooft coupling (default 0) 00259 double KD; 00260 00261 /// Return string denoting type ("cfl6_eos") 00262 virtual const char *type() { return "cfl6_eos"; }; 00263 00264 /** \brief The absolute value below which the CSC 't Hooft coupling 00265 is ignored(default \f$ 10^{-6} \f$) 00266 */ 00267 double kdlimit; 00268 00269 protected: 00270 00271 #ifndef DOXYGEN_INTERNAL 00272 00273 /// Set the quark effective masses from the gaps and the condensates 00274 int set_masses(); 00275 00276 /// The size of the matrix to be diagonalized 00277 static const int mat_size=36; 00278 00279 /// Storage for the inverse propagator 00280 omatrix_cx iprop6; 00281 00282 /// The eigenvectors 00283 omatrix_cx eivec6; 00284 00285 /// The derivative wrt the ds gap 00286 omatrix_cx dipdgapu; 00287 00288 /// The derivative wrt the us gap 00289 omatrix_cx dipdgapd; 00290 00291 /// The derivative wrt the ud gap 00292 omatrix_cx dipdgaps; 00293 00294 /// The derivative wrt the up quark condensate 00295 omatrix_cx dipdqqu; 00296 00297 /// The derivative wrt the down quark condensate 00298 omatrix_cx dipdqqd; 00299 00300 /// The derivative wrt the strange quark condensate 00301 omatrix_cx dipdqqs; 00302 00303 /// Storage for the eigenvalues 00304 ovector eval6; 00305 00306 /// GSL workspace for the eigenvalue computation 00307 gsl_eigen_hermv_workspace *w6; 00308 00309 private: 00310 00311 cfl6_eos(const cfl6_eos &); 00312 cfl6_eos& operator=(const cfl6_eos&); 00313 00314 #endif 00315 00316 }; 00317 00318 #ifndef DOXYGENP 00319 } 00320 #endif 00321 00322 #endif
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