Some Aspects of Particle Physics in Thermal Background
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SOME ASPECTS OF PARTICLE PHYSICS IN THERMAL BACKGROUND THESIS SUBMITTED TO JADAVPUR UNIVERSITY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY (SCIENCE) arXiv:hep-ph/0205110v1 11 May 2002 By INDRAJIT MITRA THEORY GROUP SAHA INSTITUTE OF NUCLEAR PHYSICS KOLKATA 2002 ACKNOWLEDGMENTS I am indebted to Prof. Palash Baran Pal for guidance, encouragement, collaboration and numerous stimulating discussions. I am also grateful to Prof. Jos´eF. Nieves for collaboration and many valuable discussions via e-mail. Interactions with many other members, both past and present, of the Theory Group of SINP, including Dr. Ananda DasGupta, Prof. Asit K. De, Prof. Binayak Dutta-Roy, Dr. Abhee K. Dutt-Mazumder, Prof. A. Harindranath and Prof. Samir Mallik, have helped me learn quantum field theory and thermal field theory. I thank them all. I express my gratitude to all members of the Theory Group for their help and cooper- ation. The Research Fellows, in particular, have helped me in every possible way during my stay here. Last, but not the least, I thank my mother for her support and understanding, without which this work could not have been done. ii TABLE OF CONTENTS Page Acknowledgments..................................... ii ListofFigures ...................................... vi Chapters: 1. Introduction..................................... 1 1.1 Importance of particle physics in thermal background . .... 1 1.2 The problems considered in this thesis and their motivations . .. 2 1.3 Outlineofthethesis ............................. 5 1.4 Somenotationsandconventions. 6 2. Thermalfieldtheory ................................ 7 2.1 Introduction ................................. 7 2.2 Thermodynamic equilibrium . 7 2.3 Time-orderedthermalpropagators . 10 2.3.1 Scalarfield .............................. 10 2.3.2 Fermionfield ............................. 12 2.3.3 Photonfield.............................. 14 2.4 Additionalvertexandpropagators . 15 3. Linearizedtheoryofgravity ........................... 17 3.1 Introduction ................................. 17 3.2 Graviton ................................... 18 3.3 Photonsincurvedspace ........................... 22 3.4 Spinorsincurvedspace ........................... 23 3.5 Interactionvertices.............................. 28 3.5.1 Photon-photon-gravitonvertex . 29 3.5.2 Fermion-fermion-gravitonvertex . 30 iii 3.5.3 Fermion-fermion-photon-graviton vertex . 32 3.5.4 Importantfeaturesofthevertexfactors . 33 4. ChargedleptonsinthermalQED . 35 4.1 Dispersion relation and inertial mass . 35 4.2 Spinor and wavefunction renormalization factor . .. 39 4.3 Effectofchargeconjugationinvariance . 43 4.4 Relations when background is not C invariant............... 47 4.5 Calculations at O(e2)............................. 48 4.5.1 Self-energy .............................. 48 4.5.2 Calculation of inertial mass . 51 4.5.3 Calculation of wavefunction renormalization factor . 53 5. Gravitational couplings of charged leptons in a medium . .... 55 5.1 Introduction ................................. 55 5.2 Gravitationalmass .............................. 55 5.3 Operational definition at O(e2) ....................... 59 5.4 Gravitationalvertex ............................. 62 5.4.1 Irreduciblediagrams ......................... 62 5.4.2 Bubble diagrams given in Fig. 5.1 . 64 5.4.3 Tadpole diagrams given in Fig. 5.2 . 66 5.4.4 Additionalremarks.......................... 69 5.5 Calculation of the gravitational mass . 70 5.5.1 Terms with the photon distribution from Fig. 5.1 . 70 5.5.2 Terms with the electron distribution from Fig. 5.1 . 72 5.5.3 TermsfromFig.5.2 ......................... 74 5.5.4 Summary ............................... 75 5.6 Resultsforparticularcases . .. .. .. 76 5.6.1 Classical electron gas and classical proton gas . 77 5.6.2 Degenerate electron gas and classical proton gas . 80 6. Someaspectsofhigh-temperatureQED . ... 83 6.1 Introduction ................................. 83 6.2 One-loop electron self-energy in a general linear covariant gauge..... 84 6.3 Equation governing gauge dependence of dispersion relation at one loop, andgeneralconsiderations. 88 6.4 Gauge independence of one-loop dispersion relation at momenta much larger than eT ................................ 89 6.5 Gauge independence of effective mass at one-loop . .. 91 6.5.1 Subtletiesatzeromomentum . 91 6.5.2 Formulafortheeffectivemass. 91 iv 6.5.3 Investigationofgaugeindependence . 93 7. Conclusions ..................................... 96 Appendices: A. Field quantization using discrete momenta . .. 