Trabajo de Fin de M´asteren F´ısica Avanzada

Modified dispersion relations and the anomalies in 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 . 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 model with modified dispersion relations...... 29 6.3 Numerical implementation...... 31

7 Conclusions 34

2 1 Neutrino physics

The Standard Model of elementary (SM) is the most accurate theory in 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 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 . 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 and, therefore, they could acquire mass a la Majorana or a la Dirac. Should neutrinos be their own , 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 |ναi denote the flavour eigenstates

and |νii 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 ∗ |ναi = Uαi |νii 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 |να(t)i = Uαie |νii i

and the probability of measuring a flavour eigenstate different from the originally produced is

N 2 X ∗ ∗ −iEit iEj t Pνα→νβ =| hνα|νβ(t)i | = 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 ' |p| + i . i i 2|p|

Consequently, m2 − m2 ∆m2 E − E = i j = ij . i j 2|p| 2|p| 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. Hence, one phase can be reabsorbed in the field definitions (one per neutrino and one per charged lepton) except a global one which takes into account the lepton number conservation. If neutrinos were Dirac fermions, which have 4 degrees of freedom, the number of phases needed would be N(N + 1)/2 − (2N − 1) = (N − 1)(N − 2)/2. In the case of three neutrinos, the mixing matrix

   −iδCP    1 0 0 c13 0 s13e c12 s12 0       U = 0 c23 s23  0 1 0  −s12 c12 0 iδCP 0 −s23 c23 −s13e 0 c13 0 0 1 is parametrised using 3 angles and one phase. Note that the shorthand notation cij = cos θij and sij = sin θij has been used. A Majorana mass term in the Lagrangian is of the form

1 X −L = m ν C ν + h.c. M 2 i iL iL i

It is not invariant under U(1) transformations and, consequently, all charges are broken in two units.

iθi T † T † i2θi νiL → νiLe ; νiLC νiL → νiLC νiLe Therefore, only truly neutral particles can be of Majorana type. If neutrinos were Majorana fermions, which have 2 degrees of freedom, only N phases could be reabsorbed in the definitions of the fields, the ones from the charged leptons, leaving N(N + 1)/2 − N = N(N − 1)/2 physical phases. Then, the mixing matrix has two additional Majorana phases with respect to the Dirac case

   −iδCP      1 0 0 c13 0 s13e c12 s12 0 1 0 0        iα  U = 0 c23 s23  0 1 0  −s12 c12 0 0 e 0  . iδCP iβ 0 −s23 c23 −s13e 0 c13 0 0 1 0 0 e

Nonetheless, these do not play any role in neutrino oscillations.

7 2.2 Matter effects as a modification of the dispersion relation

Elastic scattering of neutrinos in matter changes their effective masses. The

interaction through charged currents only affects νe since there are no muons or taus present in matter; whereas the neutral current contribution is the same for all neutrino flavours.

For simplicity, we will consider only two neutrino flavours, νe and νµ. The Hamiltonian in vacuum for the mass eigenstates is of the form ! ! E 0 1 m2 0 H = 1 ' |p|I + 1 0 2 0 E2 2|p| 0 m2

In the following calculations, in all the terms proportional to the identity matrix, the symbol I will be omitted. † For νe and νµ, the Hamiltonian in the flavour basis, H0,F = UH0U

2 2 2 ! m1 + m2 ∆m − cos 2θ sin 2θ H0,F = |p| + + 4|p| 4|p| sin 2θ cos 2θ

The Hamiltonian in matter will be HF = H0,F + HNC + HCC . The value of the terms related to the neutral and charged current interactions can be estimated [5]:

1 HNC = −√ GF ρn 2 √ ! √ √ ! 2GF ρe 0 2GF ρe 2GF ρe/2 0 HCC = = + √ 0 0 2 0 − 2GF ρe/2

where ρn and ρe are the local density of neutrons and electrons respectively. √ 2 Defining A = 2 2GF ρe|p| , the Hamiltonian can be rewritten as

2 2 2 2 ! m1 + m2 1 A 1 A − ∆m cos 2θ ∆m sin 2θ HF = |p| + − √ GF ρn + + . 4|p| 2 4|p| 4|p| ∆m2 sin 2θ −A + ∆m2 cos 2θ

This Hamiltonian can be diagonalised with a rotation matrix Um, with angle θm.

The diagonalising angle of the matrix, θm is given by the expression

2 m 2H12 ∆m sin 2θ tan 2θ = = 2 H22 − H11 ∆m cos θ − A Thus, the effective mixing angle changes inside matter. The eigenvalues of H, when

2One could also drop the terms proportional to the identity since they are common to the two neutrinos and eventually, they will not play any role. The same argument holds in the 3ν picture.

8 considering matter effects, are

2 1 mi Ei = |p| − √ GF ρn + 2 2|p|

where

1  q  m2 = (m2 + m2 + A) ± (∆m2 cos 2θ − A)2 + (∆m2)2 sin2 2θ 1,2 2 1 2 Therefore, due to their interaction with matter, neutrinos acquire an effective mass.

From the previous expressions, three different regimes can be identified:

• For A = ∆m2 cos 2θ, even if the mixing angle in vacuum is very small, θm = π/4. This situation is called resonant conversion.

• For A << ∆m2 cos 2θ, which corresponds to almost vacuum, the lighter eigenstate

corresponds to νe whereas the heavier one is almost purely νµ.

• For A >> ∆m2 cos 2θ, the effective mixing angle θm = π/2. This means that the

lighter eigenstate is almost purely νµ and the heavier one, νe.

The phenomenon of νe oscillation into another flavour in matter is known as level-crossing.

Matter effects play a crucial role in the determination of the mass hierarchy of the neutrinos. The resonant effect happens when A = −∆m2 cos 2θ for antineutrinos and A = ∆m2 cos 2θ for neutrinos, since the potential inducing the matter effects has a different sign for particles and antiparticles. As a consequence, only neutrino or antineutrino mixing would experience a resonant behaviour and this fact allows to determine the sign of the mass splitting.

