ABSTRACT. PATTON, KELLY MARIE. Investigating Nuclear and Astrophysical Systems Using Neutrinos. (Under the direction of Gail C. McLaughlin.

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1 ABSTRACT PATTON, KELLY MARIE. Investigating Nuclear and Astrophysical Systems Using Neutrinos. (Under the direction of Gail C. McLaughlin.) Neutrinos are one of the most mysterious particles in the universe, and at the same time one of the most important. Recent experimental efforts place us at an exciting time in the field of neutrino physics. Instead of simply studying the properties of neutrinos themselves, we can now use neutrinos as probes of other complex systems. Here, we study two such systems: the nucleus and supernovae. We first study the possibilities of using coherent elastic neutrino-nucleus scattering (CENNS) to probe the neutron distribution in the nucleus. We use an expansion of the form factor into moments to show that neutrinos from stopped pions can measure the second and fourth moments of the neutron distribution. Particularly, the second moment or RMS radius can be measured to a few percent uncertainty in tonne-scale detectors made of argon, germanium, or xenon. In order to achieve this, the energy shape uncertainty of the detector must be understood at the percent level. We also investigate the effects on neutrino oscillations of turbulent matter densities, such as those found in a supernova. We have developed an analytic formula that correctly predicts the transition wavelength and amplitude for neutrinos traveling through turbulence with up to fifty Fourier modes. Using this formula, we have identified two important wavelength scales. The first important scale stimulates transitions known as parametric resonances, and corresponds to the mass-splitting scar of the neutrino system. The second important scale is a much longer wavelength, and causes a suppression of transitions. These long wavelengths correspond to modes with a ratio of amplitude to wave number of order, or greater than, the first root of the Bessel function J 0. We have expanded this analytic approach to a 1D supernova model, and show that we can predict where transitions will occur as the neutrino propagates. We also investigate the effects of changing different parameters of the turbulence, such as the RMS amplitude and cutoff wavelengths.

2 Copyright 2014 by Kelly Marie Patton All Rights Reserved

3 Investigating Nuclear and Astrophysical Systems Using Neutrinos by Kelly Marie Patton A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Physics Raleigh, North Carolina 2014 APPROVED BY: James P. Kneller Carla Frohlich Mohamed Bourham Gail C. McLaughlin Chair of Advisory Committee

4 DEDICATION This dissertation is dedicated to Garth Fowler and John Lindner, who first opened my eyes to the wonder of physics. ii

5 BIOGRAPHY Kelly Marie Patton grew up in Wooster, Ohio. After graduation from Wooster High School in 2004, she went on to attend The College of Wooster. While at The College of Wooster, she completed a Senior Independent Study project under the direction of Dr. John F. Lindner entitled Experimental observation of solitons propagating in a hydro-mechanical array of one-way couple oscillators. Patton obtained a Bachelor of Arts degree in physics in May Patton continued her study of physics at North Carolina State University in the fall of While at NCSU, she worked closely with Drs. Gail C. McLaughlin and James P. Kneller. In addition, she had the opportunity to collaborate with other physicists in the area, particularly Drs. Jon Engel (UNC-Chapel Hill) and Kate Scholberg (Duke University). Patton hopes to continue to work in neutrino physics or a related field after graduation. iii

6 ACKNOWLEDGEMENTS I would first like to thank my advisors, Gail McLaughlin and Jim Kneller, for all of their help, guidance and support throughout my years at NCSU. These projects would not have been possible without them. Next, I would like to thank the rest of my committee, Carla Frohlich and Mohammed Bourham. Thank you for your support. To my family, thank you for all of your encouragement and for your belief in me. A special thank you to my mother Beth, who was there for me through all the ups and downs that make up the graduate school experience. I never would have made it but for you. To Michelle Snyder, my best friend from NCSU, thanks for all the Thursday night dinners and the commiseration. Jeff Rutherford, my best friend, I have to thank you for listening to me whenever I needed it, whether I was excited about some new discovery you didn t understand or sure that I d never graduate. Finally, I must thank the women of the Perfection Trap and especially Angel Bowers. You all made me realize that I was not alone in this process. Sharing my struggles and triumphs with you and hearing about yours made my final years at NCSU richer and much healthier. I go on in my career knowing that what I learned from you will keep me strong. iv

7 TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES vii viii Chapter 1 Introduction Coherent Elastic Neutrino Nucleus Scattering Theory CENNS as an Experimental Tool CENNS in Nuclear Physics Neutrino Oscillations History of Neutrino Oscillations Vacuum Oscillations Other Types of Oscillations Neutrinos in Turbulence References Chapter 2 Neutrino-nucleus coherent scattering as a probe of neutron density distributions Abstract Introduction Coherent Scattering and the Form Factor Kinematics Form-factor expansion Effective Moments Density Functional Theory Calculations of Moments Results and Discussion Monte-Carlo simulations Discussion Conclusions References Chapter 3 Prospects for using coherent elastic neutrino-nucleus scattering to measure the nuclear neutron form factor Abstract Introduction Nuclear Neutron Form Factor Experimental Prospects Possible Detectors Possible Sources Simulations Discussion Conclusion References v

