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Implications of two-flavour colour superconductivity on compact star physics Al-Quaderi, Golam Dastegir 2007

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Implications of Two-Flavour Colour Superconductivity on Compact Star Physics by Golam Dastegir Al-Quaderi  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in The Faculty of Graduate Studies (Physics)  The University of British Columbia April 2007 © Golam Dastegir Al-Quaderi 2007  Abstract In this thesis, we study the thermodynamics of two-flavour colour superconductivity and the effects of its occurrence on the structure and cooling of compact stars. We consider pure two-flavour colour superconductors (2SC), gapless 2SC (g2SC), mixed phase of non-superconducting (normal) quark matter and 2SC or g2SC phase, with possibly free strange quarks present. We employ the BCS pairing interaction through a four-quark operator with the quantum numbers of a single gluon exchange. Low density matter on the crust of the star is modeled by the neutral hadronic phase in the relativistic mean field approximation. The equations of state of the quark and the hadronic phases are found, as well as the phase diagram of the quark matter. The quark matter is seen to undergo second order phase transitions from the mixed phase to pure 2SC or g2SC phase and in between the 2SC and g2SC phases. The quark matter also undergoes a second order phase transition from normal to pure colour superconducting phase with increase of the baryon chemical potential. The mass and structure of the stars are found by solving the TolmanOppenheimer-Volkoff equations together with the equation of state of the particular phase. We find a transition from pure quark to hybrid and ultimately pure hadronic stars with increase of bag constant. For certain bag constants, with increase of central pressure, we see an unstable branch in the mass versus radius plot of the hybrid stars which later changes into a stable branch. We find the difference of mass and total baryon number between configurations of the compact stars at zero and finite central temperatures as functions of the central pressure. The cooling stars are considered at fixed total baryon number. We find the thermal capacity or the ratio of the energy lost to the change of central temperature for quark stars with the colour superconducting or normal quark phase. The quark stars without free strange quarks in 2SC phase loose more energy while cooling compared to stars with non-zero strangeness. The cooling quark stars however shrink more if they contain free strange quarks.  ii  Table of Contents Abstract  ii  Table of Contents  iii  List of Figures  vi  Acknowledgements  xx  1 Introduction 1.1 Physical Overview 1.2 Thesis Outline  1 1 4  2 Phases of Quantum Chromodynamics within Compact Stars 2.1 The QCD Lagrangian and its Symmetries 2.2 Nuclear Matter Phase of QCD 2.2.1 Hadronic Composition of Neutron Stars 2.2.2 Effective Field Theoretical Model of Neutron Star Matter 2.2.3 The Mean Field Approximation 2.3 Colour Superconductivity of Quark Matter 2.4 The 2SC Phase 2.4.1 The 2SC Lagrangian Density 2.4.2 The Chemical Potentials of the Quarks 2.4.3 Dispersion Relation and Partition Function of the 2SC Phase 2.5 The Gapless 2SC Phase 3 Thermodynamics of QCD Phases 3.1 Thermodynamics of Normal Quark Matter 3.2 Thermodynamics of Hadronic Phase 3.3 Thermodynamics of Two-Flavour Colour Superconductors  6 7 11 12 14 17 21 29 29 32 33 35  37 37 41 . . 43 iii  Table of Contents  3.4  3.5  3.6  4  3.3.1 The Thermodynamic Potential in 2SC Phase 3.3.2 Regularization of Thermodynamic Potential 3.3.3 Gap Equation for the 2SC Phase 3.3.4 Electric and Colour Charge Neutrality in 2SC Phase . Mixed Phase of Colour Superconducting Quark Matter and Normal Matter 3.4.1 Mixed Phase of 2SC Phase and Normal Quark Matter 3.4.2 Thermodynamic Quantities in the Mixed Phase . . . . Computation and Numerical Evaluation 3.5.1 Evaluation of Hadronic Thermodynamic Quantities in Terms of Fermi-Dirac Functions 3.5.2 Quasi-particle Dispersion Relations and Evaluation of Quark Integrals 3.5.3 Gaussian Quadrature Method for Evaluating Fermion Integrals Results 3.6.1 Thermodynamic Potential for the Pure 2SC and the Mixed Phase with Normal Quark Matter 3.6.2 Superconducting Gap for the 2SC Phase and Mixed Phase of 2SC and Normal Quark Matter 3.6.3 Dependence of the Coupling Constant on the Momentum Cut-off 3.6.4 Electric and Colour Charge Chemical Potentials for 2SC and Mixed Phase with Normal Quark Matter . . 3.6.5 Thermodynamic Properties of the Neutral Hadronic Phase 3.6.6 Phase Diagram of the Quark Phase 3.6.7 Equation of state of Quark Matter and Hadronic Phase  Compact Stars with Colour Superconducting Cores . . . . 4.1 The Tolman-Oppenheimer-Volkoff Equations for Stellar Structure 4.1.1 Equation of State of Stellar Matter 4.2 Solution of TOV Equations 4.2.1 Scaled TOV Equations 4.3 Mass and Radius of Cold Compact Stars with Colour Superconducting Matter 4.3.1 Mass and Radius of Hybrid Stars  44 46 47 48 49 50 52 53 53 55 59 59 59 62 64 67 69 75 79 83 83 85 88 90 91 94 iv  Table of Contents 4.4  Temperature Dependence of Compact Star Structure 4.4.1 Hot Quark Stars 4.4.2 Binding Energy of Quark Stars 4.4.3 Change of Mass, Radius and Total Baryon Number of Quark Stars with Temperature 4.4.4 Error in the Determination of Radius, Mass and Total Baryon Number of Compact Stars 4.4.5 Hot Hybrid and Hadronic Stars 4.4.6 Change of Mass, Radius and Total Baryon Number of Hybrid and Hadronic Stars with Temperature  97 98 100 102 122 124 127  5 Cooling of Quark Stars 130 5.1 Compact Stars with and. without Trapped Neutrinos 130 5.2 Cooling of Quark Stars 134 5.2.1 Cooling of Quark Star with 2SC Phase and Free Strange Quarks 139 5.2.2 Cooling of Quark Star with Two Flavours 145 6 Summary and Conclusions 6.1 Summary of Results 6.2 Comparison with Other Studies 6.2.1 Quark Phase Diagram 6.2.2 Maximum Mass of Compact Stars 6.3 Outlook  153 153 159 160 162 162  Bibliography  164  Appendices A Frequency Summation of the Free Energy  170  B General Thermodynamic Relations  172  C Generalized Fermi-Dirac Functions and Thermodynamic Quantities 174 C . l Free Fermion Thermodynamic Quantities 174 C.2 Hadronic Thermodynamic Quantities 176  List of Figures 3.1  The quasi-particle dispersion relations in the two-flavour colour superconducting phase (2SC/g2SC): E = E^ = [{Ef ± p) + (A) ] ± 6\i. The left plot is the excitation spectrum in the gapped 2SC phase in which there is a finite gap along the energy direction. The second plot is the magnitude of the excitation energy in the g2SC phase as a function of momentum p. At some momenta, gapless modes exist 3.2 Comparison of different integration schemes based on subdivision of the integration range of the momentum p for baryon chemical potential IXB = 1050.0 MeV.In the first method, we integrate over ranges of p where the integrand is non-zero at T = 0 MeV and add the other parts on either side within the integration range.The second scheme consists of one-step integration over the whole range of the momentum.In the third scheme, we divide the range of momentum into 100 equal subdivisions and integrate 3.3 Grand thermodynamic potential for two-flavour colour superconductor in pure (solid line) and mixed phase (dashed line) with normal quark matter and no strange quarks present as a function of the gap parameter A . The figure corresponds to a baryon chemical potential of 1050.0 MeV and three temperatures as shown, in units of MeV. The mixed phase has two degenerate minima at a lower value than the minimum for the pure phase when it is favoured. As the temperature increases, the minima for the mixed phase are less pronounced and at some higher temperature, no mixed phase is possible. For T=0 MeV we get the same graphs as in reference [55]. The horizontal lines are drawn for comparison of the degenerate minima in the mixed phase ±  2  2  1//2  ree  List of Figures 3.4  3.5  3.6  3.7 3.8  3.9  Grand thermodynamic potential Q surface for two-flavour coloursuperconductor with no net colour charges present as a function of gap parameter A and charge chemical potential at baryon chemical potential of 1050.0 MeV and temperature T = 0 MeV. A = 0 line corresponds to the normal quark matter phase. The surface has a minimum with respect to the A and a maximum with respect to fiQ. The negative of the slopes in the \IQ direction gives the charge densities. If at some particular /XQ, the normal and 2SC phases have opposite (but not necessarily equal) slopes and same value of Q, then we have a mixed phase The contour plot of the colourless thermodynamic potential for the values between [—95, —85] MeV for the same values of the parameters as in plot (3.4). We see that at some particular charge chemical potential £LQ, the normal quark matter phase and the 2SC phase have the same value of the thermodynamic potential but opposite slopes ( as seen from the contour lines). This corresponds to a mixed phase since the thermodynamic potential at this phase is more negative than the saddle point where the contours have a broad region of equal levels Temperature Dependence of the superconducting gap parameter for the 2SC phase with and without free strange quarks at baryon chemical potential [XB of 1050.0, 1075.0 and 1100.0 MeV's Dependence of superconducting gap at temperature T = 0 MeV in the gapless/gapped 2SC phase on the baryon chemical potential for different values of the cut-off parameter Plot of |/XQ/2 ± n 1 vs. A for gapless or gapped 2SC phase. At some value of the superconducting gap Ao ~ 148 MeV, the colour-superconducting quark-matter changes phase from gapless to gapped 2SC. The cut-off parameter value is A = 1500.0 MeV. The irregularities in the plot beyond the phasechange point, arises from errors in /Z3 (see Fig. 3.14) which are small compared to the value of /JLQ Dependence of the superconducting gap in the 2SC superconducting component of the mixed phase on the baryon chemical potential at T = 0 MeV and A = 1500.0 MeV 3  61  62  63 65  0  66 67  List of Figures 3.10 Variation of the coupling constant G on the cut-off A as a result of fixing the minimizing gap parameter at 100.0 MeV at zero chemical potentials and temperature at different reference baryon chemical potentials 3.11 Variation of electric- and colour-charge chemical potentials for the pure 2SC and mixed phase with respect to the superconducting gap parameter A . For the pure phase, the thermodynamic potential has an extremum at some A = A which is the physically relevant gap-value 3.12 Variation of electric charge chemical potential /XQ with respect to baryon chemical potential /x^. For the range in \IB where the gap A is zero, the chemical potential /XQ corresponds to that of free normal quark matter. After the onset of superconductivity, the slope of the graph for \IQ changes at the phase transition point from gapless to gapped 2SC phase 3.13 Variation of colour-8 charge chemical potential ix with respect to baryon chemical potential JJLB- For the range in xx# where the gap A is zero, the chemical potential xx is essentially zero within numerical errors and corresponds to that of free normal quark matter. After the onset of superconductivity, the slope of the graph for xx changes at the phase transition point from gapless to gapped 2SC phase 3.14 Variation of colour-3 charge chemical potential /X3 with respect to baryon chemical potential xx^. For the range in /J,B where the gap A is zero, the chemical potential /X3 is essentially zero and corresponds to that of free normal quark matter. After the onset of superconductivity, the value for /x remains very close to zero. At the phase-change point from gapless to gapped 2SC, the thermodynamic potential Q is very nearly flat near xx ~ 0.0 with irregularities due to numerical errors along the /X3 direction. The minimum finding routine powell goes to one of these local valleys along the irregularities. . . . 3.15 Surface plot for pressure of the electric charge-neutral hadronic phase with the hyperons present. The pressure is seen to increase with the baryon chemical potential as well as with temperature  68  69  70  8  8  8  71  C  3c  3c  C  72  73  viii  List of Figures 3.16 Surface plot for energy density of the electric charge neutral hadronic phase with the hyperons present. The energy density is seen to increase with the baryon chemical potential as well as with temperature 3.17 Surface plot for baryon number density of the electric charge neutral hadronic phase with the hyperons present. The baryon number density is seen to increase with the baryon chemical potential as well as with temperature 3.18 Ratio of the number densities of baryonic and leptonic species with respect to total baryon number density at any baryon chemical potential at zero temperature. Charge neutrality requires the relative population of protons and electrons to be equal when there is no other charged particles present. The appearance of particle species depends on the effective chemical potentials and effective masses 3.19 Fractional population of hadronic species and leptons with respect to total baryon number density at finite temperature T = 10 and T = 30 MeV 3.20 Phase diagram of quark phase in the /Lts-T plane. The mixed phase occurs at low temperature and through the range of the baryon chemical potential [900.0,1500.0] MeV. The gapped 2SC phase occurs only at a certain region of the /I/B-T plane, appearing after the mixed phase in the T direction and then changing into the gapless 2SC phase. At high temperature the phase is that of free non-superconducting quarks 3.21 The volume fraction of the 2SC phase X2SC within the mixed phase of 2SC and normal quark matter, in the /J-B-T plane for different values of the cut-off parameter A. The fraction changes continuously within the mixed phase from low values to unity giving way to the occurrence of pure 2SC phase. The irregularities in the graphs come from the finite number of grid points on the /L^B-T plane at which the phase has been determined .  74  74  75 76  77  78  List of Figures 3.22 Phase diagram of quark matter with free strange quarks present in the ^£-T plane. In the presence of free starnge quarks, there is no mixed phase of 2SC and normal quark matter. The occurrence of the colour superconducting phases depend strongly on the cut-off parameter A. For A — 1500 MeV, at high temperature we have a narrow region of gapless 2SC phase. 79 3.23 Plot of equations of state of the mixed phase of 2SC and normal quark matter with no strange quark present and for the pure 2SC phase with free strange quarks at different values of the cut-off parameter A = 650,1300,1500 MeV. The presence of free strange quark makes the equation of state softer and results in the absence of mixed phase between 2SC and normal quark matter. The plots corresponds to baryon chemical potential /xj3 varying from 900 to 1500 MeV for the quark phase and from 940 to 1500 MeV for the hadronic phase 80 3.24 Plot of equations of state (energy density vs. baryon number density) of the neutral hadronic phase and the pure 2SC phase with free strange quarks at different values of the cut-off parameter A = 650,1100,1300,1500 MeV. The bag constant used is B — (195.0MeV) . The presence of free strange quark makes the equation of state softer. The plot corresponds to baryon chemical potential fig varying from 900 to 1500 MeV for the quark phase and from 940 to 1500 MeV for the hadronic phase 81 4  Mass vs. radius relation for cold quark-, hybrid- and hadronicstars with strange particles present. The structure and the composition of the stars depend on the bag constant B as well as the equation of states of the quark and hadronic phases. . . 91 Mass vs. radius relation for cold quark-, hybrid- and hadronicstars without any strange particles present. As the bag constant is increased, we see a transition from quark- to hybridand finally purely hadronic-stars. The maximum mass of the stars depend on the bag constant as well as on the equation of the state of the non-strange quark and hadronic phases. . . 92  x  List of Figures 4.3 Mass and phase change radius of hybrid stars with strange particles present at T = 0 MeV. Hybrid stars result only for a certain range of the bag constant B, since within this range, the pressure of the quark- and the hadronic phases are comparable, with the bag constant subtracted from the the quark phase pressure 4.4 Mass and phase change radius of hybrid stars without any strange particles present at T = 0 MeV. Hybrid stars are seen to result for the bag constant within a certain range, similar to the case of quarks stars with strange particles 4.5 Mass-radius relationship for quark stars with 2SC phase with and without strange quarks present. The upper panel represents 2SC and free strange quarks phase. The lower panel represents the case for either pure 2SC phase or a mixed phase of 2SC and normal quark matter in two flavours (u and d quarks). The temperatures need to increase ~ 10 MeV for any noticeable change to occur 4.6 Temperature dependence of the mass M, total baryon number N and central baryon chemical potential \x of quark stars in the 2SC phase with free strange quarks present. The variation in mass and total baryon number are not noticeable until the temperature increase A T ~ O(10) MeV. The central baryon chemical potential for low central pressures is obtained by extrapolation and is seen to have values that are below the lower limit (900 MeV) of the tabulated values in the equation of state table 4.7 Mass, total baryon number, radius and central baryon chemical potential for quark stars with 2SC quark matter with free strange quarks as a function of the central baryon number denisty. For most part of the range of n^c, the quark stars are gravitationally stable. The part of the plots after the maximum mass corresponds to unstable stars. The central baryon chemical potential has very little dependence on either the temperature or the bag constant B  93  95  96  Bc  97  99  xi  List of Figures 4.8 The binding energy, binding energy per unit gravitational mass and compactness parameter of quark stars in the 2SC+S phase as a function of central pressure at central temperatures T = 0,10,20 MeV. As the star becomes more massive the binding energy in general increases.The dependence on bag constant and temperature is also noticeable on the plots of binding energies 4.9 The relative binding energy of incompressible fluid (BE /M) and the internal part of the relative binding energy (BE /M) of quark stars in the 2SC+s phase. The relative internal binding energy is seen to decrease with increase of central pressure while BE[ /M increases. As the central temperature increases BE /M is clearly seen to decrease 4.10 The change of mass M, total baryon number NB and radius R of quark stars in 2SC+S phase at an increase of temperature from 0 to 5,10,20 MeV plotted against a range of central pressure. In general all three of M,NB and R increase with T and for each final temperature, the change peaks at some particular pressure. For high central pressures and for some bag constants we see a decrease of NB 4.11 The change of mass M, total baryon number NB and radius R of quark stars in 2SC+S phase at an increase of temperature from T = 0 to T = 0.5,1 MeV plotted against a range of central pressure. Mass- and radius-change are always positive while NB is seen to decrease at high central pressures for some bag constants 4.12 Change of gravitational mass and total baryon number of quark stars with 2SC+S phase, per unit gravitaionnal mass as a function of the central pressure at increase of the central temperature of the stars. The relative change of mass is always positive while that of the total baryon number becomes negative at high central pressures. In general, the higher the central pressure the less are the magnitudes of the relative changes  101  INC  INT  NC  INT  103  104  105  106  List of Figures 4.13 Relative change of gravitational mass and total baryon number of quark stars with 2SC+S phase at increase of central temperature from T = 0 to T = 0.5,1 MeV's. The relative mass-change is always positive while the relative change of N becomes negative for some bag constant and central pressures. At higher bag constants the the relative change in Ng becomes negative at low central pressures 4.14 Relative change of radius of quark stars with 2SC+S phase as a function of central pressure at change of central temperature. The change AR/Ro is always seen to be positive but decreases in magnitude as the central pressure increases. For higher bag constants, the change is less 4.15 Relative change of radius of quark stars with 2SC+S phase versus central pressure at increase of central temperature from T = 0 to T = 0.5,1 MeV's. We see similar temperature dependences of radius as in the plot (4.14) for the case of larger temperature increases 4.16 Mass, total baryon number and central baryon chemical potential for quark stars with 2SC or 2SC+NQ phase, versus central pressure at different central temperatures. The portion of the plots beyond the peaks of M and NB correspond to gravitationally unstable configurations. For higher bag constants the mass and total baryon number decreases. The central baryon chemical potential fiBc at low central pressures have values low enough to correspond to unphysical configurations of the quark stars 4.17 Mass, total baryon number and radius of quark stars with 2SC or 2SC+NQ phase as function of central baryon number density UB at different central temperatures. The mass and the total baryon number peaks at the higher end of the range of UBC while the radius peaks at much lower n . Configurations of the stars before the mass peaks are gravitationally stable. As the bag constant is increased, all of M, n and R are seen to decrease B  107  108  109  110  C  B c  Bc  112  List of Figures 4.18 Difference of mass and total baryon number and the central baryon chemical potential [XB of quark stars with 2SC or 2SC+NQ phase versus central pressure at change of central temperature to T = 5,10,20 MeV's. At low central pressures the central baryon chemical potential is low enough that quark stars can not be produced and hence correspond to unphysical configurations. The change in the mass and total baryon number peak at some central pressure values and decrease with increase of bag constant at higher central pressures 4.19 Difference of mass and total baryon number as functions of the central baryon chemical potential jiBc of quark stars with 2SC/g2SC or (2SC/g2SC)+NQ phase at change of central temperature to T = 5,10,20 MeV's. For physically relevant values of the central baryon chemical potential the mass difference is always positive. The differences AM and ANB peak at some central baryon chemical potential values for each temperature difference 4.20 Difference of mass and total baryon number of quark stars with 2SC or 2SC+NQ phase as a function of central pressure at change of central temperature to T = 0.5,1 MeV's. The changes AM and AN have similar behaviour as for the case of higher temperature changes in the plot (4.18) 4.21 Difference of mass and total baryon number as functions of the central baryon chemical potential \i at change of central temperature from zero to T = 0.5,1 MeV's. For physically relevant values of the central baryon chemical potential the mass difference is always positive. The changes AM and ANB have similar behaviour as for the case of higher temperature changes in the plot (4.19) 4.22 Relative change of mass and total baryon number for quark stars with 2SC or 2SC+NQ phase versus central pressure at different central temperatures. At physically relevant values of the central pressure, where quarks stars are realizable, the changes decrease with increase of central pressure and bag constant. At the higher end of the pressure range, the dependence on bag constant is less C  113  114  B  115  Bc  116  117  xiv  List of Figures 4.23 Relative change of mass and total baryon number of quark stars with 2SC or 2SC+NQ phase as a function of central pressures at change of central temperature to T — 0.5,1 MeV's. The relative changes have similar behaviour as for the case of higher temperature changes in the previous plot (4.22) 4.24 The absolute and relative changes of radius of quark stars with two flavours in the 2SC or 2SC+NQ phase, as a function of central pressure at temperature increase of A T = 5,10,20 MeV's. As the central pressure increases, the change in radius decreases. At higher central pressures the dependence of the changes on the bag constant becomes less 4.25 Absolute and relative change of the radius of quark stars with 2SC or 2SC+NQ phase versus central pressure at temperature increase of A T = 0.5,1 MeV's. The changes of radius show similar behaviour as in the case of higher temperature changes in the previous plot (4.24) 4.26 Mass versus radius relationships of hadronic stars and hybrid stars with quark matter cores, with and without strange particles present, at zero and finite central temperatures. The structure and nature of the stars depend on the bag constant used. Non-strange stars in general have larger mass and radius compared to stars with non-zero strangeness at the same bag constant and central pressure. The temperature dependence is stronger at lower central pressures and for stars with large hadronic regions 4.27 Mass, total baryon number and central baryon chemical potential of hybrid stars with quark cores in the 2SC+s phase with different bag constants and at central temperatures T = 10,20 MeV's. At low central pressures and bag constant of B = (160 MeV) we have unstable configuration of stars at temperature T = 10,20 MeV's. At higher bag constant.B = (180MeV) , we see another unstable branch at central pressures where the quark core just starts to appear  118  119  120  121  4  4  122  xv  List of Figures 4.28 Mass, total baryon number and central baxyon chemical potential of pure hadronic stars as a function of central pressure at zero and finite central temperatures. As the central pressure increases both mass and total baryon number attain maximum values and then decrease. The branch beyond the peak in the mass corresponds to unstable configurations of the star. The temperature dependence of the quantities becomes less as the pressure becomes higher 4.29 Difference of mass and total baryon number of hybrid and hadronic star configurations as a function of central pressure at zero and finite temperatures. In general the mass-change AM decreases with increase of central pressure and bag constant. For certain bag constants AM shows one or more local maximum depending on the degree of hybridness.The shape of the change ANB follows closely that of AM 4.30 Relative change of mass and total baryon number of hybrid and hadronic stars versus central pressure at change of stars' central temperature from T = 0 to T = 10,20 MeV's.The mass changes are always positive while that for total baryon number may become negative at high central pressures for some bag constants used 4.31 Absolute and relative change of radius of hybrid stars with quark cores in 2SC+S phase and hadronic stars versus central pressure at increase of central temperature from T = 0 to T = 10,20 MeV's . For bag constant of B = (160 MeV) as the temperature increases the star's configuration at low central pressures, changes as its radius shrinks. These correspond to unstable configurations of hybrid stars  123  125  126  4  5.1  128  Change of gravitational mass at constant total baryon number of quarks stars with quark matter in the 2SC+S phase. Within the range of the central baryon number density, the quark star is in a stable branch. The data-points represent star configurations with the same central pressure but at different central temperatures 131  xvi  List of Figures 5.2  Gravitational mass of quark stars with quark matter in the 2SC+S phase as a function of total baryon number of the star. As the central temperature increases, the gravitational mass increases. The plots corresponds to a gravitationally stable configurations of the stars. The nearby data-points corresponds to the same central pressure but at different central temperatures 5.3 Change of gravitational mass of quark stars with quark matter in the 2SC+S phase while cooling from a initial temperature of T — 10 MeV to T = 0 MeV when the total baryon number of the star is kept fixed within a margin of error. The variation of the total baryon number is also shown in the bottom panel. Compare with the total baryon number of quark stars which are of the order of 1 ~ 10 x 10 . The pressure values are in dyne/cm 5.4 Change of gravitational mass of quark star with 2SC+S phase for cooling from an initial temperature T = 1 MeV to T = 0.05 MeV at fixed total baryon number NB of the star. We consider different initial central pressures in dyne/cm and bag constants in MeV. The third panel shows the error in keeping NB constant 5.5 The dependence of the central pressure P and the radius of a quark star with quark matter in the 2SC+S phase as it cools down from T = 10 MeV to T = 1 MeV at fixed total baryon number. Depending on the initial central pressure at finite initial central temperature the central pressure at a lower temperature is seen to increase and the radius decrease 5.6 Relative difference of central pressure and radius with respect to those at central temperature T=0 MeV for a cooling quark star in the 2SC+S phase. The star cools down from T=10 MeV to T=0 MeV at fixed total baryon number corresponding to the hot star  132  56  3  133  3  134  C  135  136  xvn  List of Figures 5.7 Relative change of central pressure and the relative radius of quark stars with quark matter in the 2SC+S phase as it cools down from initial central temperature T = 1 MeV to T = 0.05 MeV at fixed total baryon number. Depending on the initial central pressure, the central pressure at a lower temperature may increase or decrease to keep NB constant. The radius of the quark star with this phase of matter, is however seen to decrease as it cools 137 5.8 Relative difference of central pressure and radius with respect to those at initial temperature T=0.05 MeV for quark stars with quark matter in the 2SC+s phase. The star cools down from initial central temperature T = 1 MeV to T = 0.05 MeV at fixed total baryon number corresponding to the hot star. . . 138 5.9 Mass and total baryon number of quark stars with quark matter in the 2SC or 2SC+NQ phase at different central temperatures as a function of the central baryon number density. In the range of the central baryon number density considered, the quark stars are gravitationally stable. At the same total baryon number the masses at different central temperatures give the difference of mass of the stars as they cool 140 5.10 Mass of the quarks stars with two-flavour quark matter in the 2SC or 2SC+NQ phase as a function of their total baryon number for bag constants of B = (135 MeV) , (140 MeV) . The stars considered here are in gravitationally stable configurations and as the stars cool down the gravitational masses are seen to decrease. At the same total baryon number, the change of mass correspond to energy lost due to cooling. . . . 142 5.11 The mass versus the total baryon number of quark stars with quark matter in the two-flavour color superconducting phase for a bag constant of B = (145 MeV) 143 5.12 The absolute and relative change of mass versus the central temperature of quark stars with matter in the 2SC or 2SC+NQ phase as they cool down from central temperature of T = 10 MeV to T = 0 MeV at constant total baryon number. As the temperature decreases, the change of mass increases indicating energy loss due to cooling. The initial central pressures axe in dyne/cm and bag constants in MeV. The third panel shows the error in the total baryon number while keeping it fixed. . . 144 4  4  4  3  xviii  List of Figures 5.13 The absolute and relative changes of mass versus central temperature of quark stars with 2SC or 2SC+NQ phase for cooling from a central temperature T = 1 MeV to zero temperature. The initial central pressures are in dyne/cm and bag constants in MeV. We keep the total baryon number of the star fixed to the value at that of the initial hot star. The third panel shows the error in holding NB fixed 5.14 Relative central pressure and relative radius of quark stars with 2SC or 2SC+NQ phase versus central temperature. As the star cools down from initial hot state with T — 10 MeV to T = 0 MeV, the central pressure and radius changes depending on the initial central pressure and the equation of state used. For most of the initial configurations of the star, the central pressure at T < 10 MeV is seen to increase while the radius decrease 5.15 Relative change of central pressure and radius of quark stars with 2SC or 2SC+NQ phase with respect to those at temperature T=0 MeV. At increased temperature the star becomes larger and the central pressure decreases 5.16 Relative change of central pressure and relative radius of quark stars with 2SC or 2SC+NQ phase versus the central temperature as it cools down from T = 1 MeV to T = 0 MeV at constant total baryon number. The change in P depends on the initial central pressure. The radius of the stars are seen to decrease. Contrast the change of P and R with those at cooling from a higher initial central temperature T — 10 MeV shown in the previous plot (5.14) 5.17 Relative change of central pressure and radius of quark stars with 2SC or 2SC+NQ phase with respect to those at temperature T=0. The star cools down from T = 1 MeV to T = 0 MeV at constant total baryon number. The initial total baryon number corresponds to that of the hot star. The star srinks as it cools down while the central pressure may decrease or increase depending on the initial value 3  146  c  147  148  c  c  150  151  xix  Acknowledgements I would like to thank my supervisor Professor Jeremy S. Heyl for providing research guidance and support to carry out this research. I am also grateful to Professor Ariel Zhitnitsky for kindly agreeing to be my second reader. Thanks to my parents who, despite loving me dearly, have encouraged me to pursue my study at UBC staying so far away from them. To my friends Shahed, Asfak, Tuhin and many others who had been helpful in difficult times.  xx  Chapter 1 Introduction 1.1  Physical Overview  The fate of a dying main sequence star having a mass a few times the solar mass ( ~ 8M ) is the gravitational collapse of the core followed by a fiery supernova explosion. The end product is a remnant star made out of mainly neutrons and hence called a neutron star. Depleted out of nuclear fuel to burn and shining meekly in the sky, neutron stars are supported against the gravitational pull of the matter by neutron degeneracy pressure. Requiring a huge number of neutrons at close distance to each other to become degenerate, neutron stars are closely bound by strong gravity making them compact in nature. The typical size of neutron stars is of the order of 10 km while the mass is of the order of the mass of the sun. This makes the surface potential GM/Rc ~ 10 , warranting general relativity to govern the structure and stability of these stars. The collapsed core also acquires high angular velocities through the conservation of angular momentum if the progenitor was a rotating star. Together with the collapse of matter, the magnetic field is also compressed and amplified to up to ~ 10 G due to conservation of magnetic flux. Being the densest directly observable objects in the universe, neutron stars thus provide a testing ground for physical phenomena under extreme conditions of high densities, pressure and strong electromagnetic and gravitational fields, a unique condition that is not possible to be generated in the laboratory environment. Although predicted in the 1930s by Baade and Zwicky [9] on theoretical grounds, neutron stars were not observed until 1967 (Hewish et.ai, 1968)[33]. The rotation of the strong magnetic dipole field generates a beamed periodic signal, which helped the pulsating neutron stars i.e. the pulsars to be detected. Observational astronomy has probed many properties of these extreme objects, resulting in two Nobel prizes in physics to be awarded; one for the discovery of pulsars (Hewish 1974) and the other for the discovery of the first neutron star-neutron star binary pulsar PSR1913+16 (Hulse and TayQ  2  _1  13  1  Chapter 1. Introduction lor, 1993) [41]. The decay of the orbital period of PSR1913+16, confirmed the emission of gravitational radiation in full agreement with/vindicating the prediction of Einstein's General theory of Relativity. Since 1967, nearly one thousand neutron stars have been discovered and many of their properties have been measured to constrain the physical phenomena that may occur in these stars. Timing observations set an upper limit of the possible rotational frequency sustainable by neutron stars - the Kepler frequency, which is the absolute limit of rotation. Measurement of the maximum mass of the neutron star companion in a binary system also constraints the properties of matter in the star. Small irregularities superimposed on the otherwise highly stable rotational frequencies of the stars provide hints of the presence of superfluid matter in the interior. Matter inside a neutron star is probably more complex than a simple mixture of mostly highly degenerate neutrons , small portion of protons and electrons, to make them stable against beta decay and be charge neutral. The outermost region is made up of a highly compressed ionized nuclei, mostly Fe and electrons. This is covered by the a thin "atmosphere" made up of H, He and other light elements, ions and/or molecules. Radiation coming from the neutron stars may contain transition lines of the elements in this atmosphere and provide information of its composition. With increasing density in the outer crust (10 gcm~ < p < 10 gcm~ ) matter becomes neutron rich and a sequence of high-mass-number nuclei, stable at the ground state of cold dense nuclear matter occurs ( N i , K r , Se, Ge, Zn, R u , M o , Z r , Sr and other neutron-rich isotopes [32]). Below the melting temperature T ~ 10 K , the matter consists of crystal lattice formed by different species of nuclei depending on the density at the particular layer, together with the conduction electrons. Heat flow through this outer crust is impeded due to scattering of the electrons off the nuclei and the phonons; the elementary excitations of the lattice vibrations and off impurities. Thus the outer quarter of a kilometer of the crust insulates the core of a neutron star from the atmosphere. After the density approaches the nuclear drip point (p ~ 4 x 1 0 g c m ) , neutrons start to leak out of the nuclei and a mixture of very neutron-rich nuclei immersed in a gas of dripped neutrons and relativistic electrons is formed. The two mostly populated species of nuclei are the closed Z-shell nuclei Zr and Sn [32]. The free neutrons are believed to be in a superfluid state having consequences on the generation of glitches in pulsars and on thermal properties of the inner crust. 