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Univerzita Karlova v Praze Matematicko-fyzikální fakulta BAKALÁŘSKÁ PRÁCE Tomáš Nosek Fyzika oscilací neutrin na experimentu NOνA Ústav částicové a jaderné fyziky Vedoucí bakalářské práce: RNDr. Karel

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Univerzita Karlova v Praze Matematicko-fyzikální fakulta BAKALÁŘSKÁ PRÁCE Tomáš Nosek Fyzika oscilací neutrin na experimentu NOνA Ústav částicové a jaderné fyziky Vedoucí bakalářské práce: RNDr. Karel Soustružník, Ph.D. Studijní program: Fyzika, Obecná fyzika Praha 013 Charles University in Prague Faculty of Mathematics and Physics BACHELOR THESIS Tomáš Nosek Physics of Neutrino Oscillations at NOνA Experiment Institute of Particle and Nuclear Physics Supervisor of the bachelor thesis: RNDr. Karel Soustružník, Ph.D. Study programme: Physics, General Physics Prague 013 I would like to thank my supervisor Karel Soustružník for introducing me kindly and with patience to the problematics of neutrino oscillations and for sharing his experience from Fermilab and the NOνA experiment. I would also like to thank Dalibor Nosek for technical suggestions and a lot of critical comments. i I declare that I carried out this bachelor thesis independently, and only with the cited sources, literature and other professional sources. I understand that my work relates to the rights and obligations under the Act No. 11/000 Coll., the Copyright Act, as amended, in particular the fact that the Charles University in Prague has the right to conclude a license agreement on the use of this work as a school work pursuant to Section 60 paragraph 1 of the Copyright Act. In... date... signature of the author ii Název práce: Fyzika oscilací neutrin na experimentu NOνA Autor: Tomáš Nosek Katedra: Ústav částicové a jaderné fyziky Vedoucí bakalářské práce: RNDr. Karel Soustružník, Ph.D. Abstrakt: Bakalářská práce se zabývá fyzikou oscilací neutrin a základními možnostmi měření některých jejich parametrů pomocí experimentu NOνA: určení hierarchie hmot neutrin, zda dochází během oscilací k CP narušení a velikost směšovacích úhlů θ 13 a θ 3. V práci je shrnut standardní formalismus oscilací v modelu 3 neutrin a vysvětlen vliv hmotného prostředí na jejich propagaci prostorem. Objasněny jsou způsoby, jimiž lze získat informace o dosud neznámých parametrech. Pro různé možné realizované kombinace hierarchie hmot, CP narušení a směšovacích úhlů, jsou spočteny očekávané pravděpodobnosti oscilací mionových na elektronová neutrina v konfiguraci experimentu NOνA s použitím analytické aproximace. Ukázány jsou důležité vlastnosti experimentu, možné výsledky experimentu a jeho senzitivita. Klíčová slova: neutrino, oscilace neutrin, hmotová hierarchie neutrin, hmotový efekt, CP narušení Title: Physics of Neutrino Oscillations at NOνA Experiment Author: Tomáš Nosek Department: Institute of Particle and Nuclear Physics Supervisor: RNDr. Karel Soustružník, Ph.D. Abstract: This thesis goes into the physics of neutrino oscillations and into possibility of measuring some of its parameters in the NOνA experiment: the mass hierarchy determination, whether there is a CP violation in neutrino oscillations, or the value of mixing angles θ 13 and θ 3 in particular. The standard formalism of 3 neutrinos model is summarized and the effect of matter to neutrino propagation through space is clarified. It is explained how to get information on yet unknown oscillation parameters. Using analytical approximations expected probabilities of muon to electron neutrino oscillation in NOνA configuration are calculated for different combinations of the mass hierarchy, CP violation, or angles that could be realized in nature. Main features of the NOνA experiment, its sensitivity and basic character of the results that can be obtained are described. Keywords: neutrino, neutrino oscillations, neutrino mass hierarchy, matter effect, CP violation iii Contents 1 Introduction 