8 Chapter 4 Stimulated neutrino transformation with sinusoidal density profiles Abstract Introduction Stimulated Neutrino Transformation Constant Potentials with Sinusoidal Perturbations Comparison with numerical results Applications Non-constant Density Profiles: A Supernova Test Problem Summary and Conclusions Acknowledgements References Chapter 5 Stimulated neutrino transformation through turbulence Abstract Introduction A Numerical Solution Predictions for the Wavelength and Amplitude Suppressed Transitions Conclusions Acknowledgments References Chapter 6 Stimulated neutrino transformation through turbulence on a changing density profile Methods Predicting Transitions Effect of RMS Amplitude Effect of Length Scale Cutoff Conclusions References Chapter 7 Conclusions Using CENNS to Probe the Nuclear Neutron Radius Conclusions Future Work Stimulated Transitions Conclusions Future Work References vi

9 LIST OF TABLES Table 2.1 Isotopes and abundances of germanium and xenon Table 2.2 Effective moments of germanium and xenon Table 2.3 Numerical Monte Carlo results for 40 Ar, Ge, and Xe detectors, L ν varied Table 2.4 Numerical Monte Carlo results for 40 Ar, Ge, and Xe detectors, L ν constant Table 3.1 Characteristics of past, current and planned stopped-pion neutrino sources worldwide 48 vii

10 LIST OF FIGURES Figure 1.1 Neutrino spectra resulting from stopped pion decay Figure 2.1 Neutron form factors for 40 Ar, 74 Ge, and 132 Xe Figure 2.2 Event rates and differences for 40 Ar Figure 2.3 Monte Carlo results for 40 Ar Figure 2.4 Monte Carlo results for Ge Figure 2.5 Monte Carlo results for Xe Figure 3.1 Figure 3.2 Figure 3.3 Figure 3.4 Figure 3.5 Figure 3.6 Energy distributions for the different neutrino flavors produced by a stopped-pion source Approximate duty factor and power of various past, current and future stopped-pion sources Contours of uncertainty in R 2 n 1/2 at the 90% confidence level as a function of detector shape uncertainty and detector size in 40 Ar Contours of uncertainty in R 2 n 1/2 at the 90% confidence level as a function of detector shape uncertainty and detector size in germanium Contours of uncertainty in R 2 n 1/2 at the 90% confidence level as a function of detector shape uncertainty and detector size in xenon Contours of uncertainty in R 4 n 1/4 at the 90% confidence level as a function of detector shape uncertainty and detector size in xenon Figure 4.1 Comparison between a numerical result and the analytic solution for a particular test problem Figure 4.2 Comparison between the Rotating Wave Approximation prediction and numerically derived oscillation wavelength Figure 4.3 Comparison between the Rotating Wave Approximation prediction and numerically derived oscillation amplitude Figure 4.4 Wavelength associated with the mass splitting compared to the perturbing wavelength 68 Figure 4.5 Comparison of the reduced wavelength 1/q n with the density scale height, r ρ Figure 4.6 Transition probabilities between matter eigenstates, P 12, P 13 and P 23, as calculated numerically Figure 5.1 A turbulent MSW potential Figure 5.2 The numerical solution for the transition probability between the unperturbed neutrino states through the potential shown in figure (5.1) Figure 5.3 Two turbulent neutrino potentials Figure 5.4 The numerical and analytical solutions for the neutrino evolution through the turbulent potentials shown in figure (5.3) Figure 6.1 The probability of transition for the Fogli et al. profile with no perturbations Figure 6.2 Example of the transitions caused by a perturbation with four modes Figure 6.3 Example of transitions caused by a perturbation with 20 modes Figure 6.4 Close up of the last km of propagation for the example in figure (6.3) viii

11 Figure 6.5 Effects of changing the RMS amplitude of the turbulence Figure 6.6 Effects of varying the cutoff scale of the turbulence ix

12 Chapter 1 Introduction Neutrinos are among the most fascinating and elusive particles in the universe. These ghostly particles are chargeless, nearly massless, and only weakly interacting. Even so, they can teach us about topics as varied as atoms and the subatomic world to supernovae and the Big Bang. The neutrino was first postulated by Wolfgang Pauli in 1930 as a solution to an energy conservation problem in beta decay. Pauli initially apologized for theorizing a particle that might never be observable. In fact, it took over twenty-five years before the first experimental detection of a neutrino. Since that Nobel Prize winning detection by Reines and Cowan [1], the field of neutrino physics has expanded dramatically. We now know that there are three flavors of neutrino, plus their antiparticles. While first thought to be massless, we have learned that neutrinos actually have a very small mass, which allows for oscillations between flavors as the neutrinos propagate. Many experiments have been performed to measure the mixing angles and mass differences that govern these oscillations [2 11]. However, for all that we do know, there are still many mysteries to be solved. For example, we still do not know if neutrinos are their own antiparticles. While we have measured the difference in mass between neutrino states, we still do not know the absolute mass scale of the neutrino or the origin of the neutrino masses. A related problem is the neutrino hierarchy, the ordering of the three mass states. From the measured mass differences, we know that mass state 1 is lighter than mass state 2, but not if mass state 3 falls above or below the others. We also have yet to determine if neutrinos are CP violating and their role in the matter-antimatter asymmetry in the Universe. In fact, many of the experiments designed to solve problems in neutrino physics have instead created their own unknowns, such as whether a sterile neutrino exists [12 17]. Experiments are being planned and constructed to investigate these new mysteries. Even with the remaining unknowns, we are at an exciting point in the field of neutrino physics. We now know enough about neutrinos themselves to use them as probes to study other complex systems, from the atomic scale to the astrophysical. This dissertation will seek to understand two such systems. 1