56  7  3  n  62  124  122  86  3  84  82  80  126  120  7  s  u  40  -3  50  2  Chapter 1. Introduction At the edge of inner crust (p ~ 10 g c m ) , neutrons no longer remain in the clustered forms of nuclei and a mixture of protons, neutron and electron gas is formed. In this outer core, protons and neutrons condense into superconducting and superfluid states, respectively, below their critical temperatures at the density, while the electrons remain as a relativistic Fermi gas. Depending on the chemical potential of the neutrons, muons (p), other baryon species than the nucleons (the hyperons A, S ' ° ' , E~) and the mesons (n, K) will be populated [30]. The mesons may form Bose-Einstein condensate (BEC) due to effective nucleon-meson interaction. This has the effect of softening the equation of state and changing the neutrino reaction rates ([67, 68]). As the density of matter increases to a few times the nuclear saturation density(p ~ 3-4 xp ), Fermi momenta of the baryonic species increase since k,B ~ (67T PB/(2JB + l ) ) / [30]. This introduces a natural high-energy scale in the matter, the chemical potential ~ lOOMeV. The building blocks of the hadrons are the quarks and the gluons, whose behaviour are governed by the fundamental theory of strong interactions, the quantum chromodynamics (QCD). QCD at these energies is weakly coupled resulting in the deconfined phase of the quarks and we expect the formation of quark matter (QM). Since the densities reached at the centre of neutron stars is assumed to be of this order, it is likely that the central core of neutron stars are composed of the up(u), down (d) and possibly the strange (s) quarks. These almost free quarks will form Fermi spheres in the momentum space of individual species. However, it is known that such a system of free fermions, e.g. of the quarks, is unstable, in the presence of an attractive interaction, even arbitrarily small [13, 14]. Since in QCD, the gluon exchange in the colour anti-triplet 3 channel is attractive, we expect the formation of coherent state of Cooper pairs. Because of the bosonic nature of the Cooper pairs, they occupy the same lowest energy level and form a Bose condensate . In the presence of such a condensate of Cooper pairs, the ground state of quark matter becomes a (colour) superconductor [63]. This is very similar to the ground state of the electronic systems in the Bardeen-Cooper-Schrieffer theory of low temperature superconductivity [13, 14]. This was first realized several decades ago [12, 15]. The novel feature in these colour superconductors is that, unlike electrons, the quarks come in various flavours and carry non-Abelian colour charges, which is responsible for the pairing, instead of the electric charge, although it is present on the quarks. Hence the name colour superconductivity. 14  -3  _  +  0  2  1  3  3  Chapter 1. Introduction Recent works based on the effective models for the quark-quark interactions predicted a superconducting gap of A ~ 100 MeV for a quark chemical potential of p ~ 400 MeV [3, 54]. The presence of such a high superconducting gap, comparable to the natural energy scale of QCD, A.QCD — 200 MeV , gives rise to a rich phase structure in the matter. The superconducting gap in the dispersion relation modifies the kinetic as well as thermodynamic properties of quark matter and hence possibly affects the properties of the compact star containing colour superconducting quark core. One of the probable phases of colour superconductivity at such baryon number density involves Cooper pairing between only the up(u) and the down (d) quarks, with the strange (s) quark, either not present or not involved in Cooper pairing. This is the two-flavour colour superconducting (2SC) phase [3, 15, 54]. Ever since Witten's seminal paper on strange quark matter hypothesis [70], a lot of interest has been devoted to the stars with non-zero strangeness. Among the earlier works, the strange stars were considered by Alcock, Farhi, Olinto, Jaffe and others [21, 22, 27, 28]. The interest with strange matter lies in the possibility of phase transition to it as hypothesized by Witten. Colour superconducting phase with strange quarks include the colour-flavour-locked phase (CFL), the gapless CFL phase and possibly the 2SC phase with free strange quarks. In the context of the present thesis we consider only the latter.  1.2  Thesis Outline  In the present thesis we focus on the 2SC phase of colour superconductors with the possible presence of free strange quarks (the 2SC+S phase). We also study the neutral hadronic phase to model the outer region of the compact stars. In the second chapter we review the general theory of colour superconductors especially the 2SC phase and of the neutral hadronic phase. We get the equations of motion, the dispersion relations and the partition functions of the phases. In chapter 3 we study the thermodynamic properties of the relevant phases starting our analysis from the partition function. We emphasize on getting the phase diagram of the quark matter including colour superconducting phases considered and the equation of state to be used in building compact star models in the next chapter. In chapter 4, we focus on the astrophysical aspects of the compact stars 4  Chapter 1. Introduction by studying their structure and composition. We consider pure quark stars, hybrid stars and pure hadronic stars by using different equations of state obtained by varying the parameters of the theory. We also consider the thermal effects on the structure and composition of these stars by varying the central temperature and pressure. In the penultimate chapter, we study the cooling of quark stars containing the two-flavour colour superconductors. We investigate the amount of energy lost due to cooling and the change of the size and central pressure of the cooling stars. Our results are relevant for mature quark stars at low central temperature which cool by the emission and immediate dissipation of photons and/or neutrinos. In the last chapter, we summarize our results and deal with the possible avenues of extending the work for future studies.  5  Chapter 2 Phases of Quantum Chromodynamics within Compact Stars The physical universe that we know is composed of the material particles and the particles of the mediators of the four fundamental types of forces, the socalled gauge particles. The material particles come in two main categories; the ones that are capable of interacting via the strong forces, the hadrons, and the particles which do not have any strong interaction, the leptons. The hadrons are composite particles made of further elementary massive quarks and the massless gluons (the mediators of the strong force), while the leptons are fundamental in the true sense that they have no sub-structure. Quarks carry the colour charges and at low energy scales, they exist only in colourless clusters of either quarks and anti-quarks, the mesons, or of a number of quarks, the baryons, both bound together by the gluons. This phenomenon is called the confinement of colour and is fundamental to the theory of strong interactions. On the other hand, at high enough momentum transfer, the quarks are weakly interacting and can freely roam into the space. This is due to the weakening of the strength of the interaction at high energy scales and is called the property of asymptotic freedom. At high enough temperature, a "soup" of quarks and gluons can be formed which we call the quark-gluon-plasma (QGP). This has been observed at the Relativistic Heavy Ion Collider (RICH) at ~ 10 K temperature and was believed to be present in the first few moments after the creation of the universe in the Big Bang [47]. At high enough density, e.g. a few times the nuclear density, the quarks are highly energetic due to the fundamental quantum uncertainty relation (AxAp > h/2n), and hence are free to "leak" out of the hadrons and can sample a much larger "colourless" region of space called quark matter. This may exist in the central core of neutron stars where the density may be as high as ten times the nuclear saturation density no ~ 0.15 fm~ [30]. 12  3  6  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars As the fundamental theory of strong interactions, it is hoped that Quantum Chromodynamics (QCD) will be able to describe all the manifestations of quarks and gluons at the appropriate regions of its phase diagram. The phases relevant to the neutron stars are however present only at relatively low temperatures, compared to the Fermi temperature and in the range of baryon chemical potential corresponding to nuclear matter to the densest possible quark matter at the central core of the stars.  2.1  The Q C D Lagrangian and its Symmetries  The elementary degrees of freedom in Quantum Chromodynamics (QCD) are the quarks and the gluon fields appearing in the Lagrangian density: CQCD = Qt(i^D^  - mjqf  -  \F%F^  (2.1)  where JTIJ are the current masses of the quark fields qf entering explicitly in the Lagrangian density. The bar denotes the Dirac conjugation and the quark fields are labeled by the flavour index i € (u, d, s, c, b, t) and the colour index a £ (r,g,b). The colour gauge-covariant derivative is defined by K  = 9"6  0  a0  + igA^  = d»6  aP  + i g A ^ ^  (2.2)  where g is the strong coupling constant and the gluon fields are denoted by A%, with the index a — 1... 8 , running over the eight colours of the gluons. The matrices A are the Gell-Mann matrices which act as the generators of the colour SU(3) gauge group satisfying the Lie algebra a  where f are the structure constants of the colour group SU(3) . The field strength tensor F£ is defined in terms of the gluon fields as abc  C  V  F%  = %K ~ d Al v  igfabcA^Al  (2.4)  7  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars Under the colour-gauge group SU(3) , form as C  q?{x) -  the quarks and the gluons trans-  r(x) = e [-i^6 (x)')qf(x)  q  W  A'^UAJJ-  1  = U(e(xM(x)  a  + -(d»U)Ul  (2.5) (2.6)  1  while the QCD Lagrangian density remains invariant. The last term in the field strength tensor in (2.4) arises because of the non-commuting nature of the generators of the group 4r, (c.f. 2.3) rendering the theory non-Abelian. Thus, even without any matter field, i.e. with pure gluonic fields, the Lagrangian density contains interactions, because of self-couplings of the gauge fields. We get a cubic-interaction term linear in the coupling and also a quartic-interaction term, which is quadratic in the coupling g. Because of the non-Abelian nature, quantitative predictions from QCD are difficult to get at strong or intermediate coupling strengths. At certain energy scales, where the masses of the lighter quarks are too small to be accounted for while the heavier quarks are too massive to appear, we can take the masses of the lighter quarks to be equal m = rrid « m (or even equal to zero).Then the theory becomes invariant under arbitrary flavour rotations of the quark fields u  A  3  a  ft -> VijQj = [exp(-i-^p£)]y9.j (-) where A^ are Gell-Mann matrices'in the flavour space, Vij € SU(3)fi and the symmetry group is enlarged to SU(3) i <8> SU(3)fi . This is the flavour iso-spin symmetry of QCD. If the quark masses are not just equal, but can effectively be taken to zero, we have an even larger symmetry group. Defining the left- and right-handed fields by q = ^{l± )q =P q (2.8) 2  7  avour  co our  i  i  RtL  avour  i  lb  RtL  where PR,L are the chiral projection operators, we get the fermionic part of the Lagrangian density as Cguark  =  ?L + QR^QR  + QRMQL  + q Mq L  R  (2.9)  where the mass matrix M — diag(m , m</, m ) and q = (q ) with i as the flavour index. Thus for massless quarks, the left- and the right-handed fields l  u  s  8  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars become decoupled. As a consequence, the QCD Lagrangian density becomes invariant under the global unitary transformations of the UL(N;) ® U (Nf) group, where Nf = 3 is the number of flavours under consideration, given by R  U , qi ,  QIR -  L R  I i 53 PL^R-J  U , = exp  tR  L R  I  e  (2.10)  U , {N ) L R  f  where A = y/2/Nf \ and 0L,R are the parameters of the transformations. While the kinetic term is invariant under arbitrary transformations of the UL(Nf) ® Uji(Nf) group, the mass term is invariant only when 9L = OR'0  q Mq L  + q Mq  R  R  L  -> q MU[U q L  R  R  + q MU U q R  R  L  (2.11)  L  We thus can introduce the vectorial subgroup Uv(Nf) C U^Nf) 0 U (Nf), with the property U = U , denoted by U {Nf) = U (Nf). At very high energies, if the quark masses can be taken to be approximately equal, the symmetry is an approximate £/y(3) symmetry. The coset group UA{Nf) = U(Nf)/Uv{Nf) is generated by all the transformations with the property UL = Ul Since any unitary transformation can be decomposed into a product of a special unitary transformation (SU(Nf)) and a complex phase (U(l)), we can decompose the direct product of the unitary groups into semi-simple groups giving R  L  U {3) ® U {3)^ L  R  R  v  U {1)  ® U -R(1)  L+R  L  L+R  ® SU (3) ® SU (3) L  R  (2.12)  The independent transformations of flavour of the left- and right-handed fields under the subgroup SUL(3) <g> SU (3) are R  QR  R,L  -» RijQR, qL-> ij<lL, = exp(-iP T e ), L  a  RtL  a  RtL  (i,j)eu,d,s (R,L)eSU(3) ,  {R L)  (2.13) (2.14)  where 2T's are Gell-Mann matrices in flavour space and P L are the chiral projection operators. This is called the chiral symmetry of QCD with the total symmetry group extended to Rt  SU(3)  colour  ® UL+ (1) ® UL- (\) r  R  ® SU{3) ® SU(3) L  R  (2.15) 9  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars The corresponding iV - 1 = 8 chiral currents are given by 2  <R  = ?H7"y9* = ? 7 ^ p g  withVa,* = 0  (2.16)  <L  =  with ^ J £ = 0  (2.17)  M  y ^ L  = q i ^ - ^ q  L  while the conserved charges are  which satisfy commutation relations identical to those of one of the generators of the group SU(3)L ® SU(3)R. However, these charges mix under parity transformations VQiV = Q . The currents conserved under parity are given by the those of the diagonal subgroup SUL+R(3) and the coset group (SU(3) ® SU(3) )/SU R(3). The equivalent symmetry transformations, a vector one and another, an axial vector one are defined by the following R  L  r  L+  q  u  =  exp(-iT°#) ( q  |  d  (2.19)  q  u  = eM-il^X)  I Qd )  (2.20)  with the vector and axial vector currents given by V?  = JtR + J^L = 9f^  K  =  <R-<L  =  withW = 0  ^ Y  q  w i t h  »«  d  A  (2.21)  =°  (' ) 2  22  Thus the symmetry group of QCD in the chiral limit can equivalently be expressed as SU{3) ® C/ (3) ® C^_ (3)'® S£/(3)y ® 5£/(3). The mass term that we had put to zero can be rewritten as m  u  q M q = ( q  u  q  d  A  fl  L+H  colour  q ){ a  0 0  0 m  d  0 0 ][  0 m,  q  d  ]  (2.23)  10  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars with M =  m +m +m u  d  g  i  3  x  3  H  m -m u  d  A  3  m +  H  u  m -2m -^-j= A d  s  8  l/-^4J  Hence, we see that all the three terms in M break the axial symmetry SU(3) A while the vector symmetry SU(3)v remains intact even if the masses are not zero but the same (m = m = m ). With the small explicit breaking due to the (current) quark masses, the axial-vector current is not exactly conserved. This is known as partial conservation of the axial-vector current (PCAC). We notice that, in equation (2.12), the phase multiplication of the group UL+R{X) = UviX) corresponds to an exact symmetry and is independent of the masses of the quarks. The QCD Lagrangian density is invariant under the global transformations u  q  l  q  i  d  -> e x p H f l * ) ? * -> e x p ( - i 7 5 ^ ) 9 *  3  W I T H  <VB = d^q-fq  with d^  A  =  = d^lsQ  0 =0  (2.25) (2.26)  where the first symmetry is called the baryon number conserving UB{1) symmetry and the second one is called the axial symmetry UL-R(1) = UA(1)f/s(l) is conserved both at the classical and quantum levels even if the quarks are not massless while the axial symmetry is exact at the classical level but is broken at the quantum level. This is called the axial anomaly in QCD.  2.2  Nuclear Matter Phase of Q C D  Our expectation that Quantum Chromodynamics (QCD) is the theory of strong interactions with quarks and gluons as its degrees of freedom, is primarily based on the experimental results at high energy scales and/or temperature. The strong coupling constant at these regimes assumes small values rendering perturbative methods applicable for calculating the properties of strongly interacting matter and comparing them with the observations. A l though high-energy probes show ample evidence that hadrons are made of quarks (and gluons) we do not see any quarks as free asymptotic states. This leads to one of the fundamental assumptions about QCD, which has not yet been proved in any rigorous manner, that the quarks are confined within small regions of space and this is termed as the confinement of colour. The absence of any free colour-charges leads us to believe that the low energy 11  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars states of quarks and gluons are in fact colour singlet states of the fundamental representations of the quark fields. At large distances (compared to sub-nuclear lengths) and/or low temperatures, i.e. at low energy scales, strong coupling constant is no longer small compared to unity (a > 1) rendering the perturbative methods of QCD inapplicable. The way around this may be to use numerical simulations using Lattice Gauge Theory or non-perturbative methods. Lattice QCD has its own problem with putting the fermions on the lattice at finite chemical potentials and being extremely time consuming. Non-perturbative methods are notoriously difficult to handle giving only limited results so far. Thus any practical method for describing low energy phases of QCD has to rely on effective theories for QCD, which should be motivated by some of the properties of QCD and those of the low energy states. A few such effective theories have been studied extensively which includes, the Chiral Perturbation Theory (xPT), MIT Bag Model, Nambu-Jona-Lasinio (NJL) model, and so on. s  2.2.1  H a d r o n i c C o m p o s i t i o n of N e u t r o n Stars  The naive picture of neutron stars as made of solely neutrons is an oversimplification as this is not the lowest possible state of energy of dense matter. After the birth of a hot protoneutron star (T « 10 ~ 10 K) in a supernova core collapse, many different reactions occur in the star and it evolves into a cold ground state. The neutrons can undergo /3-decay (as well as inverse /3-decay), to produce protons and electrons (or convert back into neutrons) n <-> p + e~ + P and in the process, a portion of the energy of the particles is carried off by the neutrinos (or anti-neutrinos). The neutrinos can easily diffuse through the surface and leave the star since at these temperatures (T < 1 MeV), the star is transparent to neutrinos. The proton and electron populations in the neutron stars are determined by the chemical equilibrium condition as well as the charge neutrality condition of the star. Since the neutrinos (antineutrinos) escape the star, their chemical potential has to be set to zero to get the chemical equilibrium condition for the nucleons as 11  12  e  MP = M« - Me  (2-27)  At high enough densities the nucleons will have sufficient Fermi momenta and strong interactions between them may produce other higher-mass baryon 12  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars species. The typical reactions for their production include [30]  N + N <-> N + A + K n + n <-> n + ^ + K*  (2.28) (2.29)  where N stands for the nucleons. The kaons, if produced, generally decay into photons and neutrinos unless they undergo phase transition to condense. Typical decay channels include  K° -» 2 K--yfi~  7  (2.30)  +v  (2.31)  p~ + K -y pT + p + v -> 27 + v  (2.32)  +  +  Whether the hadronic species are populated is determined by the condition of chemical equilibrium in terms of the chemical potentials of the species involved in the reactions between them. The threshold for the population of a species in the non-interacting Fermi gas model (at zero temperature), is the mass of the species. The vanishing populations of the photons and the neutrinos imply the chemical equilibrium conditions for the baryon species other than the nucleons (c.f. 2.27) as  po pp+ PA K  K  K  = = = =  0 p -p Mn e  e  PY,± = PnTHe  (2.33) (2.34) (2.35) (2-36) (2-37)  If the electron chemical potential is greater or equal to the mass of the muons p~, the leptonic reaction e~~ <-> p~ + v + produces the muons giving the the equilibrium condition e  AV- = Me  (2-38)  In general, because of the conservation of electric charge and the baryon number conservation, the chemical potential of any species can be expressed as the sum of the chemical potentials associated with the electric charge and baryonic charge and put in the form Pi = bip + QiPQ b  (2.39)  13  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars where is the baryon quantum number of the particle and Qi is the electric charge. This can be expressed in terms of the familiar chemical potentials by noting that fib = \i and \IQ — —fj, . Any theory of nuclear matter within the neutron star thus has to be general enough to describe the baryonic composition subject to the constraints of zero net charge for a given number of baryons. The electric charge neutrality of the neutron matter in the bulk forces the fraction of protons in the star to be much smaller than the neutron fraction. This is seen by the comparison of the relative magnitude of the electrostatic repulsive force to that of the gravitational attractive force that binds the star n  (Z e)e  G{Am)m  net  —^2—  e  <  ^2  ^ ^  Z N E T  36  ~ >P ~ Z  Z E  \  <  W  A  (2.4U)  where Z is the proton number and Z is the electron number. This implies that the net proton number in the star is smaller than A/2 where A is the net baryon number of the star (c.f. 2.27). Thus the neutron star matter tend to be highly iso-spin asymmetric [30]. v  2.2.2  e  Effective F i e l d Theoretical M o d e l of N e u t r o n Star M a t t e r  We take the particles that appear in the low energy phase (i.e. the hadrons) as the degrees of freedom themselves and describe their interactions by the mediation of mesons (such as the scalar, vector and vector-isovector mesons) whose exchange between the hadrons within the neutron stars mimic the nature of the strong or nuclear force. Pioneered by J. D. Walecka [23, 69], this method is valid until the density or energy scale, where the quark degrees of freedom become more important than the hadronic ones. This is similar to a description of ordinary matter in terms of atoms, molecules and electrons instead of neutrons, protons and electrons or quarks, gluons, electrons and the mediators of forces. The method applied is the relativistic mean field method (RMF) in which the equations of motion for the baryons and the meson fields are solved self-consistently in the mean-field approximation. In an iso-spin asymmetric hadronic matter inside a neutron star, the dominant force comes from the exchange of scalar meson (cr), the vector meson (LO) and the vector-isovector meson (p). The scalar meson field reduces the effective mass of the baryons and contributes to the reduction of the energy per nucleon near saturation, while the vector meson is responsible for 14  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars repulsive interaction. With increase in density, the repulsion grows while the attraction saturates until a minimum of the binding energy is reached at the saturation point. However, if we include up to only quadratic terms in the meson fields, it turns out that one has no control over the compressibility or the effective mass of the matter. To get a handle over these, we include self interaction terms, proposed by Boguta and Bodmer and Rafelski [18, 19], having cubic and quartic terms in the scalar field and getting two more parameters at our disposal. This allows us to fit the compressibility and nuclear effective mass results from the theory to the values of those at saturation density of the nuclear matter. The iso-vector meson (p) field is introduced to take into account of the iso-spin asymmetry of neutron star matter. Lagrangian Density of Neutron Star Matter The Lagrangian density in a Walecka-type relativistic field theoretical model with interactions between the baryons being mediated by the exchange of a, LO and p mesons is given by [30, 31]: C  =  C  + Ci  H  = ^ B(x) i7 a - (m - g„B) - 9uBl^ B M  M  ~  \p^-r  +  B  ~ ^9 B^^T.p^ B(x) P  \rnfr.(?-U(v)  + ^2l(i^-m )l  (2.41)  x  i  Here B{x) denotes the Dirac spinor field of the baryons of species B. The sum is over all the states in the baryon octet (p,n,A,S ,S ,E°,E! ,S°) and possibly over the A quartet. The field strength operators of the u and p — (p~,p°, p ) mesons are given by — d^i -d u^ and — d^,p -d p^ respectively. We note the following about the Lagrangian density: +  _  _  +  v  v  v  v  1. The scalar meson field couples to the scalar density BB via the coupling g while the vector meson field w and the iso-vector field p^ to the vector and iso-vector currents Bj^B and B^^TB, via the couplings M  aB  15  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars g^B and g B, respectively. This makes the interaction part of the Lagrangian density d a Lorentz scalar. The form of the Lagrangian is motivated by the fact that at the static limit, the one boson exchange of the cr and LO* mesons lead to the Yukawa type of potential. P  nt  1  2. The p vector mesons couples to the iso-spin breaking current \B^^fB, the form of which is motivated by the fact that at the mean field level, this is the only contributing term for iso-spin current [30]. Here r is the 2 x 2 matrix vector whose components are the Pauli spin matrices. 3. The potential U(u) represents the self-interaction of the scalar field and is taken to be of the form U{o) = -bm {g af + \c{g aY  (2.42)  l  n  a  a  introduced for fitting compressibility and effective mass data. 4. The leptonic fields I = e~ and p" are taken to be non-interacting, since their interactions give small contributions compared to those of their free Fermi gas part. The Field Equations The Euler-Lagrange equations computed from the Lagrangian C of the neutron star matter are i? ~ 9*B</ ~ \gpBTy  1  (a  + ml)a(x)  + (bm gla  2  n  + cgy)  - [m - g„ a)\ B = 0 B  (2.43)  B  =  (2.44)  J2^BB(X)B(X)  B {U + ml)^{x)-d^d u {x) v  u  =  (2.45)  Y,g B(x)rB(x) UB  B (D + m ^ l - W p . l i )  l^2B(x)g rr B(x)(2AQ)  =  I  pB  B where • is the D'Alembertian operator and slash / represents contraction of a four-tensor with the Dirac 7 matrices i.e. = l^A^,. 1  16  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars  2.2.3  The Mean Field Approximation  We consider a system of neutron star matter which is static and uniform in composition, near its ground state. The baryon number is conserved in the strong and weak interactions and as the density gets large, the source terms in the right hand sides of the equations of motion above, become large and we get many quanta of the meson fields present in the ground state. Under these circumstances, the deviations of the fields from their ground state expectation values are negligible and we may replace the meson fields by their mean values at the ground state: (2.47) (2.48) (2.49) For a uniform system at rest, we have the following considerations: 1. Since the matter is assumed to be uniform and isotropic, there is no preferred spatial direction of the fields. Thus the vector and iso-vector fields can have only time or the zeroth component (2.50) (2.51) 2. Furthermore, since the system is at rest, the expectation values of the fields LOQ and p°, in the ground state, must be constant in space and time i.e. space-time independent. As a consequence, the derivative terms of the mean fields in the equations of motion vanish. Furthermore, we note that, in the mean field approximation, the ground state expectation value of the first two components of the iso-spin vector p vanish as they can be written in terms of raising and lowering operators and only the third component contributes (B-fnB)  = (B-friB)  (BI°T B)  ±  3  0  =0  (2.52) (2.53)  17  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars Lagrangian Density and Field Equations in Mean-Field Approximations Under the simplifications of mean-field approximations, the Lagrangian density becomes 1 -m:I J o + tmlpt  'MFA  + ^2 ( ) B  ~ 9uBlov° - gpBhBlopl ~ (m - 9aBO-) B(x)  x  + Y ^ ^ 1 ^  1  Jm»*a - U(a  -  2  B  (2.54)  - mi)l  The field equations satisfied by the mean meson fields and the equation of motion of the Dirac fields in the mean-field approximation are m cr 2  - [bm g {g a)  2  n  a  0  + cg {g„df] +  g {BB)  a  aB  (2.55)  B ^guB(BloB), B my  \ ^2g (B °T B) B pB  7  3  m£<D* = 0  (2.56)  = ^g I (B °B) B  (2.57)  pB 3B  7  if? — duBlou - g hB-yof - (m - g a)\ B{x) = 0 0  pB  z  B  aB  Here, we have used the iso-spin quantum number I defined by  for the baryons  3B  hp — ^~  hn —  1 ^  (2.58)  and similarly for other baryons  (2.59)  In the following, we denote, for simplicity, the mean fields by the fields a , tu° and p\ themselves, without a bar over them. From the Dirac equation (c.f. 2.58), the energy eigenvalues for the baryons (and anti-baryons) are found to be CB(JP) = g^BtOo + gpBPoshB + VP + m*  (2.60)  &B{V) =  (2.61)  2  -9WBU  - g BpmhB + VP + m* 2  0  2  P  2  where the bar denotes the anti-particle energy and iso-spin quantum number and the effective mass of the particle (anti-particle) species are given by m* = rri - goBO B  B  18  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars Determination of the Source Currents The ground state expectation values of the baryon currents in the mean field approximation above i.e. (BB) and (B^B) act as source currents for the meson fields. To evaluate these we note that in the single-particle state, the baryons are characterized by the momentum k and spin and iso-spin projections, which we denote by £ for brevity. Thus the expectation value of any operator T, in the ground state of many particle system has to obtained by multiplying the probability density of that state to be occupied by at a particular energy (and temperature) i.e. by the appropriate statistical factor: <> =  E  F  £  y=B,anti-B  / 7S3 (*mk,cf D(e*(k)  - fly)  F  (2.62)  C  where, generically, the sum extends over the baryons (as well as the antibaryons at finite temperatures) with the appropriate chemical potentials and energy dispersion relations used. The expectation value in the single-particle state characterized by k and £, can be found by calculating the expectation value of the Dirac Hamiltonian density operator in such a state as (B^H B) B  kX  = (0\a B^H Bal \0) kt(  =  B  c  = e (k) B  9UJB^O + 9 Bp03hB + Vk + m* 2  (2.63)  2  P  which is independent of ( and here is the creation operator in the singleparticle state. The Hamiltonian density H for the baryon species B is given by: (2.64) HB = 1O\JZ' + 9U>B</ + 9 BposhB + m* B  P  The energy eigenvalue depends on a number of fields and variables and in general we can take a derivative with respect to any variable a of the above expectation value to get ^(BlH B) B  kX  = (B^B)  kx  =  (B^B)  kx  +  k (k)^B) 0  kx  (2.65)  where the second term vanishes since the state is an eigenstate of the Hamiltonian. The normalization condition can be found by taking the derivative 19  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars with respect to the time-like components of the meson field e.g. LU or p , to 0  (B^B)  0  =1  kA  (2.66)  We consider a derivative with respect to k , using the equations (2.64) and (2.65) as: l  (B B) 7i  = ^e (fc)  kx  (2.67)  B  Hence, the baryonic source current is (B-nB) =  (2J + 1) J~^-^-e (k)f (e (k) B  (  =  2Js  =  B  +  FD  ) /  -  B  J  1  p) B  te (k ,k )f (e (k)-pB) j  k  B  FD  B  0  (2.68)  where the integration is over energy vanishes since the integrand is isotropic in the momentum space i.e. depends only on the magnitude \k\ of the momentum. From the above, we get that the space-like components of the fields CJ and p^ vanishes identically, m  3  u = pi = 0  (2.69)  To find the baryon number density, we use the normalization condition = n -n  n-B-tot  B  =  E/  B  = (B^B)  ^{B B) {fF {e {k)-p )-f {e {k)  +  ]  kA  =  D  J dk(f  B  B  (e (k)  FD  FD  B  - p)  B  F  B  + Ms))  - f D(e (k)  B  p^  B  (2.70)  where we have included the number density of the anti-baryons as well. The derivative of the single-particle expectation value with respect to the mass of the baryon is (BB)  k  C  EE  = om  ™B-9°B° y/k? + (m - g a)  B  B  aB  *B  M = 2  Jfe  +  ( 2 m  ?  1  )  ^2  20  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars which gives the scalar density as n  = (BB) =  sB  fdk * (f (e (k)-p ) J Jk + m* m  FD  n  +  B  B  B  f (e (k)+fx )) FD  B  B  2  B  (2.72) In terms of these densities, the field equations for the mesons become ° "°  = -^\ n(9aNO-) bm  -  2  + c(g af]+y2^§n a  (2.73)  aB  ^ = 0  J2 f B> 9j n  (2-74)  B  Pi = E ^ f ^ n  (2.75)  B  B  where the meson fields represent their values in the mean-field approximation and we have written the net baryon number density n ^ , simply as n . B  2.3  tot  B  Colour Superconductivity of Quark Matter  Colour superconductivity is essentially a Fermi surface phenomena of the quarks, having energies near the Fermi energy. At zero temperature and sufficiently high density, because of the asymptotic freedom, quarks are free to move around and follow the Fermi statistics given by f (k) = e(u. F  quark  - E(k)),  at  T =0  (2.76)  where \i ark = f> is the quark chemical potential and E(k) is the energy of a free quark. All the states up to that with the Fermi momentum k = y/u, — m , are occupied and quarks form a spherical Fermi surface in momentum space. Only the quarks at the Fermi surface are able to interact and exchange momenta. The cost in the free energy, in creating a quark with Fermi momentum, \E (k ) — fj\, is zero. This results in an instability in the presence of an attractive interaction, lowering the free energy and making the Fermi surface unstable. This is the essence of the famous Cooper instability [13, 14]. At temperatures small compared to the Fermi energy, T <C E , most of the quarks still reside near the Fermi surface and a few qU  F  2  2  F  F  F  F  21  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars are thermally excited creating holes inside the Fermi sea. The cost of creating quarks having Fermi momentum, is still small and we expect Copper instability to persist up to a critical temperature T above which there will be no instability. This instability is resolved spontaneously by modifying the dispersion relation by creating a gap in energy of the excitation spectrum. Schematically, the excitation spectrum of a superconducting particle looks like q  Ei(k) = \J{\Jp + m?