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Modified Dispersion Relations and the Anomalies in Neutrino Oscillation

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Modified Dispersion Relations and the Anomalies in Neutrino Oscillation Trabajo de Fin de M´asteren F´ısica Avanzada Modified dispersion relations and the anomalies in neutrino oscillation experiments July 2019 Pablo Mart´ınezMirav´e Tutoras: Gabriela Barenboim Mar´ıa Amparo Tortola´ Baixauli Abstract In this work, an overview of the formalism describing neutrino oscillations is presented in a way that allows a direct extension to include additional neutrinos. Relevant experimental results are discussed, including the anomalies that lead to the proposal of sterile neutrinos. An exhaustive study of the implications of sterile neutrinos with modified dispersion relations and their potential to account for the anomalous results is conducted. Since none of the studied scenarios involving oscillations into sterile neutrinos can successfully reconcile the experimental results, the hypothesis of sterile neutrinos is undermined. 1 Contents 1 Neutrino physics3 2 Neutrino oscillations5 2.1 Oscillations in vacuum............................5 2.2 Matter effects as a modification of the dispersion relation.........8 3 Experiments on neutrino oscillations 10 3.1 Solar experiments............................... 10 3.2 Atmospheric neutrinos............................ 12 3.3 Long Baseline Accelerator Experiments (LBL)............... 13 3.4 Short baseline reactor experiments..................... 14 3.5 Global fit of the neutrino oscillation parameters.............. 14 4 Experimental anomalies and sterile neutrinos 16 4.1 Appearance anomalies............................ 16 4.2 Disappearance anomalies........................... 17 4.3 Oscillations in the presence of sterile neutrinos and the 3+1 formalism.. 19 5 Modified dispersion relations in the 3+1 picture 24 5.1 Intrinsic modified dispersion relation.................... 24 5.2 Modified dispersion relations from effective potentials........... 26 6 Modified dispersion relations through effective potentials in a 3+3 picture 29 6.1 GLoBES, snu.c and modifying the probability engine........... 29 6.2 A 3 sterile neutrino model with modified dispersion relations....... 29 6.3 Numerical implementation.......................... 31 7 Conclusions 34 2 1 Neutrino physics The Standard Model of elementary particles (SM) is the most accurate theory in particle physics. Most of its predictions have been successfully tested by a large variety of experiments. Nonetheless, there are several hints indicating that its description of nature is still incomplete. Neutrinos are one of the fundamental pieces of the theory, not only because they are some of of the elementary particles that conform the matter content of the SM, but also because they are essential to understand some of the most relevant processes in nature, such as beta decays. Neutrinos are also incredibly abundant. It is roughly estimated that the neutrino density in the Universe is 330 neutrinos/m3, which makes them the most abundant massive particles. Moreover, only the density of photons is larger. In the last second, more than 1014 neutrinos have travelled across the reader's body, and they did so without causing any damage. This fascinating fact leads us to the next exciting property of neutrinos: their elusive character. These particles barely interact with matter and this is precisely why, in spite of having studied them for decades, some of their basic properties are still unknown. Designing experiments with neutrinos is utterly challenging and major technological advances have resulted from the development of the state-of-the art detectors needed. This is a clear example of how fundamental research pushes the improvement of tools and techniques that will eventually have a social impact. Relatively recent experiments measuring the oscillation between different types of neutrinos have shown that they are also chamaeleonic particles [1]. These flavour oscillations are a well-known quantum effect resulting from the fact that neutrinos do have non-degenerate masses and oscillate among each other. This is one of the aforementioned pieces of evidence pointing towards the existence of physics beyond the Standard Model. In this theory, neutrinos are massless, which is contradicted by the experiments measuring neutrino oscillations. Although the actual values of the masses remain unknown, it is clear that they are very light, around 100 000 times lighter than electrons. Current experiments such as KATRIN [2], which studies the spectrum of the tritium beta decay, aim to measure their masses. The great variety of processes in which neutrinos are involved and their diverse origins make the field of neutrino physics a promising and interdisciplinary area of research. Neutrinos are produced in nuclear power plants, in particle accelerators and as a consequence of natural radioactivity (so it is true, eating bananas turns you into an antineutrino source, a very weak one though). Atmospheric showers from cosmic rays are partially formed by neutrinos and antineutrinos, too. They are relevant in many astrophysical phenomena such as the nuclear reactions powering the stars, like the Sun, and supernova explosions. They have also played an important role in the evolution of the Universe. 