13 First, we will examine the possibility of using neutrino scattering to understand the nucleus. Then, we will study the effects of turbulence, specifically in a supernova, on the oscillations of neutrinos. 1.1 Coherent Elastic Neutrino Nucleus Scattering In Chapters 2 and 3, we discuss the potential of using coherent elastic neutrino nucleus scattering (CENNS) to measure the nuclear neutron distribution. Presented here is a short introduction to the theory, experimental interest in and uses for CENNS. In addition, a brief overview of past and current measurements of neutron radii and distributions is discussed Theory In coherent elastic scattering, we consider the neutrino scattering off the nucleus as a whole, rather than scattering from individual nucleons. To calculate the cross section, the contribution from each nucleon is summed together, then that total is squared. The CENNS cross section for a spin zero nucleus can be written as [18] [ dσ dt (E, T) = G2 F 2π M 2 2T ( T ) 2 E + MT ] Q 2 W E E 2 4 F2 (Q 2 ). (1.1) Here, E is the energy of the incoming neutrino, T is the recoil energy of the nucleus, and M is the mass of the nucleus. The Fermi constant is given by G F, and the weak charge of the nucleus is Q W = N (1 4 sin 2 θ W )Z, where N is the number of neutrons, Z the number of protons, and sin 2 θ W is the Weinberg angle. The final piece of the cross section is the form factor, given by F(Q 2 ), where Q 2 = 2E 2 T M/(E 2 ET) is the momentum transfer of the scattering. The form factor is defined such that F(0) = 1. This element of the cross section contains all of the nuclear structure information, and corrects for scattering at higher energies that is not completely coherent. As the energy of the neutrino and the momentum transfer increase, the neutrino interacts more with individual nucleons, causing the structure of the nucleus to become apparent. The form factor, assuming a spherical nucleus, is calculated through a Fourier transform of the nuclear densities, given by [19] F(Q 2 ) = 1 Q W [ρn (r) (1 4 sin 2 θ W )ρ p (r) ] sin (Qr) r 2 dr, (1.2) Qr where ρ n,p (r) are the neutron and proton densities. The form factor is the largest uncertainty in the cross section, mainly due to the uncertainty in the neutron density. This will be addressed further in Chapters 2 and 3. In order to use CENNS as a detection method, the number of scatterings occurring in the detector must be calculated. This is accomplished by folding the neutrino spectrum, f (E), with the cross section, 2

14 Flux (ν/mev) ν e ν µ ν µ Energy (MeV) Figure 1.1: Neutrino spectra resulting from stopped pion decay, as described in the text. via the integral dn Emax dt (T) = N tc E min (T) f (E) dσ (E, T)dE. (1.3) dt The detector information is included through the parameters N t, the number of target nuclei in the detector, and C, the flux of neutrinos of a given flavor arriving at the detector. These two parameters together give the size of the detector and the distance from the neutrino source. The neutrino spectrum will depend on what type of source is being used. In this work, we have considered a stopped pion source. In a stopped pion source, pions decay at rest through π + ν µ + µ +. The muons resulting from this initial decay then decay at rest by µ + e + + ν e + ν µ. This entire process results in three neutrinos: a mono-energetic 29.9 MeV muon neutrino from the pion decay, and electron and anti-muon neutrinos that range from 0 to 52 MeV from the muon decay. The continuous spectra can be written as f νe = 96 m 4 (m µ Eν 2 e 2Eν 3 e )de νe, µ f νµ = 16 m 4 (3m µ E 2 ν µ 4E 3 ν µ )de νµ, (1.4) µ where m µ is the mass of the muon. The shape of these spectra are shown in figure (1.1). These neutrinos result in recoil energies on the order of tens of kev. Another option for a neutrino source is reactor neutrinos. Reactor neutrinos range from 0 to 8 MeV, resulting in much smaller recoil energies [20]. Many experiments use reactor neutrinos because they are produced in very high quantities, and there are currently several collaborations looking into 3

15 this source for CENNS detectors [21, 22]. We did not consider reactor neutrinos in our study for several reasons. First, the low energy recoils can often be overwhelmed by background, making sensitive measurements difficult. Second, our method for measuring the neutron form factor requires discerning between slight differences in recoil curves, as will be explained in later chapters. For reactor neutrinos, these slight differences lie in an energy region inaccessible to near future detector technology CENNS as an Experimental Tool There is great interest in applying CENNS as a tool for neutrino studies. Although this process has not yet been experimentally observed, the possibilities for its use are many and varied. They range from measurements of fundamental particle physics quantities to Beyond the Standard Model physics. A selection of these applications is presented here. One scenario where CENNS is important is in supernovae (SNe) and the detection of neutrinos released during a SN [19]. Approximately 99% of the energy released in a core-collapse SN is in the form of neutrinos, resulting in a very high density of neutrinos traveling out from the protoneutron star. Although CENNS is a rare process, it does have a large cross section compared to other neutrino scattering channels. The large cross section along with the high number of neutrinos means that CENNS will happen often in a SN, and must be considered when studying the dynamics of the explosion [23]. Furthermore, CENNS offers a method for detecting those neutrinos once they reach the earth. Unlike water Cherenkov detectors, which are sensitive to only electron antineutrinos through inverse beta decay, as a neutral current process CENNS is sensitive to all flavors of neutrinos. Combining CENNS detection with existing water Cherenkov detectors will increase the amount of information obtainable from SN neutrinos [19, 24]. CENNS could also be used to probe particle and nuclear physics parameters such as the weak mixing angle. The absolute cross section of CENNS is directly proportional to the weak charge of the nucleus, defined as Q W = N (1 4 sin 2 θ W )Z. The only unknown in Q W is the weak mixing angle θ W since the nucleus being used in the detector is known. Therefore, a value of the weak mixing angle can be produced from a measurement of the absolute cross section of CENNS [25, 26]. Although this parameter has been measured using other methods, and with a lower uncertainty, CENNS would provide an alternate procedure to confirm past results. Neutrino physics also offers a view into possible Beyond the Standard Model (BSM) physics, and CENNS is a possible way to accomplish this. One example is the magnetic moment of the neutrino. In the standard model, the magnetic moment is predicted to be zero. However, if the magnetic moment is nonzero, a small distortion in the CENNS signal would be one result. Measuring such distortion would be a difficult experimental task, but would be a clear signal of new physics [25]. Another BSM problem that could be solved with CENNS is the search for a sterile neutrino. Anomalous measurements from LSND, MiniBooNE, reactor neutrino experiments and others provide poten- 4