1 Neutrino oscillations.1 General formalism Three neutrinos model, flavors and masses Propagation in vacuum Oscillation probabilities Mass hierarchy and oscillation parameters Propagation in matter The ν µ ν e channel ν µ ν e appearance probabilities Determination of the mass hierarchy θ 3 octant CP and T violation The NOνA experiment Introduction NuMI beam Off-axis concept Detectors Physics capabilities Measurement of θ Determination of the mass hierarchy Measurement of CP violation and δ θ 3 ambiguity Conclusions 5 Bibliography 7 List of Tables 8 List of Figures 9 List of Abbreviations 30 Notation 31 iv 1. Introduction It was the problem of a continuous energy spectrum of electrons emitted in β- decay and a conservation of angular momentum that lead W. Pauli to propose an existence of a new neutral particle, neutrino (ν). Its role, as it seems nowadays, is of much greater importance than a missing piece of puzzle. A vast number of high energy physics phenomena involves neutrinos, e.g. nuclear reactions that power the sun and stars, cosmic rays, searching for double β-decay, imbalance between matter and antimatter and so on [1]. Neutrinos and photons are by far the most abundant particles in the universe. Hence, if we would like to uncover the secrets of the universe, we must understand neutrinos. Unfortunately they interact only through weak force and gravity making it almost impossible to detect them. Experimental results imply that there are three so called flavor states of a neutrino in accordance to three flavors of charged leptons: electron (e ), muon (µ )andtauon(τ ). Togethertheyareclassifiedinthreegenerations(ofparticles and three of antiparticles) []: ( ) νe e, ( ) ( ) νµ ντ µ, τ, (1.1) ordered in increasing masses of leptons m e = MeV m µ = MeV m τ = MeV [3]. Assuming that the weak interactions do not mix these families we can hypothetically identify the flavor of a neutrino through a process (W-boson decay) in which corresponding lepton takes part, so that for instance a e (e + ) is created in coincidence with a ν e (ν e ). On the other hand neutrino oscillations have been observed, i.e. a flavor transmutation as neutrino travels through space. It seems that reasonable explanation of this fact is leptonic mixing. There are three neutrino mass eigenstates ν i with well defined masses m i which are distinct from the flavor eigenstates. The point is that flavor states are nothing but a linear combination of mass eigenstates and/or vice versa. The greater part of oscillation parameters have been measured by a number of experiments(seeref.[4]). Bothmass-splittings m 1 and m 3 aredetermined, so are mixing angles θ 1, θ 3 and most recently θ 13 thanks to Daya Bay [5], Double Chooz and RENO [6] reactor neutrino oscillation experiments. CP-violating δ phase, sign of m 3 and whether θ 3 or 45 still remain unknown. Theaimofthisthesisistosumupthebasiccharacteristicsofneutrinooscillationswithemphasisonν µ ν e channel,propagationofneutrinosthroughmatter, CP violation and different mass hierarchies. In the second part it will turn its attention to NOνA experiment at Fermilab, USA, depicting main features, goals and possible or expected results that can be obtained. The motivation is to provide a description of neutrino oscillations by measuring values of all fundamental parameters and to explain role of second-generation long-baseline experiments, NOνA in particular, in searching for them. Reader shall notice that the system of natural units is used throughout the text, c = = k B = 1, and we assume that CPT invariance holds. 