9 3 Experiments on neutrino oscillations

3.1 Solar experiments

The nuclear chain reactions that take place in the Sun produce neutrinos in a wide range of energies and the spectrum depends on the reaction the neutrinos are coming from. The measurement of the neutrino flux can give an insight into the processes powering the Sun. Different experiments aimed to measure the flux of these particles coming from the pp chains. The main reactions happening in the Sun and the maximum energy for the neutrinos emitted are [6]

+ p + p → d + e + νe Eν ≤ 0.42 MeV

7 − 7 Be + e → νe + Li Eν = 0.86 MeV

8 8 ∗ + B → Be + e + νe Eν ≤ 15 MeV.

For the detection of solar neutrinos one can use Cherenkov techniques or neutrino capture reactions. Some of the most relevant solar neutrino experiments are listed below. The Homestake solar neutrino experiment was built in South Dakota to measure the flux of solar neutrinos above 0.814 MeV using a radiochemical technique based on 37 37 − the inverse beta reaction νe + Cl → Ar + e . It was a counting experiment, which means that one could know the number of neutrinos that had interacted by counting the amount of 37Ar in the detector. Its results, a 30% deficit with respect to the predicted flux, could only be understood once neutrino oscillations were considered [7]. The GALLEX experiment at the Gran Sasso Underground Laboratories (Laboratori Nazionali del Gran Sasso) was a radiochemical neutrino experiment which aimed to detect solar neutrinos with energies above 233 keV via the inverse beta decay reaction 71 − 71 Ga + νe → e + Ge [8]. The SAGE experiment, located in the Baksan Neutrino Observatory, in Russia, used a target of Gallium metal to measure solar neutrinos using inverse beta decay of 71Ga too [9]. The Borexino experiment, located also in Gran Sasso, aimed to measure low energy neutrinos, focusing particularly on the monoenergetic neutrinos coming from the pp chain involving 7Be. Neutrinos of all flavours could be detected by elastic scattering of electrons [10]. The Sudbury Neutrino Observatory (SNO), measured mainly neutrinos from 8B. It was sensitive to all the neutrino flavours since it measured the different interactions of neutrinos through charged currents, neutral currents and elastic scattering in a tank filled with heavy water in Sudbury, Canada [11]. Its contribution played an essential role to understand the solar flux of neutrinos and neutrino oscillations. Its director, A. McDonald was awarded with the Nobel Prize in Physics in 2015.

10 Figure 2: Measurement of reactor neutrinos in KamLAND as a function of L/E [12]

2 These solar neutrino experiments are sensitive to θ12 and also to ∆m21. Another experiment is sensitive to these very same parameters although it does not detect neutrinos from the Sun. KamLAND used anti-neutrinos form 16 different nuclear reactors in Japan to prove the disappearance of νe with L ∼ 150−180km. This fact made 2 it sensitive in the range preferred by solar neutrino experiments for ∆m21. As it is shown in Figure2, KamLAND results allowed to clearly determine the oscillatory dependence on L/E of the phenomenon. 2 The determination of the solar parameters θ12 and ∆m21 from a global fit is presented in Figure3. A small tension between the value of the mass splitting determined by solar experiments and KamLAND has been reported.

2 2 Figure 3: Confidence regions in the sin θ12 - ∆m21 plane from the solar neutrino experiments and KamLAND [13].

11 3.2 Atmospheric neutrinos

Cosmic rays produce showers of particles in the atmosphere. Among them, charged are produced. These pions decay to muons and later on, to electrons and neutrinos

− − − + + + π → µ + νµ → e + νe + νµ + νµ π → µ + νµ → e + νe + νµ + νµ

The Super-Kamiokande detector consists on a water Cherenkov detector in Japan, sensitive to atmospheric and solar neutrinos, although it was initially conceived as an experiment to detect proton decay. The atmospheric Super-Kamiokande results showed

a zenith angle dependence for νµ which can be understood in the framework of neutrino oscillations, due to the different distance travelled by neutrinos created in the atmosphere near the detector and those travelling across the Earth before arriving at the detector [14]. T. Kajita was awarded with the Nobel Prize in Physics for the discovery of neutrino oscillations in 1998. The IceCube Neutrino Observatory is located in the Antarctica. Its inner detector, DeepCore, allows IceCube to extend the measurement of the neutrino flux from 100 TeV down to below 10 GeV. The ANTARES neutrino telescope is also sensitive to

the parameters ruling the oscillation of atmospheric neutrinos. These are mainly θ23 and 2 2 2 |∆m31|. Figure4 presents the fit to the oscillation parameters in the sin θ23 − |∆m23| plane from the atmospheric sector for the normal and inverted ordering.

2 2 Figure 4: Confidence regions in the sin θ23 - |∆m31| plane from atmospheric neutrino experiments, for the normal (NO) and inverted ordering(IO) [13].

12 3.3 Long Baseline Accelerator Experiments (LBL)

Neutrinos are also produced in accelerators, where a beam of protons hits a target producing pions, kaons, muons and other particles. From the decay of these particles it is possible to generate a beam of νµ or νµ. In these experiments, it is of great importance to understand the production mechanism to determine the composition of the beam, as well as the background in the detector. Long baseline experiments are sensitive to the same parameters as the atmospheric experiments, but also to θ13 and δCP . The T2K experiment [15] is a long baseline neutrino oscillation experiment. It uses a proton beam generated in Tokai, Japan, at the J-PARC complex. It is composed of a near detector and a far detector 2.5o off-axis, Super-Kamiokande, in Kamioka. The length of the baseline is 295 km, which makes it sensitive to some of the oscillation 2 2 parameters, particularly sin θ23 and |∆m32|. It studies both the disappearance (νµ → νµ) and appearance (νµ → νe) channels. MINOS was a long baseline experiment which measured neutrino oscillations using charged-current (CC) and neutral-current (NC) interactions in a far detector (FD) and a near detector (ND) separated by 734 km. MINOS+ is the upgrade of MINOS, which continued running 3 more years. Neutrino energies range between 1 GeV and 10 GeV approximately. MINOS and MINOS+ are sensitive to oscillations involving sterile neutrinos[16]. Another experiment, NOνA, uses two detectors, one in Fermilab and another in Minnesota, 810km away, to study oscillations in the NuMI neutrino beam. In Figure5, the values of the parameters determined by MINOS, T2K and NO νA are presented.