- pf + AiA (2.77) where Aj is a coefficient that specifies the phase of the material and index i specifies the species of the particles (colour and flavour for the quarks). A is the gap in the energy spectrum and for the special case of Xi = 0, the particle is un-gapped. Thus the gapped particles possess a finite amount of energy even at the Fermi surface, E^kp) = y/X~iA. This makes the free energy cost of creating a pair of (gapped) quarks at the Fermi surface nonzero, i.e. 2%/A^A and removes the instability. The energy can be thought as the binding energy of the Cooper pairs. At quark chemical potential p ]» m, the typical momentum scale of the quarks is p.For p » A.QCD, QCD becomes weakly coupled and quark-quark interaction is dominated by single gluon exchange. The interaction amplitude in this channel is essentially proportional to the Quantum Electrodynamics (QED) amplitude by photon exchange multiplied by the SU(3) group theoretic factor 2  }2  T&T* = - ^ ^ ( 6 ^ - 5 ^ , ) + ^ ^ ( 6 ^ + 6 ^ )  A=l  c  (2.78)  c  where the indices a, b are the colour indices of the incoming quarks while a',b' are those of the outgoing quarks. The first term is antisymmetric in the interchange of colour indices of the incoming quarks (as well as those of the outgoing quarks) and corresponds to the attractive colour anti-triplet channel, while the second symmetric term corresponds to the repulsive sextet channel. The tensorial nature comes from the decomposition of the direct product of two colour triplet representations 3 of the group SU(3) into a direct sum of antisymmetric colour anti-triplet 3 and a symmetric colour sextet 6 representations: C  C  a  S  3 ®3 = 3 ©6 C  C  a  S  (2.79) 22  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars At the baryon densities corresponding to the interior of the compact stars, 71B < 10n , the QCD coupling constant a ~ 1 and weak coupling methods are inapplicable. The quark-quark interaction is still attractive in the colour anti-symmetric channel of single gluon exchange interactions and in the instanton-motivated models [3, 54]. We can understand the form of the interaction by looking at the colour and flavour structures of the condensate of Cooper pairs. In the absence of any condensate and the quark masses taken to be approximately equal, the normal quark matter is invariant under the symmetry group 0  a  SU{3)  C  ® SU(N ) F  ® U(l)  ® U(1)  V  ® SU(2)j  em  B  (2.80)  where SU(3) and U(l) are the gauged groups of colour and electromagnetic interactions, SU(Nf) is the global flavour symmetry group and U(1)B and SU(2)j are the global baryon number conservation and total spin (J = L + S) symmetry groups. In the presence of the condensate of Cooper pairs the symmetry of the ground state is reduced to that of the condensate or the order parameter $ ~ (i/j T i/). For attractive interaction, the order parameter, must be a representation of the colour gauge group 3 and antisymmetric in colour indices. Depending on the Dirac (spin) structure of the condensate, the flavour symmetry is determined by the requirement of Pauli exclusion principle. If the condensate is a spin singlet with J = 0, it will be an element of the antisymmetric representation l j of the spin symmetry group SU{2)j whereas if the condensate is a spin triplet with J = 1, it will be an element of the symmetric representation 3 . Thus the flavour structure must be antisymmetric if J — 0 and symmetric for J = 1. For a spin-0 condensate, antisymmetry in flavour requires that there should be more than one flavour of quarks in the pairs. For Nf = 2, the phase of the colour superconducting matter is called the 2SC phase, with the structure of the condensate: em  C  v  c  i  a  a>  S  2SC: = ((^ )hM) = ^ ° ^ c (2.81) which is an element of the flavour singlet representation l j. Here and abc totally anti-symmetric rank -2 and -3 tensors in the flavour and colour spaces, respectively. The presence of 7 makes it a Lorentz scalar while a condensate similar to that but without 7 , which will be a pseudo-scalar, is disfavoured in QCD because of instanton effects [63]. The symmetry of the 2SC condensate is given by: C  a  h  a  e  a r e  5  5  SU(2)  C  ® SU(2)  V  ® SU(2)j ® U(l) c B+  ® U(l)  em+C  (2.82) 23  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars The order parameter is a 3-vector in the fundamental colour space having any arbitrary orientation. By making a global colour rotation, we can transform the condensate to point in the third colour (blue) direction having the form ({$ )il ' l j) ~ £i?e - Then Cooper pairs in the 2SC phase consists only of the red and green (colours 1 and 2) quarks, while the blue quarks do not participate in pair formation. Thus, in the 2SC phase, the colour gauge group is broken down from SU(3) to SU(2) and the condensates carry colour charges. Due to Anderson-Higgs mechanism [6, 34, 35, 36], five of the eight gluons acquire Meissner masses in the 2SC phase while the three remaining gluons, which are the gauge particles of the reduced colour gauge group, do not interact with the Copper pairs and remain massless. The respective chromomagnetic fields of the five massive gluons are expelled from the bulk of the colour superconducting material leading to colour Meissner effect for those gluons. The remaining gluons give rise to pure gluonodynamics of SU(2) . The 2SC condensate (2.81) is a singlet representation of the global SU(2) ® SU(2)R chiral group in the massless quark limits of the pairing flavours. Thus chiral symmetry is not broken in the ground state of the 2SC phase. For nonzero but equal quark masses of the two participating flavours, the symmetry group is the vectorial subgroup SU(2)yThe 2SC condensate is found to be invariant under the joint rotation of the SU(2) and U(1)B group as well as another joint rotation of the SU(2) and U(l) groups [20, 57]. Thus the colour degrees of freedom are separately "locked" with the electromagnetic and baryon number conservation symmetries. Since both the colour and the electromagnetic symmetries are gauged, the photon is only rotated but remains massless and thus the 2SC phase is no electromagnetic superconductor. The rotated baryon number and charge operators are [63] C  5 t jl  a63  C  C  e  L  C  C  em  l ®diag f  coiour  (0,0,1)  (2.83)  2 1 1 1 2 diag/fa^ourlg. - 3 ) ® l - 1/ ® diag ; (-, - , - - ) c  diag  / 8 c  co  our  ( ^ , 1,-^,-^,0)  (2.84)  corresponding to the new symmetry groups U(1)B — U(1)B+ and U(l) — U(l) + - It is interesting to note that only the (anti-)blue quasi-particles C  em  em  C  24  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars carry non-zero baryon number in the 2SC phase. Because of the the presence of unpaired blue quarks and conservation of (rotated) baryon number conservation symmetry, 2SC phase does not show baryon superfluidity. Noticing the Q charge of the quark quasi-particles we get that the unpaired blue up quark is charged while the blue down quark (also unpaired) is chargeless. Thus the 2SC phase is a Q-conductor with non-zero electrical conductivity. Thus to get electrically neutral 2SC phase, it is necessary to include negatively charged electrons or strange quarks. For energy scales much greater than the strange quark mass, the up, down and the strange quarks may be treated on equal footing and they participate in the pairing equally. The flavour symmetry structure is found from the decomposition of the direct product of two flavour SU(3)v representations: 3  /,v ® 3  =3  } y  ay  0 6,,v  (2.85)  where the indices / and V denote that the representations are the vector representations of the flavour SU(3) group. For a spin-0 condensate, it should be anti-symmetric in flavour being an element 3 y- Thus for Nf — 3, the structure of the 3-flavour colour superconducting phase is given by a  3  ~ £  ew^&j  (2-86)  I,J=I  which is anti-symmetric in both colour and flavour indices of the constituent quarks. The coefficient matrix <3>j is determined by globally minimizing the free energy of the phase and is found to be diagonal <$j = 5 P [4]. The corresponding phase has the order parameter: I(  3  CFL:  ^ = <(Vi )?7 V# c  5  =  E^ * £  i=i  eaW  (' ) 2 8?  Using the property of the Levi-Civita tensors ^ e ^ '  =  6^-6^;  (2.88)  we see that the flavour and colour rotations are "locked" since the Kronecker delta tensors link these two types of indices. Hence the phase is called colourflavour-locked (CFL) phase. The condensate (2.87) breaks the colour gauge 25  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars symmetry and hence by Anderson-Higgs mechanism, all eight gluons get Meissner masses. The chromo-magnetic fields is completely expelled from the interior of the C F L matter and hence it is called a colour superconductor. Since all three flavours and colours of quarks equally participate in the formation of Cooper pairs, CFL phase is the most symmetric of colour superconducting phases. To see the symmetry breaking pattern explicitly, we rewrite the CFL condensate in the chiral basis in terms of the left- and the right-handed fields: <(«7M#> ~ m  a  ) i ^ Y ) L )  +m n i ^ Y ) * )  (- ) 2  89  where the first term breaks the chiral SU(3)L symmetry and the second term breaks chiral SU(3)R symmetry. The colour gauge symmetry is broken by both the condensates resulting in the reduced symmetries of the diagonal subgroups SU(3) L and SU(3) R. This is because the condensates remain invariant under simultaneous rotation in the flavour space and equal but opposite rotation in the colour space. In the presence of both the condensates SU(3) ® SU(3) = SU{3) R is left unbroken. Note that we get two separate condensates of the left and the right handed fields and no mixing between the opposite chirality fields [63]. The symmetry group of the CFL condensate is thus given by: c+  C+L  c+  C+R  c+l+  CFL:  SU(3)  ® SU(2)j ® U(l)  C+V  em+C  ®Z  2  (2.90)  Thus, at asymptotically large densities, since the chemical potential p^$> m , chiral symmetry is not broken explicitly by the presence of mass terms or by chiral condensates, but spontaneously broken by CFL condensates. The discrete symmetry Z indicates that the condensate is invariant under change of sign of the both quark fields. The baryon number conservation symmetry U{1)B is broken in the CFL phase and hence it exhibits superfluidity. Hence, because of the Goldstone theorem, a massless Nambu-Goldstone boson (<fr) appears in the low-energy effective theory of C F L matter. The electromagnetic gauge symmetry is "locked" with the colour SU(3) symmetry leaving the diagonal subgroup unbroken. Thus the photon is only rotated and remains massless and the CFL matter does not become an electromagnetic superconductor. The residual q  2  C  26  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars symmetry is a gauge symmetry of the rotated charge operator Q given by Q  = Q-T --±=T  (2.91)  =  (2.92)  3  8  diag (0,1,1,-1,0,0,-1,0,0) /8c  Because of the gapped dispersion relations of all the quarks, C F L phase is a Q-insulator with small electrical conductivity due to thermally excited electrons and positrons. The U(1)A axial symmetry is broken in the CFL phase similar to the vacuum case by the presence of instantons. However, at high density , instanton effects are negligible. Formation of Cooper pairs require the Fermi momenta of the pairing particles (quarks) to be equal or nearly equal. For the case of non-equal Fermi momenta, the phenomena is referred as the stressed pair-formation. Requirement of electrical neutrality and non-zero strange quark mass creates a mismatch between the Fermi momenta of different flavours of quarks. For large enough mismatch between the Fermi momenta, different quark flavours will not be able to create Cooper pairs between them. In this case, it is likely that quarks of the same flavour create Cooper pairs in the spin-1 channel [10, 11, 12, 43, 49, 52, 53, 59, 63]. For a single flavour of quarks N = 1, the condensate is an element of the spin symmetric representation 3 J. The general structure of the condensate (i.e. gap) matrix can be expressed as: F  3i  3  A  a b  = iA  ^Acift  cos6 + *_ sin0}  (2.93)  7  where the vectors k = k/k and 7x = 7* — k (y.k) are the longitudinal and transverse vectors. The 3 x 3 matrix A with elements A i, where c and 1 are indices in the colour and ordinary 3-spaces, respectively, have different structures giving rise to different phases. The angular parameter 0 and the elements of the matrix i.e. A i, in a particular structure, are determined by minimizing the free energy of the quark matter in the relevant phase. A few of the possible phases have the following structure of the matrices [63] C  C  / 1 0 0 ^(A-phase)  = _L  ^(poiar)  =  J0  0  0  I,  [ 0 0 0 ) ,  A  {CSh)  A  iplanar)  = - = = 0 1 0 ) ,  v/3  0 0 1  =  (2.94)  ( 0 1 0 ) (2.95)  v  2  \  0  0  0  27  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars Thus the condensates A  a b  ~e  1  A*, can be expressed as  A-Phase CSL: Polar Planar  8 (5 iz  cr  +  (2.96) (2.97) (2.98) (2.99)  i5 )A cg  51A 65 A iz  cb  (S^F* + 5 ^ )  A  In the CSL phase, the colour gauge group SU(3) and the spin rotation group SU(2)j are simultaneously broken down to the diagonal subgroup SU(2) +j corresponding to the joint rotations in the colour and spin space [60] and hence a "locking" between the colour and spin degrees of freedom. Hence the phase is called CSL or colour-spin-locked phase. This "locking" can be thought of as associating a direction in ordinary space to a linear combination of directions in the colour space. For example, we may think that the red and green quarks pair in the z-direction, green and blue quarks pair in the x-direction, etc. Thus, CSL phase is essentially isotropic in spite of choosing particular directions in ordinary space for pairing between specific colours. The phase is the colour superconducting equivalent of the B-phase of superfluid helium-3. The residual symmetry of the condensate is C  c  CSL:  SU{2)  (2.100)  ® U{l) ® Z  C+J  v  2  Both the electromagnetic U(\) and the baryon number conservation U(1)B symmetries are broken and hence the CSL phase is a superfluid as well as an electromagnetic superconductor. The polar phase (sometimes called the ISC phase), is similar to the 2SC phase in that the full colour symmetry group SU(3) is broken down to the reduced colour group SU(2) . The residual symmetry of the condensate is EM  c  C  Polar:  SU(2) ® U(l)j ® U(l) C  v  ® U(l)  B+9  ® U(l)  em+C  (2.101)  which shows that the photon gets rotated and remains massless. Thus the polar phase, with only one quark flavour present, is not an electromagnetic superconductor. The spin symmetry group is broken to U(l)j revealing that the condensate is invariant only under rotations about a fixed space direction i.e. has axial symmetry. However, quark matter having more than one flavour forming spin-1 Cooper pairs can be shown to be electromagnetic superconductor [7]. The residual symmetries of the planar and the A-phase 28  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars are similarly found to be: SU(2) ® U{1)  A-phase: Planar:  C  J+C  (2.102)  ® U(l),em+c  (2.103)  U(l) ®U{l) J+c  em+c  The gap parameter in the one flavour superconductors is found to be of the order of 1 ~ 10 keV, which is much smaller than the typical gaps in the 2SC or C F L phase (~ 100 MeV). Thus the presence of such a small gap will have small effect on the thermodynamic and other properties of the quark matter.  2.4  The 2SC Phase  For intermediate baryon densities i.e. corresponding to the quark chemical potential in the regime p, < 550MeV we may consider the strange quarks to be too heavy to be populated or the strange quark Fermi momenta too far away from that of the up and down quarks that no u-s or d-s pairing is be possible. In this case, the quark matter will be composed of only the up (u) and the down (d) quarks forming Cooper pairs and possibly unpaired strange quarks. It may be possible that the strange quarks pair up with themselves in one of the single flavour Cooper pairing channels but we do not consider that possibility in this thesis. Our analysis below concerns only the up and down quarks forming Cooper pairs. For simplicity we consider the masses of the up and down quarks to be equal ~ 5MeV (c.f. current masses m _ ~ 2MeV, md_ ~ 5MeV [26]). The system can then be described effectively by an effective model for QCD with SU(2) flavour symmetry. u  2.4.1  current  current  The 2SC Lagrangian Density  We consider a Lagrangian density of the form (2.104)  C sc = qiil^dp + 7°M - ™o)<? + Ant 2  where rho = TUQI is the current quark mass matrix and q = qta are the Dirac spinors with i as the flavour index of the flavour doublet and a as the colour index of the colour triplet. The quark chemical potential matrix p, is a diagonal 6 x 6 matrix in the flavour and colour space and can be expressed as p — diag® (p , fx , n , /j, , fj, , p ) (2.105) f  c  UR  ua  UB  dR  da  dB  29  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars The interaction Lagrangian density C is motivated from the single gluon exchange interaction obtained by replacing the gluon propagator with a fourfermion coupling constant i n t  Cint  =  -lG(qTlq)(qTZq)  (2.106)  where T£ = 7 A° , with A 's as the standard Gell-Mann matrices (a = 1... 8). This gives the interaction the quantum numbers of single gluon exchange. We may take the current-current interaction term in the general form a  M  jr = G}2 a,{qT™q ){q T™ ) T  (2.107)  T  int  q  v where rj labels the different di-quark pairing channels and T^ is the matrix that specifies the form of the di-quark pairing with 9  f™ = ( 7 r ^ 0  r 7 o  )  (2.108)  t  We can perform different Fierz transformations to get the terms that appear in the 2SC pairing ansatz (c.f. 2.81) and add them to get all the 2SC interaction terms. This can be done as there are no cross-terms between the different channels in the binding energy [5]. For the 2SC pairing we get Cint  The matrix  = \G{qP scq ){q P2Scq) T  (2.109)  = C^r X  (2.110)  T  2  has the form  P2SC  P2SC  2  A  where the Pauli matrix r acts in the flavour space. We rewrite in terms of the components, 2  Cint  = lG(t C (T UX Uq )(q^ T j  l5  a  2  A  0  7o(C ) 7o(T ) (A^) ,^) t  75  2  ij  7  (2.111)  and use the identity for the Gell-Mann matrices  X  A  X  A  (-^-)ap(-2~)yS  1  1  — - g ( £ a / 3 V ~~ ^ a i V y ) (  +  g(^a/3<^ + £a<5<Vy)  (2.112)  where the first term gives rise to attractive interaction in the colour antisymmetric anti-triplet channel while the second term results into repulsive 30  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars interaction in the colour-symmetric sextet channel. Since we are interested in only the attractive interaction between the quarks resulting in Cooper pair formation, we keep only the first term. Thus the interaction Lagrangian density becomes £»„  -  = G(q C^ T qp)(ci jo-flC^oT2q )(8 f36 s r  t  a  5  2  s  a  y  5 5 ) a5  0J  = G(q T Cq^)(q C T q )e e = G(iqee q ){ifee q) T  al5  b  2  ba0  l5  2  6  blS  (2.113)  c  l5  bl5  where we have made use of the property of the totally anti-symmetric LeviCivita tensors (e ) = e in the colour space and used (e) = e as the totally anti-symmetric tensors in the flavour space. q and q° are the charge conjugate spinors and C = Z7 7° is the charge-conjugation matrix. We use the results q = Cq , q = q C and C = i 7 7 ° Thus the Lagrangian density with the scalar di-quarks becomes b  afd  a0b  lk  lk  c  2  c  T  £ = q(i^d  c  T  2  + 70A - rh )q + G[(iq^ee q)(iqee q )] b  p  0  b  lb  (2.114)  c  l5  The theory represented by the Lagrangian (2.114) is not renormalizable since the canonical dimension of the coupling constant G is -2. Thus the integration over the momenta in the theory is done up to a cut-off A which we determine in Chapter 3. We introduce a complex scalar field A whose expectation value will be A sci the energy gap of the pairing in 2SC channel. Following a bosonization procedure by Hubbard-Stratonovich transformation [16, 25, 40, 45, 56, 66], the Lagrangian density becomes 6  2  I = q(ird„  + oA - ™ )q ~ ^ A ' ^ f e e S ? ) - \ A ( i q e e q ) b  7  0  b  c  l5  -  (2.115) where the di-quark fields A and A* are the colour anti-triplet and the isoscalar singlets under the chiral SU(2) (8) SU(2) group. We can take the di-quark to condense only in the third colour direction breaking the colour symmetry spontaneously: 6  b  L  A =A =0 , 1  2  R  A = A = A*V 0 b  3  (2.116)  where we have taken the gap to be real. Introducing the Nambu-Gorkov spinors  31  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars where * = ( QuR QuG QdR q~dG )  (2.118)  the bosonized Lagrangian density may be conveniently rewritten as  C  + V)*  = ¥ \i(T b  2SC  ( B )  7  - ^ £ - \^  9  )  Ci^  9  (2.119)  )  where we have defined ^  =  (M + ^ ) - 2 M 8 + ^ T  C  =  ( V  ^ ~ ° ^( 9) 75T cr A  {  m  +7 o  = O-2/X -^T M  8  (2.120)  3  -75r CT A \ iff -m 7oM(r ) /  r  (2121)  b  2  6  2  3  2  2  0  S  where M(r ) S  =  +p a  Mo + ^-T  3c  3  3  +p  (2.122)  8c  Mo = M+^r  (- ) 2  123  6 and the Pauli matrices r^'s act in theflavourspace and CTJ'S act in the redgreen colour space.  2.4.2  T h e C h e m i c a l P o t e n t i a l s of t h e Q u a r k s  In general, the chemical potential of a species of quark or any other particle is the sum of the chemical potential associated with the conserved quantities of the particle. We write the quark chemical potential matrix as P = P - QePe  +S p 3  3c  (2.124)  +Szp.g.c  where p = (l/3)/is is the quark chemical potential (one-third of the baryon chemical potential), Q = Q l is the generator of the U(1)Q symmetry of the electric charge, 53 and S are related to the generators of the U(l) and U(l)s subgroups of the colour SU(3) symmetry group and are defined in terms of the Gell-Mann matrices as [55] e  e  8  3  C  S = 2A = diag(l,-l,0), 3  3  S = 2V3\ = diag(l, 1,-2) 8  S  (2.125)  Hence we see, for the same flavour of quarks, the difference of the chemical potentials between the first two coloured quarks is induced by p while that 3c  32  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars between the first two colours and third colour (i.e. the blue quark) is induced by the chemical potential p . For the same colour of quarks, the difference in the chemical potential between the up (u) and down (d) quarks is induced by the chemical potential corresponding to the electric charge, —p . Thus explicitly, the chemical potentials of different species of quarks are given by 8c  e  Mi = Mu = P ~ (2/3)p + P8c + M3 H  e  M2 = MUG ~ M (2/3)Me + M8c - M3 p = = - (2/3K - 2 —  3  M u b  (2.126)  M8c  M4 = Md = M + (l/3)Me + M8c + M3 Ms = Md = M+ (V3)Me + M8c - M3 M4 = Md = M + (l/ )Me - 2^ H  G  3  s  8c  We also define the mean chemical potential p for the pairing of quarks (UR, do) and (UCCIR) and the difference of chemical potentials between the up (u) and down (d) quarks as _ 6p  e  2.4.3  =  p  UR  +p  dG  p  =  +p  UG  = -5p = -5p  Q  =  ^ _  ~^  =  dR  =^  UR  (  1  /  6  K  ^  +  ~^  ( 2 1 2 ? )  = y  G  (2-128)  D i s p e r s i o n R e l a t i o n a n d P a r t i t i o n F u n c t i o n of the 2 S C Phase  Expressing the bosonized Lagrangian density (2.121) in the momentum space, using the Nambu-Gorkov spinors, we get at finite temperature [5, 55] i C  <?  A* A  _1  b  6  = 2 * - T * - M -  <2  129)  where the inverse full propagator is S'  1  = ( ^  V  ~ ° m  +  70M(r ) 9  7 r cr A ' 5  2  -75T 0- A 2  \  b  2  <  2  1 3 Q  ,  / - m - 7oM(r ) /  i  2  0  S  with p( ) defined in equation (2.122). The dispersion relations of the quasi-particles can be found from the zero's of the inverse propagator S~ (iu ,p) i.e looking at the solutions of the equation D e t [ 5 (iw p)] =0 (2.131) rg  l  n  -1  n>  33  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars Here p = iui with to — (2n + l)nT are the Matsubara frequencies arising from Fourier transform of the quark fields: 0  n  n  n=+oo  *to)( ) x  =  * \^  53exp(-iKr-p.f))*(pl  (2.132)  71= — OO p Using the identity for determinant of a matrix D e t  ( C D)  (-  = Det  +  BC  (2.133)  ~ )  BDB  lA  we get,  Det(5 ) = Det(A l+(75T a )(/-mo-7VX75r2CT )-V- o + 7 V ) _1  2  TO  2  2  2  (2.134) with Ji = /j,/ = Mo + ^ 3 + M3c03 + Msc and setting A = A . This gives the dispersion relation in the 2SC phase as b  rg)  =  (2.135)  ^A + e ±5v 2  2  where i = ± and Si = £,(p) = y V + o ± ( ^ + ^ + A 8 c ) and = ^Q±/X . To get the partition function of the 2SC phase, we have to integrate over the fermionic fields. We also make the mean-field approximation by neglecting the fluctuations of the bosonic filed A and set its value equal to the expectation or mean value A s c = A . The result for the partition function following [55] is, i  m  3 C  6  2  1/2  Z = Det  S  {iu ,P)  l  n  e x p (  -r4G  }  (2.136)  The determinantal operation Det in the partition function is carried over the Dirac, colour, flavour and momentum frequency space. An alternative derivation may be found in the references [38, 39].  34  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars  2.5  The Gapless 2SC Phase  The. two-flavour colour superconductor (2SC) has the general quasi-particle dispersion relation given by (2.135) where the mean chemical potential for the quarks p (c.f. 2.127) depends on the electron chemical potential p which is equal to the difference of the chemical potentials of the pairing up (u) and down (d) quarks by virtue of /3-equilibrium. The di-quark gap (A) and other parameters of the theory assume values which extremize the thermodynamic potential at any quark chemical potential and temperature (/i,T). Depending on the relative magnitude of the di-quark condensate A and the mismatch chemical potential p , the dispersion relation (2.135) shows interesting features and the 2SC quark matter can have unconventional form of superconductivity, where gapless modes of excitation around the effective Fermi surface for certain values of the momenta of the quasi-particle exist. The phase corresponding to these gapless modes is called the gapless 2-flavour colour-superconducting (g2SC) phase [62]. To illustrate further, consider the quasi-particle dispersion relation (2.135) recast in the following way e  e  E± /6v ±±  = Wiv/mf  + 1 ± p/m)\m/5v)  + (A/<Ji/) J  2  2  ±1  (2.137)  where E = V P + rn 2  _  _  ^  ~ =  Pu  R  2  + Pdg _ Pug +  2 / i l  ~  ^ ±p 2  3c  Pd  R  2 = 5HQ ±  /_i  3c  (2.138) The value of the colour chemical potential /x is found to to be small (c.f. Fig. 3.14 ) compared to the quark chemical potential p,. We thus may assume that, — 5p = 5p ~ 8u and p, > 0 and 8p > 0. It is obvious that for A/\5v\ > 1, there are no real values of the momentum p for which E^ will have a zero value and hence there are no gapless excitations of the quasiparticles. This will correspond to the conventional 2SC phase where all the four quasi-particle spectrums are gapped. This is not always the case for A/\5v\ < 1, where gapless modes exist around the effective Fermi surface at 3c  e  ±±  35  Chapter 2. Phases of Quantum Chromodynamics within Compact Stars momenta [46] p± = J(ji± y/{5vf~^~&) - m (2.139) If fl/m is sufficiently large, both the momenta p± will be real and there will be two effective Fermi surfaces having gapless excitations. The corresponding phase is called the gSC(2) phase and the typical excitation spectrum is illustrated in the figure (3.1). For a smaller values of p,/m, only p will be real (p_ being imaginary) and there will be only one effective Fermi surface with gapless mode. The corresponding phase is called gSC(l) phase. Finally, for further smaller values of p,/m , both p and p_ will be imaginary and there will be no gapless excitations in the quasi-particle spectrum. The corresponding phase will be like a regular gapped 2SC superconducting phase (as in the case of A/|5i/| > 1). 2  2  +  +  36  Chapter 3 Thermodynamics of Q C D Phases In this chapter we study the thermodynamic properties of the QCD phases relevant to compact stars and outline their calculational procedures for numerical evaluation. The general thermodynamic relations used here are discussed in detail in the appendix B.  3.1  Thermodynamics of Normal Quark Matter  In this section, we consider the deconfined phase of quark matter without any superconductivity present, called normal quark matter. Such a phase may occur within the cores of neutron stars where charge and colour neutrality and weak equilibrium effects may change the Fermi momenta of quarks such that Cooper pairing is not possible or if possible, its effects are too small to be accounted for. Since quarks have spin quantum number of one-half, their thermodynamic properties are governed by the Fermi-Dirac statistics. The number density in phase space for any species of particles, is in general, given by  This provides a complete description of the system of particles in phase space. This number density can be represented in terms of the dimensionless distribution function in phase space f(k, T) as dT?  d xd k 3  3  =lf(k,T) h' 3  (3.2)  where h = volume of a unit cell in phase space, 7 is the statistical weight or degeneracy factor of the particles, / is the probability distribution function 3  37  Chapter  3.  Thermodynamics  of QCD Phases  of the particles to be in a state, k is the momentum of the particle and T is the absolute temperature. For fermions, / is the Fermi-Dirac distribution function / =m  = znr^TT^  (3-3)  where p is the chemical potential of a fermion and e is its energy. The baryon number density, energy density, pressure and entropy density of a completely degenerate system of quarks are given by [30] d A; =  47Tfcdfc  3  n  d xd A; 3  B  3  2  /quark  /anti -quark  —  k dk\f(k ,p )-f(k ,-p )\  (3.4)  2  f  7/  roo  / E(k)ti 2TT2Jo  f  f  f  mf{k n )+f{k ,-p,+i f)  f  f  f  B  (3.5) A;  4  d k-B3  3 d xd fc 3  3  dA; f(kf,pf)+f(k ,-p ) f  £(fc)  f  - £  fc dfcf(kf,p )+f(kf,-p ) J  5fc  f  f  (3-6) s =  v  (3.7)  \dTJn  where the sum runs over different flavours of the quarks and the degeneracy for a flavour / of the quarks is given by 7/ = 2 x 3 ; .The factor | in the formula for the baryon number density comes from the fractional baryon number for a quark. We also have used E(k) = (m + k )? and the natural system of units in which we take c = h = 1. In the above, we have included B, the bag constant, or energy density difference between the true (confining) vacuum and the perturbative (nonconfining) vacuum: spin  co  2  B — Chadron  ^perturbative — (-^bag  our  2  ^o)/^bag  (3.8)  We assume that in the Bag model [1], the hadrons are represented by bubbles (bags) of perturbative vacuum surrounded by the non-perturbative (confin38  Chapter 3. Thermodynamics of QCD Phases ing) vacuum. Since hadrons have finite size i.e. Vbag < oo> we require that the bag constant B > 0 (otherwise the system could lower its energy by increasing the size of the bag without bound). Thus it costs an energy BV^ to create a bubble in the normal vacuum in which the quarks can move freely. The pressure in the above represents the pressure relative to the hadronic (confining) vacuum as the bag constant acts as a negative pressure, stabilizing the bag. We use values of Bag constant in the range ag  135 < B /  MeV < 200  1/4  (3.9)  for the different phases considered later. For free quarks, the value of B ^ = 154.5 MeV corresponds to the case where strange quark matter is absolutely stable, but barely so, compared to the hadronic matter in the ground state (confined state) with the strange quark mass m = 150.0 MeV [30]. At zero temperature, since the Fermi-Dirac distribution function becomes a Heaviside step function 1  s  /(«,r) =  e ( e  _ ^ M  +  1  ^8(M-6)  (3.10)  we get closed-form analytic expressions for the thermodynamic quantities as: P =  -* +E^[  W  * 0.}-|m}) /  +  | }.n(a±*£)](3.11) m  ?= ^E^k/W-r?)-H'"(^r)]  <-> 32 1  /  s EE 0  (3.14)  where kf — \J(n, f — rnj) is the Fermi-momentum for the quarks of flavour / . We note that, there are no anti-quarks present at zero temperature as they are not thermally excited. In the chiral limit (m k —> 0), another simplification occurs, giving analytic expressions for the thermodynamic quantities for both the quarks 2  quar  39  Chapter 3. Thermodynamics of QCD Phases as well as the anti-quarks as 1 7/  n  (3.15)  3 2TT  2  B  15  24TT L 2  e  =  s =  - B  (3.16)  +B  (3.17) (3.18)  15  In the grand canonical ensemble, the thermodynamic potential Q is defined as the negative of the pressure and the other thermodynamic quantities are derived from the potential as  Q = 2>({ },T) = -p M/  BE  k dk s  V  6 7 T  2  J dk  f(k,fl)+f(k, -fJL) + B  (3.19)  0  (3.20)  n  f  dfif  (3.21)  n  B  /—u,d,a  s =  —  (;yr)W}  Yl\n({nf},T)  [n(jif,T) +  + n n + sT f  (3.22)  f  iM nf  + sT + B  (3.23) f f We can express the thermodynamic potential in an alternate form which is useful for numerical calculations and for comparison with the results of the  40  Chapter 3. Thermodynamics of QCD Phases two-flavour colour superconducting (2SC) phase, by noting that  -  2L  ~  WJo  = =  ^  r * Tk  T  - JL  1 dv  e-t -»fV E  ( [  l  n  roo  -JLL / 27T J 0  r  T  + l~wJo  T  k  ^ v  ^ - o - / ° ° ^ l n , d / c ) r  _  A; Tln l + e - ( - ^ ) / dA; I i 2  B  (3.24)  r  where we have defined v(k) = e~^ ~^ + 1 and E(k) = \/k + m . Thus adding the contribution of the anti-quarks as well, we get, for a particular flavour / of quarks and anti-quarks, the thermodynamic potential as E  Q = F  3.2  v  J  /T  2  2  ^ { T l n [ l + e-^-^/ ]+Tln[l + -( r  e  £ +  ^/ ]} r  (3.25)  Thermodynamics of Hadronic Phase  For the hadronic phase, we consider the mean-field Lagrangian density of hadronic matter CMFA (see equation 2.54). The chemical potentials p of the baryons are introduced as the energy cost of increasing the number of baryons of that particular species (B — n,p, E, E*, K , Aetc.) by one unit. The partition function in the finite temperature field theory is given by B  ±fi  Z = Z Zi = J[dB}[dB}[dl][dl} exp ( J dr jd?x\c A 0  H  MF  +  ^2^BN  B  (3.26) where r = it is the Euclidean time and B — ( / C B T ) is the inverse temperature. In the mean field Lagrangian density (2.54), the hadronic and the leptonic parts are quadratic in the fields. Hence, the hadronic and the leptonic fields B, B, I, I respectively can be integrated out in the partition function. For the hadronic degrees of freedom, the partition function is found (using -1  41  Chapter 3. Thermodynamics of QCD Phases 2.54) to be U(a)  lnZ  H  +V^(2J  B  1)  +  y^[ln(l + -«^-^) C  + In ( l +  (3.27)  e~^ °- - ~ * - ^) E  nti B  >1  anti B  where V is the volume of the space. Here we have introduced the effective masses and the effective chemical potentials of the baryon and anti-baryon species by m  =  B  m  -  B  (3.28)  g a aB  (3.29)  E* = E, anti—B B  #  PB - 9WBV° -  PB P*B  g plh pB  B  =  Pb +  = =  Ms ~ 9u,BU° 9 BplhB -p - q p + 5w.su/ + g pl{-hB)  <?BM<J - 9UBU° ~ 9 BplhB  (3.30)  P  P  (3.31)  3  b  B  q  pB  Note that the third component of the iso-spin quantum number for the antibaryons is opposite to that of the baryons (I = —h ) and all other additive quantum numbers change sign for the anti-particles. Also note that the effective mass enters into the partition function only in the quadratic form. Thus, in calculating the thermodynamic quantities, we need to consider the absolute value of the effective mass only. The meson field equations can be obtained by extremization of the partition function with respect to the field variables to yield the same equations as found in chapter 2 (see Eqs. 2.75). Recast in a form using the effective chemical potentials and effective masses, the scalar densities and the baryon number densities are given by 3B  n  sB  —  2J + 2?r  1  \ B\ m  B  E*  2  B  exp (J3(E - p )) + 1 B  B  B  +exp  1  - p )) + 1 B  (3.32) n  B  =  2J  +  B  2TT  2  2J  B  n  I/a,  exp ((3(E - p )) + 1 B  + fe  B  (3.33)  B  I  exp(P(E%-p )) R  +l  (3.34) 42  Chapter 3. Thermodynamics of QCD Phases Using the partition function Z, the grand canonical thermodynamic potential Q and hence the other thermodynamic quantities, are found as fi  =  -Tk ^  = -p  l  B  1 , ,  1 m~„o" U(o) 2 k dk + m^ L e ^ ^ - ^ ) + 1 +  1  m  1 2J + 1 3 2TT J 2  B  ^  + i  2  1 2 Jj + 1 /" /• fc dfc •r 12J1 1 1 j] + , 3 2TT y y ^ + m L e ^ - « > + 1 e^ <+«) + 4  2  2  2  = \m y  1,1  + U(o) + \mlul  + \mlpf i l P(E* -»h)  B  B  e  J  B  + 1  1 e  S B  (3-35)  £  B,B 2  =  (  £B+ P B  -^  si = ^+pi-^)f  3.3  1  4  B  n s  /3(£,- ) _)_ 1  )r  e  -(5r) ^ {  =  -(w)  + i  1 0(Ei+m)  +  w  =  + e^S-Mfl) +  (3 }  (3.37)  i  '  38)  ( 3 M  -  3 9 )  Thermodynamics of Two-Flavour Colour Superconductors  In the two-flavour colour superconducting (2SC) phase of quark matter, the system is relativistic, so the particle number and the energy density of the material are not constant. Rather, the temperature and the quark chemical potential of the material remain the same locally. The thermodynamic description of such a system is again given in terms of the grand canonical thermodynamic potential, for the 2SC phase fi2Sc(T, p, V) which is related 43  Chapter 3. Thermodynamics of QCD Phases to the partition function by n c(T i,V) 2S  = -k T\n{Z )  }f  B  (3.40)  2SC  The pressure, energy density, baryon number density, entropy density etc. of the system can be evaluated using the above grand potential in a similar manner as in the case of hadronic matter.  3.3.1  The Thermodynamic Potential in 2SC Phase  We start with the partition function of the 2SC phase, derived in chapter two (see 2.136) in the NJL-model description as Det  Z sc — 2  S  1/2  (iui ,p)  1  (3.41)  n  e x p (  -r4G  }  where po — iu and ui = (2n + l)7rT are the Matsubara frequencies. The determinantal operation Det in the partition function, is carried over the Dirac, colour, flavour and momentum frequency spaces. Using the identity Det(A) = exp(Tr(lnA)) and the identification of the summation over discrete momenta with the integration over the momentum space, n  n  J  4<  3^  (3.42)  (2n}  we get the quark thermodynamic potential as V =  -Tki  In Z-isc  n=+oo  V «  1  3  I  T  A2  r E / ^ M k ' ^ ' O + S;  (343)  Since, the zeros of the determinant of the inverse propagator, i.e. T>et(S~ (ioj ,j: are given by the dispersion relation pi = E (p) =*> u + E (p) = 0, using the fact that the trace of a matrix is the sum of the eigenvalues of the matrix, we have 1  n  2  Trln(^S-V ,p))=2j>( n  2  rp2  2  (3.44) 44  Chapter 3. Thermodynamics of QCD Phases where the factor 2 arises because of the doubling of the degrees of freedom introduced by the Nambu-Gorkov formalism and j denotes the different forms of the dispersion relation Ej = E^ , etc (see.2.135). Using the result for the summation over the Matsubara frequencies from the appendix (A) ±±  (3.45) the grand thermodynamic potential for the quarks in the 2SC phase is found to be fW-quark  ~ j ^ 3  =  +^A ~  2  +  e\ -  [\J^  (8p +  i +  +  2  +  e\ -  (5p  -  p) 3  2 / ^ ^ r ( l n [ l + e x ( - ^ ) ] + ln[l + exp(- % ) ] P  K  i  + ln[l +exp(A quark  2  3  p ) + \JA 3  + E + (5p - ^3)  + P ) + \J^  £  + exp — (E _ ) A 0-i  -)] + In [1  T  2  (3.46)  T  A + £? 