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Neutrinos: recent developments and origin of neutrino mass matrix

Certainly one of the most exciting areas of research at present is neutrino physics. The neutrinos are fantastically numerous in the universe and as such they have bearing on our understanding of the universe. Therefore, we must understand the neutrinos, particularly their mass. There is compelling evidence from solar and atmospheric neutrinos and those from reactors for neutrino oscillations implying that neutrinos mix and have nonzero mass but without pinning down their absolute mass. This is reviewed. The implications of neutrino oscillations and mass squared splitting between neutrinos of different flavor on pattern of neutrino mass matrix is discussed. In particular, a neutrino mass matrix, which shows approximate flavor symmetry where the neutrino mass differences arise from flavor violation in off-diagonal Yukawa couplings is elaborated on. The implications in double beta decay are also discussed.


Introduction
Certainly one of the most exciting areas of research at present is neutrino physics. Neutrinos are fantastically numerous in the universe and as such to understand the universe we must understand neutrinos. It is fair to say that the results of the last decade on neutrinos from the sun, from the atmospheric interaction of cosmic rays, and from reactors provide a compelling evidence that the neutrinos have nonzero mass and mix.
In 1930's protons, neutrons and electrons were considered as elementry particles. Such a picture was confronted with two fundamental problems: conservation of energy and angular momentum (A.M.) in β-decay n → p + e − This is because experimentally seen continuous β spectrum can not be explained for 2-body final state if energy is conserved, since in that case E e would have an unique energy. Further the final state would necessarily have integral A.M. while initial state has half integeral A.M.
To solve these problems Pauli assumed that there exists a new electrically neutral elementry particle, with spin 1/2, mass less than electron mass and an interaction much weaker than photon interaction. Thus n → p + e − +ν e leading to continuous β spectrum and conservation of A.M. This was the first particle postulated by a theoretician.
Direct observation ofν e was made much later in 1950's, when high flux fission reactors as source of neutrons become available.ν e 's, electron-type antineutrinos are produced in the decay of pile neutrons in a fission reactor. These can be captured in hydrogen giving the reaction ν e + p → n + e − whose cross-section was measured by Reines and Cowan: σ exp = (11 ± 2.5) × 10 −44 cm 2 to be compared with the theoretical value σ th = (11 ± 1.6) × 10 −44 cm 2 Note the extreme smallness of the cross-section (nuclear cross sections are of order 10 −24 cm 2 ). It is a reflection of the fact that neutrino has only weak interaction. It is remarkable that neutrinos which have almost no interaction with matter have contributed to some of the most important discoveries in physics given below: • 1950'sν e : electron type anti-neutrino discovered in experiments of Reines and Cowan (1995 Nobel Prize).
• 1956: Parity non-conservation in β decays was discovered (Wu et al.) after its conservation in weak interaction was questioned by Lee and Yang (1957 Nobel Prize).
• 1957: It was proved that neutrino (antineutrino) is left handed (right handed) particle (Goldharber etal), after Salam, Landau and Lee and Yang proposed the 2-component neutrino theory. This is an impressive list of discoveries.

Neutrino Mass
Neutrino occurs in one helicity state (Left handed). This togather with lepton number L conservation implies m ν = 0. However there is no deep reason that it should be so. There is no local gauge symmetry and no massless gauge boson coupled to lepton number L, which therefore is expected to be violated. Thus one may expect a finite mass for neutrino. Moreover, all other known fermions, quarks and charged leptons, are massive. But the intriguing question is: why m(ν e ) ≪ m(e), which needs to be understood, even though we do not understand why e.g. electron mass is what it is and why muons and tauons are heavier than electron see Fig. 1 [1]. This is the so called flavor problem which has so far eluded us. Neutrino mass has added importance for two other reasons: • The interesting phenomena of neutrino oscillations is possible if one or more of neutrinos have non vanishing mass.
• Non-vanishing of neutrino mass has important implications in Astrophysics and Cosmology. It is a candidate for hot dark matter.