2 2 Figure 5: Confidence regions in the sin θ23 - |∆m31| from long baseline experiments, for the normal (NO) and inverted ordering(IO) [13].

13 3.4 Short baseline reactor experiments

Neutrinos generated at nuclear plants can also be used to study the phenomenon of

neutrinos oscillations using νe beams. These experiments are particularly sensitive to θ13 2 and |∆m31|. The Daya Bay Reactor Experiment consists of six nuclear reactors and

eight different detectors performing precision measurements of the oscillation of νe. Other experiments such as RENO in South Korea, and Double Chooz in France, follow a similar set up. In all of them, the baseline length is of the order of a few kilometres. In Figure6, the results from the fits at the reactor sector are shown.

2 2 Figure 6: Confidence regions in the sin θ13 - |∆m31| plane from reactor experiments, for the normal (NO) and inverted ordering(IO) [13].

3.5 Global fit of the neutrino oscillation parameters

Neutrino oscillation experiments have already reached the precision era and consequently, it is important to perform a joint analysis of the experimental results. 2 Nowadays, short baseline (SBL) reactor experiments can measure θ13, ∆m31, solar 2 experiments and KamLAND are sensitive to θ12, ∆m21 and θ13; and atmospheric and 2 long baseline experiments (LBL) measure θ23, ∆m31, θ13 and δCP . The current status of the global fit to the oscillation parameters is summarised in Table1. Despite the precision of the measurements of some of the parameters, a few questions remain unanswered and future experiments such as JUNO [17], the Deep Underground Neutrino Experiment (DUNE) [18] and Hyper-Kamiokande [19] will focus on them.

• Is the mixing angle θ23 maximal? If not, which octanct does it lie in? Although current data seems to show a preference for a non-maximal mixing, better precision is needed.

14 2 • What is the sign of ∆m31? JUNO will try to measure the sign directly whereas other experiments could do it as well taking advantage of matter effects.

• Is there CP violation in the neutrino sector? That is, what is the value of δCP ? A large enough source of CP violation is needed in order to explain the baryon asymmetry of the Universe, and it is often claimed that it could come from the neutrino sector.

Table 1: Results from the global fit to the neutrino oscillation parameters [13]

Parameter best fit ± 1σ 2σ range 3σ range

2 −5 2 +0.20 ∆m21[10 eV ] 7.55−0.16 7.20–7.94 7.05–8.14 2 −3 2 |∆m31|[10 eV ] (NO) 2.50±0.03 2.44–2.57 2.41–2.60 2 −3 2 +0.03 |∆m31|[10 eV ] (IO) 2.42−0.04 2.34–2.47 2.31-2.51 2 −1 +0.20 sin θ12/10 3.20−0.16 2.89–3.59 2.73–3.79 2 −1 +0.20 sin θ23/10 (NO) 5.47−0.30 4.67–5.83 4.45–5.99 2 −1 +0.18 sin θ23/10 (IO) 5.51−0.30 4.91–5.84 4.53–5.98 2 −2 +0.083 sin θ13/10 (NO) 2.160−0.069 2.03–2.34 1.96–2.41 2 −2 +0.074 sin θ13/10 (IO) 2.220−0.076 2.07–2.36 1.99–2.44 +0.21 δCP/π (NO) 1.32−0.15 1.01–1.75 0.87–1.94 +0.13 δCP/π (IO) 1.56−0.15 1.27–1.82 1 –1.94

Nevertheless, some experimental results seem not to fit in this 3 ν picture. A detailed description of this fact and its consequences is provided in the next section.

15 4 Experimental anomalies and sterile neutrinos

Anomalous results have been found in both (anti)neutrino appearance and disappearance channels. These signals lead to the proposal of the existence of additional neutrinos. They are the so-called sterile neutrinos.

4.1 Appearance anomalies

LSND

Using neutrinos produced at Los Alamos Meson Physics Facility (LAMPF) beam stop, the Liquid Scintillator Neutrino Detector (LSND) took data between 1993 and 1998. The neutrino source is mainly produced by the reaction

+ + + π → µ νµ → e νeνµνµ.

The detector was a tank filled with a liquid scintillator and surveilled by photomultiplier tubes (PMTs), located about 30 m from the neutrino source.

Since the neutrino beam has no νe fraction, the measurement of the reaction νep → + e n is a signature of νµ → νe oscillation. This event can be identified by the detection of + a e and a 2.2 MeV from the reaction np → γd. The flux of νe coming + + − from π and µ decays in flight is very small and, therefore, the reaction νeC → e N can be utilised in order to measure νµ → νe oscillations. An excess of events with e+ with energies between 20 and 60 MeV was reported, indicating that it was extremely unlikely that it was due to statistical fluctuations [20, 21, 22]. When interpreted as a signature of neutrinos oscillations, and after taking into account the results from contemporary experiments (KARMEN, Bugey...), the data from LSND indicated that the oscillation happened for values of ∆m2 in the range 0.2 − 10 eV2.

MiniBooNE

The booster neutrino beam (BNB) at Fermilab delivers to the MiniBooNE experiment a flux of neutrinos and antineutrinos. The BNB is produced by 8 GeV protons from the Fermilab booster interacting on a beryllium target inside a magnetic focusing horn that allows to produce a nearly pure beam of νµ or νµ. The detector consists of a sphere filled with pure mineral oil (CH2) and is located 541 m away from the beryllium target. The detector is covered by photomultiplier tubes. The MiniBoone experiment was designed to test the LSND signal. It runed in the neutrino and antineutrino mode. The last results report that a νe charged-current quasielastic (CCQE) event excess (4.5σ) is observed in the energy range QE 200 < Eν < 1250 MeV.

16 Figure 7: LSND anomalous signal [20] (left) and MiniBooNE anomaly [24] (right).

Combining these data with the νe appearance data in the antineutrino mode, a total

νe plus νe CCQE event excess (4.7σ) is observed [23, 24]. When these two anomalies (see Figure7) are analysed in a two neutrino framework, the best fit occurs at (∆m2, sin2 2θ) = (0.041 eV2, 0.96), which would imply the existence of a sterile neutrino [24]. However, even when the best fit values of the parameters are considered, the excess of events is not fully understood.

