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A Model of Composite Dark Matter in Light-Front Holographic QCD

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A Model of Composite Dark Matter in Light-Front Holographic QCD Master's thesis Theoretical physics A Model of Composite Dark Matter in Light-Front Holographic QCD Jere Remes 2015 Advisor: Kimmo Tuominen Examiners: Kimmo Tuominen Aleksi Vuorinen UNIVERSITY OF HELSINKI DEPARTMENT OF PHYSICS PL 64 (Gustaf H¨allstr¨omin katu 2) 00014 University of Helsinki HELSINGIN YLIOPISTO HELSINGFORS UNIVERSITET UNIVERSITY OF HELSINKI Tiedekunta/Osasto Fakultet/Sektion Faculty Laitos Institution Department Faculty of Science Department of Physics Tekijä Författare Author Jere Remes Työn nimi Arbetets titel Title A Model of Composite Dark Matter in Light-Front Holographic QCD Oppiaine Läroämne Subject Theoretical physics Työn laji Arbetets art Level Aika Datum Month and year Sivumäärä Sidoantal Number of pages Master's thesis August 2015 87 Tiivistelmä Referat Abstract Even after 50 years, there is still no standard, analytically tractable way to treat Quantum Chromodynamics (QCD) non-numerically besides perturbation theory. In the high-energy regime perturbation theory agrees with experimental results to a great accuracy. However, at low energies the theory becomes strongly coupled and therefore not computable by perturbative methods. Therefore, non-perturbative methods are needed, and one of the candidates is light-front holography. In this thesis, the basics of light-front quantisation and holography are discussed. Light-front quantisation takes light-cone coordinates and the Hamiltonian quantisation scheme as its basis and the resulting eld theory has many benecial properties like frame-independence. Still, to extract meaningful results from the light-front QCD, one needs to apply bottom-up holographic methods. Last part of this work focuses on the applicability of light-front holographic QCD in the area of dark matter. We nd that one can build a secluded SU(3) sector consisting of a doublet of elementary particles, analogous to quarks and gluons. Due to a global symmetry, the lightest stable particle is analogous with ordinary neutron. It meets the basic requirements for being a WIMP candidate when its mass is higher than 5 TeV. Avainsanat Nyckelord Keywords holography, light-front holography, QCD, dark matter Säilytyspaikka Förvaringsställe Where deposited Kumpula campus library Muita tietoja Övriga uppgifter Additional information Contents Notations and Conventions 1 1 Introduction 3 2 Basics of Gauge/Gravity Duality 7 2.1 The Holographic Principle . .7 2.2 Type IIb Strings on AdS5 ....................8 2.2.1 Anti-de Sitter Space . .8 2.2.2 Type IIb String Theory . .9 2.3 AdS/CFT . 10 2.3.1 Conformal Field Theory . 10 2.3.2 AdS/CFT Correspondence . 11 2.4 Holography Bottom-up . 12 3 Light Front Holography 15 3.1 The Problem with Strong Interactions . 15 3.1.1 Experimental Evidence of QCD . 15 3.2 Light-Cone Quantisation . 18 3.2.1 The Hamiltonian Quantisation Scheme . 19 3.3 Modelling Hadrons . 23 3.3.1 Light-Front Wave Equation for Mesons . 23 3.3.2 The Duality . 26 3.3.3 Effective Potential . 27 3.3.4 Light Front Wave Equation for Baryons . 30 3.4 Hard- and Soft-Wall Models for Hadrons . 35 3.4.1 Meson Masses . 36 3.4.2 Baryon Masses . 39 3.5 Form Factors for Hadrons . 42 3.5.1 Meson Form Factors . 43 3.5.2 Nucleon Form Factors . 51 3.6 Improving the Model . 59 4 A Model of Composite Dark Matter 61 4.1 A Word About Dark Matter . 61 4.1.1 Observational Evidence for Existence . 62 4.1.2 The Dark Matter Problem . 62 4.1.3 Exclusions from Direct Detection Experiments . 63 4.2 Proposing a New Strongly Interacting Sector . 64 4.3 Results . 68 5 Conclusions And Discussion 71 Bibliography 73 A Tables of Particles 85 Kiitokset Opiskeluvuodet Helsingin yliopistolla ovat olleet elämäni rikkaimpia vuosia sekä ympäristön, matkustamisen, taiteen, tieteen että ennen kaikkea ihmisten takia. Siksi onkin tarpeen osoittaa kiitokseni teille yhteisesti. Ensiksi haluaisin kiittää ohjaajaani Kimmo Tuomista tämän työn mai- niosta ohjauksesta, aiheen esittämisestä ja valaisevista keskusteluista. On myös aihetta kiittää Aleksi Vuorista tämän työn tarkastamisesta. Lisäksi tämän tutkielman sisältöön vaikuttivat maininnan arvoisesti omilta osiltaan Tommi Tenkanen ja Eemeli Tomberg auttaessaan erään matemaattisen kyn- nyksen ylittämisessä ja Johann Muszynski antaessaan tärkeää palautetta tästä behemotista. Jarkko Järvelää ja Alexander Meaneyta taas on kiittäminen vertaistuesta työn synkempinä hetkinä. Tuntuisi vähättelyltä alkaa eritellä opiskelutovereitani nimin kiitokseen: mikäli olette joskus pudonneet seinältä seuranani tai istuneet pöytään kanssani miettimään olevaisen luonnetta missään muodossa – fyysisessä kuin metafyy- sisessä – olette ansainneet syvimmät kiitokseni, sillä mikään ei ole niin elähdyttävää kuin älyllinen keskustelu. Lisäksi