1 . Neutrino oscillations This chapter describes the formalism of neutrino oscillations in Sections.1 and.. Propagation of neutrinos through vacuum and oscillation probabilities are derived in Section.3 using 3 flavors model. More on these basic topics can be found in Ref. [1, 7, 8, 9, 10] or [11]. Section.4 briefly reviews yet unresolved question of the mass hierarchy and gives a summary of present oscillation parameters status. More details in Ref. [3] or [4]. Section.5 roughly explains the effect of matter in neutrino oscillations. Exhausting description is provided by Ref. [1]. Another information can be found in Ref. [7, 9, 11, 13, 14]. The last Section.6 introduces main features of ν µ ν e appearance channel at long-baseline experiments including approximative formulae to calculate oscillation probabilities in matter, possibility of the mass hierarchy determination, θ 3 ambiguity and CP and T violation effects. See Ref. [1, 9, 1, 14, 15] for details..1 General formalism In general there could be an arbitrary number of n orthonormal neutrino eigenstates. Since the flavor eigenstates and mass eigenstates are not the same, the neutrino state ν α of flavor α is a quantum superposition of neutrino mass eigenstates ν i 1) ν α = i U αi ν i, ν i = α U αi ν α, ν α ν i = U αi, (.1) with ν α ν β = δ αβ, ν i ν j = δ ij and [1] U αi Uβi = δ αβ, i U αi Uαj = δ ij, (.) α where the coefficients U αi form a mixing matrix U. In the case of antineutrinos all U αi have to be replaced by their complex conjugates ν α = U αi ν i. (.3) i The n n unitary complex matrix U has n parameters (n minus conditions of unitarity and orthogonality of columns). The n 1 relative phases of the n neutrino states can be fixed in such a way that (n 1) independent parameters remain. But if neutrinos are Majorana particles, which means ν i = ν i, then physically significant phases cannot be eliminated by phase redefinition and we have n(n 1) independent parameters in the end [9].. Three neutrinos model, flavors and masses Recently all experimental data and observations can be explained by three neutrinos model [4] as indicated in Introduction. Three active neutrino flavors 1) Specification of complex conjugates is conventional, see Ref. [1, 9, 10, 11] for comparison. ν α = (ν e,ν µ,ν τ ) are connected with three mass eigenstates ν i = (ν 1,ν,ν 3 ) via unitary MNS ) (Maki-Nakagawa-Sakata) matrix U MNS [4, 7, 8, 9, 10], [etc.] ν α = 3 (U MNS ) αi ν i, ν i = (U MNS ) αi ν α. (.4) i=1 α=e,µ,τ U MNS is usually parametrized as follows U e1 U e U e3 U MNS = U µ1 U µ U µ3 = (.5) U τ1 U τ U τ3 c 13 c 1 c 13 s 1 s 13 e iδ = c 3 s 1 s 13 c 1 s 3 e +iδ c 3 c 1 s 13 s 1 s 3 e +iδ c 13 s 3 s 3 s 1 s 13 c 1 c 3 e +iδ s 3 c 1 s 13 s 1 c 3 e +iδ c 13 c 3 with s ij = sinθ ij, c ij = cosθ ij, θ ij are mixing angles and δ is the Dirac phase. a and b are known as Majorana phases and are nonzero only if neutrinos are Majorana particles. This representation of the mixing matrix is often rewritten in other useful factorized form of three rotational matrices ) U MNS = U 3 U 13 U 1 diag (e ia,e i b,1 = (.6) 1 = c 13 s 13 e iδ c 1 s 1 c 3 s 3 1 s 1 c 1 s 3 c 3 s 13 e +iδ c 13 1 These three matrices are associated with different regimes of mixing that have been explored. The sector (3) with atmospheric 3) ( m atm, θ 3 ) and the (1) sector with solar neutrinos 4) ( m sol, θ 1) [4]. (13) is responsible for ν e transitions at atmospheric scales and, including the δ phase, also for CP violation. Recent best fit values of the angles θ ij as found in Ref. [3] are summarized in Table.1. Unlike mixing angles, masses of neutrinos m i are yet unknown. The mass of the heaviest one is expected to be less than 1 ev. Despite this fact, oscillation experiments are sensitive to the differences between masses as will be theoretically shown in next Sections..3 Propagation in vacuum Let us assume three neutrinos model mentioned earlier. The time evolution of an arbitrary neutrino state ν is given by standard Schrödinger equation i d dt ν(t) = Ĥ0 ν(t), ν(t) = e iĥ0t ν(0), (.7) ) Or PMNS (Pontecorvo-Maki-Nakagawa-Sakata) matrix. 3) Involved atmospheric and accelerator experiments: KK, MINOS, LSND, KARMEN etc., see Ref. [4] for details. 4) SNO, Homestake, SAGE, GALLEX, KamLAND etc., see Ref. [4] or [16] for details. e ia e ia e ib e ib 1 1,. 