Neutrino Mass and Neutrino Oscillations

Gelo Noel M. Tabia1, ∗ 1Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada Department of Physics and Astronomy and Institute for Quantum Computing, University of Waterloo, 200 University Avenue West, Waterloo, ON, N2L 3G1, Canada Measurements performed in solar neutrino experiments in the 1960s and atmospheric neutrino experiments in the 1980s exhibited a deficit in the flux of neutrinos when compared to the predictions of the Standard Model. To explain this unexpected ‘disappearance’ of momentum and energy, neutrinos were hypothesized to transition between flavors in a mixing process known as neutrino oscillations, which was first proposed by Bruno Pontecorvo in 1957 as a distinct possibility for massive neutrinos. We discuss a few phenomenological models that fit known observational and experimental data and attempt to explain neutrino masses and flavor mixing. In particular, they generally provide bounds on the primary oscillation parameters—the mixing angles and the squared mass differences—in the context of various mass-generating mechanisms for neutrinos. In this project, we provide a basic overview of the theory of neutrino mass and neutrino oscillations. We also present some of the existing observational evidence from solar and atmospheric neutrinos as well as the experimental results from reactor and accelerator laboratory searches.

I. INTRODUCTION density can be written as

GF µ In 1911, Lisa Meitner and Otto Hahn observed in beta Hweak = √ ψpγµψnψeγ ψνe (2) decay experiments that the energies of the emitted elec- 2 trons exhibited a continuous spectrum, confirmed in later where G is the Fermi coupling constant. The Hamilto- experiments by Charles Drummond Ellis and colleagues F nian in low-energy weak decays essentially retains the in the 1920s. To reconcile these experimental observa- form as Fermi’s interaction except for some modifications with energy conservation, Wolfgang Pauli suggested tions: (i) fundamental fields for proton and neutron are a third unobserved particle in the decay process: a neu- now in terms of quarks, (ii) before electroweak theory, trino. For this purpose, the neutrino should carry no Robert Marshak and George Sudarshan, and Richard electric charge and have virtually no interactions with Feynman and Murray Gell-Mann determined the correct matter. Pauli himself pointed out that its mass should V -A form for the currents, and (iii) weak interactions be no larger than 1 percent of the proton mass—the ear- include both charged current (W ±) and neutral current liest known limit to neutrino mass. Thus, in standard ra- (Z0) interactions. Although Fermi’s theory of weak in- dioactive decay, nuclei mutate to a different species when teractions matches well with tree Feynman diagram cal- neutrons transform into protons according to culations, it fails with loop calculations as the theory is n → p + e + ν . (1) not renormalizable in four dimensions–it requires modi- e fications that were introduced later in electroweak the- The energy range for the emitted electron corresponded ory, first discovered in the 1960 when Sheldon Glashow perfectly to the various ways energy can be distributed extended electroweak unification models due to Julian in a radioactive decay into a three-particle final state. Schwinger by including a short range neutral current 0 In 1956, Fred Reines and Clyde Cowan Jr. first directly Z , and improved upon by Mohammad Abdus Salam observed neutrinos produced by a nuclear reactor [1], a and Steven Weinberg when they incorporated the Higgs discovery that was awarded the Nobel Prize in 1995. The mechanism. For the most part, we know that the Standard Model muon neutrino νµ was discovered in 1962 by Leon Le- derman, Melvin Schwartz and Jack Steinberger (Nobel provides a remarkable picture of the vast array of phenomena in particle physics, with the exception of gravi- Prize, 1988) and the tau neutrino ντ almost forty years later in 2000 by the DONUT Collaboration at Fermilab tational effects. It is the most general theory consistent in an experiment built specifically to detect it. with general physical principles like Lorentz invariance To explain Pauli’s neutrino hypothesis, Enrico Fermi and unitarity plus two assumptions: (i) renormalizability developed a theory of beta decay [2] in terms of the and (ii) observed particle content, all of which have been weakly interacting fields of a neutron, proton, electron observed experimentally or has indirect experimental evidence (gluons, quarks) save for the yet-unseen Higgs bo- and antineutrino. If we denote the fermion fields by ψj son. for j = n, p, e, n¯e then Fermi’s interaction Hamiltonian There is however one phenomenon where it is clear that the Standard Model fails: explaining the phenomenon of neutrino oscillations. The Standard Model asserts sep- ∗Electronic address: [email protected] arate conservation laws for the three lepton numbers Le,Lµ and Lτ . Even taking into account electroweak 2 anomaly suggests quantities like Le − Lµ or Lµ − Lτ II. NEUTRINO MASS AND THE STANDARD should be exactly conserved. Consequently, the theory MODEL predicts stable, massless neutrinos where charged current − weak interactions only involve specific pairs (ν`, ` ). One of the most beautiful aspects of modern theo- In 1957, Bruno Pontecorvo realized that non-zero neu- ries of particles physics is the relation between forces trino masses imply the possibility of neutrinos oscillating mediated by spin-1 particles and local gauge symme- from one flavor to another [3]. He proposed that the tries. Within the Standard Model, the strong, weak and neutrino state produced in weak interactions is a super- electromagnetic interactions are related to, respectively, position of states of two Majorana neutrinos with definite SU(3), SU(2) and U(1) gauge groups. Many features of mass. Such a phenomenon is familiar in the quark sector the various interactions are then explained by the symme- of the Standard Model, where we have neutral kaon mix- try to which they are related. In particular, the way that ing. At the time, only the electron neutrino was known the various fermions are affected by the different types of so this hypothesis of neutrino mixing was highly specu- interactions is determined by their representations under lative. the corresponding symmetry groups. The problem of searching for neutrino mass and study- ing the oscillation phenomenon has been faced in the past through many different experimental techniques. The A. Basic neutrino properties first studies, based on the Fermi-Perrin method [4] of observing near the end point of the β-spectrum, obtained Neutrinos are fermions that have neither strong nor the limit mν = 500 MeV [5], later improved in the 1950s electromagnetic interactions. In group theory language, to m 250 MeV. Therefore, it became evident that if ν . they are singlets of SU(3)C × U(1)EM. Active neutrinos neutrinos had any mass at all, it would be lighter than have weak interactions, that is, they are not singlets of an electron. After the discovery of parity violation in SU(2)L. Sterile neutrinos, if they exist, have none of β-decay, proponents of the two-component spinor theory the Standard Model gauge interactions and they are sin- of neutrinos [6] suggested that the neutrino is a massless glets of the associated gauge group. It has half-integer particle with a chiral field νL or νR. The experiment of spin with a very tiny mass as implied by the phenom- Goldhaber et al. in 1958 [7] supported this hypothesis, ena of neutrino oscillations. The existence of a neutrino establishing that neutrinos are left-handed particles. mass also strongly suggests the existence of a tiny neu- At present, the effects of neutrino masses and mixing −19 trino magnetic moment of the order of 10 µB, which are investigated in various experimental setups. There might allow for small electromagnetic interactions. Neu- are three kinds of experiments involving small neutrino trinos have been shown experimentally to always have masses: (i) neutrino oscillation experiments, (ii) search left-handed chirality. for neutrinoless double-beta decay, and (iii) measure- The Standard Model has three active neutrinos. They ments of electron neutrino mass in the high-energy β reside in lepton doublets, spectrum of 3H. There are also those experimental results that favor massive neutrinos and neutrino mixing in νL,` (i) solar neutrinos (Homestake, Kamiokande, GALLEX, L` = − where ` = e, µ, τ. (3) `L SAGE, Super-Kamiokande), (ii) atmospheric neutrinos (Super-Kamiokande, Kamiokande, IMB, Soudan, Here e,µ and τ are the charged lepton mass eigenstates. MACRO), and (iii) the accelerator LSND experiment. The three neutrino interaction eigenstates νe,νµ and ντ In this project, we will provide a review of the basic are defined as the SU(2)L partners of the mass eigen- theory and phenomenology of massive neutrinos and neu- states. The charged current interaction for leptons goes trino oscillations. In Section II, we will consider how cat- like egorize the place of neutrinos in the Standard Model and g X µ − + + µ − how extensions of it can accommodate neutrino masses. −LC = √ νL`γ ` W + ` γ νL`W (4) 2 2 L µ L µ In Section III we describe the fundamental phenomenol- ` ogy of neutrino oscillations, where we show the relevant equations for explaining the experimental observations Additionally, there are neutral current interactions for without going into any rigorous field theoretic deriva- neutrinos of the form tion. In Section IV, we discuss the neutrino oscillation g X µ 0 −LNC = νL`γ νL`Zµ. (5) experiments and their key results. In particular, we are 2 cos θW also interested in the upper and lower bounds for neu- ` trino masses from direct searches and double beta decay The charged and neutral current interactions yield all experiments. In Section V, we discuss models that ex- neutrino interactions in the Standard Model. tend the Standard Model and generate neutrino masses. The measurement of the decay width of the Z0 bo- There is a significant amount of literature on the subject son into neutrinos makes the existence of exactly three but in this report, our main references are [8–13]. light (mν . mZ /2) active neutrinos an experimental fact. Expressed in terms of the Standard Model prediction for 3 single neutrino generation, the observed data [14] implies term of the Majorana type, which requires just one he- that licity state; however, since the νL is part of the SU(2)L doublet field and possesses lepton number +1, the Majo- Nν = 2.994 ± 0.012 (Fit to LEP data) T −1 (6) rana mass term νL C νL, where C is the Lorentz con- Nν = 3.00 ± 0.06 (Invisible Z width) jugation matrix, transforms as an SU(2)L triplet—it is not gauge invariant. It also violates the lepton number symmetry by 2 units. B. The Standard Model predicts mν = 0 Initially, we might think it is possible to induce the neutrino mass from non-perturbative effects. In the Stan- The Standard Model is based on the gauge group dard Model, the only known source of lepton number violation involves the so-called weak instanton effects. GSM = SU(3)C × SU(2)L × U(1)Y (7) However, even though the lepton number current conservation is indeed broken non-perturbatively though a with three fermion generations, each generation consist- chiral anomaly, it turns out an identical contribution ing of 5 different representations of the gauge group: can be found for the baryon number current in such a way that the B − L current is conserved to all orders in QL (3, 2) 1 ,UR (3, 1) 2 ,DR (3, 1)− 1 , 6 3 3 (8) the gauge coupling. Because the neutrino mass operator LL (1, 2) 1 ,ER (1, 1) . above also violates B − L, this shows that neutrino mass − 2 −1 vanishes even in the presence of non-perturbative correc- The notation above reads in the following way: for ex- tions. Consequently, we expect neither mixing nor CP ample, a left-handed lepton field LL is a singlet of the violation in the leptonic sector. SU(3)C group, a doublet (2) of the SU(2)L group, and carries hypercharge -1/2 under the U(1)Y group. The electroweak gauge symmetry is broken by the non- C. Extensions allow mν 6= 0 vanishing vacuum expectation value of the Higgs field, through the Higgs mechanism. Using a doublet Higgs There are many good reasons to think that the Stan- representation φ dard Model provides an inadequate description of all known physical phenomena: the problem of a fine-tuned φ φ = + , (9) Higgs mass might be solved by supersymmetry; gauge φ0 coupling unification and the variety of gauge representations may be explained well by grand unified theo- with Higgs potential ries; and the existence of gravity suggests extensions like string theories are relevant to Nature. If any of the pro- V (φ) = −µφ†φ + λ(φ†φ)2 (10) posed modifications is realized in Nature, the Standard then for µ2 > 0, the vacuum expectation Model becomes an effective field theory, valid as a low- energy approximation to a more general theory up to r µ2 v some energy scale ΛNP characterizing new physics. hφ i = ≡ √ (11) Even as an effective low energy theory, the Standard 0 2λ 2 Model would still retain the gauge group, the spectrum gives mass to the charged bosons and Z0 of fermions, and the pattern of spontaneous symmetry breaking as valid ingredients to describe Nature at en-