Astrophysical Constraint on Neutrino mass
The total mass-energy of the universe is composed of several constituents, each of which is characterized by its energy density, which is expressed in terms of critical density ρ 0i ≡ Ω oi ρ c0 Critical density is the minimum density required for the expansion of the Universe to be turned around by the gravitational attraction of all the matter in it and is defined as where H 0 is the Hubble constant and G N is the Newton's gravitational constant Using the present value of H 0 (Hubble constant), namely H 0 = 100h 0 km s −1 Mpc −1 , Mpc=3 × 10 19 km so that This gives What is the neutrinos contribution to hot dark matter (since relic light ν's had relativistic velocity) ? Neutrinos are fantastically numerous n ν = 3 10 n γ = 112 cm −3 n γ = 400 cm −3 so if they have even a tiny mass, they can outweigh all the stars and galaxies in the universe. The neutrinos contribution to energy density is Unfortunately there is no direct particle physics evidence on i m νi . We shall come back to this question later. Here we simply note that This is the astrophysical constraints on light neutrino masses.

Double β-Decay
The double β-decay is another way to look for a finite mass of neutrino. Two kinds of double β-decay can be considered: Usually the neutrinos are assumed to be Dirac particles: neutrino ν and antineutrinoν ≡ ν c are distinct.In Majorana picture, they are identical. Thus Thus (0ν)ββ-decay is especially interesting: where decay Q value ≈ T e1 + T e2 . Here where λ i is a possible sign since Majorana neutrinos are CP eigenstates; as shown the expectation value is weighted by neutrino's electron couplings.
There is direct evidence of (2ν) ββ decay (2ν) ββ 82 Se → 82 Kr For neutrinoless double β decay One recent result [2] has claimed the evidence for this decay with the best value T 1/2 = 1.5×10 25 yrs. This analysis claims m ν = 0.39 +0.17 −0.28 eV which has been commented upon [3]. If the above finding were to be confirmed, it would be the first indication of lepton number violation in nature and that Majorana neutrino can exist in nature. We shall come to other implications of above value of m ν later.

Cosmological Constraints
We can see the universe 300,000 years after Big Bang by studying the cosmic microwave background radiation (CMB), which is a direct relic of the universe when it became transparent to electromagnetic radiation. Fluctuations in the CMB radiation (at the level of a few parts in 10 5 ) have been detected with angular resolutions from 7 o to a few arc minutes in the sky [4]. These indicate the first clumping of matter particles into cosmic structures, which is resisted by the repulsive pressure of photons. The net result was gravity driven acoustic-like oscillations. These oscillations left their signature in the anisotropy of the CMB. Since the amplitude and position of the primary and secondary peaks are directly determined by the sound speed (and hence the equation of state) and by the geometary and expansion of the universe, they can be used as powerful test of the density of baryons and dark matter (DM) and other cosmological parameters.
Recent measurements of the fluctuations by an orbiting observatory called the Wilkinson Microwave Anisotropy Probe (WMAP) and their analysis have settled a number of issues about the universe, its age, its expansion rate and its composition. The results are summarized below [4].
Note that visible baryon density is only about 4.6 percent. The situation is summarized in Figure 2 [5]. On the composition of the universe there is dramatic observation that the fraction of cosmic mass-energy residing in ordinary matter is only about 4 %. Around 23 % of the universe is made up of another substance, called dark matter, proposed 25 years ago when it became clear that all the galaxies behaved as if they were more massive than they seemed to be. The remaining 72 % is a new discovery, called dark energy, that work against gravity on large scales implying that the expansion of the universe is speeding up, rather than decelerating. In essence what we have learned about the universe is largely restricted to 4 %. The nature of 96 % is essentially unknown. One thing is certain that we have to go beyond the ordinary matter and radiation we already know. For the dark matter we have a real chance of learning within the next 5 to 10 years when we might discover a new type of matter at CERN, Geneva where world's largest accelerator is being developed. Such a matter is predicted by a new symmetry in particle physics, called supersymmetry. For dark energy, we have to wait unless or until there is a unified theory of space-time, trying to bring gravity within the same framework as other interactions.