2  AG  V ^ + W + O ^ + Ms)  + M [ 1+ e  ^ _^±2+(^z^ T (  ^A + ln [1 + exp(-  )]  )]  + et - (5p + ^ ) )] T v A + ^ - Ms) + ln [1 + exp()1) T /  2  3  2  2  (3.47)  where we write the quark part of the thermodynamic potential fi 5c-quark simply as fi rk- Introduce the function u>(E, T) having the property 2  qua  u(E,T) = Tln(l + exp(-E/T))  = -E + Tin (1 + exp (E/T))  (3.48)  45  Chapter 3. Thermodynamics of QCD Phases Hence the thermodynamic potential can be rewritten as 'quark  3.3.2  R e g u l a r i z a t i o n of T h e r m o d y n a m i c P o t e n t i a l  Regularization of the thermodynamic potential depends on the choice of the zero of the potential. We choose the zero of the potential such that Q s c ( A = 0) gives the free energy of the system of non-interacting electrons and quarks 2  Q SC(M> A,/J.Q, ^3, Ms, T) 2  = Q (A) quark  - Q (A quark  -  0) + Q e fre  + P> (3.50)  This cancels the last term in (3.49) allowing us to use the equivalent thermodynamic potential for the quarks for convenience of numerical calculation, as quark  - ^1 ^rt  The bag constant B is introduced in (3.50) with the free part for comparison with the hadronic phase. Hence, for a non-zero value of the gap parameter A , we can compare the the thermodynamic potential or pressure of the free normal quark phase or of the full 2 S C phase, with the hadronic phase, using the same formula (3.50) by noting that ^free-quark = ^free = ^2Sc(M> A =  0, /J.Q, ^3, M8,  T)  (3.52)  which is true by construction. 46  Chapter 3. Thermodynamics of QCD Phases Free Thermodynamic Potential Note that, in the NJL-model, the gluonic degrees of freedom have been frozen out (i.e. there are no gluons in the model) and hence the free part represents only the non-interacting electrons, muons and quarks. Using the result for the thermodynamic potential for the free quark matter (see 3.25), we can express the free thermodynamic potential as ^free  ^free—quark  =  =  _2T J I n  [1 + - ^ - ^ ' ] + In [l + - ^ ' ^ ] } +  T  T  e  E  _ 2 T J { In [1 + - ( ^ ) A ] + In [1 + "(^+^)/r] j e  _  2T^J-0^  e  In [1 + e~^-^ ] T  +ln [l + e~^ ^ ] +  T  } (3.53)  where the sum runs over all the three colours (r, g, b) of the up (u), down (d) and possibly strange (s) quarks.  3.3.3  G a p E q u a t i o n for t h e 2 S C P h a s e  In the grand canonical ensemble for the 2SC phase, the thermodynamic potential is a function of the superconducting gap A which is an order parameter of the system. As A —> 0, we expect the system to be in the normal quark matter phase. For a colour superconducting phase, the gap parameter A should assume a unique value rather than being a free parameter of the theory assuming any arbitrary value. Thermodynamics requires that the physical pressure should be the negative of the thermodynamic potential, at the extremizing value of such order parameters like A : p =-n(fx ,fiQ,H ,fi ,T;A) B  3  8  (3.54)  such that  @ u  ='  <> SM  This is called the gap equation for the 2SC phase. In the field theoretic context, the gap equation is a self-consistency requirement arising from defining the gap parameter A, in terms of the mean value of the di-quark fields, which in turn depend on the gap [63]. 47  Chapter 3. Thermodynamics of QCD Phases The explicit form of the gap equation for the 2SC phase, using the thermodynamic potential from equation (3.51) is 1 -  [2 + f(E )  AG  i  ++  +  f(E\  = 0  f(-Et )-f(-Ei_)} +  (3.56)  where, f(x) = l/[exp(x/T) + 1] is the Fermi-Dirac distribution function. Alternatively, using the form of the quark thermodynamic potential Sl .rk (see 3.47), we get qua  1  ^/|f,E^{ -/(^)-/(^) 2  /(£i )-/(£!_)}  = 0  +  (3.57)  which is a non-linear equation in the gap parameter and has trivial as well as non-trivial solutions for A . In the latter case, the corresponding ground state is a colour superconductor. The superconducting gap depends implicitly on the temperature as well as on the baryon chemical potential p = 3p. At some critical temperature, we expect a phase transition from the superconducting phase to the normal quark matter phase as A sc —» 0. The dependence of the gap parameter on temperature and the baryon chemical potential is discussed in the results section of this chapter. B  2  3.3.4  Electric and Colour Charge Neutrality in 2SC Phase  For bulk matter to be stable, it should be neutral with respect to electromagnetic as well as with net colour charges. Otherwise, large electric or colour-electric fields would be set up creating a long-range effect and making the energy non-extensive [48]. For matter within a neutron star, total electric charge of the star must be zero, otherwise the much stronger electric force will act against the gravitational attractions and the star will not be 48  Chapter 3. Thermodynamics of QCD Phases stationary. For electromagnetic gauge symmetry, charge neutrality requires zero charge whereas, as there are more than two types of colour charges, bulk matter should be at least in a colour singlet state. However, if the colour charges themselves are zero, the matter will obviously be colour neutral. At the thermodynamic limit, it is sufficient to impose the zero individual colour charges as the free energy cost to get it from a colour singlet state has been shown to be zero [51]. Thus we seek the colour charge densities with respect to the two global symmetries ( for the subgroups C/(l) and U(l) of SU(3)c) i.e. n and n to be zero, besides the zero electric charge density (i.e. n = 0). The electric charge and colour charge densities, which are the derivatives of the grand canonical thermodynamic potential with respect to the respective chemical potentials must thus be zero: 3  3  s  8  Q  n  Q  n  3  = =  dQ,2SC  = 0  (3.58)  0O.2SC  - 0  (3.59)  = 0  (3.60)  dp  3  n  Q  =  dps  In full QCD colour charge neutrality is a consequence of the dynamics of the gauge fields, whose zeroth components act as the chemical potentials. Since, in the NJL-model, there are no gluons nor photons present, we have to introduce the chemical potentials for the electric charge and colour charges "by hand" and demand the corresponding charge densities to be zero as extra constraints [48].  3.4  Mixed Phase of Colour Superconducting Quark Matter and Normal Matter  The matter inside a neutron star generally has baryonic charge (i.e. total baryon number is non-zero) as well as electric charge. The requirement of stability of the star puts a constraint on the total electric charge of the neutron star that it should be very close to zero, as we found in chapter 2 (see eq. 2.40). However, the local charge neutrality of the neutron star matter (or quark-matter within neutron star) is too restrictive a requirement. This 49  Chapter 3. Thermodynamics of QCD Phases is beacuse, this state might not be the most energetically favourable state of matter. If it is possible to have more than one phase of matter within a compact star, leading to so called "hybrid stars", then all the components may not necessarily be charge neutral. In that case we have a mixed phase of more than one kind of matter of opposite charge densities and the ground state of the system might not be locally charge neutral and only globally so. Examples of this occur in the familiar case of an atom, in which the ground state has positively charged nucleus at the centre and negatively charged electron clouds around it. Thus if it is energetically favourable to have charges separated, the system will rearrange itself to be in such a state. Theoretically, we may consider the mixed phase of different colour-charged matter such that the net global colour charge is zero, but due to the nature of the strong interactions, it will involve greater forces (compared to the electromagnetic force) to separate colour charges over large distances which will cost large amount of energy, making a heterogeneous phase in colour charges unfavourable. The colour Debye screening length ADebye-coiour — 11Pq at strong coupling is expected to be short and comparable to the inter-particle distance [55]. Under these conditions, colour neutrality will essentially be a local constraint.  3.4.1  M i x e d Phase of 2 S C Phase a n d N o r m a l Q u a r k Matter  We consider the possible mixed phase of two-flavour colour superconducting matter (2SC phase) and normal quark matter comprising of free up and down quarks. Both of these phases will be colour neutral and separately electrically charged. Thus we consider the thermodynamic potential of the 2SC phase that will be a function of the baryon chemical potential ps, electric charge chemical potential PQ as well as the temperature but evaluated at the extremizing values of the colour chemical potentials and the gap parameter (fac-CL,  M8c-CL, ACL)'^2SC-colourless =  Q{pB,  A C L , MQJ  fac-CL,  M 8 c - C L , T)  (3.61)  This is to be compared with the free-quark thermodynamic potential at the same value of the baryon chemical potential p and charge chemical potential PQ, respectively, but again at extremizing values of the colour chemical potentials at which the normal quark matter will be colour neutral. One obvious choice is to set the colour chemical potentials p and p,& to zero, but B  3c  c  50  Chapter 3. Thermodynamics of QCD Phases even if we do not set those to zero, and demand that the normal quark matter be colour neutral, the solution found for the colour chemical potentials at A = 0, are numerically very close to zero: ^free-colourless = ^ Sc(f-B,  A = 0, U.Q, ^Sc-fq, f-8c-fq,T)  2  where  fac-fq ~ 0  and  pzc-fq  (see Figs.  ~ 0  3.12, 3.13, 3.14)  and  (3.62) /J,Q = —fi . e  Gibbs Criteria for Phase Equilibrium The thermodynamic conditions for the mixed phase to occur are given by the Gibbs criteria: 1.  Thermal equilibrium - T sc  2.  Mechanical equilibrium - p sc  3.  Chemical equilibrium - PB-2SC  = TNQ  2  2  = PNQ  and  = PB-NQ  and  J1Q- SC 2  =  PQ-NQ  where the bar denotes the value of the charge chemical potential (^Q) at which the charge densities of the two phases are such that the net charge of the material is zero. Note that, we do not require the colour chemical potentials of the two phases to be equal as the colour charge densities are locally zero. The colour charges are not felt over macroscopic distances (comparable to separation of the two phases) and should not affect the stability and dynamics of the mixed phase. In terms of the thermodynamic potential, the Gibbs criteria can be expressed as Q sc 2  {f-B, A C L ,p.Q,T)  = a {p , 2SC  B  A = 0,fl ,T') Q  (3.63)  at the respective values of the colour chemical potentials which make the phases colour-less. The value of fiQ is uniquely determined from the above. However, the Gibbs criteria do not incorporate the constraint of global charge neutrality which comes from the astrophysical consideration of electric charge neutrality of the star. It is easy to see that, as long as the charge densities of the two phases are opposite, it is always possible to find a mixed phase of the superconducting and normal quark matter with appropriate volume  51  Chapter 3. Thermodynamics of QCD Phases fractions since the condition for global charge neutrality can be written as Xn sc 2  +  (1-X)"JVQ  C  ^  ™ = NQ  (3.64)  d^2Sc{jJ-B,A L,PQ,T)  n  n  0  =  =  _ ^ c ( ^ . A dp  =  „  (3fi6)  0, ,T) w  p  _  6 6 )  Q  where 0 < x < 1> is the volume fraction of the 2SC phase. Even if there is a solution for the mixed phase, it might not be the most energetically favourable phase. For this to be the most stable state, the pressure has to be the larger or the thermodynamic potential (which is negative in value since pressure is positive) has to be smaller than the colour and electric charge neutral phases. Thus, to find the most favourable phase at any particular baryon chemical potential and temperature, we have to evaluate all charge neutral phases and possible mixed phases and compare the respective pressures. In our analysis, we have ignored the effect of surface tension between the two different phases in the mixed-phase solution since we assume that it will be small [55].  3.4.2  T h e r m o d y n a m i c Q u a n t i t i e s i n the M i x e d Phase  The pressure in the mixed phase, according to the Gibbs criteria, has to be equal to the common pressure of the component phases while the other thermodynamic quantities are found by averaging over the volume occupied by the phases. The volume-weighted average of the energy density, baryon number density, charge density and entropy density are: () e  =  (1 -X2Sc)eNQ(VB,fiQ,T')  + X2Sce2Sc{PB,P'Q,T')  (3.67)  (n )  = (l-X2Sc)K (pB^Q,T')  +  X2Scn (p ,p ,T')  (3.68)  (n )  = (l-X2Sc)n (pB,t ,T')  + 2Scn (p ,p ,T')  (3.69)  + X2SCS2SC{^B^Q,T')  (3.70)  B  Q  Q  JQ  Q  Q  (s) = {^-X2SC)SNQ(PB,PQ,T')  2 sc B  B  Q  sc  X  Q  B  Q  We note the following: 1. For each volume fraction X2SC, as the baryon chemical potential p varies, so also the charge chemical potential PQ must vary to satisfy the B  52  Chapter 3. Thermodynamics of QCD Phases charge neutrality condition (UQ) — 0. We can write this dependence as PQ =  AQ(MB,  X2sc, T')  (3.71)  If for some value of the baryon chemical potential /J, , there exists a mixed phase, then can get / 2 Qfora given X2SC and T . However, it is not guaranteed that there will be real solution of the above relation for p,Q for each value of HB,X2SC or T'. Also note that because of the relation (3.71), the pressure p varies continuously even if the volume fraction remains the same. B  2. Conversely, for each baryon chemical potential, the volume fraction and the pressure varies continuously throughout the mixed phase as a function of charge chemical potential: X2SC  =  p =  X2Sc(tJ'B,t Q T  = PQ,T')  p(fj, ,p, ,T') B  Q  (3.72) (3.73)  where p represents the pressure value in the mixed phase. 3. Hence we get an important observation: Because of the two independent chemical potentials, and the global charge neutrality condition, the phase transition from pure to the mixed phase is not a constant pressure transition [30].  3.5  Computation and Numerical Evaluation  In this section we discuss the various computational issues and numerical methods applied to evaluate the physical quantities for the different phases of QCD, covered earlier in this chapter.  3.5.1  E v a l u a t i o n of H a d r o n i c T h e r m o d y n a m i c Quantities i n Terms of F e r m i - D i r a c F u n c t i o n s  In the effective field theoretical description of the hadronic phase, the thermodynamic quantities can be put in the form of free-particle thermodynamic quantities with the use of effective mass and effective chemical potentials. It  53  Chapter 3. Thermodynamics of QCD Phases is convenient to express the thermodynamic quantities of free particles in terms of the generalized Fermi-Dirac functions defined by f°°xx (l + \(3x) ' dx exp (x — 77) + 1 kk  1 2  (3.74)  where kT  (3.75)  B  P  mc' (7 — l)mc x = E /k T = kT kin  (7 — 1)  2  B  (3.76)  P  B  _M_  (3.77)  kT B  In terms of the Fermi-Dirac functions, the baryon number density, pressure and energy density of degenerate free fermions and anti-fermions can be expressed in the following form (derived in the appendix C): / [f (k,u.*)-f (k,u:*)]k dk Jo m\ ^/2f¥ Fi(r),P) + PF (f),(3)-Fi(fj,p) 2  n  FD  B  Ideg  2^2  FD  3  3  = = ^  - PF (fj,(3) (3.78) 3  l^fk^^[f (k,^-fF (k,fI*)}k dk 2  FD  m 4  l  2 3 / 2  P  5 / 2  D  [ l %P) +  (V,P)  F  (3.79)  +F {f ,(3)+ -(3F {f,,P) l  l  =  2? I  l  *( )[fMk,V*)-  E  k  l  f (k,p*)}k dk 2  FD  +Fx (f,, P) + 2PF, (fj, (3) + P F, (fj, 13)(3.80) 2  where r7 = ^ - ^ = 77-^andr7 = ^ - g § r = 7 7 - ± where n and fj are the reduced particle and anti-particle chemical potentials, respectively. We treat the particle and anti-particles in a similar way using the same formulae but with the appropriate chemical potentials and quantum numbers. 54  Chapter 3. Thermodynamics of QCD Phases  3.5.2  Quasi-particle Dispersion Relations and E v a l u a t i o n of Q u a r k Integrals  The general temperature-dependent term in the thermodynamic potential has the function u(E,T) = k T\n[l + e~ ] which is integrated over the momentum p in the range (0, A), where E = E(p) is the dispersion relation which is now more complicated than the simple case of the hadronic dispersion relation or even the simpler normal quark matter dispersion relation. We have (see 2.135), E/kBT  B  E*sc = Ei  ±±  = yjA* +  s ±(5p±p ) 2  ±  3  = • \JA + ( V P + m ± p) ± 5u 2  2  2  2  (3.81)  where 5u = 5p±p and the chemical potentials 5p, p and the superconducting gap A have been defined before (see 2.127, 2.128). The first ± on the gap A refers to the sign in front of the potential bv while the second refers to the ± in the definition of 5u — 5p± p . The form of the dispersion relation depends on the positive or negative values of ±p = ±(p + Ps) where in general p > 0 at high enough quark chemical potential (see 2.127). We note that, 3  3  0  1. For ±p > 0, E is a monotonically increasing function of the momentum p and the quasi-particle energy will have a minimum at p = 0 given by E  = ^(m±p)  2  min  + A ± (Sp ± p ) 2  3  (3.82)  which can be shifted up and down by changing the electron chemical potential ( since 5p — ^ = — ^). 2. For ±.p < 0 however, E, has a local maximum at p = 0 and a global minimum at p j„ = \Jp? — m and ultimately increases monotonically beyond p in- The minimum of the quasi-particle energy is 2  m  m  JS  roin  = |A|±<Jj/  (3.83)  If the minimum is less than or equal to zero, we have a gapless superconductor. There are quasi-particle modes (with finite momentum) but zero energy (no gap). This is important in the low temperature limit because unless the superconductor is gapless, the integrals over u(E, T) will vanish. 55  Chapter 3. Thermodynamics of QCD Phases 400  Quasi-particle excitations in Gapped and Gapless 2 S C  300  E or | E | (MeV)  200  100 -f  0  100  200  300  400 500 p (MeV)  600  700  800  Figure 3.1: The quasi-particle dispersion relations in the two-flavour colour superconducting phase (2SC/g2SC): E = E^ = [(E ± p) + (A) ] / ± 5p. The left plot is the excitation spectrum in the gapped 2SC phase in which there is a finite gap along the energy direction. The second plot is the magnitude of the excitation energy in the g2SC phase as a function of momentum p. At some momenta, gapless modes exist. 2  ±  2  1  2  free  Because of the modified form of the dispersion relation and for the finite upper limit of the momentum-integration, we have to numerically integrate over to(E,T). For convenience of integration we approximate u(E,T) in the form below (ks = 1): u{E,T)  = Tln[l + e=  £/T  (3.84)  ] .  (3.85)  -E + Tln[l + e ] E/T  ' -E B{-E) -E + T\n(l + e l ) ' -E + Te ' T l n [ l + e- / ] B T  ~  E T  £  T  for T = 0 for E/T » 1 for E/T <^ - 1 otherwise for T ^ 0  (3.86)  where 6(x) is the Heaviside theta function. For numerical evaluation of the quark integrals, we note the followings: 1. It is convenient to get the ranges of the momentum p in which the function u(E(p),T) is non-zero analytically, and integrate over that range only. Otherwise, a numerical integrator routine, using for example the Romberg integration procedure, will try to subdivide the range many times to see whether there is a contribution of the integrand 56  Chapter 3. Thermodynamics of QCD Phases p u(E(p),T) and at most subdivisions, will find no contribution. This will result in a wastage of computational cycles. As an alternative, we use the routine gaus squad, using Gaussian Quadrature method for numerical integration to avoid many subdivisions. However, it is still convenient to integrate only in the range where we expect the integrand to be non-zero. 2  2. However, since u(E(p),T) is a smoothly varying function, going to zero with a tail, over most part of which , the contribution from the integrand may be very small, we first find the ranges of p where the zero temperature limit of the distribution function u(E(p),T) > —E8(—E), has non-zero contribution. 3. Since we are looking for the thermal effect for temperature in the range ksT ~ 0 — 50 MeV, while the quark chemical potential is in the range M = n = Ms/3 ~ 300 — 500 MeV, we typically expect a contribution due to thermal excitation of the order O(^Z-) ~ 1 0 in the tail part of uj(E(p), T) . We thus add the tail part, due to thermal excitation to the either side of the zeros of —E9(—E). In doing so, we again use the Gaussian Quadrature method. Thus, we necessarily cover the whole range of the integration (0, A), once each time, but give importance to the range where we expect the most contribution. q  _1  4. For the purpose of comparison and to have an idea of how accurate the above two integration methods are, we first integrate the whole range of momentum (0, A) in a direct step using quadrature. Secondly, we divide it into 100 equal subdivisions and integrate in each part using Gaussian Quadrature. Lastly, we adopt the scheme of selecting of integration range on the expectation of most contribution and compare the results in the three different integration procedures. As a representative result to compare, we plot the gap parameter as a function of temperature at the baryon chemical potential of MB = 1050 MeV for temperature in the range 0—120 MeV. We find that at low temperature, the direct one-step integration gives less contribution than the 100 subdivision integration while the judicious choice of integration range agrees well with the 100-subdivision method. This may be because, since the Gaussian Quadrature method makes use of fixed sampling points, there may be many small ranges of the momentum , where the integrand will 57  r  Chapter 3. Thermodynamics of QCD Phases Gap Parameter A vs Temperature using different integration schemes 50  Integration scheme 1 Integration scheme 2 Integration scheme 3  45 40  \f"  35 30  > <  25 20 15 10  10  30  20  40  50  60  T (MeV)  Figure 3.2: Comparison of different integration schemes based on subdivision of the integration range of the momentum p for baryon chemical potential PB = 1050.0 MeV.In the first method, we integrate over ranges of p where the integrand is non-zero at T = 0 MeV and add the other parts on either side within the integration range.The second scheme consists of one-step integration over the whole range of the momentum.In the third scheme, we divide the range of momentum into 100 equal subdivisions and integrate  have contribution but which fall entirely between the sampling points of the quadrature. At high temperature, however, all three methods agree well. This justifies our choice of importance sampling, as it is applicable for the whole range of temperature without the use of too many computational cycles (see figure 3.2). 5. For p = po+psc < 0 and E — \&\ — \5v\ < 0, there is no contribution from the integrand at T = 0. However, at finite temperature, we may still have the distribution function having nonzero value in some range of the momentum p. We thus integrate in the whole range of the momentum and since we expect contribution only at high temperature, we adopt one-step Gaussian Quadrature integration which is sufficient max  58  Chapter 3. Thermodynamics of QCD Phases for our numerical purposes.  3.5.3  G a u s s i a n Q u a d r a t u r e M e t h o d for E v a l u a t i n g F e r m i o n Integrals  For evaluating thermodynamic potentials for the hadronic or the quark phases, we encounter integrals of the Fermi-Dirac distribution function or the function uj(E(p),T). In section 3.5.1, we saw how the thermodynamic quantities can be expressed in terms of the Fermi-Dirac functions. The Fermi-Dirac functions are evaluated numerically using the method developed by Josep Aparicio in the reference [8] and implemented in C following the FORTRAN code, from the web page of F. X . Timmes and J.D. Maldonado [29]. For evaluating the quark integrals, we use an 80-point Gauss-Legendre quadrature method to integrate between the finite lower and upper momentum limits. The abscissae and weights used were evaluated with 100 decimal digit arithmetic [29].  3.6  Results  In this section we discuss the results for the thermodynamic properties of the two flavour colour superconducting matter, normal quark matter, hadronic matter and a mixed phase of normal and colour superconducting quark matter.  3.6.1  T h e r m o d y n a m i c P o t e n t i a l for the P u r e 2 S C a n d the M i x e d Phase w i t h N o r m a l Q u a r k M a t t e r  The charge- and colour-neutral thermodynamic potential for the 2SC phase as a function of the gap-parameter must have a minimum at some value of the gap parameter for the phase to be thermodynamically stable. Since the expression for the thermodynamic potential is even in the gap A , and since the charge-neutral normal quark matter corresponds to the case of A —> 0, the thermodynamic potential must have an extremum at A = 0. For a colour superconducting phase, we must have another extremum of the potential along the A direction. From the figure (3.3) we indeed see that the thermodynamic potential in the pure 2SC phase has a minimum at the  59  Chapter 3. Thermodynamics of QCD Phases 2SC and Mixed Thermodynamic Potential \ j  2SC at T=0 - 8 9 _ Mixed at T=0 2SC at T=10 - 9 0 Mixed at T=10 2SC at T=15 Mixed at T=15 -91 /  -92  >  — •*• —  -93  \\  /  -94  /  /  /  /  /  j  / i i / t i l I I '  JL  / I  //  /  /  /-  -95 -96  ! ~ ~ \ ) 20  -97  y y : i 40  A  i  60 (MeV)  i 80  100  Figure 3.3: Grand thermodynamic potential for two-flavour colour superconductor in pure (solid line) and mixed phase (dashed line) with normal quark matter and no strange quarks present as a function of the gap parameter A . The figure corresponds to a baryon chemical potential of 1050.0 MeV and three temperatures as shown, in units of MeV. The mixed phase has two degenerate minima at a lower value than the minimum for the pure phase when it is favoured. As the temperature increases, the minima for the mixed phase are less pronounced and at some higher temperature, no mixed phase is possible. For T=0 MeV we get the same graphs as in reference [55]. The horizontal lines are drawn for comparison of the degenerate minima in the mixed phase. extremizing value of the gap A . At A = 0, the potential has a local extremum corresponding to the normal phase with zero electric- and colour-charges. The value of the gap parameter for which the potential has a minimum and the minimum value itself depend on the temperature. For increasing temperature, the superconducting gap in general decreases and the pressure p = —Q.(p , A,pQ,p3 ,pg ,T) increases as expected for a thermodynamic system in chemical equilibrium. At some critical temperature, we expect the gap parameter to vanish completely, resulting in a phase transition to normal B  C  c  60  Chapter 3. Thermodynamics of QCD Phases quark matter (see section 3.6.2). Colourless Thermodynamic Potential  300 250 200 150 100 50 0 -50 -100 -150 -200  Figure 3.4: Grand thermodynamic potential Q surface for two-flavour coloursuperconductor with no net colour charges present as a function of gap parameter A and charge chemical potential at baryon chemical potential of 1050.0 MeV and temperature T = 0 MeV. A = 0 line corresponds to the normal quark matter phase. The surface has a minimum with respect to the A and a maximum with respect to /J,Q. The negative of the slopes in the fiQ direction gives the charge densities. If at some particular U,Q, the normal and 2SC phases have opposite (but not necessarily equal) slopes and same value of fi, then we have a mixed phase. In the mixed phase, we span the 2-D space of the colour-neutral thermodynamic potential ^l iouriess as a function of the gap parameter and the charge chemical potential. The surface has a local minimum along the A direction at the global maximum value in the (XQ direction. This corresponds to the pure gapped or gapless 2SC phase. For the mixed phase to occur, there must be degenerate minima of Q. iouriess along the gap direction at A j d , having the same value of the thermodynamic potential as at A = 0 but also necessarily having opposite electric charge densities UQ = The oppoco  co  m  Xe  61  Chapter 3. Thermodynamics of QCD Phases site signs of the slopes with respect to the PQ axis put the respective volume fractions in the range (xlsc >XNQ ) € [0, !]• ]  C)  u  Q  (MeV)  Figure 3.5: The contour plot of the colourless thermodynamic potential for the values between [-95, —85] MeV for the same values of the parameters as in plot (3.4). We see that at some particular charge chemical potential ft,Q, the normal quark matter phase and the 2SC phase have the same value of the thermodynamic potential but opposite slopes ( as seen from the contour lines). This corresponds to a mixed phase since the thermodynamic potential at this phase is more negative than the saddle point where the contours have a broad region of equal levels.  3.6.2  S u p e r c o n d u c t i n g G a p for the 2 S C Phase a n d M i x e d Phase of 2 S C a n d N o r m a l Q u a r k M a t t e r  We calculate the superconducting gap A as a function of temperature, for different baryon chemical potentials. For smaller values of the baryon chemical potential, the gap parameter is seen to increase first, attain a maximum value and then decrease with temperature T. For higher baryon chemical potential, the maximum value is seen to occur at the zero temperature and we 62  Chapter 3. Thermodynamics of QCD Phases get a monotonic decrease of A with respect to temperature. At some particular temperature, called the critical temperature T , the gap sharply drops to zero value resulting in transition to normal (i.e. non-colour superconducting) quark matter. c  Temperature Dependence of Gap Parameter A at different Baryon Chemical Potential  0  20  40  60  T  80  100  120  (MeV)  Figure 3.6: Temperature Dependence of the superconducting gap parameter for the 2SC phase with and without free strange quarks at baryon chemical potential p of 1050.0, 1075.0 and 1100.0 MeV's. B  The plot for the variation of A with respect to the baryon chemical potential shows some interesting features: 1. We see an onset of colour superconductivity for the pure 2SC phase, at some non-zero baryon chemical potential at zero temperature, at around p = 967.0 MeV for A = 1500.0 MeV. Note that, before this threshold, the phase is that of free (non-colour-superconducting) up and down quarks as these are the degrees of freedom of our theory. The threshold for p exists because a certain amount of baryon number density is required for the quark matter to attain energy density high enough to excite the pseudo-particles over the energy gap to undergo transition to colour superconductor at T = 0. B  B  In terms of the topology of the colour-neutral thermodynamic potential surface (see Figs. 3.3, 3.4), the second extremum along the gap direction does not occur or is too shallow enough, for the system to be in 63  Chapter 3. Thermodynamics of QCD Phases the colour-supercondicting phase if the baryon chemical potential ps lies below a certain value. Before this threshold, the thermodynamic potential Q has the minimum value at A = 0. 2. Note that, with the increase of baryon chemical potential, since we have a phase change from normal quark matter (A = 0) to pure 2SC phase (A > 0), and since the gap parameter changes continuously with respect to p , the gapless 2SC colour superconducting phase occurs for low values of the baryon chemical potential (see Fig. 3.8). B  3. Although A increases linearly beyond p ~ 1040.0 MeV, there is a change of slope in the graph at around p ~ 1407.0 MeV. At this value, the superconducting gap has attained a high enough value that the condition for gapped phase to occur is satisfied A < \PQ/2.0 + p \. Thus we have a phase transition from gapless to gapped 2SC phase. This is illustrated in the plot of \PQ/2 ± p \ vs. A for the gapped or gapless 2SC phase (see figure 3.8). This transition is also seen very markedly in the plots for the charge chemical potential and colour-8 chemical potentials. 0  B  B  3  3  0  4. It is interesting to note that the threshold value for the baryon chemical potential, ^B-ths for the onset of colour superconductivity depends significantly on the cut-off parameter used. As the cut-off parameter is increased, the threshold baryon chemical potential p ~ths at temperature T = 0 increases towards a limiting value of p -ths ^ 970 MeV. The threshold for the onset of colour superconductivity on the cut-off parameter decreases as the cut-off A increases. B  B  In contrast, the plot for the variation of gap parameter of the coloursuperconducting component in the mixed phase A d (figure 3.9), is seen to vary almost linearly with the baryon chemical potential starting from p « 900.0 MeV for the cut-off A = 1500.0 MeV. Before this value of p , the conditions for mixed phase to occur are not satisfied. mixe  B  3.6.3  B  Dependence of the C o u p l i n g C o n s t a n t o n the M o m e n t u m Cut-off  In the thermodynamic quantities, the momentum integration is done up to the cut-off A since the theory is non-renormalizable. The coupling constant 64  Chapter 3. Thermodynamics of QCD Phases Gap Parameter A in pure 2SC phase at T = 0 vs baryon chemical potential p.  B  180  A with A = 1500 A with A = 1300 A with A = 1100 A with A = 650 0 0  0  0  0 800  900  1000  MeV MeV MeV MeV  1100  1200  1400  1300  1500  Me ( M e V )  Figure 3.7: Dependence of superconducting gap at temperature T = 0 MeV in the gapless/gapped 2SC phase on the baryon chemical potential for different values of the cut-off parameter G is fixed at a particular value of the cut-off A by the requirement that the colour superconducting gap A sc = 100 MeV for the quark chemical potential p — 350 MeV and other chemical potentials equal to zero (i.e. HQ = ^ = p = 0 MeV) at zero temperature, following the reference [55]. The rationale behind this is that, first principle calculations using weak coupling QCD at asymptotically high baryon number density extrapolated to moderate densities, indicate that the gap parameter will assume a value of the order ~ 100 MeV [3, 54]. The superconducting gap A sc found from the solution of the gap equation 2  3  s  IS  2  (3.87) and the corresponding extremized thermodynamic potential OA, is being evaluated at the values of the chemical potential mentioned above. The value of the coupling is then changed such that the gap parameter becomes equal to the reference value of 100.0 MeV. This gives a relationship between the coupling constant G and the cut-off A. The figure (3.10) shows the variation of the coupling constant on the 65  Chapter 3. Thermodynamics of QCD Phases |/2Q/2  0  + / i | vs A in gapless or gapped pure 2SC phase at T = 0. 3  I  1  1  20  40  60  —  —I  80 A  0  1  1  1  100  120  140  1—  160  180  (MeV)  Figure 3.8: Plot of \PQ/2 ± p \ vs. A for gapless or gapped 2SC phase. At some value of the superconducting gap A ~ 148 MeV, the coloursuperconducting quark-matter changes phase from gapless to gapped 2SC. The cut-off parameter value is A = 1500.0 MeV. The irregularities in the plot beyond the phase-change point, arises from errors in p (see Fig. 3.14) which are small compared to the value of PQ . 3  0  0  3  cut-off for different reference quark chemical potentials. We see that, for higher values of the cut-off, the value of the coupling becomes less dependent on different reference quark chemical potentials but monotonically decreases with the cut-off. Since, we are shooting for the minimizing gap parameter A = 100.0 MeV, and since the coupling constant appears only through the term A /AG , the contribution from the integrals within the thermodynamic potential should essentially be increasing linearly with the gap parameter (see Eq. 3.51): 2  fW„(A  +  *) ~  -4xA/ ||g- =i== {2 A  ?  f  +  7( i ) ...} £ +  +  (A + x) /4G ~ x /AG + x A + A /4G » x A + C 2  2  2  and monotonically increases with the cut-off A. That is why the minimum of the thermodynamic potential in the direction of the gap parameter, occurs near the same place (A ~ 100.0 MeV). This particular procedure should 66  Chapter 3. Thermodynamics of QCD Phases Gap Parameter in the mixed phase A ,„ix at T = 0 vs Baryon Chemical Potential 0  180 ,  800  900  1000  1100  1200 Ms (MeV)  1300  1400  1500  1600  Figure 3.9: Dependence of the superconducting gap in the 2SC superconducting component of the mixed phase on the baryon chemical potential at T = 0 MeV and A = 1500.0 MeV. make the physical results independent of the cut-off and can be thought of as a renormalization procedure. From the form of the plot, we expect that at high enough cut-off, the coupling will assume an asymptotic small value. The decrease of the coupling constant is consistent with the asymptotic freedom of QCD, which is what we expect since the NJL model for the quarks should mimic QCD in certain regimes. Also note that, since the whole procedure is done at the zero values of the electric- and colour-charge chemical potentials, changing any chemical potential of any particle will not change the dependence of the coupling constant on the cut-off.  3.6.4  Electric and Colour Charge Chemical P o t e n t i a l s for 2 S C a n d M i x e d P h a s e w i t h Normal Quark Matter  The electric charge chemical potential fj,Q d the colour-8 charge chemical potential figc for the pure two-flavour colour-superconductor (gapless/gapped 2SC) phase are both negative and seen to increase in magnitude becoming a n  67  Chapter 3. Thermodynamics of QCD Phases Dependence of coupling constant on cutoff A  1000 1200 Cutoff A (MeV)  1800  Figure 3.10: Variation of the coupling constant G on the cut-off A as a result of fixing the minimizing gap parameter at 100.0 MeV at zero chemical potentials and temperature at different reference baryon chemical potentials more negative as the gap A increases. For zero gap i.e. for charge neutral normal quark matter, PQ ^ 0 while p —> 0. The colour-3 charge chemical potential p & 0 for all A (c.f. figure 3.14). These imply that there is a finite energy cost involved in creating an electron for both the normal quark matter and pure 2SC phase, while the energy cost in converting a red to green (and vice-versa) is vanishing, but not for changing a red/green into a blue, for both the up or down quarks (see equation 2.127). For the mixed phase, for each baryon chemical potential and temperature, the mixed phase, if it occurs, is possible only at one electric charge chemical potential PQ. The chemical potential ps-mixed is however different for different A and seen to decrease from zero, becoming more negative with the increase of the gap. Since at zero temperature, the colour superconducting phase persists up to the highest baryon chemical potential considered (ps — 1500.0 MeV), we see a decrease of electric- and colour-8 charge chemical potentials with the increase of ps, since A is seen to increase, too (see figure 3.7). The phase-change point from gapless to gapped phase is seen on both the figures (3.12,3.13) as the point where the slopes of the chemical potential plots Sc  3c  0  68  Chapter 3. Thermodynamics of QCD Phases Chemical potential for the colour-8 charge  ""fj  20  40 CO A (MeV)  80  Electric charge chemical potential  100  0  20  40 60 A (MeV)  80  100  Figure 3.11: Variation of electric- and colour-charge chemical potentials for the pure 2SC and mixed phase with respect to the superconducting gap parameter A . For the pure phase, the thermodynamic potential has an extremum at some A = A which is the physically relevant gap-value. change.  