16 tial evidence for a sterile neutrino. However, all of the methods used in these measurements required a complicated analysis to obtain the possible sterile signal. Mixing from electron flavor to muon or tau neutrinos must be distinguished from mixing to a sterile neutrino. CENNS, however, is a neutral current method, so any oscillations seen must be to a non-active flavor [27]. This straightforward measurement could provide clear evidence of the existence of sterile neutrinos. Finally, CENNS is also important when considering the search for dark matter. Dark matter signals are very small, and easy to overwhelm with background. One of the potential backgrounds in dark matter detectors is neutrino scattering, in particular, CENNS. A full understanding of CENNS, its cross section, and scattering rate will greatly aid in the search for elusive dark matter signals [28 32] CENNS in Nuclear Physics We are interested in using CENNS to probe the nucleus, and specifically the neutron distributions in nuclei. As mentioned above, the form factor is the largest uncertainty in the cross section, and most of that uncertainty comes from the relatively unknown neutron density. Although proton densities have been measured with an impressive amount of certainty, neutron densities are very difficult to measure experimentally. This fundamental property of the nucleus could have ramifications in both nuclear and astrophysics. The prediction of nuclear properties, including the neutron radius, is done today using energy density functional theory (DFT). These theories typically involve parameters that are found by fitting to data, such as atomic masses, nuclear separation energies, and proton radii [33 36]. Improved measurements of such data will help to optimize existing and future functionals. The neutron radius is an important quantity in the optimization of these functionals, as it plays a part in many other parameters, such as the symmetry energy. The predictions made by these theories would be greatly enhanced by a measurement of neutron radii. The most significant impact in astrophysics is in the study of neutron stars. Currently, measurements of the masses of neutron stars and the orbital period of pulsars exist, but the radius is not well known [37 40]. Determining the radius must be done using equations of state of neutron-rich matter. These equations of state depend on the symmetry energy which, as noted above, depend on the neutron radius. Measuring the neutron radius of a large nucleus, such as lead, would provide information on the equation of state, and thus could be used to refine our understanding of neutron stars [41 43]. Previous experimental efforts to measure the neutron RMS radius have used various hadronic scattering methods. A typical example of such an effort is presented by Ozawa et al. in [44]. In that experiment, argon nuclei were scattered off a target composed of 12 C, and the interaction cross section was measured. A simple form for the nuclear density was assumed, and used to calculate an interaction cross section theoretically. Various parameters in the assumed density were altered until the theoretical cross section matched the measured value. Once the parameters were set, a value for the neutron ra- 5

17 dius was extracted. Other methods involve pion scattering, as in [45], or analysis of antiprotonic nuclei [46]. Errors on these experiments are quoted as low as 1%. In all cases, assumptions about the form of the density and potentially the hadron-nucleus interaction itself are required. These assumptions and approximations lead to complicated analyses that are difficult to understand and interpret. A different method, recently used by the Lead Radius Experiment (PREX) at Jefferson Laboratory, involves parity-violating electron scattering. Polarized electrons are scattered off a 208 Pb target, and the difference in cross section between positive- and negative- helicity electrons is measured. This parity violating asymmetry is proportional to the weak form factor, which, like the form factor in equation 3.2, is a Fourier transform of the weak charge density, where Q W = ρ W (r) d 3 r. This type of measurement needs no assumptions about the form of the nuclear densities; instead it is completely model independent. The current measurement reports the neutron radius of 208 Pb to be about ±2.5%, but an improved uncertainty of ±1% may be possible [47]. Following the ideas put forth in [48], we propose CENNS as a new complementary method that can be used to measure the neutron radius. This alternate method is model independent and has a much cleaner analysis than hadronic scattering. In Chapter 2, the theoretical framework is laid out and a preliminary investigation into the necessary detector size is presented. Chapter 3 continues with the same theoretical framework, and expands on the investigation into necessary detector characteristics. 1.2 Neutrino Oscillations Another important aspect of neutrino physics is the neutrino s ability to oscillate between flavors. The way neutrinos oscillate provides a wealth of information about the environments in which they were created, propagated through, and were detected. In Chapters 4, 5 and 6, a specific type of neutrino oscillation, parametric resonance, is discussed. The following sections include a short discussion on the history of neutrino oscillations, the basic theory behind the phenomenon, and various types of oscillations that are possible History of Neutrino Oscillations The first discussion of possible neutrino oscillations came from Bruno Pontecorvo in 1957 [49]. Instead of oscillations between flavors, the type of transformation we are familiar with today, Pontecorvo proposed that neutrinos may oscillate between their particle and antiparticle states, building upon the known K 0 K 0 transition. There were barriers to this theory, because it was unknown if neutrino charge had to be conserved, or whether neutrinos and their antiparticles were in fact the same particle. Pontecorvo merely pointed out that if neutrino charge was not conserved, there was nothing to keep a neutrino antineutrino conversion from taking place. The possibility of multiple types of neutrino was theorized as early as The basic idea was that 6