1 MW ergies much smaller than ΛNP. The Standard Model MW = gv, cos θW = (12) predictions, however, are modified with corrections pro- 2 MZ portional to powers of E/ΛNP. This means that a more where θW is known as the Weinberg angle for neutral general field theory than the Standard Model would have gauge bosons. The gauge boson orthogonal to Z0 is the to incorporate non-renormalizable terms. massless photon. Every particle that couples to the Higgs A more general theory of new physics won’t have to re- field acquires a mass. spect the accidental symmetries of the Standard Model. Chiral symmetry forbids a bare mass term for the Indeed it is known that there is a set of dimension-5 terms fermions; fermion masses arise from the Yukawa inter- made of Standard Model fields consistent with gauge actions, symmetry but violates

u ∗ d −L = Y Q iτ φ U + Y Q φD Gglobal = U(1)B × U(1)e × U(1)µ × U(1)τ , (14) Yukawa ij Li 2 Rj ij Li Rj (13) +Y ` L φE + (h.c.) ij Li Rj the accidental global symmetry corresponding to baryon after spontaneous symmetry breaking. The Yukawa in- number and lepton ﬂavor numbers. These terms are of teractions give masses to charged fermions but neutrinos the form remain massless. In general, neutrinos can have a mass ν Zij φφLLiLLj. (15) ΛNP 4

In particular, eq. (15) violates L by two units and leads be directly related to ψ(x)∗. However, if we just naively to neutrino masses: impose ψ = ψ∗ we eventually run into terms that are not Lorentz covariant. Instead we deﬁne a conjugate ﬁeld Zν v2 (M ) = ij . (16) ν ij ˆ ∗ ΛNP 2 ψ(x) ≡ γ0Cψ (x) (21)

This provides a generic extension of the Standard Model where the specific form of the conjugation matrix C = which, among other things, would natural explain why µ iγ2γ0 depends on the representation used for γ . the neutrino mass is small (it scales with the inverse of The condition for the Majorana field in terms of the the large energy scale of new physics), and how the lepton Dirac field is then obtained by setting flavor symmetries are broken, allowing for lepton mixing and CP violations. ψ(x) = ψˆ(x). (22) The best known scenario that leads to eq. (16) is the see-saw mechanism [27], where we assume the ex- If we express the Dirac bispinor as istence of heavy sterile neutrinos with bare mass terms and Yukawa interactions that result in a mass matrix χ(x) ψD(x) = 2 ∗ (23)  v  σ φ (x) 0 Y ν √ M =  2 (17) where χ(x) is a left-handed Weyl spinor then the Majo- ν  ν T v  (Y ) √ MN rana spinor is obtained in the case when φ = χ. In this 2 way, the Dirac fermion is shown to be equivalent to two

Whenever the eigenvalues of MN are much larger than Majorana fermions of equal mass. the electroweak breaking scale v, diagonalization of Mν If we introduce a two-component spinor ρ such that leads to three light mass eigenstates with a mass matrix 1 1 (16). We will see more details of ways to go beyond the χ = √ (ρ2 + iρ1), φ = √ (ρ2 − iρ1), (24) Standard Model in Section V. 2 2 then the U(1) symmetry of the Dirac Lagrangian

III. THEORY OF NEUTRINO OSCILLATIONS µ LD = ψγ ∂µψ − mψψ, (25)

A. Dirac and Majorana fermions which for ψ given by eq. (23) yields X m X Massive fermions can either be of the Dirac or Majo- L = −i ρ †σµ∂ ρ − ρT σ2ρ + (h.c.), (26) D α µ α 2 α α rana type. If they carry electric charge, then they are α α necessarily Dirac fermions. Electrically neutral particles such as neutrinos are expected to by Majorana fermions corresponds to a continuous rotation symmetry between on rather general grounds, no matter how they acquire components of ρ, that is, their mass. Phenomenological differences between Dirac 0 0 iα ρ1 cos θ sin θ ρ1 and Majorana neutrinos are tiny for most purposes be- ψD = e ψD =⇒ 0 = (27) cause neutrinos are very light and the chiral weak inter- ρ2 − sin θ cos θ ρ2 actions are well described in the V-A form. This knowl- which result from the mass degeneracy between ρ , ρ , edge is basic material, however, and it is useful to briefly 1 2 indicating that the concept of fermion number is not ba- review them here. sic. Consider the free Dirac field, whose field operator is The mass term in eq. (26) vanishes unless ρ and ρ∗ given by anti-commute, so the quantized field of interest is the Z 3 Majorana fermion at the onset. The solutions to eq. (26) d p 1 X −ip·x ip·x † ψ = use as + vse a (18) (2π)3/2 p 0 s is identical in form to the Dirac field in eq. (18) 2p s Z 3 † d p 1 X −ip·x ip·x † where s = ±1/2, a (p), as(p) are creation and annihila- ψM = √ √ use as + vse a (28). s ( 2π)3 2E s tion operators, and the coefficients us(p), vs(p) satisfy s

µ µ but with u = CvT . The expression in eq. (28) diﬀers (γ pµ − m)us(p) = 0, (γ pµ + m)vs(p) = 0 (19) from the usual Fourier expansion for the Dirac spinor in with γµ being the usual Dirac gamma matrices that obey eq. (18) in two signiﬁcant ways: (i) the spinor is two- the anti-commutation relations component, as there is a chiral projection acting of us(p) and vs(p) and (ii) there is only one Fock space since par- {γµ, γν } = 2ηµν . (20) ticle and anti-particle coincide.