Origion of Neutrino Masses
The minimal standard model involves 3 chiral neutrino states, but it does not admit renormalizabile interactions that can generate neutrino masses. If there is no SU L (2) × U Y (1)-singlet fermion in nature, then neutrino masses are necessarily Majorana However, even if such a field exist, its mass is naturally much greater than the weak scale in which case light neutrinos are Majorana fermions as we shall see. In the SM, Majorana neutrino masses are forbidden by a global Baryon-Lepton (B − L) symmetry but there is no reason to expect that this symmetry is fundamental. If one allows right-handed neutrinos ν R which are SU L (2) × U Y (1) singlets, then one can write Yukawa interactions: where the SM places the left-handed components of charged leptons and associated neutrinos into SU L (2) doublets ℓ L . φ is the usual Higgs doublet under SU L (2). The lepton number violation is induced by the third term, which is allowed by the gauge symmetry. M is the Majorana mass matrix while h are Yukawa couplings. After spontaneous symmetry breaking the vacuum expectation value of the Higgs field φ ≡ v = 175 GeV generates the Dirac mass term (m D ) ij = h ij v and 6×6 neutrino mass matrix After diagonalization M ν has 6 mass eigenstates ν k that represent Majorana neutrinos (ν k =ν k ). One can consider some useful limits: In the seesaw limit, the diagonalization of M ν gives This also yields light and heavy neutrino mass eigen states where V ν is the neutrino mixing matrix. Thus

Oscillations in vacuum
Neutrinos are produced in weak interactions as flavor eigenstates, characterized by e, µ, τ . The flavor eigenstates |ν α need not coincide with mass (energy) eigenstate |ν i and are generally coherent superposition of such states where the mixing matrix is unitary. This matrix is characterized by 3 angles, θ 12 = θ 3 , θ 13 = θ 2 , θ 23 = θ 1 , one CP violating phase δ and two Majorana phases, which we put equal to zero.
In vacuum, the mass eigenstates propagate as plane waves where The probability at time t that ν ℓ is converted into ν ℓ ′ is For oscillations involving two neutrinos, it takes a simple form It is convenient to write it as P ν ℓ →ν ℓ ′ = sin 2 2θ sin 2 1.27 where L is the distance (measured in meters) travelled after ν ℓ is converted into ν ℓ ′ . ∆m 2 = m 2 1 − m 2 2 in units of eV 2 while E ν ≃ k is measured in MeV. Thus the oscillations in this simple case are characterized by the oscillation length in vacuum and by the amplitude sin 2 2θ. To look for the oscillations, the above formula also shows that one needs low energy ν's, long path length and large flux. In the flavor basis the Hamiltonian is where H is diagonal in ν 1 − ν 2 base. This gives, ignoring a trivial diagonal term not relevent for oscillations, Lv sin 2θ 2Lv sin 2θ 2Lv 0 (25)

