Neutrino Masses and Mixing: Evidence and Implications

REVIEWS OF MODERN PHYSICS, VOLUME 75, APRIL 2003 Neutrino masses and mixing: evidence and implications

M. C. Gonzalez-Garcia Theory Division, CERN, CH-1211, Geneva 23, Switzerland; Instituto de Fı´sica Corpuscular, Universitat de Vale`ncia–C.S.I.C, Ediﬁcio Institutos de Paterna, Apt. 22085, 46071 Vale`ncia, Spain; C.N. Yang Institute for Theoretical Physics, State University of New York at Stony Brook, Stony Brook, New York 11794-3840

Yosef Nir Department of Particle Physics, Weizmann Institute of Science, Rehovot 76100, Israel (Published 26 March 2003)

Measurements of various features of the fluxes of atmospheric and solar neutrinos have provided evidence for neutrino oscillations and therefore for neutrino masses and mixing. The authors review the phenomenology of neutrino oscillations in vacuum and in matter. They present the existing evidence from solar and atmospheric neutrinos as well as the results from laboratory searches, including the final status of the Liquid Scintillator Neutrino Detector (LSND) experiment. The theoretical inputs that are used to interpret the experimental results are described in terms of neutrino oscillations. The allowed ranges for the mass and mixing parameters are derived in two frameworks: First, each set of observations is analyzed separately in a two-neutrino framework; Second, the data from solar and atmospheric neutrinos are analyzed in a three-active-neutrino framework. The theoretical implications of these results are then discussed, including the existence of new physics, the estimate of the scale of this new physics, and the lessons for grand unified theories, for models of extra dimensions and singlet fermions in the bulk, and for flavor models.

CONTENTS 4. Global analysis 368 V. Atmospheric Neutrinos 369 A. Atmospheric neutrino fluxes 371 I. Introduction 346 1. Cosmic-ray spectrum 372 II. The Standard Model and Neutrino Masses 347 2. Geomagnetic effects 372 A. The standard model implies m␯ϭ0 348 3. The neutrino yield 373 B. Extensions of the standard model allow m␯Þ0 348 4. The neutrino fluxes 373 C. Dirac and Majorana neutrino mass terms 349 B. Interaction cross sections 374 D. Lepton mixing 350 C. Two-neutrino oscillation analysis 375 III. Neutrino Oscillations 351 1. Predicted number of events 375 A. Neutrino oscillations in vacuum 351 B. Neutrinos in matter: Effective potentials 352 2. Conversion probabilities 376 C. Evolution equation in matter: Effective mass 3. Statistical analysis 376 ␯ →␯ and mixing 354 4. ␮ e 377 ␯ →␯ ␯ →␯ D. Adiabatic versus nonadiabatic transitions 355 5. ␮ ␶ and ␮ s 378 E. Propagation in the sun: Mikheyev-Smirnov- VI. Laboratory Experiments 379 Wolfenstein effect 357 A. Short-baseline experiments at accelerators 379 IV. Solar Neutrinos 358 B. LSND and KARMEN 379 A. Solar neutrino experiments 360 C. Disappearance experiments at reactors 381 1. Chlorine experiment: Homestake 360 D. Long-baseline experiments at accelerators 382 2. Gallium experiments: SAGE and E. Direct determination of neutrino masses 382 GALLEX/GNO 360 VII. Three-Neutrino Mixing 383 3. Water Cerenkov experiments: Kamiokande A. Probabilities 384 and SuperKamiokande 360 B. Allowed masses and mixing 386 4. SNO 361 VIII. Implications of the Neutrino Mass Scale and Flavor 5. Future: Borexino and low-energy Structure 387 experiments 362 A. New physics 387 B. The solar neutrino problem 363 B. The scale of new physics 388 C. Solar neutrino oscillation probabilities 364 C. Implications for flavor physics 388 1. Quasivacuum oscillations and the dark side 364 1. The flavor parameters 388 2. Evolution in the Earth 365 2. Special features of the neutrino flavor D. Two-neutrino oscillation analysis 366 parameters 389 1. Predictions 366 3. Large mixing and strong hierarchy 390 2. Analysis of total event rates: allowed IX. Implications for Models of New Physics 393 masses and mixing 367 A. Grand unified theories 393 3. Day-night spectra: Excluded masses and B. Extra dimensions 393 mixing 368 1. Coupling to bulk fermions 394

2. Lepton number breaking on a distant brane 395 into a muon and a neutrino. The material in the dump 3. The warp factor in the Randall-Sundrum will eventually absorb the muons, photons, and any scenario 395 other charged particles, so that only neutrinos exit at the X. Conclusions 395 Note added 396 opposite end, forming an intense and controlled beam. KamLAND results 396 Neutrinos are also produced in natural sources. From Consequences for solar neutrinos 397 the 1960s on, neutrinos produced in the sun and in the Effect on three-neutrino mixing scenarios 398 atmosphere have been observed. As we shall see in this Acknowledgments 398 review, these observations play an important role in ex- References 398 panding our understanding of the detailed features of neutrinos. In 1987, neutrinos from a supernova in the I. INTRODUCTION Large Magellanic Cloud were also detected. The properties of the neutrino and in particular the In 1930 Wolfgang Pauli postulated the existence of the question of its mass have intrigued physicists ever since neutrino in order to reconcile data on the radioactive it was proposed (Kayser, Gibrat-Debu, and Perrier, decay of nuclei with energy conservation. In radioactive 1989; Ramond, 1999). In the laboratory neutrino masses decays, nuclei of atoms mutate into different nuclei have been searched for in two types of experiments when neutrons are transformed into slightly lighter pro- (Boehm and Vogel, 1987): (i) direct kinematic searches tons with the emission of electrons: of neutrino mass, of which the most sensitive is the study neutron→protonϩelectronϩantineutrino. (1) of tritium beta decay, and (ii) neutrinoless double-␤ decay experiments. Experiments have achieved higher and Without the neutrino, energy conservation would re- higher precision, reaching upper limits for the electron quire that the electron and proton share the neutron’s Ϫ neutrino mass of 10 9 of the proton mass, rather than energy. Each electron would therefore be produced with the 10Ϫ2 originally obtained by Pauli. This raised the a fixed energy, while experiments indicated conclusively that electrons were not monoenergetic but were ob- question of whether neutrinos are truly massless like photons. served with a range of energies. This energy range cor- Ϫ9 responded exactly to the many ways the three particles Can one go further below the eV scale (that is, 10 of in the final state of the reaction above could share en- the proton mass) in the search for neutrino masses? This ergy while satisfying the conservation law. The postu- is a very difficult task in direct measurements. In 1957, lated neutrino had no electric charge and, for all practi- however, Bruno Pontecorvo realized that the existence cal purposes, did not interact with matter; it just served of neutrino masses implied the possibility of neutrino as an agent to balance energy and momentum in the oscillations. This phenomenon is similar to what hap- above reaction. In fact, Pauli pointed out that for the pens in the quark sector, where neutral kaons oscillate. neutrino to do the job, it had to weigh less than one Flavor oscillations of neutrinos have been searched for percent of the proton mass, thus establishing the first using either neutrino beams from reactors or accelera- limit on neutrino mass. tors, or natural neutrinos generated at astrophysical Observing neutrinos is straightforward—in principle. sources (the Sun giving the largest flux) or in the atmo- Pauli had to wait a quarter of a century before Fred sphere (as the by-products of cosmic-ray collisions). The Reines and Clyde Cowan, Jr. observed neutrinos pro- longer the distance that the neutrinos travel from their duced by a nuclear reactor. In the presence of protons, production point to the detector, the smaller the masses neutrinos occasionally initiate the inverse reaction of ra- that can be signalled by their oscillation. Indeed, the dioactive decay: solar neutrinos allow us to search for masses that are as Ϫ Ϫ ¯␯ϩp→nϩeϩ. (2) small as 10 5 eV, that is, 10 14 of the proton mass! In recent years, experiments studying natural neutrino Experimentally, one exposes a material rich in protons fluxes have provided us with the strongest evidence of to a neutrino beam and simply looks for the coincident neutrino masses and mixing. Experiments that measure appearance of an electron and a neutron. In the alternative possibility where the incident neutrino carries muon the flux of atmospheric neutrinos have found results that flavor, a muon appears in the final state instead. suggest the disappearance of muon-neutrinos when By the 1960s neutrino beams were no longer a futur- propagating over distances of the order of hundreds of istic dream, but had become one of the most important kilometers (or more). Experiments that measure the flux tools of particle physics. Neutrino physics contributed in of solar neutrinos (Bahcall, 1989) suggest the disappear- important ways to the discovery of quarks, the constitu- ance of electron neutrinos while propagating within the ents of protons and neutrons. The technique is concep- Sun or between the Sun and the Earth. The disappear- ␯ ␯ tually simple, though technologically very challenging. A ance of both atmospheric ␮’s and solar e’s is most eas- very intense beam of accelerated protons is shot into a ily explained in terms of neutrino oscillations. As con- beam dump which typically consists of a kilometer-long cerns experiments performed with laboratory beams, mound of earth or a 100-meter-long block of stainless most have given no evidence of oscillations. One excep- steel. Protons interact with nuclei in the dump and pro- tion is the Liquid Scintillator Neutrino Detector duce tens of pions in each collision. Charged pions decay (LSND) experiment, which has observed the appear-

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 347

ance of electron antineutrinos in a muon antineutrino and V we discuss the evidence from solar and atmo- beam. This signal has not been confirmed so far by any spheric neutrinos (for further details we refer the reader other experiment. to the recent review on atmospheric neutrino observa- What can we learn from measurements of neutrino tions by Kajita and Totsuka, 2001). We review the theo- masses about our theories of particle physics? The stan- retical modeling that is involved in interpreting the ex- dard model of particle physics is a mathematical descrip- perimental results in terms of neutrino oscillations and tion of the strong, weak, and electromagnetic interac- briefly describe the techniques used in the derivation of tions. Since it was conceived in the 1960s by Glashow, the allowed ranges for the neutrino flavor parameters Salam, and Weinberg, it has successfully passed numer- which, in these sections, is performed in the framework ous experimental tests. In the absence of any direct evi- of mixing between two neutrinos. Section VI is devoted dence for their mass, neutrinos were introduced in the to the results from searches in laboratory experiments standard model as truly massless fermions for which no concentrating on the final status of the LSND experi- gauge-invariant renormalizable mass term could be con- ment (a detailed review of the status of neutrino oscilla- structed. Consequently, in the standard model there is tion searches employing nuclear reactors as sources can neither mixing nor CP violation in the lepton sector. be found in Bemporad, Gratta, and Vogel, 2002). The Therefore experimental evidence for neutrino masses or two most robust pieces of evidence, from solar and at- mixing or leptonic CP violation provides an unambigu- mospheric neutrinos, can be accommodated assuming ous signal of new physics. masses and mixing of three standard neutrinos: in Sec. The standard-model prediction of massless neutrinos VII we derive the allowed ranges of parameters in this is accidental: unlike photons, no profound principle pre- case. In Secs. VIII and IX we discuss the various impli- vents them from having a mass. On the contrary, modern cations for theory based on this evidence, focusing on elementary particle theories anticipate ways in which models that explain the atmospheric and solar neutrino they might have small but definitely nonvanishing data through mixing among three active neutrinos. In masses. In fact, there are good theoretical reasons to particular, we discuss flavor models, GUTs, and models expect that neutrinos are massive but much lighter than of extra dimensions. Our conclusions are summarized in all the charged fermions of the standard model. Specifi- Sec. X. cally, it is very likely that neutrino masses are inversely proportional to the scale of new physics. Consequently, if neutrino masses are measured, we can estimate the II. THE STANDARD MODEL AND NEUTRINO MASSES relevant new scale. All dimensionless flavor-blind parameters of the stan- One of the most beautiful aspects of modern theories dard model, that is, the three gauge couplings and the of particles physics is the relation between forces medi- quartic Higgs coupling, are of order one. In contrast, ated by spin-1 particles and local (gauge) symmetries. most of the flavor parameters—quark and charged- Within the standard model, the strong, weak, and elec- lepton masses (except for the mass of the top quark), tromagnetic interactions are related to, respectively, and the three Cabibbo-Kobayashi-Maskawa (CKM) SU(3), SU(2), and U(1) gauge groups. Many features of mixing angles—are small and hierarchical. This situation the various interactions are then explained by the sym- constitutes the flavor puzzle: Within the standard model, metry to which they are related. In particular, the way the hierarchical structure can be accommodated but is that the various fermions are affected by the different not explained. If neutrinos have Majorana (Dirac) types of interactions is determined by their representa- masses, then there are nine (seven) flavor parameters tions (or simply their charges in the case of Abelian beyond the thirteen of the standard model. If some of gauge symmetries) under the corresponding symmetry these extra parameters are measured, we will be able to groups. test and refine theories that try to solve the flavor Neutrinos are fermions that have neither strong nor puzzle. electromagnetic interactions. In group theory language, ϫ In grand unified theories (GUT’s), lepton masses are they are singlets of SU(3)C U(1)EM . Active neutrinos often related to quark masses. Measurements of neu- have weak interactions, that is, they are not singlets of trino parameters provide further tests of these theories. SU(2)L . Sterile neutrinos have none of the standard The values of neutrino parameters that explain the model gauge interactions and they are singlets of the anomalies observed in atmospheric and solar neutrino standard model gauge group. fluxes can be used to address the many theoretical ques- The standard model has three active neutrinos. They tions described above: they imply new physics, they sug- reside in lepton doublets, gest an energy scale at which this new physics takes ␯ ϭ Lᐉ ᐉϭ ␮ ␶ place, they provide stringent tests of flavor models, and Lᐉ ͩ ᐉϪ ͪ , e, , . (3) they probe GUT’s. L In this review we first present the low-energy formal- Here e, ␮, and ␶ are the charged-lepton mass eigen- ␯ ism for adding neutrino masses to the standard model states. The three neutrino interaction eigenstates, e , ␯ ␯ and the induced leptonic mixing (Sec. II) and then de- ␮ , and ␶ , are defined as the SU(2)L partners of these scribe the phenomenology associated with neutrino os- mass eigenstates. In other words, the charged-current cillations in vacuum and in matter (Sec. III). In Secs. IV interaction terms for leptons read

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 348 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

␯ g Y ϪL ϭ ␯ ␥␮ᐉϪ ϩϩ ij ␾␾ CC ͚ Lᐉ W␮ H.c. (4) LLiLLj . (11) & ᐉ L v

In addition, standard model neutrinos have neutral- This, however, cannot happen, as can be easily under- current interactions, stood by examining the accidental symmetries of the standard model. (It often happens that, as a conse- g quence of the symmetries that define a model and of its ϪL ϭ ␯ ␥␮␯ 0 NC ␪ ͚ Lᐉ LᐉZ␮ . (5) particle content, all renormalizable Lagrangian terms 2 cos W ᐉ obey additional symmetries, which are not a priori im- Equations (4) and (5) give all the neutrino interactions posed on the model. These are called accidental symme- within the standard model. tries.) Within the standard model, with the gauge sym- The measurement of the decay width of the Z0 boson metry of Eq. (7) and the particle content of Eq. (8), the into neutrinos makes the existence of three, and only following accidental global symmetry arises (at the per- Շ three, light (that is, m␯ mZ/2) active neutrinos an ex- turbative level): perimental fact. In units of the standard model predic- globalϭ ͑ ͒ ϫ ͑ ͒ ϫ ͑ ͒ ϫ ͑ ͒ tion for a single neutrino, one gets (Groom et al., 2000) GSM U 1 B U 1 e U 1 ␮ U 1 ␶ . (12) Here U(1) is the baryon number symmetry, and N␯ϭ2.994Ϯ0.012 B U(1)e,␮,␶ are the three lepton flavor symmetries, with ϭ ϩ ϩ ͑standard model fits to LEP data͒, total lepton number given by L Le L␮ L␶ . Terms of the form of Eq. (11) violate Gglobal and therefore cannot ϭ Ϯ SM N␯ 3.00 0.06 be induced by loop corrections. Furthermore, the global ͑direct measurement of invisible Z width͒. U(1)BϪL subgroup of GSM is nonanomalous. Terms of the form of Eq. (11) have BϪLϭϪ2 and therefore can- (6) not be induced even by nonperturbative corrections. It follows that the standard model predicts that neu- A. The standard model implies m ϭ0 ␯ trinos are precisely massless. Consequently, there is nei- The standard model is based on the gauge group ther mixing nor CP violation in the leptonic sector. G ϭSU͑3͒ ϫSU͑2͒ ϫU͑1͒ , (7) SM C L Y B. Extensions of the standard model allow m␯Þ0 with three fermion generations, where a single generation consists of five different representations of the There are many good reasons to think that the stan- gauge group, dard model is not a complete picture of Nature. For example, the fine-tuning problem of the Higgs mass can ͑ ͒ ͑ ͒ ͑ ͒ QL 3,2 ϩ1/6 , UR 3,1 ϩ2/3 , DR 3,1 Ϫ1/3 , be solved by supersymmetry; gauge coupling unification and the variety of gauge representations may find an ͑ ͒ ͑ ͒ LL 1,2 Ϫ1/2 , ER 1,1 Ϫ1 . (8) explanation in GUT’s; baryogenesis can be initiated by Our notation here means that, for example, a left- decays of heavy singlet fermions (leptogenesis); and the existence of gravity suggests that string theories are rel- handed lepton field LL is a singlet (1) of the SU(3)C evant to Nature. If any of these (or many other pro- group, a doublet (2) of the SU(2)L group, and carries Ϫ posed) extensions is indeed realized in nature, the stan- hypercharge 1/2 under the U(1)Y group. The vacuum expectations value of the single Higgs dard model must be thought of as an effective low- energy theory. That means that it is a valid doublet ␾(1,2)ϩ breaks the symmetry, 1/2 ⌳ approximation up to the scale NP which characterizes 0 new physics. ͗␾͘ϭ ⇒ → ͑ ͒ ϫ ͑ ͒ By thinking of the standard model as an effective low- ͩ v ͪ GSM SU 3 C U 1 EM . (9) energy theory, we still retain the gauge group (7), the & fermionic spectrum (8), and the pattern of spontaneous symmetry breaking [Eq. (9)] as valid ingredients to de- Since the fermions reside in chiral representations of scribe nature at energies EӶ⌳ . The standard model the gauge group, there can be no bare mass terms. Fer- NP predictions are, however, modified by small effects that mions masses arise from the Yukawa interactions, ⌳ are proportional to powers of E/ NP . In other words, ϪL ϭYd Q ␾D ϩYu Q ␾˜ U the difference between the standard model as a com- Yukawa ij Li Rj ij Li Rj plete description of Nature and as a low-energy effective ϩ ᐉ ␾ ϩ YijLLi ERj H.c. (10) theory is that, in the latter case, we must also consider nonrenormalizable terms. ␾˜ ϭ ␶ ␾Ã (where i 2 ) after spontaneous symmetry break- There is no reason for generic new physics to respect ing. The Yukawa interactions of Eq. (10) lead to charged the accidental symmetries of Eq. (12). Indeed, there is a fermion masses but leave the neutrinos massless. One single set of dimension-five terms that is made of might think that neutrino masses would arise from loop standard-model fields and is consistent with gauge sym- corrections if these corrections induced effective terms, metry, and this set violates Eq. (12). It is given by

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 349

␯ Z Ramond, 1979; Yanagida, 1979) and left-right symmetry ij ␾␾ ⌳ LLiLLj . (13) (Mohapatra and Senjanovic, 1980). NP While these terms have the same form as Eq. (11) they do not have radiative corrections as their source, but C. Dirac and Majorana neutrino mass terms rather some heavy fields, related to new physics, which could induce such terms by tree or loop diagrams. In If the only modification that we make to the standard particular, Eq. (13) violates L (and BϪL) by two units model is to assume that it is a low-energy effective and leads, upon spontaneous symmetry breaking, to theory, that is, allowing for nonrenormalizable terms neutrino masses: that are consistent with gauge symmetry and the fermi- ␯ onic content of the standard model, then the only way Z v2 ͑ ͒ ϭ ij that neutrinos can gain masses is through terms of the M␯ ij ⌳ . (14) 2 NP form of Eq. (13). These are Majorana mass terms which, in particular, violate lepton number by two units. A few comments are in order, regarding Eq. (14): One can, however, open up other possibilities by add- (a) Since Eq. (14) would arise in a generic extension of ing new fields. The most relevant extension is to add an ␯ the standard model, we learn that neutrino masses arbitrary number m of sterile neutrinos si(1,1)0 to the are very likely to appear if there is new physics. three standard generations of Eq. (8). Now there are, in (b) If neutrino masses arise effectively from nonrenor- general, two types of mass terms that arise from renormalizable terms, we gain an explanation not only malizable terms: for their existence but also for their smallness. The 1 ϪL ϭM ␯ ␯ ϩ M ␯c ␯ ϩH.c. (17) scale of neutrino masses is suppressed, compared M␯ Dij Li sj 2 Nij si sj to the scale of charged fermion masses, by v/⌳ . NP ␯c ␯cϭ ␯T (c) The terms of Eq. (14) break not only total lepton Here indicates a charge-conjugated field, C¯ , number but also the lepton flavor symmetry and C is the charge conjugation matrix. The first term is a Dirac mass term. It has the follow- U(1) ϫU(1)␮ϫU(1)␶ . Therefore we should ex- e ing properties: pect lepton mixing and CP violation. (i) Since it transforms as the doublet representation The best known scenario that leads to Eq. (13) is the of SU(2)L , it is generated after spontaneous elec- seesaw mechanism (Gell-Mann et al., 1979; Ramond, troweak symmetry breaking from the Yukawa in- 1979; Yanagida, 1979). Here one assumes the existence ␯ ␾˜ ␯ teractions YijLLi sj , similarly to the charged of heavy sterile neutrinos Ni . Such fermions have, in fermion masses discussed in Sec. II.A. general, bare mass terms and Yukawa interactions: (ii) Since it has a neutrino field and an antineutrino 1 field, it conserves total lepton number [though it ϪL ϭ c ϩ ␯ ␾˜ ϩ N MNijNi Nj YijLLi Nj H.c. (15) breaks the lepton flavor number symmetries of 2 Eq. (12)]. ϫ The resulting mass matrix (see Sec. II.C for details) in (iii) MD is a complex 3 m matrix. the basis The second term in Eq. (17) is a Majorana mass term. ␯ ͩ Liͪ It is different from the Dirac mass terms in many impor- Nj tant aspects: has the following form: (i) It is a singlet of the standard model gauge group. Therefore it can appear as a bare mass term. [Had v 0 Y␯ we written a similar term for the active neutrinos, & it would transform as a triplet of SU(2) . In the ϭ L M␯ ͩ ͪ . (16) absence of a Higgs triplet, it cannot be generated v ␯ T by renormalizable Yukawa interactions. Such ͑Y ͒ MN & terms are generated for active neutrinos from the nonrenormalizable Yukawa interactions of Eq. If the eigenvalues of MN are all well above the elec- (13).] troweak breaking scale v, then the diagonalization of (ii) Since it involves two neutrino fields, it breaks lep- M␯ leads to three light mass eigenstates with a mass ton number conservation by two units. More gen- matrix of the form of Eq. (14). In particular, the scale erally, such a term is allowed only if the neutrinos ⌳ NP is identified with the mass scale of the heavy sterile carry no additive conserved charge. This is the neutrinos, that is, the typical scale of the eigenvalues of reason that such terms are not allowed for any MN . charged fermions which, by definition, carry Two well-known examples of extensions of the stan- U(1)EM charges. dard model that lead to a seesaw mechanism for neu- (iii) MN is a symmetric matrix (as follows from simple trino masses are SO(10) GUT’s (Gell-Mann et al., 1979; Dirac algebra) of dimension mϫm.

