Aristotle University of Thessaloniki School of Physics B.Sc. Thesis in Relativistic Mean-Field Theory In Quantum Hadrodynamics Author: Theodoros Soultanis Supervisor: Theodoros Gaitanos September, 2016 1 Abstract The goal of this Thesis is to introduce the reader to the main ideas regarding relativistic mean-field theory and modeling of nuclear matter. The description of nuclear matter is studied in the framework of the nuclear model called as Quantum Hadrodynamics (QHD). In QHD, we treat protons and neutrons as elementary particles where their interaction is mediated through the exchange of sigma-, omega- and rho- mesons. The prescription follow is simple, firstly we present the most important nuclear properties any nuclear model has to reproduce. In the later chapters we study the basic concepts of classical field theory and then we proceed on the quantization of neutral Klein-Gordon and Dirac fields. Consequently, we compute the energy-momentum tensor for each of the most important relativistic fields. The energy momentum tensor provides the energy density and pressure expressions of the system. Finally, we deploy all the previous knowledge and apply it to QHD. We derive the Euler-Lagrange equations of motion for the QHD-I parameter set and in the next chapter we study more advanced QHD parameter sets. In both cases we apply the mean-field approximation to simplify the complicated equations of motion and derive the equation of state of the system. Finally we compute the theoretical expressions of the symmetry energy coefficient. 2 PerÐlhyh O stìqoc thc Diplwmatik c ergasÐac eÐnai na eisaggoume ton anagn¸sth stic basikèc idèec ìson afor thn sqetikistik jewrÐa tou mèsou pedÐou kai thn montelopoÐhsh thc purhnik c Ôlhc. H perigraf thc purhnik c Ôlhc epitugqnetai sta plaÐsia tou purhnikoÔ montèlou gnwstì kai wc Kbantik Adrodunamik (QHD). 'Opou antimetwpÐzoume ta prwtìnia kai netrìnia wc stoiqei¸dh swmtia twn opoÐwn h allhlepÐdrash pragmatopoieÐtai me thn antallag twn sÐgma-, wmèga- kai ro- mesìniwn. H dom thc ergasÐac eÐnai apl , arqik parajètoume tic pio shmantikèc idiìthtec thc purhnik c Ôlhc tic opoÐec opoiod pote purhnikì montèlo ofeÐlei na anapargei. Sta epìmena keflaia meletoÔme tic ènnoiec thc klassik c jewrÐac pedÐou kai Ôstera proqwroÔme sthn kbntwsh tou pragmatikoÔ Klein-Gordon kai Dirac pedÐwn. En suneqeÐa, upologÐzoume ton tanust enèrgeiac-orm c gia kje èna apì ta pio shmantik sqetikistik pedÐa. O tanust c enèrgeiac-orm c parèqei tic ekfrseic thc puknìthtac enèrgeiac kai pÐeshc tou sust matoc. En katakleÐdi, epistrateÔoume ìlec tic prohgoÔmenec gn¸seic kai tic efarmìzoume sthn QHD. Exgoume tic Euler-Lagrange exis¸seic kÐnhshc gia to QHD-I set paramètrwn kai sto epìmeno keflaio meletoÔme akìmh pio exeidikeumèna QHD set paramètrwn. Kai stic dÔo peript¸seic efarmìzoume thn prosèggish tou mèsou pedÐou gia na aplopoi soume tic polÔplokec exis¸seic kÐnhshc ¸ste na exgoume thn katastatik exÐswsh tou sust matoc. Tèloc, upologÐzoume tic jewrhtikèc ekfrseic tou suntelest thc enèrgeiac summetrÐac. 3 Contents 1 Introductory Concepts 5 1.1 Notation . 5 1.2 Nuclear Matter . 6 1.3 Nuclear Properties . 8 2 Field Theory 9 2.1 Lagrangian Formalism . 9 2.2 Hamiltonian formalism . 11 2.3 Noether's Theorem and Conservation Laws in Field Theory . 12 2.3.1 Invariance under translation . 12 2.3.2 Internal Symmetries-Noether Charge . 13 2.4 Relativistic Fields . 14 2.4.1 Klein-Gordon Field . 14 2.4.2 Dirac Field . 16 2.4.3 Photon Field . 19 2.4.4 Massive Vector Boson Field . 20 3 Quantum Field Theory 23 3.1 Harmonic Oscillator . 23 3.2 Neutral Klein-Gordon Field . 24 3.3 Dirac Field . 28 4 Energy-Momentum Tensor 35 4.1 Relativity-EMT-Perfect Fluid . 35 4.2 EMT- Klein-Gordon Field . 36 4.3 EMT- Dirac Field . 36 4.4 EMT- Massive Vector Boson Field . 37 5 Quantum Hadrodynamics - QHD 39 5.1 QHD-I . 40 5.2 Relativistic Mean-Field Approximation . 41 5.3 Baryon Field - Expectation values . 42 5.4 Parameter Set, Equation of State, Observables . 47 5.5 Alternative Method - Expectation Values . 49 6 Advanced QHD Parameter Sets 51 6.1 Formalism and Conventions . 52 6.2 Relativistic Mean-Field Approximation . 54 6.3 Expectation Values . 56 6.4 Parameter Sets, Equation of State, Observables . 