Since the particle and antiparticle are identical for a Ma- jorana neutrino, it feels like ψ(x) in that situation would 5

For completeness, it might be worth noting that two surviving in the beam is independent propagators follow from eq. (26): 2 Pr[ν`|ν`0 (t)] = | hν`|ν`0 (t)i | ∗ µ h0| ρ(x)ρ (y) |0i = iσ ∂µ∆F (x − y), X ∗ ∗ = U`αU`0αU`βU`0β 2 (29) h0| ρ(x)ρ(y) |0i = mσ ∂µ∆F (x − y), α,β

× cos [(Eα − Eβ)t − φ``0αβ] (34) where ∆F (x − y) is the usual Feynman propagator. The first one is the “normal” propagator that intervenes in to- ∗ ∗ where φ 0 = arg U U 0 U U 0 . tal lepton-number-conserving (∆L = 0) processes, while `` αβ `α ` α `β ` β the other one describes the virtual propagation of Majo- In all situations of practical interest, neutrinos are rana neutrinos in ∆L = 2 processes such as neutrino-less highly relativistic, so we are allowed to make the ap- double beta decay. proximation 2 mα Eα = |~p| + (35) B. The PMNS mixing matrix 2|~p| and the time t to be replaced by the distance x traveled To include neutrino mixing in the Standard Model, one by the beam. Thus, should first decide whether new neutrino flavors need to be introduced. For now we suppose that there are 0 X ∗ ∗ 2πx no more neutrino states beyond those in the Standard Pr[ν`|ν`] = U`αU`0αU`βU`0β cos − φ``0αβ Lαβ Model. We can then give a phenomenological descrip- αβ tion of how neutrinos convert from one species to another (36) by adding an explicit mass term in the Standard Model where the oscillation lengths Lαβ, Lagrangian 4π|~p| L ≡ , ∆m2 = m2 − m2 (37) 1 αβ ∆m2 ab a b L = (M ) (ν P ν ) + (h.c.) (30) ab Mν 2 ν ij i L j determine the relevant length scale for which oscillation for some arbitrary complex symmetric 3×3 matrix, where effects stay appreciable. 1 PL = 2 (1+γ5). With massive neutrinos, neutrino oscilla- Observe that if x = κLαβ for some positive integer κ, tions follow when the different neutrino states are allowed 0 to mix. Pr[ν`|ν`] = δ``0 . (38) Consider a neutrino beam created in a charged current interaction along with an antilepton `. We call the neu- This means that the interesting non-trivial effects we trino ν`. In general, this is not a physical particle but a want to observe experimentally happen in that regime superposition of physical fields νa with different masses in between multiples of Lαβ. ma In the simplest case where only two flavors of neutrinos participate in oscillations, we have a rather simple mixing X |νì = U`α |ναi . (31) matrix α cos θ sin θ U = (39) For simplicity, we assume that the 3-momentum of the − sin θ cos θ beam are the same but because the masses are different, the energies of all components can’t all be equal. In this so the survival probability of ν` reduces to case, after time t, the state evolves into 2 0 2 2 x∆m12 Pr[ν`|ν ] = 1 − sin 2θ sin . (40) X −iEαt ` |ν`(t)i = e U`α |ναi (32) 4|~p| α Of course, we know that there are (at least) three neu- where we suppose for the moment that the neutrinos are trinos in Nature so to be more realistic we should con- stable. sider mixing across three generations of neutrinos. For Eq. (32) represents a different superposition of states simplicity, we start with the CP-preserving case where U compared to eq. (31). The probability amplitude of find- is a real-valued matrix. Here the oscillation probabilities ing the original state in the new beam is then given by are

X −iEαt ∗ 0 X 2 hν`|ν`0 (t)i = e U`0αU`α (33) Pr[ν`|ν`] = (U`αU`0α) + α α 2 ! X x∆mαβ since the mass eigenstates are orthonormal, i.e. 2 U`αU`0αU`βU`0β cos 2|~p| hνβ|ναi = δαβ. Thus, at any time t, the probability of ν` α>β 2 ! X x∆mαβ = δ 0 − 4 U U 0 U U 0 sin . `` `α ` α `β ` β 4|~p| α>β (41) 6

The most general mixing matrix excluding CP- C. Oscillations with unstable neutrinos violation effects is given by   The analysis above changes quite dramatically if neu- c12c13 −s12c13 s13 trinos are unstable [15]. To see how the probabilities are U = s12c23 + c12s23s13 c12c23 − s12s23s13 −s23c13 modified, we consider the simple case of two-flavor mix- s12s23 − c12c23s13 c12s23 + s12c23s13 c23c13 ing. Suppose the neutrino states are νe and νµ (42) where cαβ = cos θαβ and sαβ = cos θαβ. This mixing |νì = U`1 |ν1i + U`2 |ν2i , ` = e, µ (48) matrix, which looks similar to the Cabibbo-Kobayashi- Maskawa matrix for quark-mixing weak decays, is called where ν1, ν2 are the mass eigenstates and U is the two- the PMNS matrix—after Pontecorvo, Maki, Nakagawa flavor mixing matrix (39) as before. Suppose that the and Sakata—and it describes the amplitude in which mass mν1 > mν2 then we can consider the decay process a particular neutrino type ν0 participates in a charged- ` −1 current interaction with a charged lepton `. This matrix ν1 → ν2 + X, any X with decay lifetime Γ . (49) differs from the mass matrix in the Lagrangian because of the appearance of extra phases in the charged current Thus, suppose we have some initial state mixing term which introduce CP-violation in neutrino interactions. These Majorana phases disappear in the |νini = cos α |νei + sin α |νµi (50) expressions for probabilities, which is why the PMNS ma- then after time t this state is generally expected to evolve trix is sufficient for neutrino oscillation observations even into though it can not distinguish between the possibilities of

−iE1t −Γt/2 Dirac and Majorana fermions. |ν(t)i = e e sin(θ + α) |ν1i It is interesting to look at a couple of limiting cases, −iE2t X (51) the ﬁrst one being +e cos(θ + α) |ν2i + ck |νk,Xi k x∆m2 12 1 (43) where the last term corresponds to the ﬁnal state of the 2|~p| ν1-decay. Hence, the survival probabilities for both the ν and which leads to e νµ components of the initial beam are given by 0 Pr[ν |ν ] = δ 0 − 4U U 0 ` ` `` `β ` β Pr[ν |ν ] = cos2(θ + α) cos2 θ + e−Γt sin2(θ + α) sin2 θ 2 e in 2 x∆m32 2 × (δ 0 − U U 0 ) sin . (44) 1 t∆m `` `3 ` 3 4|~p| + e−Γt/2 sin 2θ sin 2(θ + α) cos , 2 2|~p| In this case, the survival probability `0 = ` is just like in 2 2 −Γt 2 2 Pr[νµ|νin] = cos (θ + α) sin θ + e sin (θ + α) cos θ eq. (40). 1 t∆m2 The other limiting case of interest is − e−Γt/2 sin 2θ sin 2(θ + α) cos . 2 2|~p| x∆m2 3j 1, (j = 1, 2) (45) In the limit of fast decays, Γt 1, these expressions 2|~p| simplify into where the oscillatory terms in the probability average out 2 2 Pr[νe|νin] = cos (θ + α) cos θ, when we integrate over the energy spectrum of ν`, leading 2 2 (52) Pr[νµ|νin] = cos (θ + α) sin θ, to

0 0 which can correspond to long distances traveled. Pr[ν |ν (t)] = δ ` − 2U U 0 (δ 0 − U U 0 ) ` ` ` `3 ` 3 `` `3 ` 3 To see how this compares with the scenario with stable 2 2 x∆m21 neutrinos, take the case of an initial beam of electron −4U`1U`01U`2U`02 sin 4|~p| neutrinos with α = 0. In the fast decay limit, (46) 4 If we had an electron neutrino, the survival probability Pr[νe|νe(t)] = cos θ, (53) would be given by which can become very small close to π/2. Compare this x∆m2 to eq. (40), which averaging over the distance x gives Pr[ν |ν (t)] = cos4 θ 1 − sin2 2θ sin2 21 e e 13 12 4|~p| + sin4 θ 1 2 1 13 Pr[νe|νe(t)] = 1 − sin 2θ > (54) (47) 2 2 0 which for small θ13 reduces to eq. (40) with ` = ` = e. for all values of θ. Therefore, the physical impact of unstable neutrinos is rather signiﬁcant. 7

D. Oscillations in matter and the MSW effect where terms proportional to the unit matrix are gone because they don’t affect oscillation probabilities. So far our treatment of neutrino oscillations assumes An important special case is when the medium- that the neutrino propagates in vacuum. Because neutri- dependent term becomes maximal, i.e., nos hardly interact, this is a generally good approxima- n G ∆m2 tion. To include medium effects, the standard approach √e F = cos 2θ (62) is to consider changes in the interactions as a change in 2 4E effective neutrino mass, similar to how the refractive in- dex is modified for photons. a condition that defines the resonance conditions, also Consider neutrinos propagating in an environment like called the MSW effect, after Mikheyev, Smirnov and the interior of the sun. We can approximate this environ- Wolfenstein [16]. ment as a uniform background medium if its fluctuations In essence, neutrino propagation in uniform matter is are tiny enough. In this case, a mean field approximation an exercise in degenerate perturbation theory: the de- is sufficient to describe the charged current interactions generacy in the vacuum states of massless νe and νµ is between say the electrons and neutrinos broken by two small effects: the neutrino mass matrix and a medium-dependent term. Therefore, large mixings are possible even with small mixing angles provided that GF µ LC,eff = √ [ieγ (1 + γ5)νe][iνγµ(1 + γ5)e] . (55) 2 these two effects are roughly the same order of magnitude. Using a Fierz transformation, this can be rewritten as When the medium is non-uniform, the situation is not significantly different provided the following adiabaticity GF µ condition is fulfilled: LC,eff = √ [iνeγµ(1 + γ5)νe][ieγ (1 + γ5)e] 2 (56) ˜ 2 +(other terms). dθ ∆m21 (63) dx 2E where we can easily read off the electron 4-current ˜ µ µ where θ is the effective mixing angle. Je = ieγ e. (57) For solar neutrinos, adiabatic evolution through res- µ onance is a fair approximation of what happens during There is also an axial current ieγ γ5e that vanishes in propagation. When these neutrinos hit earth, the total any parity-invariant environment like the Sun’s interior. survival probability for ν goes like The neutrino propagation term in the effective La- e grangian becomes 2 2 2 2 Pr[νe,⊕|νe, ] = sin θp sin θd + cos θp cos θd (64)