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 350 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

It is convenient to define a (3ϩm)-dimensional neu- beta decay will the question of Majorana versus Dirac trino vector ␯ជ , neutrinos be crucial. ␯ From the theoretical model-building point of view, ␯ជ ϭ Li however, the two cases are very different. In particular, ͩ ␯ ͪ . (18) sj the seesaw mechanism provides a natural explanation That allows us to rewrite Eq. (17) in a unified way: for the lightness of Majorana neutrinos, while for Dirac neutrinos there is no such generic mechanism. (The situ- 1 c ation can be different in models of extra dimensions; see ϪL ϭ ␯ជ M␯␯ជ ϩH.c., (19) M␯ 2 Sec. IX.B.) where 0 M ϭͩ D ͪ D. Lepton mixing M␯ T . (20) MD MN We briefly review here our notation for lepton mixing. The matrix M␯ is complex and symmetric. It can be di- We denote the neutrino mass eigenstates by ϩ ␯ ␯ ␯ ␯ ϭ ϩ agonalized by a unitary matrix of dimension (3 m). ( 1 , 2 , 3 ,..., n) where n 3 m, and the charged- ␯ The resulting mass eigenstates, k , obey the Majorana lepton mass eigenstates by (e,␮,␶). The corresponding ␯c ϭ␯ I ␮I ␶I ␯ជ condition, k k . interaction eigenstates are denoted by (e , , ) and ϭ ␯ ␯ ␯ ␯ ␯ There are three interesting cases, differing in the hier- ( Le , L␮ , L␶ , s1 ,..., sm). In the mass basis, lep- archy of scales between MN and MD : tonic charged-current interactions are given by (1) One can assume that the scale of the mass eigen- ␯ 1 values of MN is much higher than the scale of elec- ␯ 2 troweak symmetry breaking ͗␾͘. This is the natural situ- g ϩ ϪL ϭ ͑e ␮ ␶ ͒␥␮U ␯ W ϪH.c. (21) ation in various extensions of the standard model that CC & L L L ͩ 3 ͪ ␮ are characterized by a high energy scale. We are back in ] ␯ the framework of the seesaw mechanism discussed in n the previous section. If we simply integrate out the ster- Here U isa3ϫn matrix (Schechter and Valle, 1980a, ile neutrinos, we get a low-energy effective theory with 1980b). three light, active neutrinos of the Majorana type. Given the charged-lepton mass matrix Mᐉ and the (2) One can assume that the scale of some eigenvalues neutrino mass matrix M␯ in some interaction basis, of M is not higher than the electroweak scale. Now the I N e standard model is not even a good low-energy effective R 1 I I I ␮I c ϪL ϭ͑e ␮ ␶ ͒ Mᐉͩ R ͪ ϩ ជ␯ M␯␯ជ ϩH.c., theory: there are more than three light neutrinos, and M L L L ␶I 2 they are mixtures of doublet and singlet fields. These R light fields are all of the Majorana type. (22) ϭ ᐉ ␯ (3) One can assume that MN 0. This is equivalent to we can find the diagonalizing matrices V and V : imposing lepton number symmetry on this model. ᐉ† † ᐉ 2 2 2 V MᐉM V ϭdiag͑m ,m ,m ͒, Again, the standard model is not even a good low- ᐉ e ␮ ␶ ␯† † ␯ϭ ͑ 2 2 2 2 ͒ energy theory: both the fermionic content and the as- V M␯M␯V diag m1 ,m2 ,m3 ,...,mn . (23) sumed symmetries are different. (Recall that within the Here Vᐉ is a unitary 3ϫ3 matrix while V␯ is a unitary standard model lepton number is an accidental symme- nϫn matrix. The 3ϫn mixing matrix U can be found try.) Now only the first term in Eq. (17) is allowed, which from these diagonalizing matrices: is a Dirac mass term. It is generated by the Higgs ϭ ᐉ† ␯ ͑ ͒ mechanism in the same way that charged fermions Uij Pᐉ,iiVik Vkj P␯,jj . (24) masses are generated. If indeed it is the only neutrino A few comments are in order: mass term present and mϭ3, we can identify the three sterile neutrinos with the right-hand component of a (i) Note that the indices i and k run from 1 to 3, four-component spinor neutrino field (actually with its while j runs from 1 to n. In particular, only the ␯ charge conjugate). In this way, the six massive Majorana first three lines of V play a role in Eq. (24). neutrinos combine to form three massive neutrino Dirac (ii) Pᐉ is a diagonal 3ϫ3 phase matrix that is conven- states, equivalent to the charged fermions. In this par- tionally used to reduce by three the number of ticular case the 6ϫ6 diagonalizing matrix is block diag- phases in U. onal and it can be written in terms of a 3ϫ3 unitary (iii) P␯ is a diagonal matrix with additional arbitrary matrix. phases (chosen to reduce the number of phases in From the phenomenological point of view, it will U) only for Dirac states. For Majorana neutrinos, make little difference for our purposes whether the light this matrix is simply a unit matrix. The reason for neutrinos are of the Majorana or Dirac type. In particu- that is that if one rotates a Majorana neutrino by lar, the analysis of neutrino oscillations is the same in a phase, this phase will appear in its mass term, both cases. Only in the discussion of neutrinoless double which will no longer be real.

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 351

We conclude that the number of phases that can be TABLE I. Characteristic values of L and E for various neu- absorbed by redefining the mass eigenstates depends on trino sources and experiments. whether the neutrinos are Dirac or Majorana particles. Experiment L (m) E (MeV) ⌬m2 (eV2) In particular, if there are only three Majorana neutrinos, U isa3ϫ3 matrix analogous to the CKM matrix for the Solar 1010 1 10Ϫ10 quarks (Maki, Nakagawa, and Sakata, 1962; Kobayashi Atmospheric 104 – 107 102 – 105 10Ϫ1 – 10Ϫ4 and Maskawa, 1973), but due to the Majorana nature of Reactor 102 – 103 1 10Ϫ2 – 10Ϫ3 the neutrinos it depends on six independent parameters: Accelerator 102 103 – 104 տ0.1 three mixing angles and three phases. The two Majorana Long-baseline Accelerator 105 – 106 104 10Ϫ2 – 10Ϫ3 phases do not affect neutrino oscillations (Bilenky, Hosek, and Petcov, 1980; Langacker, Petcov, Steigman, and Toshev, 1987). ͓CP conservation implies that the n n 2 three lepton phases are either zero or ␲ (Schechter and ϭ͉͗␯ ͉␯ ͑ ͉͒͘2ϭͯ ͗␯ ͑ ͉͒␯ ͑ ͒ͯ͘ P␣␤ ␤ ␣ t ͚ ͚ U␣*iU␤j j 0 i t . Valle, 1981; Wolfenstein, 1981).] This is to be compared iϭ1 jϭ1 to the case of three Dirac neutrinos, in which the num- (27) ber of physical phases is one, similarly to the CKM ma- ͉␯͘ trix. Note, however, that the two extra Majorana phases We use the standard approximation that is a plane affect only lepton-number-violating processes. Such ef- wave (for a pedagogical discussion of the possible quantum-mechanical problems in this naive description fects are suppressed by m␯ /E and are very hard to mea- of neutrino oscillations we refer the reader to Lipkin, sure. ͉␯ 1999, and Kim and Pevsner, 1993), i(t)͘ If no new interactions for the charged leptons are Ϫ ϭ iEit͉␯ ͘ present, we can identify their interaction eigenstates e i(0) . In all cases of interest to us, the neutri- with the corresponding mass eigenstates after phase re- nos are relativistic: 2 definitions. In this case the charged-current lepton mix- mi ϫ E ϭͱp2ϩm2Ӎp ϩ , (28) ing matrix U is simply given by a 3 n submatrix of the i i i i 2E unitary matrix V␯. i where Ei and mi are, respectively, the energy and the ␯ mass of the neutrino mass eigenstate i . Furthermore, Ӎ ϵ Ӎ we can assume that pi pj p E. Then we obtain the following transition probability (we include here only III. NEUTRINO OSCILLATIONS the CP-conserving piece): Ϫ A. Neutrino oscillations in vacuum n 1 n ϭ␦ Ϫ ͓ ͔ 2 P␣␤ ␣␤ 4 ͚ ͚ Re U␣iU␤* U␣* U␤j sin xij , ϭ ϭ ϩ i j Neutrino oscillations in vacuum would arise if neutri- i 1 j i 1 (29) nos were massive and mixed. In other words, the neu- ϵ⌬ 2 ⌬ 2 ϵ 2Ϫ 2 ϭ trino state that is produced by electroweak interactions where xij mijL/(4E) with mij mi mj . L t is is not a mass eigenstate. This phenomenon was first the distance between the source (that is, the production pointed out by Pontecorvo in 1957 (Pontecorvo, 1957), point of ␯␣) and the detector (that is, the detection point while the possibility of arbitrary mixing between two of ␯␤). In deriving Eq. (29) we used the orthogonality ␯ ͉␯ ϭ␦ massive neutrino states was first introduced by Maki, relation ͗ j(0) i(0)͘ ij . It is convenient to use the Nakagawa, and Sakata (1962). following units: From Eq. (21) we see that if neutrinos have masses, ⌬ 2 mij L/E the weak eigenstates ␯␣ produced in a weak interaction ϭ xij 1.27 2 . (30) (i.e., an inverse beta reaction or a weak decay) are, in eV m/MeV ␯ general, linear combinations of the mass eigenstates i , The transition probability [Eq. (29)] has an oscillatory n behavior, with oscillation lengths ͉␯ ϭ ͉␯ ␣͘ ͚ U␣* i͘, (25) ϭ i 4␲E i 1 Loscϭ (31) 0,ij ⌬m2 where n is the number of light neutrino species and U is ij the mixing matrix in Eq. (24). (Implicit in our definition and amplitude that is proportional to elements in the of the state ͉␯͘ is its energy-momentum and space-time mixing matrix. Thus, in order to have oscillations, neu- ⌬ 2 Þ dependence.) After traveling a distance L (or, equiva- trinos must have different masses ( mij 0) and they Þ lently for relativistic neutrinos, time t), a neutrino origi- must mix (U␣ U␤i 0). ␣ i nally produced with a flavor evolves as follows: An experiment is characterized by the typical neu- n trino energy E and by the source-detector distance L.In ͉␯ ͑ ͒͘ϭ ͉␯ ͑ ͒͘ ⌬ 2 ␣ t ͚ U␣*i i t . (26) order to be sensitive to a given value of mij , the ex- iϭ1 Ϸ⌬ 2 ϳ osc periment has to be set up with E/L mij (L L0,ij). It can be detected in the charged-current interaction The typical values of L/E for different types of neutrino ␯␣(t)NЈ→ᐉ␤N with a probability sources and experiments are summarized in Table I.

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 352 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

ӷ⌬ 2 Ӷ osc If (E/L) mij (L L0,ij), the oscillation does not 2 have time to give an appreciable effect because sin xij Ӷ Ӷ⌬ 2 ӷ osc 1. The case of (E/L) mij (L L0,ij) requires more careful consideration. One must take into account that, in general, neutrino beams are not monochromatic. Thus, rather than measuring P␣␤ , the experiments are sensitive to the average probability d⌽ ͵ ␴ ͑ ͒ ͑ ͒⑀͑ ͒ dE␯ CC E␯ P␣␤ E␯ E␯ dE␯ ͗P␣␤͘ϭ d⌽ ͵ ␴ ͑ ͒⑀͑ ͒ dE␯ CC E␯ E␯ dE␯

nϪ1 n ϭ␦ Ϫ ͓ ͔͗ 2 ͘ ␣␤ 4 ͚ ͚ Re U␣iU␤*iU␣*jU␤j sin xij , iϭ1 jϭiϩ1 (32) ⌽ ␴ where is the neutrino energy spectrum, CC is the cross section for the process in which the neutrino is detected (in general, a charged-current interaction), and ⑀ ӷ osc (E␯) is the detection efficiency. For L L0,ij , the oscillating phase goes through many cycles before the detec- 2 ϭ tion and is averaged to ͗sin xij͘ 1/2. For a two-neutrino case, the mixing matrix depends on a single parameter, FIG. 1. The characteristic form of an excluded region from a cos ␪ sin ␪ negative search with fixed L/E and of an allowed region from Uϭͩ ͪ , (33) a positive search with varying L/E in the two-neutrino oscilla- Ϫsin ␪ cos ␪ tion parameter plane. and there is a single mass-squared difference ⌬m2. Then P␣␤ of Eq. (29) takes the well-known form 2 to be in the region of averaged oscillations, ͗sin xij͘ 2 2 ϭ ⌬ 2 P␣␤ϭ␦␣␤Ϫ͑2␦␣␤Ϫ1͒sin 2␪ sin x. (34) 1/2. Consequently, no upper bound on m can be ⌬ 2у achieved by such an experiment. For negative searches The physical parameter space is covered with m 0 that set an upper bound on the oscillation probability, and 0р␪р ␲/2 (or, alternatively, 0р␪р ␲/4 and either ͗ ͘р ⌬ 2 P␣␤ PL , the excluded region always lies on the up- sign for m ). ⌬ 2 2 ␪ ⌬ 2 per right side of the ( m ,sin 2 ) plane, limited by the Changing the sign of the mass difference, m following asymptotic lines: →Ϫ⌬m2, and changing the octant of the mixing angle, ⌬ 2ӷ 2 ␪ϭ ␪→ ␲/2 Ϫ␪, amounts to redefining the mass eigenstates, • for m 1/͗L/E͘, a vertical line at sin 2 2 PL ; ␯ ↔␯ ⌬ 2Ӷ ͗ ͘ 1 2 : P␣␤ must be invariant under such transforma- • for m 1/ L/E , the oscillating phase can be ex- tions. Equation (34) reveals, however, that P␣␤ is actu- panded and the limiting curve takes the form ⌬ 2 ␪ϭ ͱ ally invariant under each of these transformations sepa- m sin 2 4 PL/͗L/E͘, which in a log-log plot rately. This situation implies that there is a twofold gives a straight line of slope Ϫ1/2. discrete ambiguity in the interpretation of P␣␤ in terms of two-neutrino mixing: the two different sets of physical If, instead, data are taken at several values of ͗L͘ parameters, (⌬m2,␪) and (⌬m2,␲/2 Ϫ␪), give the same and/or ͗E͘, the corresponding region may be closed, as transition probability in vacuum. One cannot tell from a it is possible to have direct information on the charac- measurement of, say, Pe␮ in vacuum whether the larger teristic oscillation wavelength. ␯ component of e resides in the heavier or in the lighter neutrino mass eigenstate. Neutrino oscillation experiments measure P␣␤ .Itis B. Neutrinos in matter: Effective potentials common practice for the experiments to interpret their results in the two-neutrino framework. In other words, When neutrinos propagate in dense matter, the inter- the constraints on P␣␤ are translated into allowed or actions with the medium affect their properties. These excluded regions in the plane (⌬m2, sin2 2␪) by using effects are either coherent or incoherent. For purely in- Eq. (34). An example is given in Fig. 1. We now explain coherent inelastic ␯-p scattering, the characteristic cross some of the typical features of these constraints. section is very small: When an experiment is taking data at fixed ͗L͘ and 2 2 ͗ ͘ GFs E E , as is the case for most laboratory searches, its result ␴ϳ ϳ10Ϫ43 cm2ͩ ͪ . (35) can always be accounted for by ⌬m2 that is large enough ␲ 1 MeV

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 353

The smallness of this cross section is demonstrated by As an example we derive the effective potential for 10 ␯ the fact that if a beam of 10 neutrinos with E the evolution of e in a medium with electrons due to its ϳ1 MeV was aimed at the Earth, only one would be charged-current interactions. The effective low-energy deﬂected by the Earth’s matter. It may seem then that Hamiltonian describing the relevant neutrino interac- for neutrinos matter is irrelevant. However, one must tions is given by take into account that Eq. (35) does not contain the contribution from forward elastic coherent interactions. In coherent interactions, the medium remains unchanged G and it is possible to have interference of scattered and ϭ F ͓␯ ͑ ͒␥ ͑ Ϫ␥ ͒ ͑ ͔͒ HW e x ␣ 1 5 e x unscattered neutrino waves which enhances the effect. & Coherence further allows one to decouple the evolution ϫ͓ ͑ ͒␥ ͑ Ϫ␥ ͒␯ ͑ ͔͒ equation of the neutrinos from the equations of the me- ¯e x ␣ 1 5 e x . (36) dium. In this approximation, the effect of the medium is described by an effective potential which depends on the density and composition of the matter (Wolfenstein, And the effective charged-current Hamiltonian due to 1978). the electrons in the medium is

G (e) F 3 ␣ H ϭ ͵ d p f͑E ,T͒͗͗e͑s,p ͉͒¯e͑x͒␥ ͑1Ϫ␥ ͒␯ ͑x͒␯ ͑x͒␥␣͑1Ϫ␥ ͒e͑x͉͒e͑s,p ͒͘͘ C & e e e 5 e e 5 e

G F 3 ϭ ␯ ͑x͒␥␣͑1Ϫ␥ ͒␯ ͑x͒ ͵ d p f͑E ,T͒͗͗e͑s,p ͉͒¯e͑x͒␥␣͑1Ϫ␥ ͒e͑x͉͒e͑s,p ͒͘͘, (37) & e 5 e e e e 5 e

where s is the electron spin and pe its momentum. The energy distribution function of the electrons in the medium, ͐ 3 ϭ ͗ ͘ f(Ee ,T), is assumed to be homogeneous and isotropic and is normalized as d pef(Ee ,T) 1. By ¯ we denote the averaging over electron spinors and summing over all electrons in the medium. Notice that coherence implies that s,pe are the same for initial and final electrons. Expanding the electron fields e(x) in plane waves we find

1 † ͗e͑s,p ͉͒¯e͑x͒␥␣͑1Ϫ␥ ͒e͑x͉͒e͑s,p ͒͘ϭ ͗e͑s,p ͉͒u ͑p ͒a ͑p ͒␥␣͑1Ϫ␥ ͒a ͑p ͒u ͑p ͉͒e͑s,p ͒͘, (38) e 5 e V e s e s e 5 s e s e e

where V is a volume normalization factor. The averaging GFNe gives H(e)ϭ ␯ ͑x͒␥ ͑1Ϫ␥ ͒␯ ͑x͒. (41) C & e 0 5 e 1 1 ͗ ͑ ͉͒ †͑ ͒ ͑ ͉͒ ͑ ͒͘ ϭ ͑ ͒ ͚ Š e s,pe as pe as pe e s,pe ‹ Ne pe , ␯ V 2 s The effective potential for e induced by its charged- (39) current interactions with electrons in matter is then given by where Ne(pe) is the number density of electrons with momentum pe . We assumed here that the medium has G N 2 ϩ Ϫ ϭ͗␯ ͉ ͵ 3 (e)͉␯ ͘ϭ F e ͵ 3 † equal numbers of spin 1/2 and spin 1/2 electrons, VC e d xHC e d xu␯u␯ † ϭN (s) & V and we used the fact that as (pe)as(pe) e (pe) is the number operator. We thus obtain ϭ& GFNe . (42) ͑ ͉͒ ͑ ͒␥ ͑ Ϫ␥ ͒ ͑ ͉͒ ͑ ͒ ͗e s,pe ¯e x ␣ 1 5 e x e s,pe ͘ ␯ Š ‹ For e the sign of V is reversed. This potential can also ␳ 1 be expressed in terms of the matter density : ϭ ͑ ͒ ͑ ͒␥ ͑ Ϫ␥ ͒ ͑ ͒ Ne pe ͚ u(s) pe ␣ 1 5 u(s) pe 2 s ␳ ϭ& Ӎ ␣ VC GFNe 7.6 Ye 14 3 eV, (43) N ͑p ͒ m ϩp p 10 g/cm ϭ e e ͫ e ” ␥ ͑ Ϫ␥ ͒ͬϭ ͑ ͒ e Tr ␣ 1 N p . (40) 2 2E 5 e e E ϭ ϩ e e where Ye Ne /(Np Nn) is the relative number density. Three examples that are relevant to observations are the Isotropy implies that ͐d3p pជ f(E ,T)ϭ0. Thus only the e e e following: p0 term contributes upon integration, with ͐ 3 ϭ ␳ϳ 3 ϳ Ϫ13 d pef(Ee ,T)Ne(pe) Ne (the electron number den- • At the Earth’s core 10 g/cm and VC 10 eV; ␳ϳ 3 ϳ Ϫ12 sity). Substituting Eq. (40) in Eq. (37) we obtain • At the solar core 100 g/cm and VC 10 eV.

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 354 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

␾ Following the same procedure we can obtain the ef- So i(x) has the form of a free-particle solution with E ϭ Ϫ fective potentials for any flavor neutrino or antineutrino local energy i(x) E Vii(x): due to interactions with different particles in the me- ␹ dium (for a list, see, for instance, Kim and Pevsner, E ϩ 1/2 ␯ ␯ ϭ i mi 2 2 1993). For ␮ and ␶ , VC 0 for most media, while for ␾ ͑ ͒ϭͫ ͬ ϫ ͱE Ϫ i x ͫ m ͬ , (48) 2E i i ␴ ␹ any active neutrino the effective potential due to i E ϩ x i mi neutral-current interactions in a neutral medium is VN ϭϪ & 1/ GFNn , where Nn is the number density of neu- where ␹ is the Pauli spinor. We make the following ap- trons. One can further generalize this analysis to other proximations: types of interactions (Bergmann, Grossman, and Nardi, (i) The scale over which V changes is much larger than 1999). the microscopic wavelength of the neutrino: .x)/VӶបm/E2ץ/Vץ) C. Evolution equation in matter: Effective mass and ␣ ␾ (ii) Expanding to first order in V implies that V12 x 2 mixing Ӎ␾ ␣ ␾ Ӎ␾ ͕͓ Ϫ ͔2Ϫ 2͖1/2 1 , V12 x 1 2 , and E Vii(x) mi Ӎ Ϫ Ϫ 2 There are several derivations in the literature of the E Vii(x) mi /2E. evolution equation of a neutrino system in matter (see, From (i) we find that the Dirac equations take the form for instance, Halprin, 1986; Mannheim, 1988). We follow ץ ប C1 here the discussion in Baltz and Weneser (1988). Con- EC ␾ ϭ ␣ ␾ ϩ͑␤m ϩV ͒C ␾ ϩV C ␾ , x 1 1 11 1 1 12 2 2ץ sider a state that is an admixture of two neutrino species 1 1 i x ͉␯ ͘ and ͉␯ ͘ or, equivalently, of ͉␯ ͘ and ͉␯ ͘: ץ e X 1 2 ប C2 ͉⌽͑x͒͘ϭ⌽ ͑x͉͒␯ ͘ϩ⌽ ͑x͉͒␯ ͘ EC ␾ ϭ ␣ ␾ ϩ͑␤m ϩV ͒C ␾ ϩV C ␾ . x 2 2 22 2 2 12 1 1ץ e e X X 2 2 i x ϭ⌽ ͑ ͉͒␯ ϩ⌽ ͑ ͉͒␯ 1 x 1͘ 2 x 2͘. (44) (49) ⌽ ␣ The evolution of in a medium is described by a system Then multiplying by x and using the equation of mo- ␾ of coupled Dirac equations: tion of i and (ii), we can drop the dependence on the ␾ spinor and obtain ץ ប ⌽ ϭͫ ␣ ϩ␤ ϩ ͬ⌽ ϩ ⌽ 2 ץ m1 V11 1 V12 2 , ប ץ E 1 x i x C1 m1 ϭͩ EϪV ͑x͒Ϫ ͪ C ϪV C , x 11 2E 1 12 2ץ i ץ ប ⌽ ϭͫ ␣ ϩ␤ ϩ ͬ⌽ ϩ ⌽ (m2 V22 2 V12 1 , (45 ץ E 2 x C m2ץ i x ប 2 ϭͩ Ϫ ͑ ͒Ϫ 2 ͪ Ϫ ␤ϭ␥ ␣ ϭ␥ ␥ E V22 x C2 V12C1 . (50) x 2Eץ where 0 and x 0 1 . The Vij terms give the ef- i fective potential for neutrino mass eigenstates. They can →␯ បϭ be simply derived from the effective potential for inter- Changing notations Ci,␣(x) i,␣(x) (and 1), re- moving the diagonal piece that is proportional to E, and action eigenstates [such as Vee of Eq. (42)]: rotating to the flavor basis, we can rewrite Eq. (50) in ϭ͗␯ ͉ ͵ 3 medium͉␯ ͘ϭ matrix form (Wolfenstein, 1978): Vij i d xHint j Ui␣V␣␣Uj*␣ . (46) ␯ 2 ␯ ץ ⌽ ϭ ␾ e Mw e We decompose the neutrino state: i(x) Ci(x) i(x). Ϫi ͩ ͪ ϭͩ Ϫ ͪͩ ͪ , (51) x ␯ 2E ␯ץ ␾ Here i(x) is the Dirac spinor part satisfying X X ␣ ͕͓ Ϫ ͑ ͔͒2Ϫ 2͖1/2ϩ␤ ϩ ␾ ͑ ͒ϭ ␾ ͑ ͒ „ x E Vii x mi mi Vii… i x E i x . where we have defined an effective mass matrix in mat- (47) ter

2ϩ 2 ⌬ 2 ⌬ 2 m1 m2 m m ϩ2EV Ϫ cos 2␪ sin 2␪ 2 e 2 2 M2 ϭ . (52) w ͩ ⌬ 2 2ϩ 2 ⌬ 2 ͪ m m1 m2 m sin 2␪ ϩ2EV ϩ cos 2␪ 2 2 X 2

⌬ 2ϭ 2Ϫ 2 m m Here m m2 m1. ␯ ␯ cos ␪ sin ␪ ␯ ͩ e ͪ ϭ ͑␪ ͒ͩ 1 ͪ ϭͩ m m ͪͩ 1 ͪ We deﬁne the instantaneous mass eigenstates in mat- ␯ U m ␯m Ϫ ␪ ␪ ␯m . (53) ␯m X 2 sin m cos m 2 ter, i , as the eigenstates of Mw for a ﬁxed value of x (or t). They are related to the interaction eigenstates through a unitary rotation, The eigenvalues of Mw , that is, the effective masses in

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 355

Ϫ Notice that once the sign of Ve VX (which depends on the composition of the medium and on the state X)is known, this resonance condition can only be achieved for a given sign of ⌬m2 cos 2␪, i.e., for mixing angles in only one of the two possible octants. We learn that the symmetry present in vacuum oscillations is broken by matter potentials. Also, if the resonant condition is achieved for two neutrinos, it cannot be achieved for antineutrinos of the same flavor, and vice versa. The ␪ mixing angle tan 2 m changes sign at AR . As can be seen Ͼ ␪ ӷ␪ in Fig. 3, for A AR we have m . We define an oscillation length in matter FIG. 2. Effective masses acquired in the medium by a system osc⌬ 2 L0 m of two massive neutrinos as a function of the potential A [see oscϭ Eq. (54)]. L , (58) ͱ͑⌬m2 cos 2␪ϪA͒2ϩ͑⌬m2 sin 2␪͒2 osc where the oscillation length in vacuum, L0 , was de- matter are given by (Wolfenstein, 1978; Mikheyev and fined in Eq. (31). The oscillation length in matter pre- Smirnov, 1985) sents a resonant behavior. At the resonance point the 2ϩ 2 oscillation length is m1 m2 ␮2 ͑x͒ϭ ϩE͑V ϩV ͒ 1,2 2 e X Losc oscϭ 0 LR ␪ . (59) 1 sin 2 ϯ ͱ͑⌬ 2 ␪Ϫ ͒2ϩ͑⌬ 2 ␪͒2 m cos 2 A m sin 2 , ␦ 2 The width (in distance) of the resonance, rR , corresponding to ␦A ϭ2⌬m2 sin2 2␪, is given by (54) R ␦ ␪ AR 2 tan 2 1 dA while the mixing angle in matter is given by ␦r ϭ ϭ , h ϵͯ ͯ , (60) R dA h R A dr ⌬m2 sin 2␪ ͯ ͯ R R tan 2␪ ϭ . (55) dr m ⌬m2 cos 2␪ϪA R The quantity A is defined by where we have defined the resonance height hR . For constant A, i.e., for constant matter density, the ϵ ͑ Ϫ ͒ A 2E Ve VX . (56) evolution of the neutrino system is described just in In Figs. 2 and 3 we plot, respectively, the effective terms of the masses and mixing in matter. But for vary- masses and the mixing angle in matter as functions of ing A, this is in general not the case. the potential A, for AϾ0 and ⌬m2 cos 2␪Ͼ0. Notice that even massless neutrinos acquire nonvanishing effec- D. Adiabatic versus nonadiabatic transitions tive masses in matter. Taking time derivative of Eq. (53), we find The resonant density (or potential) AR is defined as ␯ ␯m ␯˙ m ץ the value of A for which the difference between the ͩ e ͪ ϭ ˙ ͑␪ ͒ͩ 1 ͪ ϩ ͑␪ ͒ͩ 1 ͪ (␯ U m ␯m U m ␯m . (61 ץ :effective masses is minimal t X 2 ˙ 2 A ϭ⌬m2 cos 2␪. (57) R Using the evolution equation in the flavor basis, Eq. (51), we get ␯˙ m 1 ␯m ͩ 1 ͪ ϭ †͑␪ ͒ 2 ͑␪ ͒ͩ 1 ͪ i ␯m U m MwU m ␯m ˙ 2 2E 2 ␯m Ϫ † ˙ ͑␪ ͒ͩ 1 ͪ iU U m ␯m . (62) 2 ␪ For constant matter density, m is constant and the second term vanishes. In general, using the definition of the ␮ effective masses i(t) in Eq. (54) and subtracting a di- ␮2ϩ␮2 ϫ agonal piece ( 1 2)/2E I, we can rewrite the evolution equation as m Ϫ⌬͑ ͒ Ϫ ␪˙ ͑ ͒ m ␯˙ 1 t 4iE m t ␯ iͩ 1 ͪ ϭ ͩ ͪ ͩ 1 ͪ , (63) FIG. 3. The mixing angle in matter for a system of two massive ␯m ␯m ˙ 2 4E 4iE␪˙ ͑t͒ ⌬͑t͒ 2 neutrinos as a function of the potential A for two different m ⌬ ϵ␮2 Ϫ␮2 values of the mixing angle in vacuum [see Eq. (55)]. where we defined (t) 2(t) 1(t).