18 one type of neutrino was associated with electrons, ν e, and another associated with the muon, ν µ. This was later confirmed by Danby et al. in 1962 [50], an achievement which resulted in the 1988 Nobel Prize for Leon Lederman, Melvin Schartz and Jack Stienberger. The two-component neutrino theory was built upon by Maki, Nakagawa and Sakata in 1962 [51]. The states ν e and ν µ were called the weak neutrinos, while a second set of states, the true neutrinos, were defined as a superposition of the weak states: ν 1 = ν e cos θ + ν µ sin θ, ν 2 = ν e sin θ + ν µ cos θ, (1.5) where θ is some real constant. The difference between these two sets of states is that ν e and ν µ are unstable in relation to some types of interactions, such that the transformation ν e ν µ will occur. During these same interactions, the true neutrinos remain stable. Pontecorvo noted in 1968 that in order for these flavor oscillations to occur, neutrinos must have a non-zero mass [52], and idea that has since been proven true. The concept of weak and true states has been built upon for the last several decades, creating the field of neutrino oscillation research as we know it today. Since then, we have learned there are actually three types of weak neutrinos, each associated with either the electron, muon, or tau lepton. We have realized that the weak states are involved when neutrinos interact with matter, while the particle propagate through vacuum in their true states. One of the early problems solved by neutrino oscillations is the solar neutrino problem, first noted by Raymond Davis. Davis used a large chlorine detector to measure the neutrinos coming from the Sun, and compared the result to predictions made by Bahcall [53 55]. A large deficit was noted, although at the time there were still large uncertainties in both the prediction and the experiment. As both theory and experiment were refined, the deficit persisted [56]. The problem was not resolved completely until 2001, by the Sudbury Neutrino Observatory (SNO) experiment. SNO was able to detect all types of neutrinos coming from the Sun, and showed that the total number matched the original predictions [57]. However, only about 35% of these neutrinos were electron flavor. Since Davis experiment was only sensitive to electron flavor neutrinos, all neutrinos which had oscillated away to other flavors were missed, resulting in the observed deficit. Since oscillations were first proposed, many experiments have observed the phenomenon and measured parameters associated with it. We now know that neutrino oscillations are governed by a mixing matrix, which can be written as [58] c 12 c 13 s 12 c 13 s 13 e ıδ e ıα U = s 12 c 23 c 12 s 23 s 13 e ıδ c 12 c 23 s 12 s 23 s 13 e ıδ s 23 c 13 s 12 s 23 c 12 c 23 s 13 e ıδ c 12 s 23 s 12 c 23 s 13 e ıδ c 23 c 13 0 e ıα e ıα 3, (1.6) 7

19 where c i j = cos θ i j and s i j = sin θ i j. In addition to the three mixing angles θ 12, θ 13, and θ 23, there is the CP-violating phase δ and the Majorana phases α 1,2,3. These phases are the last unknown elements of the mixing matrix. All three angles have been measured experimentally. In addition, there are two mass splittings, given by δm 2 12 and δm2 23, which have also been measured in various experiments [2 11]. The current global values for these mixing parameters are as follows [59]: sin 2 (2θ 12 ) = ± (1.7) δm 2 21 = (7.50 ± 0.20) 10 5 ev 2 (1.8) sin 2 (2θ 23 ) > 0.95 (1.9) δm 2 32 = ev2 (1.10) sin 2 (2θ 13 ) = ± (1.11) Currently, the absolute mass scale is not known, nor is the sign of δm This unknown sign is known as the hierarchy problem, and is a topic of great interest to many in the neutrino community Vacuum Oscillations The basic theory of neutrino oscillations is very similar to that laid out by Pontecorvo, Maki, Nakagawa and Sakata. The simplest case to understand is that of neutrinos traveling in vacuum. The following discussion is adapted from reference [60]. In vacuum, the neutrino mass states are eigenvalues of the Hamiltonian, such that H ν k = E k ν k, (1.12) where k = 1, 2, 3... Since these states are eigenvalues of the Hamiltonian, the time evolution of the mass states can be written as ν k (t) = exp ( ıe k t) ν k (0). (1.13) In general, we are more interested in the flavor state as a function of time, since the neutrinos will be detected as a either electron, muon or tau. Therefore, we rewrite equation 1.13 as ν α (t) = Uαk exp ( ıe kt) ν k, (1.14) k where α = e, µ, τ and ν α (0) = ν α. In the last equation, we have used the fact that the flavor states are a superposition of the mass sates, written as ν α = k Uαk ν k. The coefficients U αk are elements of the mixing matrix U, defined for three flavors in equation 1.6. The matrix U is a unitary matrix, and as such we can write the mass states as a superposition of flavor states as ν k = U αk ν α. (1.15) α 8