4.2 Disappearance anomalies

Reactor anomaly

Historically, very short baseline experiments did not observe any νe disappearance, since the observed antineutrino flux was consistent with the theoretical predictions. New and improved predictions of the reactor antineutrino fluxes for 235U, 238U,239Pu, and 241Pu, have had a relevant impact on previous reactor experiments with distances from the reactor to the detector shorter than 100 m. The reevaluation of the predicted fluxes resulted on an increased mean flux and a shift on the observed event rate to predicted rate to 0.943 ± 0.023, which deviates from unity at 98.6% C.L. This is the so-called Reactor Antineutrino Anomaly (RAA) [25]. In Figure8, it is shown how the observed number of events is on average smaller than expected. Even when the latest theoretical prediction of the fluxes is considered for experiments such as Daya Bay or Double Chooz, an excess at around 5 MeV is found. It seems that our understanding of the fluxes of reactor antineutrinos is still incomplete. Therefore, a new series of experiments are being conducted with flux independent measurements, using the ratios of events at two different distances.

17 Figure 8: Ratio between the observed number of events and the theoretical calculations as a function of the length of the baseline for different reactor experiments. The average value with the corresponding uncertainties is also presented [26].

Gallium anomaly (GA)

The GALLEX and SAGE Gallium solar neutrino experiments have been callibrated with intense artificial 51Cr and 37Ar radioactive sources placed inside the detectors [27].

The ratio between the number of νe measured and the theoretical prediction is lower than one and it shows a deviation of around 3σ(see Figure9). In this case, L ' 1-2 m and E ' 0.4-0.8 MeV, which gives an L/E ratio of the same order as the reactor anomaly.

Figure 9: Ratio between measured and expected number of events for the GALLEX and SAGE experiments [28, 29]

18 4.3 Oscillations in the presence of sterile neutrinos and the 3+1 formalism

The 3ν standard picture can be extended including additional neutrinos. Those neutrinos are called sterile since they can not feel weak interactions or they have a mass larger than the mass of the Z boson. The formalism previously presented (see Equation1) for neutrino oscillations is valid for N neutrino families. The mixing matrix in a general case is parametrised using N(N − 1)/2 mixing angles and (N − 1)(N − 2)/2 phases. Only N-1 mass differences are relevant since the other can be written as a combination of them. The three experimental anomalies presented (LSND and MiniBooNE, GA and 2 RAA) could be interpreted in terms of an additional mass difference and mixings sin θee 2 and sin θµe. This fact would imply the existence of extra sterile neutrinos, at least one of them. However, it is not trivial to accommodate them in the neutrinos oscillations formalism, since additional neutrinos would have an impact on different experiments. The simplest case is the 3+1 formalism.

LSND and MiniBooNE

2 2 In a 3+1 framework, assuming that ∆m41  ∆m31 and ignoring the terms involving

δCP , the electron appearance probability for short baseline experiments (MiniBooNE, LSND, KARMEN, OPERA) is

∆m2 L ∆m2 L P ' 4|U |2|U |2 sin2 41 = sin2 θ sin2 2θ sin2 41 (3) νµ→νe e4 µ4 4E 24 14 4E

2 2 2 where sin θ24 sin 2θ14 is often referred as sin 2θµe. Figure 10 shows the allowed regions for this model, obtained from data taken by different experiments.

Reactor and Gallium experiments

The appearance channel in reactor and Gallium experiments, when studied in a 3+1 framework, is sensitive to the neutrino survival probability

2 2 2 2 ∆m41L Pνe→νe ' 1 − 4|Ue4| (1 − |Ue4| ) sin 4E (4) ∆m2 L = 1 − sin2 2θ sin2 41 14 4E

If the Gallium anomaly (GA) is studied as an oscillatory phenomenon related to 2 2 2 sterile neutrinos, the region with (sin 2θ14 > 0.07 , ∆m41 > 0.35 eV ) is allowed at 99% 2 2 2 CL and the best-fit values correspond to sin 2θ14 = 0.50 , ∆m14 = 2.24 eV [28]. The allowed region for this anomaly and the corresponding one for the RAA can be found in Figure 11.

19 2 2 Figure 10: Excluded and allowed regions in the ∆m41 − sin θeµ plane. The point corresponding to the best fit for MiniBooNE is in a region excluded by OPERA [24].

The reactor anomaly can be studied as a signature of the existence of a 2 2 sterile neutrino, being the parameters ruling this oscillation ∆m41 > 1.5 eV and 2 sin 2θ14 = 0.14 ± 0.08. The fact that this anomaly is due to some unknown reactor effects is not ruled out though [30]. Multiple reactor experiments have conducted searches for signatures of sterile neutrino oscillations:

• The DANSS detector is placed under an industrial reactor of the Kalinin Nuclear

Power Plant. It measures νe spectra at 3 distances from the reactor core centre to the detector centre (10.7 m, 11.7 m, and 12.7 m) [31]. The detection strategy consists in comparing relative spectral shapes instead of their absolute values. Hence, the uncertainties due to the detector efficiency and the initial neutrino flux do not play a role. Antineutrinos are detected using the inverse beta decay reaction + + νe + p → e + n with Eν = Ee + 1.80 MeV. The excluded area covers a large fraction of regions indicated by the GA and RAA (see Figure 11). DANSS results show a preference for the 3+1 formalism at 3σ 2 2 2 approximately and the best fit corresponds to ∆m14 = 1.4 eV and sin 2θ14 = 0.05 [32].

• The NEOS detector was installed in Yeonggwang, Korea. This is the same reactor complex being used for the RENO experiment. Their last report states that no 2 2 apparent parameter set of (sin 2θ14, ∆m41) that shows significant preference for 2 the 3+1 hypothesis was found. However, it excludes the value of sin 2θ14 to less 2 2 2 than 0.1 for the 0.2 eV < ∆m41 < 2.3 eV range.

20 Figure 11: Comparison between the 2σ and 3σ allowed regions for the Gallium Anomaly and the Reactor Antineutrino Anomaly, together with the fit for the combined data from DANSS and NEOS [35].

• The STEREO (ILL) experiment, located in Grenoble, has recently reported no deviation from the standard three neutrino picture neither [33], excluding the RAA best fit at 99% C.L.