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 356 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

The evolution equations (63) constitute a system of ␮2͑tЈ͒ ͑ ͒Ӎ ϩ i coupled equations: the instantaneous mass eigenstates Ei tЈ p . (69) ␯m 2p i mix in the evolution and are not energy eigenstates. The importance of this effect is controlled by the rela- Thus the transition probability for the adiabatic case is ␪˙ given by tive size of the off-diagonal piece 4E m(t) with respect ⌬ ⌬ ӷ ␪˙ ͑␯ →␯ ͒ to the diagonal one (t). When (t) 4E m(t), the P e e ;t instantaneous mass eigenstates, ␯m , behave approxi- i i t 2 mately as energy eigenstates and they do not mix in the ϭ͚ͯ U ͑␪͒U*͑␪ ͒expͩ Ϫ ͵ ␮2͑tЈ͒dtЈͪͯ . ei ei m,0 2E i evolution. This is the adiabatic transition approximation. i t0 ␪ From the definition of m in Eq. (55), we find (70) ⌬m2 sin 2␪ For the case of two-neutrino mixing, Eq. (70) takes the ␪˙ ϭ A˙ . (64) m 2⌬͑t͒2 form ͑␯ →␯ ͒ϭ 2 ␪ 2 ␪ϩ 2 ␪ 2 ␪ The adiabaticity condition then reads P e e ;t cos m cos sin m sin 2 1 ␦͑t͒ 2EA⌬m sin 2␪ A˙ ϩ ␪ ␪ ͩ ͪ ⌬͑t͒ӷ ͯ ͯ. (65) sin 2 m sin 2 cos , (71) ⌬͑t͒2 A 2 2E ␪˙ where Since for small mixing angles the maximum of m occurs at the resonance point (as can be seen in Fig. 3), the t ␦͑t͒ϭ ͵ ⌬͑tЈ͒dtЈ strongest adiabaticity condition is obtained when Eq. t (65) is evaluated at the resonance (the generalization of 0 the condition of maximum adiabaticity violation to large t ϭ ͵ ͱ͓⌬ 2 ␪Ϫ ͑ Ј͔͒2ϩ͑⌬ 2 ␪͒2 Ј mixings can be found in Friedland, 2001 and Lisi et al. m cos 2 A t m sin 2 dt , t0 2001). We define the adiabaticity parameter Q at the resonance as follows: (72) ⌬m2 sin2 2␪ 4␲␦r which, in general, has to be evaluated numerically. There ϭ ϭ R are some analytical approximations for specific forms of Q ␪ osc , (66) E cos 2 hR LR A(tЈ): exponential, linear, etc. (see, for instance, Kuo ␦ and Pantaleone, 1989). For ␦(t)ӷE the last term in Eq. where we used the definitions of AR , rR , and hR in Eqs. (57) and (60). Writing it in this form, we see that (71) is averaged out and the survival probability takes the adiabaticity condition, Qӷ1, implies that many os- the form cillations take place in the resonant region. Conversely, 1 Շ ͑␯ →␯ ͒ϭ ͓ ϩ ␪ ␪͔ when Q 1 the transition is nonadiabatic. P e e ;t 1 cos 2 m cos 2 . (73) ␯ 2 The survival amplitude of a e produced in matter at Ͼ t0 and exiting the matter at t t0 can be written as fol- In Fig. 4 we plot isocontours of constant survival prob- lows: ability in the parameter plane (⌬m2,tan2 ␪) for the particular case of the Sun’s density, for which AϾ0. Notice ͑␯ →␯ ͒ϭ ͓␯ ͑ ͒→␯ ͑ ͔͒ 2 ␪ 2 ␪ A e e ;t ͚ A e t0 i t0 that, unlike sin 2 , tan is a single-valued function in i,j the full parameter range 0р␪р␲/2. Therefore it is a ϫA͓␯ ͑t ͒→␯ ͑t͔͒A͓␯ ͑t͒→␯ ͑t͔͒ more appropriate variable once matter effects are in- i 0 j j e cluded and the symmetry of the survival probability with (67) respect to the change of octant for the mixing angle is ␪Ͻ␲ ␯ with lost. As can be seen in the figure, for /4, P( e →␯ ) in matter can be larger or smaller than 1/2, in ͓␯ ͑ ͒→␯ ͑ ͔͒ϭ͗␯ ͑ ͉͒␯ ͑ ͒͘ϭ ͑␪ ͒ e A e t0 i t0 i t0 e t0 Uei* m,0 , contrast to the case of vacuum oscillations where, in the vacϭ Ϫ 1 2 ␪Ͼ 1 A͓␯ ͑t͒→␯ ͑t͔͒ϭ͗␯ ͑t͉͒␯ ͑t͒͘ϭU ͑␪͒, averaged regime, Pee 1 2 sin 2 2. j e e j ej In Fig. 4 we also plot the limiting curve for Qϭ1. To ␪ where Uei*( m,0) is the (ei) element of the mixing matrix the left of and below this curve, the adiabatic approxi- ␪ in matter at the production point and Uej( ) is the (ej) mation breaks down and the isocontours in Fig. 4 devi- element of the mixing matrix in vacuum. ate from the expression in Eq. (73). In this region, the In the adiabatic approximation the mass eigenstates ␪˙ off-diagonal term m cannot be neglected, and the mix- do not mix, so ing between instantaneous mass eigenstates is impor- ͓␯ ͑ ͒→␯ ͑ ͔͒ϭ␦ ␯ ͑ ͉͒␯ ͑ ͒ A i t0 j t ij ͗ i t i t0 ͘ tant. In this case we can write ͓␯ ͑ ͒→␯ ͑ ͔͒ϭ ␯ ͑ ͉͒␯ ͑ ͒ ␯ ͑ ͉͒␯ ͑ ͒ t A i t0 j t ͗ j t j tR ͗͘ j tR i tR ͘ ϭ␦ expͭ i ͵ E ͑tЈ͒dtЈͮ . (68) ij i ϫ͗␯ ͑ ͉͒␯ ͑ ͒͘ t0 i tR i t0 , (74)

Note that Ei is a function of time because the effective where tR is the point of maximum adiabaticity violation, ␮ mass i is a function of time, which, for small mixing angles, corresponds to the reso-

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 357

FIG. 5. The solar matter density proﬁle: solid line, for the model of Bahcall, Pinsonneault, and Basu (2001); dashed line, the exponential approximation.

FIG. 4. Isocontours of the survival probability P in the Sun. ee ⌬m2 cos 2␪ Also shown is the limit of applicability of the adiabatic ap- ␥ϵ␲ ϭ␲Q . (77) ϭ ͉ ˙ ͉ sin2 2␪ proximation Q 1 (dashed line). E A/A R ␯ ӷ ␪ nant point. The possibility of this level crossing can be When e is produced at A AR and is small, ␪ ϳ ␥ described in terms of the Landau-Zener probability m 90°. In this case is very large and PLZ Ӎ Ϫ␥ 2 ␪ Ӎ Ϫ ␲ (Landau, 1932; Zener, 1932): exp( sin ) exp͓ ( /4) Q͔, and the survival probability is simply given by P ϭ͉͗␯ ͑t ͉͒␯ ͑t ͉͒͘2 ͑iÞj͒. (75) LZ j R i R ␲ ͑␯ →␯ ͒Ӎ Ӎ ͩ Ϫ ͪ Introducing this transition probability in Eq. (67) we P e e ;t PLZ exp Q , find that in the nonadiabatic regime (after averaging out 4 the oscillatory term), the survival probability can be where Q is the adiabaticity parameter defined in Eq. written as (66). Since Qϳ⌬m2 sin2 2␪/E, the isocontours of con- 1 stant probability in this regime correspond to diagonal P͑␯ →␯ ;t͒ϭ ͓1ϩ͑1Ϫ2P ͒cos 2␪ cos 2␪͔. lines in the (⌬m2,tan2 ␪) plane in a log-log plot, as illus- e e 2 LZ m trated in Fig. 4. (76) The physical interpretation of this expression is straight- Ͼ E. Propagation in the sun: Mikheyev-Smirnov-Wolfenstein forward. An electron neutrino produced at A AR con- ␯ 2 ␪ ␯ effect sists of an admixture of 1 with fraction cos m and 2 2 ␪ with fraction sin m . In particular, for very small mixing As an illustration of the matter effects discussed in the ␪ ϳ␲ ␯ angles in vacuum, m /2 (see Fig. 3) so e is almost a previous section we describe now the propagation of a ␯ pure 2(t0) state. When the neutrino state reaches reso- ␯ Ϫ␯ neutrino system in the matter density of the Sun. ␯ ␯ ␯ ␯ ͓ e X nance, 2 ( 1) can become 2 ( 1) with probability 1 For the sake of concreteness we assume that X is some Ϫ ͔ ␯ ␯ PLZ or 1 ( 2) with probability PLZ . So after pass- superposition of ␮ and ␶. ␯ ␯ ing the resonance, the e flux contains a fraction of 1 : The solar density distribution decreases monotoni- P ϭsin2 ␪ P ϩcos2 ␪ (1ϪP ), and a fraction of ␯ : e1 m LZ m LZ 2 cally (see Fig. 5). For RϽ0.9R᭪ it can be approximated ϭ 2 ␪ ϩ 2 ␪ Ϫ ␯ Pe2 cos mPLZ sin m(1 PLZ). At the exit, 1 con- ϭ Ϫ 2 by an exponential Ne(R) Ne(0)exp( R/r0), with r0 sists of ␯ with fraction cos ␪, and ␯ consists of ␯ with 7 14 Ϫ1 e 2 e ϭR᭪/10.54ϭ6.6ϫ10 mϭ3.3ϫ10 eV . 2 ␪ fraction sin so (Haxton, 1986; Parke, 1986; Petcov, After traversing this density, the dominant component ϭ 2 ␪ ϩ 2 ␪ 1987) Pee cos Pe1 sin Pe2 which reproduces Eq. of the exiting neutrino state depends on the value of (76). the mixing angle in vacuum and the relative size of The Landau-Zener probability can be evaluated in the ⌬ 2 ␪ ϭ m cos 2 versus A0 2EGFNe,0 (at the neutrino pro- WKB approximation (for details see, for instance, Kim duction point): and Pevsner, 1993). The general form of the Landau- ⌬ 2 ␪ӷ (1) m cos 2 A0 : matter effects are negligible and Zener probability for an exponential density can be the propagation occurs as in vacuum. The survival prob- written as (Krastev and Petcov, 1988; Petcov, 1988a) ability at the sunny surface of the Earth is 2 exp͑Ϫ␥ sin ␪͒Ϫexp͑Ϫ␥͒ 1 1 ϭ 2 2 PLZ , P ͑⌬m cos 2␪ӷA ͒ϭ1Ϫ sin 2␪Ͼ . (78) 1Ϫexp͑Ϫ␥͒ ee 0 2 2

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 358 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

⌬ 2 ␪տ (2) m cos 2 A0 : the neutrino does not pass through the resonance point but its mixing is affected by the matter in the Sun. This effect is well described by an adiabatic propagation: 1 P ͑⌬m2 cos 2␪տA ͒ϭ ͑1ϩcos 2␪ cos 2␪͒. (79) ee 0 2 m ␪ Since the resonance point is not crossed, cos 2 m has the same sign as cos 2␪ and the corresponding survival probability is also larger than 1/2. ⌬ 2 ␪Ͻ (3) m cos 2 A0 : the neutrino can cross the resonance point on its way out. In this case, as discussed in the previous section, for small mixing angle in vacuum, ␯ ϳ␯m ␯m e 2 at the production point and remains 2 till the resonance point (for larger mixing but still in the first ␯ ␯m ␯m ␯m octant, e is a combination of 1 and 2 with larger 2 ␯ component). It is important in this case to find whether FIG. 6. The survival probability for a e state produced in the ⌬ 2 the transition is adiabatic. For the solar density, Qϳ1 center of the Sun as a function of E/ m for various values of corresponds to the mixing angle. ͑⌬m2/eV2͒sin2 2␪ ϳ3ϫ10Ϫ9. (80) IV. SOLAR NEUTRINOS ͑E/MeV͒cos 2␪ Solar neutrinos are electron neutrinos produced in the For Qӷ1 the transition is adiabatic and the neutrino thermonuclear reactions which generate the solar en- state remains in the same linear combination of mass ␪ ergy. These reactions occur via two main chains, the pp eigenstates after the resonance determined by m .As ␪ ␯ chain and the CNO cycle, shown in Figs. 7 and 8, respec- can be seen in Fig. 3, m (that is, the e component of ␯ tively. There are five reactions which produce e in the the state) decreases after crossing the resonance and, pp chain and three in the CNO cycle. Both chains result consequently, so does the survival probability Pee .In in the overall fusion of protons into 4He: ␯ particular, for small mixing angle, 2 at the exit point is ϩ ␯ 4p→4Heϩ2e ϩ2␯ ϩ␥, (83) almost a pure X and, consequently, Pee can be very e ϭ small. Explicitly, where the energy released in the reaction, Q 4mp Ϫ Ϫ Ӎ 1 m 4He 2me 26 MeV, is mostly radiated through the P ͑⌬m2 cos 2␪ϽA ,Qӷ1͒ϭ ͑1ϩcos 2␪ cos 2␪͒ photons and only a small fraction is carried by the neu- ee 0 2 m,0 trinos, ͗E ␯ ͘ϭ0.59 MeV. (81) 2 e In order to determine precisely the rates of the differ- ␪ can be much smaller than 1/2 because cos 2 m,0 and ent reactions in the two chains, which would give us the cos 2␪ can have opposite signs. Note that the smaller the final neutrino fluxes and their energy spectrum, a de- mixing angle in vacuum the larger is the deficit of elec- tailed knowledge of the Sun and its evolution is needed. tron neutrinos in the outgoing state. This is the Solar models (Bahcall and Ulrich, 1988; Turck-Chieze, Mikheyev-Smirnov-Wolfenstein (MSW) effect (Wolfen- Cahen, Casse, and Doom, 1988; Bahcall and Pinson- stein, 1978; Mikheyev and Smirnov, 1985). This behavior neault, 1992, 1995; Bahcall, Basu, and Pinsonneault, is illustrated in Fig. 6, where we plot the electron sur- 1998; Bahcall, Pinsonneault, and Basu, 2001) describe vival probability as a function of ⌬m2/E for different values of the mixing angle. For smaller values of ⌬m2/E (right side of Fig. 6) we approach the regime where QϽ1 and nonadiabatic effects start playing a role. In this case the state can jump ␯ ␯ from 2 into 1 (or vice versa) with probability PLZ . For ␯ ϳ␯ ␯ small mixing angle, at the surface 1 e and the e component of the exiting neutrino increases. This can be seen from the expression for Pee , ͑⌬ 2 ␪Ͻ Ӷ ͒ Pee m cos 2 A0 ,Q 1 1 ϭ ͓1ϩ͑1Ϫ2P ͒cos 2␪ cos 2␪͔, (82) 2 LZ m and from Fig. 6. For large mixing angles this expression is still valid. FIG. 7. The pp chain in the Sun.

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 359

FIG. 9. Neutrino ﬂuxes from the pp chain reactions as a function of neutrino energy.

cause the final state, 8Be, is a wide resonance. On the other hand, the 7Be neutrinos are almost monochromatic, with an energy width of about 2 keV, which is FIG. 8. The CNO cycle in the Sun. characteristic of the temperature in the core of the Sun. As discussed in Sec. III, to describe the evolution of the properties of the Sun and its evolution after entering neutrinos in solar matter, one needs to know other quan- the main sequence. The models are based on a set of tities that are predicted by the SSM, such as the density observational parameters: surface luminosity (3.844 and composition of solar matter, which give the solar ϫ1026 W), age (4.5ϫ109 yr), radius (6.961ϫ108 m) and electron number density. As discussed in Sec. III and mass (1.989ϫ1030 kg), and on several basic assump- shown in Fig. 5, the density of solar matter decreases tions: spherical symmetry, hydrostatic and thermal equi- monotonically and can be approximated by an exponen- librium, equation of state of an ideal gas, and present tial profile. Furthermore, in order to determine precisely surface abundances of elements similar to the primordial the evolution of the neutrino system one also needs to composition. Over the past four decades, the solar mod- know the production point distribution for the different els have been steadily refined as the result of increased neutrino fluxes. In Fig. 10 we show the prediction of observational and experimental information about the Bahcall, Pinsonneault, and Basu (2001) for the distribu- input parameters (such as nuclear reaction rates and the tion probability of the pp chain. surface abundances of different elements), more accu- The standard solar models have had notable suc- rate calculations of constituent quantities (such as radia- cesses. In particular, the comparison between the ob- tive opacity and equation of state), inclusion of new served and the theoretically predicted sound speeds and physical effects (such as element diffusion), and devel- opment of faster computers and more precise stellar evolution codes. We use as Standard Solar Model (SSM), the most updated version of the model developed by Bahcall, Pin- sonneault, and Basu, (2001, sometimes referred to as BP00). (At present, the various solar models give essentially the same results when interpreting the bulk of experimental data in terms of neutrino parameters. In this sense, the implications for particle physics discussed in this review are independent of the choice of solar model.) In Fig. 9 we show the energy spectrum of the fluxes from the five pp chain reactions. In what follows we refer to the neutrino fluxes by the corresponding source reaction, so, for instance, the neutrinos produced from 8B decay are called 8B neutrinos. Most reactions produce a neutrino spectrum characteristic of ␤ decay. For 8B neutrinos the energy distribution presents devia- FIG. 10. Production-point distribution of pp chain neutrinos tions with respect to the maximum allowed energy be- as a function of the distance from the solar center.

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 360 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

pressures has tested the SSM and provided accurate in- The special properties of this target include a low formation on the solar interior. The solar quantities that threshold (0.233 MeV) and a strong transition to the have been determined by helioseismology include the ground level of 71Ge, which gives a large cross section sound velocity and matter density as a function of solar for the lower-energy pp neutrinos. According to the radius, the depth of the convective zone, the interior SSM, approximately 54% of the events are due to pp rotation rate, and the surface helium abundance. The neutrinos, while 26% and 11% arise from 7Be and 8B excellent agreement between helioseismological obser- neutrinos, respectively. The extraction of 71Ge takes vations and solar model calculations has made a con- place every 3–4 weeks and the number of 71Ge decays ϭ vincing case that the large discrepancies between solar (t1/2 11.4 days) is measured in a proportional counter. neutrino measurements and solar model predictions The event rates measured by SAGE (Abdurashitov cannot be due to errors in the solar models. et al., 1999, 2002) and GALLEX (Hampel et al., 1999) are ϭ ϩ5.3ϩ3.7 A. Solar neutrino experiments RSAGE 70.8Ϫ5.2Ϫ3.2 SNU, ϭ Ϯ ϩ4.3 RGALLEX 77.5 6.2Ϫ4.7 SNU, 1. Chlorine experiment: Homestake while the prediction of the SSM is 130 SNU. The first result on the detection of solar neutrinos was The GALLEX program was completed in fall of 1997 announced by Ray Davis, Jr. and his collaborators from and its successor, GNO, started taking data in spring of Brookhaven in 1968 (Davis, Harmer, and Hoffman, 1998. The latest combined GALLEX/GNO result is 1968). In the Homestake gold mine in Lead, South Da- (Kirsten, 2002) ϳ kota, they installed a detector consisting of 615 tons of ϭ Ϯ ␯ RGALLEXϩGNO 70.8 5.9 SNU. C2Cl4 . Solar e’s are captured via Ϫ Since the pp flux is directly constrained by the solar 37Cl͑␯,e ͒37Ar. luminosity, in all stationary solar models there is a the- The energy threshold for this reaction is 0.814 MeV, so oretical minimum of the expected number of events of the relevant fluxes are the 7Be and 8B neutrinos. 7Be 79 SNU. Furthermore, the largest uncertainties on the neutrinos excite the Gamow-Teller transition to the capture cross section for 7Be neutrinos were consider- ground state with strength known from the lifetime of ably reduced after direct calibration using neutrinos the electronic capture of 37Ar. 8B neutrinos can excite from 51Cr decay as a source. higher states of 37Ar, including the dominant transition to the isobaric state with transition energy 4.99 MeV. 3. Water Cerenkov experiments: Kamiokande and The strengths of the Gamow-Teller transitions to these SuperKamiokande excited states are determined from the isospin-related ␤ decay, 37Ca(␤ϩ)37K. For the SSM fluxes, 78% of the Kamiokande and its successor SuperKamiokande expected number of events are due to 8B neutrinos (SK) in Japan are water Cerenkov detectors that are while 13% arise from 7Be neutrinos. The produced 37Ar able to detect in real time the electrons scattered from was extracted radiochemically approximately every the water by elastic interaction of the solar neutrinos, 37 ␯ ϩ Ϫ→␯ ϩ Ϫ three months, and the number of Ar decays (t1/2 a e a e . (84) ϭ34.8 days) measured in a proportional counter. The average event rate measured during the more The scattered electrons produce Cerenkov light, which than 20 years of operation is is detected by photomultipliers. Notice that, while the detection process in radiochemical experiments is purely ϭ Ϯ Ϯ RCl 2.56 0.16 0.16 SNU, a charged-current (W-exchange) interaction, the detec- where 1 SNUϭ10Ϫ36 captures/atom/sec. This corre- tion process of Eq. (84) goes through both charged- and sponds to approximately one third of the SSM predic- neutral-current (Z-exchange) interactions. Conse- tion (for the latest publications, see Cleveland et al., quently, the detection process (84) is sensitive to all ac- ␯ 1998, and Lande et al., 1999). tive neutrino flavors, although e’s (which are the only ones to scatter via W exchange) give a contribution that is about six times larger than that of ␯␮’s or ␯␶’s. 2. Gallium experiments: SAGE and GALLEX/GNO Kamiokande, with 2140 tons of water, started taking data in January 1987 and was terminated in February In January 1990 and May 1991, two new radiochemi- 1995. SK, with 45 000 tons of water (of which 22 500 are cal experiments using a 71Ga target started taking data, usable in solar neutrino measurements) started in May SAGE and GALLEX. The SAGE detector is located in 1996, and it has analyzed so far events corresponding to Baksan, Kaberdino-Balkaria, Russia, with 30 tons (in- 1258 days. The detection threshold in Kamiokande was creased to 57 tons from July 1991) of liquid metallic Ga. 7.5 MeV while SK started at a 6.5-MeV threshold and is GALLEX is located in Gran Sasso, Italy, and consists of currently running at 5 MeV. This means that these ex- 30 tons of GaCl -HCl. In these experiments the solar 3 periments are able to measure only the 8B neutrinos neutrinos are captured via (and the very small hep neutrino flux). Their results are 71Ga͑␯,eϪ͒71Ge. presented in terms of measured 8B flux.