Oscillations in Matter
In traversing matter neutrinos interact with electrons and nucleons of intervening material and their forward coherent scattering induces an effective potential energy √ 2G F N e modifying H ν given in Eq. (21) to Thus the evolution of the flavor eigenstates in matter is governed by the where H M is given in Eq. (26) and there is the corresponding matter oscillation length. Here N e denotes the number of electrons per unit volume: where Y e is the number of electrons per nucleons ≈ 1/2 in ordinary matter. The effective oscillation length in matter is θ m is new mixing angle in matter. Thus, resonance sin 2 2θ m = 1 occures when cos 2θ is equal to The transition point between the regime of vacuum and matter oscillations is determined by the ratio L v /L 0 . If it is greater than 1, matter oscillations dominate. If it is less than cos 2θ, vacuum oscillation dominate Generally there is a smooth transition between these two regimes. Matter effects become maximum at resonance L v /L 0 = cos 2θ. This is the basis of Mikheyev-Smirnov-Wolfenstein (MSW) effect. The survival probability P (νe→νe) averaged over the detector position L (from the solar surface) [6] is where θ m (ρ max ) is the initial mixing angle, usually cos 2θ m (ρ max ) ≃ −1 and P x is a finite probability for jumping from one eigenstate to the other one and convertion might be incomplete. The survival probability P (νe→νe) as a function of E ν is displayed for various mixing angles in Fig. 3 [7]. For the parameters corresponding to prefered solution for neutrino oscillations (see below) sin 2 2θ ≈ 0.8, ∆m 2 ≈ 7 × 10 −5 eV 2 and ρ = 100 g/cm 3 at the center of the sun, Y e ≈ 1/2, L v /L 0 = 0.11E v /1 MeV, cos 2θ = 0.45.
Due to different reaction thersholds, solar neutrinos with energy E ν > 0.814 MeV can be detected in 37 Cl and those with E ν > 0.233 MeV in 71 Ga. Note that for pp neutrinos (E ν < 0.42 MeV) and 7 Be neutrinos (E ν ≈ 0.86 MeV), L v /L 0 < cos 2θ and they undergo vacuum oscillations, while the neutrinos with E ν > 4.5 MeV, ( 8 B neutrinos) undergo MSW matter oscillations.

Evidence for Oscillations
One looks for oscillations in two types of experiments.

Appearance experiments:
Here one searches for a new neutrino flavor, absent in the initial beam, which can arise from oscillations.

Atmospheric neutrino anomaly:
Atmospheric neutrinos are produced in decays of pions (kaons) that are produced in the interaction of cosmic rays with the atmosphere: These neutinos are detected in and beneath underground detectors through the reactions ν µ + n → µ − + p ν µ + p → µ + + n and ν e + n → e − + p ν e + p → e + + n These are respectively called µ-like and e-like events. The observed ratios of these events was found to be substantially reduced from the expected value ∼ 2. There is compelling evidence that atmospheric neutrinos change flavor as the Super-Kamiokande experiment clearly indicated a deficit of up-ward µ-like events (produced about 10 4 km away at the opposite side of earth) relative to the downward going events (produced about 20 km above). The e-like events showed a normal zenith angle dependence. The data is described by ν µ → ν τ oscillations. The conversion probability P νµ→ντ fits the data quite well for [8] ∆m 2 23 = 2.0 × 10 −3 eV 2 , sin 2 2θ 23 ≈ 1.0 (33)

Solar Neutrinos
Particularly compelling evidence that the solar neutrinos change flavor has been reported by the Sudbury Neutrino Observatory (SNO). SNO measures the high energy part of the solar neutrino flux ( 8 B neutrinos). The reactions νd → νnp → epp νe → νe were studied by SNO. SNO measured arriving ν e + ν µ + ν τ flux, φ e + φ µτ , and the ν e flux, φ e . From the observed rates for the first two reactions, which involve respectively neutral current and charge current, SNO finds that the ratio of the two fluxes φ e and φ e + φ µτ is 0.306 ± 0.050. This implies that the flux φ µτ is not zero. Since all the neutrinos are born in nuclear reactions that produce only electron neutrinos, it is clear that neutrinos change flavor. Corroborating information comes from the direct reaction νe → νe, studied by both SNO and Super-Kamiokande. The strongly favored explanation of solar neutrino flavor change is the Large Mixing Angle version of the MSW effect, with the best fit parameters [9] ∆m 2 12 = 7.1 × 10 −5 eV 2 , sin 2 2θ

Disappearance experiments:
Reactors are source ofν e 's through the neutron β-decay n → p + e − +ν e and experiment looks for a possible decrease in theν e flux as a function of distance from the reactor,ν e → X [if converted toν µ , say, one would see nothing,ν µ could have produced µ + but does not have sufficient energy to do so]. Kamland experiment [10] confirms thatν e do indeed disappear when the reactorν e have travelled ≈ 200 km.ν e flux is only 0.611 ± 0.085 ± 0.041 of what it would be if none of it were disappearing. Interestingly this reactor ν e disappearance and the solar neutrino results can be described by the same neutrino mass and mixing parameters (see Fig. 4 [7]). This gives confidence that the physics of both phenomenon has been correctly identified.