• PROSPECT, the Precision Reactor Oscillation and Spectrum Experiment, in 2 2 Tennessee, reported the exclusion of a large region on the sin θ14 − ∆m41 plane at 95% C.L and the best fit point from the RAA is disfavoured at 2.2σ [34].

Constraints from long baseline experiments

Sterile neutrinos have an impact on long baseline and atmospheric neutrino experiments too and these are indeed the ones that are setting strong bounds on sterile neutrino oscillations. Experiments such as MINOS/MINOS+ [16] and IceCube [36] are constraining the sterile neutrino oscillation parameters. The disappearance probability in long baseline experiments (T2K, NOνA, MINOS/MINOS+) in the presence of a sterile neutrino is

21 ∆m2 L P ' 1 − |U |2(|U |2 + |U |2) sin2 31 νµ→νµ µ3 µ1 µ2 4E ∆m2 L −|U |2(1 − |U |2) sin2 41 µ4 µ4 4E ∆m2 L ' 1 − sin2 2θ cos4 θ sin2 31 (5) 23 24 4E ∆m2 L − sin2 θ sin2 θ sin2 2θ sin2 31 14 14 24 4E ∆m2 L − sin2 2θ sin2 41 24 4E where the fact that θ13 is known to be small has been used. In long baseline experiments, the appearance probability is given by

∆m2 L P ' 4U U (U U + U U ) sin2 31 νµ→νe µ3 e3 µ1 e1 µ2 e2 4E ∆m2 L +4U U (U U + U U + U U ) sin2 41 µ4 e4 µ1 e1 µ2 e2 µ3 e3 4E 2   = 2 sin 2θ13 cos θ14 cos θ24 sin θ13 sin θ23 + cos θ13 sin θ14 sin θ24 × (6)   ∆m2 L − cos θ cos θ sin θ + sin θ sin θ sin θ sin2 31 13 24 23 13 14 24 4E ∆m2 L + sin2 θ sin2 2θ sin2 41 24 14 4E

Assuming small mixing angles θ14 and θ24, it is easy to recover the expressions for long baseline experiments often found in literature

∆m2 L ∆m2 L P ' 1 − sin2 2θ cos4 θ sin2 31 − sin2 2θ sin2 41 (7) νµ→νµ 23 24 4E 24 4E

∆m2 L P ' sin2 2θ cos2 θ sin2 θ sin2 31 (8) νµ→νe 13 24 23 4E However, for the purpose of the following discussion, all the terms are kept. Recent fits in the 3+1 neutrino picture drew a conclusion of great relevance. MiniBooNE and LSND results are in strong tension with those from the disappearance channel in long baseline experiments. As one can see in Figure 12, the allowed regions for the appearance and disappearance experiments are in strong tension. This tension has also been reported in other studies [37, 38] and it is still present in more recent fits [39, 40]. Models beyond the 3+1 scenario are being studied in order to reconcile the results from the different experiments. The fact that the anomalies in MiniBooNE and LSND happen at a very similar value of L/E motivates the search for models involving oscillations.

22 Among the proposals following this trend, models with 3+N neutrinos have been proven to be incapable of solving the tension. Moreover, these models are in tension with cosmological measurements [41].

Figure 12: Global fit on appearance and disappearance data [42].

23 5 Modified dispersion relations in the 3+1 picture

The 3+1 picture has been proven to be fail at reconciling the anomalous signals from LSND, MiniBooNE, the reactor anomaly and the Gallium anomaly with the result from the experiments that agree with the predictions of the three neutrino standard picture. Models with sterile neutrinos and modified dispersion relations have been invoked as a solution. In this section, this family of models is studied in detail as an extension of the 3+1 formalism.

5.1 Intrinsic modified dispersion relation

A modification of the dispersion relation per se occurs when the energy momentum relation E2 = p2 + m2 does not hold. In the case of neutrino oscillations, if a Lorentz violating term is associated to the 4th mass eigenstate, which is effectively a modified dispersion relations, the kinematic phase in the oscillation probability is affected. This is

∆m2 L ∆m2  φ = 4i −→ φ = 4i + f(E) L (9) 4i 4E 4i 4E for i = 1, 2, 3 and where f(E) = αEβ, in order to conduct a general, yet simple, study.

Let us remember the fact that the bounds from MINOS on θ24 come from the disappearance channel

∆m2 L ∆m2 L P ' 1 − sin2 2θ cos4 θ sin2 31 − sin2 2θ sin2 41 . (10) νµ→νµ 23 24 4E 24 4E In the case of MINOS, since the kinematic phase is very large, the last term in the 2 expression above is averaged to 1/2 sin 2θ24.

Figure 13: The kinematic phase φ a function of the energy for MiniBooNE (L = 0.541 2 2 km) and MINOS (L = 731 km) for ∆m41 = 1.4eV . Different values of α are also presented.

24 If the function f(E) is positive, the kinematic phase is larger than its corresponding value in the 3+1 neutrino standard framework, as it is shown in Figure 13. This 2 translates in the fact that probability terms controlled by ∆m41 get smeared out at smaller energies. Such a behaviour has no impact in the bounds set by MINOS/MINOS+

in θ24, since the term depending φ41 is already averaged to 1/2. Adding a modified dispersion relation that makes the kinematic phase grow with the energy would only result on this term getting averaged to 1/2 at a lower energy.

Figure 14: The kinematic phase φ as a function of the energy for MiniBooNE (L = 0.541 km) and MINOS (L = 731 km). Different values of α and β are also presented.

25 If the function f(E) is negative, the kinematic phase can eventually reach very small values and even get to zero. In that case,

∆m2 L P ' 1 − sin2 2θ cos4 θ sin2 31 − sin2 2θ 2φ2 . (11) νµ→νµ 23 24 4E 24 41

A very small kinematic phase φ41 along the energy range of MINOS could weaken the 2 2 2 bounds on θ24, since the bound would no longer apply to sin 2θ24 but to sin 2θ24φ41.