GF 0 Lν = √ ne iνeγ (1 + γ5)νe (58) where θp is the medium-dependent mixing angle during 2 solar production and θd is the vacuum mixing angle during detection. The numbers obtained from this analysis where n is the local electron number density. Note that e √ match reasonably well to experimental data to date. Lν shifts the neutrino energies by neGF / 2. None of the other charged-current interaction terms for neutrino vanish. E. Sterile neutrinos To see how the medium-dependent terms affect neutrino propagation, we can look at the two-flavor case. Since the new term merely shift the electron neutrino Another possibility for explaining neutrino oscillations energy, the correction to electron neutrino Hamiltonian come from supplementing the Standard Model neutrinos H is with additional states. In general one might add N Ma- νe jorana neutrino fields s , x = 1, 2, ..., N. Additional fields √ x δH = 2G n |ν i hν | . (59) means more freedom in the kinds of interactions which νe F e e e can be entertained. We assume that the new fermions couple only through the mass matrix and so typically in- This is added to Hν in vacuum, here understood to be e teract more weakly with matter than even the regular in the flavor basis {|νei , |νµi}: neutrinos—this lack of interaction with ordinary matter m2 + m2 ∆m2 − cos 2θ sin 2θ is why they are deemed sterile neutrinos. H = E + 1 2 + (60) νe 4E 4E sin 2θ cos 2θ Sterile neutrinos lead to new terms in the Standard Model Lagrangian of the form where E ≈ |~p| since neutrinos are highly relativistic in 1 µ 1 h all practical situations. Ls = − 2 sxγ ∂µsx − 2 MxysxPLsy The resulting effective mass matrix leads to i (65) +mabνaPLνb + 2µaxνaPLsx + (h.c.) 2 neGF 1 0 ∆m − cos 2θ sin 2θ Heff = √ + (61) 2 0 −1 4E sin 2θ cos 2θ 8

2 where the neutrino mass matrix Neutrino type |Usν | . x ∆m14 . y Solar 0.001 10−4 m µ (66) −3 µT M Reactor 0.1 10 Atmospheric 0.2 10−3 is an arbitrary symmetric (3 + N) × (3 + N) matrix. Supernova 0.01 10 The freedom to choose µ, m, M leads to various mech- Nucleosynthesis 0.1 10−8 anism for neutrino physics so we will only briefly sum- CMB 0.001 10 marize a few possibilities: If lepton number is conserved, then we have N = 3 with neutrino states such that TABLE I. Currently known data imposing limits on existence of sterile neutrinos. |Usν | is the sterile-active mixing element iω −iω 2 PLνa → e PLνa, and PLsa → e PLsa. (67) and ∆m14 is the relevant squared mass difference range in units of eV2 In this case, provided m = M = 0, the mass term is invariant for any choice of µ. Then, it becomes convenient 3. m ≈ µ ≈ M: When all the mass matrices are to group the 6 Majorana fields νa, sa into 3 Dirac fields comparable, the observed data suggests light ster- −2 νa iω ile neutrinos such that m, µ, M ≈ 10 eV form ψa = s. t. ψa → e ψa. (68) sa which we can expect large oscillation effects among all neutrino eigenstates. Because of the lack of evi- We get the usual charged current mixing matrix dence for sterile neutrinos, this model is not favor- able. ig µ LC = √ UaiWµ àγ γLψi + (h.c.) (69) 2 4. m µ M: In this scenario, there are√ N heavy eigenstates with eigenvalues given by M †M as where Uai is the usual PMNS matrix (42). This is iden- √well as 3 mass eigenstates with eigenvalues from tical to the scenario with 3 Majorana neutrinos except M†M, with without the CP-violating phases. Furthermore, there are no off-diagonal couplings in the neutral-current interac- M = µM −1µT + m. (70) tions, which implies that the sterile neutrinos don’t couple to Z0. The heavy eigenstates have mixing angles with ster- Without the CP-violating phases, the Dirac neutrino ile neutrinos of the order O(µ/M). The light eigen- model works precisely the same as the Majorana neu- states are almost pure flavor eigenstates but also trinos because of helicity suppression: the ratio of am- have order O(µ/M) sterile mixing. This way of ob- plitudes for detecting νa to sa is typically of the order taining neutrino masses is called the see-saw mecha- p0/m, which in the lab-frame corresponds to the proba- nism because increasing the mass from M decreases 12 bility of νa detection at least 10 times more likely than the mass from M, and vice-versa. s scattering. It is this very small probability of getting a Table (I) summarizes the constraints on the existence a right-handed sterile neutrino that makes them practi- of sterile neutrinos based on different experiments and cally irrelevant in the neutrino phenomenology. observations [9]. More generic sterile neutrinos can be distinguished according to the relative sizes of µ, m, M: 1. µ m, M: Sterile neutrinos do not mix with the IV. EXPERIMENTAL STUDIES OF NEUTRINO ordinary neutrinos in any significant way; therefore, MASS AND MIXING for all practical purposes they don’t couple at all with the usual Standard Model particles. Here we categorize the experiments aiming to measure the neutrino mass and to test neutrino oscillations 2. µ m, M: This case reduces to the Dirac neu- according to type of experiment or observations being trinos above when m = M = 0. If N > 3, this made. First, there are direct kinematic searches for neu- scenario leads to 3 Dirac neutrinos whose squared trino mass and experiments looking for evidence of neu- masses are the eigenvalues of µµ†, plus (N − 3) trinoless double beta decays. The present limits on the massless sterile neutrinos which don’t participate neutrino masses obtained from such studies, assuming in weak interactions. If m, M are not exactly zero, the normal hierarchy of masses, are: then we have a perturbation of the Dirac case. The

almost degenerate pairs of neutrinos are close in mν2 < 190 keV and mν3 < 18.2 MeV. (71) mass that should become particle-antiparticle pairs in the vanishing limit. For the same reason, the The best limits for the mass of the lightest neutrino, in- mixing angle of these nearly degenerate states is stead, have been obtained from the Mainz and the Troitsk also almost maximal. From available data, such experiments, where pseudo-Dirac neutrinos are ruled out for m, M & −9 10 eV. mν1 < 2.2 eV. (72) 9

The search for neutrinoless double beta decays is impor- For more detail, let’s consider the matrix element for tant because the observation of these decays would be the decay, a clear indication in favor of a Majorana neutrino, as- −iγ upµ + m suming CPT invariance holds. The Heidelberg-Moskow M ∼ 2G2 W µν uγ P m P γT uT collaboration published a result in 2001 and later in 2004 F µ L p2 + m2 L ν showing at least a 2.2σ signal for neutrinoless double beta (76) decay [17], but this result has been strongly contested and where u, uT are external electron spinors and W µν is the T T its validity is still an open question. weak current matrix element. The factor PLγν u comes The second group of experiments are designed to in- from antiparticle considerations. Because vestigate neutrino ﬂuxes from natural sources: solar and m atmospheric. These observations are historically impor- PLγ uPL = 0 (77) tant for being providing the earliest evidence for neutrino mixing and they remain highly relevant in astrophysical only the mass term in the middle factor contributes, giv- and cosmological research. ing rise helicity ﬂips and ∆L 6= 0. Lastly, we also look at experiments that employ neu- For small neutrino mass, the decay rate is suppressed 2 2 trino beams produced at accelerators and nuclear re- by a factor |Uejmj| , which represents a factor of at least actors. They are usually divided in long- and short- 1012 smaller in probability compared to its 2ν counter- baseline, according to the distance between the neutrino part . Despite being partially compensated by other fac- production point and the detector. Many short base- tors such as having a larger phase space to work with, line accelerator experiments weren’t able to detect any this tiny relative probability of neutrinoless double beta signal for neutrino oscillations. Nonetheless, they are decay makes it hard to observe experimentally. important, because they provide us with constraints on The neutrinoless mode can be distinguished experi- the possible values of the neutrino mixing parameters. mentally from the 2ν mode by examining the energy The most important limits have been obtained by NO- spectra of the daughter electrons. In neutrinoless dou- MAD and CHORUS at CERN, which had experiments ble beta decay designed to detect τ production from a νµ beam, indi- E + E = Q, (78) cating νµ–ντ oscillations [24]. 1 2 where Q ≈ 2 MeV is the typical decay energy for the A. Neutrino-less double beta decay mode, whereas in the 2ν case,