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 361

FIG. 12. The zenith-angle dependence of the solar neutrino flux (statistical error only). The width of the nighttime bins was chosen to separate solar neutrinos that pass through the ␪ у FIG. 11. The electron recoil energy spectrum measured at Su- Earth’s dense core (cos z 0.84) from those that pass through Ͻ ␪ Ͻ perKamiokande normalized to the Standard Solar Model pre- the mantle (0 cos z 0.84). The horizontal line shows the diction, and the expectations for the best-fit points for the so- average flux. lutions from Fig. 16 below: LMA, large mixing angle; SMA, small mixing angle; LOW, low mass ratio; VAC, vacuum oscillations. words, of the position of the Sun in the sky. The results are presented as a nadir angle (the angle the Sun forms with the vertical) distribution of events. In Fig. 12 we The final result of Kamiokande (Fukuda et al., 1996) show their nadir angle distribution, corresponding to and the latest result of SK (Smy et al., 2002) are 1258 days of data. SK have also presented their results ⌽ ϭ͑ Ϯ Ϯ ͒ϫ 6 Ϫ2 Ϫ1 Kam 2.80 0.19 0.33 10 cm s , on the day-night variation in the form of a day-night ⌽ ϭ͑ Ϯ Ϯ ͒ϫ 6 Ϫ2 Ϫ1 asymmetry, SK 2.35 0.02 0.08 10 cm s , NϪD corresponding to about 40–50 % of the SSM prediction. ϵ ϭ Ϯ ͑ ͒Ϯ ͑ ͒ ANϪD 2 ϩ 0.021 0.020 stat. 0.013 syst. , There are three features unique to the water Ceren- D N kov detectors. First, they are real-time experiments. (85) Each event is individually recorded. Second, for each where D (N) is the event rate during the day (night) event the scattered electron keeps the neutrino direction period. The SK results show a small excess of events within an angular interval which depends on the neu- during the night but only at the 0.8␴ level. ͱ trino energy as 2me /E␯. Thus it is possible, for ex- In order to simultaneously account for the energy and ample, to correlate the neutrino detection with the posi- day-night variation, SK present the observed energy tion of the Sun. Third, the amount of Cerenkov light spectrum during the day and during the night separately produced by the scattered electron allows a measure- as well as in several night bins. This is the most conve- ment of its energy. In summary, the experiment provides nient observable for including both time and energy de- information on the time, direction, and energy for each pendence simultaneously in the analysis. The reason for event. As we discuss below, signatures of neutrino oscil- that is that the separate distributions—energy spectrum lations might include distortion of the recoil electron en- averaged in time and time dependence averaged in ergy spectrum, a difference between nighttime and day- energy—do not correspond to independent data samples time solar neutrino fluxes, or a seasonal variation in the and therefore have nontrivial correlations. neutrino flux. Observation of these effects would be Finally, the SK collaboration has also measured the strong evidence in support of solar neutrino oscillations seasonal dependence of the solar neutrino flux. Their independent of absolute flux calculations. Conversely, results after 1258 days of data are shown in Fig. 13. The nonobservation of these effects could constrain oscilla- points represent the measured flux, and the curve shows tion solutions to the solar neutrino problem. the expected variation due to the orbital eccentricity of Over the years, the SK collaboration has presented the Earth (assuming no new physics, and normalized to information on the energy and time dependence of their the measured total flux). As can be seen from the figure, event rates in different forms. In Fig. 11 we show their the data are consistent with the expected annual varia- spectrum, corresponding to 1258 days of data, relative to tion. the predicted spectrum (Ortiz, et al., 2000) normalized to the prediction of Bahcall, Pinsonneault, and Basu 4. SNO (2001). As seen from the figure, no significant distortion is observed and the data are compatible with a horizon- The Sudbury Neutrino Observatory (SNO) was first tal straight line. proposed in 1987 and it started taking data in November The SK collaboration has also measured the event 1999 (McDonald et al., 2000). The detector, a great rates as a function of the time of day or night or, in other sphere surrounded by photomultipliers, contains ap-

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 362 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

April 2002 they published their results from the first phase of the experiment, which included the day-night spectrum data in the full energy range above a threshold Ͼ Te 5 MeV. Their spectrum is the result of the combination of the three possible signals. Assuming an undistorted energy spectrum, they extract the individual rates ⌽CC ϭ͑ ϩ0.06Ϯ ͒ϫ 6 Ϫ2 Ϫ1 SNO 1.76Ϫ0.05 0.09 10 cm s , ⌽ES ϭ͑ Ϯ Ϯ ͒ϫ 6 Ϫ2 Ϫ1 SNO 2.39 0.24 0.12 10 cm s , ⌽NC ϭ͑ ϩ0.44ϩ0.46͒ϫ 6 Ϫ2 Ϫ1 SNO 5.09Ϫ0.43Ϫ0.43 10 cm s , ϭ Ϯ and a day-night asymmetry ANϪD 0.07 0.049(stat.) Ϯ0.013(syst.). FIG. 13. Seasonal variation of the solar neutrino flux (statisti- ⌽NC cal errors only). The curve shows the expected seasonal varia- Notice that the neutral-current measured flux SNO is tion of the flux induced by the eccentricity of the Earth’s orbit. in excellent agreement with the prediction of the SSM, ⌽BP00ϭ ϩ0.20 ϫ 6 Ϫ2 Ϫ1 8B (5.05Ϫ0.16) 10 cm s , which confirms that the observed flux of active neutrinos is consistent with proximately 1000 tons of heavy water, D2O, and is located at the Creighton mine, near Sudbury in Canada. It the SSM. is immersed in ultrapure light water to provide shielding from radioactivity. SNO was designed to give a model- independent test of the possible explanations of the observed deficit in the solar neutrino flux by having sensi- 5. Future: Borexino and low-energy experiments ␯ tivity to all flavors of active neutrinos and not just to e . This sensitivity is achieved because energetic neutri- The Borexino experiment (Oberauer, 1999) is de- nos can interact in the D2O of SNO via three different signed to detect low-energy solar neutrinos in real time reactions. Electron neutrinos may interact via the through the observation of the elastic-scattering process ␯ ϩ Ϫ→␯ ϩ Ϫ charged-current reaction a e a e . The energy threshold for the recoil ␯ ϩ → ϩ ϩ Ϫ electrons is 250 keV. It will use 300 tons of liquid scintil- e d p p e (86) lator in an unsegmented detector with 2000 photomulti- and can be detected above an energy threshold of a few plier tubes. The event rate predicted by the SSM for a Ͼ ␯ MeV (presently Te 5 MeV). All active neutrinos ( a fiducial volume of about 100 tons is about 50 events per ϭ␯ ␯ ␯ e , ␮ , ␶) interact via the neutral-current reaction day, mostly generated by the 0.86-MeV monoenergetic 7 ␯ ϩ → ϩ ϩ␯ line of Be solar neutrinos. Since this line gives a char- a d n p a , (87) acteristic spectral signature in the elastic-scattering pro- with an energy threshold of 2.225 MeV. Nonsterile neu- cess, the flux on the earth of 7Be solar neutrinos will be ␯ ϩ Ϫ trinos can also interact via elastic scattering, a e determined and it will be possible to check whether it is →␯ ϩ Ϫ a e , but with a smaller cross section. suppressed with respect to the flux predicted by the The experimental plan of SNO consists of three SSM, as is suggested by the results of current experi- phases, of approximately one year duration each. In its ments. The Borexino experiment is under construction first year of operation, SNO has concentrated on the in the Laboratori Nazionali del Gran Sasso in Italy and measurement of the charged-current reaction rate, while is scheduled to start data taking in the near future. in a following phase, after the addition of MgCl2 salt to A new generation of experiments aiming at a high- enhance the neutral-current signal, it will also perform a precision real-time measurement of the low-energy solar precise measurement of the neutral-current rate. In the neutrino spectrum is now under study (Lanou et al., third phase, the salt will be eliminated and a network of 1993; Arpesella, Broggini, and Cattadori, 1996; Ragha- proportional counters filled with 3He will be added with van, 1997; de Bellfon, 1999). Some of them, such as the purpose of directly measuring the neutral-current HELLAZ, HERON, and SUPER-MuNu, intend to de- rate 3He(n,p)3H. tect the elastic scattering of electron neutrinos with the SNO can also perform measurements of the energy electrons of a gas and measure the recoil electron energy spectrum and time variation of the event rates. But the and its direction of emission. The proposed experiment uniqueness of SNO lies in its ability to test directly LENS plans to detect the electron neutrino via its ab- ␯ whether the deficit of solar e is due to changes in the sorption in a heavy nuclear target, with the subsequent flavor composition of the solar neutrino beam, since the emission of an electron and a delayed gamma emission. charged-current/neutral-current ratio compares the The expected rates of these experiments for the pro- ␯ ϳ number of e interactions with those from all active fla- posed detector sizes are of the order of 1 – 10 pp neu- vors. This comparison is independent of the overall flux trinos a day. Consequently, with a running time of two normalization. years, they can reach a sensitivity of a few percent in the In June 2001, SNO published their first results (Ah- total neutrino rate at low energy, provided that they can mad, 2001) on the charged-current measurement, and in achieve sufficient background rejection.

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 363

TABLE II. Standard solar model predictions: solar neutrino ﬂuxes and neutrino capture rates for the different experiments, with 1␴ uncertainties.

Flux Cl Ga SK SNO(CC) SNO(NC) Source (1010 cmϪ2 sϪ1) (SNU) (SNU) (106 cmϪ2 sϪ1) (106 cmϪ2 sϪ1)

ϩ0.01 pp 5.95 (1.00Ϫ0.01) 0.0 69.7 0.0 0.0 ϫ Ϫ2 ϩ0.015 pep 1.40 10 (1.00Ϫ0.015) 0.22 2.8 0.0 0.0 hep 9.3ϫ10Ϫ7 0.04 0.1 0.0093 0.0093 7 ϫ Ϫ1 ϩ0.10 Be 4.77 10 (1.00Ϫ0.10) 1.15 34.2 0.0 0.0 8 ϫ Ϫ4 ϩ0.20 B 5.05 10 (1.00Ϫ0.16) 6.76 12.2 5.05 5.05 13 ϫ Ϫ2 ϩ0.21 N 5.48 10 (1.00Ϫ0.17) 0.09 3.4 0.0 0.0 15 ϫ Ϫ2 ϩ0.25 O 4.80 10 (1.00Ϫ0.19) 0.33 5.5 0.0 0.0 17 ϫ Ϫ4 ϩ0.25 F 5.63 10 (1.00Ϫ0.25) 0.0 0.1 0.0 0.0 ϩ1.3 ϩ9 ϩ1.01 ϩ1.01 Total 7.6Ϫ1.1 128Ϫ7 5.05Ϫ0.81 5.05Ϫ0.81 Measured 2.56Ϯ0.226 70.8Ϯ4.3 2.35Ϯ0.083 1.76Ϯ0.11 5.09Ϯ0.64 Measured 0.337Ϯ0.065 0.55Ϯ0.048 0.465Ϯ0.094 0.348Ϯ0.073 1.01Ϯ0.23 SSM

B. The solar neutrino problem recent determination of these parameters can be found in Bahcall, 2002b). Since there are at present six experi- Table II summarizes our present knowledge of the so- ments, one cannot use all seven of the neutrino fluxes as lar neutrino fluxes, their contribution to the expected free parameters, and some additional constraints must rates, and the data from measurements of solar neutrino be used. For instance, the ratio of the pep to pp fluxes is experiments. The predicted event rates are linear func- usually taken to be the same as in the SSM. Also, the tions of the seven important neutrino fluxes: p-p, pep, CNO nuclear reactions are assumed to be in equilib- hep, 7Be, 8B, 13N, and 15O. We can make the following rium, and the 13N and 15O neutrino fluxes are taken to statements (see the last row of the table): be exactly equal (or, in some analyses, both are taken to • Before the neutral-current measurement at SNO, all be zero). Finally the contribution from hep neutrinos is experiments observed a flux that was smaller than the usually neglected. SSM predictions, ⌽obs/⌽SSMϳ0.3– 0.6. A fit of this type (Bahcall, 2002b) shows that, previous • The deficit is not the same for the various experi- to the SNO measurement, the best fit always corre- ments, which may indicate that the effect is energy sponded to an unphysical, negative 7Be flux. Forcing all dependent. fluxes to be positive, the hypothesis of no new physics was rejected at the effective 2.5␴ level (99% C.L.). Fur- These two statements constitute the solar neutrino prob- thermore, it required that 7Be neutrinos be entirely lem (Bahcall, Bahcall, and Shaviv, 1968; Bahcall and missing, a result which many authors have argued is not Davis, 1976). More quantitatively, a fit to the data using physically or astrophysically reasonable (for instance, the model of Bahcall, Pinsonneault, and Basu (2001) Bahcall and Bethe, 1990; Hata, Bludman, and Lan- ␴ shows a disagreement of 6.7 , which means that the gacker, 1994; Parke, 1995; Heeger and Robertson, 1996; probability of explaining this result as a consequence of Bahcall, Krastev, and Smirnov, 1998). a statistical fluctuation of standard particle physics is P The results of SNO have provided further model- ϭ ϫ Ϫ11 3 10 (we follow here the latest discussion by Bah- independent evidence on the problem. Both SNO and call, 2002a). SK are sensitive mainly to the 8B flux. Without new One may wonder about the model dependence of the physics, the measured fluxes in any reaction at these two solar neutrino problem. To answer this question one can experiments should be equal. Villante, Fiorentini, and allow all the solar neutrino fluxes, with undistorted en- Lisi (1999) showed how one can choose the energy ergy spectra, to be free parameters in fitting the mea- thresholds for the SK and SNO experiments in such a sured solar neutrino event rates, subject only to the con- way that the response functions for the two experiments dition that the total observed luminosity of the Sun is are made approximately equal (see also Fogli, Lisi, produced by nuclear fusion. This luminosity constraint Montanino, and Palazzo, 2001). The advantage of this can be written as a convenient linear equation in the method is that some of the systematic errors are re- neutrino fluxes: duced, but there is a slight loss of statistical power (Bah- ␣ ␾ call, 2002a; Berezinski, 2001). The first reported SNO ϭ ͩ i ͪͩ i ͪ 1 ͚ Ϫ Ϫ , (88) charged-current result compared with the elastic- 10 MeV 8.532ϫ1010 cm 2 s 1 i scattering rate from SK showed that the hypothesis of ␾ ϭ ϳ ␴ where i are the individual neutrino fluxes (i pp, pep, no flavor conversion was excluded at 3 . 7 8 13 15 ␣ hep, Be, B, N, and O) and i is the luminous Finally, with the neutral-current measurement at SNO energy released in the corresponding reaction (the most one finds that

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 364 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

SNO NC,SNO CC,SNO ␾␮␶ ϵ␾ Ϫ␾ expression for this phase can be found in Lisi, Marrone, 8B 8B Montanino, Palazzo, and Petcov, 2001). In the evalua- ϭ͑ Ϯ ϩ0.48͒ϫ 6 Ϫ2 Ϫ1 3.41 0.45Ϫ0.45 10 cm s . (89) tion of both Pi and Pie , the effect of coherent forward interaction with the matter of the Sun and the Earth This result provides evidence for neutrino flavor transi- must be taken into account. tion (from ␯ to ␯␮ ␶) at the level of 5.3␴. This evidence e , From Eq. (91) one can recover more familiar expres- is independent of the solar model. sions for Pee that hold in certain limits: (1) For ⌬m2/EՇ5ϫ10Ϫ17 eV, the matter effect sup- C. Solar neutrino oscillation probabilities presses flavor transitions in both the Sun and the Earth. Consequently, the probabilities P1 and P2e are simply The most generic and popular explanation of the solar ␯ the projections of the e state onto the mass eigenstates: neutrino anomaly is given by the introduction of neu- ϭ 2 ␪ ϭ 2 ␪ ␯ P1 cos ,P2e sin . In this case we are left with the trino masses and mixing, leading to oscillations of e ␯ ␯ ␯ standard vacuum oscillation formula, into an active ( ␮ and/or ␶) or a sterile ( s) neutrino. vacϭ Ϫ 2 ␪ 2͓⌬ 2͑ Ϫ ͒ ͔ We now discuss some issues that have been raised in the Pee 1 sin 2 sin m L r /4E , (93) literature concerning the computation of the corresponding neutrino survival probabilities in the full range which describes the oscillations on the way from the sur- of mass and mixing relevant to the solar neutrino prob- face of the Sun to the surface of the Earth. The problem. ability is symmetric under the change of octant, ␪↔ ␲/2 Ϫ␪, and change of sign ⌬m2↔Ϫ⌬m2 (see Sec. 1. Quasivacuum oscillations and the dark side III). This symmetry implies that each point in the ⌬ 2 2 ␪ The presence of neutrino mass and mixing implies the ( m ,sin 2 ) parameter space corresponds to two possibility of neutrino oscillations. Solar electron neutri- physically independent solutions, one in each octant. nos can undergo oscillations either in vacuum or via the Averaging Eq. (93) over the Earth’s orbit, L(t) ϭ ͓ Ϫ␧ ␲ ͔ matter-enhanced MSW mechanism, depending on the L0 1 cos(2 t/T) , one gets actual values of mass-squared differences and mixing ⌬ 2 1 m L0 angles. However, this simplified picture of solar neutrino ͗Pvac͘ϭ1Ϫ sin2 2␪ͫ1Ϫcosͩ ͪ ee 2 2E oscillations contains a set of approximations that are not ␧⌬ 2 always valid in the context of solar neutrinos. In order to m L0 ϫJ ͩ ͪͬ, (94) clarify this issue let us first review the calculation of the 0 2E solar neutrino survival probability in the two-neutrino ␧ϭ case. where 0.0167 is the orbit eccentricity and J0 is the ␯ The survival amplitude for a solar e of energy E at a Bessel function. In Fig. 14(a) we display the value of ͗ vac͘ ⌬ 2 terrestrial detector can be written as follows: Pee as a function of 4E/ m . As can be seen in the 2 2 figure, for large values of ⌬m the probability averages ϭ S E ͓Ϫ 2͑ Ϫ ͒ ͑ ͔͒ out to a constant value 1Ϫ 1 sin2(2␪). Aee ͚ AeiAieexp imi L r / 2E . (90) 2 iϭ1 (2) For ⌬m2/Eտ10Ϫ14 eV, the last term in Eq. (91) S ␯ →␯ vanishes and we recover the incoherent MSW survival Here Aei is the amplitude of the transition e i from E probability. In this case, P1 and P2e must be obtained by the production point to the surface of the Sun, and Aie ␯ →␯ solving the evolution equation of the neutrino states in is the amplitude of the transition i e from the surface of the Earth to the detector. The propagation am- the Sun and the Earth, respectively. As discussed in Sec. plitude in vacuum from the surface of the Sun to the III.D, the approximate solution for the evolution in the surface of the Earth is given by the exponential term, Sun takes the well-known form where L is the distance between the center of the Sun 1 1 and the surface of the Earth, and r is the radius of the P ϭ ϩͩ ϪP ͪ cos 2␪ . (95) 1 2 2 LZ m,0 Sun. The corresponding survival probability Pee is then given by Here PLZ denotes the standard Landau-Zener probabil- P ϭP P ϩP P ϩ2ͱP P P P cos ␰. (91) ity of Eq. (75) which, for the exponential profile, takes ee 1 1e 2 2e 1 2 1e 2e ␪ the form shown in Eq. (77), and m,0 is the mixing angle ϵ͉ S ͉2 Here Pi Aei is the probability that the solar neutri- in matter at the neutrino production point given in Eq. ͉␯ ͘ nos reach the surface of the Sun as i , while Pie (55). For the approximation of an exponential density ϵ͉ E ͉2 ␯ 2 Aie is the probability of i arriving at the surface of profile, ␥ϭ␲ (⌬m /E) r , which is independent of the ␯ 0 the Earth to be detected as a e . Unitarity implies P1 point in the Sun where the resonance takes place. Im- ϩ ϭ ϩ ϭ ␰ P2 1 and P1e P2e 1. The phase is given by provement over this constant-slope exponential density ⌬m2͑LϪr͒ profile approximation can be obtained by numerically ␰ϭ ϩ␦ , (92) deriving the exact Ne(r) profile at the resonant point. In 2E ␥ϭ␲ ⌬ 2 this case ( m /E) r0(rres). where ␦ contains the phases due to propagation in the During the day there is no Earth-matter effect. The Sun and in the Earth and can be safely neglected (an survival probabilities Pie are obtained by simple projec-

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 365

(Fogli, Lisi, and Montanino, 1994, 1996; de Gou- vea, Friedland, and Murayama, 2000; Fogli, Lisi, Montanino, and Palazzo, 2000a; Gonzalez-Garcia and Penã-Garay, 2000). (iii) The analysis of the survival probability of solar neutrinos would be simplified if there were a region where E/⌬m2 were small enough that vacuum oscillations were averaged out and large enough to be in the extremely nonadiabatic MSW regime. This, however, is not the case, as can be seen from comparison of panels (a) and (b) of Fig. 14. In the intermediate range, 2ϫ1014Շ4E/⌬m2 Շ1016 eVϪ1, adiabaticity is violated and the cos ␰ coherent term should be taken into account. The result is similar to vacuum oscillations, but with small matter corrections. We define this case as quasivacuum oscillations (Petcov, 1988b; Petcov and Rich, 1989; Pakvasa and Pantaleone, 1990; Pantaleone, 1990; Fogli, Lisi, Montanino, and Palazzo 2000b; Friedland, 2000; Lisi, Marrone, Montanino, Palazzo, and Petcov, 2001). The range of E/⌬m2 for the quasivacuum oscillation regime depends on the value of E/⌬m2 for which the MSW probability in Eq. (96) acquires the Ϫ 1 2 ␪ ⌬ 2 asymptotic value 1 2 sin 2 : the smaller E/ m , the more separated the MSW and vacuum regimes are, and the narrower the quasivacuum oscillation region.

⌬ 2 It is clear from these considerations that in order to FIG. 14. Pee as a function of 4E/ m . In panel (a) matter effects are ignored [see Eq. (94)]. Panels (b) (large mixing compute a survival probability for solar neutrinos that angle) and (c) (small mixing angle) take into account the MSW would be valid for any experiment and any value of the effect [see Eq. (96)]. neutrino mass and mixing, one has to evaluate the full expression (91). The results that we show here were ob- ␯ ϭ tained using the general expression for the survival tion of the e state onto the mass eigenstates: P2e,D 1 Ϫ ϭ 2 ␪ probability in Eq. (91), with P1 and P2e found by nu- P1e,D sin . One obtains the expression in Eq. (76), merically solving the evolution equation for the Sun and 1 1 the Earth. For the Sun, the evolution equation has to be PMSWϭ ϩͩ ϪP ͪ cos 2␪ cos 2␪. (96) ee,D 2 2 LZ m,0 solved from the neutrino production point to the edge of the Sun and averaged over the production point distri- In Figs. 14(b) and (c) we plot the survival probability as bution shown in Fig. 10. a function of 4E/⌬m2 for a large and a small mixing angle, respectively. Let us make some remarks concerning Eq. (96): (i) In both the limits of large and very small E/⌬m2, 2. Evolution in the Earth MSW→ Ϫ 1 2 ␪ Pee 1 2 sin 2 [see Fig. 14(b)], which is the In this section we discuss matter effects in the Earth same expression as for averaged vacuum oscilla- (for some of the original work see Bouchez et al., 1986; tions. This result, however, comes from very dif- Baltz and Weneser, 1987, 1988; Mikheyev and Smirnov, ferent reasons in the two regimes. For large 1987). The density of the Earth’s matter, unlike that of ⌬ 2 ϭ 2 ␪ ␪ ϭϪ E/ m , PLZ cos and cos 2 m,0 1. For small the Sun, is not strongly varying. It consists mainly of two ⌬ 2 ϭ ␪ ϭ ␪ E/ m , PLZ 0 and cos 2 m,0 cos 2 . layers, the lower-density mantle and the higher-density MSW (ii) Due to matter effects, Pee is only symmetric un- core, each with approximately constant density. The core der the simultaneous transformation (⌬m2,␪) radius is Lϭ2896 km. →(Ϫ⌬m2,␲/2 Ϫ␪). For ⌬m2Ͼ0, the resonance When a neutrino crosses only the mantle, the effects is only possible for ␪Ͻ␲/4, and MSW solutions of the Earth’s matter are well approximated by evolu- are usually plotted in the (⌬m2,sin2 2␪) plane as- tion with a constant potential. In this case, the resulting suming that now each point on this plane repre- probability simpliﬁes to sents only one physical solution with ␪ in the ﬁrst 4EV ␲L octant. But in principle, nonresonant solutions are ϭ 2 ␪ϩ e 2 ␪ 2 P2e sin ⌬ 2 sin 2 E sin , (97) also possible for ␪Ͼ␲/4, the so-called dark side m Lm

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 366 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

␪ where E is the mixing angle in the Earth [Eq. (55)], L is For absorption in chlorine and gallium, the cross sec- ␴ the distance traveled by the neutrino within the Earth, tions e,i(E) can be found in Bahcall (1997). The cross and Lm is the oscillation length in matter [Eq. (58)]. sections for SNO and SK (elastic scattering) can be ob- The right-hand side of Eq. (97) consists of two terms. tained from the differential cross sections of Nakamura The first term gives a simple projection of the mass to et al. (2001) and Bahcall, Kamionkowsky, and Sirlin flavor state in vacuum (which corresponds to the prob- (1995), respectively, by integrating with the correspond- ability during the daytime). The second is the regenera- ing detector resolutions and for the given detection tion term and will be denoted by freg (for details see thresholds. In particular, for the SK and SNO energy ˜ Gonzalez-Garcia, Pena-Garay, and Smirnov, 2001). It spectrum data, to obtain Rth in a given energy bin one contains the Earth matter effects and it is always posi- has to integrate within the corresponding electron recoil tive. Averaging out the distance-dependent term, we can energy bin and to take into account that finite energy now rewrite Eq. (91): resolution implies that the measured kinetic energy T of N ϭ D Ϫ͑ Ϫ ͒ ␪ Pee Pee 1 2PLZ cos 2 m,0freg , (98) the scattered electron is distributed around the true ki- Ј D N ␯ netic energy T according to a resolution function where Pee and Pee are the e survival probabilities dur- Ј Ͼ Res(T,T ) of the form (Bahcall, Krastev, and Lisi, 1997) ing the day and night, respectively. Since freg 0 and, for the interesting cases of MSW transitions in the Sun, 1 ͑TϪTЈ͒2 ␪ Ͻ D N Res͑T,TЈ͒ϭ expͫϪ ͬ, (100) cos 2 m,0 0, the relative magnitude of Pee and Pee de- ͱ 2 Ϫ Ͻ 2␲s 2s pends on the sign of (1 2PLZ). For PLZ 0.5 (and, in ϭ ϭ ͱ ϭ particular, for adiabatic MSW transitions, PLZ 0) the where s s0 TЈ/MeV, and s0 0.47 MeV for SK (Faid survival probability during the night is larger than dur- et al., 1997). On the other hand, the distribution of the ing the day. The opposite holds for large adiabaticity true kinetic energy TЈ for an interacting neutrino of en- Ͼ violations, PLZ 0.5. ergy E␯ is dictated by the differential cross section When neutrinos cross the core, adiabaticity may be d␴␣(E␯ ,TЈ)/dTЈ and the kinematic limits are violated in their evolution. The abrupt density change E␯ between the mantle and the core may induce a new form р Јр¯ Ј͑ ͒ϭ 0 T T E␯ ϩ . (101) of resonance, different from the MSW resonance (Pet- 1 me/2E␯ cov, 1998; Akhmedov, 1999). This parametric resonance For assigned values of s0 , Tmin , and Tmax , the corrected (Ermilova, Tsarev, and Chechin, 1986; Akhmedov, 1988) cross section ␴ (E)(aϭe,x) is given as is relevant mainly for small mixing angles. a Tmax ¯ Ј ␴ ͑ ͒ϭ ͵ ͵ T (E␯) Ј a E␯ dT dT D. Two-neutrino oscillation analysis Tmin 0

d␴␣͑E␯ ,TЈ͒ 1. Predictions ϫRes͑T, TЈ͒ . (102) dTЈ In an experiment i, the expected event rate in the th For data taken within a given zenith angle period i, presence of oscillations, Ri , can be written as follows: the expected number of events in the presence of oscillations is th R ϭ ͚ ␾ ͵ dE␯ ␭ ͑E␯͒ϫ͕␴ ͑E␯͒͗P ͑E␯ ,t͒͘ i k k e,i ee ␶ ⌽ kϭ1,8 1 (cos max,i) thϭ ͵ ␶ ␾ ͵ ␭ ͑ ͒ Ri d ͚ k dE␯ k E␯ ϩ␴ ͑ ͓͒ Ϫ ͑ ͒ ͔ ⌬␶ ␶(cos ⌽ ) kϭ1,8 x,i E␯ 1 ͗Pee E␯ ,t ͘ ͖, (99) i min,i ␾ ϫ ␴ ͑ ͒ ͑ ␶͒ where E␯ is the neutrino energy, k is the total neutrino ͕ e,i E␯ ͗Pee E␯ , ͘ ␭ flux, and k is the neutrino energy spectrum (normalized ϩ␴ ͑ ͓͒ Ϫ͗ ͑ ␶͔͖͒͘ ␴ ␴ x,i E␯ 1 Pee E␯ , , (103) to 1) from the solar nuclear reaction k. Here e,i ( x,i) ␯ ␯ ϭ␮ ␶ is the e ( x ,x , ) interaction cross section in the where ␶ measures the annual averaged length of the pe- ⌬␶ ϭ␶ ⌽ Ϫ␶ ⌽ standard model with the target corresponding to experi- riod i normalized to 1: i (cos max,i) (cos min,i). ␯ ment i, and ͗Pee(E␯ ,t)͘ is the time-averaged e survival For instance, if the period i corresponds to an entire day ϭ ϭ ⌬␶ ϭ probability. The expected signal in the absence of oscil- (i D) or night (i N) then D(N) 0.5. From these BP00 lations, Ri , can be obtained from Eq. (99) by substi- predictions one can easily build the corresponding ex- ϭ tuting Pee 1. In Table II we give the expected rates at pected day-night asymmetry. In Fig. 15 we show the iso- ⌬ 2 2 ␪ the different experiments which we obtain using the contours of expected ANϪD at SK in the ( m ,tan ) fluxes of Bahcall, Pinsonneault, and Basu (2001). plane for active neutrino oscillations (the results are For the chlorine, gallium, and SNO (charged-current) very similar for sterile neutrinos). As discussed in Sec. measurements, only the electron neutrino contributes IV.C, in most of the parameter space, the effect of trav- ␴ ␯ and the x,i term in Eq. (99) vanishes. For elastic scat- eling through the Earth is the regeneration of the e tering at SK or SNO, there is a possible contribution component, resulting in a positive day-night asymmetry from the neutral-current interaction of the other active with the exception of the region where the nonadiaba- neutrino flavors present in the beam. For the neutral- ticity of the oscillations in the Sun is important (tan2 ␪ current rate at SNO, all active flavors contribute equally. ϳ10Ϫ3,⌬m2ϳ3ϫ10Ϫ6 eV2). As discussed above, SK

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 367

FIG. 16. Allowed oscillation parameters (at 90%, 95%, 99%, and 99.7% C.L.) from the analysis of the total event rates of the chlorine, gallium, SK, and SNO charged-current experiments. The best-ﬁt point is marked with a star.