20 Substituting this into equation 1.14, we see that ν α (t) is in fact a superposition of flavor states when t > 0: ν α (t) = Uαk exp ( ıe kt)u βk ν β. (1.16) β k Using the fact that the flavor states are orthogonal to one another ( ν β ν α = δ βα ), the amplitude of a transition from ν β ν α can be written A νβ ν α (t) = ν β ν α (t) = Uαk U βk exp ( ıe k t) (1.17) The probability of a transition, P νβ ν α (t), is the absolute square of the amplitude. This definition allows us to write k P νβ ν α (t) = A νβ ν α (t) 2 = Uαk U βku α j Uβ j exp ( ı ( ) ) E k E j t. (1.18) k, j Assuming ultrarelativistic neutrinos, the energy can be written E k = E + m 2 k /2E. Since these ultrarelativistic neutrinos are moving near the speed of light, the distance traveled L is equal to the time t (c = 1). These assumptions leave us with the probability P νβ ν α (t) = Uαk U βku α j Uβ j exp ıδm2 k j 2E L, (1.19) k, j where δm 2 k j = m2 k m2 j is the mass-squared difference. At this point, we will assume that we have only two neutrino flavors, for simplicity. The mixing matrix U for two flavors is defined as cos θ sin θ U = sin θ cos θ, (1.20) where θ is the mixing angle. Using this definition in equation 1.19, we find the probability of transition to be P νβ ν α (t) = sin 2 (2θ) sin 2 δm2 21 4E L. (1.21) This simple form indicates that the amplitude of the probability is controlled by the mixing angle, θ, while the oscillation length is controlled by the mass splitting, δm The expressions in three flavors are similar, but generally involve multiple mixing angles Other Types of Oscillations The vacuum oscillations discussed above are the simplest case, and happen because the flavor eigenstates of the neutrino are not equal to the mass eigenstates. However, there are other types of oscillations 9

21 that can occur. These are due mainly to the interactions of neutrinos with their environments. For instance, neutrinos often move through matter, such as the Sun, the Earth, or supernovae. This matter can enhance the oscillations, an effect known as the Mikheyev-Smirnov-Wolfenstein (MSW) effect [61, 62]. When moving through matter, an extra term is added to the vacuum Hamiltonian. This leaves us, in the two flavor case, with [60] H ( f ) MS W = δm2 4E cos (2θ) + V( f ) δm (r) 2 4E sin (2θ) δm 2 4E sin (2θ) δm 2 4E cos (2θ), (1.22) where δm 2 is the vacuum mass difference, θ is the vacuum mixing angle, and V ( f ) (r) = 2G F N e (r) is the matter potential. The matter potential results from charged current interactions between electron neutrinos and the surrounding electrons, so it depends on the electron density, N e (r). A corresponding potential due to neutral current interactions applies equally to all flavors and causes a common phase, so it is removed from the problem. The Hamiltonian here is written in the flavor basis, indicated by the superscript ( f ). We now define a new basis in addition to the flavor and mass bases. We call this the matter basis, indicated by a superscript (m), and define it as the basis in which the above Hamiltonian is diagonalized. This requires the definition of a new mixing matrix, U (m), such that U (m) H MS W U (m) = H (m) MS W = 1 4E δm2 m 0 0 δm 2 m, (1.23) where δm 2 m is the effective mass-squared difference in matter. The matrix U (m) is defined as U (m) = cos θ m sin θ m sin θ m cos θ m, (1.24) where θ m is the effective mixing angle. These effective quantities are calculated from the vacuum mass difference and mixing angle using the equations δm 2 m = tan (2θ m ) = (δm 2 cos (2θ) 2EV ( f ) (r) )2 + ( δm 2 sin (2θ) )2, (1.25) tan (2θ) 1 2EV( f ) (r) δm 2 cos (2θ). (1.26) The MSW resonance is defined as the location of maximal mixing, or when θ m = π/4. At this point, if the resonance region is large enough a complete transition between flavors is possible. The resonance condition is met when N R e = δm2 cos (2θ) 2 2EG F. (1.27) 10

22 MSW transitions are an important consideration when calculating the transformations of neutrinos in most astrophysical and terrestrial environments. The transition probabilities in matter can be calculated in a manner similar to that of vacuum oscillations. Again, this discussion is adapted mainly from [60]. We first transform our evolution equation, ı dψ e(r) dr = H ( f ) MS W Ψ e(r), (1.28) where Ψ e = (ψ ee, ψ eµ ) T, and Ψ e (0) = (1, 0) T, to the matter basis. In this definition, the first component of Ψ e is the amplitude of electron neutrino survival probabilities, and the second component is the amplitude of transitions from electron to muon neutrino. The transformation uses the matrix U (m) from equation 1.24 to define a new wave function, Φ e, where Ψ e = U (m) Φ e. The new evolution equation in the matter basis is ı dφ e(r) dr = 1 4E δm2 m 4Eı dθ m dr 4Eı dθ m dr δm 2 m Φ e. (1.29) Note that we have not discarded the off-diagonal terms dependent on the derivative of θ m. If the neutrino is traveling through matter with a constant density, the off-diagonal terms will disappear, since the effective mixing angle and mass-squared differences will be constant as well. After solving this differential equation for Φ e, we transform back to the flavor basis wave function Ψ. In that case, just as in the vacuum, the probability of transition is ( δm P νβ ν α (t) = ψ 2 ee = sin 2 (2θ m ) sin 2 2 ) m 4E L. (1.30) The only difference is that the vacuum mixing parameters have been replaced by the effective values. If instead the matter density is not constant, we cannot ignore the off-diagonal terms as the matter mixing angle θ m will change. There are two cases to consider now, and they are differentiated by the adiabaticity parameter, which is defined as γ = δm 2 m 4E dθ m /dr, (1.31) where the derivative of θ m can be found using equation If γ 1, the evolution is called adiabatic. The density is changing slowly enough that the neutrino system can adjust, and the transitions between states are small. If we neglect the off-diagonal terms because they are much smaller than the on-diagonal components, we can once again solve the differential equation for Φ e and transform back to Ψ e, now using the matrix U (m) (r), where r is the detection point. Since the mass-squared difference is changing, this time we must integrate from the starting point to our detection point r. We find the 11