E1 + E2 < Q (79) Double beta decay is a higher-order nuclear process wherein a nuclei with charge Z changes into another with since there is a distribution of electron energies depending Z + 2 without modifying its atomic mass A. In nature, on how much energy goes to the neutrinos. To observe there are 35 isotopes known to have the right ground this diﬀerence, we would need to examine the electron state conﬁguration for the decay reaction double beta decay spectrum near its end point at high

− energy resolution. The neutrinoless decay mode is also (Z,A) → (Z + 2,A) + 2e + 2νe, (73) sensitive to the different CP-violating phases eiα that appear in the charged current interaction for Majorana which can be seen as two simultaneous neutron beta de- neutrinos. cays. Another decay mode for double beta decay was Experimentally, the search for neutrinoless double beta proposed by Giulio Racah in 1937 and Wendell Furry in decay relies on finding a peak in the region below 4.3 1939: MeV. Common to all experiments is the low-background (Z,A) → (Z + 2,A) + 2e−, (74) environment due to long expected half-lives. The types of experiments being made are a process which clearly violates lepton number conserva- 1. Ge semiconductor devices: Heidelberg-Moscow col- tion and is forbidden by the Standard Model. It can be laboration, IGEX; viewed as a two-step process 2. Cd-Zn-Te detectors: COBRA in the Gran Sasso (Z,A) → (Z + 1,A) + e− + ν , e (75) Underground Laboratory; (Z + 1,A) + ν → (Z + 2,A) + e− e 3. Cryogenic bolometers: CUORINCO at Gran Sasso; where a neutron first undergoes ordinary beta decay and 4. time-projection chambers: NERMO-3 in the Frejus then if νe = νe indicating we have Majorana neutrinos, Underground Laboratory. then the second process can occur after that. Moreover, to allow helicity matching, it is necessary to have mν > 0 So far, there is no compelling evidence to suggest that so that the neutrinos don’t possess fixed helicities. neutrinoless double beta decay does indeed occur. Esti- mates on the decay lifetime lead to corresponding limits on the neutrino mass, and some of the key experimental results available are shown in table (II). 10

limit Isotope Half-life (yrs) mν (eV) because Homestake found a discrepancy in the solar neu- 48 48 21 20Ca→22Ti 9.5 × 10 8.3 trino flux of more than 60% that predicted by the standard solar model [20]. The updated value of the double 76Ge→76Se 1.9 × 1025 0.35 32 34 ratio R for the chlorine experiment, 76 76 25 32Ge→34Se 0.7 × 10 0.6 82 82 23 (µ/e)data 34Se→36Kr 2.1 × 10 2.3 R = , (82) 100 100 23 (µ/e)MC 42 Mo→44 Ru 5.8 × 10 1.2 116 116 23 48 Cd→50 Sn 1.7 × 10 1.7 where MC stands for Monte Carlo predictions of the ratio 128 128 24 52 Te→52 Xe 7.7 × 10 1.1 (µ/e) of muon-electron events, is 130 130 24 52 Te→54 Xe 3.0 × 10 1.0 R = 0.34 ± 0.03. (83) 136 136 23 54 Xe→56 Ba 4.4 × 10 2.3 150 150 21 The advantage of considering (µ/e) is that the flux uncer- 60 Nd→62 Sm 2.1 × 10 4.1 tainties do not affect the result. If there are no neutrino TABLE II. Decay lifetime estimates and the corresponding oscillations, then R = 1 [21]. neutrino mass limits. The Homestake result was later verified in similar experiments using gallium, which has a lower energy thresh- −2 −1 Chain Reaction Φ⊕ (cm s ) old (Ethr ≈ 233 keV), making them sensitive to the main pp 5.95 × 1010 pp component of the solar neutrino flux. The gallium results indicated that pep 1.40 × 108 pp 3 hep 9.30 × 10 R = 0.60 ± 0.05 (SAGE), 7Be 4.77 × 109 (84) R = 0.58 ± 0.05 (GALLEX, GNO). 8B 5.05 × 106 13N decay 5.48 × 108 These observations confirmed the existence of the solar CNO cycle 15O decay 4.80 × 108 neutrino puzzle. 17F decay 5.63 × 106 Essential improvement in solar neutrino observations came with the water Cerenkov experiments, TABLE III. Solar process that produce neutrinos and the cor- Kamiokande, and Super-Kamiokande that probed into − − responding fluxes. the elastic scattering νe + e → νe + e and confirmed the electron neutrino deficit [18]. The real breakthrough, however, with the SNO deterium Cerenkov experiment B. Solar neutrinos in 2001, which studied three processes simultaneously: charged-current, neutral current and elastic scattering Neutrinos play an essential role in stellar evolution be- reactions. During its first phase, SNO observed the cause of reactions such as charged current and elastic scattering events, with an energy threshold for electron detection of 6.75 MeV. The + 4p → α + 2e + 2νe + 28 MeV (80) flux measured from the charged current was

6 −2 −1 which represents a series of successive processes that lie Φνe,C = 1.75 ± 0.07 × 10 cm s (85) at the core of neutrino astronomy. Because neutrinos hardly react with any intervening matter, it is more ad- which corresponded to a double ratio of vantageous to examine solar neutrinos as opposed to solar radiation, since the former carry valuable information R = 0.35 ± 0.03. (86) about the core of stars, allowing us indirect access to whatever happens in there. The main processes respon- The observations at Super-Kamiokande didn’t agree with sible for neutrino production [8] are listed in table (III). the νe flux found at SNO, which found Φ = 3.69 ± 1.13 × 106 cm−1s−1. (87) The earliest study of solar neutrinos was the Homes- ν take experiment [19] headed by Raymond Davis, Jr. and This result was the first evidence (at 3σ level) of the John Bahcall in the late 1960s, which used the inverse presence of different flavored neutrinos in the solar neu- decay on chlorine trino beam, providing the most robust evidence of into 37 37 − other active neutrinos. It is also remarkable that the Cl + νe → Ar + e , (81) sum of the νe and νµ fluxes give a total flux value that agrees well with Standard Model predictions, which ac- with threshold energy Ethr ≈ 0.81 MeV. Hence, it was sensitive to the pep, 7Be, 8B and hep components of the tually strongly disfavors the hypothesis that these active solar neutrino flux. The results were really surprising, neutrinos oscillate into sterile ones. 11

Experiment Double ratio R Experiments indicate that for the electron, αe = 0 while Kamiokande 0.60 ± 0.06 ± 0.05 αµ < 0 for high momenta. This evidence suggests that IMB 0.54 ± 0.05 ± 0.12 R < 1 is a consequence of a reduced muonic neutrino flux, Sudan2 0.69 ± 0.19 ± 0.09 which could be due to oscillations whose probabilities are enhanced by the interaction with matter. Frejus 0.87 ± 0.21 NUSEX 0.99 ± 0.40 D. Reactor experiments TABLE IV. Double ratios for atmospheric neutrino observations. Nuclear reactors have long played a pivotal role in neu- sub-GeV 0.638 ± 0.16 ± 0.050 trino research, from their first detection to the most re- Super-Kamiokande cent neutrino oscillation experiments. Reactors are high multi-GeV 0.658 ± 0.030 ± 0.078 − intensity, isotropic sources of νe, a product of β decay TABLE V. Best available data for R. of neutron-rich fragments of uranium-plutonium fission. Almost all reactor experiments detect antineutrinos via inverse beta decay C. Atmospheric neutrinos + νe + p → e n. (90) Atmospheric neutrinos are useful both in the direct The observed energy spectrum for inverse beta decay has study of neutrino oscillations and as the background and a peak around 3.6 MeV. Inverse beta decay occurs only calibration beam in the search for neutrinos from astro- for electron antineutrinos with sufficiently high energies physical sources [22]. Interactions of cosmic ray protons ( 1.8MeV) so only roughly a quarter of the fission events with nuclei in the upper atmosphere leads to various pro- & with ν is detected from the reactor core. cesses that produce electron and muon-type neutrinos e When Reines and Cowan first detected the neutrino, that belong to the following chain of reactions: they were looking for antineutrinos from a nuclear reac- p + X → π± + Y tor [23]. In an initial experiment in 1953, they saw an antineutrino signal from inverse beta decay as the de- π± → µ± + ν (88) µ layed coincidence between a positron and neutrino cap- ± ± µ → e + νe + νµ ture in cadmium, releasing ∼ 8 MeV in gamma rays. where we don’t distinguish between neutrinos and an- Because the experiment had a high background(signal- tineutrinos, as what is done in experiments. The π± can to-noise: 1/20) the results were inconclusive: 0.41 ± 0.20 be replaced by K±. A simple counting argument shows events/minute. In 1956 the experiment at Savannah River yielded bet- that r = νµ/νe = 2. The values for the double ratio R in early experiments (82) can be found in table (IV). The ter observations. There, the inverse beta decay produced error bars account for the statistical and systematic er- two signals: (i) a prompt signal from the positron anni- rors, respectively. The first two experiments in the list hilation, creating 2 gamma rays with energy 0.51 MeV employ Cerenkov detectors while the last three used iron and (ii) a delayed neutron capture with the ∼ 8 MeV bar calorimeters. In the water Cerenkov detectors, the gammas. With a signal-to-background ratio of 3 to 1, muons are distinguished from the electrons either (i) by they reported a cross-section of the size of the Cerenkov light rings or (ii) by observing σ = 6.3 × 10−44cm2 (91) the products of muon decay. The flux observations for atmospheric neutrinos which agreed with the theory at the time. In 1959 parity were verified later on with experiments by the Super violation was found and Reines and Cowan revised their Kamiokande collaboration in the 1990s, which obtained analysis to include parity violation effects and arrived at the best known results for the double ratio, shown in table (V). +7 −44 2 σ = 12−4 × 10 cm (92) Notice that the first three experiments in table (IV) implies either muon depletion or an excess of electron which matched the new theoretical predictions. events. To decide which, we can examine the up-down As was established before, neutrino oscillations are el- symmetry of detection events. Up (U) refers to events egantly described by a picture where flavor eigenstates ν` where the neutrino crossed the earth before hitting the are viewed as admixtures of mass eigenstates νj, which detector while down (D) refers to events where the neu- undergo mixing according to the unitary matrix (42). trino directly comes from the atmosphere to the detector. The PMNS matrix is typically expressed in terms of Euler This leads to an observable quantity angles θ12, θ13, θ23 and depending on model, may include CP-violating phases αi as well. U − D The oscillation probability (40) shows the θ depen- α` = . (89) U + D dence, which in almost all neutrino studies typically involve the disappearance of electron or muon neutrinos. 12