3 where Ĥ0 is the free Hamiltonian and ν is a linear combination of ν i. For mass eigenstates ν i are eigenstates of Ĥ 0 following equation holds ν i (t) = e iĥ0t ν i (0) = e ie it ν i. (.8) As long as we are in the frame of ν i (τ i stands for proper time), we can write ν i (τ i ) = e im iτ i ν i (.9) and switching to the laboratory frame with time t and position x Thus the initial state ν develops with time into ν i (t) = e i(e it p i x) ν i. (.10) ν(t) = 3 e i(e it p i x) ν i ν i ν(0). (.11) i=1 Imagine that ν is emitted by a source at t = 0 with energy E. Momentum p i of an ultra relativistic neutrino (m i E) is approximately [7] p i = E i m i E m i E (.1) and t x distance from the source. Therefore 5) ν(t x) = 3 i=1 e im i E x ν i ν i ν(0). (.13).3.1 Oscillation probabilities Neutrino is born at the source in an eigenstate of flavor α [1, 7, 11], then travels through space a distance L to a detector, where it interacts with a target and produces a charged lepton of flavor β. Thus, by definition, it is identified as a ν β. Using Eq. (.1) and Eq. (.13) the amplitude of such a process is Amp(ν α ν β ;L) = ν β ν α (L) = 3 i=1 ( ) UαiU βi exp i m i E L. (.14) 5) Let ν be a mass eigenstate ν i, then using Eq. (.13) ν i (t) = e im i E t ν i with m i /E being eigenvalues E i from Eq. (.8). Ĥ 0 can be written in ν i basis as Ĥ 0 = 1 m 1 m. E m 3 4 Squaring it, we find the probability of detecting ν β, P(ν α ν β ;L). This is often referred to the oscillation probability of ν α ν β 6) : P(ν α ν β ;L) = ν β ν α (L) = Amp(ν α ν β ;L) = (.15) 3 ( ) = U αiu βi exp i m i E L = i=1 3 3 ( ) = UαiU αj U βi Uβjexp i m ij E L = with = i=1 j=1 3 U αi U βi +Re i=1 i j ( ) UαiU αj U βi Uβjexp i m ij E L, m ij = m i m j. (.16) To obtain corresponding probability for antineutrinos we can repeat the procedure and replace all U αi by complex conjugates in terms of Eq. (.3), or we observe that ν α ν β is the CPT-mirror image of ν β ν α. Hence, if CPT invariance holds [9] P( ν α ν β ;U MNS ) = P(ν β ν α ;U MNS ) = (.17) = P(ν α ν β ;U MNS) = P( ν β ν α ;U MNS)..4 Mass hierarchy and oscillation parameters Itisclearnowthatbesidethreemixinganglesθ ij neutrinooscillationprobabilities depend on the mass differences Eq. (.16) and CP-violating phase δ. There are two independent squared-mass splittings m 1 and m 3 and two possible mass hierarchies that raise from the unknown sign of m 3: 1. m 1 m m 3 (normal hierarchy). m 3 m 1 m (inverted hierarchy) Due to the tremendous difference between m 1 and m 3(31), these have been measured in experiments observing oscillations in solar, or atmospheric mode and are referred to m sol and m atm, respectively: m sol = m 1 m 31 = m 3 = m atm. (.18) Unlike θ 1 and θ 13, which both are smaller than 45, θ 3 45 leading to maximal mixing of ν µ and ν τ in ν 3. However, sign of θ 3 45 is unknown. Is θ 3 or 45? Current results best fit implies sin θ 3 = Fig..1 shows the scheme of two allowed three neutrino-mass spectra as indicated. The relative content of flavors in particular mass eigenstates is determined by U αi, theelementsofu MNS matrix. Summaryofalltheoscillation parameters can be found in Table.1. 6) We note that the probability P(ν α ν β ) is independent of Majorana phases a and b as can be deduced from the inner product ν β ν α [17]. 5 Figure.1: Flavor content of neutrino mass eigenstates showing the dependence on the Dirac phase δ. Fractional content of flavor eigenstates (ν e - black, ν µ - light blue, ν τ - red) is determined by U αi. Left diagram is for normal and right for inverted mass hierarchy. This figure was taken from Ref. [13]. Table.1: Current (Jul 01) best fit values of oscillation parameters [3]. parameter best fit (±1σ) sin θ a) sin θ ±0.013 sin θ ±0.04 m 1 (7.50±0.0) 10 5 ev m ev δ [0;π] a) Including MINOS and TK results [3, p. 193]..5 Propagation in matter When neutrinos propagate in matter the oscillations are modified by the coherent interactions with the medium. These