Given a modified dispersion relation f(E) such that for a given energy E0 in the 2 spectrum of MINOS, f(E0) = ∆m4i/4E0, that is φ41(E0) = 0. If along the energy

spectrum of MINOS, φ4i was very small, it would be possible to weaken its bounds on the 3+1 framework, as it was previously explained. However, since f(E) is multiplied by

a large L, the condition of φ4i being small would be valid only for a short range of the energy spectrum. After a certain value of the energy, the modulus of kinematic phase

φ4i becomes very large and one recovers the 3+1 ν picture (see Figure 14), in which the 2 contribution from the sterile neutrino is to the appearance probability is 1/2 sin 2θ24. Therefore, modified dispersion relations whose origin is a Lorentz violation, together with sterile neutrinos, can not reconcile the LSND and MiniBooNE anomalies with other long baseline experiments.

5.2 Modified dispersion relations from effective potentials

As it was previously explained when discussing the effect of neutrino propagation in matter, interactions can be parametrised through an effective potential. This additional term in the Hamiltonian affects both the mixing angles and the effective masses of the neutrinos. Let us study the impact of adding a dispersion relation to the sterile neutrino in the 3+1 framework through an effective potential in the flavour basis 3.

 2    m1 0 0 0 0 0 0 0  2     0 m2 0 0  †  0 0 0 0  HF = U   U +   (12)  0 0 m2 0   0 0 0 0   3    2 α 0 0 0 m4 0 0 0 E

2 2 2 2 In the limit where m4 is much heavier than the other masses (m1 = m2 = m3 = 0), 3In the 3ν picture, when matter effects are considered, the interaction term due to neutral currents is irrelevant, since it is proportional to the identity. However, NC interactions become important when sterile neutrinos are included, since only the active flavours feel them .

26 4 and if we consider the mixing matrix to be slightly simplified U = UPMNSU14 ,then,

2 2  ∆m41 ∆m41  − 4E cos 2θ14 0 0 4E sin 2θ14    0 0 0 0  † HF ' UPMNS   U (13)  0 0 0 0  PMNS  2 2  ∆m41 ∆m41 α 4E sin 2θ14 0 0 4E cos 2θ14 + E

The effective mixing angle that diagonalises the Hamiltonian is given by

2 sin 2θ tan 2θ˜ = 14 (14) 14 2Eα+1 cos 2θ14 + 2 ∆m41 2 2 ˜ sin 2θ14 sin 2θ14 = 2 (15) 2  2Eα+1  sin 2θ14 + cos 2θ14 + 2 ∆m41

This model was initially proposed for α = 1, which would correspond to sterile neutrinos travelling through extra dimensions [43]. In this case, the different regimes are:

2 2 2 2 • sin 2θe14 ' sin 2θ14 when 2E   ∆m14 cos 2θ14

• Maximal mixing between active and sterile neutrinos at the resonance energy, when 2 2 2E  = −∆m14 cos 2θ14

• The effective angle goes to zero and the mixing becomes irrelevant for larger energy values, which allows to recover the three standard neutrino picture in this energy range.

Such a behaviour does not differ significantly from the resonant effect due to matter effects, except for the fact that the energy dependence is on E2, not on E. Therefore, in this case, the resonance is narrower. It is important to point out that the effective masses, which can be written from the eigenvalues of the Hamiltonian λi,

2 mi,eff = 2Eλi(E) become energy dependent and so do the effective mass splittings. Since the order of the rotation matrices is unconventional, the expressions in terms of the mixing angles that were obtained previously do not hold. However, they are still valid when written as a function of the elements of the mixing matrix.

4This parametrisation does not follow the usual convention. It is important to underline the fact that the value of the mixing angles depend on the order on which the rotation matrices are multiplied. Hence, if this unusual parametrisation is chosen, the values obtained from fits and the existing bounds do only apply at the level of the matrix elements. This choice was made so that it can be directly compared to the initial proposal for this type of effective potentials [43].

27 Let us study how the resonant behaviour of the mixing angles could potentially relax the tension between appearance (MiniBooNE) and disappearance experiments (MINOS), and how it can be constrained by other experiments.

˜ Resonant mixing in θ14

The electron appearance probability in MiniBooNE is given by

∆m2 L P ' 4|U |2|U |2 sin2 41 (16) νµ→νe e4 µ4 4E

According to the way we have defined the mixing matrix,

4 ˜ 2 2 2 2 4 ˜ 2 2 sin 2θ14 |Ue4| |Uµ4| = |Ue1| |Uµ1| tan θ14 = |Ue1| |Uµ1| . ˜ 4 (1 + cos 2θ14)

2 ˜ Hence, one could think that if sin 2θ14 ∼ 1 at the energy of the MiniBooNE anomalous signal (E ∼ 0.2 − 0.3 GeV), this mechanism could give rise to a significant appearance probability in MiniBooNE. A crucial fact is missing, though. In the analysis of any predictions made by these models the fact that the neutrino beam from T2K lies in the same region of the energy spectrum as the anomalies in the MiniBooNE experiment has to be taken into account. As a consequence, any deviation from the L/E determined by standard neutrino oscillations, namely additional energy-only dependencies, are expected to impact the predictions for both experiments. In order to appreciate an oscillatory effect, MiniBooNE’s kinematic phase has to be of order one. Then, the corresponding term in the appearance probability for T2K is averaged to 1/2. In T2K, the probability in the appearance channel is

∆m2 L 1 P ' 4U U (U U + U U ) sin2 31 + · 4|U |2|U |2. (17) νµ→νe µ3 e3 µ1 e1 µ2 e2 4E 2 µ4 e4

The first term is the standard signal whereas the second one is analogous to the one in MiniBooNE’s prediction, but with the kinematic phase averaged to 1/2. ˜ Note that if the angle θ14 is resonant at 0.2-0.5 GeV, or at least large enough so that the MiniBooNE signal can be explained, the second term in the expression of the appearance probability in T2K would have manifested as a significant deviation in the experiment. However, no evidence for such signal has been reported. Some additional problems arise in these types of model from the mass differences in the high energy regime of accelerator and atmospheric neutrinos, as it was pointed out in previous studies [43].

28 6 Modified dispersion relations through effective potentials in a 3+3 picture

In this section, a model with 3 sterile neutrinos and modified dispersion relations arising from an effective potential is studied using numerical tools.