3 Despite the fact that many outstanding physicists have devoted years of research to neutrino physics, there are some puzzling aspects that are still not completely understood. As an example, anomalous results reported by some experimental collaborations could be interpreted in terms of additional neutrinos which would not feel the electroweak interactions. They are the so-called sterile neutrinos. These deviations from the theoretical predictions are a current subject of research since they are susceptible of many interpretations. Another open question related to neutrinos which enhances their mysterious character is whether they are Dirac or Majorana particles. If the leptonic number were not conserved, neutrinos would be truly neutral particle and, therefore, they could acquire mass a la Majorana or a la Dirac. Should neutrinos be their own antiparticles, a whole new plethora of physical processes would be allowed. Neutrinos offer one of the most auspicious opportunities to explore physics Beyond the Standard Model. Their masses are a clear indication that our picture is yet to be completed and, together with Dark Matter, motivate the search for new physics. The more we learn about neutrinos, the more promising we find them. The number of hot topics they could be related to seems endless. Are they a subcomponent of Dark Matter? Could they help understand the baryon asymmetry of the Universe and be part of some leptogenesis mechanism? Are they somehow related to Dark Energy? These and many other questions are being addressed in a great variety of experiments and studies worldwide and, hopefully, some of them will be answered in the near future. 4 2 Neutrino oscillations 2.1 Oscillations in vacuum If neutrinos were massless, there would be no point in making a difference between flavour eigenstates and mass eigenstates. If they had mass and their masses were degenerate, they would be mass eigenstates too. Let jναi denote the flavour eigenstates and jνii the mass eigenstates. Note here that Greek indices are used for flavour eigenstates and Latin indices for mass eigenstates. The interaction modes (flavour eigenstates) can be written as a function of the propagation modes (mass eigenstates) as follows: X ∗ jναi = Uαi jνii i where Uαi are the elements of the unitary mixing matrix UPNMS, also known as Pontecorvo { Maki { Nakagawa { Sakata matrix [3]. After a certain amount of time t, the initial state has evolved 1 X ∗ −iEit jνα(t)i = Uαie jνii i and the probability of measuring a flavour eigenstate different from the originally produced is N 2 X ∗ ∗ −iEit iEj t Pνα!νβ =j hναjνβ(t)i j = UαiUβiUαjUβje e i;j One can assume an ultra-relativistic behaviour and that all the neutrinos are produced with the same momentum p, so that q m2 E = p2 + m2 ' jpj + i : i i 2jpj Consequently, m2 − m2 ∆m2 E − E = i j = ij : i j 2jpj 2jpj One can also approximate the fact the distance travelled by the neutrino is L ∼ t. 1From now on, natural units will be used all along the manuscript. 5 Then, the oscillation probability from the flavour α to β is N 2 X ∆mijL P = δ − 4 Re U ∗ U U U ∗ sin2 (1) να!νβ αβ αi βi αj βj 4E i>j N 2 X ∆mijL −2 Im U ∗ U U U ∗ sin (2) αi βi αj βj 2E i>j It is clear to see that neutrino oscillations are not sensitive to the absolute value of 2 2 2 the masses, but to the difference ∆mij = mi − mj . Therefore, in the standard 3 neutrino picture, only two mass differences are relevant, since the third one is just a combination of the other two. Actually, although the absolute value of the mass differences is known 2 with a reasonable accuracy, the absolute sign of ∆m31 remains unknown. This question is of great importance since it will determine the ordering of the mass eigenstates in the 2 spectrum, the neutrino mass ordering. If ∆m31 > 0, it is said that neutrinos are normally ordered, since ν1 would be the lightest mass eigenstate and ν3 the heaviest one. Otherwise, it is called inverted ordering and ν3 would be the lightest state, whereas ν2 would be the heaviest one, as it is shown in Figure1. Figure 1: Graphical representation of the two possible ordering fro the neutrino mass 2 spectrum depending on the sign of ∆m31 [4]. Mixing matrix parametrisation Another relevant point that needs to be considered is the parametrisation of the mixing matrix. From algebraic arguments, it is clear that a N × N matrix can be parametrised in terms of N(N − 1)=2 mixing angles and N(N + 1)=2 phases. However, depending on the nature of neutrinos, namely its Majorana or Dirac character, some of these phases are non-physical and can be removed by redefining the fields. 6 A Dirac mass term in the Lagrangian is of the form X −LD = miνiRνiL + h:c: i The whole Lagrangian, including the CC interaction term, is invariant under the transformation iθi iθi iφα νiL ! e νL; νiR ! e νR; α ! e α; which means that lepton number is conserved.
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