23 survival probability in this case to be P να ν α (t) = cos ( 2θm) 0 cos (2θm (r)) sin ( ( r 2θm) 0 sin (2θm (r)) cos 0 δm 2 m(r ) ) 2E dr, (1.32) where θ 0 m is the initial matter mixing angle. We have essentially neglected the off-diagonal terms, but included the fact that the initial and final matter densities may not be equal. If we assume the distance from the neutrino production to the detector is large, we can average the final cosine term to zero, and are left with P να ν α (t) = cos ( 2θ 0 m) cos (2θm (r)). (1.33) The more interesting case to consider is that of non-adiabtic evolution, where large transitions are possible. These large transitions occur when the adiabticity parameter γ reaches a minimum. From equation 1.31, this minimum occurs when d 2 cos(2θ m ) dr = 0. (1.34) This condition is similar to the condition for an MSW resonance, which can be written as cos(2θ m ) = 0. In reality, the resonance point and the minimum of γ are not at the same location, but in practice that approximation is often made. For non-adiabatic evolution, we actually break the propagation distance into pieces to find the probability. The basic prescription is to evolve the neutrinos adiabatically up to the resonance point, include some hopping probability across the resonance, then continue to evolve adiabatically. The solution for Φ e is a product of all of these pieces. Once again, we must integrate over the propagation distances since the effective mass-splitting will depend on position. If we once again assume the distance from production to detection is large, we can integrate out similar phase terms as we did in equations 1.32 and The averaged survival probability turns out to be P να ν α (r) = cos 2 θ cos 2 θ 0 m A R cos 2 θ sin 2 θ 0 m A R sin 2 θ cos 2 θ 0 m A R sin 2 θ sin 2 θ 0 m A R 11 2, (1.35) where we have assumed θ m (r) = θ. The terms A R i j 2 are the probabilities of a transition from ν i ν j across the transition. We define them as A R 11 2 = A R 22 2 = 1 P c (1.36) A R 12 2 = A R 21 2 = P c, (1.37) where P c is the hopping probability at the resonance. Using these definitions, we can simplify equation 1.35 to P να ν α (r) = 1 ( ) P c cos ( 2θm) 0 cos (2θ). (1.38) 12

24 This survival probability is known as the Parke formula [60]. The difficult part of using this formula is determining the hopping probabilities P c. This can be done analytically for some simple density profiles, such as linear and exponential forms, but often must be calculated numerically. From this discussion, it is clear that an analytic description of neutrinos propagating through turbulence in the MSW picture would be very difficult to obtain. In some environments, such as supernovae and accretion disks, the number of neutrinos is so high that they begin to interact with each other. These self-interactions can also influence the neutrino transformations. In contrast to the MSW effect which occurs in either neutrinos or antineutrinos, selfinteractions can cause neutrinos and antineutrinos to undergo transformations at the same time. As with the MSW effect, self-interactions require an addition to the vacuum Hamiltonian, involving the density of neutrinos and antineutrinos, as well as the angles of interaction and the energy of the neutrinos. These complicated nonlinear interactions are written for a specific neutrino with momentum p as H νν (t) = 2G F (1 cos θ pq ) [ˆρ q (t) ˆ ρ q (t) ] d 3 q, (1.39) where q is the momentum of a background neutrino, and cos θ pq is the angle of interaction between the test neutrino and the background neutrino [63]. Calculation of these effects is difficult because the evolution histories of all of the neutrinos are coupled. In addition, in a SN some neutrinos travel radially outward while others move tangentially to the neutrino sphere. These two trajectories can have dramatically different evolution histories, and all of this must be taken into account. Self-interactions have been the source of a wide variety of research in the last decade [63 72]. A lot of this work is done using simplified models, such as the single angle approximation where dependence on the angle of interaction is integrated out. Collective effects are not considered in this work, but need to be fully understood to accurately describe the SN neutrino problem. In this work, we are interested in a specific type of oscillation known as parametric resonances. Like the MSW effect, parametric resonances are caused by interactions of the neutrino with the matter around it. However, unlike the MSW effect which requires a specific resonant density, the enhancement is due to density fluctuations and can happen at any density scale. The resonances in this type of oscillation occur when the frequency of fluctuation corresponds to the mass splitting scale of the neutrino system, or harmonics of that scale. [58, 73 78]. Parametric resonances are very similar to stimulated transitions by lasers in atomic systems. Comparison of the theory used to describe both systems, neutrino and atomic, shows a high correspondence between the phenomena [79]. Many of the environments neutrinos propagate through contain fluctuating densities or even turbulence containing many frequencies, some of which might correspond to the correct scale to cause a large transition. This third type of neutrino oscillation must be considered in conjunction with the others to fully understand the histories of neutrinos arriving at detectors. Parametric resonances in neutrinos are further discussed in Chapters 4, 5 and 6. 13