6.1 GLoBES, snu.c and modifying the probability engine

GLoBES (”General Long Baseline Experiment Simulator”) is a software package to simulate reactor and long baseline neutrino oscillation experiments. It allows one to describe most of the experiments in a flexible way using an Abstract Experiment Definition Language (ADEL) [44, 45]. The neutrino flux, the oscillation channels the experiment is sensitive to and other important features can be specified for each experiment. Given the values of the oscillation parameters, it solves the system numerically in order to compute the oscillation probabilities for each channel at a given distance and for a given energy. If the technical features of the experiments are known, it allows computing the number of events expected after a certain running time and one can perform sensitivity studies to the different parameters. Although it is a very powerful tool to perform these sort of studies, the fact that it is originally limited to the 3ν standard picture prevents its application to other scenarios. Nevertheless, there is an extension of the code for new physics called snu.c. It consists of a redefinition of the function which calculates the probability (also called probability engine) in order to adapt it to scenarios with up to 6 additional neutrinos and non-standard interactions affecting the production, propagation and detection [46, 47]. Taking the aforementioned code as a starting point, a further extension to the so-called probability engine was developed. It includes up to 6 sterile neutrinos and a particular modification of the dispersion relation which made it suitable for the analysis of a potentially interesting model. It was tested against similarly structured codes written independently on Python and Fortran, to ensure that the results were reliable.

6.2 A 3 sterile neutrino model with modified dispersion relations

A model including 3 sterile neutrinos and an effective potential in the flavour space altering their dispersion relation was proposed in Ref. [43]. Such potential

29   0 0 0 0 0 0    0 0 0 0 0 0     0 0 0 0 0 0    Veff =    0 0 0 E 0 0     0 0 0 0 κE 0    0 0 0 0 0 ξE can be constructed in a scenario in which the sterile neutrino takes a short-cut via an asymmetrically warped extra dimension. This potential gives rise to the resonant behaviour proposed as the explanation to the anomalies.  is the short-cut parameter, related to the running time difference between the active and the sterile neutrino. The other two parameters, κ and ξ, are included to generate a resonance capable of explaining MiniBooNE and LSND results. In order to simplify the model, each of the sterile neutrinos mixes with just one of the active neutrinos, and the elements of the mixing matrix are all considered to be real. The three active-to-sterile mixing angles are assumed to be equal since otherwise, additional oscillations should have been measured in atmospheric neutrino experiments. The mixing matrix can be parametrised as follows 5

6x6 U = UPMNS · U14 · U25 · U36.

The aim of this unconventional choice is to offer a direct comparison to the initial proposal for these family of models [43]. The mass differences are 2 2 2 ∆m51 = ∆m41 + ∆m21

2 2 2 ∆m61 = ∆m41 + ∆m31 The idea of the model is to generate a resonant-like structure to explain the excess of events observed in MiniBooNE and LSND. For energies above the resonance, although the sterile neutrinos decouple from the active ones, experiments should have been sensitive to a significant signal in the form of an oscillation. Since this situation is not observed, the parameters in the effective potential are chosen ad hoc so that the mixing is suppressed for atmospheric neutrino experiments and MINOS.

5Again, this parametrisation does not follow the usual convention. Since the value of the mixing angles depends on the order on which the rotation matrices are multiplied, the values obtained from fits and the existing bounds do only apply at the level of the matrix elements. In this case, new bounds on the mixing angles would apply.

30 6.3 Numerical implementation

In order to numerically address the model, GLoBES software was used with a modification of the probability engine, as it was explained previously. In a 3+3 framework, with the oscillation parameters

2 −7 2 −3 ∆m21 = 7.55 · 10 ∆m31 = 2.50 · 10 δCP = 0 2 2 2 sin θ12 = 0.320 sin θ23 = 0.547 sin θ13 = 0.0216

2 2 different sets of the parameters (∆m41, sin θ14, , κ, ξ) are studied. Normal ordering is assumed. The values of the parameters in the effective potential (, κ and ξ) are chosen so that there is a resonance for E ∼ 20 MeV and E ∼ 10 − 20 MeV (one for MiniBooNE and the other one for LSND).

Problems with the mixing angle

The aforementioned tension between T2K and MiniBooNE arising from the energy dependence of the mixing angles (see Section 5.2) is still present in models with modified dispersion relations coming from effective potentials and is not solved by adding additional sterile neutrinos. The values of the parameters can be chosen so that a resonant behaviour happens in the energy range in which MiniBooNE and LSND signals deviate from the predictions of the 3ν picture. It is possible to achieve the desired resonant effect in MiniBooNE but it is clear to see in Figure 15 that this result is in conflict with T2K appearance measurements.

MiniBooNE (L = 0.541km) T2K (L = 295km) 0.7 0.7 6 + MDR (1) 6 + MDR (1) 0.6 6 + MDR (2) 0.6 6 + MDR (2) standard 3 standard 3 0.5 0.5

e 0.4 e 0.4

P 0.3 P 0.3

0.2 0.2

0.1 0.1

0.0 0.0 0.10 0.15 0.20 0.25 0.30 0.35 10 1 100 E (GeV) E (GeV) Figure 15: Comparison between the predictions of the three neutrino standard picture and a 3+3 model with modified dispersion relations for two sets of parameters: 2 2 2 −16 −17 −17 (1) (∆m41, sin θ14, , κ, ξ) = (1.59 eV , 0.05, 8 · 10 , 4 · 10 , 4 · 10 ) and (2) 2 2 2 −15 −17 −17 (∆m41, sin θ14, , κ, ξ) = (1.59 eV , 0.05, 5 · 10 , 5 · 10 , 5 · 10 ).

31 2 Problems with the running effective masses ∆mij After diagonalising the Hamiltonian, one can calculate the effective masses and their differences from the eigenvalues, λi(E), which depend on the energy

2 mi,eff (E) = 2Eλi(E).

These, together with the mixing angles, are the parameters that rule the oscillatory phenomena. The wanted behaviour regarding the mass splittings is the following:

• For energies below the one for which the LSND anomaly happens, E < ELSND ∼ 7 2 2 10 eV, the standard mass differences must recover ∆m21 and ∆m31, while the three additional neutrinos must be considerably heavier.