FIG. 15. Isocontours of the day-night asymmetry at SK.

S34 , S1,14 , and S17), the solar luminosity, the metalicity Z/X, the Sun’s age, the opacity, the diffusion, and the has observed a very small day-night asymmetry. This ob- 7 servation implies that some of the oscillation parameter electronic capture of Be, CBe . These errors are space can be excluded, as we discuss below. strongly correlated as the same astrophysical factor can In the same fashion, integrating Eq. (103) for the dif- affect the different neutrino production rates. Another ferent electron recoil energy bins and for the day and source of theoretical error arises from uncertainties in night periods separately, one can obtain predictions for the neutrino interaction cross section for the different SK and SNO day-night spectra. detection processes. For a detailed description of how to include all these uncertainties and correlations in the ␴ construction of ij we refer the reader to the work of 2. Analysis of total event rates: allowed masses and mixing Fogli and Lisi (1995a). Updated uncertainties can be found in Fogli, Lisi, Montanino, and Palazzo (2000a) The goal of analyzing of the solar neutrino data in and Gonzalez-Garcia and Penã-Garay (2001). terms of neutrino oscillation is to determine which range The results of the analysis of the total event rates are of mass-squared difference and mixing angle is respon- shown in Fig. 16, where we plot the allowed regions that sible for the observed deficit (see, for instance, Hata and correspond to 90%, 95%, 99%, and 99.73% (3␴) C.L. Langacker, 1997; Bahcall, Krastev, and Smirnov, 1998; ␯ for e oscillations into active neutrinos. Fogli, Lisi, and Montanino, 1998; Gonzalez-Garcia, de As can be seen in the figure, for oscillations into active Holanda, et al., 2000). In order to answer this question neutrinos there are several oscillation regimes which are in a statistically meaningful way, one must compare the compatible within errors with the experimental data. predictions in the different oscillation regimes with the These allowed parameter regions are labeled as MSW observations, including all the sources of uncertainties small mixing angle (SMA), MSW large mixing angle ␹2 and their correlations, by building, for instance, the (LMA), MSW low mass (LOW), and vacuum oscilla- function tions (VAC). Before including the SNO (charged- current) data, the best fit corresponded to the SMA so- ␹2 ϭ ͑ thϪ exp͒␴Ϫ2͑ thϪ exp͒ R ͚ Ri Ri ij Rj Rj . (104) lution, but after SNO the best fit corresponds to the i LMA solution. For the LMA solution, oscillations for th exp 8 Here Ri is the theoretical expectation (99), Ri is the the B neutrinos occur in the adiabatic regime, and the ␴ observed number of events in the experiment i, and ij survival probability is higher for lower-energy neutrinos. is the error matrix, which contains both the theoretical This situation fits well with the higher rate observed at uncertainties and the experimental statistical and sys- gallium experiments. For the LOW solution, the situa- tematic errors. The main sources of uncertainty are the tion is opposite, but matter effects in the Earth for pp theoretical errors in the prediction of solar neutrino and 7Be neutrinos enhance the average annual survival fluxes for the different reactions. These errors are due to probability for these lower-energy neutrinos. The com- uncertainties in the 12 basic ingredients of the solar bination of these effects still allows a reasonable descrip- model, which include the nuclear reaction rates (param- tion of the gallium rate. etrized in terms of the astrophysical factors S11 , S33 , Oscillations into pure sterile neutrinos are strongly

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 368 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

disfavored by the SNO data since they would imply very similar neutrino fluxes for the neutral-current, charged- current, and elastic-scattering rates in SNO as well as the elastic-scattering rate at SK. Schematically, in the presence of oscillations, ⌽CCϭ⌽ e , ⌽ESϭ⌽ ϩ ⌽ e r ␮␶ , (105) ⌽NCϭ⌽ ϩ⌽ e ␮␶ , ϵ␴ ␴ Ӎ ␯ Ϫ ␯ where r ␮ / e 0.15 is the ratio of the e e and ␮ Ϫe elastic scattering cross sections. The flux ⌽␮␶ of ac- ␯ tive no-electron neutrinos in the beam would vanish if e oscillate into a purely sterile state. Thus, if the beam ␯ ␯ comprises only e’s and s’s, the three observed rates should be equal (up to effects due to spectral distor- FIG. 17. Excluded oscillation parameters at 95%, 99% tions), a hypothesis that is now ruled out at more than (shaded region), and 99.7% C.L. from the analysis of the day- 5␴ by the SNO data. Oscillations into an admixture of night spectrum data. active and sterile states are still allowed provided that the 8B neutrino flux is allowed to be larger than the SSM ϫ Ϫ5Ͻ⌬ 2Ͻ ϫ Ϫ7 2 ␪Ͼ ϫ Ϫ3 expectation (Bahcall, Gonzalez-Garcia, and Penã-Garay, (2 10 m 3 10 ,tan 3 10 ) is excluded 2002b; Barger, Marfatia, and Whisnant, 2002). due to the small observed day-night variation (compare to Fig. 15). The rest of the excluded region is due to the absence of any observed distortion of the energy spec- 3. Day-night spectra: Excluded masses and mixing trum. Superimposing Figs. 17 and 16, we can deduce the Further information on the different oscillation remain consequences of adding the day-night spectrum in- gimes can be obtained from the analysis of the energy formation to the analysis of the total event rates: and time dependence data from SK (for an early sugges- tion of the use of the day-night spectra see Maris and • SMA: within this region, the part with larger mixing Petcov, 1997), which is currently presented in the form angle fails to comply with the observed energy spec- of the observed day-night or zenith spectrum (Fukuda trum, while the part with smaller mixing angles gives a et al., 2001: Smy et al., 2002). The way to treat these data bad fit to the total rates. statistically has been discussed, for instance, by Bahcall, • VAC: the observed flat spectrum cannot be accommo- Krastev, and Smirnov (2000), Gonzalez-Garcia, de dated. Holanda, et al. (2000), Fukuda et al. (2001), and • LMA and LOW: the small ⌬m2 part of LMA and the Gonzalez-Garcia and Penã-Garay (2001). large ⌬m2 part of LOW are reduced because they The observed day-night spectrum in SK is essentially predict a day-night variation that is larger than ob- undistorted in comparison to the SSM expectation and served. Both active LMA and active LOW solutions shows no significant differences between the day and the predict a flat spectrum in agreement with the observa- night periods. Consequently, a large region of the oscil- tion. lation parameter space where these variations are expected to be large can be excluded. In Fig. 11 we show the SK spectrum corresponding to 1258 days of data 4. Global analysis relative to the Ortiz et al. (2000) spectrum normalized to the predictions of Bahcall, Pinsonneault, and Basu In order to quantitatively discuss the combined analy- (2001), together with the expectations from the best-fit sis of the full bulk of solar neutrino data, one must de- points for the large-mixing-angle, small-mixing-angle, fine a global statistical function. How to implement such low-mass, and vacuum oscillation solutions. a program is a question that is being discussed inten- The various solutions give different predictions for the sively in the literature.1 The situation at present is that, day-night spectrum. For large mixing angle and low although one finds slightly different globally allowed re- mass, the expected spectrum is distorted very little. For gions, depending on the particular prescription used in small mixing angle, a positive slope is expected, with the combination, the main conclusions are independent larger slope for larger mixing angle within the small- of the details of the analysis. mixing-angle region. For vacuum oscillations, large dis- tortions are expected. The details are dependent on the precise values of the oscillation parameters. 1See, for example, Bahcall, Gonzalez-Garcia, and Penã- In Fig. 17 we show the excluded regions at the 99% Garay, 2001, 2002a, 2002c; Creminelli, Segnorelli, and Strumia, C.L. from the analysis of SK day-night spectrum data, 2001; Garzelli and Giunti, 2001; Barger, Marfatia, Whisnant, together with the contours corresponding to the 95% and Wood, 2002; de Holanda and Smirnov, 2002; Fogli, Lisi, and 99.73% (3␴) C.L. In particular, the central region Marrone, Montanino, and Palazzo, 2002.

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 369

muon neutrinos and antineutrinos. The predicted absolute fluxes of neutrinos produced by cosmic-ray interactions in the atmosphere are uncertain at the 20% level. The ratios of neutrinos of different flavor are, however, ␯ expected to be accurate to better than 5%. Since e is produced mainly from the decay chain ␲→␮␯␮ followed ␮→ ␯ ␯ ␯ by e ␮ e , one naively expects a 2:1 ratio of ␮ to ␯ e . (For higher-energy events, the expected ratio is smaller because some of the muons arrive at the Earth before they have had time to decay.) In practice, however, the theoretical calculation of the ratio of muonlike interactions to electronlike interactions in each experiment is more complicated. Atmospheric neutrinos are observed in underground experiments using different techniques and leading to different type of events, which we briefly summarize here. They can be detected by the direct observation of FIG. 18. Allowed oscillation parameters (at 90%, 95%, 99%, their charged-current interaction inside the detector. and 99.7% C.L.) from the global analysis of the solar neutrino These are the contained events. Contained events can be data. The best-fit point (large-mixing-angle active) is marked with a star. further classified into fully contained events, when the charged lepton (either electron or muon) that is produced in the neutrino interaction does not escape the We show in Fig. 18 and Table III the results from the detector, and partially contained muons, when the pro- global analysis of Bahcall, Gonzalez-Garcia, and Penã- duced muon exits the detector. For fully contained Garay (2002c). In Fig. 18 we show the allowed regions events, the flavor, kinetic energy, and direction of the ␴ which correspond to 90%, 95%, 99%, and 99.73% (3 ) charged lepton can be best determined. As discussed ␯ C.L. for e oscillations into active neutrinos. In Table III later, some experiments further divide the contained we list the local minima of the allowed regions, the value data sample into sub-GeV and multi-GeV events, ac- ␹2 of min in each local minimum. cording to whether the visible energy is below or above The results show that at present the most favored so- 1.2 GeV. On average, sub-GeV events arise from neutri- lution is the large-mixing-angle oscillation, while the nos of several hundreds of MeV, while multi-GeV events low-mass solution provides the second best fit. There are are originated by neutrinos with energies of the order of some small allowed islands for vacuum oscillations. The several GeV. Higher-energy muon neutrinos and an- active small-mixing-angle solution does not appear at 3␴ tineutrinos can also be detected indirectly by observing as a consequence of the incompatibility between the ob- the muons produced in their charged-current interac- served small charged-current rate at SNO, which would tions in the vicinity of the detector. These are the so- favor larger mixing, and the flat spectrum, which favors called upgoing muons. Should the muon stop inside the smaller mixing. Oscillations of solar neutrinos into a detector, it is classified as a stopping muon (which arises ␴ pure sterile state are also disfavored at the 5.4 level. from neutrinos E␯ϳ10 GeV), while if the muon track crosses the full detector the event is classified as a V. ATMOSPHERIC NEUTRINOS through-going muon (which is originated by neutrinos with energies of the order of hundreds of GeV). Down- Atmospheric neutrinos are produced in cascades initi- going muons from ␯␮ interactions above the detector ated by collisions of cosmic rays with the Earth’s atmo- cannot be distinguished from the background of cosmic ␯ sphere. Some of the mesons produced in these cascades, ray muons. Higher-energy e’s cannot be detected this mostly pions and some kaons, decay into electron and way as the produced e showers immediately in the rock.

TABLE III. Best-ﬁt global oscillation parameters with all solar neutrino data. This table corresponds to the global solution illustrated in Fig. 18. The last four solutions do not appear in Fig. 18 because ␹2 their min is too large. ⌬ 2 ͓ 2͔ 2 ␪ ␹2 Solution m eV tan fB min (46 degrees of freedom) Large mixing angle 5.0ϫ10Ϫ5 4.2ϫ10Ϫ1 1.07 45.5 Low mass 7.9ϫ10Ϫ8 6.1ϫ10Ϫ1 0.91 54.3 Vacuum oscillations 4.6ϫ10Ϫ10 1.8ϫ100 0.77 52.0 Small mixing angle 5.0ϫ10Ϫ6 1.5ϫ10Ϫ3 0.89 62.7 Just So2 5.8ϫ10Ϫ12 1.0ϫ100 0.46 86.3 Sterile vacuum oscillations 4.6ϫ10Ϫ10 2.3ϫ100 0.81 81.6 Sterile small mixing angle 3.7ϫ10Ϫ6 4.7ϫ10Ϫ4 0.55 89.3

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 370 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

␯ ␯ FIG. 20. The double ratio of ␮ to e events, data divided by theoretical predictions, for various underground atmospheric neutrino detectors.

1992) and multi-GeV neutrinos (Fukuda et al., 1994), FIG. 19. Event rates as a function of neutrino energy for fully which showed the same deficit. This was the original for- contained events, stopping muons, and through-going muons mulation of the atmospheric neutrino anomaly. In Fig. at SuperKamiokande. From Engel, Gaisser, and Stanev, 2000. MC 20 we show the values of the ratio R␮/e /R␮/e , which denotes the double ratio of experimental-to-expected In Fig. 19 we display the characteristic neutrino energy ratio of muonlike to electronlike events. Whether MC ␯ distribution for these different type of events (taken R␮/e /R␮/e was small because there was ␮ disappear- ␯ from Engel, Gaisser, and Stanev, 2000). ance or e appearance or a combination of both could Atmospheric neutrinos were first detected in the not be determined. Furthermore, the fact that the 1960s by the underground experiments in South Africa anomaly appeared only in water Cerenkov and not in (Reines et al., 1965) and the Kolar Gold Field experi- iron calorimeters left the window open for the suspicion ment in India (Achar et al., 1965). These experiments of a possible systematic problem as the origin of the measured the flux of horizontal muons (they could not effect. discriminate between downgoing and upgoing direc- Kamiokande also presented the zenith angular depen- tions) and although the observed total rates were not in dence of the deficit for multi-GeV neutrinos (Fukuda full agreement with theoretical predictions (Zatsepin et al., 1994). The zenith angle, parametrized in terms of and Kuzmin, 1962; Cowsik et al., 1965; Osborne, Said, cos ␪, measures the direction of the reconstructed and Wolfendale, 1965), the differences were not statisti- charged lepton with respect to the vertical of the detec- cally significant. tor. Vertically downgoing or upgoing particles corre- Other experiments were proposed and built in the spond to cos ␪ϭϩ1orϪ1. Horizontally arriving par- 1970s and 1980s. The original purpose was to search for ticles come at cos ␪ϭ0. Kamiokande results seemed to nucleon decay, for which atmospheric neutrinos consti- indicate that the deficit was mainly due to the neutrinos tute background, although the possibility of using them coming from below the horizon. Atmospheric neutrinos to search for oscillation was also known (Ayres et al., are produced isotropically at a distance of about 15 km 1984). Two different detection techniques were em- above the surface of the Earth. Therefore neutrinos ployed. In water Cerenkov detectors, the target is a coming from the top of the detector have traveled ap- large volume of water surrounded by photomultipliers proximately those 15 km before interacting, while those which detect the Cerenkov ring produced by the coming from the bottom of the detector have traversed charged leptons. The event is classified as an electron- the full diameter of the Earth, ϳ104 km, before reach- like or muonlike event according to whether the ring is ing the detector. The Kamiokande distribution sug- diffuse or sharp. In iron calorimeters, the detector is gested that the deficit increases with the distance be- composed of a set of alternating layers of iron, which act tween the neutrino production and interaction points. as a target, and some tracking element (such as plastic In the last five years, the case for the atmospheric drift tubes) which allows the reconstruction of the neutrino anomaly has become much stronger with the shower produced by the electrons or the tracks pro- high precision and large statistics data from SK, and it duced by muons. Both types of detectors allow for flavor has received important confirmation from the iron calo- classification of the events. rimeter detectors Soudan2 and MACRO. The two oldest iron calorimeter experiments, those of SK is a 50-kiloton water Cerenkov detector con- Fre´jus (Daum et al., 1995) and NUSEX (Aglietta et al., structed under Mt. Ikenoyama located in the central 1989), found atmospheric neutrino fluxes in agreement part of Japan, giving it a rock overburden of 2700 m with the theoretical predictions. On the other hand, two water equivalent. The fiducial mass of the detector for water Cerenkov detectors, IMB (Becker-Szendy et al., atmospheric neutrino analysis is 22.5 kilotons. In June 1992) and Kamiokande, detected a ratio of ␯␮-induced 1998, at the Neutrino98 conference, SK presented evi- ␯ ␯ events to e-induced events smaller than that expected dence of ␮ oscillations (Fukuda et al., 1998) based on by a factor of about 0.6. Kamiokande performed sepa- the angular distribution for their contained event data rate analyses for both sub-GeV neutrinos (Hirata et al., sample. Since then SK has accumulated more statistics

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 371

because at higher energy the direction of the charged lepton is more aligned with the direction of the neutrino. It can also be described in terms of an up-down asymmetry: UϪD A␮ϵ ϭϪ0.316Ϯ0.042͑stat.͒Ϯ0.005͑syst.͒, UϩD (106) where U(D) are the contained ␮-like events with zenith angle in the range Ϫ1Ͻcos ␪ϽϪ0.2 (0.2 Ͻcos ␪Ͻ1). It deviates from the standard-model value, A␮ϭ0, by 7.5 standard deviations. (iii) The overall suppression of the flux of stopping ⌽ muons, ST , is by a factor of about 0.6, similar to that of contained events. However, for the flux of ⌽ through-going muons, TH , the suppression is weaker, which implies that the effect is smaller at larger neutrino energy. This effect is also parametrized in terms of the double flux ratio: ⌽ /⌽ ͉ ST TH obs ϭ Ϯ ͑ ͒Ϯ ͑ ͒ ⌽ ⌽ ͉ 0.635 0.049 stat. 0.035 syst. ST / TH MC Ϯ0.084͑theo.͒ (107) which deviates from the standard-model value of 1 by about 3 standard deviations.

These effects have been confirmed by the results of the iron calorimeters Soudan2 and MACRO, which removed the suspicion that the atmospheric neutrino anomaly is simply a systematic effect in water detectors. In particular, Soudan2, which has mainly analyzed contained events (Allison et al., 1999), has measured a ratio MCϭ Ϯ Ϯ R␮/e /R␮/e 0.68 0.11 0.06, in good agreement with FIG. 21. Zenith-angle distribution of 1289 days’ worth of Su- the results from the water Cerenkov experiments. The perKamiokande data samples: dotted line, data; solid line, MC main results from MACRO concern the angular distri- with no oscillation; dashed line, MC with best-fit oscillation bution for through-going muons [see Ambrosio et al. parameters. From Toshito et al., 2001. (2001) for their latest data] and show good agreement and has also studied the angular dependence of the up- with the results from SK. The Baksan experiment (Bo- going muon sample (Fukuda et al., 1999a, 1999b). In Fig. liev et al., 1999) has also reported results on the angular 21 (from Toshito et al., 2001) we show their data as of distribution of through-going muons, but their data are summer 2001, corresponding to a 79-kiloton year (1289 less precise. days) exposure for their contained sub-GeV (2864 one- To analyze the atmospheric neutrino data in terms of ring e-like events and 2788 one-ring ␮-like events) and oscillations, one needs to have a good understanding of multi-GeV (626 one-ring e-like events, 558 one-ring the different elements entering into the theoretical pre- ␮-like events, and 754 events that were partially con- dictions of the event rates: the atmospheric neutrino tained within the detector), as well as stopping muons fluxes and their interaction cross section, which we de- and through-going muon samples (1416 events). scribe next. In the figure, we show the angular zenith distribution of the different samples. Comparing the observed and A. Atmospheric neutrino fluxes the expected Monte Carlo (MC) distributions, we can Modern calculations of atmospheric neutrino fluxes make the following statements: consist of a Monte Carlo procedure that combines the ␯ (i) e distributions are well described by the MC measured energy spectra and chemical composition of while ␯␮ presents a deficit. Thus the atmospheric the cosmic-ray flux at the top of the atmosphere with the neutrino deficit is mainly due to disappearance of properties of their hadronic interaction with light atmo- ␯ ␯ ␮ and not the appearance of e . spheric nuclei followed by neutrino production from sec- (ii) The suppression of contained ␮-like events is ondary ␲, K, and ␮ decay. stronger for larger cos ␪, which implies that the Present experiments use the neutrino flux calculations deficit grows with the distance traveled by the mainly from Honda et al. (1990, 1995, 1996), the Bartol neutrino from its production point to the detector. group (Gaisser, Stanev, and Barr, 1988; Barr, Gaisser, This effect is more obvious for multi-GeV events and Stanev, 1989; Agrawal et al., 1996; Lipari, Gaisser,

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 372 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing and Stanev, 1998) and Fiorentini, Naumov, and Villante (2001). These calculations have in common a one- dimensional picture, in which all secondary particles in the showers, neutrinos included, are considered collinear with the primary cosmic rays (see also Bugaev and Naumov, 1989). The three-dimensional picture was first considered by Lee and Koh (1990) and most recently by Tserkovnyak et al. (2003) and Battistoni et al. (2000). [For detailed comparisons between the various simula- tions, see Gaisser et al. (1996), Lipari (2000), and Battis- toni (2001).] The flux of neutrinos of flavor i coming from the direction ⍀ can be schematically written as ␾␯ (⍀) i ϭ͚ ⌽ R Y →␯ . Here ⌽ is the primary cosmic- A A A p,n i A ray spectrum, RA is the geomagnetic cutoff for protons or light nuclei incident on the atmosphere from the direction ⍀, and Y →␯ is the yield per nucleon of ␯ . p,n i i The separation into different nuclear species of the cosmic-ray spectrum is necessary because the neutrino yield depends on the energy per nucleon of the incident cosmic rays, but the geomagnetic cutoff depends on the magnetic rigidity (Rϭpc/Ze) which, at the same energy per nucleon, depends on A/Z. We now briefly summarize the uncertainties in each of these three ingredients of the calculation, namely, the primary cosmic-ray flux, the geomagnetic effects, and the hadronic interactions on light nuclei.

