Mass Gap and its Applications VAKHTANG GOGOKHIA GERGELY GABOR BARNAFOLDI Institute for Particle & Nuclear Physics, Wigner Research Centre for Physics, Hungarian Academy of Sciences, Hungary World Scientific • • • • NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI • HONG KONG • TAIPEI • CHENNAI Contents Preface vii Acknowledgments ix Theory of the Mass Gap 1 1. Quantum Chromodynamics and the Mass Gap 3 1.1 Quantum Chromodynamics 3 1.2 The Jaffe-Witten theorem on the Mass Gap 5 2. Color Gauge Invariance and the Origin of the Mass Gap 9 2.1 Introduction 9 2.2 The gluon Schwinger-Dyson equation 11 2.3 Transversality of the full gluon self-energy 14 2.4 Slavnov-Taylor identity for the full gluon propagator . 19 2.5 The general structure of the full gluon propagator .... 21 2.6 Non-perturbative vs. Perturbative QCD 24 2.7 The Mass Gap 26 2.8 Subtraction at the fundamental gluon propagator level . 28 2.9 Discussion 31 2.A Appendix: Application for Abelian case 34 3. Formal Exact Solutions for the Full Gluon Propagator at Non-zero Mass Gap 39 3.1 Introduction 39 3.2 Singular solution 41 xi xii The Mass Gap and Its Applications 3.3 Massive solution 46 3.4 Conclusions 49 3.A Appendix: The dimensional regularization method in the perturbation theory 51 3. B Appendix: The dimensional regularization method in the distribution theory 53 4. Renormalization of the Mass Gap 59 4.1 Introduction 59 4.2 The intrinsically non-perturbative gluon propagator ... 60 4.3 Confining gluon propagator 61 4.4 The renormalized running effective charge 64 4.5 The general criterion of gluon confinement 65 4.6 The general criterion of quark confinement 68 4.7 The general criterion of dynamical/spontaneous breakdown of chiral symmetry 69 4.8 Physical limits 72 4.9 Asymptotic freedom and the mass gap 73 4. A Appendix: The Weierstrass-Sokhatsky-Casorati theorem 76 5. General Discussion 79 5.1 Discussion 79 5.2 Subtractions 83 5.3 Conclusions 86 Applications of the Mass Gap 91 6. Vacuum Energy Density in the Quantum Yang-Mills Theory 93 6.1 Introduction 93 6.2 The vacuum energy density 94 6.3 The intrinsically non-perturbative vacuum energy density 97 6.4 The bag constant 99 6.5 Analytical and numerical evaluation of the bag constant . 101 6.6 The trace a,nomaly relation 105 6.7 Comparison with phenomenology 107 6.8 Numerical values for Bym in different units 109 6.9 Contribution of Bym to the dark energy problem 110 Contents xiii 6.10 Energy from the QCD vacuum Ill 6.11 Conclusions 114 6. A Appendix: The general role of ghosts 117 7. The Non-perturbative Analytical Equation of State for the Gluon Matter I 121 7.1 Introduction 121 7.2 The gluon pressure at zero temperature 123 7.3 The gluon pressure at non-zero temperature 125 7.4 The scale-setting scheme 127 7.5 The PNp{T) contribution 127 7.6 Conclusions 131 7. A Appendix: The summation of the thermal logarithms . 133 8. The Non-perturbative Analytical Equation of State for the Gluon Matter II 137 8.1 Introduction 137 8.2 Analytic thermal perturbation theory 137 8.3 Convergence of the perturbation theory series 144 8.4 The gluon pressure, Pg(T) 147 8.5 Low-temperature expansion 148 8.6 High-temperature expansion 156 8.7 Discussion and conclusions 162 9. The Non-perturbative Analytical Equation of State for SU{3) Gluon Plasma 171 9.1 Introduction ' 171 9.2 The gluon pressure Pg(T) 171 9.3 The full gluon plasma pressure 174 9.4 Main thermodynamic quantities 189 9.5 The Stefan-Boltzmann limit 190 9.6 Analytical formulae for the gluon plasma thermodynamic quantities 191 9.7 Double-counting in integer powers of as problem 193 9.8 Numerical results and discussion 196 9.9 The dynamical structure of SU(3) gluon plasma 207 9.10 Conclusions 212 xiv The Mass Gap and Its Applications 9.A Appendix: Analytical and numerical evaluation of the latent heat 217 9.B Appendix: The /3-function for the confining effective charge at non-zero temperature 218 9.C Appendix: Least Mean Squares method and the definition of the average deviation 219 9.D Appendix: Restoration of the lattice pressure below 0.9TC 220 Bibliography 225 Index 233.