3.6.5  T h e r m o d y n a m i c P r o p e r t i e s of the N e u t r a l Hadronic Phase  In the electric charge neutral hadronic phase, evaluated in the mean field approximation, as discussed in chapter 2, we calculate the pressure, energy density, baryon number density and the fractional populations of the hadronic species as functions of the baryon chemical potential HB and temperature T. The fundamental degrees of freedom in the effective theory applied are the hadrons and leptons and not the quarks and the theory does not show any phase change to quark matter. Since, we expect at high baryon number density or equivalently at high baryon chemical potential, a tran69  Chapter 3. Thermodynamics of QCD Phases Chemical potentials MQ at T = 0 vs baryon chemical potential M B  -350 I 900  1  1  1000  1100  1  MB  1200 (MeV)  1  1  1  1300  1400  1500  Figure 3.12: Variation of electric charge chemical potential fiQ with respect to baryon chemical potential p . For the range in fi where the gap A is zero, the chemical potential [XQ corresponds to that of free normal quark matter. After the onset of superconductivity, the slope of the graph for LIQ changes at the phase transition point from gapless to gapped 2SC phase. B  B  sition from hadronic matter to quark matter, the effective theory for the hadronic phase is physically relevant only at lower baryon number densities or baryon chemical potentials. We however show the results in the range of PB € [940.0,1500.0], i.e. up to the same higher limit of p as used for the quark phase but ignore the physically irrelevant part for our application to compact star physics. B  Parameters of the Theory for Hadronic Matter The parameters used for the solution of the theory are chosen such that they reproduce the nuclear properties at the stauration density. We choose the  70  Chapter 3. Thermodynamics of QCD Phases Colour 8-charge chemical potential /xg at T = 0 vs baryon chemical potential fig  s >  00  a.  -14 h  -18  900  1000  1200  1100  1300  1400  1500  Ms ( M e V )  Figure 3.13: Variation of colour-8 charge chemical potential p with respect to baryon chemical potential ps- For the range in ps where the gap A is zero, the chemical potential ps is essentially zero within numerical errors and corresponds to that of free normal quark matter. After the onset of superconductivity, the slope of the graph for p changes at the phase transition point from gapless to gapped 2SC phase. s  8  coupling constants as: 9OBJ9ON 9UJB/9U)N 9pBJ9pN  = = = = =  (3.88) (3.89) (3.90) (3.91) (3.92)  x = 0.6 x = 0.568 x = x — 0.6 8.569 x 10~ -2.421 x 10~ a  u  p  a  3  3  following the reference [30]. These choices reproduce the binding energy per nucleon B/A = -16.3 MeV, the saturation density n = 0.153fm , nucleon effective mass at the saturation m* = Q.78m , the symmetry energy co-efficient a = 32.5 MeV at the nuclear matter compression modulus K = 240 MeV. The ratio of the nucleon couplings to the mass of the mesons -3  0  N  sym  71  Chapter 3. Thermodynamics of QCD Phases Colour 3-charge chemical potential M3 at T = 0 vs baryon chemical potential ji  B  900  1000  1100  1200  1300  1400  1500-  Ms (MeV)  Figure 3.14: Variation of colour-3 charge chemical potential LI3 with respect to baryon chemical potential LI . For the range in p where the gap A is zero, the chemical potential /i is essentially zero and corresponds to that of free normal quark matter. After the onset of superconductivity, the value for / i remains very close to zero. At the phase-change point from gapless to gapped 2SC, the thermodynamic potential Q is very nearly flat near /J ~ 0.0 with irregularities due to numerical errors along the fx direction. The minimum finding routine powell goes to one of these local valleys along the irregularities. B  B  3c  3 c  3C  3c  are used as 9aN/m  guN/mu  w  9 N/m P  = a = 3.15071 fm = a = 2.19545 fm = a = 2.18884 fm a  a  p  p  (3.93) (3.94) (3.95)  Pressure, Energy Density and Baryon Number Density of Hadronic Matter The surface plot for the pressure in the neutral hadronic phase has positive slope along the temperature as well as the baryon chemical potential directions. If we extrapolate to lower chemical potentials at non-zero temperature, we get positive pressure at a baryon chemical potential lower than 940.0 MeV. This is beacuse of the thermal pressure and shows that at high 72  Chapter 3. Thermodynamics of QCD Phases Pressure of neutral hadronic phase as a function of JIB and T 0  200  400  600  800  1000  1200  1400  900  P (MeV//m ) , 3  MB  (MeV)  1000-^QO  Figure 3.15: Surface plot for pressure of the electric charge-neutral hadronic phase with the hyperons present. The pressure is seen to increase with the baryon chemical potential as well as with temperature temperature, the energy cost at fixed pressure to increase unit baryon number is less than that at zero temperature. The baryon number density and the energy density plots show similar behaviour along the T and LIB directions. Fractional Population of Baryonic and Leptonic Species in Neutral Hadronic Phase The fractional population of the baryonic species depends on the requirement of local charge neutrality and on the values of the effective chemical potentials and the effective masses of the baryons. At low baryon number density, the dominant species is the neutron. The presence of protons created via the weak interaction between nucleons, requires that the electrons be present in the same number as protons. The next species of particles that appear are the muons and the ^ 's being negatively charged, the electron number density falls behind the proton number density. At zero temperature, the appearance of particle species depends on the threshold conditions u* > m* or pi > mi. Since both the effective mass m* and the effective chemical _  B  B  B  73  Chapter 3. Thermodynamics of QCD Phases Energy density of neutral hadronic phase as a function of HB and T ^ 0  — 500 1000 1500 2000 2500 3000 3500 4000 4500 5000  4 5e4-03 4e+03  Figure 3.16: Surface plot for energy density of the electric charge neutral hadronic phase with the hyperons present. The energy density is seen to increase with the baryon chemical potential as well as with temperature. Baryon number density of neutral hadronic phase as a function of HB and T  Figure 3.17: Surface plot for baryon number density of the electric charge neutral hadronic phase with the hyperons present. The baryon number density is seen to increase with the baryon chemical potential as well as with temperature.  74  Chapter 3. Thermodynamics of QCD Phases Relative Population of Hadrons and Leptons in Neutral Hadronic Phase at T=0 MeV 1  0.0001  P n  0.2  0  0.4  0.6  0.8  1.2  J  i_  1.4 1.6 1.8 n (fm- )  2  2.2  2.4  2.6  2.8  3  B  Figure 3.18: Ratio of the number densities of baryonic and leptonic species with respect to total baryon number density at any baryon chemical potential at zero temperature. Charge neutrality requires the relative population of protons and electrons to be equal when there is no other charged particles present. The appearance of particle species depends on the effective chemical potentials and effective masses. potential p* depend on the meson fields (see equations 3.28, 3.30, 3.31), on the mass of the baryons and on the charge and iso-spin quantum numbers, the dependence of the number densities on the baryon chemical potential is non-linear. At non-zero temperature, due to thermal effects, we see many particles appear at lower baryon number density or chemical potential than those at T = 0 MeV. B  3.6.6  Phase Diagram of the Quark Phase  We construct the phase diagram of the color superconducting quark matter with and without free strange quarks and muons present. The phase diagram shows the possible presence of a mixed phase of 2SC and normal quark matter, gapped and gapless pure 2SC phase and a normal quark-matter phase, in the baryon chemical potential and temperature space. Depending on the 75  Chapter 3. Thermodynamics of QCD Phases Relative Population of Hadrons and Leptons in Neutral Hadronic Phase at T=10 MeV  0.01  0.001 t-  0.0001  0  2.2 2.4 2.6 1.i 1.4 1.6 1.8 n (fm~) Relative Population of Hadrons and Leptons in Neutral Hadronic Phase at T=30 MeV 0.2  0.4  0.6  0.1  1  1.2  3  a  0.001  0  0.2  0.4  0.6  0.  Figure 3.19: Fractional population of hadronic species and leptons with respect to total baryon number density at finite temperature T = 10 and T = 30 MeV. choice of the cut-off parameter A, in the momentum integration and using the scheme for determination of the coupling parameter, we see some general trends in the structure of the phase diagram: 76  Chapter 3. Thermodynamics of QCD Phases 1  1  1  LT  1  1  1  1  1  1  100  -  A = 1300.0  NQ Phase  -  NQ Phase  -  80  g2SC Phase/  -  g2SC Phase,  -  60  -  40  -  A = 1500.0  -  2SC Phase "  —  '  S  2SC Phase  T  (MeV)  20 f  i  Mixed Phase 1 l  i  ! ! A = 650.0  !  !  1 r  NQ Phase  i  1  1  900  1000  i  I  1100  1200  1300  MB  L  (MeV)  1400  i  ur  -  NQ Phase  -  2 S C Phase,  -  g  S  Mixed Phase  i  A = 1100.0  jr  i  i  -  -  g2SC Phase  i  Mixed Phase i i  f 1500 900  2SC Phase  /  -  T (MeV)  -  j  Mixed Phase i i  i  i  1000  1100  M B  1200  1300  i 1400  0 1500  (MeV)  Figure 3.20: Phase diagram of quark phase in the / ^ B - T plane. The mixed phase occurs at low temperature and through the range of the baryon chemical potential [900.0,1500.0] MeV. The gapped 2SC phase occurs only at a certain region of the LI -T plane, appearing after the mixed phase in the T direction and then changing into the gapless 2SC phase. At high temperature the phase is that of free non-superconducting quarks. B  1. The mixed phase is in general present almost right from the lower limit considered (LIB — 900.0 MeV), of the baryon chemical potential, depending on the cut-off A used. 2. The mixed phase continues to be present for higher baryon chemical potentials but gives way to pure gapped 2SC or gapless 2SC phase at higher temperatures. At even higher temperatures, we have a phase 77  Chapter 3. Thermodynamics of QCD Phases  T (MeV)  T (MeV)  900  1000  1100  |X  1200  B  1300  1400  1500900  1000  1100  (MeV)  U B  1200  1300  1400  1500  (MeV)  Figure 3.21: The volume fraction of the 2SC phase X2SC within the mixed phase of 2SC and normal quark matter, in the p -T plane for different values of the cut-off parameter A. The fraction changes continuously within the mixed phase from low values to unity giving way to the occurrence of pure 2SC phase. The irregularities in the graphs come from the finite number of grid points on the PB-T plane at which the phase has been determined. B  transition to normal quark matter as the superconducting gap becomes vanishingly small. 3. For 2SC+s phase, the presence of free strange quarks changes the thermodynamic potential of the normal-quark and of the colour-neutral 2SC phase, Jlf and f2 oiouriess, respectively. Since the strange quarks carry electric charge, the electric charge chemical potential has to change from that of the pure 2SC (or of free quarks case) and hence changes the slope of the surface of colour-neutral thermodynamic potential in the PQ — A space. This results in the absence of the mixed phase. ree  C  78  Chapter 3. Thermodynamics of QCD Phases  T (MeV)  4 20 I  i  i  l  900  1000  1100  1200  fiB  _J  1300  (MeV)  I  1400  I  I  1500 900  l  l  l  I  l  1000  1100  1200  1300  1400  ii  B  I o  1500  (MeV)  Figure 3.22: Phase diagram of quark matter with free strange quarks present in the U-B-T plane. In the presence of free starnge quarks, there is no mixed phase of 2SC and normal quark matter. The occurrence of the colour superconducting phases depend strongly on the cut-off parameter A. For A = 1500 MeV, at high temperature we have a narrow region of gapless 2SC phase. 4. Furthermore for the 2SC+S phase, the global minimum of f2 iouriess is shifted from A w 0, for small values of the baryon chemical potential LI , to some finite A , resulting in the pairing between the up and the down quarks. The 2SC phase is seen to be present right from the beginning of the range of LIBco  B  The fraction of the 2SC phase in the mixed phase of normal quark matter and pure 2SC phase is plotted in thefigure3.21. The fraction X2SC is seen to vary from small values to unity depending on the baryon chemical potential and temperature. X2SC increases in general with fiB at zero temperature. At higher temperature, the mixed phase gives way to the pure 2SC phase with X2SC ->  1-0.  3.6.7  Equation of state of Quark Matter and Hadronic Phase  The plots of energy density against the pressure of the different phases studied show how the assumption of colour superconductivity changes the equation  79  Chapter 3. Thermodynamics of QCD Phases Equation of State of Neutral Hadronic, Pure 2SC+S and Mixed phase of 2SC+NQ Quark Matter at T=0 MeV  ~\ 2500  r  .o MeV  SV4= 190  A = 650 MeV.2SC+NQ A = 650 MeV,2SC+s A = 1300 MeV, 2SC+NQ A = 1300 MeV,2SC+s //' A = 1500 MeV, 2SC+NQ A = 1500 MeV,2SC+s Neutral hadronic Phase e= P A = 650 MeV,2SC+s with Bag A = 1500 MeV,2SC+s with Bag  /  2000  1500  /#/B  1 / 4  = 0.0 MeV  e= P  1000  J  0  100  200  300  400  l_  J  500 600 700 800 900 Pressure (MeV-fm )  L  1000 1100 1200 1300 1400  -3  Figure 3.23: Plot of equations of state of the mixed phase of 2SC and normal quark matter with no strange quark present and for the pure 2SC phase with free strange quarks at different values of the cut-off parameter A = 650,1300,1500 MeV. The presence of free strange quark makes the equation of state softer and results in the absence of mixed phase between 2SC and normal quark matter. The plots corresponds to baryon chemical potential PB varying from 900 to 1500 MeV for the quark phase and from 940 to 1500 MeV for the hadronic phase. of state considerably. We have the following features observed in the equation of state plots: 1. The colour superconducting states have stiffer equation of state than the neutral hadronic phase but within the causal limit of e = P. The dependence of e on the pressure P is very nearly linear for both the pure 2SC and the mixed phase (2SC+NQ). The bag constant used is independent of the thermodynamic variables and changes the origins 80  Chapter 3. Thermodynamics of QCD Phases 3500,  ~0  Equation of State of pure 2SC+S phase and neutral hadronic phase at T=0 MeV , , , , r  0.2  0.4  0.6  0.8  1 1.2 1.4 1.6 1.8 2 Baryon Number Density (1/fm )  2.2  2.4  2.6  2.8  3  Figure 3.24: Plot of equations of state (energy density vs. baryon number density) of the neutral hadronic phase and the pure 2SC phase with free strange quarks at different values of the cut-off parameter A = 650,1100,1300,1500 MeV. The bag constant used is B = (195.0MeV) . The presence of free strange quark makes the equation of state softer. The plot corresponds to baryon chemical potential u. varying from 900 to 1500 MeV for the quark phase and from 940 to 1500 MeV for the hadronic phase. 4  B  but not the slopes of the plots for the equations of state in the quark phase. Thus the introduction of a bag constant independent of the baryon chemical potential and temperature does not affect the stiffness of the equations of state. 2. The introduction of the free strange quark makes the equation of state softer than that of the mixed phase. Also, the pressure and energy densities have higher values for the same baryon chemical potential at the end of the range considered LIB = 1500.0 MeV. This is because, more degrees of freedom are available in the 2SC+S phase than those in pure 2SC (u-d paired)) or free u,d phase. 81  Chapter 3. Thermodynamics of QCD Phases 3. Within the theory of the 2SC colour superconductors, there is a clear dependence of the thermodynamic state (P,£,%,..), on the cut-off A used. We see that the equation of state becomes softer with the higher cut-off value. Furthermore, with higher cut-off parameter, the pressure and the energy densities have higher values at the same baryon chemical potential. The difference, however, becomes smaller at the higher limit of A.  82  Chapter 4 Compact Stars with Colour Superconducting Cores In this chapter we study the structure and properties of compact stars with the central densities reaching a few times the nuclear saturation density (no ~ 0.15 fm~ ) [30], such that the inner core of these stars contain degenerate quarks in the normal phase or the colour superconducting phase. If the compact star is composed purely of quark matter it is called a quark star. On the other hand, stars with quark matter core but a nuclear matter mantle are called a hybrid stars where a phase transition occurs at some radial distance from the centre. The quark matter phases considered are colour- and electriccharge neutral state of two-flavour colour superconducting matter in 2SC or g2SC phase, a mixed phase of the 2SC and normal quark matter (2SC+NQ) or the 2SC phase in the presence of free strange quarks (2SC+s). 3  4.1  The Tolman-Oppenheimer-Volkoff Equations for Stellar Structure  The matter inside a compact star is packed in a small region; it gravitates strongly enought to change significantly the curvature of the spacetime around it. The compact nature of such a star necessitates the use of Einstein's general theory of relativity for the study of its structure. The corresponding spacetime geometry of a static, spherically symmetric relativistic star is found by solving the Einstein's equation (in conjunction with the equation of state of the matter): =  R  ^  _  l-g^R  =  -SirGT^  (4.1)  where G^ is the Einstein curvature tensor, W is the Ricci tensor (R = R£), g^" is the metric tensor and is the energy momentum tensor of the matter v  v  83  Chapter 4. Compact Stars with Colour Superconducting Cores inside the star that represents the properties of the matter. The metric inside a static spherically symmetric star is found from the symmetry considerations as = dlag(e , - e « - r , - r sin 9) (4.2) Mr)  2 A  2  2  2  9fiu  where e ^^ — ^1 — 2l2M^ . Using the above ansatz, the Einstein's equation takes the form of the Tolman-Oppenheimer-Volkoff (TOV) equations, expressed as 2A  dm — dr dv dr f  9 2  _  m(r) + 4ur p(r) r(r — 2m(r))  .  3  = -W)  r  ., „. 4.3)  = 47rr e  + <r)]^  .  (4.5)  Here we have used the gravitational or geometric system of units in which we set G = c = 1 such that mass and length have the same dimension. The Newtonian limit corresponds to the case ] ) C e and m <^r. The "mass" m(r), is interpreted as the "mass inside radius r" which represents the total mass-energy of the star including the gravitational potential energy of the matter inside r. The proper volume element for the metric is given by, / 2m\ / - 1  dV = yf^gVrdr x 4?rr = ( l - — J 2  2  47rr dr  (4.6)  2  From equation (4.3), the total gravitational mass of the star is R  f  M = m{R) = 4TT / e(r )r' dr Jo  (4.7)  2  which not only adds up the local contributions of mass-energy but also includes the global contribution of the negative gravitational potential energy of the star [61]. By the same token, the total number of baryons inside a star is given by N = B  f  f  R  n dV = 4n B  f / n ( r ) e V d r = 4TT / n ( r ) f l R  B  B  2m(r)\- /2 1  r dr 2  (4.8)  84  Chapter 4. Compact Stars with Colour Superconducting Cores This gives the differential form of the equation for total baryon number inside a radius r as dN (r) , ./ 2m(r)\-V2 —fr = 47rr n (r) ( l J (4.9) B  A  2  2  B  The baryon number density n (r) is found locally from the equation of state as a function of either the pressure or the energy density at a radial distance r and we can integrate the differential equation for the baryon number density (4.9), concurrently with the TOV equations (4.3, 4.4, 4.5), to get the total baryon number N of the star. We define the "total rest mass" or the "baryon mass" of the star in terms of the total baryon number as B  B  M = Nm Q  B  n  = 47r / n ( r ) m „ e V d r Jo B  (4.10)  which is the mass of the N number of neutrons, dispersed at rest at infinity, such that there is no interaction between them. For a stable star, the gravitational mass must be less than its baryon mass. B  4.1.1  Equation of State of Stellar Matter  We consider the matter inside a compact star to be a perfect relativistic fluid with fixed chemical composition. The non-zero components of the energy momentum tensor T^" for a such a matter are given by Tj? = diag(e, -p, -p, -p)  (4.11)  where e and p are the total energy density (including mass) and the pressure in the local inertial comoving frame of the matter. We further make the assumption that microscopic interactions in the matter (e.g. weak or strong interactions) are not affected by the gravity as the length scales associated with these interactions are much smaller than the gravitational length scale. Since the system is relativistic, the total number of particles as well as the total mass-energy inside a volume is not fixed, in general. Under these assumptions, the thermodynamic properties of the matter are well described, in the context of the grand canonical ensemble, by the grand canonical thermodynamic potential ft(/z , T). The other thermodynamic quantities, which can be derived from ft (see appendix B ), include the pressure p = —ft, the baryon number density n = —dft/<9/i , the entropy density s = —DQ/dT B  B  B  85  Chapter 4. Compact Stars with Colour Superconducting Cores and the energy density e = + p n + Ts whereas, the temperature and baryon chemical potential are considered as independent variables. The equation of state of the matter is a relation which gives one of the thermodynamic quantities in terms of any other two, e.g. B  B  e = e{n ,s), P = p{n ,s),  or e = e{p ,T) or p = p(p ,T)  B  B  B  B  (4.12) (4.13)  which encodes the microscopic physics in a locally inertial comoving frame. Given any two variables, we get a complete thermodynamic description of the matter provided that the composition of the fluid element is unchanged. Barotropic Equation of State and Variation of Chemical Potential within a Star A perfect fluid is not necessarily isentropic, i.e. s is not necessarily the same everywhere. In the case of cold star, however the temperature of the star can essentially be taken to zero (k T <S Ep => T « 0). Thus, the entropy density is very nearly zero s « 0. Again, the entropy of a super-massive star can be uniform because of convection [61]. In these special cases, the equation of state does not depend on the temperature or the entropy density and is called a barotropic equation of state. This can be represented as B  P = p(n-B),  e = e{n )  P  e = e(p )  =  B  P(PB),  B  or,  (4.14)  (4.15)  The pressure equation can be inverted to get the energy density as a function of pressure to be used in the solution of the TOV equation e = e(p)  (4.16)  At T — 0, the relationship between chemical potential and pressure is given by p- np B  where p = de/dn potential is B  B  B  - e  (4.17)  Thus, the relationship between pressure and chemical *  = » B ^  dn  B  dn  (4.18)  B  86  Chapter 4. Compact Stars with Colour Superconducting Cores Hence, from the last two of the Tolman-Oppenheimer-Volkoff equations we get, dp  _ n dp B  B  p+ e  p nB  ln[p (r)  - p (r')}  B  B  B  _  dp  B  PB  (4.19)  where r and r are any two radii inside the star (r,r < R). Hence the chemical potential of the material of the star obeying barotropic equation of state, satisfies /a (r)e" = (r')e ^ = constant (4.20) (r)  fl  i/(r  /Us  Thus the baryon chemical potential at any radial distance of a barotropic star is not arbitrary but equal to the central baryon chemical potential times a factor e"<> [30]. We note that, the restriction on the baryon chemical potential is not a constraint that has to be imposed on the solution of the Tolman-VolkoffOppenheimer equation, rather it is automatically satisfied by matter obeying barotropic equation of state. Matter is arranged inside the star by gravity so as to satisfy the above [30]. For a non-barotropic equation of state, because of an extra degree of freedom of temperature in the equation of state, the chemical potential does not obey such a relation. In our case, for both the finite temperature quark matter at the star's core as well as the hadronic matter in the crust, the energy density e and the pressure p depend on the temperature (which we may take to be non-zero). Hence we do not have a barotropic equation of state for matter within a hot compact star. r  Thermal Equilibrium of a Compact Star We consider a compact star in thermal equilibrium such that at any radial distance from the centre, there is no net amount of energy transfer through a shell. In a locally inertial frame, a chunk of material will have to have the same temperature for thermal equilibrium to be achieved. However, in the case of compact star, because of strong gravitational field, the spacetime is curved and we may expect on general grounds (since gravity couples universally to all matter and energy), that any form of energy in transition will feel the effects of strong gravitational field. 87  Chapter 4. Compact Stars with Colour Superconducting Cores In thermodynamics, the condition for thermal equilibrium is that the net energy-flux due to thermal conductivity vanishes. From microscopic theory, the thermal conductivity is proportional to the mean free path of a particle, e.g. of an atom in a neutral gas or of an electron in a plasma or of the photon in a thermal bath. Hence to find the effects of gravitational field on the thermal conductivity, we have to examine how the field affects the motion of free material particles or of photons. The frequency of a photon emitted inside a compact star and moving outside is shifted by an amount — 1/2  proportional to g , while the motion of collisionless particles is affected such that its energy is also red-shifted by the same amount [71]. Hence, the condition for thermal equilibrium inside a compact star is given by: T(r)e" = T(r){g o) = constant (4.21) 00  (r)  1/2  0  in contrast to the non-relativistic condition T = constant [30]. The condition above is consistent with the non-relativistic limit where g —> 1. 00  4.2  Solution of T O V Equations  The set of Tolman-Oppenheimer-Volkoff (TOV) equations (4.3, 4.4, 4.5), together with the differential equation for the baryon number density (4.9) can be solved numerically provided we use the equation of state to get the energy density as a function of pressure and temperature. The basic steps for the numerical computation involve: 1. Pick a value of either the central energy density e or the central pressure p and then find e from the local equation of state. The boundary conditions at the centre are c  c  c  m(r n (r p(r u(r B  = 0) = 0) = 0) = 0)  = = = =  0 0 p , e{r = 0) = e v c  c  c  (4.22) (4.23) (4.24) (4.25)  where v can be taken arbitrarily since only the first and second derivatives of v(r) occur in the Einstein's equation, not v(r) itself. c  2. Integrate the TOV equations and the equation for total baryon number enclosed, starting from the centre at r = 0 out to some radial distance 88  Chapter 4. Compact Stars with Colour Superconducting Cores using the initial conditions. At each radial value, a new value of the mass enclosed m(r), pressure p(r), total baryon number enclosed NB{T) and v(r) are obtained. These values are used as the initial conditions for the next step of integration. 3. In the equation of state, we make use of the red-shifted temperature (c.f. Eq. 4.21) at that radial distance T' = T(r = 0)e " \ to get the energy density and the baryon number density as implicit functions of the pressure p(r): _  (r  e(r) = e( (p(r),T'),T')  (4.26)  Ms  n (r) B  = n (u, (p(r),T'),T') B  (4.27)  B  This is done by a root finding routine which finds the baryon chemical potential fi , that would produce the given pressure at the given temperature using the equation of state i.e. by inverting p — p(u-B,T). B  4. At the surface of the star, the pressure must be zero p(R) = 0 while the mass m(R) = M will be the total gravitational mass of the star. 5. The metric function u(r) should match smoothly with the Schwarzschild metric at the surface: VSC(T  (4.28)  = R) = \ \ T L ( \ - ™ )  which can be achieved by changing the central value of u, appropriately. In numerical work, we can choose an arbitrary value of u[r = 0) (e.g. f (r = 0) = 0) to get a value at the surface v[r — R) — v j and then add an appropriate constant to v(r) to satisfy the boundary condition (4.28) above. sur  However, the value of v does not affect the mass M and radius R of the star; it does not affect the surface temperature either since it depends only on the change in the function v(r) and we may redefine it for our convenience: c  T(r = R) = T{r = v(Q)-v(R)  ti)e v{0)  (4.29)  v{R)  = (i/(0) + v ) - (i/(R) + v ) c  c  Hence for the purpose determining M — R relation of a star, we need not worry about v . c  89  Chapter 4. Compact Stars with Colour Superconducting Cores  4.2.1  Scaled T O V Equations  In solving the Tolman-Oppenheimer-Volkoff equations numerically it is convenient to scale the different variables in the differential equations by their representative values, as expected for a compact star. This will ensure that the different physical variables do not assume values in hugely different orders of magnitude. We scale the variables according to the following scheme: r -* r/R i , m -> m/M i ,  R = 1.0 x 10 cm M i = M© = Mass of the sun = 1.4776 x 10 cm N = 1.0 x 10 5  sm e  scaU  sca e  sca e  5  n  B  -> n /N , e -> e/e i B  P  ->  56  scale  sca  scale  e  .  P/Cscale  (4.30) (4.31) (4.32) (4.33) (4-34)  We note that R cale/M = 1/Ro S  (4.35)  Q  where R = Schawrzschild radius of the sun in km = 1.4776(km). Here the scaling value of the energy density and pressure e / , is arbitrary and wefixit such that the right hand side of the differential equation for the mass (Eq. 4.3) is equal to 47rr e, where r and e are now the scaled radial distance and scaled energy density, respectively. Hence we have 0  sca  e  2  (M i /R ie)— 3ca e  =  sca  477T e 2  R ee ale 2  scal  3C  (l/Mscale)^scaleR s ale ~ 1 descale = Ro/RLle 3  C  (4-36)  The scaled form of the Tolman-Oppenheimer-Volkoff equations and the differential equation for the total baryon number of the star are thus ^ = 47rr e dr dp / ->  (4.37)  2  M m m(r) + 4nr p(r) 3  r  I dv dr ^  = -  ^ ^r(r-2R m(r)) m(r) + 47rr p(r) r{r — lRom{r)) +  e  0  ^  3  = < A ( * , ( l - ^ r "  (4.40)  90  Chapter 4. Compact Stars with Colour Superconducting Cores where all the physical variables m, r, p, e represent scaled forms. The baryon number density rig and the total baryon number NB need not be scaled as scaling factor N i can be factored out. We also do not need to scale u(r) or lnpoo — 2u(r). We solve the above scaled TOV equations numerically to get the mass, total baryon number and radius of the compact stars. sca e  Mass vs Radius of Compact Stars at central temperature T=0 MeV 1.8  B B 1.6 B B 1.4 - B B 1.2 - B  Mass  1  = / = = / = = / = / =  1 / 4 1  4  1 / 4 1  4  1 / 4 1 1  4  4  140 150 160 170 180 190 200  ,\  /  -  /  0.8  1  ^1  1  MeV MeV MeV MeV MeV MeV MeV  1  /  i  /  |  0.6  '  /  0.4 0.2 0  i  0  2  4  6  8 Radius (km)  i  i  i  10  12  14  Figure 4.1: Mass vs. radius relation for cold quark-, hybrid- and hadronicstars with strange particles present. The structure and the composition of the stars depend on the bag constant B as well as the equation of states of the quark and hadronic phases.  4.3  Mass and Radius of Cold Compact Stars with Colour Superconducting Matter  We solve the Tolman-Oppenheimer-Volkoff (TOV) equations with the equation of state of matter in the colour-superconducting or normal quark matter phase or in the neutral hadronic phase in thermal equilibrium throughout the star's interior. During the integration of the equations, at each radial value, 91  Chapter 4. Compact Stars with Colour Superconducting Cores Mass vs Radius for non-strange Compact Stars at central temperature T=0 MeV 2.25 2 1.75  135 MeV 140 MeV 145 MeV BV4 146 MeV 147 MeV B •• 148 MeV B ' •• 150 MeV B l* •• 155 MeV 160 MeV B' 165 MeV J3V4  :  :  1/4  1.5  1 4  l  1.25  1 4  :  1 0.75 0.5 0.25 0  10 Radius (km  Figure 4.2: Mass vs. radius relation for cold quark-, hybrid- and hadronicstars without any strange particles present. As the bag constant is increased, we see a transition from quark- to hybrid- and finally purely hadronic-stars. The maximum mass of the stars depend on the bag constant as well as on the equation of the state of the non-strange quark and hadronic phases. we have picked the favoured phase based on the pressure of the different possible phases. From thermodynamics, the most probable phase is the one in which the pressure is the highest at the same value of the baryon chemical potential and temperature. We get a family of configurations of the compact stars by solving the TOV equations for the same equation of state but different central pressure values, giving the mass versus radius relationship of the stars. The M-R relationship of the compact stars shows the following features in the plots: 1. Depending on the bag constant chosen, we get a range of compact stars from that made of pure quark matter, hybrid stars with a quarkmatter core and a hadronic matter mantle, to purely hadronic stars. For matter with strange quarks present, at low value of the bag constant (for B = 140,150 MeV), we get stars made of pure 2SC and free 1/4  92  Chapter 4. Compact Stars with Colour Superconducting Cores strange quark phase (2SC+s). 2. The maximum mass of the compact star depends on the stiffness of the equation of state and since the equation of state of 2SC+S phase at a low bag constant is stiffer than that of the hadronic phase (c.f. figure 3.23), the maximum mass of quark stars with 2SC+S quark matter is seen to be considerably higher than maximum mass of purely hadronic stars. Mass and Phase Change Radius vs Central Pressure for Cold (T=0 MeV) Hybrid Stars 1.8 B< MeV 1.6 B' eV J3 / = 160 MeV 1.4 B / = 170 MeV 1.2 B = 180 MeV 1 B > = 190 MeV B' 200 MeV 0.8 0.6 0.4 0.2 H—I I I 1 I I i —r^T'TTi'lTl i—I I I I I | 18 B = 140 MeV B = 150 MeV / j = 160 MeV B l = 170 MeV B ' = 180 MeV B = 190 MeV B / = 200 MeV -l  4  =  1 4 0  =  1 5 0 M  x 4 1  1  4  4  1/4  l  4  1 4  =  1 / 4  l/i  1/4  x 4 1 4 1/4 1  4  10  100  1000  P (10 dyne/cm ) 33  2  c  Figure 4.3: Mass and phase change radius of hybrid stars with strange particles present at T = 0 MeV. Hybrid stars result only for a certain range of the bag constant B, since within this range, the pressure of the quark- and the hadronic phases are comparable, with the bag constant subtracted from the the quark phase pressure. 3. The equation of state of the quark matter in the 2SC+NQ phase (with two flavours) is even stiffer than that in the 2SC+s phase. This results in a bigger maximum mass for the quark stars with the mixed phase.  93  Chapter 4. Compact Stars with Colour Superconducting Cores 4. Stars made of purely non-strange hadronic matter are seen to have a lower maximum mass M « 1.65M©, which is higher than that of the star with pure hadronic matter with non-zero strangeness. max  5. The radius corresponding to the maximally massive quark, hybrid or hadronic stars having strangeness, is however seen to be smaller or comparable to that of the non-strange stars (R trange ~ Rnon-strange ~ 10 km). s  The hybrid stars that result for certain ranges of the bag constant, show some interesting features in their structure which are discussed in the next section.  