99 B. Full propagator and wavefunction renormalization factor in vacuum . .101 C. Charge conjugation and the 1PI gravitational vertex . .......104 D. Transversalityofthegravitationalvertex . ......107 E. Apparently ill-defined contributions to the gravitational vertex .........111 E.1 TheB1contribution .............................111 E.2 TheA2contribution .............................114 E.3 TheXcontribution..............................117 E.4 Alternative evaluation of the B1 and A2 contributions . 119 F. Thermal contribution to the equation for the effective mass at high temperature123 Bibliography .......................................126 ListofPublications....................................130 v LIST OF FIGURES Figure Page ′ 3.1 Feynman rule for the coupling of the photon with the graviton. Cµνλρ(k,k ) isgiveninEq.(3.79). ............................. 29 ′ 3.2 Feynman rule for the tree-level gravitational vertex of a fermion. Vλρ(p,p ) isgiveninEq.(3.95). ............................. 32 3.3 Diagrams of O(eκ) with two fermions, one photon and one graviton in the external lines. These are the tree-level diagrams for the photoproduction ofgravitonsoffelectrons. .. .. .. 33 4.1 One-loop diagrams for the self-energy of a charged lepton ℓ in a medium. .. 48 5.1 Bubble diagrams for the one-loop gravitational vertex of charged leptons. These diagrams contribute in a background of photons and electrons. The braided line represents the graviton. ...................... 63 5.2 Tadpole diagrams for the one-loop gravitational vertex of charged leptons. These diagrams contribute to the vertex of any charged lepton in a background of electrons and nucleons. ........................... 63 5.3 The one-loop diagrams that contribute to order κ to the photon tadpole in a medium, in the presence of a static and homogeneous (q = 0) gravitational potential. The fermion loop involves a sum over all the species of fermions present in the medium. 67 vi CHAPTER 1 Introduction 1.1 Importance of particle physics in thermal background The study of elementary particles in vacuum is based on quantum field theory. How- ever, there are situations in which it is necessary to consider these particles in a non-trivial background consisting of a large number of particles at finite temperature and/or density. Such temperatures and densities of significance occur in stellar interiors and were also present in the early universe. Another relevant situation occurs in the laboratory during heavy-ion collisions. The formalism needed for the study of these situations is provided by thermal field theory, also known as finite temperature field theory [1, 2, 3]. From a theoretical point of view (as we will see in the course of the present work), two major features are induced by the background medium. These are the occurrence of an additional four-vector in the form of the four-velocity of the medium, and the breaking of discrete symmetries like C, P and T even when the underlying Lagrangian respects these symmetries. As a result, self-energies, vertex functions and other Green’s functions involve new tensor structures and associated form factors, together with a relaxation of some of the constraints restricting the form factors. A third important feature induced by the medium (also to be encountered in the present work) is the breaking of flavour symmetry. All these features give rise to rich physics which was absent in the vacuum. A modified self-energy, for example, gives rise to modification of the dispersion relation (energy-momentum relation). This, in turn, among other things, can open up the phase space, and allow processes which were kinematically forbidden in the vacuum to take place in the medium. Let us now turn to some specific illustrations of changes in the properties of elementary particles due to a thermal background. Gauge symmetry spontaneously broken in vacuum can be restored by finite temperature corrections. The masses of the fermions and gauge 1 bosons of the Standard Model vanish when this happens [4]. In the phase of restored chiral invariance, one finds the generation of thermal effective masses of the particles, different effective masses of left-handed and right-handed leptons, and different dispersion relations of transverse and longitudinal modes of gauge bosons [5, 6]. Modification of photon dispersion relations leads to Cerenkov radiation (emission of a photon) by even a single neutrino in a medium [7]. The dispersion relation of neutrinos become flavour dependent in a medium containing electrons but not the other charged leptons [8, 9]. In such a flavor asymmetric background, the electromagnetic [10] and gravitational [11] interactions of the neutrinos in response to external fields are also flavor dependent, while the radiative decay of a massive neutrino is strongly enhanced compared to the decay in vacuum (as the leptonic GIM mechanism becomes inoperative) [12]. When neutrinos travel in a background of electrons with a static external magnetic field also present, the latter causes an