produce effective interaction potentials ˆV int and lead to new effective neutrino masses and mixing angles that differ from their vacuum counterparts. The Standard-Model interactions between neutrinos andotherparticlesdonotchangeflavor[]andthematrixofpotentialsisdiagonal in the flavor basis [1]. This matrix 7) includes interaction energies mediated by W ± (charged current scattering - CC) or Z 0 (neutral current scattering - NC) exchange. Stressing the fact that surrounding matter contains only (or almost only) electrons, protons and neutrons, neither ν µ, nor ν τ can undergo CC interactions and it gives rise to an extra potential energy V e possessed by ν e in matter [7, 11, 1] V e = G F N e, (.19) 7) Elements of the matrix are ν α ˆV int ν β = δ αβ V α. 6 - νe e να να ± W 0 Z - e ν e - e, p, n - e, p, n Figure.: Feynmann diagram of neutrino interactions in matter. Left: W ± exchange between electron and ν e, CC interaction; Right: Z 0 exchange between any neutrino flavor ν α and electron, proton or neutron, NC interaction. where G F is the Fermi coupling constant and N e the number of electrons per unit of volume. Since neutral scattering potentials for electrons and protons are equal and opposite, they cancel each other in electrically neutral media [11, 1]. On the other hand neutrons contribute to the potential with a term V n = G FN n [7, 1]. But, as illustrated in Fig.., all neutrino flavors are treated in the same manner in NC interactions. For oscillation probabilities depend on relative phases of the different neutrino eigenstates (θ ij, m ij and δ), they will not be affected by interaction that shifts the eigenvalues of the Hamiltonian Ĥmatter by the same amount. Therefore we may omit the contribution of Z 0 exchange [9]. In order to obtain potentials for antineutrinos, we can formally exchange ˆV int ˆV int [7, 11, 1]. Summarizing preceding paragraphs the effective Hamiltonian Ĥmatter in flavor basis can be written as [1, 13] Ĥ matter = Ĥ0 + ˆV int = 1 E U MNS m 1 m m 3 e U V MNS + 0, (.0) 0 where we have omitted all irrelevant parts proportional to the unit matrix. The neutrino mixing is defined with respect to the eigenstates of the Hamiltonian in matter Ĥmatter. In order to find them we have to solve a cubic eigenproblem arising from Eq. (.0). Its exact solution is rather complicated and exceeds the main subject of this text. Two neutrino case of propagation in matter is quite simpler model but still provides a good description and understanding of neutrino oscillations in media, see Ref. [7, 10] or [1]. Only one mixing angle θ and mass-splitting m take place. The evolution equation for two flavors ν e and ν a in matter is i d ν f dt [ m = 4E ( ) ( cosθ sinθ Ve + sinθ cosθ 7 )] ν 0 f, ν f = ( νe ν a ). (.1) The vacuum Hamiltonian in flavor approximation with interaction extension in flavor basis from Eq. (.1) can be expressed in a more symmetrical form Ĥ (f) (f) 0 + ˆV int = m 4E = m 4E ( ) cosθ sinθ + sinθ cosθ ( ) cosθ sinθ + 1 sinθ cosθ ( Ve ( Ve ) = (.) 0 ) + 1 ( ) Ve. V e V e Again, we can effectively drop all parts proportional to the unitary matrix, which do not affect neutrino oscillations[7], and write the matter Hamiltonian as follows ( Ĥ (f) matter = m cosθ + EV e ) sinθ m. (.3) 4E sinθ cosθ EVe m We are ready to find the eigenvalues λ of Ĥ (f) matter. Using standard and straightforward procedure we have λ m +V e 4E cosθ V e 4 and solving this quadratic equation and ( ) m ( sin θ +cos θ ) = 0, (.4) 4E λ = V ( ) e 4 V m m e 4E cosθ + ( sin θ+cos θ ) = (.5) 4E ( ) [ m (EVe ) = 4E m cosθ +sin θ]. We can define new matter quantities of neutrino oscillations (EVe ) m M = m m cosθ +sin θ (.6) sin θ M = Then Ĥ(f) matter has the form [7, 9] Ĥ (f) matter = m M 4E sin θ ( EVe cosθ ) m +sin θ. (.7) ( ) cosθm sinθ M sinθ M and with ν 1M, ν M being eigenstates of Ĥ(f) matter cosθ M (.8) ν e = cosθ M ν 1M +sinθ M ν M, (.

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