ideal For reactor studies, experiments are often designed to be Detector M(kton) νe? νµ? ντ ? Eν most sensitive to the parameter θ13, that is, lightest to Liquid Ar 0.6 Y Y - huge heaviest mass transitions. H O Cereknov 50 Y Y - < 2 GeV 2 2 The best current limit on sin 2θ13 comes from the Emulsion/Pb/Fe 0.27 Y Y Y > 0.5 GeV CHOOZ collaboration [25]. The CHOOZ experiment Scintillator 1.0 Y Y - huge uses a single detector with a 5 ton target of GF-loaded liquid scintillator at a distance 1.05 km from two 4.2 Steel 5.4 - Y - > 0.5 GeV GW reactors. It observed a prompt signal from the th TABLE VI. Detector types used or planned in accelerator positron and delayed signal from neutron capture that neutrino studies. M indicates largest mass to date. ν`? indi- yielded κ = 4, 5 gammas with energies ∼ 8 MeV: cates event identification. n +m Gd → m+1Gd∗ → Gd + κγ. (93) small electron neutrino component from kaon and muon The Gd serves to shorten the neutron capture time to decay. The π themselves are obtained by first produc- 300 µs, increasing the neutron signal while simultane- ing an intense beam of protons then putting a long, thin ously reducing background effects. target material in the path of the protons—long so the The ratio of observed to predicted flux is interaction time for creating π is increased while thin so that secondary interactions of the π with the target are r = 1.01 ± 0.028(stat) ± 0.027(sys). (94) reduced. Focusing a neutrino beam requires a strong magnetic For ∆m2 = 2.5 × 10−5 eV2, the CHOOZ result corre- field with an integrated path length proportional to the sponds to radius at which the particles enter the focusing sys- 2 tem. The standard design for a focusing system fea- sin 2θ13 < 0.15 at 90% confidence level. (95) tures a parabolic horn: an inner conductor shaped like a parabola holding the magnetic field between inner and E. Accelerator-based studies outer conductors such that the length of the field at radius r is proportional to r2. Typical currents running 5 The pioneering accelerator experiments on neutrino os- through such a horn are ∼ 10 A. cillations were designed to look for oscillations similar Conventional neutrino beams are produced containing to quark mixing, that is for small mixing angles. CHO- νµ and a small contamination of νe. To measure νµ → νe RUS and NOMAD are two recent experiments that found transitions, we must distinguish between not only between electrons and muons but also between electrons no oscillations between νµ and ντ , setting the oscillation 0 probability limit to below 1 percent. There have also and π , which constitute a large fraction of neutrino in- been experiments (K2K, MINOS) that unambiguously teractions at & 100 MeV. Neutrinos that come from the measured the disappearance of ν , which would corre- decay of muons present an entirely different challenge be- µ cause we should also keep track of the muon charge—a spond to ∆m2 = 3 × 10−3 eV2. flip in the charge indicates ν → ν —so it is crucial that The accelerator LNSD experiment has reported evi- e µ the muon charge is not misidentified. There are various dence of muon to electron neutrino transition at ∆m2 = detectors used to observe neutrinos and we summarize 0.1 eV2. If this result is verified, it would call into ques- the most important ones for current and future experi- tion the current number of neutrinos and strongly suggest ments in table (VI). the existence of other neutrino flavors. What is known at With beamlines and detectors in place, oscillation the moment is that the LNSD signature can be excluded probabilities are measured based on the following rela- as being due to a simple oscillation scenario, as shown by tion: the MiniBooNE experiment. One of the key issues in accelerator neutrino studies Nfar = Φν σν Pr[ν`|νµ]`Mfar + Bfar (96) wishes to address is the existence of sterile neutrinos, So- µ ` lar and atmospheric evidence already suggest that two of whereNfar is the number of events at a far detector, Φνµ the three mixing angles are large. The third mixing angle is the muon neutrino flux, σν` is the ν` cross-section, Mfar hasn’t been measured but the results from the CHOOZ is the far detector mass, ` is the detector efficiency for ◦ experiment imply that it has to less than 8 . If the third signal ` and Bfar is the predicted background events. mixing angle is non-zero, it follows that there will be To reduce systematic uncertainties, it is conventional CP-violation in the lepton sector. to place an additional detector close to the neutrino In general, neutrino beams originate from charged par- source, at a distance before oscillations can occur. Then ticles that decay into neutrinos. To isolate the neutrinos, one predicts the number of events in the far detector the charged particles are magnetically focused to travel in terms of the ratio between the near and far detector in one direction and given enough space in which to de- background events: cay while being in flight. Conventional beams start with pions that yield muon neutrinos or antineutrinos, with a Sfar Bfar = Nnear (97) Snear 13

Experiment νµCC NC νe ντ CC Signal F Parameter Best ﬁt 2σ 3σ 2 −5 2 K2K (2006) 0 1.3 0.4 0 1 0.6 ∆m12 (10 eV ) 7.6 7.3 – 8.1 7.1 – 8.3 2 −3 2 MINOS (2006) 5.6 39 8.7 4.7 29.1 3.1 |∆m13| (10 eV ) 2.4 2.1 – 2.7 2.0 – 2.8 2 OPERA (2003) 1 5.2 18 4.5 10 1.6 sin θ12 0.32 0.28 – 0.37 0.26 – 0.40 2 T2K (2001) 1.8 9.3 11.1 0 103 9 sin θ23 0.50 0.38 – 0.63 0.34 – 0.67 2 NOvA (2004) 0.5 7 11 0 148 11.5 sin θ13 0.007 ≤ 0.033 ≤ 0.050

TABLE VII. Signal and background rates for typical long TABLE VIII. Three-flavor neutrino oscillation parameters baseline νe search. νCC, νe and NC refer to various possible from global analysis of solar, atmospheric, reactor (Kam- background events. F is the figure of merit for sensitivity to LAND and CHOOZ) and accelerator (K2K and MINOS) ex- oscillation events, defined as signal divided by the square root periments. Includes best fit estimate and 2σ and 3σ ranges. of sum of signal and background events.

Model corresponds to massless neutrinos but evidence P 0 where Sj = ` Φν` σν` `` Mj for j = near, far and `` for neutrino oscillations tells us neutrinos must possess describes the efficiency of identifying ν` as the signal ν`0 . mass. If we stay with the same gauge group, it is still pos- The most important information gathered from sible to conjecture extra fermions or Higgs bosons that accelerator-based experiments has been on νe → νµ os- will predict massive neutrinos. This provides us with the cillations. The standard exposure for optimal analysis of simplest extensions of the Standard Model and is usually electron neutrino appearance in a muon neutrino beam is achieved via the see-saw mechanism. shown in table (VII). The signal rates indicated assume Of course, one may choose to expand the gauge group 2 2 −3 2 sin θ13 = 0.1 and ∆m ∼ 2.75 × 10 eV . into something where symmetry breaking leads to the Standard Model gauge. This is achieved in models at- tempting to unify all forces at short distance, the so- F. Summary of current experimental data called grand unified theories, the most popular of which involve SU(5), SO(10) and E6. Due to space limitations, The developments in neutrino oscillation experiments we will briefly cover only how neutrino masses are gen- in the last few years has given us a rough picture of the erated using the first two gauge groups. parameters governing three-flavor oscillations: There are For completeness, we mention that there is also a two squared mass differences separated by a factor ≈ class of Standard Model extensions that lead to non- 30, there are two large mixing angles (θ23, which could vanishing neutrino masses by introducing a symmetry ◦ be as much as 45 , and θ12, which is large but almost between bosons and fermions known as supersymmetry, certainly smaller than 45◦, according to data known at which is desirable because it provides a direct solution to high significance level), and one mixing angle which must the hierarchy problem in the Standard Model. be small (θ13). Present data is consistent with 2 possible ordering of masses for neutrinos, typically parameterized 2 A. The see-saw mechanism by the sign of ∆m13: 2 1. In the normal hierarchy (∆m13 > 0) the mass state One strange property of the Standard Model is that it which contains predominantly νe has the smallest includes left and right chiral projections of all fermions mass; except neutrinos. Thus, it seems that a natural way to 2 extend it is to add N right-handed neutral fields N`R with 2. In the inverted hierarchy (∆m13 < 0), νe is part of a nearly degenerate doublet of mass states which zero hypercharge and which are assumed to be SU(2)L is separated from the lightest neutrino mass by singlets. These fields have no interaction with gauge bosons but will have non-trivial effects due to their other |∆m13|. properties. A global analysis of presently available neutrino oscil- The simplest model considers N`R to be the right- lation data is depicted in fig. (1) and summarized in handed components of a Dirac neutrino: table (VIII). ! ν`L ν` = . (98) N`R V. MODELS INCORPORATING NEUTRINO MASS The N`R fields give rise to extra mass terms in the La- grangian The Standard Model is based on the gauge group SU(2) × U(1) . But this only fixes the gauge bosons X L Y −LM = M``0 ν`LN`0R + (h.c.) (99) in the theory—fermions and Higgs content are chosen `,`0 somewhat arbitrarily. The choice made in the Standard 14