25 1.2.4 Neutrinos in Turbulence Neutrinos offer a unique view into the center of an exploding supernova. To fully take advantage of this, we first must understand the history of SN neutrinos detected on earth. These neutrinos will undergo some combination of the types of oscillations discussed above. The situation becomes even more complicated, however, by the fact that the matter the neutrinos will move through and interact with will have a decaying density profile and turbulent fluctuations. As we have seen, changing densities can cause MSW-type oscillations and fluctuations can cause parametric resonances, in addition to the high number of neutrinos present inducing collective effects. The effects of density fluctuations on neutrino oscillations has been studied widely over the last several decades. Small scale fluctuations, such as those in the Earth and Sun, have been examined [80 84]. In 1990, Sawyer examined the effects of lumpy media on the efficiency of MSW transitions [81]. He found that there were regions where a large decrease in conversion efficiency due to an increase in charged current scattering, would occur for density fluctuations on the order of 1% on the scale of 1000 km. However, Sawyer notes that this size of fluctuation was much larger than expected in the Sun. This conclusion, that the parameters required to cause a noticeable effect in the solar neutrino flux, was seen in later work as well [82]. Because of the importance of neutrinos in understanding SN dynamics, large scale turbulence in SN has been extensively studied. Previous work has shown the importance of turbulent modes with frequencies corresponding to the mass splitting scale [85]. This result matches well with what is known about parametric resonance. In our work, we have seen the importance of this scale as well. However, we have also seen, and will discuss in Chapters 5 and 6, that the other modes in the turbulence have an effect as well. Often, MSW calculations can be split based on resonant densities, which are different depending on the mass-splitting and mixing angle being considered. The typical split is into a high density or H resonance channel, and a low density or L resonance channel. Each of these channels includes transitions only between two states, for instance the H resonance involves matter states 2 and 3 in the normal hierarchy. This HL factorization allows a single three-flavor calculation to be broken into two two-flavor calculations. However, large amplitude turbulence can break this factorization and induce three-flavor effects [86, 87]. These results show the importance of performing full three flavor calculations, as well as the complexity of the MSW picture when turbulence is involved. There has also been investigation into the effects of turbulence on a possible fourth flavor: a sterile neutrino [88]. A sterile neutrino would result in extra changes to the neutrino spectrum from turbulence and shock waves that could be seen in detectors on Earth. Such changes would be a clear indication of the existence of a sterile neutrino. Neutrinos can have a strong effect on nucleosynthesis, and the implications of turbulence on nucleosynthesis processes have also been studied [88]. During the explosion, electron neutrinos and antineutrinos interact with the surrounding protons and neutrons through beta and inverse beta decays. These 14

26 interactions set the proton-to-neutron ratio, an essential parameter in the calculation of nucleosynthesis. Turbulence induced oscillations can alter the number of electron neutrinos and antineutrinos, affecting the final proton-to-neutron ratio and allowing or disallowing, for instance, r-process nucleosynthesis. A full understanding of the histories of the neutrinos could rule in or out the SN a possible r-process site. Density fluctuations can also have an effect on collective oscillations. This is a relatively new area of research, but there has been interesting work done already. Fluctuations in the density of neutrinos was seen to have a large effect on these self-interactions, while matter density fluctuations were not as influential [89]. The specific effects of a bump in the matter density of O-Ne-Mg stars were seen to alter the neutrino signal enough to potentially hamper the ability of researchers to reverse engineer the density profile of the star [90, 91]. Much more work is needed to fully understand the interplay between collective effects and density fluctuations. As computational power increases, and SN simulations become ever more realistic, it is important that neutrino researchers use models which resemble that seen in those SN simulations. There has been investigations into the types of neutrino transformations seen in simulated profiles [92], and such collaboration is sure to continue as both communities advance. In Chapters 4, 5 and 6, we look at the problem of turbulence through the lens of parametric resonance. We develop an analytic approach to identify the most important modes in a given turbulent spectrum, and predict the resulting wavelength and amplitude of the neutrino oscillation. References [1] C. L. Cowan Jr., F. Reines, F. B. Harrison, H. W. Kruse, and A. D. McGuire, Science 124, 103 (1956). [2] S. Adrian-Martinez et al. (The ANTARES Collaboration), Phys. Lett. B 714, 224 (2012). [3] G. Bellini et al. (The Borexino Collaboration), Phys. Rev. Lett. 107, (2011). [4] F. P. An et al. (The Daya Bay Collaboration), Phys. Rev. Lett. 108, (2012). [5] Y. Abe et al. (The DOUBLE-CHOOZ Collaboration), Phys. Rev. Lett. 108, (2012). [6] A. Gando et al. (The KamLAND Collaboration), Phys. Rev. D 83, (2011). [7] P. Adamson et al. (The MINOS Collaboration), Phys. Rev. D 86, (2012). [8] J. K. Ahn et al. (The RENO Collaboration), Phys. Rev. Lett. 108, (2012). [9] B. Aharmim et al. (The SNO Collaboration), Phys. Rev. C 81, (2010). [10] K. Abe et al. (The Super-Kamiokande Collaboration), Phys. Rev. D 83, (2011). [11] K. Abe et al. (The T2K Collaboration), Phys. Rev. D 85, (2012). [12] C. Athanassopoulos et al. (The LSND Collaboration), Phys. Rev. Lett. 75, 2650 (1995). 15