haluan muistaa perhettäni opintoihin kannustamisesta ja kasvat- tamisestani. On ollut etuoikeus saada kasvaa niin moninaisessa ja tiiviissä perhepiirissä kuin muodostamamme on ollut. Suurimmat kiitokset kuuluvat Katjalle sietämisestäni ja järjissä pitämi- sestäni. Enempää en osaa tekstualisoida tähän. Minä sanon teille: täytyy vielä sisältää kaaosta voidakseen synnyt- tää tanssivan tähden. Minä sanon teille: teissä on vielä kaaosta. F. Nietzsche, Näin puhui Zarathustra Notations and conventions In this thesis, natural units are used, i.e. ~ = c = 1. Also, the Minkowski metric tensor η is realised in the mostly minus form ηµν = diag(1, −1, −1, −1). (1) The basic AdSd+1 metric is in the form R2 ds2 = η dxµ dxν − dz2 , (2) z2 µν where the Greek letters denote the Lorentz indices of the d-dimensional spacetime µ, ν = 0, . , d. The vielbeins for AdSd+1 are R eA = δA , (3) M z M where M = 0, . , d are the AdS indices and A = 0, . , d are the AdS tangent space indices. For the rank-J tensors in AdSd+1 the notations Φ{N} and Φ{LN/j } are used, where Φ{N} = ΦN1N2...NJ (4) Φ{LN/j } = ΦLN1...Nj−1Nj+1...NJ , (5) and a similar notation is used for the products of components of the metric gMN : g{MN} = gM1N1 ··· gMJ NJ . (6) In light-front the Lepage-Brodsky (LB) convention [1] is used: µ + − 1 2 + − x = (x , x , x , x ) = (x , x , x⊥), (7) x+ = x0 + x3, (8) x− = x0 − x3, (9) 1 where the x0 and xi are the Minkowski coordinates (i = 1, 2, 3). The flat metric is then 1 0 2 0 0 0 0 0 2 2 0 0 0 1 µν 0 0 0 g = , gµν = 2 . (10) 0 0 −1 0 0 0 −1 0 0 0 0 −1 0 0 0 −1 In the LB convention the new Dirac matrices in Dirac representation become γ+γ+ = γ−γ− = 0 (11) and the projection operators are thus 1 1 Λ = γ0γ+ = γ−γ+, (12) + 2 4 1 1 Λ = γ+γ− = γ+γ−. (13) − 2 4 The volume integral in the LB convention is Z 1 Z Z dω = dx− d2x = dx d2x , (14) + 2 ⊥ + ⊥ where the raising and lowering of the Lorentz indices is done with + +12 = 1 (15) 1 ⇒ = . (16) +12− 2 The light-front spinors are 1 + 0 0 k χ(↑) for λ = +1 u(p, λ) = √ p + γ m + γ γ p⊥k × (17) p+ χ(↓) for λ = −1 1 + 0 0 k χ(↓) for λ = +1 v(p, λ) = √ p + γ m − γ γ p⊥k × (18) p+ χ(↑) for λ = −1, where k = 1, 2, 3 and 1 0 1 0 1 1 χ(↑) = √ , χ(↓) = √ . (19) 2 1 2 0 0 −1 To avoid confusion between Euler’s constant e and electromagnetic coupling constant e, the latter is always in italics. 2 Chapter 1 Introduction Quantum chromodynamics (QCD)– the SU(3) gauge field theory of quarks and gluons – has been proven by time to be the correct theory of strong interactions. Perturbative QCD predicts a great variety of high-energy scattering results with great success. However, because of its strongly coupled nature, no perturbative predictions can be made for low energies, and therefore there is still no analytical demonstration of colour confinement, hadron mass spectrum or other crucial details of the theory. Important aspects on that end have been shown by numerical Euclidean lattice methods. But even as lattice QCD has been hugely successful in many cases, aspects of QCD, such as the excitation spectrum of light hadrons, is still a formidable challenge because of its computational complexity. It would be of a great theoretical advantage to have an analytically calculable form of the theory for closer examination of its properties like the mass spectrum. There is still no standard, analytically tractable way to handle quantum field theories besides perturbation theory. To find one, it is probable that we need to make an analytical, relativistic, non-perturbative approximation of the wanted model, which can then be improved upon. One of the possible methods for achieving an approximation of QCD is using light-front holography – a light-cone quantised formulation of QCD dual to a gravity theory. Light-cone quantisation was found by P. A. M. Dirac in 1949 [2]. It takes light-cone coordinates x+ = x0 − x3 as the new time coordinate, and uses a 3 Hamiltonian quantisation scheme instead of the usual action quantisation. These simple changes result in a fully relativistic, frame-independent, ghost- free formulation of the field theory. Because the vacuum has a simple structure, the light-front wave functions (LFWFs) can be defined unambiguously. The wave functions |ψi of a bound state can be obtained from the eigenvalue 2 equation HLF |ψi = M |ψi, which becomes an infinite set of coupled integral equations for the Fock-state expansion. Here an approximation is made to solve the equation, and only the valence state is taken into account. Still, the solution is lacking the interaction term, which cannot be retrieved from pure QCD. This is where holographic methods are needed. Holographic models utilise the AdS/CFT duality. It is a conjectured 5 duality between type IIb string theory in AdS5× S and conformal N = 4 super Yang-Mills gauge field theory, first put forward by Juan Maldacena in his seminal 1997 article [3].
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