FIG. 1. Best ‘solar’ (left) and ‘atmospheric’ (middle) neutrino oscillation parameters determined from a combined analysis of various neutrino data sources as of 2007. Also, a global analysis of the neutrino data established some constraints on θ13 (right). Image courtesy of [26].

√ where M``0 = v/ 2 f``0 is an N × N mass matrix. In leads to a mass matrix of the form general, M``0 isn’t diagonal in the flavor basis so that the ! fields ν ,N 0 do not correspond to physical fermion 0 M `L ` R M = . (106) fields. MB The physical fields are obtained from the eigenvectors of M. If we diagonalize M such that The matrix M has 2N Majorana neutrinos in general since M and B are N × N matrices. † U MV = m, m = diag(m1, ..., mN ) (100) To see further implications, suppose we look at the simplest case, N = 1. Then assuming M,B ∈ R, B > 0, then we can write we can choose some rotation matrix X X ν = U v , ν = U v . (101) ! `,L `α αL `,L `α αL cos θ − sin θ 2M α α O = , tan 2θ = (107) sin θ cos θ B This allows us to rewrite eq. (99) as such that X LM = ναLmαναR + (h.c.), (102) ! −m 0 α OMOT = 1 (108) 0 m2 which shows that να are the mass eigenstates with mass mα. and Neutrino mixing follows from eq. (4) for our Dirac neutrino here when we write it in terms of the mass eigen- 1 p m = B2 + 4M 2 ∓ B . (109) states: 1,2 2

g X X + µ − This isn’t quite the diagonalization we want because of −LC = √ `L γ U`αναLWµ + (h.c.) (103) 2 ` α the minus sign so if we introduce K = diag(i, 1) then we can write which shows an expression analogous to quark mixing. The problem with the Dirac neutrino model is that it M = OT mK2O. (110) makes no restrictions on the size of f``0 , which consequently means it doesn’t guarantee that neutrino masses Introducing column vectors will be small. This can be ﬁxed by considering instead ! ! ! ! ˆ n ν n νˆ Majorana neutrinos. If we also introduce the ﬁelds N`L 1L ≡ O L , 1R ≡ K2O R (111) conjugate to N`R, then we get extra bare mass terms n2L NˆL n2R NR 1 X −L = B 0 Nˆ N 0 + (h.c.), (104) we get a mass term involving M in the Lagrangian that B 2 `` `L ` R `,`0 can be reduced into which with the identity −LM = m1n1Ln1R + m2n2Ln2R + (h.c.) (112)

where n1 = −nˆ1, n2 =n ˆ2, showing n1 and n2 are Majo- ν N 0 = Nˆ 0 νˆ (105) `L ` R ` L `R rana neutrinos. In general, we get 2N Majorana neutrinos. 15

This model now explains why active neutrino masses With B − L = −2 the electric charge of the triplet ∆~ is should be small. Recall that M comes from Higgs cou-   pling and so it is natural to assume it is the same order of ∆0 magnitude as the other fermions in the same generation, ~  −  ∆ =  ∆  (118) i.e. the charged leptons in the case of neutrinos. Suppose ∆−2 B M, then and additional Yukawa coupling outside those already in M 2 m ≈ , m ≈ B, (113) the Standard Model: 1 B 2 X 1 ~ ˆ from which it follows that m M. Thus, if the active −LY = f``0 ψ`L √ ~σ · ∆ψ`0R + (h.c.) (119) 1 2 neutrino masses are small, the sterile neutrino masses `,`0 have to be big in compensation. This is exactly the see- saw mechanism at work. where ~σ denotes the vector of Pauli matrices. ~ However, there are cosmological arguments which limit The Higgs potential now involves both ∆ and the usual doublet φ. Let us assume that the parameters in this the mass of any stable neutrinos to . 1 eV. If we believe potential are such that its minimum corresponds to ντ to be stable then we can take M ∼ mτ and ﬁnd

9 v2 v3 B 5 × 10 GeV. (114) hφ0i = √ , h∆0i = √ , (120) & 2 2 2 Comparing this to the weak scale of ∼ 10 GeV, we see a which after symmetry breaking leads to the mass term huge gap in energy scales. This is known as the hierarchy problem and it appears in any conventional grand unified X −L = ν M 0 νˆ 0 + (h.c.) (121) theory. M `L `` ` R `,`0 √ where M 0 = v f 0 / 2. B. Expanding the Higgs sector `` 3 `` Using the conjugate field we can rewrite eq. (119) as If no new fermions are included in the Standard Model, X ˆT −1 1 ~ ˆ then we only have two degrees of freedom that correspond −LY = f``0 ψ`RC √ ~σ · ∆ψ`0R + (h.c.). (122) `,`0 2 to uncharged fermions ν`L, νˆ`R, which if they have mass must necessarily be Majorana particles. This also implies ˆ B − L violation. B − L can be recovered is new Higgs In this form, it is clear that the bilinear involves ψR twice bosons are introduced to compensate for the B − L due and therefore it must obey Fermi statistics. Note that to neutrinos. f``0 = f`0` makes the mass matrix symmetric. Given the lepton fields The mass eigenvalues and eigenstates are obtained from diagonalizing M. The couplings f``0 , however, are ! unspecified so the definite pattern of neutrino mixing ν`L ψL = (115) can’t be uniquely determined. In this case, neutrinos `R are light because v3 v2. This can be seen from the observation that since ∆ couple to W and Z with antiparticles given by 0 M 2 1 + 2(v /v )2 ˆ ∗ ˆ ∗ ρ ≡ W = 3 2 (123) ψR = γ0CψL (2, 1), `L = γ0C`R (1, 2) (116) 2 2 MZ cos θW 1 + 4(v3/v2) where = iσ2 is a 2 × 2 antisymmetric matrix, we can where using experimental bounds on ρ: form fermion bilinears that have non-vanishing B − L quantum numbers: v3 < 0.07. (124) v2 ψ ψˆ (1, 2) ⊕ (3, 2), `ˆ ` (1, −4). (117) L R L R Going back to the question of B − L symmetry, the The Higgs multiplets that can directly couple to these Yukawa coupling eq. (119) as such does not show B − L bilinears to form gauge invariant Yukawa couplings are violation. However, once this quantum number is car- ried by ∆,~ it is broken when ∆0 acquires a non-vanishing 1. triplet field ∆~ (3, −2), vacuum expectation value. In fact, it breaks B − L by 2 units, exactly what is needed for generating the Majo- 2. single-charge singlet (1, −2), and rana mass terms. 3. double-charge singlet (1, 4). Next we consider the second option: a model with a single-charged singlet h− first proposed by Anthony Zee in 1980 [29], which involves the Yukawa coupling

X ˆ − −LY = f``0 `Lψ`0Rh + (h.c.). (125) `,`0 16

Because h has an electric charge, its vacuum expectation where value must vanish, leading to a prediction of massless ! ! neutrinos. The situation changes, however, if there are Z d4p Z d4q 1 1 I`1`2 = 4 4 2 2 2 2 more than one Higgs doublets in the theory. For instance, (2π) (2π) p − m` q − m` 0 1 2 if there is a second doublet φ then the trilinear coupling 1 1 1 × p2 − m2 p2 − m2 (p − q)2 − m2 µφT φ0h + (h.c.) (126) h h k (133) yields the B − L violation we want. so that the eigenvalues of M are naturally small because The mass terms are particularly simple if we assume of the suppression factors in such two-loop integration that only one of the Higgs doublets couples to leptons. terms. In this case, An interesting prediction from this model follows from the asymmetry of f``0 : det(M) = 0 =⇒ m3 = 0 if there 2 02 are three generations. The third mass isn’t necessarily M``0 = κf``0 m` − m` (127) zero because there are higher-order perturbation terms to where the coupling constant is f``0 = −f`0` so M is sym- consider but at this level it already suggests that m3 metric. m1, m2 for three neutrino mass eigenstates. Interestingly, this model comes with a pattern of neutrino mixing, unlike the previous models discussed above. If we deﬁne C. Grand uniﬁed theories