4.3.1  Mass and Radius of Hybrid Stars  At low central pressure the star is made completely of hadronic matter. As the central pressure increases the matter at the core changes to either one of the normal (NQ) or pure colour superconducting quark phase (2SC or 2SC+s) or a mixture (2SC+NQ) phase. At some outer radius, the pressure decreases enough such that the favoured phase is that of the hadronic matter. Although we have continuity of pressure, in general the energy density and the baryon number density have discontinuities at the phase-change radius. We observe two interesting phenomena for the hybrid stars: Unstable Branch in between Stable Branches For hybrid star with strange quarks, the M-R plot branches out of the purely hadronic star M-R plot (see Fig.4.1). For B ' = 170,180 MeV, we have a stable branch for mass up to « 0.65M , then the mass decreases with decrease of radius and then increases again. The local maximum mass of at ~ 0.65M© (cf Fig. 4.3) is a critical or turning point in the M-R plot and the clockwise bend at it signals instability with increasing central density. Thus, we have an unstable branch of the M-R plot which later becomes gravitationally stable as the central pressure increases. The latter is a new stable branch of the M-R plot for hybrid stars: However, for hybrid stars without any strange particles, we do not see such a behaviour in the M-R plot (see Fig.4.2). l A  Q  94  Chapter 4. Compact Stars with Colour Superconducting Cores Mass and Phase Change Radius for Cold (T=0 MeV) Non-strange Hybrid Stars  10 (10 dyne/cm ) 33  1000  2  Figure 4.4: Mass and phase change radius of hybrid stars without any strange particles present at T = 0 MeV. Hybrid stars are seen to result for the bag constant within a certain range, similar to the case of quarks stars with strange particles. Maximum Size of Quark Core We expect the appearance of quark phase in the core of the compact stars with the increase of central pressure. This is seen in the plot of radius of phase change i?phase-change vs. the central pressure P . At higher bag constants, the quark phase becomes favoured at higher pressures. However, at the same bag constant, the radius of phase change, in general increases with increase of central pressure since the quark-core becomes larger. For bag constant B = (160MeV) , we see a maximum value of the phase-change-radius at R « 8.4 km. This implies the existence of a maximum size of the quark core, even when the central pressure increases. Note that, the star itself becomes c  4  95  Chapter 4. Compact Stars with Colour Superconducting Cores Mass vs Radius for strange and non-strange Quark Stars at central temperature T=0,5,10,20 MeV 1.8 , B ^ = 140, T = 0 MeV 1.6 h B ' = 140 'T = 5 MeV 1.4 B ' = 140, T = 10 MeV 1  4  1 4  1 4  B > = 140, T = 20 MeV B ' = 150, T = 0 MeV 1.2 , B ' = 150. T = 5 MeV 1 l-B ' = 150, T = 10 MeV B ' = 150, T = 20 MeV l  4  1 4  1 4  1 4  0.8  1 4  0.6 0.4 0.2 0 B ' = 135, T'= 0 MeV 2.25 B ' = 135 'r = 5 MeV 2 B / = 135, T = 10 MeV |_B / = 135,T = 20 MeV 1.75 B = 140, T = 0 MeV B ' = 140J = 5 MeV 1.5 B / = 140, T = 10 MeV 1.25 B / = 140, T = 20 MeV - B / = 145,T = 0MeV 1 B = 145. T = 5 MeV 0.75 -BV4 145 T = 10 MeV B / = 145, r = 20 MeV 0.5 0.25 00 1 2 3 4 5 1  4  1 4  1  4  1  4  1 / 4  1  1 4  1 1  4 4  1  4  1 / 4  1  4  =  6  7 8 Radius (km)  i  i  i  10  11  12  13  14  Figure 4.5: Mass-radius relationship for quark stars with 2SC phase with and without strange quarks present. The upper panel represents 2SC and free strange quarks phase. The lower panel represents the case for either pure 2SC phase or a mixed phase of 2SC and normal quark matter in two flavours (it and d quarks). The temperatures need to increase ~ 10 MeV for any noticeable change to occur. unstable after this central pressure since the mass of the star assumes a peak at this value of P . For pure quark stars, as well as for pure hadronic stars, since there is no phase change, i?hase-change is set to the value of zero. For non-strange hybrid stars, we see a similar behaviour of a maximum quarkcore-size for the equation of state with bag constant B = (150 MeV) . c  p  4  96  Chapter 4. Compact Stars with Colour Superconducting Cores  Temperature Dependence of Compact Star Structure  4.4  Mass, N and )J,BC VS Central Pressure for Quark Star at T=0,10,20 MeV B  1.8 1.6 1.4 ^ ^ 1.2 o 1 S 0.8 % TO 0.6 s 0.4 0.2 0 30 • B '* = 140 'T = 0 Me B = 140, T = 10 MeV 25 -jgi/t 140, T = 20 MeV B / = 150 'T = 0 MeV 20 •B = 150, T = 10 MeV 10 O B = 150, T = 20 MeV X 15 m 2. 10 1  l/i  = 4  1  1/A  1 / 4  5  > 0)  a.  0 1500 1400 1300 1200 1100 1000 900 800  - £i/4 = 140.T = 0Me. j? / = 140, T = 10 MeV B = 140, T = 20 MeV - B / = 150 'T = 0 MeV B = 150, r = 10 MeV "B / = 150, T = 20 MeV \x = 900 MeV 1  4  1 / 4 1  4  1 / 4  1 4  Bc  1  10 100 Central Pressure P (10 dyne/cm ) 33  1000  2  c  Figure 4.6: Temperature dependence of the mass M, total baryon number N and central baryon chemical potential p of quark stars in the 2SC phase with free strange quarks present. The variation in mass and total baryon number are not noticeable until the temperature increase A T ~ O(10) MeV. The central baryon chemical potential for low central pressures is obtained by extrapolation and is seen to have values that are below the lower limit (900 MeV) of the tabulated values in the equation of state table. B  Bc  97  Chapter 4. Compact Stars with Colour Superconducting Cores For finding the temperature dependence of the structure of the compact stars we solve the TOV equations at non-zero central temperature and in thermal equilibrium. We focus on the equation of state of matter without any trapped neutrinos. The scenario is discussed in greater detail in the chapter on cooling of quark stars (see section 5.1). As discussed in the section (4.1.1), we use the red-shifted local temperature at any radial distance (see Eq. 4.21). We consider the equations of state of quark and hadronic matter, with and without strange particles present, at various values of the bag constant. Our results for central temperatures of T = 0,0.5,1,5,10,20 MeV are plotted.  4.4.1  Hot Quark Stars  Solution of the TOV equations for the hot compact stars shows that for small bag constants, the favoured phase is of pure quarks for the whole star; at intermediate bag constants, we get hybrid stars and at even higher bag constants, purely hadronic stars result. This is similar to the case of cold compact stars. To get the equation of state at finite temperature we interpolate/extrapolate the tabulated equations of state with the baryon chemical potential u. 6 [900,1500] MeV and temperature T € [0,109] MeV, as the independent variables and all other thermodynamic quantities as dependent variables. B  Mass, Radius and Total Baryon Number of Hot Quark Stars We present the mass versus radius relationship of quark stars in the plots (4.5), for temperatures T — 0,5,10,20 MeV only, since changes in M or R are not clearly visible in the plots, for smaller temperature increases. As the central temperature and hence the temperature of the star as a whole increases, both the the mass and the radius of the quark stars increase. Since, the mass represents the total gravitational mass-energy, we expect an increase with the increase of temperature. Similarly, we expect the star to be puffed up as it becomes hot. For both the cases of, with and without strange quarks present, the masses of the quark stars peak at some particular values of the central pressure (see Figs. 4.6, 4.16). The maximum mass and the corresponding central pressure depend on the equation of state employed and hence on the chosen bag constant. For bag constant of B / = 140 MeV, we get higher values of the 1 4  98  Chapter 4. Compact Stars with Colour Superconducting Cores Mass, Radius, Total Baryon Number and Central Baryon Chemical Potential vs Central Baryon Number Density for Quark Stars at T=0,10,20 MeV  o  1.8 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 30  B l* = 140, T = 0 MeV B ' = 140, T = 10 MeV x  1 4  B / B / B'/" B ' 1  1  = 140, T = 20 MeV = 150, T = 0 MeV = 150, T = 10 MeV = 150, T = 20 MeV  4  1  4  4  1  1  h  25 o  20  X  15  B ' = 140, T = 0 MeV B'/" = 140, T = 10 MeV B 1 4 = 140, T= 20 MeV B ' = 150, T = 0 MeV B ' / = 150, T= 10 MeV B = 150, T = 20 MeV 1 4 1  1 / 4  2  10  4  5  If 3 1  1 / 4  0 11 10 9 g 7 6 5 4 3 2 1500 • B ' / = 140, T = 0 Me# = B / = 140, T = 10 MeV 1400 "B / = 140,T = 20 MeV 1300 - B / = 150, T = 0 MeV B / = 150, T = 10 MeV 1200 'B'/ = 150, T = 20 MeV 1100 4  1  s  CO  a.  1  H  B / = 140, T = 0 MeV B = 140, T= 10 MeV = 140.7'.. 20 MeV B ' = 150, T = 0 MeV B1'4 = 150, T = 10 MeV B / = 150, T = 20 MeV 1  4  1 / 4  1  1  7  -  4  4  1  1  /i  = 900 MeV  1  4  1  1  1  4  1  :> CD  j  4  4  4  1000 900  Bc  800 0.2  0.4  0.6  0.8  1  1.2  1.4  1.6  1.8  Central Baryon Number Density UB (fm ) -3  C  Figure 4.7: Mass, total baryon number, radius and central baryon chemical potential for quark stars with 2SC quark matter with free strange quarks as a function of the central baryon number denisty. For most part of the range of UB , the quark stars are gravitationally stable. The part of the plots after the maximum mass corresponds to unstable stars. The central baryon chemical potential has very little dependence on either the temperature or the bag constant. C  maximum mass and maximum total baryon number, than those with a bigger bag constant (e.g. v3 = 150 MeV). 1/4  99  Chapter 4. Compact Stars with Colour Superconducting Cores The central baryon chemical potential as a function of the central pressure, however does not vary much with the variation of the bag constant.  4.4.2  Binding Energy of Quark Stars  The binding energy of a compact star is defined as the difference of the energy equivalent of the baryonic mass and the total gravitational mass: BE = m N B  B  -M  G  =m N B  - M  B  (4.41)  where, m is the mass per unit baryon number, N is the total baryon number of the star and MQ = M is the gravitational mass of the star. Since the total gravitational mass represents the mass energy of the star including the gravitational binding energy and the internal binding energy of the star's matter, we expect the baryonic mass M = m N to be greater than the total gravitational mass. The gravitational potential energy of a spherical non-rotating massive star is E ~ — f^T °^ incompressible fluids, the binding energy per unit gravitational mass is given by [50] B  B  B  B  B  I n t  n  e c a s e  G  * w * - £ ( ^ - V n * ) - . «  — + — + — + ... 5 14 6  (4.42) '  K  where B = M/R is the compactness parameter. For quark stars we see the following features in the plots of binding energy of the star: 1. The total binding energy of the quark stars in 2SC+S phase increases with the increase of the central pressure. This is because, as the central pressure increases, the star becomes more massive and hence more binding energy is required to keep the star intact. The binding energy is always positive and depends more profoundly on the bag constant used, as the central pressure increases. 2. The binding energy per unit gravitational mass (BE/M) of the quark stars with matter in 2SC+S phase is seen to depend on the bag constant throughout the range of the central pressure (see Figs. 4.8, 4.9). For a lower value of the bag constant (B — (140 MeV) ), BE/M changes 4  100  Chapter 4. Compact Stars with Colour Superconducting Cores Binding Energy, Relative Binding Energy and MIR vs Central Pressure for Quark Star at T=0,10,20 MeV 0.6  B ' = 140 J* = 0 MeV B / = 140,r = 10 MeV B' = 140, T = 20 MeV 0.4 . B / = 150 'T = 0 MeV , B = 150, T = 10 MeV 0.3 L B / = 150,T = 20 MeV x 4  0.5  1  4  1 4  1  4  1 / 4  1  4  0.2 0.1 0 0.35 0.3 0.25  BI X  B  A  1 / 4  = 140. T" = 0 M e ,  = 140, T = 10 MeV  B ' -• B' , B ? = 150 1 4  1 4  1  4  0.2 L B / = 150 1  4  0.15 0.1 0.05 0 0.3 . B  = 140 'T = 0 = 140, T = 10 0.25 L B / = 140, T = 20 B' = 150 'T = 0 = 150, T = 10 0.2 "BB / = 150,T = 20 0.15 , B '1  1 / 4  1 4  4  1 4  1 / 4 1  4  Mev MeV MeV MeV MeV MeV  0.1 0.05 0  1000 P (10 dyne/cm ) 33  2  c  Figure 4.8: The binding energy, binding energy per unit gravitational mass and compactness parameter of quark stars in the 2SC+S phase as a function of central pressure at central temperatures T = 0,10,20 MeV. As the star becomes more massive the binding energy in general increases.The dependence on bag constant and temperature is also noticeable on the plots of binding energies. from « 15% to a maximum of « 32% and slightly decreases with further increase of the central pressure. For higher bag constant (B — (150 MeV) ), the BE/M changes from w 15% to 25%. 4  101  Chapter 4. Compact Stars with Colour Superconducting Cores 3. The star becomes more massive at higher temperatures, so also its radius increases in general. However the compactness parameter (5 — M/R does not change much with the temperature. 4. The total binding energy is seen to decrease with increase of temperature. We expect this as higher temperatures result in more thermal energy and hence less bound quarks in the star's interior. 5. The internal energy part of the star's binding energy per unit gravitational mass (BE — BEi )/M is also seen to decrease with increase of, temperature. This part is seen to be highly dependent on the bag constant used. nc  4.4.3  Change of Mass, Radius and Total Baryon Number of Quark Stars with Temperature  We plot the change of mass, radius and total baryon number (AM,Ai?,A/Y ) of hot quark stars with increase of temperature over those of a cold quark star at T = 0 MeV, for various bag constants used. In general, both the mass, radius are seen to increase with temperature while and the total baryon number decreases for some bag constants and at high central pressures. We discuss the quark stars in 2SC+S, two- or three-flavour NQ and 2SC or 2SC+NQ phases below. B  Hot Quark Stars with free strange quarks We note the following features in the plots of change of mass, total baryon number and radius for quark stars without trapped neutrinos, in the 2SC+S phase for increase of central temperature: 1. For quark stars with 2SC+S phase, the change of mass A M has a maximum at certain central pressure for each final temperature. The peak is seen to shift towards higher central pressures as the temperature is increased, for the same bag constant. 2. As the bag constant is increased, the peak of the maximum increase of mass, with rise of temperature, occurs at higher central pressures.  102  Chapter 4. Compact Stars with Colour Superconducting Cores Binding Energy of Incompressible Fluid and Internal Binding Energy per unit Mass vs Central Pressure for Quark Star at T=0,10,20 MeV 0.25  B'  1 4  B / 0.2 B / ' B B / B / 0.15  = 140, T = 0 MeV = 140, T = 10 MeV = 140,T = 20 MeV / = 150 'T = 0 MeV = 150, T = 10 MeV = 150, T = 20 MeV  1  4  1  4  1  4  1  4  1  4  0.1 0.05 0  B / B / 0.25 B / , B / r-B / 0.2 B / 1  1  4  1  4  1  1  1  4  4  4  4  = 140,T = 0 M e , = 140, T = 10 MeV = 140, T = 20 MeV = 150 'T = 0 MeV = 150, T = 10 MeV = 150, T = 20 MeV  0.15 0.1 0.05 1000  10 100 Central Pressure P (10 dyne/cm ) 33  2  c  Figure 4.9: The relative binding energy of incompressible fluid (BEi /M) and the internal part of the relative binding energy (BE /M) of quark stars in the 2SC+S phase. The relative internal binding energy is seen to decrease with increase of central pressure while BEi /M increases. As the central temperature increases BEi /M is clearly seen to decrease. nc  int  nc  nt  3. Although, the mass- and radius-increase are not constant, they are seen to be always positive for quark stars with strange quarks present within the range of central pressure of the star (see Figs. 4.10,4.11). 4. The change of total baryon number AN with increase of temperature over that at T = 0 MeV, is also seen to have a maximum at some central pressure for each final temperature. The maximum is more pronounced at higher final temperatures, such as at T = 5,10,20 MeV's, than at B  103  Chapter 4. Compact Stars with Colour Superconducting Cores Change of Mass, Total Baryon Number and Radius with central temperature vs Central Pressure for Quark Stars with 2SC+S Phase 0.05 0.045  r  0.04  B ' / = 140, M - Mo B = 140, Mm - M B = 140, M - M B / = 150, M - M B ' / = 150, Mio - M B'/" = 150, M - M 4  5  1 / 4  0  1 / 4  2 0  1  0  4  5  0.035  <  0.03  %  0.025  0) 60 Ci J3  0.015  s .s O  0  2 0  -—. —' o  1  Z.  <  If  0  0.02 0.01 0.005 0 0.6 . B 0.55 YB\I* 0.5 B'/ B' 0.45 B\ 0.4 B' 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 -0.05 0.45  4  = 140,  N s-N o B  =  4  1 4  to O X  0  4  14  / 4  B  140,NBW-N  = 140,7V =  B20  -iV  B0  BO  150,N -N o B5  B  = 150, NBW - iVflo = 150, N  B20  -  N  BA  l I l I 11,14  l l I l I [  B '  B;/ B  0.4  3  0.25  a  0.15 0.1  Radi  0.3  M  §  JO  O  5  B ' / = 150, fl - •Ro B = 150, Rio - Ro B ' / = 150, R o • -Ro 4  0.35  <  = 140, R • •Ro  = 1 4 0 , ^ 0 - •Ro = 140,7?2o- •Ro  4  1 / 4  5  1 / 4  4  2  0.2  0.05 0 -0.05  10  1000  100  Central Pressure P (10 dyne/cm ) 33  2  c  Figure 4.10: The change of mass M, total baryon number N and radius R of quark stars in 2SC+S phase at an increase of temperature from 0 to 5,10,20 MeV plotted against a range of central pressure. In general all three of M,N and R increase with T and for each final temperature, the change peaks at some particular pressure. For high central pressures and for some bag constants we see a decrease of N . B  B  B  lower temperatures. In the plot (Fig. 4.11), we see as the temperature is increased, the peak shifts towards higher central pressures. 104  Chapter 4. Compact Stars with Colour Superconducting Cores Change of Mass, Total Baryon Number and Radius with central temperature vs Central Pressure for Quark Star 2.5  B = 140, M - M B = 140, Mi - M B' = 150, M .5 - M B ^ = 150, Mi - M 1/4  0  1/4  x  5  0  0  L A  1  0  4  0  0  1.5 1 0.5 bo S3 OS  o 05  O  <  J3  0 1.5 1 0.5 0 -0.5 -1 -1.5 -2 17.5 15 12.5 10 7.5 5 2.5 0  H—I  i l l I |  i  II  H  1—i  I  I I  -I  1  I  I I I | J,  1—i  1  i  M  I I  B / = 140,^0.5-^0 • B I = 140, NBI — NBO .B l*1 = 150, iVflo.6 - iVflo 1  4  l  B /" = 150, AT - A^ fll  H  H  -  B0  1  1  1  1  I  I I I I  B /* = 140;/J .5--Ro 1  0  B 4/  4  B /  4  1  = 140, R - R o x  BV = 150,Bo. -i?o 1  1000  10 100 Central Pressure P (10 dyne/cm )  1  33  "  5  = 150, Rx-Ro  2  c  Figure 4.11: The change of mass M , total baryon number NB and radius R of quark stars in 2SC+S phase at an increase of temperature from T = 0 to T = 0.5,1 MeV plotted against a range of central pressure. Mass- and radius-change are always positive while NB is seen to decrease at high central pressures for some bag constants. 5. Also, the lower the bag constant, the more the maximum change of total baryon number (see the cases for B — (140 MeV) and B = (150 MeV) in the Figs. 4.11 and 4.10 ). 4  4  6. At higher temperature increases, the shapes of the plots of the change of 105  Chapter 4. Compact Stars with Colour Superconducting Cores Relative Change of Mass and Total Baryon Number with central temperature vs Central Pressure for Quark Stars with strange quarks present B  = 140, (M - M )/M = 140, (Mio - M / M B / = 140, M - M / M By = 150. ( M - M n /Mo B ,, — 150, (M - Mo)/Mo B I = 150, (M - M )/M 0.004  0.12 |-  B  1 / 4  6  1  4  2 0  4  0.1  0  0  l / l  0  0  0  0  0  o  5  10  20  0.08 0.06 h 0.04 J§  - O.OIJk  loqo  0.02 0  H  1—I  I I IIJ  B / = 140, N - N Bj = 140,7V - N 1  B5  0.12  B0  B10  B0  B . = 140, N N BV = 150, N -N B / = 150, N -N B' = 150, N -N B20  B0  4  0.1  S  B  B  1  0  B  0 0  B  2  0  B  0  1 4  0.004 0.003 0.002 0.001 0  0.08 0.06  B  4  No.ooi  -&.Q02 -100  0.04  loqo  v  0.02 0  10 100 Central Pressure P (10 dyne/cm ) 33  1000  2  c  Figure 4.12: Change of gravitational mass and total baryon number of quark stars with 2SC+S phase, per unit gravitaionnal mass as a function of the central pressure at increase of the central temperature of the stars. The relative change of mass is always positive while that of the total baryon number becomes negative at high central pressures. In general, the higher the central pressure the less are the magnitudes of the relative changes. total baryon number, are very similar to those of the mass-change. The difference in the shapes is more noticeable at low temperatures. The peaks in the change of baryon number occur at central pressure values, 106  Chapter 4. Compact Stars with Colour Superconducting Cores in  Relative Change of Mass and Total Baryon Number with central temperature vs Central Pressure for Quark Stars with strange quarks present  Central Pressure P (10 dyne/cm ) 33  2  c  Figure 4.13: Relative change of gravitational mass and total baryon number of quark stars with 2SC+S phase at increase of central temperature from T = 0 to T = 0.5,1 MeV's. The relative mass-change is always positive while the relative change of N becomes negative for some bag constant and central pressures. At higher bag constants the the relative change in NB becomes negative at low central pressures. B  slightly below the corresponding central pressure values, at which the mass-change peaks. 7. Unlike the case of mass-change, the total baryon number does not always have an increase with central temperature for the same central pressure. As the central pressure is increased, the difference AN deB  107  Chapter 4. Compact Stars with Colour Superconducting Cores Relative Change of Radius with central temperature vs Central Pressure for Quark Stars with strange quarks present 0.06 F  B/  1 4  B  4  B^  0.05  = 140, {R - Ro) Ro = 140,fk -i?o)Ro 5  10  = 140, (R  4  20  &J*  B'  1 4  - Ro}R  0  = 150.(^5 -  = 150, (R  w  Ro  - RoVRo  BV = 156;(R -R»)/R« 4  q  0.04  <  "\ 0.03  q  20  0.0015 0.001 \0.0005 "\ 0 -0.0005 soo  100  v  bo C  0.02  1  c3  o  0.01  10 100 Central Pressure P (_10 dyne/cm ) 33  1000  2  c  Figure 4.14: Relative change of radius of quark stars with 2SC+S phase as a function of central pressure at change of central temperature. The change AR/Ro is always seen to be positive but decreases in magnitude as the central pressure increases. For higher bag constants, the change is less. creases and eventually becomes negative. This is seen at even small increase of temperature (see Figs. 4.11, 4.12, 4.13) such as A T = 0.5,1,5 MeV's. However, as AT becomes larger, the point at which AN becomes negative, shifts towards the higher pressures. B  Naively, as the temperature increases, we would expect an increase of the total gravitational mass and also of the baryonic mass of the star. However at finite temperature, we also expect anti-baryons to be populated. The plots show that, comparing two configurations of the star at the same central pressure but at zero and finite temperature, we may get the difference of the net number of baryons (A(N _ ), written as AN ), as negative. This is despite the fact that the net baryon number density n for the quark phases increases with temperature. Also note that the binding energy in the quark phase is never negative (see section 4.4.2). B  B  B  B  108  Chapter 4. Compact Stars with Colour Superconducting Cores Relative Change of Radius with central temperature vs Central Pressure for Quark Stars with strange quarks present  Central Pressure P (10 dyne/cm ) 33  2  c  Figure 4.15: Relative change of radius of quark stars with 2SC+S phase versus central pressure at increase of central temperature from T = 0 to T = 0.5,1 MeV's. We see similar temperature dependences of radius as in the plot (4.14) for the case of larger temperature increases. 8. The plots for change of radius with increase of central temperature at the same central pressure, are also seen to have peaks for each final temperature with respect to central pressure. The peak for each final temperature however occurs at different central pressure values than those for the change of mass or NB- In general, the increase of radius is smaller for higher bag constants employed. In the plots of the relative change of mass and total baryon number of quark stars, we see that the change becomes smaller as the central pressure increases. Since we expect the quark stars to be formed with higher baryon chemical potentials and central pressures, the higher end of the plots are more physically relevant than the end with low central pressures. For central pressures of 1.0 x 10 dynes/cm or higher, the relative change of mass in an increase of temperature from T = 0 to T = 20 MeV is AM/M < 3.5%. We see a similar change in NB- In the plot for the relative change in 35  2  0  109  Chapter 4. Compact Stars with Colour Superconducting Cores Mass, Total Baryon Number and Central Baryon Chemical Potential vs Central Pressure for non-strange Quark Star at T=0,10,20 MeV 2.5  B' B'/ 2 B / B'/ B'/i 1.5 | _ BB' /' , B' 1 LB'/ 1 4  4  1  4  4  1 4  1 4  I  1— — i— i i III  =  4  |  i i 1111  = 135, T = 0 MeV = 135,T = 10 MeV = 135,T = 20 MeV = 140,T = 0 MeV 140, T = 10 MeV = 140, T = 20 MeV = 145, T = 0 MeV = 145, T = 10 MeV = 145,T = 20 MeV  4  0.5 0 35 30 to lO  S  x ^  z  «  1  » 'l'«l'<<"»  HBrJwTJn  B ' = 135, T = 0 MeV B '* = 135, T = 10 MeV B = 135, T = 20 MeV 1 4  1  114  B'  / 4  = 140,T = 0 MeV  B = 140, T = 10 MeV B>l = 140,T = 20 MeV 20 j_ B ' = 145, T = 0 MeV , B = 145, T = 10 MeV 1>4  4  1 4  1 / 4  15 | _ B  = 145, T = 20 MeV  1/4  10  o 1500  1  B / = 135, T = 0 MeV B = 135, T = 10 MeV . • 1 / 4 _ 135, T = 20 MeV B = 140, T = 0 MeV £ i / 4 _ 140, T = 10 MeV R I / 4 _ 140 T = 20 MeV B ' = 145, T = 0 MeV B = 145, T = 10 MeV IB = 145, T = 20 MeV ^ ^ = 900 MeV 1  4  1—I  I I I I 11  1  1—H  1 / 4  1 / 4  >  1  4  1 / 4  1 / 4  0)  a.  1  10 100 Central Pressure P (10 dync/cm ) 33  1000  2  c  Figure 4.16: Mass, total baryon number and central baryon chemical potential for quark stars with 2SC or 2SC+NQ phase, versus central pressure at different central temperatures. The portion of the plots beyond the peaks of M and N correspond to gravitationally unstable configurations. For higher bag constants the mass and total baryon number decreases. The central baryon chemical potential p at low central pressures have values low enough to correspond to unphysical configurations of the quark stars. B  Bc  110  Chapter 4. Compact Stars with Colour Superconducting Cores radius, at the higher end of the central pressure range, we see a change of AR/Ro « 0.01%, 0.1%, 0.05% for increase in temperature of AT = 5,10,20 MeV's, respectively. As the central pressure increases, the relative change becomes less dependent on the bag constant used. Hot Quark Stars without strange quarks For quark stars with quark-matter in two flavours only (e.g in the 2SC, noncolour-superconducting up and down quark (u,d NQ) or the mixed phase (2SC+NQ) ), we see the following features in the plots of the change of mass and total baryon number due to change of temperature: 1. For a bag constant with B ^ = 135 MeV, at low central pressure, we see decrease of mass with increase of temperature in the plots. However, the values of the central pressure at which the mass is seen to decrease, correspond to central baryon chemical potential UBC 900 MeV, at which we do not expect the matter to be in the quark phase. Thus, we may ignore the results for quark stars with such low central pressures. Furthermore, since 900 MeV is the lower limit of the tabulated values of [iBc, errors may creep in, while getting the equation of state by extrapolation beyond the lower limit of JIBC1  2. We see a decrease of the total baryon number NB of the star, for low central pressures and for bag constant 5 / = 135 MeV, at temperatures T = 5,10,20 MeV and for B ' = 140 MeV, at temperatures T = 20 MeV. The corresponding central baryon chemical potential is however not large (< 940 MeV) and hence we may ignore the results as unphysical. 1  4  1 4  3. The change of mass A M has a maximum at some central pressure for each final temperature, similar to the case of quark stars with strange quarks. The peak is seen to shift towards higher central pressures as the temperature increases, but at a fixed bag constant. However, as the bag constant is increased, the peaks of the change of mass (AM), occur at slightly lower central pressures. 4. The change of total baryon number ANB is also seen to have a maximum at some central pressure for each final temperature, similar to the case of quark stars with strange particles. The peaks are seen to have 111  Chapter 4. Compact Stars with Colour Superconducting Cores Mass, Radius, Total Baryon Number and Central Baryon Chemical Potential vs Central Baryon Number Density for non-strange Quark Stars at T=0,10,20 MeV  1.5 1 0.5 0 35 30 25 = 135 'T = = 0 MeV 135, T = 10 MeV 135, T = 20 MeV -- 140. T := 0 MeV 140, T = 10 MeV 140, T = 20 MeV 145 'T ••= 0 MeV 145, T = 10 MeV 145, T = 20 MeV  20 15 10 5  H—  0  -  12 10 B B / B / B / B / B'/ B'/ B' B /  = 135,7 = 0 MeV = 135, T = 10 MeV = 135, T = 20 MeV = 140.T = 0 MeV = 140, T= 10 MeV = 140, T = 20 MeV = 145. T = 0 MeV = 145, T= 10 MeV = 145,T = 20 M e V I 1 1.1 0.8 0.9 l / i  1 1  4 4  1  1  4  4  4  4  1 1 1  0.1  0.2  0.3  0.4  0.5  0.6  0.7  4  Central Baryon Number Density ns  1.2  (fm~ ) 3  c  Figure 4.17: Mass, total baryon number and radius of quark stars with 2SC or 2SC+NQ phase as function of central baryon number density ns at different central temperatures. The mass and the total baryon number peaks at the higher end of the range of ns while the radius peaks at much lower ns Configurations of the stars before the mass peaks are gravitationally stable. As the bag constant is increased, all of M, ns and R are seen to decrease. c  c  c  c  the same kind of shift towards the higher (or lower) central pressures with the change of temperature (or bag constant employed). 112  Chapter 4. Compact Stars with Colour Superconducting Cores Change of Mass and Total Baryon Number with central temperature and Central Baryon Chemical Potential vs Central Pressure for non-strange Quark Star 0.035  B '" = 135, M - M 0.03 _B'' = 135, M i o - M B ' = 135, M - M S'/" = 140, M - M 0.025 'B '" = 140, M - M B / = 140, M - M 0.02 1  5  0  2 0  0  S  1 4  5  1  5 <  1  1 0  4  0  0  0  2 0  0  0.015 0.01 0.005 43  o  0 -0.005 0.2  1  1—I—I I I I I |  B /* = 135,7Vs5 - N B / = 135,JV -7V B '" = 135, N - N B){ = U0,N -N B J = 140, 7V - N B = 140, N o - N 1  B0  4  B10  1  0.15  4  B20 m  0.1  B0  B10  1 / 4  B2  B0  B0  —t—I—I I I I I  • • • -  B0 B0  0.05  z <  0 -0.05 -0.1 B -B ' B' B "B B  1/4  1050  1  4  1 4  1 / 4  1000  1/4  1 / 4  . H—1 I I I I I = 135, T = 0 MeV = 135, T= 10 MeV = 135, T = 20 MeV = 140, T = 0 MeV = 140, T = 10 MeV = 140, T = 20 MeV  950 900 850  1  10 100 Central Pressure P (10 dyne/cm ) 33  1000  2  c  Figure 4.18: Difference of mass and total baryon number and the central baryon chemical potential JIBC of quark stars with 2SC or 2SC+NQ phase versus central pressure at change of central temperature to T = 5,10,20 MeV's. At low central pressures the central baryon chemical potential is low enough that quark stars can not be produced and hence correspond to unphysical configurations. The change in the mass and total baryon number peak at some central pressure values and decrease with increase of bag constant at higher central pressures.  113  Chapter 4. Compact Stars with Colour Superconducting Cores Difference of Mass and Total Baryon Number with change of Central temperature versus Central Baryon Chemical Potential for non-strange Quark Stars 0.045  B ' = 135, M B '1 /4 = 135, M B '1 4 = 135, M B ' = 140, M B ' / = 140, M •-flj/ = 140, M 1 4  5  0.04  Co  1 0  2 0  0.035  1  0.03 0.025  S a  0.015  0  0  4  2  o  0  5  4  <  - M — -M — - M — - Mo — -M -- M 0  4  0  0  0.02  V  bO  J3 O  0.01 0.005 0 -0.005 0.2  1  H  h  h  0.15 0.1  to  O X  «  Z  <  0.05 0  B' := 1 4  135, N - N 135, JV - N N -N B = 135, B ' / : -- 140, N - N 140, JV N B'/ = B / = 140, Af - N  -0.05  Bh  B0  fll0  B0  1 / 4  B20  -0.1  ji  Bc  = 940  4  MeV  4  1  -0.15 900  950  4  B0  m  B0  BI0  B0  B20  B0  1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 UBC  (MeV)  Figure 4.19: Difference of mass and total baryon number as functions of the central baryon chemical potential ps of quark stars with 2SC/g2SC or (2SC/g2SC)+NQ phase at change of central temperature to T = 5,10,20 MeV's. For physically relevant values of the central baryon chemical potential the mass difference is always positive. The differences A M and ANB peak at some central baryon chemical potential values for each temperature difference. In the plots of the relative change of mass and total baryon number, we see the following features: c  1. We see a maximum relative increase of mass AM/MQ < 2.75% for bag constant _B = (140MeF) at A T = 20 MeV. For a lower temperature increase of A T = 1,0.5 MeV's, the peaks in the relative mass-change are AM/M < 1.8 x 10~ %, 5.0 x 10" % respectively, for bag constant of B = (135MeV) . For an increased bag constant, B = (140MeV) the relative change is higher ( AM/M < 2.8 x 10~ %, 1.0 x 1 0 %), 4  2  3  0  4  4  2  -2  0  114  Chapter 4. Compact Stars with Colour Superconducting Cores Change of Mass and Total Baryon Number with central temperature and Central Baryon Chemical Potential vs Central Pressure for non-strange Quark Star  Central Pressure P (10 c  33  dyne/cm ) 2  Figure 4.20: Difference of mass and total baryon number of quark stars with 2SC or 2SC+NQ phase as a function of central pressure at change of central temperature to T = 0.5,1 MeV's. The changes AM and AN have similar behaviour as for the case of higher temperature changes in the plot (4.18). B  but of the same order of magnitude as for B = (135 MeV) . 4  2. At physically relevant values of the central pressure where we can expect the stars to be made of quarks only, the relative change is much lower, typically 1 ~ 2% for increase of temperature A T = 10,20 MeV and 1 x 10~ % ~ 1 x 10~ % for A T = 1,0.5 MeV. 3  2  115  Chapter 4. Compact Stars with Colour Superconducting Cores Difference of Mass and Total Baryon Number with small change of Central Temperature versus Central Baryon Chemical Potential for non-strange Quark Stars  -0.0005 -0.001 900  950  1000 1050 1100 1150 1200 1250 1300 1350 1400 1450 1500 UBC  (MeV)  Figure 4.21: Difference of mass and total baryon number as functions of the central baryon chemical potential UBC at change of central temperature from zero to T = 0.5,1 MeV's. For physically relevant values of the central baryon chemical potential the mass difference is always positive. The changes AM and ANB have similar behaviour as for the case of higher temperature changes in the plot (4.19). 3. The maximum relative change of total baryon number is however smaller {AN /NBO < 1% for B = 140 MeV). Typical values of N /N for physically relevant central pressures is NB/NBO ~ 0.5% at T = 20 L  /  I  B  B  MeV, NB/N O B  » 0.35% a t T = 10 MeV and N /N B  B0  B0  « 0.1% at T = 5  MeV. 4. The relative change of both mass and total baryon number are seen to be less dependent on the bag constant employed as the central pressure 116  Chapter 4. Compact Stars with Colour Superconducting Cores Relative Change of Mass and Total Baryon Number with central temperature vs Central Pressure for non-strange.Quark Star  850 I 10  1  ^-1—'  '—''''I 100 Central Pressure P (10 c  ' 33  1  '  1—>  ' • 'i 1000  dyne/cm ) 2  Figure 4.22: Relative change of mass and total baryon number for quark stars with 2SC or 2SC+NQ phase versus central pressure at different central temperatures. At physically relevant values of the central pressure, where quarks stars are realizable, the changes decrease with increase of central pressure and bag constant. At the higher end of the pressure range, the dependence on bag constant is less. becomes large.  117  Chapter 4. Compact Stars with Colour Superconducting Cores Relative Change of Mass and Total Baryon Number with central temperature vs Central Pressure for non-strange Quark Star  100 Central Pressure P (10 c  1000 33  dyne/cm ) 2  Figure 4.23: Relative change of mass and total baryon number of quark stars with 2SC or 2SC+NQ phase as a function of central pressures at change of central temperature to T — 0.5,1 MeV's. The relative changes have similar behaviour as for the case of higher temperature changes in the previous plot (4.22). The absolute as well as the relative change of the radius of the nonstrange quark stars, both are seen to have peaks at some values of the central pressure, for each final temperature and bag constant used. The maximum 118  Chapter 4. Compact Stars with Colour Superconducting Cores Absolute and Relative Change of Radius with central temperature vs Central Pressure for non-strange Quark Star B / = 135, (Rs B\l* = 135, (R B = 135, fljo B = 140CiFk B { = 140, (R «••'' - 140, (R 1 4  l0  1  4  x  4  10  20  Ro)/Ro RO)/RQ Ro) Ro RSRO — Ro) Ro Ro) Ro —  B / = 135, R, - Ro • B J = 135,fi -Ro B ' = 135,fi o-flo Bl 4 = 140, i? - Ro • 4  10  1 4  2  1 / 4  5  B J = 140, /Jio - Ro  - - J3»/ = 140, fi o - Ro 4  2  B l" = 135, T = 0 MeV B / = 135, T = 10 MeV B = 135,T = 20 MeV B ' = 140, T = 0 MeV B ' = 140, T = 10 MeV B = 140, T = 20 MeV = 940 MeV l  1  1050  4  1 / 4  1 4  1000  1 4  1 / 4  950  )i  Bc  900  Central Pressure P (10 c  33  dyne/cm ) 2  Figure 4.24: The absolute and relative changes of radius of quark stars with two flavours in the 2SC or 2SC+NQ phase, as a function of central pressure at temperature increase of A T = 5,10,20 MeV's. As the central pressure increases, the change in radius decreases. At higher central pressures the dependence of the changes on the bag constant becomes less. change of the radius is seen to be of the order of 100 m for a increase of temperature of 20 MeV. The relative change in radius is typically AR/R < 0.8%, 0.2%, 0.1% for high central pressure at T = 20,10,5 MeV temperatures, 0  119  Chapter 4. Compact Stars with Colour Superconducting Cores Absolute and Relative Change of Radius with central temperature vs Central Pressure for non-strange Quark Star  1000 Central Pressure P (10 c  33  dyne/cm ) 2  Figure 4.25: Absolute and relative change of the radius of quark stars with 2SC or 2SC+NQ phase versus central pressure at temperature increase of A T = 0.5,1 MeV's. The changes of radius show similar behaviour as in the case of higher temperature changes in the previous plot (4.24). respectively. The relative increase is however seen to be less dependent on the bag constant used as the central pressure becomes high.  120  Chapter 4. Compact Stars with Colour Superconducting Cores Mass vs Radius of Hybrid and Hadronic Stars with and without strange quarks at central temperature T=0,10,20 MeV 1.6 - B / B / 1.4 B / 'B ' B / 1.2 - B  = 160, T = 0 MeV = 160. T = 5 MeV = 160, T = 10 MeV = 160,T = 20 MeV = 180, T = 0 MeV = 180, T = 5 MeV B ' = 180, T= 10 MeV - £ / = 180,T = 20 MeV B = 200. T = 0 MeV B' = 200, T = 10 MeV B = 200, T = 20 MeV 1  4  1  4  1  4  1 4  1  4  1 / 4  2  1 4  1  1  4  1 / 4  1 4  i  °-  ^  0.6  8  1 / 4  0.4 0.2 0  - B / = 150, T = 0 Me B / -= 150. T = 5 MeV = 150, T = 10 MeV 1.75 B - i/4 150, T = 20 MeV B / = 155,7 = 0 MeV 1.5 B / -= 155 'T = 5 MeV B' = 155, T = 10 MeV 155, T = 20 MeV 1.25 B / " == 160,T = 0 MeV - B /" = 160, T = 5 MeV B B / = 160, T = 10 MeV 160, T = 20 MeV 1 - i/4 1  4  1  4  1 / 4  B  =  1  4  1  4  1 4  1  1  I  1 / 4  1  B  4  =  0.5 0.25 0  10 Radius (km)  Figure 4.26: Mass versus radius relationships of hadronic stars and hybrid stars with quark matter cores, with and without strange particles present, at zero and finite central temperatures. The structure and nature of the stars depend on the bag constant used. Non-strange stars in general have larger mass and radius compared to stars with non-zero strangeness at the same bag constant and central pressure.The temperature dependence is stronger at lower central pressures and for stars with large hadronic regions.  