Neutrino Mass Matrix
As discussed the data from solar and atmospheric neutrino and reactor antineutrinos experiments provide evidence for neutrino mass and mixing with two different mass scales and large mixing angles: ∆m 2 atm ≡ ∆m 2 23 = (2.0 ± 0.5) × 10 −3 eV 2 sin 2 θ 23 ≡ sin 2 θ 1 = 1.00 ± 0.4 ∆m 2 solar ≡ ∆m 2 12 = (7.1 ± 0.6) × 10 −5 eV 2 tan 2 θ 12 ≡ tan 2 θ 3 = 0.45 ± 0.06 Further the CHOOZ experiment [11] gives We would interpret these results in terms of small off-diagonal perturbations of a degenerate diagonal mass matrix in flavor basis for light Majorana neutrinos [12]. In this approach there is no fundamental distinction between masses of neutrinos of different flavors; the mass differences arise from small flavor violation of off-diagonal Yukawa coupling constants. Further the neutrino mass differences do not in anyway constraint the absolute value of neutrino mass. The constraint on it will come from neutrinoless double β-decay experiments, cosmology and direct laboratory experiments, e.g. tritium β-decay.
Let us consider a Majorana mass matrix in (e, µ, τ ) basis a ee a eµ a eτ a eµ a µµ a µτ a eτ a µτ a τ τ    It is convenient to define the neutrino mixing angles as follows where U is given in Eq. (18). We shall put δ as well as Majorana phases to be zero. In view of mixng angles given above, we shall take s 13 ≡ s 2 = 0, c 13 ≡ c 2 = 1. and c 1 = 1/ √ 2, s 1 = ∓1/ √ 2. The diagonalization gives In view of we can take m 1 ≃ ±m 2 .
For the degenerate neutrino mass pattern m 1 ∼ m 2 ∼ m 3 ≫ ∆m 2 32 = 0.045, the effective mass in neutrinoless double β-decay is larger than ∼ 0.05 eV, constrained from above by the mass limit from tritium β-decay. If the effective Majorana mass is confirmed to be 0.39 +0.17 −0.28 eV [2], it would strongly indicate that neutrinos follow degenrate mass pattern (see Fig. 5 [7]), when ∆m 2 m 2 ≪ 1. Finally for two modest extensions of the standard model in which the neutrino mass matrix advocated in this section can be embedded, see Ref. [12].

Conclusion
To conclude various neutrino mass patterns and corresponding neutrino mass matrix types are possible. Further the absolute value of neutrino mass is not yet determined. However, one thing is certain that neutrinos are providing an evidence for new physics but the scale of new physics is not yet pinned down. The heavy right handed neutrinos at new physics scale may provide an explanation for baryogenesis through leptogenesis. If past is of any guide, neutrinos will enrich physics still further.  4. Ratio of observed to expected rates (without neutrino oscillations) for reactor neutrino experiments as a function of distance, including the recent result from the Kamland experiment. The shaded region is that expected due to neutrino oscillations with large mixing parameters as determined from solar neutrino data.
5. Dependence of effective Majorana mass m ν derived from the rate of neutrinoless double β-decay on the absolute mass of the lightest neutrino. The stripes region indicates the range related to the unknown Majorana phases, while the cross hatched region is covered if one σ errors on the oscillation parameters also included. The arrows indicate the three possible neutrino mass patterns.