• For energies larger than the one for which the MiniBooNE excess of events is found, 8 E > EMiniBooNE ∼ 10 eV, the active and sterile neutrinos decouple. In this range 2 2 2 2 ∆m64 and ∆m54 have to recover the values of ∆m31 and ∆m21 In these two regimes, far from the resonances, the active-sterile mixing angles are very small.

• For LSND and MiniBooNE, mass splitting of order ∼ 1 eV2 are needed, and mixing angles are large due to the fact that the energy is near the resonant energy.

3 2 10 m21 2 m31 2 2 10 m41 2 m51 1 10 2 m61

) m2 2 54 100

V 2 m64 e (

1 j 2

i 10 m 10 2

10 3

10 4

10 5 103 104 105 106 107 108 109 1010 1011 1012 E (eV)

Figure 16: Running of the mass squared differences with the energy for 2 2 2 −17 −15 −15 (∆m41, sin θ14, , κ, ξ) = (1.59 eV , 0.05, 5 · 10 , 5 · 10 , 5 · 10 )

32 The running of the mass differences in this particular model is presented in Figure 16. Two experiments are particularly sensitive to these types of model. First of all, Daya Bay can set strong constrains in this family of models, since its measurements of both 2 θ13 and ∆m31 are very accurate. The second one is KamLAND, which was capable of 2 measuring the ∆m21 mass splitting with an excellent accuracy. The energy spectrum of these two experiments is similar, E ∼ 1 − 12MeV , since they are both reactor experiments. In this range, the dependence with the energy of the mass splitting is very relevant. As it is shown in Figure 16, in this energy range, the value of 2 ∆m21, which is the one contributing in KamLAND, differs significantly from the value 2 −5 2 measured (∆m21 = 7.55 · 10 eV ). This causes the prediction to deviate dramatically from the standard three neutrinos framework, as one can see in the Figure 17. The same 2 argument applies also to Daya Bay and ∆m31.

KamLAND (L = 180km) DayaBay (L = 1.3km) 1.0 1.0 6 + MDR 6 + MDR standard 3 standard 3 0.8 0.8

e 0.6 e 0.6 e e

P 0.4 P 0.4

0.2 0.2

0.0 0.0 0.002 0.004 0.006 0.008 0.010 0.012 0.014 0.002 0.004 0.006 0.008 0.010 0.012 E (GeV) E (GeV) Figure 17: Comparison between the predictions of the three neutrino standard picture 2 2 and the 3+3 model with modified dispersion relations for (∆m41, sin θ14, , κ, ξ) = (1.59 eV2, 0.05, 5 · 10−15, 5 · 10−17, 5 · 10−17) for Daya Bay (right) and KamLAND (left).

33 7 Conclusions

As a consequence of neutrinos being massive, flavour oscillations have been experimentally measured in a large variety of experiments, using neutrinos from diverse origins and different approaches for their detection. In this project, the theoretical framework required to understand the experimental results has been presented. In the determination of the oscillation parameters from experimental measurements, global fits are of great importance. This is due to the fact that experiments from different sectors are capable of measuring the same parameters. However, some important aspects

like the value of δCP and the mass hierarchy remain unknown. Future experiments will aim to measure them. Although the 3ν picture has been proven to provide an accurate description of the phenomenon, anomalous signals have been reported by different experimental collaborations. These anomalies are not understood and could be the signature of Beyond a Standard Model physics. Nevertheless, in some cases their origin could still be due to an incomplete description of the processes taking place in the experiment and might not be due to new physics. The sterile neutrino hypothesis arose as an explanation to the anomaly reported by LSND, later on confirmed by MiniBooNE. Both experiments have in common the fact that the deviation from the theoretical prediction happens for a similar value of L/E. This variable, L/E, happens to be the one on which neutrino oscillations depend. The Gallium anomaly and the Reactor Antineutrino Anomaly also seem to share a comparable L/E dependence. Nonetheless, it has been proven that the 3+1 formalism and extensions with additional neutrinos do not reconcile these anomalous signals with the negative results from other experiments. Models with sterile neutrinos and modified dispersion relations were proposed as a way to include additional energy dependencies to the sterile neutrino hypothesis. These types of models have been the subject of an extensive study in this project, considering a variety of potentially interesting scenarios. In the case of the 3+1 formalism, modified dispersion relations originated from Lorentz violating terms or through effective potentials have been studied. Neither of the cases can reconcile MiniBooNE’s low energy excess with some long baseline experiments without leading to inconsistent predictions for other experiments. In the line of some proposals claiming to explain the anomalies, the study of a 3+3 scenario with an effective potential has been conducted. Although some of the pathologies present in the 3+1 case can be fixed when considering two additional sterile neutrinos, the model is not capable of consistently explaining the anomalies while giving a correct description of the non-anomalous experiments. A number of powerful numerical tools have been utilised in this analysis, namely the General Long Baseline Experiment Simulator (GLoBES) package. Nonetheless, since it

34 is mainly devoted to simulations in the 3ν picture, an extension of the probability engine has been required to accommodate the additional number of neutrinos as well as the modification of the dispersion relation. Recent evidence disfavours the sterile neutrino hypothesis as the explanation to the anomalous signals. When considering modified dispersion relations, no matter the physical mechanisms responsible for them, not only the number of parameters needed to phenomenologically explain the results increases substantially, but also a strong energy dependence arises. Therefore, these models do not provide a good description for neutrino oscillations. There is no doubt that the anomalous signals reported are an indication of the fact that our understanding of neutrino experiments and its theoretical framework is not complete, and that there are some underlying physical processes playing a role which is not understood. Solutions in the Standard Model are diverse and could be due to a wrong evaluation of the initial fluxes. Experiments taking flux independent measurements will address that possibility. Among the Beyond the Standard Model explanations proposed, the sterile neutrino hypothesis and its variations are clearly disfavoured, whereas some others including non-standard interactions or production and detection processes involving dark matter are being developed. A whole new generation of experiments, including DANSS, NEOS and the Short Baseline Neutrino Program at Fermilab, will be focusing on finding an explanation to the anomalies. Signatures of new physics could also be present in a near detector in DUNE or in T2K. No matter what the final outcome turns out to be, it will definitely help us shed some light on the neutrino sector.

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