121  Chapter 4. Compact Stars with Colour Superconducting Cores Mass and Total Baryon Number and Central Baryon Chemical Potential vs Central Pressure for Hybrid Star at 1=0,10,20 MeV  100 Central Pressure P (10 c  33  1000  dyne/cm ) 2  Figure 4.27: Mass, total baryon number and central baryon chemical potential of hybrid stars with quark cores in the 2SC+S phase with different bag constants and at central temperatures T = 10,20 MeV's. At low central pressures and bag constant of B = (160 MeV) we have unstable configuration of stars at temperature T = 10,20 MeV's. At higher bag constant B = (180MeV) , we see another unstable branch at central pressures where the quark core just starts to appear. 4  4  4.4.4  Error in the Determination of Radius, Mass and Total Baryon Number of Compact Stars  In the integration of the TOV equation, as we go towards the edge of the star, the local pressure decreases and corresponding to this pressure, in the extrapolated equation of state, the baryon chemical potential gets below the 122  Chapter 4. Compact Stars with Colour Superconducting Cores Mass and Total Baryon Number and Central Baryon Chemical Potential vs Central Pressure for Pure Hadronic Star at T=0,10,20 MeV 1.4  900  I  """ 1  '  ' —  '  1  —  '''''I  10 100 Central Pressure P (10 dyrie/cm ) 1  33  1  '—'  ' ' i i '  1000  2  c  Figure 4.28: Mass, total-baryon number and central baryon chemical potential of pure hadronic stars as a function of central pressure at zero and finite central temperatures. As the central pressure increases both mass and total baryon number attain maximum values and then decrease. The branch beyond the peak in the mass corresponds to unstable configurations of the star. The temperature dependence of the quantities becomes less as the pressure becomes higher. lower limit of the table. Thus the corresponding values of other thermodynamic variables like the energy density or baryon number density, obtained from extrapolation may have significant amount of error and we can not trust their values very much. The termination criteria for the integration loop along the radial direction used here is, any one of the energy density, 123  Chapter 4. Compact Stars with Colour Superconducting Cores the baryon number density or pressure becoming zero or negative first. Due to error in the extrapolation, the energy density or baryon number density may become negative before the pressure becomes so. Hence, in general, we may get significant error in the estimate of the radius of the star. However, for the determination of the mass of the compact star, since near the edge of the star, the baryon number density becomes very small, so also the mass density of the matter and hence the mass of the outer part of the compact star is only a small fraction of the total mass of the star. Thus, we may ignore the error in the estimate of the mass and the total baryon number of the compact star due to error in extrapolation.  4.4.5  Hot Hybrid and Hadronic Stars  As the bag constant increases, within a hybrid star the hadronic phase becomes more and hence the size of the quark core shrinks. At some high bag constant, e.g. B l « 194 MeV at T = 0 MeV, for stars with strange particles, the quark core disappears completely. For non-strange stars, this happens at a much lower bag constant for B / « 160 MeV at T — 0 MeV. For low central pressure, even for lower bag constants, the compact stars are made only of the hadronic phase. The same kind of behaviour persists at finite temperatures. For hybrid and hadronic stars, we see stronger temperature dependence of the mass and radius as well as the total baryon number as compared to those of the quark star. We see the following features in the plots of mass vs. radius and mass vs. central pressures of hot hybrid and hadronic stars: l 4  1 4  1. Large radius at low central pressure: At bag constants near the value where the compact stars start- to have a hadronic mantle (B « (lQOMeV) for stars with strange particles and B « (146MeV) for non-strange stars), we see a large radius of the star for small central pressures. As the mass of the quark core at low central pressures is relatively small, (and hence the total mass of the star M ~ 0.005M ), the integration along the radial direction continues for a long distance until the pressure becomes negative. This is due to the nature of the soft equation of state of the hadronic phase. Hence, the hadronic mantle becomes very fat and the radius of the hybrid star becomes large compared to that of a pure quark star, with the same central pressure. 4  4  star  o  124  Chapter 4. Compact Stars with Colour Superconducting Cores Change of Mass and Total Baryon Number with increase of Central Temperature vs Central Pressure for Hybrid and Hadronic Star 0.3  B B B B B B /  1 / 4 1 / 4  0.25  1 / 4 1 / 4  0.2  1 / 4  1  0.15  4  = = = = = =  160, 160, 170, 170, 180, 180,  Mm M Mio M Mio M 2 0  2 0  2 0  -  M Mo M M M M  -  M M M M  0  0  0  0  0  0.1 0.05 0 0.2  : 190, Mio B > - : 190, M B ' •- 200, M • 200, M l  4  2 0  0.15  1 4  :  w  2 0  0 0  0 0  0.1 0.05 0  7++f  -H-H-  3.5  B/ BV B B B ,B  4  3  B  4  2  B0  B 1 0 B2  B0  B  Bl0  1 / 4  I I I I  O  B20  4  14  2.5  1—I  H  = 160,JV io-7Vfl = 160, JV - N = 170, / V - N = 170, JV o - N o = 180, N - N = 180, JV o - N  B0  B2  B 0  1.5 1 0.5 0 2.25 2 1.75 1.5 1.25 1 0.75 0.5 0.25 0  B  = 190,JV  4  B  4  fllo  -iV o B  =  190,^20-^0  =  200,^20-^0  B / = 200, iV o - WHO 1  4  B1  B'/  1  4  1000  100  10  Central Pressure P (10 dyne/cm ) 33  2  c  Figure 4.29: Difference of mass and total baryon number of hybrid and hadronic star configurations as a function of central pressure at zero and finite temperatures. In general the mass-change AM decreases with increase of central pressure and bag constant. For certain bag constants AM shows one or more local maximum depending on the degree of hybridness.The shape of the change AN follows closely that of AM. B  125  Chapter 4. Compact Stars with Colour Superconducting Cores Relative Change of Mass and Total Baryon Number with increase of central temperature vs Central Pressure for Hybrid and Hadronic Stars Bl B'/" B B"* B" Bl i i  10  i/4  4  l 4  6h  = 160, (Mio = 160, ( M = 170, M = 170, ( M = 180, ( M = 180, (Mm  20 1 0  20 10  - M )/M - M )/M„ -M )/M . - M )/M - M )/M -Mo)/M 0  0  0  0  0  0  0  o  0 9  2 0  H—I  H—I  III  I I I I |  B'/" = 190, B ' / i = 190, B / = 200, B l* = 200,  0.8  1  4  i  ( M - Mo)/M • (M - M )/M • (A/,0 - Mo)/Mo • \M -Mo) I Mo • 10  0  20  0  0  m  0.6  5r  0.4 0.2 0 _ B\l* B\l* B ' B]' B '*  10 8  I  1  4  6  0.5 0.4 0.3 0.2 0.1 0  2 0  4  1  m,N N N AT N „ JV  - iV -  BK  B2a Bla  B20  B1  H?n  BI  1000 H  1 I I I I  I  B\'* B / B\' B"  0.8  4  4  0.6  "is5  = 160, = 170, = 170, = 180, = 180,  = 190, N = 190, N = 200, JV = 200,iV  B10 S 2 0 BI0 B20  - iV • - Nso • - WHO • B0  0.4  <  0.2 0  10 100 Central Pressure P (10 dyne/cm ) 33  1000  2  c  Figure 4.30: Relative change of mass and total baryon number of hybrid and hadronic stars versus central pressure at change of stars' central temperature from T = 0 to T = 10,20 MeV's.The mass changes are always positive while that for total baryon number may become negative at high central pressures for some bag constants used. 126  Chapter 4. Compact Stars with Colour Superconducting Cores This is clearly seen at zero central temperature T ~ 0 MeV stars in the figure (4.26). c  2. Decrease of radius at finite temperature: At non-zero central temperature of the hybrid star, we expect a more massive compact star. Hence for the star to be gravitationally stable we expect the star to become more compact i.e. the radius of the star to decrease. We see it in the figure 4.26 of mass vs. radius of the hot hybrid and hadronic star. As a result we see a decrease of the radius although the temperature of the star as a whole increases. This is only true for stars whose total gravitational mass is relatively small. For more massive compact hybrid/hadronic stars (like the quark stars), we see increase of radius with temperature. 3. Unstable branch at finite temperature: For stars with strange particles, at bag constants near the transition value from pure quark stars to hybrid stars (B (160MeV) for stars with strange particles), we see that the radius at the low central pressures starts to become smaller as the central temperature increases. In the mass vs. radius relation, we see the appearance of an unstable branch at finite temperature T = 5,20, 20 MeV's and at low central pressures, which become stable as the central pressure increases (c.f. Figs. 4.26,4.27). This temperature driven unstability is absent if higher bag constants are used. For non-strange hybrid stars, we do not see any such unstability in the M — R plots. Rather, as the central temperature increases, we see decrease of the radius from to that at zero temperature and without any unstable branch appearing. 4  4.4.6  Change of Mass, Radius and Total Baryon Number of Hybrid and Hadronic Stars with Temperature  Hot Hybrid and Hadronic Stars with Strange Particles We note the following features in the plots of change of mass, total baryon number and radius of hybrid and hadronic stars with strange particles present: 1. In general, hybrid and hadronic stars with strange particles are seen to increase in mass and total baryon number as the temperature increases. 127  Chapter 4. Compact Stars with Colour Superconducting Cores Absolute and Relative Change of Radius with increase of central temperature vs Central Pressure for Hybrid and Hadronic Stars 4  B  1 / 4  B ' B\l 1  4 4  3  B / 1  B ' B '  2  1  4  1  4  4  = 160, = 160, = 170, = 170, = 180, = 180,  fiio - Ro R R R R R  20 l 0  20 w  20  -  Ro Ro Ro Ro Ro  1 0 -1 1.4  _ / J  1.2  B ' B ' B  1  1  4  1  4  4  20  10  1 / 4  0.8 0.6 0.4 0.2 0 0.8  H  1—I  I  I I I I  I I I B ' B ' B ' B J 1  0.6  1  4  4  4  0.4  B / B' 1  4  I I = = = =  = 190,i?io -Ro • = 190, R - Ro • = 200, R - Ro •  = 200, 7?20 - Ro •  1  (R [R o [R [R20 = 180, [Rio 160, 160, 170, 170,  10  2  l0  = 180, fl  1 4  20  1 1—I I I I - Ro)/Ro - R o ) / R o - RoSlRo - Ro)/Ro - Ro) Ro  - Ro)/Ro  0.2 0 -0.2 0.12  B B'/ B' B'/  1 / 4  0.1  4  / 4  0.08  4  = = : :  190, (Rio - Ro)/Ro 190, [R o - Ro)/Ro 200, Bio - Ro)/Ro 200, fl-20 - Ro)/Ro 2  ~— —  0.06 0.04 0.02 0 100  10  1  Central Pressure P (10 c  33  1000  dyne/cm ) 2  Figure 4.31: Absolute and relative change of radius of hybrid stars with quark cores in 2SC+S phase and hadronic stars versus central pressure at increase of central temperature from T = 0 to T = 10,20 MeV's . For bag constant of B = (160 MeV) as the temperature increases the star's configuration at low central pressures, changes as its radius shrinks. These correspond to unstable configurations of hybrid stars. 4  For bag constant with B ^ = 160 MeV and at low central pressures, 1  128  Chapter 4. Compact Stars with Colour Superconducting Cores the hybrid stars having large hadronic outer region, the mass change is very large, A M < O.3M , at T = 20 MeV, resulting in a relative change of A M / M < 10. This is a very strong temperature dependence, probably due to the small mass of the star and we expect this to be unphysical. At higher central pressures, the relative mass change reduces ( A M / M < 2%), which we can expect to be realistic for a more relativistic star. 0  0  0  2. For bag constants with JE? / = 170,180 MeV, at low central pressures, we see a decrease of mass-change at T = 10 MeV, while there is a local maximum in the A M , for T = 20 MeV. The relative changes are however large A M / M ~ 50%, 100%, respectively. At higher central pressures, when the quark cores begin to appear within the stars, we have another local maximum the the plot for mass change both at T — 10,20 MeV's. As the bag constant increases, the local maxima for the hybrid stars shift towards the higher pressure end and become smaller in magnitude. 1 4  0  3. The plot of the change of the total baryon number follows closely that of the mass-change. However, for purely hadronic stars, at the higher end of the central pressure, P « 4.0 x 10 dyne/cm , we see a decrease of the total baryon number, similarly to the case of hot quark stars. 35  2  c  4. In the plots of absolute and relative change of radius with temperature, we see maximum in the change, at certain central pressures for each final temperature. For bag constant with f? / = 160 MeV, we see a decrease of radius at low central pressures, as discussed in the section 4.4.5. The corresponding maximum increase in the radius is seen to be large. For hybrid stars at bag constants B ^ = 170,180 MeV, we see another local maximum in the plots as the central pressure becomes larger. At the higher end of the central pressure, the relative change in the radius is AR/R » 0.5 - 1.0% for bag constants of B = 170,180 MeV's depending less on the temperature increase, while it is larger {AR/RQ « 3%) at T = 20 MeV, for higher bag constants. 1  4  1  L/4  0  129  Chapter 5 Cooling of Quark Stars Compact Stars with and without Trapped Neutrinos  5.1  It is believed that stars of the main sequence, with a mass of M > 8M© at the end of their nuclear burning life, undergo core collapse and supernova explosion to become either a black hole or a protoneutron star [30]. During the process of formation, the final protoneutron star becomes lepton-rich as neutrinos are produced by inverse beta-decay of protons. When the density of the collapsing star reaches above the critical density for neutrino trapping p -tra.p ~ 10 g/cm , the neutrinos produced are essentially trapped inside the collapsing star. This is because the diffusion time of the neutrinos becomes larger than the collapse time [17]. During the collapse, the core temperature rises due to infall of material and contraction. The protoneutron star, thus produced, has a high temperature TPNS ~ 30 — 50 MeV [58]. The neutrino mean-free-path at these temperatures is shorter than the radius of the protoneutron star r„ < RPNS ~ 10 km. Thus most of the neutrinos remain trapped while the star cools by diffusion of some neutrinos. As the star's temperature becomes ~ 1 MeV, the neutrino mean free path becomes comparable to the star's radius and the star becomes transparent to the neutrinos. After the neutrino transparency, the star cools via neutrino emission from the core. The crust of the star however remains relatively warm as it cools relatively slowly. The crust acts as an insulating layer that keeps the star warm for up to 100 years after which the interior of the star cools down enough to come to thermal equilibrium. If during the collapse or immediately after that, the central density of the compact star is high enough for a phase transition from hadronic matter to complete quark matter, quark stars may be produced. We assume that for a quark star, most of the neutrinos have diffused through the star resulting in 12  3  u  130  Chapter 5. Cooling of Quark Stars Change of Mass at Constant Total Baryon Number for Quark Stars with free strange quarks at increase of temperature to T = 10,20 MeV 1.4  £i/" = 140, T = d MeV B> X  4  = 140, T = 10  MeV  fli/i = 140, T = 20 MeV 1.35  1.3  B '-  AM'a4)»0.0bl8Mo  1.25  ]  « 0.01635M '' A M _ o ~ O.OO312M AM20-0  1.2  10  Q  0  1.15  1.1 20 G-  19  2  18  x  ~o z  17  £i/4 B' £V4 1 4  16 0.425 15  0.4375  = 140, T = 0 MeV = 140, T = 10 MeV 140 T = 20 MeV  =  0.45 0.4625 0.475 0.4875 0.5 Central Baryon Number Density nu (fm )  0.5125  0.525  -3  c  Figure 5.1: Change of gravitational mass at constant total baryon number of quarks stars with quark matter in the 2SC+S phase. Within the range of the central baryon number density, the quark star is in a stable branch. The data-points represent star configurations with the same central pressure but at different central temperatures. a neutrino chemical potential p sa 0. In this chapter, we focus our attention to such cold and mature compact stars, which have central temperatures of the order of 1 MeV. However, for the purpose of exploring the nature of the two-flavour colour-superconducting phase and its possible consequences on the compact star's properties, we also consider hypothetical states of the compact stars with no trapped neutrinos, but at temperatures T , higher v  c  131  Chapter 5. Cooling of Quark Stars Change of Mass at Constant Total Baryon Number for Quark Stars with free strange quarks at increase of temperature to T=10,20 MeV  16 16.5 17 17.5 18 18.5 Total Baryon Number of Star N (x 1.0e56)  19  19.5  B  Figure 5.2: Gravitational mass of quark stars with quark matter in the 2SC+S phase as a function of total baryon number of the star. As the central temperature increases, the gravitational mass increases. The plots corresponds to a gravitationally stable configurations of the stars. The nearby data-points corresponds to the same central pressure but at different central temperatures. than 1 MeV (e.g. T ~ 5,10,20 MeV). c  132  Chapter 5. Cooling of Quark Stars Absolute and Relative Change of Mass at Constant Total Baryon Number of Quark Stars for Cooling from a temperature of T=10 MeV to T=0 MeV vs Central Temperature 0.0045 PcT=W = 2.42 k l O , ^ / = 140 0.004 hfcr=io = 2.42 x 10 , S / = 150 = 8.47 x 10 , B = 140 0.0035 , PcT=\0 8.47 x 10 , B ' = 150 \PcT= 3.63 x 10 , B\' = 140 0.003 P PcT=10 3.63 x 10 , B ' = 150 • 0.0025 \-PcT= 10= 9.68 x 10 , B ' = 140 PcT=lO = 9.68 x l O , ^ / = 150 0.002 3 3  1  4  33  1  33  1 / 4  33  1 4  4  34  c T = 1 0  4  1 4  34  1 4  34  3 4  <  1  4  0.0015  2-  0.001 0.0005  X3  ,„  O  lu-.if*'-  =  !4Q  \t',:r = 2.42 x 10 , B ' = 150 —o— 1 4  33  10  o  = 8.47 x 10 , B»/4 .PcT.. - = 8.47 x 10 , B > 0.5 \PcT-. PcT, -io = 3.63 x 10 , B ' = 3.63 x W ,B > 0.4 -io = 9.68 x l O , ^ / YPcTPcT-\ = 9.68 x l O , ^ / 0.3 33  = ] 0  33  10  is  X  M  10  4  1 4  34  L  4  3 4  1  4  3 4  1  4  w  = 140 - - K = 150 ~-o- = 140 = 150 = 140 = 150 --j-jfr. .,-v" '~ !  J3  o ~  = 2.42 x 10 , B ' = 2.42 x 1 0 , B ' / = 8.47 x 10 , B 8.47 x 1 0 , B = 3.63 x 1 0 , S ' / = 3.63 x 10 , B ,B'/4 tPcT- = 9.68.-x-a0 4 $•68 x M ,#y PcT 1 4  33  iPcTPcT, PcT-. 4 rPcT. PcT-.  33  4  1/4  33  33  1/4  34  4  ,/4  34  34  4  = 140  = 150 = 140 = 150 = 140 — • = 150 — = 140 ~4— = 150  /»>,  // ~~~7 -N t  \ X  -2  < 10  8  6  4  2  0  Central Temperature T (MeV)  Figure 5.3: Change of gravitational mass of quark stars with quark matter in the 2SC+S phase while cooling from a initial temperature of T = 10 MeV to T = 0 MeV when the total baryon number of the star is kept fixed within a margin of error. The variation of the total baryon number is also shown in the bottom panel. Compare with the total baryon number of quark stars which are of the order of 1 ~ 10 x 10 . The pressure values are in dyne/cm . 56  3  133  Chapter 5. Cooling of Quark Stars Absolute and Relative Change of Mass at Constant Total Baryon Number of Quark Stars for Cooling from a temperature of T = l MeV to T=0.05 MeV vs Central Temperature 3.5 P c T = \ • : 2.42 k 10 , B ' = 140 : 2.42 x 10 , B ' = 150 3 P c T = \ • 9.68 x 10 , B / = 140 P c T = l •• 9.68 x l O , ^ / = 150 2.5 P c T = l : 3.63 x 10 , B ' = 140 • x = 150 • , P c T = l • 3.63 x 10 , B 1.09 x 1 0 , £ ' / = 140 2 1.09 x Vfi^B ' = 150 1.5 I 33  1 4  33  1 4  33  1  3 3  :  4  4  34  1 4  34  i/4  35  4  :  1 4  :  0.5 h  <  0  •- 2.42 x 10 , B ' = 2.42 x 10 , £ = 9.68 x 10 , B : 9.68 x l O ^ / " = 3.63 x l O , ^ / = 3.63 x 1 0 , B = 1.09 x l O ^ . S / = 1.09 x l O ^ . B / 33  0.0035  P:T=1 PcT=l  0.003 6?  0.0025  ^  0.002 0.0015  ~  0.001  1  1 / 4  33  1 / 4  3 3  PcT=l UPcT=: PcT=\ PcT=\ PcT,  4  33  1  3 4  1  34  4  1/4  1  4  1  4  = 140 = 150 = 140 = 150 = 140 = 150 = 140 = 150.  0.0005 P  c  T  =  1  = 2.42 x l O ^ . B / = 140  P  c  T  =  i  =  2.42  x  10 ,  Pcr=i  =  9.68  x  10 ,  1  4  B'  33  =  1 4  B  33  = 9.68 x l O ^ /  PCT=I  =  3 3  3-63  x  =  1 / 4  PCT=I  1  4  B'  10 , 34  =  1 4  150 140  = 150 — 140  -  = 150 h~~ L & r i i = 1.09/\10 ,B = 140''-VPcT^ = 1.09/xV0 ,B /'' = 150 — V I m . , = 3.63 x 10 , B\ 34  35  35  \^ -4  1  4  1/4  1  \ y 0.8  0.6 0.4 Central Temperature T (MeV)  Figure 5.4: Change of gravitational mass of quark star with 2SC+s phase for cooling from an initial temperature T = 1 MeV to T — 0.05 MeV at fixed total baryon number NB of the star. We consider different initial central pressures in dyne/cm and bag constants in MeV. The third panel shows the error in keeping NB constant. 3  5.2  Cooling of Quark Stars  We consider a quark star at finite central temperature which is in quasithermal equilibrium. The star loses energy slowly via diffusion of neutrinos 134  Chapter 5. Cooling of Quark Stars Relative Central Pressure and Radius of Quark Stars at Cooling from a temperature of T=10 MeV to T=0 MeV vs Central Temperature 1.025  1.015  6 4 Central temperature T (MeV)  Figure 5.5: The dependence of the central pressure P and the radius of a quark star with quark matter in the 2SC+s phase as it cools down from T = 10 MeV to T = 1 MeV at fixed total baryon number. Depending on the initial central pressure at finite initial central temperature the central pressure at a lower temperature is seen to increase and the radius decrease. c  and thermal photons. At any instant, we can consider the star to be in (instantaneous) thermal equilibrium and can follow the thermal history of the star by studying a sequence of compact stars in thermal equilibrium. As the star loses energy, its gravitational mass reduces; the change of gravitational mass equals the energy lost due to cooling. Since, the chemical reactions in the matter do not violate baryon number 135  Chapter 5. Cooling of Quark Stars Relative Change of Central Pressure and Radius of Quark Stars at Cooling from a temperature of T=10 MeV to T=0 MeV vs Central Temperature at Fixed N i I  !  :  B  0.1 0 -0.1 -0.2 -0.3  -o.4  !  -0.5 -0.6 > -0.7 > -0.8 >o -0.9  j  i  ; > ; j ;  140 — • 2.42 x 10 £ 1 / 4 : 2.42 x 10 £ 1 / 4 150 —o— 140 • PcT= 10 : 8.47 x 10 £ 1 / 4 150 PcT=W '• 8.47 x 10 140 PcT=10 = 3.63 x 10 BV4 £ 1 / 4 150 PcT=W '• = 3.63 x 10 £1/4 = 9.68 x 10 140 PcT=10 £1/4 = 9.68 x 10 PcT=10 = 150 ~ 1 1 _ 140 —«— PcT=W • • 2.42 x 10 .£ 1 / 4 £1/4 = 150 PcT=\0 '• : 2.42 x 10 . 140 PcT=W : 8.47 x 10 £ 1 / 4 £ 1 / 4 150 —- PcT= 10 : 8.47 x 10 £1/4 : 3.63 x 10 140 ^cT=10 — : 3.63 x 10 £l/ 150 P=T=10 '• B r=10 '• : 9.68 x 10 £ 1 / 4 140 PcT=10 '• : 9.68 x 10 £ 1 / 4 = 150 PcT=  -i  10  PcT=10  -1.1 -1.2 -1.3 -1.4 -1.5 -1.6 0.6  33  :  =  33  :  =  33  :  =  33  :  :  : :  34  =  34  =  34  =  34  =  T  33  33  |  33  :  0.5  =  33  :  =  34  :  34  0.4  =  34  "o  g>  4  c  34  0.3  0.2 0.1  0  1  2  3  4 5 6 7 8 Central temperature T (MeV)  9  10  Figure 5.6: Relative difference of central pressure and radius with respect to those at central temperature T=0 MeV for a cooling quark star in the 2SC+S phase. The star cools down from T=10 MeV to T=0 MeV at fixed total baryon number corresponding to the hot star. conservation, the total baryon number in the star should remain the same. Hence, we seek the change of mass due to decrease of central temperature of the star subject to the condition that the total baryon number of the star NB, remains constant. We present our results for two-flavour color superconducting quark matter (2SC/g2SC phase), mixed phase of 2SC/g2SC and normal quark matter and 2SC+S phase. We estimate the change of gravitational mass A M of the quark stars from 136  Chapter 5. Cooling of Quark Stars Relative Change of Central Pressure and Relative Radius of Quark Stars at Cooling from a temperature of T=l MeV to T=0.05 MeV vs Central Temperature  c3  O  0.999998 0.999996 0.999994 0.999992 0.99999 0.999988 0.999986 0.999984 0.999982 1  0.9  0.8  0.7 0.6 0.5 0.4 0.3 Central temperature T (MeV)  Figure 5.7: Relative change of central pressure and the relative radius of quark stars with quark matter in the 2SC+S phase as it cools down from initial central temperature T — 1 MeV to T = 0.05 MeV at fixed total baryon number. Depending on the initial central pressure, the central pressure at a lower temperature may increase or decrease to keep N constant. The radius of the quark star with this phase of matter, is however seen to decrease as it cools. B  the plots of: 1. Mass and total baryon number as functions of central pressures at different temperatures: We plot the mass M and total baryon number N versus central pressure P , for different temperaB  c  137  Chapter 5. Cooling of Quark Stars Relative Change of Central Pressure and Radius of Quark Stars at Cooling from a temperature of T=l MeV to T=0.05 MeV vs Central Temperature at Fixed N  B  0.3 0.4 0.5 0.6 0.7 Central temperature T (MeV)  Figure 5.8: Relative difference of central pressure and radius with respect to those at initial temperature T=0.05 MeV for quark stars with quark matter in the 2SC+S phase. The star cools down from initial central temperature T — 1 MeV to T = 0.05 MeV at fixed total baryon number corresponding to the hot star. tures in the same plot and find those values of P that result in the same N , but at different central temperatures. The differences of the masses at finite and zero temperatures, corresponding to these values of P , give us the sought A M . C  B  C  2. Mass as a function of the total baryon number of the star at various temperatures: Since in our data table, the independent 138  Chapter 5. Cooling of Quark Stars variable is the central pressure, we use the plotting program gnuplot to solve the set of implicit equations M = M(P ),N = NB(P ) to find M = M(NB)- This method is prone to linearization error within small ranges of the total baryon number as we do not use any data fitting procedure. C  B  C  For both of the above two methods, we consider high central-temperaturestars without any neutrinos since at lower temperature increases, the graphical methods give high percentage errors as the changes in mass and total baryon number are very small. 3. Change of mass as a function of central temperature: Here, we use a numerical code to find the set of configurations of a quark star at different low central temperatures but at the same total baryon number as that, at a initial high central temperature. The routine integrates the scaled TOV equations using central pressures slightly below and above the value at the initial temperature, and uses root finding method to choose the appropriate new central pressure (at a lower central temperature) that produces the same total baryon number within a specified margin of error. This method is suitable for calculating the change of mass at small decrease of central temperature and we present our results for A T — 1 MeV as well as for higher A T = 10,20 MeV's (although those correspond to hypothetical states of the hot stars as discussed in the section 5.1).  5.2.1  Cooling of Quark Star with 2SC Phase and Free Strange Quarks  For quark stars with the 2SC+S phase, we see the following features in the cooling plots: 1. For a bag constant of B = (140 MeV) , we see mass-changes A M - o ~ 0.016M and AMio_o ~ 0.003M© at total baryon number of N = 17 x 10 , for temperature changes from 20,10 to 0 MeV, respectively. For total baryon numbers of N = 16 x 10 , the corresponding changes are A M - o ~ 0.015M© and AM10-0 ~ 0.003M©, respectively while 4  20  Q  B  56  56  B  20  139  Chapter 5. Cooling of Quark Stars Change of Mass at Constant Total Baryon Number for non-strange Quark Stars for decrease of temperature from T=10,20 to T=0 MeV  Central Baryon Number Density n  (fm ) 3  Bc  Figure 5.9: Mass and total baryon number of quark stars with quark matter in the 2SC or 2SC+NQ phase at different central temperatures as a function of the central baryon number density. In the range of the central baryon number density considered, the quark stars are gravitationally stable. At the same total baryon number the masses at different central temperatures give the difference of mass of the stars as they cool. for total baryon number of NB = 18 X 10 , the changes are AM o-o ~ O.O16M and A M - « 0.002M . 56  2  0  1 0  0  Q  2. From the cooling plots using the modified code that keeps the baryon 140  Chapter 5. Cooling of Quark Stars number fixed (within a certain error), we see a maximum mass-change of the order of AMio-o « O.OO3M which is consistent with the previous plots. However, we find that the absolute mass-change depends on the central pressure and the bag constant used, as the equation of state of matter and hence the mass of the star depend on those. 0  3. In general, the bigger the bag constant, the less is the absolute change in mass at constant total baryon number due to decrease of temperature. This was also seen for change of mass due to change of central temperature even when the total baryon number was not held fixed (see Figs. 4.10, 4.11). 4. The central pressure values we have used in solving for the mass change, correspond to the branch in the M-R plot of the quark stars, before the peak in the mass. Hence, as the central pressure is increased, the mass of the initially hot star increases so also the change of mass A M . 5. The relative mass change is also seen to depend on the initial central pressure and the bag constant used. For a fixed bag constant, e.g. B — (150MeV) , as the central pressure is increased, the relative change is seen to decrease and approaches the value AM/M « 0.25%. This is also seen for a lower bag constant B = (140 MeV) , although the change A M / M is seen to vary between 0.24% — 0.28% without showing any monotonic shift. 4  W  4  1 0  6. For lower initial temperature T ~ 1 MeV, the mass-change is much less and is roughly of the order of A M ~ 1.0 x 10~ M . We have a similar behaviour of the relative mass-change AMi-o.os/Mi, but it approaches a value of A M / M i « 0.002% at high central pressure. ini  5  Q  7. The average value of the relative thermal capacity of the quark star (AM/M )/AT is ~ 2.6 x 10- /MeV for a change from T = 10 MeV to T = 0 MeV while that from T = 1 MeV to T = 0.05 MeV is seen to be ~ 2.01 x 10 /MeV, both at high initial central pressures. The relative thermal capacity at T = 10 MeV is seen to be 6.0 x 10~ /MeV while that at T = 1 MeV is found to be 5.0 x 10~ /MeV. 4  ini  _5  4  5  To understand how the radius and the central pressure change as the star cools down, we plot the relative central pressure or relative change of the central pressure and the relative radius with respect to central temperature 141  Chapter 5. Cooling of Quark Stars Change of Mass at Constant Total Baryon Number for non-strange Quark Stars at decrease of temperature from T=20,10 MeV to T=0 MeV  Total Baryon Number of Star N (xlO ) 56  B  Figure 5.10: Mass of the quarks stars with two-flavour quark matter in the 2SC or 2SC+NQ phase as a function of their total baryon number for bag constants of B = (135 MeV) , (140 MeV) . The stars considered here are in gravitationally stable configurations and as the stars cool down the gravitational masses are seen to decrease. At the same total baryon number, the change of mass correspond to energy lost due to cooling. 4  (P -T/Pc-ini,( c-T-Pc-ini)/Pc-ini p  c  4  and R /'Rini vs. T  T).  We see the following 142  Chapter 5. Cooling of Quark Stars Change of Mass at Constant Total Baryon Number for non-strange Quark Stars at decrease of central temperature from T=20,10 to T=0 MeV  15  15.5  16 16.5 17 17.5 18 Total Baryon Number of Star N (xl.0e56)  18.5  19  B  Figure 5.11: The mass versus the total baryon number of quark stars with quark matter in the two-flavour color superconducting phase for a bag constant of B = (145 MeV) . 4  features in the plots: 1. For a temperature change from T = 10 to T = 0 MeV, the central pressure is seen to increase up to a maximum ~ 16% and the radius to decrease up to ~ 0.5%. 2. The change of central pressure depends on the bag constant and the initial central pressure. At a fixed bag constant, the change of the central pressure decreases with higher initial central pressure. 3. As the bag constant is increased, the change of the central pressure becomes less compared to the case of lower bag constant. Hence, the bigger the bag constant, the less is the change of the central pressure of a cooling star. This behaviour is also seen to persist for cooling of the star from a smaller initial temperature T — 1 MeV. 143  Chapter 5. Cooling of Quark Stars Absolute and Relative Change of Mass at Constant Total Baryon Number of non-strange Quark Stars for Cooling from a temperature of T=10 MeV to T=0 MeV vs Central Temperature 0.008 LPcT=10 : 2.42 * 1 303^ ,£ 51 /^4 = 155 : 2.42 x 33 ni/4 LP ° : 0.007 K c T = l 0 • : 8.47 x 3 3 l / 4 : 8.47 x , 0.006 PcT=W '• : 3.63 x 33 44 ££ 11 // 44 , VPcT=W ' : 3.63 x 3 4 £ 1 / 4 0.005 \PcT=W '• : 9.68 x 34 £ 1 / 4 PcT=lO ' : 9.68 x 0.004 :  1 0  =  1  4  5  CT=I  1 0  =  1 Q  I  B  =  1  3  5  1  4  5  1 0  =  1  3  5  1 0  =  1  4  5  1 Q  =  1  3  5  1 Q  =  1  4  5  0.003 ^ <  0.002 0.001 0 0.8 0.7  O  : 2.42 IlO , B 33  PcT=10 PcT=10 fPcT^W [PcT=lO PcT=10  : 2.42 : 8.47 : 8.47 : 3.63 0.6 VPcT=10 : 3.63 : 9.68 0.5 : 9.68 tPc  1 / 4  x x x x x x x  10 ,B 10 ,B'/ 10 ,B lO^.B / 10 ,B lO^.B / lO^.B /  x x x x x x  10 ,B'/ 10 ,B'<' lO^.B /" 10 ,B'/ lO^.B'/ 10 ,B'/  3 3  1 / 4  33  4  3 3  1 / 4  1  3 4  4  1 / 4  1  4  1  4  0.4  <  145 —o—• 135 145 135 145 135 145  0.3 0.2 0.1 -PcT=10  <l  7 -PcT=\Q 6 -PcT=\0 5 PcT=W P c T = 10 4 PcT= 10 3 ~PcT= 10 2 ~PcT=W 1 0 -1 -2 -3 -4 -5 10  = = = = = =  2.42 2.42 8.47 8.47 3.63 3.63  33  4  33  4  1  33  4  4  34  4  = g.ea^io ^'/ 34  = 135 = 145 = 135 = 145 = 135 = 145  4  8  6 . 4 Central Temperature T (MeV)  2  0  Figure 5.12: The absolute and relative change of mass versus the central temperature of quark stars with matter in the 2SC or 2SC+NQ phase as they cool down from central temperature of T — 10 MeV to T — 0 MeV at constant total baryon number. As the temperature decreases, the change of mass increases indicating energy loss due to cooling. The initial central pressures are in dyne/cm and bag constants in MeV. The third panel shows the error in the total baryon number while keeping it fixed. 3  144  Chapter 5: Cooling of Quark Stars 4. Similarly, for bigger bag constants and the initial central pressures, the change in the relative radius of a cooling quark star is lesser. 5. Depending on what the initial configuration of the star is, at an initial central pressure and a finite initial temperature, the central pressure may decrease or increase while the star cools down. This is consistent with the plots of AN vs. P (Figs. 4.10, 4.11) since we see both increase and decrease of N and a maximum in the change AN with respect to P . B  c  B  B  c  For a quark star cooling from T = 1 MeV to T = 0.05 MeV, we see similar dependence of the change in P , on the initial central pressure and bag constant, as a function of temperature, as for the case of cooling from a higher temperature Tj„i = 10 MeV. At high enough initial central pressure and bag constant, the change in the central pressure from the initial value may be negative. c  5.2.2  Cooling of Quark Star with Two Flavours  For quark stars with two-flavour colour superconductor or normal quark matter, without any strange quarks, we have the following features in the cooling curves: 1. In the plots of M and N versus the central baryon number density n , for a bag constant of B = (135 MeV) , we see mass-changes A M _ o « 0.025M© and A M - « 0.008M© at total baryon number N = 18 x 10 , for temperature changes from 20,10 to 0 MeV, respectively. However, in the plots of M vs. N , we see a slightly different values of A M - o ~ 0.021M and A M _ o ~ 0.0053M . In both the cases, the plotting program gnuplot uses linear approximation in between the data-points and hence we may get the differences in the values. B  4  Bc  20  1 0  0  56  B  B  20  o  10  e  For bag constant of B = (135 MeV) and at total baryon numbers of N = 16 x 10 , the corresponding changes are A M _ ~ 0.019M© and AMio-o ^ 0.0053M©, respectively, while for total baryon number of N = 17 x 10 , the changes are A M _ « 0.0203M and A M _ ~ 0.0053M©, respectively. 