1. Cosmic-ray spectrum The cosmic radiation incident at the top of the atmosphere includes all stable charged particles and nuclei. FIG. 22. The main components of the primary cosmic-ray Apart from particles associated with solar flares, cosmic spectrum. From Simpson, 1983. rays are assumed to come from outside the solar system. ϳ In Fig. 22 (from Simpson, 1983) we show a compilation 1-GeV cosmic rays, and it decreases to 10% for 10-GeV cosmic rays. The fluxes shown in Fig. 22 of the major components of cosmic rays as a function of correspond to a particular epoch of the solar energy per nucleon. The following features of the cycle. cosmic-ray spectrum are relevant to atmospheric neutrino fluxes: 2. Geomagnetic effects (i) Composition: Most of the primaries are protons, Lower-energy cosmic rays are affected by the geomag- although the chemical composition varies with en- netic field that they must penetrate to reach the top of ergy at low energies. Above a few GeV of energy the atmosphere. This effect depends on the point where per nucleon, about 79% of the primaries are pro- the cosmic rays arrive at the Earth (being stronger near tons, while helium, the next most abundant com- the geomagnetic equator), and on their direction. Near ponent, is about 3%. This fraction remains con- the geomagnetic poles almost all primary particles can stant till beyond 100 TeV. reach the atmosphere moving along field lines. In con- (ii) Energy dependence: For energy above a few GeV, trast, close to the geomagnetic equator the field restricts the cosmic-ray flux is a a steeply falling function the flux at the top of the atmosphere to particles with of the energy: d␾/dEϰEϪ3.7. energy larger than a few GeV, the exact value depending (iii) Solar modulation: At energies below ϳ10 GeV, on the direction of the particle trajectory. The relevant the incoming charged particles are modulated by quantity is the magnetic rigidity, Rϭpc/Ze, which pa- the solar wind, which prevents the lower-energy rametrizes the characteristic radius of curvature of the cosmic rays of the inner solar system from reach- particle trajectory in the presence of a magnetic field: ing the Earth. Because of this effect there is an particles with R below a local cutoff are bent away by anticorrelation between the 11-year cycle of solar the Earth’s magnetic field and do not reach the atmo- activity and the intensity of cosmic rays below 10 sphere. GeV, which makes the cosmic-ray fluxes time de- Given a map of the magnetic field around the Earth, pendent. The flux difference at solar maximum we find that the value of the rigidity cutoff can be ob- and solar minimum is more than a factor of 2 for tained from a computer simulation of cosmic-ray trajec-

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 373

tories by using a backtracking technique, in which one determines whether a given primary particle’s three- momentum and position correspond to an allowed or a forbidden trajectory. To do so, one integrates back the equation of motion of the cosmic-ray particle (actually one integrates the antiparticle equation) in the geomagnetic field and finds whether the past trajectory remains confined to a finite distance from the Earth—in which case it is a forbidden trajectory—or it originates from large enough (տ10R ) distances. The rigidity cutoff is calculated as the minimum R for which the trajectory is allowed or, in other words, the backtracked trajectory escapes from the magnetic field of the Earth. The following features of the geomagnetic effects are relevant to the atmospheric neutrino fluxes: (i) For a fixed direction, the cutoff rigidity grows monotonically from zero at the magnetic pole to a maximum value at the magnetic equator. FIG. 23. Atmospheric neutrino fluxes as a function of neutrino (ii) The cutoff rigidities for particles traveling toward energy. the magnetic west are higher than those for particles traveling toward the magnetic east.

(iii) Thus the highest cutoff corresponds to westward- (i) Energy dependence: For E␯տ1 GeV, the ﬂuxes going, horizontal particles reaching the surface of obey an approximate power law, d⌽/dEϰEϪ␥ the Earth at the magnetic equator. where ␥ϳ3 (3.5) for muon (electron) neutrinos and antineutrinos. For E␯Շ1 GeV, the depen- 3. The neutrino yield dence on energy is weaker as a consequence of the bending of the cosmic-ray spectrum by geo- As cosmic rays propagate in the atmosphere and in- magnetic effects and solar modulation. ␲ teract with air nuclei, they create and K mesons, (ii) ¯␯/␯ ratio: For EՇ1 GeV, when all pions and sub- ␯ which in turn decay and create atmospheric ’s: sequent muons decay before reaching the Earth, Ϫ ϩ ϩ →␲Ϯ Ϯ 0 the ratio ¯␯ /␯ ϭ␲ /␲ Ͻ1. At these energies, the Acr Aair ,K ,K ,...; e e cosmic rays are mainly protons. The positively Ϯ Ϯ Ϯ ϩ ␲ ,K →␮ ϩ␯␮͑¯␯␮͒, (108) charged protons produce, on average, more ␲ ’s ␲Ϫ ␮Ϯ→ Ϯϩ␯ ͑␯ ͒ϩ␯ ͑␯ ͒ than ’s in their interactions. On the other hand, e e ¯ e ¯ ␮ ␮ . ϩ ϩ since ␲ (␮ ) produces a ␯␮ (¯␯␮) in its decay, we expect ¯␯␮ /␯␮ϭ1. We have displayed only the dominant channels. Since ␮Ϯ the decay distributions of the secondary mesons and As energy increases, the secondary ’s do not muons are well known, the largest source of differences have time to decay before reaching the surface of the Earth and consequently ¯␯␮ /␯␮ decreases. between the various calculations for the neutrino yield is ␯ ϩ␯ ␯ ϩ␯ Շ the use of different models for the hadronic interactions (iii) ( ␮ ¯ ␮)/( e ¯ e) ratio: At E 1 GeV, the ratio (Gaisser et al., 1996). Since neutrinos produced by the is very close to 2, as expected from the chain decays in Eq. (108). At higher energies, the ratio decays of pions, kaons, and muons have different energy ␮ spectra, the main features of the interaction model decreases because, as mentioned above, the ’s do ␯ ␯ which affect the ␮ / e composition and the energy and angular dependence of the neutrino ﬂuxes are the K/␲ ratio and their momentum distribution.

4. The neutrino fluxes In Figs. 23 and 24 we show the atmospheric neutrino fluxes from the calculation of the Bartol group (Gaisser, Stanev, and Barr, 1988; Barr, Gaisser, and Stanev, 1989; Agrawal et al. 1996; Lipari, Gaisser, and Stanev, 1998) for the location of SK and for maximum solar activity. In the first figure we show the flux as a function of the neutrino energy averaged over the arrival direction. In the second figure we show the fluxes as a function of the zenith angle for various neutrino energies. We empha- FIG. 24. Zenith-angle dependence of atmospheric neutrino size the following points: fluxes.

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 374 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

2 2 where sϪuϭ4 ME␯ϩq Ϫmᐉ , M is the proton mass, 2 mᐉ is the charged-lepton mass, and q is the momentum transfer. For ¯␯p→ᐉϩn, a similar formula applies with →Ϫ the only change A2 A2 . The functions A1 , A2 , and A3 can be written in terms of axial and vector form factors:

2 2 2 2 mᐉϪq q q A ϭ ͫͩ 4Ϫ ͪ ͉F ͉2Ϫͩ 4ϩ ͪ ͉F1 ͉2 1 4M2 M2 A M2 V

2 2 q 4q Ã Ϫ ͉␰F2 ͉2Ϫ Re͑F1 ␰F2 ͒ͬ M2 V M2 V V

2 FIG. 25. The various contributions to the neutrino-nucleon mᐉ Ϫ ͉͑F1 ϩ␰F2 ͉2ϩ͉F ͉2͒, cross section. M2 V V A q2 ϭϪ ͓ Ã ͑ 1 ϩ␰ 2 ͔͒ not have time to decay before reaching the Earth A2 2 Re FA FV FV , (111) ␯ M and fewer e’s are produced. (iv) Up-down asymmetry: At EՇa few GeV, the 1 q2 A ϭϪ ͩ ͉F ͉2ϩ͉F1 ͉2Ϫ ͉␰F2 ͉2 ͪ , fluxes are up-down asymmetric due to geomag- 3 4 A V 4M2 V netic effects. Geomagnetic effects are very small already at Eϭ2 GeV (see Fig. 24) and the flux where we neglected second-class currents and assumed becomes up-down symmetric above that energy. conserved vector current. With this assumption all form (v) Horizontal-vertical ratio: For Eտa few GeV, the factors are real and can be written as follows: fluxes are maximal for neutrinos arriving horizon- q2 Ϫ1 q2 Ϫ2 tally and minimal for neutrinos arriving vertically 1 ͑ 2͒ϭͩ Ϫ ͪ ͩ Ϫ ͪ FV q 1 2 1 2 (see Fig. 24). This is due to the difference in the 4M MV atmosphere density, which determines whether q2 the pions decay before reinteracting with the air ϫͫ1Ϫ ͑1ϩ␮ Ϫ␮ ͒ͬ, 4M2 p n (thereby producing neutrinos). Pions arriving horizontally travel for a longer time in less dense q2 Ϫ1 q2 Ϫ2 ␰ 2 ͑ 2͒ϭͩ Ϫ ͪ ͩ Ϫ ͪ ͑␮ Ϫ␮ ͒ atmosphere and are more likely to decay before F q 1 1 n p , V 4M2 M2 reinteracting. V (112) q2 Ϫ2 F ϭF ͑0͒ͩ 1Ϫ ͪ . A A 4M2 B. Interaction cross sections A ␮ ␮ Here p and n are the proton and neutron anomalous 2 ϭ 2 In order to determine the expected event rates at the magnetic moments and MV 0.71 GeV is the vector experiment, one needs to know the neutrino-nucleon in- mass, which is measured with high precision in electron- teraction cross sections in the detector. The standard ap- scattering experiments. The largest uncertainties in this proach is to consider separately the contributions to this calculation are associated with the axial form factor. For cross section from the exclusive channels of lower mul- example, the axial mass used by various collaborations 2 ϭ 2 tiplicity (quasielastic scattering and single-pion produc- varies in the range MA 0.71– 1.06 GeV . tion), and include all additional channels as part of the So far we have neglected nuclear effects. The most deep-inelastic scattering cross section: important of these effects is related to the Pauli principle and can be included by using a simple Fermi-gas ␴ ϭ␴ ϩ␴ ␲ϩ␴ . (109) CC QE 1 DIS model. In this approximation, the cross section of a bound nucleon is equal to the cross section of a free In Fig. 25 we plot the cross sections for these processes. 2 The cross section for the quasielastic interaction is nucleon multiplied by a q -dependent factor which can given by (Smith, 1972) be found, for instance, in Smith (1972). The effect of this factor is to decrease the cross section. The decrease is d␴ M2G2 cos2 ␪ larger for smaller neutrino energy. For energies above 1 QE ͑␯ →ᐉϪ ͒ϭ F c 2 n p 2 d͉q ͉ 8␲E␯ GeV, the nuclear effects lead to an 8% decrease in the quasielastic cross section. As can be seen in Fig. 25, sϪu quasielastic interactions dominate for E␯Շ1 GeV and ϫͫA ͑q2͒ϪA ͑q2͒ 1 2 M2 induce most of the observed contained events. In the interpretation of the zenith-angle distribution Ϫ 2 ͑s u͒ of contained events, an important role is played by the ϩA ͑q2͒ ͬ, (110) 3 M4 relative angle between the direction of the incoming

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 375 neutrino (carrying the information on the neutrino path 1. Predicted number of events length) and the direction of the produced charged lep- For a given neutrino oscillation channel, the expected ton (which is measured). For energies below 1 GeV, the number of ␮-like and e-like contained events, N␣ (␣ opening angle is rather large: for sub-GeV events the ϭ␮ correlation between the measured ᐉ direction and the ,e), can be computed as տ ϭ ϩ ϭ ϩ distance traveled by the neutrinos is weak. For E a few N␮ N␮␮ Ne␮ , Ne Nee N␮e , (114) GeV, the two directions are almost aligned (for example, it is on average 1° for upgoing muons), and the ᐉ direc- where 2 tion gives a very good measurement of the neutrino path d ⌽␣ ϭ ͵ ␬ ͑ ␪ ͒ length. N␣␤ ntT ␣ h,cos ␯ ,E␯ dE␯d͑cos ␪␯͒ Single-pion production, ␯N→ᐉϪNЈ␲, is dominated by the ⌬(1232) resonance (Fogli and Nardulli, 1979; Na- d␴ ϫ ␧͑ ͒ ͑ ␪ ͒ kahata et al., 1986). It is most relevant at E␯Ӎ1 GeV P␣␤ E␤ dE␯dE␤d cos ␯ dh. (115) dE␤ (see Fig. 25). Ϫ Deep-inelastic processes, ␯N→ᐉ X where X repre- Here P␣␤ is the conversion probability of ␯␣→␯␤ for sents any hadronic system, dominate atmospheric neu- given values of E␯ , cos ␪␯ and h, that is, P␣␤ϵP(␯␣ trino interactions for E␯տa few GeV. The parton model →␯␤ ; E␯ ,cos ␪␯ ,h). In the standard model, the only cross section is given by nonzero elements are the diagonal ones, i.e., P␣␣ϭ1 for ␣ ␴ ͑␯͒ 2 all . In Eq. (115), nt denotes the number of target par- d DIS GFsx ϭ ͓ Ϫ ϩ͑ ϩ ͒͑ Ϫ ͒2͔ ticles, T is the experiment running time, E␯ is the neu- ␲ F1 F3 F1 F3 1 y , (113) dxdy 4 trino energy, ⌽␣ is the flux of atmospheric ␯␣’s, E␤ is the 2 ␧ where yϭ1ϪEᐉ /E␯ and xϭϪq /(2ME␯y). For ¯␯,a final charged-lepton energy, (E␤) is the detection effi- →Ϫ ␴ similar formula applies, with F3 F3 . F1 and F3 are ciency for such a charged lepton, is the neutrino- given in terms of the parton distributions. For isoscalar nucleon interaction cross section, and ␪␯ is the angle ϭ ͚ ϩ ϭ͚ Ϫ targets F1 2 i(qi ¯qi) and F3 i(¯qi qi). In order to between the vertical direction and the incoming neutri- avoid double counting of the single-pion production, the nos (cos ␪␯ϭ1 corresponds to the down-coming neutri- deep-inelastic contribution must be integrated in the re- nos). In Eq. (115), h is the slant distance from the pro- Ͼ ϭ gion of hadronic masses WX Wc (Wc 1.4 GeV), duction point to sea level for ␣-type neutrinos with Ϫ у 2Ϫ 2 which implies 2ME␯y(1 x) Wc M . energy E␯ and zenith angle ␪␯ , and ␬␣ is the slant distance distribution, normalized to one. (We use in our calculations ␬␣ from Gaisser and Stanev, 1998.) To obtain the expectation for the angular distribution C. Two-neutrino oscillation analysis of contained events, one must integrate the corresponding bins for cos ␪␤ , where ␪␤ is the angle of the detected The simplest and most direct interpretation of the at- lepton, taking into account the opening angle between mospheric neutrino anomaly is that of muon neutrino the neutrino and the charged-lepton directions as deter- oscillations (Barger and Whisnant, 1988; Hidaka, mined by the kinematics of the neutrino interaction. Honda, and Midorikawa, 1988; Learned, Pakvasa, and As discussed above, the neutrino fluxes, in particular Weiler, 1988). The estimated value of the oscillation pa- those in the sub-GeV range, depend on the solar activity. rameters can be easily derived in the following way: In order to take this fact into account, one uses in Eq. • The angular distribution of contained events shows (115) a linear combination of atmospheric neutrino ⌽max ⌽min that, for Eϳ1 GeV, the deficit comes mainly from L fluxes ␣ and ␣ which correspond to the most (so- ϳ102 – 104 km. The corresponding oscillation phase lar maximum) and least (solar minimum) active phases must be maximal, ⌬m2(eV2)L(km)/2E(GeV) ϳ1, of the Sun, respectively, with different weights which de- which requires ⌬m2ϳ10Ϫ4 – 10Ϫ2 eV2. pend on the specific running period. Experimental results on upgoing muons are presented • Assuming that all upgoing ␯␮’s which would lead to multi-GeV events oscillate into a different flavor in the form of measured muon fluxes. To obtain the effective muon fluxes for both stopping and through-going while none of the downgoing ones do, the up-down 2 2 muons, one must convolute the survival probabilities for asymmetry is given by ͉A␮͉ϭsin 2␪/(4Ϫsin 2␪). The ␯␮’s with the corresponding muon fluxes produced by present one-sigma bound reads ͉A␮͉Ͼ0.27 [see Eq. neutrino interactions with the Earth. One must further (106)], which requires that the mixing angle be close 2 ␪Ͼ take into account the muon energy loss during propaga- to maximal, sin 2 0.85. tion both in the rock and in the detector, and also the effective detector area for both types of events, stopping In order to go beyond these rough estimates, one must and through-going. Schematically, compare in a statistically meaningful way the experi- ϱ mental data with the detailed theoretical expectations. 1 d⌽␮͑E␮ ,cos ␪͒ ⌽ ͑␪͒ ϭ ͵ We now describe how to obtain the allowed region. We ␮ S,T ͑ ␪͒ ␪ A Lmin , E␮,min dE␮d cos consider two neutrino cases, where ␯␮ oscillates into ei- ␯ ␯ ␯ ϫ ͑ ␪͒ ther e or ␶ or a sterile neutrino s . AS,T E␮ , dE␮ , (116)

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 376 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing where where ϱ ϱ ϱ ϱ 2 d⌽␮ ⌬m ϭ ͵ ͵ ͵ ͵ ϭ Ϫ ␪ ␪ NA dE␮0 dE␯ dX dh H␮ V␮ cos 2 , dE␮d cos E␮ E␮0 0 0 4E␯ d⌽␯ ͑E␯ ,␪͒ ⌬m2 ␮ ϭ ϩ ␪ ϫ␬␯ ͑h,cos ␪,E␯͒ HX VX cos 2 , (121) ␮ dE␯d cos ␪ 4E␯ ␴͑ ͒ 2 d E␯ ,E␮0 ⌬m ϫ ␮␮ ͑ ␮ ␮ ͒ ϭ ␪ P Frock E 0 ,E ,X . H␮X sin 2 . dE␮0 4E␯ (117) The various neutrino potentials in matter are given by ϭ ϩ ϭ ϭ ϭ Here NA is the Avogadro number, E␮0 is the energy of Ve VC VN , V␮ V␶ VN , and Vs 0 where the the muon produced in the neutrino interaction, E␮ is the charged-current and neutral-current potentials VC and muon energy when entering the detector after traveling VN are proportional to the electron and neutron number a distance X in the rock, and cos ␪ labels both the neu- density (see Sec. III.B), or equivalently, to the the mat- trino and the muon directions which, at the relevant en- ter density in the Earth (Dziewonski and Anderson, ergies, are collinear to a very good approximation. We 1981). For antineutrinos, the signs of the potentials are denote by Frock(E␮0 ,E␮ ,X) the function that character- reversed. izes the energy spectrum of the muons arriving at the For Xϭ␶, we have V␮ϭV␶ and consequently these detector. The standard practice is to use an analytical potentials can be removed from the evolution equation. approximation obtained by neglecting fluctuations dur- The solution of Eq. (120) is then straightforward and the ing muon propagation in the Earth. In this case the three probability takes the well-known vacuum form [Eq. quantities E␮0 , E␮ , and X are not independent: (34)], which is equal for neutrinos and antineutrinos. For ϭ ϱ 1 X e or s, the effect of the matter potentials requires a ͵ ͑ ͒ ϭ numerical solution of the evolution equations in order to Frock E␮0 ,E␮ ,X dX , (118) ͗dE␮͑E␮͒/dX͘ 0 obtain P␣␤ which, furthermore, is different for neutrinos where ͗dE␮(E␮)/dX͘ is the average muon energy loss and antineutrinos. As a first approximation, one can use due to ionization, bremsstrahlung, eϩeϪ pair produc- a constant Earth matter density. Then (see Sec. III) the tion, and nuclear interactions in the rock (Lohmann, solutions take the same form as the vacuum probability Kopp, and Voss, 1985). [Eq. (34)], but with the mixing angle and the oscillation For SK, the pathlength traveled by the muon inside length replaced by their effective values in matter [Eqs. ϭ Ϫ the detector is given by the muon range function in wa- (55) and (58) with A 2E(V␮ VX)]. In Fig. 26 (from ter: Lipari and Lusignoli, 1998) we show the survival probability of ␯␮ for the different oscillation channels for E␮ 1 ⌬ 2ϭ ϫ Ϫ3 2 2 ␪ϭ ͑ ͒ϭ ͵ Ј m 5 10 eV , sin 2 1, and various values of L E␮ dE␮ . (119) 0 ͗dE␮͑E␮Ј ͒/dX͘ E␯ . As can be seen in the figure, matter effects damp ␪ ϭ ␪ ϩ ␪ the oscillation amplitude. For the chosen mass differ- In Eq. (116), A(Lmin , ) AS(E␮ , ) AT(E␮ , )isthe ence, they are important for neutrino energies of few projected detector area for internal path lengths longer ϭ tens of GeV and therefore are relevant mainly for upgo- than Lmin ( 7 m in SK). Here AS and AT are the effec- ing muons. We return to this point below when describ- tive areas for stopping and through-going muon trajec- ing the results of the analysis. tories. These effective areas can be computed using a simple geometrical picture given, for instance, by Lipari and Lusignoli (1998). For a given angle, the threshold 3. Statistical analysis energy cut for SK muons is obtained by equating Eq. th ϭ In order to define in a statistically meaningful way the (119) to Lmin , i.e., L(E␮ ) Lmin . In contrast to SK, MACRO present their results as regions of neutrino flavor parameters that are allowed muon fluxes for E␮Ͼ1 GeV, after correcting for detec- by a given set of atmospheric neutrino observables, one 2 tor acceptances. Therefore in this case we compute the can construct, for example, a ␹ function, in a similar expected fluxes as in Eqs. (116) and (117) but without fashion to that described in Sec. IV.D.2 for the case of the inclusion of the effective areas. solar neutrino rate analysis. Details on the statistical analysis of the atmospheric neutrino data and results 2. Conversion probabilities from the analyses with various data samples can be found in the literature.2 ␯ →␯ ϭ ␶ We consider a two-flavor scenario, ␮ X (X e, , or s). The oscillation probabilities are obtained by solving the evolution equation of the ␯␮Ϫ␯ system in the X 2See, for example, Fogli, Lisi, and Montanino (1994, 1995); matter background of the Earth (see Sec. III.C): Fogli and Lisi (1995b); Foot, Volkas, and Yasuda (1998); d ␯ H␮ H␮ ␯ Gonzalez-Garcia, Nunokwa, et al. (1998); Yasuda (1998); ␮ ϭ X ␮ i ͩ ␯ ͪ ͩ ͪͩ ␯ ͪ , (120) Akhmedov et al. (1999); Gonzalez-Garcia, Nunokwa, et al. dt X H␮X HX X (1999); Gonzalez-Garcia, Fornengo, and Valle (2000).

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 377

␯ →␯ FIG. 27. The status of the ␮ e oscillation solution to the atmospheric neutrino anomaly.

␯ (i) SK high-precision data show that the e contained events are very well described by the standard-model

FIG. 26. P␮␮ as a function of the zenith angle for maximal prediction both in normalization and in their zenith- ␯ ␯ ␯ ␯ ⌬ 2ϭ angle dependence (see Fig. 21). The ␯␮ distribution, mixing of ␮ (a) with ␶ , (b) e, and (c) s . For m 5 Ϫ3 2 ␯ →␯ ϫ10 eV the curves correspond to E␯ of 20, 40, 60, and 80 however, shows an angle-dependent deficit. ␮ e os- GeV. The dashed curves are calculated with the approximation cillations could explain the angular dependence of the of constant average densities in the mantle and in the core of ␯␮ flux only at the price of introducing angular depen- ␯ the Earth. From Lipari and Lusignoli, 1998. dence of the e flux, in contrast to the data. Further- ␯ →␯ more, even the best-fit point for ␮ e oscillations does The general strategy is to compute the ␹2 as a func- not generate the observed up-down asymmetry in the tion of the neutrino parameters. By minimizing ␹2 with multi-GeV muon sample. This is illustrated in Fig. 27, respect to sin2 2␪ and ⌬m2, one determines the best-fit where we show the predicted angular distribution of contained events at SK for the best-fit points of the dif- results, while the allowed regions are determined by the ␯ →␯ conditions: ␹2ϵ␹2 ϩ4.61, 6.1, or 9.21 for a confidence ferent oscillation channels. For ␮ e oscillations, the min asymmetry in the multi-GeV muon distribution is much level (C.L.) of 90%, 95%, or 99%, respectively. ␯ →␯ ␯ →␯ smaller than in the ␮ ␶ or ␮ s channels. ␯ →␯ ␯ →␯ (ii) Explaining the atmospheric data with ␮ e tran- 4. ␮ e ␯ →␯ sition has direct implications for the ¯ e ¯ ␮ transition. ␯ →␯ ␯ At present ␮ e is excluded with high C.L. as the In particular, there should be a ¯ e deficit in the CHOOZ explanation of the atmospheric neutrino anomaly for reactor experiment. Thus the neutrino parameters not two different reasons: only give a poor fit to the atmospheric data but are ac-

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 378 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

FIG. 29. Allowed regions (at 90%, 95%, and 99% C.L.) from the analysis of the full data sample of atmospheric neutrinos ␯ →␯ ␯ →␯ for the oscillation channels ␮ ␶ and ␮ s . The best-ﬁt points are marked with a star (see text for details). Also shown are the expected sensitivities from long-baseline experiments.

reason is that no information on the minimum oscillation length can be inferred from the data (see Sec. III.A). (iii) The allowed regions for ␯␮→␯␶ transition are symmetric with respect to maximal mixing. This must be the case because the corresponding probabilities take the vacuum expression and therefore depend on sin2 2␪. (iv) The allowed regions for ␯␮→␯ transition are FIG. 28. Allowed regions (at 90%, 95%, and 99% C.L.) from s asymmetric due to Earth matter effects. These ef- the analysis of various partial data samples of atmospheric ␯ →␯ ␯ →␯ fects are more pronounced when the condition for neutrinos for the oscillation channels ␮ ␶ and ␮ s .The ⌬ 2 ␪ϳ best-fit points are marked with a star (see text for details). maximal matter effect, m cos 2 2EVs , is fulfilled. Since in the chosen convention the poten- ϭ Ϫ Ͻ tial difference A 2E(V␮ Vs) 0, matter effects tually excluded by the negative results from the enhance neutrino oscillations for cos 2␪Ͻ0 (sin2 ␪ CHOOZ reactor experiment (see Fig. 27). Ͼ0.5). The opposite situation holds for antineutrinos, but neutrino fluxes are larger and dominate in the resulting effect. As a result the allowed regions are wider on the sin2 ␪Ͼ0.5 side of ␯ →␯ ␯ →␯ 5. ␮ ␶ and ␮ s the plot. (v) The best fit to the full data for ␯␮→␯␶ corre- In Fig. 28 we show the values of the oscillation param- ⌬ 2ϭ ϫ Ϫ3 2 2 ␪ϭ ⌬ 2 2 ␪ sponds to m 2.6 10 eV and sin 2 0.97. eters ( m ,sin ) which describe various sets of data ␯ →␯ ⌬ 2ϭ The best fit for ␮ s lies at m 3 for these two oscillation channels. The upper panels (la- Ϫ ϫ10 3 eV2 and sin2 ␪ϭ0.61. beled as FINKS) and the central panels refer to contained events. The upper panels take into account only In order to discriminate between the ␯␮→␯␶ and ␯␮ the total rates, and the central ones only the angular →␯ s options, one can use the difference in the survival distribution (from both Kamiokande and SK). The probabilities due to the presence of matter effects for lower panels correspond to the angular distribution for oscillations into sterile neutrinos (Lipari and Lusignoli, upgoing muons from SK. The three shaded regions are 1998). As discussed above, the effect is important mainly allowed at the 90%, 95%, and 99% C.L. In Fig. 29 we for the higher-energy neutrinos which lead to through- plot the allowed regions from the global analysis, includ- going muons events. In Fig. 30 we plot the expected ing all the atmospheric neutrino data. Also shown are ␯ distributions for the best-fit points for the two channels. the expected sensitivities from ␮ disappearance at the As can be seen in the figure, the distribution is steeper long-baseline experiments K2K and MINOS discussed ␯ →␯ ␯ →␯ for ␮ ␶ , while for ␮ s a flattening is observed for in Sec. VI.D. (These figures are an update of the results neutrinos coming close to the vertical due to the damp- presented by Gonzalez-Garcia, Fornengo, and Valle, ing of the oscillation amplitude (see Fig. 26). The data 2000.) We emphasize the following points: favor the steeper distributions and this translates into a (i) The allowed regions from the various data better global fit for oscillations into ␯␶ . For the global samples overlap. The oscillation hypothesis can analysis, an update of the results of Gonzalez-Garcia, then consistently explain the atmospheric neu- Fornengo, and Valle (2000) shows that for ␯␶ oscillations ␹2 ϭ ␯ trino data. min 56/63 degrees of freedom, while for s oscillations ␹2 ϭ (ii) The information from the total event rates alone min 72/63 degrees of freedom. is consistent with arbitrarily high ⌬m2 values. The SK has also used other methods to distinguish be-

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 379

relevant quantity when interpreting the results in the more-than-two-neutrino framework.