2 ! 2 fµτ mµ feµ mµ Evidence for new physics such as dark matter and tan α = 1 − 2 , τ = 2 cos α, (128) feτ mτ feτ mτ dark energy show some of the limitations of the Stan- dard Model. Various attempts at extending it include then the mass matrix M in the flavor basis is grand unification theories (GUTs) that imagine scenar-   ios wherein matter and energy are unified at some funda- 0 τ cos α mental energy scale and where this overarching symmetry   is spontaneously broken at the low energy scales where M = m0  τ 0 sin α (129) cos α sin α 0 the Standard Model holds. Here we briefly examine how GUTs incorporate neutrino masses in the two most pop- 2 ular grand unification groups, SU(5) and SO(10). where m0 = κmτ fτe/ cos α. Diagonalization of M here gives the physical masses; unfortunately, the predictions In the Georgi-Glashow SU(5) model, the Standard of the model don’t agree with the most updated oscilla- Model gauge groups are combined into a single gauge tion data. group SU(5). Fermions are assigned to 10 and 5 repre- The last option we consider is a model with a double- sentations that can be denoted by charge singlet, which can have Yukawa couplings of the  ˆ    form d1 0u ˆ3 −uˆ2 u1 d1 ˆ d2 −uˆ3 0u ˆ1 u2 d2  X 0 +2     0 ˆ  ˆ    −LY = F`` `L`Rk + (h.c.). (130) F = d3 ,T =  uˆ2 −uˆ1 0 u3 d3  `,`0    +  e  −u1 −u2 −u3 0 e  ν −d −d −d −e+ 0 The field k+2 has a B − L quantum number of 2 but it L 1 2 3 L turns out that no amount of additional Higgs doublets (134) will lead to the B − L violation needed to generate neu- wherex ˆ denotes the conjugate of x, e.g.u ˆL is the CPT- trino masses. conjugate state to the right-handed helicity state of the What is required is the addition of a single-charge up quark. scalar of the same nature as h above [28]. With h− and The breakdown of the gauge symmetry to SU(3)C × k+2, we have a trilinear coupling U(1)Q is achieved by two Higgs multiplets H ≡ 5 , Φ ≡ 24 with the vacuum expectation value of Φ is chosen so µh−h−k+2 + (h.c.) (131) that 3 3 which breaks the B − L symmetry. Neutrino masses are hΦi = diag V,V,V, − V, − V (135) generated in this scenario from 2-loop diagrams, which 2 2 give rise to a mass matrix where V is the unification scale obtained from unifying X the three gauge couplings at low energies. M 0 = 8µ m m f F f 0 I (132) `` `1 `2 ``1 `1`2 ` `2 `1`2 If we write the gauge bosons of SU(5) as `1,`2  q  √1 P λaG + 2 B Z 2 a a 15 24  q  (136) Z−1 √1 ~σ · W~ − 3 B 2 10 24 17 where would naturally explain small neutrino masses when

4 1 µS ∼ O(V ).  3 3  X Y 4 4 4 ! The next symmetry used in GUTs is SO(10), ﬁrst dis- 1 1 − 3 − 3 − 3 4 1 X X X 3 3 −1 1 2 3 covered by Howard Georgi a few hours before ﬁnding the Z = X Y  , Z = 1 1 1  2 2  − 3 − 3 − 3 4 1 Y Y Y SU(5) model in 1973, and independently developed by 3 3 1 2 3 X3 Y3 Harald Fritzsch and Peter Minkowski in 1974. An ele- (137) gant aspect of SO(10) is that it contains the left-right we get that symmetric gauge group SU(2)L × SU(2)R × SU(4)C and 25 thus, it automatically contains a right-handed neutrino, M 2 = M 2 = g2V 2. (138) Furthermore, because quarks and leptons belong to the X Y 8 same irreducible representation, neutrinos acquire masses

The final stage to get to U(1)Q occurs via H: naturally through the same mechanism as the quarks and other leptons. v T The spinor representation of SO(10) is the 16 - hHi = 0, 0, 0, 0, √ (139) dimensional SO(10) which has a maximal subgroup 2 SU(4)C × SU(2)L × SU(2)R × D where D is a discrete which leads to symmetry that essentially plays the role of charge conjugation on fermion fields. Known as D-parity, this symme- 1 gv ˆ ¯ MW = gv, MZ = (140) try maps fL → fL, interchanging the (2, 1, 4) and (1, 2, 4) 2 2 cos θW sub-multiplets of the SO(10) spinor. It plays an important role in understanding neutrino mass in left-right where θ is the Weinberg angle. With a single gauge W symmetric models. coupling constant, the unification scale is predicted to Symmetry breaking is usually achieved with a combi- correspond to nation of Higgs fields: choose 45 H or 54 H and then add either 16 and 16 or 126 and 126 . Details are r3 H H H H tan θW = . (141) too involved to present here so the interested reader is 5 referred to [31]. Now since 16 ⊗ 16 = 10 ⊕ 120 ⊕ 126 , the mass- However, it turns out that at the scale MZ , experimental data show that the three gauge couplings don’t extrap- generating Higgs bosons must belong to 10 , 120 or 126 olate to the same point at high energy scales. SU(5) dimensional representations of SO(10). Fermion mass in is not a viable GUT model unless there are more Higgs SO(10) originate from the Yukawa coupling bosons introduced, which would ruin the simplicity and X ab T −1 i predictive power of the theory. Nonetheless, SU(5) has LY = f10 ψa BC Γ ψb10 i proven to be an excellent training ground for understand- a,b ab T −1 i j k ing generic features of GUT models. +f120 ψa BC Γ Γ Γ ψb120 ijk The most general gauge-invariant Yukawa coupling in ab T −1 i j k l m +f126 ψa BC Γ Γ Γ Γ Γ ψb126 ijklm + (h.c.) the SU(5) model is (146) where Γi and B are the SO(10) analogs of Dirac matrices L = h T T C−1F iHj + h ijklmT T C−1T H + (h.c.) Y 1 ij 2 ij kl m and the spinor conjugation matrix, respectively. Note (142) that f ab and f ab are symmetric while f ab is symmetric. Once the Higgs boson H acquires a vacuum expectation 10 126 120 At the unification scale M u = M d = M ` = M 0 . value, we get ab ab ab ab Adding a 126 Higgs boson with a non-vanishing vac-

d ` v u v uum expectation value to its SU(5) singlet component M = M = h1 √ ,M = h2 √ (143) 2 2 gives a Majorana mass to the right-handed neutrino:

u,d ` M where M are the quark masses matrices and M is for NR BL T −1 L = f126 NR C NR (147) charged leptons. M g At this stage, neutrinos remain massless. To modify the model to generate neutrino mass, a 15 -dimensional where it follows that the corresponding term for the left- handed neutrino is Higgs boson Sij = Sji is included. The multiplet S allows the coupling 2 νL k T −1 LM = λf126 νL C νL (148) 0 T −1 ij MBL/g LY = fF i C FjS + (h.c.) (144) Under the Standard Model gauge group, S contains where k is the SU(2)L breaking scale. This leads to light neutrino mass given by the triplet scalar ∆.~ Assigning a non-zero vacuum expectation to S, 2 2 k 1 −1 k mν ' λf126 − f10 f126 f10 (149) 2 MBL/g 9 MBL/g u µSv hS55i = √ ' (145) 2 V 2 18 which is just a diﬀerent type of see-saw formula, where equal—speciﬁcally, this means they cannot all be zero. mν → 0 as MBL is made very large. Current experimental evidence suggests that if any neu- Assuming f126 are roughly the same order of magni- trinos have non-zero mass, probably all of them do. tude, it follows that Experimental results show that the probability of a neutrino changing type is related to the distance be-

mν` ∼ O(1 eV ), ` = e, µ, τ. (150) tween its point of production to its point of detection, with greater depletion for larger distances travelled. The On the other hand, if the direct Majorana mass vanishes oscillation probability is also a function of the neutrino (λ = 0) the model predicts, assuming mντ < 18 MeV, masses and energy, and of the mixing angles, two of which that are suspected to be relatively large and one much smaller than the other two. m ≤ 1.5 keV. (151) νµ Various phenomenological approaches have been proposed in extending the Standard Model into a more encompassing framework that incorporates neutrino VI. CONCLUDING REMARKS masses, where the models discussed here involve more neutrinos, more Higgs bosons, or larger gauge symmetry Despite the successes of the Standard Model, the phe- groups. The hope is that future experiments in reactors nomenon of neutrino oscillations presents the most con- and accelerators will achieve sub-percent precision for vincing evidence for the need to modify it, because mass- testing the neutrino mass hierarchy and establish which less neutrinos can not oscillate. Put diﬀerently, observ- theory best describes the origin of neutrino mass. ing neutrino oscillations imply neutrino masses cannot be

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