4  56  B  2 0  0  56  B  2 0  0  Q  1 0  0  145  Chapter 5. Cooling of Quark Stars Absolute and Relative Change of Mass at Constant Total Baryon Number of non-strange Quark Stars for Cooling from a temperature of T = l MeV to T=0 MeV vs Central Temperature PdT=l •PdT=l PdT=l PdT=l PdT=l PdT=l ~PcT=l Pdr=\  x  • 2.42 * l O ^ . B / : 2.42 x 10 , B / : 9.68 x 10 , B / : 9.68 x 10 , B / : 3.63 x 10 , B / : 3.63 x l O ^ / : 1.09 x l O ^ . B / : 1.09 x 10 , B 1  33  1  4  33  1  4  33  1  4  34  1  4  3 4  1  35  4  = 135 = 145 = 135 = 145 = 135  4  1 / 4  • 0.0065 r-BcT=i • 2.42 x l O ^ . B / 0.006 PdT=l : 2.42 x l O ^ . B / 0.0055 PdT=l : 9.68 x l O ^ . B / x lO^.B / 0.005 Y-P*r=i :: 9.68 3.63 x 10 , B 0.0045 PdT=l 'Pdr=i = 3.63 x 0.004 B C T = I = 1.09 x 0.0035 VP* = 1.09 x 0.003 0.0025 0.002 0.0015 0.001 0.0005 0 ~ 7 • PdT=l = 2.42 x l O ^ . B / = 2.42 x l O ^ . B ' / 2 6 P(fT=l Bcr=i = 9-68 x 10 , B x 5 9.68 x 10 , B / ~" 4 3.63 x 10 , B / 3 3.63 x l O ^ / 2 = l . O a ^ l O 35^ B 1 / 4 9..xSo B / 1 0 -1 -2 -3 1  1  4  1  4  1  4  34  1 / 4  1  4  4  33  -4  = 135 = 145 = 135 = 145 = 135 • = 145 • = 135 = 145  4  1  1 / 4  33  1  4  34  1  4  3 4  4  1  4  1  4  = 135 = 145 = 135 = 145 = 135 = 145 = 135  /\  i  = 145  0.6 0.4 Central Temperature T (MeV)  Figure 5.13: The absolute and relative changes of mass versus central temperature of quark stars with 2SC or 2SC+NQ phase for cooling from a central temperature T — 1 MeV to zero temperature. The initial central pressures are in dyne/cm and bag constants in MeV. We keep the total baryon number of the star fixed to the value at that of the initial hot star. The third panel shows the error in holding NB fixed. 3  2. From the cooling plots using the numerical code that keeps the baryon number constant (within a certain margin of error), we see a maximum 146  Chapter 5. Cooling of Quark Stars Relative Central Pressure and Radius of non-strange Quark Stars at Cooling from a Temperature of T=10 MeV to T=0 MeV vs Central Temperature 1.015 1.0125  1.01 1.0075 1.005  4  1.0025 h  0.999 h  0.998 h  0.997 h  0.996  0.995 10  8  6 4 Central temperature T (MeV)  2  Figure 5.14: Relative central pressure and relative radius of quark stars with 2SC or 2SC+NQ phase versus central temperature. As the star cools down from initial hot state with T = 10 MeV to T = 0 MeV, the central pressure and radius changes depending on the initial central pressure and the equation of state used. For most of the initial configurations of the star, the central pressure at T < 10 MeV is seen to increase while the radius decrease. c  mass-change A M _ o up to « 0.006M©. However, we expect that the 10  147  Chapter 5. Cooling of Quark Stars Relative Change of Central Pressure and Radius of non-strange Quark Stars at Change of Central Temperature from T=10 MeV to T=0 MeV at Fixed N B  0 -0.1 -0.2 -0.3 -0.4 -0.5 -0.6 -0.7 -0.8 -0.9 -1 -1.1 -1.2 -1.3 -1.4 -1.5 0.45  X. PcT=W PcT=10 ' PcT=W '• PcT=10 PcT=10 • P c T = l 0 '•  : 2.42 x 1 0 , B = 145 —» :8.47x l O , ^ / " = 145 —o  PcT=W PcT=\0 PcT=\0 PcT=\0 PcT=\0  = 2.42 x 10 , B ' = 145 —» : 8.47 x 10 , B ' = 145 - -o : 3.63 x l O ^ . B / = 135 —  :  :  0.4 0.35  PcT=10  '• ' '•  3 3  1 / 4  : : : :  3.63 3.63 9.68 9.68  x x x x  1  10 10 10 10  ,B^ ,B ' ,B^ ,B '  34  = = = =  4  M  1 4  34  A  34  1 4  33  1  4  1 4  M  1 4  = 9.68 x 1 0 , B 3 4  1  1 4  : 3.63 x 10 ,B ' • : 9.68 x W ,B ' 34  X..  135 — 145 — 135 — 145 — • •  1 4  33  :  :  X.  3 3  1 / 4  = 145 = 135  —  = 145  0.3 0.25 0.2 0.15 0.1 0.05 0 0  2  4 6 Central temperature T (MeV)  8  10  Figure 5.15: Relative change of central pressure and radius of quark stars with 2SC or 2SC+NQ phase with respect to those at temperature T=0 MeV. At increased temperature the star becomes larger and the central pressure decreases. absolute mass-change will depend on the central pressure and the bag constant used, as the mass of the star depends on those. In general, the bigger the bag constant the less the absolute value of 148  Chapter 5. Cooling of Quark Stars the mass change AM for non-strange quark stars. This is similar to the case of quark stars in the 2SC+s phase. 3. The relative mass change also depends on the central pressure and the bag constant used. For a fixed bag constant, as the central pressure is increased, the relative mass-change is seen to decrease and approaches the value AM/M » 0.32% (e.g. for B = (135 MeV) ). For bag constant of B = (145 MeV) , AM/M is also seen to vary similarly. 4  W  4  W  This monotonic decrease of relative mass-change was not seen in the case of quark stars with the 2SC+S phase, where the relative mass change AM/M was seen to both increase and decrease for different bag constant values. However, in both the cases, the relative change in mass approaches some limiting values as the central pressure increases. ini  4. For a lower initial temperature T ~ 1 MeV, the mass-change is much less and is roughly of the order of AM ~ 1.0 x 1O~ M . We have a similar behaviour of the relative mass-change A M i _ o / M i as that at higher temperatures, but it approaches a smaller value of AM/Mi s» 0.0027% at high central pressures. ini  5  0  5. The (relative) thermal capacity of the star (AM/M )/AT has an average value of ~ 3.3 x 10~ /MeV for a change from T = 10 MeV to T = 0 MeV while that from T = 1 MeV to T = 0 MeV is seen to be ~ 2.7 x lO~ /MeV, both at high initial central pressures. The relative thermal capacity at T = 10 MeV is seen to be 1.0 x 1 0 / M e V while that at T = 1 MeV is found to be 6.0 x 10~ /MeV for non-strange quark stars. ini  4  5  -3  5  In the plots of relative radius and the central pressure or its relative change, due to cooling of a non-strange quark stars (i.e. P _r/'P -mi, {PC-T — Pc-ini)/Pc-ini and Rr/Rint'^s. T), we see the following features: c  c  1. For a temperature change from T = 10 to T = 1 MeV, the central pressure is seen to increase up to ~ 14% and the radius decreases up to 0.45%. Both of these are comparably smaller than the corresponding values for quark stars with 2SC+S phase. 2. The change of central pressure is again seen to depend on the bag constant and the initial central pressure. Depending on the initial 149  Chapter 5. Cooling of Quark Stars Relative Change of Central Pressure and Relative Radius of Quark Stars at Cooling from a Temperature of T=l MeV to T=0 MeV vs Central Temperature 2.5  03  o -1.5  0.99999 0.99998 0.99997 0.99996 0.99995 0.99994 : 2.42 X lO^.B / = : 2.42 x 10 ,B = PcT=\ • PcT=l 7.26 x lO^.B / = PcT=\ : 7.26 x 10 , B / : PcT=\ : 9.68 x 10 , B !* -• PcT=l : 9.68 x 10 , B / : PiT=l : 3.63 x lO^.B / : PcT=\ : 3.63 x lO^.B / : PcT=\ : 1.09 x l O ^ / : PcT=\ : 1.09 X lO^.B'/ :  0.99993 L P c T = l  33  4  1/4  1  0.99992  :  33  4  1  33  0.9999  4  1  33  0.99991  1  4  1  4  1  35  0.99989 0.99988  1  -  1  4  4  4  1  0.9  I  0.!  I  0.4 0.3 0.7 Central temperature T (MeV)  0.2  0.1  Figure 5.16: Relative change of central pressure and relative radius of quark stars with 2SC or 2SC+NQ phase versus the central temperature as it cools down from T = 1 MeV to T = 0 MeV at constant total baryon number. The change in P depends on the initial central pressure. The radius of the stars are seen to decrease. Contrast the change of P and R with those at cooling from a higher initial central temperature T = 10 MeV shown in the previous plot (5.14). c  c  150  Chapter 5. Cooling of Quark Stars Relative Change of Central Pressure and Radius of non-strange Quark Stars at Change of Central Temperature from T=l MeV to T=0 MeV at Fixed N B  0  0.1  0.2  0.3 0.4 0.5 0.6 0.7 Central temperature T (MeV)  Figure 5.17: Relative change of central pressure and radius of quark stars with 2SC or 2SC+NQ phase with respect to those at temperature T—0. The star cools down from T = 1 MeV to T = 0 MeV at constant total baryon number. The initial total baryon number corresponds to that of the hot star. The star srinks as it cools down while the central pressure may decrease or increase depending on the initial value.  151  Chapter 5. Cooling of Quark Stars central pressure and the initial temperature, the central pressure may both inrease or decrease. 3. As the bag constant is increased, the change of the central pressure becomes less compared to the case of lower bag constant. This is similar to the case of quark star with 2SC+S phase. The bigger the bag constant, the less is the change of the central pressure of a cooling quark star. This behaviour is also seen to persist for cooling of the star from smaller initial temperatures. 4. For radius, similar to the case of quark stars with 2SC+S phase, at the high end of the central pressure range, where quark stars are more realizable, the radius of the cooling star decreases less if the bag constant used is larger.  152  Chapter 6 Summary and Conclusions In this thesis we have investigated the two main issues: the thermodynamic properties of some particular two-flavour colour superconducting phases and the effect of their presence on a compact star's structure and cooling. The colour superconducting phases considered include some of the highly plausible phases at intermediate baryon number density. Since colour superconductivity can not be realized in laboratory experiments and since the existence of such phases is most likely to be inside the compact stars, the investigations and comparing the predictions with the astronomical observations deal with the possibility of vindicating the validity of the theories of colour-superconductivity themselves.  6.1  Summary of Results  In chapter 3 we focused on the thermodynamic properties of the quark matter in two-flavour colour superconducting (2SC/g2SC), non-colour-superconducting or normal, mixed phase of normal and 2SC/g2SC and in the neutral hadronic phase. The goal of that chapter was to study the nature of the different phases and especially find the structure of the phase diagram (/zs-T space) by finding the most probable phase at any point. Our main results of chapter 3 may be summarized as below: 1. The quark matter undergoes a second order phase transition from normal quark matter to pure 2SC phase beyond a threshold baryon chemical potential at zero temperature. The threshold potential depends on the cut-off parameter A, for lower cut-off but the dependence becomes less as the cut-off is increased. 2. At low baryon chemical potential, the colour superconductor is in the gapless 2SC phase which changes via a second order transition into the gapped 2SC at some higher baryon chemical potential. The slopes of 153  Chapter 6. Summary and Conclusions the plots of electric and colour charge-8 chemical potentials change at the phase-change point, indicating different functional dependences on PB, in the gapped or gapless phases. 3. At low baryon chemical potential the colour-superconducting gap A for the pure 2SC phase (without strange quarks) increases with temperature, attains a maximum and then decreases with increasing T, unlike the monotonic decrease in conventional superconductors. This implies that the Cooper pairing of the up and down quarks is strongest at some non-zero temperature for certain baryon chemical potential. At higher baryon chemical potential for the pure 2SC phase the gap however decreases monotonically with T. 4. In the quark phase diagram, the favoured phase is the mixed phase of pure (gapless/gapped) 2SC and normal quark matter at low temperatures, from very low baryon chemical potentials. Within the mixed phase, the gapped 2SC phase is favoured at higher fi - At high u. , the pure gapped 2SC phase is favoured above the mixed phase in temperature. At even higher temperatures we see a brief region of gapless 2SC and then normal quark matter phase. B  B  5. The dependence of the structure of the quark phase diagram on the cutoff reduces as the cut-off assumes higher values. This suggests the use of high cut-off for a more accurate determination of the phase structure and properties. 6. In contrast to the pure 2SC (gapless/gapped) phase, the mixed phase of two-flavour colour superconductor occurs from very low baryon chemical potential. Since the theory of 2SC phase that we consider does not incorporate chiral phase transition (the quark current masses are independent of ps), we expect the mixed phase to appear as soon as the hadronic matter phase undergoes a transition into the quark phase. We expect our theory of colour-superconductor to be valid only at that transition value of ps and beyond. 7. The presence of free strange quarks with the 2SC phase results in the absence of mixed phase. The gapless 2SC phase is less favoured in the presence of free strange quarks and occurs only in a narrow region of the phase diagram at high temperature (~ 80-100 MeV) for cut-off 154  Chapter 6. Summary and Conclusions A = 1500.0 MeV. For lower cut-off, gapless 2SC phase is absent in the phase diagram. 8. Similar to the case of the mixed phase, the 2SC+S phase occurs from very low values of p . Hence, the baryon chemical potential where the strange quarks start to appear, we expect a phase transition to the 2SC+s phase from the mixed phase. Note that, our theory does not take into account the medium modified masses of the quarks and can not determine the threshold for strange quark's appearance. However from the hadronic phase we expect particles carrying strangeness to 3 appear first at baryon number density as low as n ~ 0.27 far (E-) (c.f. Fig. 3.18). The baryon number density for the 2SC+S phase is 3 indeed of the order of n « 0.3 far from the very low baryon chemical potential p ~ 900 MeV. B  B  B  B  9. We expect the hadronic phase to be physically relevant only at relatively low baryon chemical potential and temperature. The fractional populations of the different hadrons depend strongly on the temperature and baryon chemical potential. At high temperature, the threshold for appearance of hyperons and other massive hadrons are lowered considerably along the baryon number density axis. The knowledge of these thermodynamic properties enables us to apply the theory of the 2SC colour-superconductors and the neutral hadronic matter in the study of quark- and hybrid-stars. In chapter 4, we have performed a systematic study of the structural properties and their temperature dependence, of the quark stars with twoflavour color superconducting matter, the hybrid stars and purely hadronic (hyperonic) stars, with and without strange particles present. Our main results there may be summarized as below: 1. The structure and composition of the compact stars depend on the bag constant B for the quark phase, making the quark matter favourable or disfavoured for the whole or part of the compact star for different values of it. Since the bag constant is not a very preciously known quantity, we employ a range of bag constants for taking into consideration the different possible types of compact stars (quark , hybrid or purely hadronic). 155  Chapter 6. Summary and Conclusions 2. Our goal in the chapter was to study the effects of the presence of the two-flavour phases considered, on the compact star structure without concerning with the possibility of occurrence of many other colour superconducting phases. In this respect, the study was limited into a specific objective. 3. The maximum masses of the quark stars for both with and without free strange quarks, depend on the bag constant used. In general, the quark stars corresponding to the same central pressure are more massive and larger in radius for lower bag constants. 4. Quark stars with only two flavours of quarks forming colour superconductors have greater maximum mass since the equations of state in the 2SC or 2SC+NQ phase are stiffer than that in the 2SC+S phase. 5. With an increase of central temperature, quark stars with lower bag constants have greater change of mass, total baryon number and radius at the same central pressure. The relative changes in M , NB and R also have the same dependence on the bag constants as the absolute ones. 6. Quark stars with 2SC+S phase with low bag constants are more strongly bound than those with high bag constants. The total, gravitational and internal binding energies and the binding energy per unit mass, all have higher values for the quark stars with lower bag constants. The higher the temperature the less bound the quark stars. 7. The difference of mass, total baryon number and radius (i.e. A M , ANB and AR, respectively), for the quark stars with the same central pressure but at increase of central temperature, have peak values with respect to the central pressure. These maximum values occur at different values of the central pressures p than where M , NB or R assume maximum. c  8. The changes A M , AN and AR for same-central-pressure configuration of quark stars at increase of temperature are the less the higher the central pressure. The dependence on the bag constant also becomes less for the relative changes A M / M , ANB/N O and AR/R as the central pressure increases. B  0  B  Q  156  Chapter 6. Summary and Conclusions 9. For certain bag constants and central pressures, the total baryon number NB, for same-central-pressure configuration of quark stars, at increase of central temperature decreases in value. This results in a negative change of NB i-e. /S.N < 0 although the baryon number density of quark matter n increases locally as the temperature increases. B  B  10. With increase of bag constant the colour-superconducting quark phases are disfavoured and hadronic matter appears to be favoured on the outer region of the stars. This results in hybrid stars which converts into pure hadronic stars at further increase of the bag constant. The maximum mass of the hybrid and hadronic star configurations are comparably smaller than that of pure quark stars. 11. Hybrid stars with quark matter core in the 2SC+S phase have unstable branch in between two stable branches in the M-R plots for bag constants near the transition value from pure quark stars to hybrid stars (e.g. B £3 (170MeV) ). This is however absent for hybrid stars with quark matter cores in pure two-flavour (2SC or 2SC+NQ or u+d NQ) phase. 4  12. The quark core in a hybrid stars have a maximum size with increase of central pressure. Beyond or near the pressure value where the maximum occurs, the mass of the stars also decrease at further increase of central pressure, hence resulting in an unstable branch. 13. Hybrid stars, with bag constants near the transition value from quark to hybrid stars, at increase of temperature, shrink in size and become unstable. This instability is removed with the higher values of the bag constant used. 14. The relative change of M, N and R for hybrid and hadronic stars decreases with increase of central pressure and increase of bag constant. B  In general, we see that the thermal effects of quark and hadronic stars decrease as the central pressure and the bag constant increase. However, compared to the quark stars, the hadronic equation of state is so much softer that the "specific" thermal effects (i.e. the relative changes) are many times more. In all the analyses in chapter 4, we have compared compact stars' configurations taking the central pressure and temperature as the independent 157  Chapter 6. Summary and Conclusions quantities. However, during the thermal evolution of a compact star, as its central temperature changes, it is not the central pressure that remains constant, rather the total baryon number of the star, provided there is no baryon number violating process involved. We treat that scenario in chapter 5, in the cooling of quark stars with normal or two-flavour colour superconducting phases, with possibly strange quarks present. Because of the constraint of constant total baryon number N of the star, as the star cools down its central pressure (or energy density) should change to keep NB fixed. Our main goals in chapter 5, were to find out: B  • how the physical conditions at the centre of a cooling star change; • how much energy does a quark star loose for a certain decrease of central temperature; • how much does the star's radius change as it cools; • and the dependence of the above thermal behaviours on the equation of state of the quark phase and on the initial configuration of the star. Our main results and conclusions of chapter 5, may be summarized as below: 1. The change of mass of a cooling quark star (at fixed N ) is in general seen to be smaller than the difference of mass, of two configurations of the star with the same initial and final temperature but with a fixed central pressure. This difference between two mass-changes arises because of change of central pressure as a result of the cooling. B  2. The amount of mass change depends very much on the initial central pressure of the star. In general, for both with and without free strange quarks, but with two-flavour colour superconductors, the configuration of the stars with higher central pressures (and hence which are, in general, more massive) have greater mass-change due to cooling. This is obvious since more massive stars should loose more energy as they cool. However, the relative change of the mass depends not only on the absolute change but also on the initial mass of the hot star. The latter also depends on the central pressure (as well as on central temperature) 158  Chapter 6. Summary and Conclusions and hence their ratio or the relative change reflects dependence of both of these on the central pressure. For quark stars with free strange quarks, the relative change of mass approaches a values A M / M J O ~ 0.25% while the same for non-strange quark stars is AM/Mio ~ 0.32%. Hence, in general, the non-strange quark stars loose more fraction of its mass compared to quark stars with 2SC+S phase due to cooling. 3. For stars with a particular type of phase, the amount of mass-change depends on the bag constant used via the equation of state. For nonstrange quark stars with the 2SC/g2SC or (2SC/g2SC)+NQ phase, we see a slight decrease of the relative mass-change with increase of the bag constant. For quark stars with the 2SC+S phase, we however see an increase of relative mass-change with the bag constant increasing. It may be noticed that, the absolute changes however always decrease with increase of the bag constant. 4. The central pressure of the quark stars, during the cooling, may decrease or increase depending on the initial central pressure and initial temperature. The change in the central pressure also depends on the bag constant used. In general, for higher bag constants, the central pressure changes less during cooling. 5. A cooling quark star is in general seen to decrease in radius during cooling. The quark stars with the 2SC+S phase are seen to undergo more fractional change of size than the non-strange quark stars. However, since the determination of the radius is not as accurate as the mass determination, we mainly notice the qualitative behaviour of the star's size due to cooling.  6.2  Comparison with Other Studies  Our studies of the two-flavour colour superconducting phases and the astrophysical consequences have similarities with other studies previously done. However, in some cases we get different results than those. We discuss below the main similarities and the differences:  159  Chapter 6. Summary and Conclusions  6.2.1  Quark Phase Diagram  Absence of Chiral Phase Transition Our theory of colour superconducting quark matter is based on a pairing interaction through a four-quark operator with the quantum numbers of the single gluon exchange interaction. This results in a di-quark condensate. This is the minimal interaction to model the colour superconductors. Other interaction may include the quark-antiquark interactions in the color singlet scalar/pseudoscalar channel, as in the Nambu-Jona-Lasinio type of models [39, 62, 63]. This results in the chiral (quark-antiquark) condensate. At small temperature and density, the attractive interaction in the color singlet quark-antiquark interaction channel is responsible for the appearance of a quark-antiquark condensate and for the spontaneous breakdown of the chiral symmetry. However, as the density increases, the chiral condensate melts down and at high enough density or baryon chemical potential, the dynamical quark mass, generated by the chiral condensation is seen to decease and eventually become zero [38]. Reddy and Rupak has studied the (gapless) 2SC phase and mixed phase with the normal quark matter at zero temperature with the same interaction that we used [55]. They have found the mixed phase to persist at temperature T = 0 in the range of the quark chemical potential p rk £ [300,500]MeV. Our result agrees with this. The value of the quark chemical potential at which chiral symmetry starts to be restored is reported in reference [24] as u. - hirai ~ 365 MeV for intermediate coupling (the ratio of the the di-quark coupling to the scalar coupling GD/GS = 0.75). However, this depends on the choice of the coupling. We expect our results to be physically relevant for quark chemical potential high enough that the dynamically generated mass of the quarks to be small enough compared to the quark chemical potential. The same assumption was used in the reference [55]. qua  q  c  Presence of Mixed Phase of 2SC with Normal Quark Matter In the reference [24], the authors find the favoured phase at zero temperature and baryon chemical potential in the range ps G [1095,1360] MeV and at intermediate coupling to be the mixed phase of the normal quark matter and the 2SC phase. At the higher end, the mixed phase gives away to the colour-flavour locked (CFL) phase. The interactions considered in their 160  Chapter 6. Summary and Conclusions Lagrangian density are the quark-antiquark interaction as well as the scalar and pseudoscalar di-quark interactions. Since we assume that the strange quarks do not participate in the colour superconducting phenomenon, we expect that the mixed phase will persist for high baryon chemical potential and indeed we see that in our phase diagram. In the reference [64], the authors additionally use t'Hooft interaction term in the Lagrangian density and the constituent quark mass in the determination of the thermodynamic potential. The constituent quark masses contain one term linear in the product of quark-antiquark coupling and the condensate and another term having the t'Hooft coupling and quadratic in the quark-antiquark coupling. They find the chiral symmetry restoration at p > 368 MeV and the appearance of three-flavour colour superconductor at mu > 442 MeV for intermediate coupling (G /Gs = 0.75). In the range between these two values of the quark chemical potential, they however find normal quark matter as the favoured phase. However, their analysis is done for locally electric-charge-neutral phases only. Given that the authors included all three types of interactions (the quark-antiquark, di-quark and t'Hooft interaction), their finding of the two-flavour normal quark matter in the intermediate range of the baryon chemical potential, suggests the possibility of presence of mixed phase of (gapped/gapless) 2SC and normal two-flavour quark matter for physically relevant densities for the compact stars. q  q  D  Presence of Free Strange Quarks In the 2SC+s phase, we have considered the strange quark mass to be equal to its current mass of m = 150 MeV. Since we did not consider any neutrinos present and the strangeness to be conserved, the chemical potential for the strange quark was equal to the chemical potential of the down quarks. However, the strange quarks being electrically charged, the charge chemical potential had to change from the value from that for the pure 2SC matter only. This resulted in the decrease of electron chemical potential (p = —pQ — —25p) compared to the superconducting gap. As a result, the 2SC phase resulted was in the gapped mode. However, for high cutoff A — 1500 MeV, as the temperature increased, the gap A decreased and for a narrow region in the phase diagram, the gapless 2SC phase was realizable. Alford and Rajagopal [2] argues that the charge neutral 2SC+S phase is not a favourable phase compared to the colour-flavour-locked (CFL) phase. s  e  161  Chapter 6. Summary and Conclusions The possibility of occurrence of 2SC+S phase depends on the large mismatch of the Fermi momenta between the light (up and the down) quarks and the strange quark. The constituent mass of the up and down quarks are found to be much smaller (m -. ~ md-c<mst ~ 330 MeV) than the constituent mass of the strange quark (m ~ 500 MeV) [37]. As the baryon density increases, the constituent mass decreases and the strange quarks restore chiral symmetry at a larger baryon chemical potential than the up and the down quarks. The 2SC+S phase may occur only at a small range of the baryon chemical potential where the strange quarks are populated but still heavy enough that their Fermi momentum is much smaller than the Fermi momentum of the light quarks, hence disfavouring Cooper pairing. u  const  s  6.2.2  const  Maximum Mass of Compact Stars  The mass of the quark stars are seen to depend strongly on the equation of state and the bag constant used. For pure 2SC and mixed phase of 2SC and normal quark matter, the authors in the reference [42] find the maximum mass of the quark star to be M ^Qs ~ 1.7M©. Their theory of 2SC phase has the quark-antiquark as well as the di-quark interactions at moderate coupling strength. This is comparable to our results for the mass of nonstrange quark stars (c.f. Fig. 4.2). On the hadronic sector, we employ the theory of hadronic matter in the mean field approximation following Glendenning [30]. We find the maximum mass of a purely hadronic star to be M x-HS ~ 1.3M . For the same set of parameters, our result is comparable to the maximum mass for compact star containing neutron, proton and hadrons as reported in the reference [30]. max  ma  6.3  Q  Outlook  This project was an exercise in determining the thermodynamic properties of the two-flavour colour-superconducting phases and studying the structural and thermal behaviour of the compact stars having these phases of matter, especially during cooling of the stars. The particular focus was on some specific phases that are a few of many possible colour superconducting phases possible inside compact stars. In this respect the project was an endeavour that may be repeated for other possible colour-superconducting phases. 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Phase diagram of neutral quark matter at moderate densities. arXiv:nucl-th, /0602018, 2006. [66] R. L. Stratonovich. Doklady Akad. Nauk., 115, 1957. [Translation: Sov. Phys. Doklady 2, 416(1958)]. 168  Bibliography [67] T. Tatsumi V. Thorsson, M. Prakash and C. J. Pethick. Neutrino emission from dense matter containing meson condensates. Phys. Rev. D, 52, 1995. [68] T. Tatsumi V. Thorsson, M. Prakash and C. J. Pethick. Neutrino opacities in neutron stars with kaon condensates. Phys. Rev. D, 67, 2003. [69] J.D. Walecka. A theory of highly condensed matter. Ann. Phys., 83, 1974. [70] Edward Witten. Cosmic seperation of phases. Phys. Rev. D, 30, 1984. [71] Ya. B. Zel'dovich and I. D. Novikov. Stars and Relativity. University of Chicago Press, 1971.  169  Appendix A Frequency Summation of the Free Energy In this appendix we discuss the summation over the Matsubara frequencies in the expression of the free energy, which appear in finite temperature field theory [44]. In the general form of the free energy, we have the summation 1B2  =E E  I  n  (A.l)  +  where /3 = ^ is the inverse of temperature (ks = 1) and uj — (2n + l)iyT are the Matsubara frequencies for fermions. We consider the summation over n at a particular momentum p and differentiate the free energy with respect to E n  p  d  E  p  ~  +  2  E  p  ^  +  [{2n +  l)ir}*+p Ef 2  n=+oo  = w> E  (  -. 2  „  . <-> A2  +  1  )  W  71— — C O  where 6 = @E . The summation over n can be performed using the standard residue summation formula V  ™y^° 1 _ n(cot(nx) - cot(ny)) ^ (n — x)(n — y) y—x  ^  ^  170  Appendix A. Frequency Summation of the Free Energy to get m=+oo  n=+co  E  0 + (2n + 1) 7T 2  2  2  E  0 + (2m + 1) 7T 2  2  2  71=— OO  ^  n=+oo  1  ^ ^ o c ^ - ^ ) ^ - ^ )  1  (cot(^)-cot(*&))  n  4TT  2  1  ^f5  1  N  , ,0,  1 = e* + l  1  (A.4)  2/(0/0)]  where the function f(x) = (e + 1) is the usual Fermi Dirac distribution function. Thus the general term in the Free energy becomes 0x  lnZ  p  =  1  1  J dE 8 1 - 2 p  e  0E  v  +  1  = PE + 2 \n(e- " + 1) 0E  p  = p\E  p  + 2T\n( 1 +  (A.5)  which the result sought.  171  Appendix B General Thermodynamic Relations The composition and the thermal properties of matter inside a compact star is determined by the thermodynamic temperature and the baryon chemical potential at that point of the star.In the phase diagram of QCD, the corresponding point (p, T), gives the phase structure of the matter. The temperature and the baryon chemical potentials are determined by the central values (at r = 0) and by the gravitational field at that radial distance. The thermodynamic properties of the matter can however, be locally described, subject to the constraint of the fixed temperature and chemical potential, at the position. In such a system, the total energy as well as the particle numbers of different species are not fixed, only the ensemble averages being fixed by the conditions external to the system, i.e. by local {p,T). The proper thermodynamic descriptions of such systems entails the use of grand canonical ensemble and the state being conveniently determined by giving the grand canonical potential defined as a function of temperature T, chemical potential p and volume of the system:  n = h{T,p,v) i  =  F-J^piNi  (B.2)  i  = F-G  = -pV  (B.3)  where U is the total internal energy of the system, F = U — TS is the Helmholtz free energy, G = U — TS + pV = J2i faNi is the Gibbs free energy and the particle species are denoted by the index i with pi and Ni as their corresponding chemical potentials and numbers. At chemical equilibrium, all the chemical potentials can be determined from the single "baryon" chemical potential p. 172  Appendix B. General Thermodynamic Relations The related intensive quantity is the grand potential per unit volume  n = ncr,„) = ^fYl -  v  {BA)  v  2 ^ ^  V  i  =  e-Ts-^piUi  (B.5) i  = p  (B.6)  In the statistical mechanical description, the grand thermodynamic potential O is defined in terms of the grand partition function as  173  Appendix C Generalized Fermi-Dirac Functions and Thermodynamic Quantities Fk{n,P)  The generalized Fermi-Dirac functions, W  )  p!<l±i^  s  J where /3 = T = %T,X = E /k T kin  =  B  (  c,)  exp (x - 77) + 1  0  C.l  was defined in chapter 3 as  = ilzil  a n  d 77 =  Free Fermion Thermodynamic Quantities  Consider relativistic free fermions of mass m having a total energy E(k) = y/k c + m c = (k + m ) / . The Lorentz factor is given by 7 = 7(1;) = 2  2  2  4  2  2  1  2  (l-^j , such that E (k) = 7 m V c + m c = 7 m c => £(&) = Hence, putting c = 1,fcs= 1, we get 2  k  2  =  2  2  m ( l + fix) - ^k 2  2  2  m  2  4  2  2  4  = myJ2px(l + ^px)  £ - ^^^(i+w./. =• fcdA; = m /3(l + (3x)dx ( 7 m - / i ) / T = ((7 — l)m — (LI — m))/T = x — 77 2  7771c . 2  (C.2) (&3)  (C.4) (C.5)  To express the thermodynamic quantities in terms of the generalized Fermi-Dirac functions, consider the number density of particles and anti174  Appendix C. Generalized Fermi-Dirac Functions and Thermodynamic Quantities particles rv  f°°  n = [f (k,p)-f D(k,fl)]k dk 2  FD  1 dx exp(^-??) + l J  F  ^m J P(l  + Px)k  2  o  exp(^-J?) + l  P \l/2 X  / 0(1 + Jo  n  Px)m^2plc(l  2 '  1 Lexp (x — fj) + 1  gf m  3  ^ jf(** +  )(1 + ^ )  exp (x — 77) + 1 g | m  3  1 exp (x — rj) + 1 1 / 2  dx exp (x — 77) + 1  [ F i (77, 0),+ 0F (77, P) - Fi (i), 0) - PF\  P)\ (C6)  f  Similarly, for the expression of pressure, we use the result k , 4„2 „5 3 r = r/cdrC = 771 2202^2 (1 F(fc) 7771c  dk  0 X .  2  Hence we get, V  3 ,  (C.7)  + — ) dx 2  2  1 Ideg  =  3 27T  =  2  32^2  m  y (  2 2 / 3 2  x  +  ^  X  ^  +  + exp (x — 77) +dx Lexp (x — 77) + 1 1 1 Ideg 4 o ^ - ^ f m 2202 2  =  5  F (77,0) + §  ^,(77,0)  + Fa (77,0)  + IF (77,0)] (C.8) f  For the energy density, we use the result E(k)k dk 2  = >ymc k dk 2  2  =  m v 2 0 5 ( x 5 + 2 0 x f + 0 x i ) ( l + ^)5dx  =  m \/20§(1 +  4  4  /  2  [(x5 + 0x§) + 0(x§ + 0x§)]dx (C.9) 175  Appendix C. Generalized Fermi-Dirac Functions and Thermodynamic Quantities Hence we get, F (f ,P) l  Z7T  j  + 2PF (f,,P) +  F (?j,B) + 28F,(fj,3) +  2  B F,(fj,B) 2  h  C.2  P F,(fi,P)  3  (CIO)  Hadronic Thermodynamic Quantities  In the relativistic mean field theory, all the thermodynamic quantities have the same expressions as the free fermion quantities with the mass and chemical potentials replaced by the medium modified effective mass and effective chemical potentials, m* and p* respectively. Their expressions are given in chapter 3 (c.f. eqs. 3.28, 3.30, 3.31 )as : t  m*  B  = m -g a B  (Cll)  aB  p* = p + QBPq - 9UBU° - g BP°hB p* = - P b ~ QBPq +g^B^ + g p° (-I B) B  b  P  0  B  pB  3  3  (C12) (C.13)  Hence, we see that the effective mass can be negative depending on the value of scalar field a. In the numerical evaluation of the thermodynamic quantities, however, we can still use the same formulae as above because of the following reasons: 1. In the original formula for the baryon number density, pressure, energy density etc. mass appears in a quadratic form. Hence these are invariant under a change of sign of the mass (n ,p,e) B  m  > (n ,p,e) B  (C.14)  2. Although 6 = T/m* and x — (7—l)m*/T, depend on m*, in the numerator of the Fermi-Dirac function, their combination f3x appears which is independent of the effective mass. Furthermore, we can redefine B and x in terms of the absolute value of the effective mass. 3. We define the effective velocity of the hadrons v* such that k = 7|m*|-i7*, with 7 = (l — u* /c ) . Then, the usual relativistic dispersion relation is maintained with the effective quantities and we get E*(k) — 2  2  176  Appendix C. Generalized Fermi-Dirac Functions and Thermodynamic Quantities 7|m*|c . Hence all the above manipulations for the free fermions are valid with the mass of the fermions replaced by the absolute value of the hadronic effective mass. 2  177  

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