A. Short-baseline experiments at accelerators

Conventional neutrino beams from accelerators are mostly produced by ␲ decays, with the pions produced by the scattering of the accelerated protons on a fixed target: pϩtarget→␲ϮϩX, Ϯ Ϯ ␲ →␮ ϩ␯␮͑¯␯␮͒, (122) ␮Ϯ→ Ϯϩ␯ ͑␯ ͒ϩ␯ ͑␯ ͒ e e ¯ e ¯ ␮ ␮ . Thus the beam can contain both ␮ and e neutrinos and antineutrinos. The final composition and energy spectrum of the neutrino beam is determined by selecting FIG. 30. Zenith-angle dependence for through-going muons at ␲ SuperKamiokande compared with the predictions in the case the sign of the decaying and by stopping the produced 2 ␮ of no oscillation and for the best-fit points of ␯␮→␯␶ (⌬m in the beam line. ϭ ϫ Ϫ3 2 2 ␪ϭ ␯ →␯ ⌬ 2ϭ ϫ Ϫ3 Most oscillation experiments performed so far with 2.6 10 eV ,sin 2 0.97) and ␮ s ( m 3 10 eV2,sin2 ␪ϭ0.61) oscillations. neutrino beams from accelerators have characteristic distances of the order of hundreds of meters. We call them short-baseline experiments. With the exception of ␯ ␯ tween the ␶ and s hypotheses for explaining atmo- the LSND experiment, which we discuss below, all ␯ spheric ␮ disappearance. One is to examine events searches have been negative. In Table IV we show the likely to have been caused by neutral-current interac- limits on the various transition probabilities from the ␯ ␯ tions. While ␶’s readily undergo such interactions, s’s negative results of the most restricting short-baseline ex- do not, resulting in a relative suppression of the neutral- periments. As can be seen in the table, due to the short current signal (Hall and Murayama, 1998; Vissani and path length, these experiments are not sensitive to the Smirnov, 1998). Another method attempts to observe low values of ⌬m2 invoked to explain either the solar or ␯ the appearance of newly created ␶ , even if only on a the atmospheric neutrino data. statistical basis, by selecting enriched samples. All meth- ␯ ↔␯ ␯ ↔␯ ods strongly favor ␮ ␶ oscillations over ␮ s (Fukuda et al., 2000; Toshito et al., 2001). B. LSND and KARMEN

The only positive signature of oscillations in a labora- VI. LABORATORY EXPERIMENTS tory experiment comes from the Liquid Scintillator Neu- trino Detector (LSND) running at Los Alamos Meson Laboratory experiments to search for neutrino oscil- Physics Facility (Athanassopoulos et al., 1995, 1996, ϩ lations are performed with neutrino beams produced at 1998). The primary neutrino flux comes from ␲ ’s pro- either accelerators or nuclear reactors. In disappearance duced in a 30-cm-long water target when hit by protons experiments, using a neutrino beam primarily composed from the facility’s linac with 800 MeV kinetic energy. of a single flavor, one looks for attenuation due to mix- The detector is a tank filled with 167 metric tons of di- ing with other flavors. In appearance experiments, one lute liquid scintillator, located about 30 m from the neu- searches for interactions by neutrinos of a flavor not trino source. ϩ present in the original neutrino beam. Most of the produced ␲ ’s come to rest and decay ϩ ϩ ϩ Most of the past laboratory experiments did not have through the sequence ␲ →␮ ␯␮ , followed by ␮ → ϩ␯ ␯ ␯ an oscillation signal, nor do most of the present ones. In e e¯ ␮ . The ¯ ␮’s so produced have a maximum en- such a case, as discussed in Sec. III.A, the experiment ergy of 52.8 MeV. This is called the decay at rest flux and ␯ →␯ sets a limit on the corresponding oscillation probability. is used to study ¯ ␮ ¯ e oscillations. The energy depen- Ͻ ␯ Appearance experiments set limits ͗P␣␤͘ PL for given dence of the ¯ ␮ flux from decay at rest is very well flavors ␣Þ␤. Disappearance experiments set limits known, and the absolute value is known to within 7%. Ͼ Ϫ ␣ ͗P␣␣͘ 1 PL for a given flavor which, in the two- The open space around the target is short compared to Ͻ ␤ ␲ϩ neutrino case, can be translated into ͗P␣␤͘ PL for the pion-decay length. Thus only 3% of the ’s decay Þ␣. The results are usually interpreted in a two- in flight. The decay in flight ␯␮ flux is used to study ␯␮ →␯ neutrino framework as exclusion regions in the e oscillations. ⌬ 2 2 ␪ ␯ ( m ,sin 2 ) plane. One can take the upper bound on For decay-at-rest-related measurements, ¯ e’s are de- ⌬ 2 ␯ → ϩ the mixing angle in the asymptotic large m range and tected in the quasielastic process ¯ e p e n, in correla- 2 ␪ ϭ translate it back into the value of PL : sin 2 lim 2PL tion with a monochromatic photon of 2.2 MeV arising → ␥ (see discussion in Sec. III.A). The probability PL is the from the neutron capture reaction np d . The main

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 380 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

TABLE IV. 90% C.L. limit on the neutrino oscillation probabilities from negative searches at short-baseline experiments.

⌬ 2 2 Experiment Beam Channel Limit (90%) mmin (eV ) Ref.

CDHSW CERN ␯␮→␯␮ P␮␮Ͼ0.95 0.25 Dydak et al., 1984 ␯ →␯ Ͻ ϫ Ϫ3 E776 BNL ␮ e Pe␮ 1.5 10 0.075 Borodovsky et al., 1992 ␯ →␯ Ͻ ϫ Ϫ3 E734 BNL ␮ e Pe␮ 1.6 10 0.4 Ahrens et al., 1987 ␯ →␯ Ͻ ϫ Ϫ4 KARMEN2 Rutherford ¯ ␮ ¯ e Pe␮ 6.5 10 0.05 Wolf et al., 2001 Ϫ3 E531 FNAL ␯␮→␯␶ P␮␶Ͻ2.5ϫ10 0.9 Ushida et al., 1986 Ϫ3 CCFR FNAL ␯␮→␯␶ P␮␶Ͻ4ϫ10 1.6 McFarland et al., 1995 ␯ →␯ Ͻ e ␶ Pe␶ 0.10 20 Naples et al., 1999 ␯ →␯ Ͻ ϫ Ϫ4 ␮ e P␮e 9. 10 1.6 Romosan et al., 1997 Ϫ4 Chorus CERN ␯␮→␯␶ P␮␶Ͻ3.4ϫ10 0.6 Eskut et al., 2001 ␯ →␯ Ͻ ϫ Ϫ2 e ␶ Pe␶ 2.6 10 7.5 Eskut et al., 2001 Ϫ4 Nomad CERN ␯␮→␯␶ P␮␶Ͻ1.7ϫ10 0.7 Astier et al., 2001 ␯ →␯ Ͻ ϫ Ϫ3 e ␶ Pe␶ 7.5 10 5.9 Astier et al., 2001 ␯ →␯ Ͻ ϫ Ϫ4 ␮ e P␮e 6 10 0.4 Valuev et al., 2001

␯ ␯ ␯ ␯ background is due to the ¯ e component in the beam that identical intensities for ␮ , e, and ¯ ␮ emitted isotropi- is produced in the decay chain starting with ␲Ϫ’s. This cally. There is a minor fraction of ␲Ϫ decaying in flight background is suppressed by three factors. First, the ␲ϩ with the following ␮Ϫ decay at rest in the target station ␲Ϫ ␯ ␯ production rate is about eight times the production which leads to a very small contamination of ¯ e / e rate in the beam stop. Second, 95% of the ␲Ϫ’s come to Ͻ6.2ϫ10Ϫ4. The energy spectra of the ␯’s are well de- rest and are absorbed before decay in the beam stop. fined due to the decay at rest of both the ␲ϩ and ␮ϩ. Third, 88% of the ␮Ϫ’s from ␲Ϫ’s decay-in-flight are The neutrinos are detected in a rectangular tank filled captured from atomic orbit, a process which does not with 56 t of a liquid scintillator. The signature for the ␯ ␯ give a ¯ e . Thus, the relative yield, compared to the posi- detection of ¯ e is a spatially correlated delayed coinci- ϳ ϫ ϫ ␯ ϩ tive channel, is estimated to be (1/8) 0.05 0.12 dence of positrons from ¯p(¯ e ,e )n with energies up to ϭ ϫ Ϫ4 ϭ Ϫ ϭ ␥ 7.5 10 . For decay-in-flight-related measurements, Eeϩ E¯␯ Q 51.0 MeV and emission of either of ␯ e the e’s are observed via the detection of electrons pro- ␯ → Ϫ Ͻ duced in the process eC e X with energy 60 Ee Ͻ200 MeV. In both decay at rest and in flight measurements, an excess of events was observed as compared to the expected background while the excess was consistent with ␯ →␯ ¯ ␮ ¯ e oscillations. Further supporting evidence was ␯ →␯ provided by the signal in the ␮ e channel. In the latest results including the runs till 1998 (Aguilar et al., 2001) the total fitted excess is of 87.9Ϯ22.4Ϯ6 events, corresponding to an oscillation probability of (2.64 Ϯ0.67Ϯ0.45)ϫ10Ϫ3. In the two-family formalism these results lead to the oscillation parameters shown in Fig. 31. The shaded regions are the 90% and 99% likelihood regions from LSND. The best-fit point corresponds to ⌬m2ϭ1.2 eV2 and sin2 2␪ϭ0.003. The region of parameter space which is favored by the LSND observations has been partly tested by other experiments like the old BNL E776 experiment (Boro- dovsky et al., 1992) and more recently by the KARMEN experiment (Gemmeke et al., 1990). The KARMEN experiment is performed at the neutron spallation facility ISIS of the Rutherford Appleton Laboratory. Neutrinos are produced by stopping the 800 MeV protons in a massive beam stop target, thereby producing pions. The Ϫ ϩ ␲ ’s are absorbed by the target nuclei whereas ␲ ’s de- ␯ →␯ ϩ ϩ FIG. 31. Allowed regions (at 90% and 99% C.L.) for e ␮ cay at rest producing muon neutrinos via ␲ →␮ ␯␮ . ϩ oscillations from the LSND experiment compared with the ex- The low momentum ␮ ’s are also stopped within the clusion regions (at 90% C.L.) from KARMEN2 and other ex- ␮ϩ→ ϩ␯ ␯ massive target and decay at rest, e e¯ ␮ . The periments. The 90% C.L. expected sensitivity curve for ϩ ϩ ␲ Ϫ␮ decay chain at rest gives a neutrino source with MiniBooNE is also shown. From Wolf et al., 2001.

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 381

the two neutron capture processes: p(n,␥)d with one ␥ of E(␥)ϭ2.2 MeV or Gd(n,␥) with 3␥ quanta on average and ͚E(␥)ϭ8 MeV. The raw data presented at the EPS HEP 2001 conference (Wolf et al., 2001) correspond to ϳ9400 C protons on target. An analysis of the data results in 11 sequential events which satisfy all cuts. This number is in good agreement with the total background expectation of 12.3Ϯ0.6. Applying a Bayesian renormalization procedure, an upper limit of N(osc)Ͻ6.3 at 90% C.L. can be extracted. However, using the spectral information and a maximum-likelihood analysis, KARMEN find a best-fit value N(osc)ϭ0 within the physically allowed range of parameters, which can be translated into an upper limit of 3.8 and 3.1 oscillation events for ⌬m2Ͻ1eV2 and ⌬m2Ͼ20 eV2, respectively. These numbers are based on a complete frequentist approach as suggested by G. Feldman and R. Cousins. The corresponding exclusion curve in the two-neutrino parameter space is given in Fig. 31 together with the favored region for the LSND FIG. 32. Excluded regions at 90% C.L. for ␯ oscillations from 2 e experiment (from Wolf et al., 2001). At high ⌬m , reactors experiments and the expected sensitivity from the KARMEN results exclude the region favored by LSND. KamLAND experiment. At low ⌬m2, KARMEN leaves some allowed space, but the reactor experiments at Bugey and CHOOZ add ments. They have the advantage that smaller values of stringent limits for the larger mixing angles. This figure ⌬m2 can be accessed due to the lower neutrino beam represents the final status of the LSND oscillation signal. energy. The MiniBooNE experiment (Bazarko et al., 2000) In Fig. 32 we show the corresponding excluded re- ␯ →␯ searches for ␮ e oscillations and is specially designed gions in the parameter space for two neutrino oscilla- to make a conclusive statement about the LSND’s neu- tions from the negative results of the reactor experi- ␯ trino oscillation evidence. They use a ␮ beam of energy ments Gosgen (Zacek, 1986), Krasnoyarsk (Vidyakin 0.5–1.0 GeV initiated by a primary beam of 8-GeV pro- et al., 1994), Bugey (Achkar et al., 1995), and CHOOZ tons from the Fermilab Booster, which contains only a (Apollonio et al., 1999). Gosgen, Krasnoyarks, and ␯ small intrinsic e component (less than 0.3%). They Bugey have relatively short baselines. From the figure search for an excess of electron neutrino events in a we see that Bugey sets the strongest constraint on the detector located approximately 500 m from the neutrino allowed mixing in the ⌬m2 range that is interesting for source. The MiniBooNE neutrino detector consists of the LSND signal. CHOOZ, which can be considered the 800 tons of pure mineral oil contained in a 12.2-m- first long baseline reactor experiment (LӍ1 km), is sen- diameter spherical tank. A structure in the tank supports sitive to lower values of ⌬m2. Its 90% C.L. limits in- phototubes, which detect neutrino interactions in the oil clude ⌬m2Ͻ7ϫ10Ϫ4 eV2 for maximal mixing, and by the Cerenkov and scintillation light that they pro- sin2 2␪Ͻ0.10 for large ⌬m2. The CHOOZ results are sig- duce. nificant in excluding part of the region that corresponds The L/E ratio is similar to that of LSND, giving Mini- to the large-mixing-angle solution of the solar neutrino BooNE sensitivity to the same mode of oscillations. problem (see Sec. IV). Furthermore, the CHOOZ However, neutrino energies are more than an order of bound rules out with high significance the possibility magnitude higher than at LSND, so that the search at ␯ →␯ that ␮ e oscillations explain the atmospheric neu- MiniBooNE employs different experimental techniques. trino deficit. The CHOOZ constraint on the mixing In Fig. 31 we show the 90% C.L. limits that MiniBooNE angle is also relevant to the interpretation of the atmo- can achieve. Should a signal be found then the next step spheric neutrino anomaly in the framework of three- would be the BooNe experiment. neutrino mixing. We return to this issue in Sec. V. Smaller values of ⌬m2 can be accessed at future reac- C. Disappearance experiments at reactors tor experiments using longer baseline. Pursuing this idea, the KamLAND experiment (Piepke et al., 2001), a Neutrino oscillations are also searched for using neu- 1000-ton liquid scintillation detector, is currently in op- trino beams from nuclear reactors. Nuclear reactors pro- eration in the Kamioka mine in Japan. This under- ␯ ϳ duce ¯ e beams with E␯ MeV. Due to the low energy, ground site is conveniently located at a distance of 150– e’s are the only charged leptons which can be produced 210 km from several Japanese nuclear power stations. ␯ in the neutrino charged-current interaction. If the ¯ e os- The measurement of the flux and energy spectrum of the ␯ cillated to another flavor, its charged-current interaction ¯ e’s emitted by these reactors will provide a test to the could not be observed. Therefore oscillation experi- large-mixing-angle solution of the solar neutrino ments performed at reactors are disappearance experi- anomaly. In Fig. 32 we plot the expected 90% sensitivity

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 382 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

for the KamLAND experiment after three years of data have been observed in 22.5 kt of the fiducial volume. taking (from Piepke et al., 2001). The experiment will, The statistical probability that the observation would be for the first time, provide a completely solar-model- equal to or smaller than 44 is ϳ8%. In the presence of independent test of this particle physics solution of the oscillations the expected number of events would be solar neutrino problem. After a few years of data taking, 41.5 for ⌬m2ϭ3ϫ10Ϫ3 eV2 and maximal mixing. Al- it should be capable of either excluding the entire large- though the central value of the number of observed ␯ ↔␯ mixing-angle region or, not only establishing e other events is consistent with the data from atmospheric neu- oscillations, but also measuring the oscillation param- trinos, the discrepancy with the no-oscillation prediction eters with unprecedented precision. Data taking is exis still within the statistical error. K2K has also studied pected to commence in 2002. the energy distribution of these events compared to the expectation based on the pion monitor data and a D. Long-baseline experiments at accelerators Monte Carlo simulation, and finds good agreement with the expectation from oscillations with the atmospheric Smaller values of ⌬m2 can also be accessed using ac- mass difference and maximal mixing, but the statistics celerator beams at long-baseline experiments. In these are too poor to give any further constraint. experiments the intense neutrino beam from an accel- MINOS is designed to detect neutrinos delivered by erator is aimed at a detector located underground at a the Main Injector accelerator at Fermilab (NuMI) with ϳ distance of several hundred kilometers. The main goal average energies of 5 – 15 GeV depending on the of these experiments is to test the presently allowed so- beam configuration. Two detectors, functionally identi- lution for the atmospheric neutrino problem by search- cal, will be placed in the NuMI neutrino beam: one at ing for either ␯␮ disappearance or ␯␶ appearance. At Fermilab and the second one in Soudan iron mine, 732 present there are three such projects approved: K2K km away. The MINOS detectors are iron/scintillator (Nishikawa et al., 1997; Ahn et al., 2001; Nishikawa, sampling calorimeters with a toroidal magnetic field in ␯ et al., 2001), which runs with a baseline of about 235 km the iron. Observed interactions of ␮ can be divided into ␮ from KEK to SK, MINOS (Ables et al., 1995; Wojcicki, two classes: charged-current-like, with an identified 2001) under construction with a baseline of 730 km from track, and NC-like, muonless. The ratio of the observed Fermilab to the Soudan mine where the detector will be numbers of the charged-current- and neutral-current- placed, and OPERA (Shibuya et al., 1997; Cocco et al., like events in the two detectors provides a sensitive test 2000), under construction with a baseline of 730 km for oscillations. With its expected sensitivity, MINOS from CERN to Gran Sasso. With their expected sensi- will be able to precisely measure (roughly at the level of ␯ →␯ tivities, these experiments can cover either some frac- 10%) the oscillation parameters in the ␮ ␶ channel. tion or all of the parameter region suggested by the at- The primary measurement for this is the comparison of mospheric neutrino anomaly discussed in Sec. V. the rate and spectrum of the charged-current events in In the K2K experiment, a wide-band, almost pure ␯␮ the Far Detector with those in the Near Detector. Com- beam from ␲ϩ decays is generated in the KEK paring the neutral-current/charged-current ratios in the 12-GeV/c Proton Synchrotron and a neutrino beam two detectors, the experiment can also be sensitive to ␯ →␯ line. The detector is in SK at a distance of 250 km. Vari- the presence of ␮ s oscillations. MINOS is scheduled ous beam monitors along the beam line and two differ- to start data taking at the end of 2004. ent types of front detectors have also been constructed Both K2K and MINOS have also the capability for ␯ ␯ →␯ at the KEK site. The front detectors area1ktwater detecting the appearance of e’s due to ␮ e oscilla- Cerenkov detector, which is a miniature of the SK de- tions. This signal however suffers from large back- ␯ tector, and a so-called fine-grained detector which is grounds due to both the e’s in the beam and the NC ␯ composed of a scintillating fiber tracker trigger counters, events with a topology similar to the e interaction. lead glass counters and a muon range detector. The char- These backgrounds can be partially suppressed using the acteristics of the neutrino beam in the KEK site— information from the near detectors. In particular, direction, intensity, stability, energy spectrum and ␯ MINOS may be able to improve the CHOOZ bounds. e ␯ →␯ Ϫ␯␮ composition are examined using front detectors OPERA is designed to search for ␮ ␶ oscillations and beam monitors. They are then extrapolated to the in the Gran Sasso Laboratory. It will study the interac- SK site and used to obtain the expected number of tion of 20-GeV neutrinos produced at CERN. The goal ␯ ␯ events and the energy spectrum. is to observe the appearance of ␶’s in a pure ␮ beam. The K2K experiment had a successful start in early The detector is based on a massive lead/nuclear emul- 1999, and data were recorded during several periods in sion target. Nuclear emulsions are exploited for the di- ␶ 2000 and 2001. The accumulated beam intensity during rect observation of the decay of the in a very low back- the 2000 runs was 22.9ϫ1018 protons on target (Ahn ground environment. et al., 2001), increased to ϳ38ϫ1018 protons on target with the 2001 run till the summer (Nishikawa et al., 2001), which was about 40% of the goal of the experi- E. Direct determination of neutrino masses ment, 1020 protons on target. The no-oscillation prediction for this sample, based on the data from the front Oscillation experiments have provided us with impor- ϩ6.1 detectors, is 63.9Ϫ6.6 events, while a total of 44 events tant information on the differences between the neu-

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 383

⌬ 2 → ϩ ϩ Ϫϩ Ϫ trino masses-squared, mij , and on the leptonic mixing searches (A,Z) (A,Z 2) e e . The rate of this angles, Uij . But they are insensitive to the absolute mass process is proportional to the effective Majorana mass of ␯ scale for the neutrinos, mi . e , Of course, the results of an oscillation experiment do ⌬ 2 2 provide a lower bound on the heavier mass in mij , ϭ͚ͯ ͯ mee miUei , (127) ͉ ͉уͱ⌬ 2 ⌬ 2 Ͼ i mi mij for mij 0. But there is no upper bound on this mass. In particular, the corresponding neutrinos which, in addition to five parameters that affect the tri- could be approximately degenerate at a mass scale that tium beta decay spectrum, depends also on the three ͱ⌬ 2 is much higher than mij. Moreover, there is neither leptonic CP-violating phases. Notice that in order to in- ␤ ␯ ␯ upper nor lower bound on the lighter mass mj . duce the 2 0 decay, e must be a Majorana particle. Information on the neutrino masses, rather than mass The present strongest bound from 2␤0␯ decay is ob- differences, can be extracted from kinematic studies of tained by the Heidelberg-Moscow group (Klapdor- reactions in which a neutrino or an antineutrino is in- Kleingrothaus et al., 2001): volved. In the absence of mixing, the present limits are m Ͻ0.34 ͑0.26͒ eV, 90% ͑68%͒ C.L. (128) (Groom et al., 2000) ee Ϫ Taking into account systematic errors related to nuclear m␯ Ͻ18.2 MeV ͑95% C.L.͒ from ␶ →n␲ϩ␯␶ , ␶ matrix elements, the bound may be weaker by a factor (123) ϳ of about 3. A sensitivity of mee 0.1 eV is expected to Ͻ ͑ ͒ ␲Ϫ→␮Ϫϩ␯ be reached by the currently running NEMO3 experi- m␯␮ 190 keV 90% C.L. from ¯ ␮ , (124) ment (Marquet et al., 2000), while a series of new experiments (CUORE, EXO, GENIUS) is planned with sen- 3 3 - m␯ Ͻ2.2 eV ͑95% C.L.͒ from H→ Heϩe ϩ¯␯ , ϳ e e sitivity of up to mee 0.01 eV. (125) The knowledge of mee will provide information on the mass and mixing parameters that is independent of the where for the bound on m␯ we take the latest limit from e ⌬ 2 mij’s. However, to infer the values of neutrino masses, the Mainz experiment (Bonn et al., 2001). A similar additional assumptions are required. In particular, the bound is obtained by the Troitsk experiment (Lobashev mixing elements are complex and may lead to strong et al., 2001). A new experimental project, KATRIN, is Ӷ cancellation, mee m1 . Yet, the combination of results under consideration with an estimated sensitivity limit: from 2␤0␯ decays and tritium beta decay can test and, in m␯ ϳ0.3 eV. e some cases, determine the mass parameters of given In the presence of mixing these limits have to be schemes of neutrino masses (Vissani, 1999; Bilenky, Pas- modified and in general they involve more than a single coli, and Petcov, 2001a, 2001b; Farzan, Peres, and flavor parameter. The limit that is most relevant to our Smirnov, 2001; Klapdor-Kleingrothaus, Pas, and purposes is the most sensitive one from tritium beta de- Smirnov, 2001). cay. In presence of mixing, the electron neutrino is a combination of mass eigenstates and the tritium beta decay spectrum is modified as (Shrock, 1980) VII. THREE-NEUTRINO MIXING dN ϭ ͑ ͒ ͉ ͉2͓͑ Ϫ ͒2Ϫ 2͔1/2⌰͑ Ϫ Ϫ ͒ R E ͚ Uei E0 E mi E0 E mi , In the previous sections we discussed the three pieces dE i of evidence for neutrino masses and mixing (solar neu- (126) trinos, atmospheric neutrinos, and the LSND results) as where E is the energy of electron, E0 is the total decay usually formulated in the framework of two-neutrino os- energy, and R(E)ism␯ independent. The step function, cillations. The results are summarized in Fig. 33 where ⌰ Ϫ Ϫ (E0 E mi), reflects the fact that a given neutrino we show the ranges of masses and mixing implied by can only be produced if the available energy is larger these signals at 90% and 99% C.L. for 2 degrees of free- than its mass. According to Eq. (126), there are two im- dom, as well as relevant constraints from negative portant effects, sensitive to the neutrino masses and mix- searches in laboratory experiments. ings, on the electron energy spectrum: (i) kinks at the The three pieces of evidence correspond to three val- (i)ϭ ϳ Ϫ electron energies Ee E E0 mi with sizes that are ues of mass-squared differences of different orders of ͉ ͉2 determined by Uei ; (ii) a shift of the end point to magnitude. Consequently, there is no consistent expla- ϭ Ϫ Eep E0 m1 , where m1 is the lightest neutrino mass. nation for all three signals based on oscillations among The situation simplifies considerably if we are interested the three known neutrinos. The argument supporting in constraining the possibility of quasidegenerate neutri- this statement is very simple. With three neutrinos, there nos with mass ϳm␯ . In this case the distortion of the are only two independent mass-squared differences, spectrum can be described by a single parameter (Far- since the following relation must hold: zan, Peres, and Smirnov, 2001; Vissani, 2001a), m␤ ⌬m2 ϩ⌬m2 ϩ⌬m2 ϭ0. (129) ϭ͚ ͉ ͉2 ͚ ͉ ͉2 ϳ 21 32 13 imi Uei / i Uei m␯ . So the limit in Eq. (125) ap- ⌬ 2 plies to the unique neutrino mass scale. This relation cannot be satisfied by three mij that are Direct information on neutrino masses can also be ob- of different orders of magnitude. One may wonder if tained from neutrinoless double beta decay (2␤0␯) this naive extrapolation from the two-neutrino oscilla-