58 7 Results 61 Bibliography 64 4 Chapter 1 Introductory Concepts 1.1 Notation In this work the 3-dimensional vectors are expressed either with bold font, p, or with the indexes i; j such as, xi, while the 4-dimensional vectors are denoted by the Greek indexes, µ, ν such as xµ. The contravariant position 4-vector is given by xµ = x0; x1; x2; x3 = x0; x : (1.1) We also employ Einstein' s double-indice summation convention where two repetitive indexes (superscript, subscript) essentially mean a summation over space-time. µ The length of any 4-vector, A , is given in terms of the metric, gµν , of the flat Minkowski space as 2 µ ν A = gµν A A (1.2) µν where gµν and its inverse matrix, g is given by 01 0 0 0 1 01 0 0 0 1 B0 −1 0 0 C µν B0 −1 0 0 C gµν = B C & g = B C : (1.3) @0 0 −1 0 A @0 0 −1 0 A 0 0 0 −1 0 0 0 −1 µν µν βγ γ Note that gµν and g are symmetric. Since g is the inverse matrix of gµν the relation gαβg = δα holds. µ Given the contravariant position 4-vector, x we extract the covariant one xµ: ν 0 xµ = gµν x = x ; −x : (1.4) Therefore the product of two 4-vectors A and B is given by µ 0 0 A · B = A Bµ = A B − A · B : (1.5) The field space-time dependence, φ(xµ) is usually expressed as φ(x; t) or φ(x) (1.6) however sometimes it is omitted for the sake of simplicity. The covariant and contravariant derivatives with respect to contravariant and covariant position 4-vector respectively are hold as @ @ @ @ @ @ ≡ = ; ; = @ ; r (1.7) µ @xµ @x0 @x1 @x2 @x3 0 µ @ @ @ @ @ @ ≡ = ; ; = @0; −r : (1.8) @xµ @x0 @x1 @x2 @x3 where the properties 1.3 where employed. The choice between the two different notations of the partial derivative is usually determined by personal criteria. In addition the time derivative is often expressed as _ @0φ = φ (1.9) 5 Now, in order to simplify a lot of the expressions associated with physical formulas we adopt the natural unit system. In the latter Plank' s reduced constant, ~, and the speed of light are considered to be dimensionless constants equal to unity: ~ = c = 1 : (1.10) Under these terms every quantity is measured either to Energy units (MeV) or Length units (fm) with the following expression connecting them: ~ c = 1 = 197:33 MeV fm : (1.11) In natural units the famous relativistic energy-momentum relation is thus given as E2 = p2c2 + m2c4 ! E2 = p2 + m2 : (1.12) 1.2 Nuclear Matter It is believed that shortly after the Big Bang the universe was filled with an astonishingly hot, dense soup made of particles, such as quarks and gluons, moving with speeds near the speed of light. Quarks are the fundamental bits of matter and are bound together through gluons, the mediators of strong interaction. However in these first moments of extreme temperature they were weakly bound and free to move on their own in what is called a quark-gluon plasma. This state of matter is investigated with high-energy collisions of heavy-ions such as the ones that performed in the STAR and PHENIX experiments at Relativistic Heavy Ion Collider (RHIC) in the US. Further research is done in the ALICE, ATLAS and CMS experiments at CERN's Large Hadron Collider (LHC). Another topic of special interest for nuclear physicists is the study of the interior of neutron stars. Neutron stars are the smallest and densest stars known to exist. Their radius lies in between 11.0 to 11.5 kilometers and their mass is estimated to be between 1.1 and perhaps up to 3 solar masses while the maximum observed mass of neutron stars is about 2.01 solar masses. In general, compact stars of less than 1.39 solar masses are white dwarfs, whereas compact stars with a mass between 1.4 and 3 solar masses should be neutron stars. Based on the current theory regarding the formation of neutron stars, they are the collapsed core of of a massive star. A normal star is sustained against gravitational collapse by the thermal pressure due to the energy release of thermonuclear fusion in its core. Now, throughout this process the core becomes more iron-rich and at the point the fuel is exhausted the core must be supported by electron degeneracy pressure alone. If the electron degeneracy pressure is not strong enough, the core starts to contract under its own gravity, and thus densities where electrons become relativistic will easily be reached. At that point, protons and electrons combine through electron-capture into neutrons resulting a enormous release of neutrinos. Thus, the core becomes more neutron- rich and when the densities reach nuclear density the the gravitational collapse is halted by neutron-degeneracy pressure. The in-falling matter of the outer atmosphere of the star, due the low pressure of the contracting core, rebounds off the stiffened core thanks to the flux of neutrinos, and thus becoming a Type II or Type Ib supernova. The remnant left is the so called neutron star. If the mass of the remnant is more than five solar masses its fate is predestined and it will become a black hole.