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 384 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

trino oscillations in general, but for the purposes of this section we can set it to zero. In this case the mixing matrix can be conveniently parametrized in the standard form (Groom et al., 2000):

c13c12 s12c13 s13 Ϫ Ϫ Ϫ Uϭͩ s12c23 s23s13c12 c23c12 s23s13s12 s23c13 ͪ , Ϫ Ϫ Ϫ s23s12 s13c23c12 s23c12 s13s12c23 c23c13 (130) ϵ ␪ ϵ ␪ where cij cos ij and sij sin ij . As we have seen in the previous sections, in most of the parameter space of solutions for solar and atmospheric oscillations, the required mass differences satisfy ⌬ 2 Ӷ⌬ 2 m᭪ matm . (131) ␪ In this approximation the angles ij can be taken with- ␪ out loss of generality to lie in the first quadrant, ij ෈͓0,␲/2͔. There are two possible mass orderings which we chose as ⌬ 2 ϭ⌬ 2 Ӷ⌬ 2 Ӎ⌬ 2 ϭ⌬ 2 Ͼ m21 m᭪ m32 m31 matm 0; (132) ⌬ 2 ϭ⌬ 2 ӶϪ⌬ 2 ӍϪ⌬ 2 ϭ͉⌬ 2 ͉Ͼ m21 m᭪ m31 m32 matm 0. (133) We refer to the first option, Eq. (132), as the direct scheme, and to the second one, Eq. (133), as the inverted FIG. 33. Summary of the present pieces of evidence for neu- scheme. The direct scheme is naturally related to hierar- trino masses and mixing as obtained from 2-␯ oscillation analy- Ӷ Ӷ Ӎͱ⌬ 2 chical masses, m1 m2 m3 , for which m2 m21 and ses. The allowed regions correspond to 90% and 99% C.L. for Ӎͱ⌬ 2 Ӎ 2 degrees of freedom. We also show relevant bounds from m3 m32, or to quasidegenerate masses, m1 m2 Ӎ ӷ⌬ 2 ⌬ 2 laboratory experiments. m3 m21 , m32 . On the other hand, the inverted Ͻ Ӎ scheme implies that m3 m1 m2 . tion picture holds once the full mixing structure of a One may wonder how good an approximation it is to three-neutrino oscillation is taken into account or, con- set the CP-violating phases to zero. It turns out to be an versely, once some special configuration of the three- excellent approximation for the analysis of solar, atmo- neutrino parameters fits the three pieces of evidence. spheric, and laboratory data if Eq. (131) holds. In this The combined fit to the data performed by Fogli, Lisi, case, as discussed below, no simultaneous effect of the ␯ Marrone, and Scioscia (1999a) and by Gonzalez-Garcia two mass differences is observable in any -appearance and Maltoni (2002) shows that this is not the case. transition. Whereas in the case of the solar and atmospheric neutrino indications several experiments agree on the exis- A. Probabilities tence of the effect, the third indication is presently The determination of the oscillation probabilities for found only by the LSND experiment. In many studies both solar and atmospheric neutrinos requires that one the LSND result is therefore left out and the analysis of solves the evolution equation of the neutrino system in the solar and atmospheric data is performed in the the matter background of the Sun or the Earth. In a framework of mixing between the three known neutri- three-flavor framework, this equation reads nos. We follow this approach and discuss next the derived masses and mixing in these scenarios. Alterna- d␯ជ i ϭH␯ជ , HϭU Hd U†ϩV, (134) tively, a possible explanation of the four pieces of dt • 0 • evidence with only three neutrinos has been proposed ␯ជ ϵ ␯ ␯ ␯ T assuming that CPT is violated in the neutrino sector where U is the lepton mixing matrix, ( e , ␮ , ␶) , d (see, for instance, Barenboim, Borissov, Lykken, and H0 is the vacuum Hamiltonian, Smirnov, 2001; Murayama and Yanagida, 2001). 1 dϭ ͑Ϫ⌬ 2 ⌬ 2 ͒ The combined description of both solar and atmo- H0 diag m21,0, m32 , (135) 2E␯ spheric anomalies requires that all three known neutrinos take part in the oscillations. The mixing parameters and V is the effective potential that describes charged- are encoded in the 3ϫ3 lepton mixing matrix (Maki, current forward interactions in matter: Nakagawa, and Sakata, 1962; Kobayashi and Maskawa, Vϭdiag͑Ϯ&G N ,0,0͒ϵdiag͑V ,0,0͒. (136) 1973). The two Majorana phases do not affect neutrino F e e oscillations (Bilenky, Hosek, and Petcov, 1980; Lan- In Eq. (136), the sign ϩ (Ϫ) refers to neutrinos (an- gacker et al., 1987). The Dirac phase (that is, the analog tineutrinos), and Ne is electron number density in the of the KM phase of the quark sector) does affect neu- Sun or the Earth.

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 385

In what follows we focus on the direct scheme of Eq. the resulting survival probabilities do not depend on ⌬ 2 ␪ (132), for which the five relevant parameters are related m21 and 12 . For instance, for constant Earth matter to experiments in the following way: density, the various P␣␤ can be written as follows: ⌬ 2 ϭ⌬ 2 ⌬ 2 ϭ⌬ 2 ϭ Ϫ 2 2 m᭪ m21 , matm m32 , (137) Pee 1 4s13,mc13,mS31 , ␪ ϭ␪ ␪ ϭ␪ ␪ ϭ␪ ϭ Ϫ 2 2 4 Ϫ 2 2 2 ᭪ 12 , atm 23 , reactor 13 . (138) P␮␮ 1 4s13,mc13,ms23S31 4s13,ms23c23S21 For transitions in vacuum, the results apply also to the Ϫ 2 2 2 4c s c S32 , inverted scheme of Eq. (133). In the presence of matter 13,m 23 23 ϭ Ϫ 2 2 4 Ϫ 2 2 2 effects, the direct and inverted schemes are no longer P␶␶ 1 4s13,mc13,mc23S31 4s13,ms23c23S21 equivalent, although the difference is hardly recogniz- Ϫ 2 2 2 able in the current solar and atmospheric neutrino phe- 4c13,ms23c23S32 , (142) nomenology as long as Eq. (131) holds (Fogli, Lisi, Mon- ϭ 2 2 2 Pe␮ 4s13,mc13,ms23 S31 , tanino, and Scioscia, 1997; Gonzalez-Garcia and ϭ 2 2 2 Maltoni, 2002). Under this approximation, the results Pe␶ 4s13,mc13,mc23S31 , obtained for the direct scheme can be applied to the ϭϪ 2 2 2 2 ϩ 2 2 2 ⌬ 2 →Ϫ⌬ 2 P␮␶ 4s13,mc13,ms23c23S31 4s13,ms23c23S21 inverted scheme by replacing m32 m32 . In general the transition probabilities present an oscil- ϩ 2 2 2 4c13,ms23c23S32 . latory behavior with two oscillation lengths. However, ␪ the hierarchy in the splittings, Eq. (131), leads to impor- Here 13,m is the effective mixing angle in matter, tant simplifications. ␪ sin 2 13 Let us first consider the analysis of solar neutrinos. A sin 2␪ ϭ , 13,m 2 2 2 oscϭ ␲ ⌬ 2 ͱ͑cos 2␪ Ϫ2E␯V /⌬m ͒ ϩ͑sin 2␪ ͒ first simplification occurs because L32 4 E/ m32 is 13 e 32 13 much shorter than the distance between the Sun and the (143) osc Earth. Consequently, the oscillations related to L32 are and Sij are the oscillating factors in matter, averaged in the evolution from the Sun to the Earth. A ⌬␮2 second simplification occurs because, for the evolution ϭ 2ͩ ij ͪ ⌬ 2 Sij sin L . (144) in both the Sun and the Earth, m32 4E␯ 2 ӷ2&G N E␯ sin 2␪ . Consequently, matter effects on F e 13 In Eq. (144), ⌬␮2 are the effective mass-squared differ- the evolution of ␯ can be neglected. The result of these ij 3 ences in matter: two approximations is that the three-flavor evolution equations decouple into an effective two-flavor problem ⌬m2 sin 2␪ ⌬␮2 ϭ 32 ͩ 13 Ϫ ͪ Ϫ for the basis (Kuo and Pantaleone, 1986; Shi and 21 ␪ 1 E␯Ve , 2 sin 2 13,m Schramm, 1992) ⌬m2 ␪ ␯ ϭ ␯ ϩ ␯ ␯ ϭϪ ␯ ϩ ␯ 2 32 sin 2 13 eЈ c12 1 s12 2 , ␮Ј s12 1 c12 2 , (139) ⌬␮ ϭ ͩ ϩ ͪ ϩ 32 ␪ 1 E␯Ve , 2 sin 2 13,m with the substitution of Ne by the effective density sin 2␪ N ⇒N cos2 ␪ . (140) ⌬␮2 ϭ⌬ 2 13 e e 13 31 m32 ␪ , (145) sin 2 13,m Thus the survival probability takes the following form: and L is the path length of the neutrino within the 3␯ ϭ 4 ␪ ϩ 4 ␪ 2␯ Pee,᭪ sin 13 cos 13PeЈeЈ,᭪ , (141) Earth, which depends on its direction. We conclude that ␯ the analysis of the atmospheric data constrains three of where P2 is the two-flavor survival probability in eЈeЈ,᭪ the five independent oscillation parameters: ⌬m2 , ␪ , ⌬ 2 ␪ 32 23 the ( m21 , 12) parameter space, but with the modified ␪ and 13 . matter density of Eq. (140). We conclude that the analy- So we find that, in the approximation of Eq. (131), the sis of the solar data constrains three of the five indepen- mixing angle ␪ is the only parameter common to both ⌬ 2 ␪ ␪ 13 dent oscillation parameters: m21 , 12 , and 13 . solar and atmospheric neutrino oscillations and which Equation (141) reveals the dominant effect of a non- may potentially allow for some mutual influence. The ␪ vanishing 13 in the solar neutrino survival probability: main effect of the three-neutrino mixing is that now at- 2␯ the energy-dependent part of the probability, PeЈeЈ,᭪ , mospheric neutrinos can oscillate simultaneously in both 4 ␪ ␯ →␯ ␯ →␯ ␯ →␯ ␯ gets damped by the factor cos 13 , while an energy- the ␮ ␶ and ␮ e (and, similarly, e ␶ and e 4 ␪ ␪ independent term, sin 13 , is added. Thus increasing 13 →␯␮) channels. The oscillation amplitudes for channels ␯ 2 ␪ makes the solar neutrino survival probability more and involving e are controlled by the size of sin 13 ϭ͉ ͉2 more energy independent. Ue3 . We learn that in the approximation of Eq. Let us now consider the analysis of atmospheric neu- (131), solar and atmospheric neutrino oscillations de- oscϭ ␲ ⌬ 2 ␪ ϭ trinos. Here L21 4 E/ m21 is much larger than the couple in the limit 13 0. This angle is constrained by relevant distance scales. Consequently, the correspond- the CHOOZ reactor experiment. To analyze the ing oscillating phase is negligible. In this approximation CHOOZ constraints we need to evaluate the survival ␪ ␯ ϳ one can rotate away the corresponding angle 12 . Thus probability for ¯ e of average energy E few MeV at a

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 386 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

distance of Lϳ1 km. For these values of energy and distance, one can safely neglect Earth matter effects. The survival probability takes the analytical form ⌬m2 L CHOOZϭ Ϫ 4 ␪ 2 ␪ 2ͩ 21 ͪ Pee 1 cos 13 sin 2 12 sin 4E␯ ⌬m2 L Ϫ 2 ␪ ͫ 2 ␪ 2ͩ 31 ͪ sin 2 13 cos 12 sin 4E␯ ⌬m2 L ϩ 2 ␪ 2ͩ 32 ͪͬ sin 12 sin 4E␯ ⌬m2 L Ӎ Ϫ 2 ␪ 2ͩ 32 ͪ 1 sin 2 13 sin , (146) 4E␯ where the second equality holds under the approxima- ⌬ 2 Ӷ tion m21 E␯ /L which can only be safely made for ⌬ 2 Շ ϫ Ϫ4 2 m21 3 10 eV . Thus in general the analysis of the CHOOZ reactor date involves four oscillation param- ⌬ 2 ␪ ⌬ 2 ␪ eters: m32 , 13 , m21 , and 12 (Bilenky, Nicolo, and Petcov, 2001; Gonzalez-Garcia and Maltoni, 2002).

B. Allowed masses and mixing

There are several three-neutrino oscillation analyses in the literature which include either solar (Fogli, Lisi, and Montanino, 1996; Fogli, Lisi, Montanino, and Palazzo, 2000a, 2000b; Gago, Nunokawa, and Zukanov- ich Funchal, 2001) or atmospheric (Fogli, Lisi, Mon- tanino, and Scioscia, 1997; Fogli, Lisi, Marrone, and Sci- oscia, 1999b; Teshima and Sakai, 1999; de Rujula, FIG. 34. Allowed regions (at 90%, 95%, and 99% C.L.) in the Gavela, and Hernandez, 2001; Fogli, Lisi, and Marrone, ⌬ 2 2 ␪ ( m21 ,tan 12) plane from a global analysis of solar neutrino 2001) neutrino data. Combined studies have also been data in the framework of three-neutrino oscillations for vari- 2 ␪ performed (Barger, Whistnant, and Phillips, 1980; Fogli, ous values of sin 13 . The global best-fit point is marked by Lisi, and Montanino, 1994, 1995; Barger and Whisnant, the star. 1999). We follow and update here the results from the latest analysis of Gonzalez-Garcia, Maltoni, Penã-Garay, and Valle (2001). for C.L.ϭ90%, 95%, and 99%, respectively. The global As discussed above, in the approximation of Eq. minimum used in the construction of the regions lies in (131), solar and atmospheric neutrino oscillations de- the large-mixing-angle region and corresponds to ͉ ͉ϭ ␪ ϭ 2 ␪ ϭ couple in the limit Ue3 sin 13 0. In this limit, the tan 13 0, that is, to the decoupled scenario. allowed values of the parameters can be obtained di- As seen in the figure, the modifications to the decou- ␪ 2 ␪ rectly from the results of the analyses in terms of two- pled case are significant only if 13 is large. As sin 13 neutrino oscillations presented in Secs. IV and V. Devia- increases, all the allowed regions disappear, leading to 2 ␪ tions from the two-neutrino scenario are then an upper bound on sin 13 that is independent of the ␪ determined by the size of 13 . Thus the first question to values taken by the other parameters in the three- answer is how the presence of this additional angle af- neutrino mixing matrix. For instance, no region of pa- fects the analysis of the solar and atmospheric neutrino rameter space is allowed (at 99% C.L. for 3 degrees of 2 ␪ ϭ͉ ͉2Ͼ data. freedom) for sin 13 Ue3 0.80. This fact is also illus- In Fig. 34 we show the allowed regions for the oscil- trated in Fig. 35 where we plot the shift in ␹2 as a func- ⌬ 2 2 ␪ 2 ␪ lation parameters m21 and tan 12 from the global tion of sin 13 when the mass and mixing parameters ⌬ 2 2 ␪ ␹2 analysis of the solar neutrino data in the framework m12 and tan 12 are chosen to minimize . of three-neutrino oscillations for different values of In Fig. 36 we show the allowed regions for the oscil- 2 ␪ ⌬ 2 2 ␪ sin 13 (updated from Gonzalez-Garcia, Maltoni, Penã- lation parameters m32 and tan 23 from the analysis of Garay, and Valle, 2001). The allowed regions for a given the atmospheric neutrino data in the framework of 2 ␪ confidence level are defined as the set of points satis- three-neutrino oscillations for different values of sin 13 ␹2 ⌬ 2 2 ␪ 2 ␪ Ϫ␹2 fying the condition ( m12 ,tan 12 ,tan 13) min (updated from Gonzalez-Garcia, Maltoni, Penã-Garay, 2 2 ␪ ϭ р⌬␹ (C.L. 3 degrees of freedom) where, for instance, and Valle, 2001). In the upper-left panel tan 13 0, 2 ⌬␹ (C.L. 3 degrees of freedom)ϭ6.25, 7.83, and 11.36 which corresponds to pure ␯␮→␯␶ oscillations. Note the

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 387

tions clearly favors the ␯␮→␯␶ oscillation hypothesis. As a matter of fact, the best fit corresponds to a very small ␪ ϭ value, 13 6°, but it is statistically indistinguishable ␪ ϭ from the decoupled scenario, 13 0°. No region of parameter space is allowed (at 99% C.L. for 3 degrees of 2 ␪ ϭ͉ ͉2Ͼ freedom) for sin 13 Ue3 0.40. The physics reason ␯ →␯ for this limit is clear from the discussion of the ␮ e ␪ oscillation channel in Sec. V.C: large values of 13 imply ␯ →␯ too large a contribution of the ␮ e channel and would spoil the otherwise successful description of the angular distribution of contained events. This situation is illustrated also in Fig. 35 where we plot the shift of ␹2 for the analysis of atmospheric data in the framework of oscillations between three neutrinos as a function of sin2 ␪ when the mass and mixing parameters ⌬m2 and ⌬␹2 2 ␪ 13 32 FIG. 35. Dependence of on sin 13 in the analysis of the 2 ␪ ␹2 tan 23 are chosen to minimize . atmospheric, solar, and CHOOZ neutrino data. The dotted For any value of the mixing parameters, the mass- horizontal line corresponds to the 3␴ limit for a single param- squared difference relevant for the atmospheric analysis eter. is restricted to lie in the interval 1.25ϫ10Ϫ3 Ͻ⌬ 2 2Ͻ ϫ Ϫ3 m32/eV 8 10 at 99% C.L. Thus it is within the exact symmetry of the contour regions under the trans- range of sensitivity of the CHOOZ experiment. Indeed, ␪ →␲ Ϫ␪ 2 ␪ formation 23 /4 23 . This symmetry follows from as illustrated in Fig. 35, the limit on sin 13 becomes the fact that in the pure ␯␮→␯␶ channel, matter effects stronger when the CHOOZ data are combined with the ␪ cancel out and the oscillation probability depends on 23 atmospheric and solar neutrino results. 2 ␪ only through the double-valued function sin 2 23 (see One can finally perform a global analysis in the five- ␪ Þ Sec. V.C). For 13 0 this symmetry breaks due to the dimensional parameter space combining the full set of three-neutrino mixing structure even if matter effects solar, atmospheric, and reactor data. Such analysis leads are neglected. The analysis of the full atmospheric neu- to the following allowed 3␴ ranges for individual param- trino data in the framework of three-neutrino oscilla- eters (that is, when the other four parameters have been chosen to minimize the global ␹2): ϫ Ϫ5Ͻ⌬ 2 2Ͻ ϫ Ϫ4 2.4 10 m21/eV 2.4 10 large mixing angle, Ͻ 2 ␪ Ͻ 0.27 tan 12 0.77 large mixing angle, ϫ Ϫ3Ͻ⌬ 2 2Ͻ ϫ Ϫ3 1.4 10 m32/eV 6.0 10 , Ͻ 2 ␪ Ͻ 0.4 tan 23 3.0, 2 ␪ Ͻ sin 13 0.06. (147) These results can be translated into our present knowledge of the moduli of the mixing matrix U: 0.73– 0.89 0.45– 0.66 Ͻ0.24 ͉U͉ϭͩ 0.23– 0.66 0.24– 0.75 0.52– 0.87ͪ . (148) 0.06– 0.57 0.40– 0.82 0.48– 0.85 In conclusion, we learn that at present the large mixing- ͉ ͉ type solutions provide the best fit. As concerns Ue3 , both solar and atmospheric data favor small values, and this trend is strengthened by the reactor data.

VIII. IMPLICATIONS OF THE NEUTRINO MASS SCALE AND FLAVOR STRUCTURE

A. New physics

FIG. 36. Allowed regions (at 90%, 95%, 99%, and 99.7% The simplest and most straightforward lesson of the ⌬ 2 2 ␪ evidence for neutrino masses is also the most striking C.L.) in the ( m32 ,tan 23) plane from a global analysis of the atmospheric neutrino data in the framework of three-neutrino one: there is new physics beyond the standard model. 2 ␪ oscillations for various values of tan 13 . The global best-ﬁt This is the ﬁrst experimental result that is inconsistent point is marked by the star. with the standard model.

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 388 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

Most likely, the new physics is related to the existence models that try to incorporate the LSND data by adding of singlet fermions at some high energy scale that in- light sterile neutrinos and only comment on this possi- duce, at low energies, the effective terms Eq. (13) bility in the context of theories with new extra dimen- through the seesaw mechanism. The existence of heavy sions. singlet fermions is predicted by many extensions of the standard model, especially by GUT’s [beyond SU(5)] B. The scale of new physics and left-right symmetric theories. ϳ 2 ⌳ Given Eq. (14), m␯ v / NP , it is straightforward to There are of course other possibilities. One could in- use measured neutrino masses to estimate the scale of duce neutrino masses without introducing any new fer- new physics that is relevant to their generation. In par- mions beyond those of the standard model. This would ticular, if there is no quasidegeneracy in the neutrino ⌬ require the existence of a scalar L(1,3)ϩ1 , that is, a masses, the heaviest of the active neutrino masses can be triplet of SU(2)L . The smallness of neutrino masses estimated, would then be related to the smallness of the vacuum ͗⌬0 ͘ m ϭm ϳͱ⌬m2 Ϸ0.05 eV. (149) expectation value L (required also by the success of h 3 atm ␳ϭ the 1 relation) and would not have a generic natural (In the case of an inverted hierarchy the implied scale is explanation. ϭ ϳͱ͉⌬ 2 ͉Ϸ mh m2 matm 0.05 eV.) It follows that the scale In left-right symmetric models, however, where the in the nonrenormalizable term (13) is given by ϫ → breaking of SU(2)R U(1)BϪL U(1)Y is induced by ⌬ ⌳ ϳv2/m Ϸ1015 GeV. (150) the vacuum expectation value of an SU(2)R triplet, R , NP h there must exist also an SU(2)L triplet scalar. Further- We should clarify two points regarding Eq. (150): more, the Higgs potential leads to an order-of- (1) There could be some level of degeneracy between magnitude relation between the various vacuum expec- the neutrino masses that is relevant to atmospheric neu- ͗⌬0 ͗͘⌬0 ͘ϳ 2 tation values L R v (where v is the electroweak trino oscillations. In such a case (see Sec. VI.E), Eq. ͗⌬0 ͘ breaking scale), and the smallness of L is related to (149) is modified into a lower bound and, consequently, the high scale of SU(2)R breaking. This situation can be Eq. (150) becomes an upper bound on the scale of the thought of as a seesaw of vacuum expectation values. In new physics. this model there are, however, also singlet fermions. The (2) It could be that the Zij couplings of Eq. (13) are light neutrino masses arise from both the seesaw mecha- much smaller than 1. In such a case, again, Eq. (150) nism and the triplet vacuum expectation value. becomes an upper bound on the scale of the new phys- Neutrino masses could also be of the Dirac type. ics. On the other hand, in models of approximate flavor Here, again, singlet fermions have to be introduced, but symmetries, there are relations between the structures lepton number conservation needs to be imposed by of the charged lepton and neutrino mass matrices that տ 2 2ϳ Ϫ4 hand. This possibility is disfavored by theorists since it is give quite generically Z33 m␶ /v 10 . We conclude ⌳ likely that global symmetries are violated by gravita- that the likely range of NP that is implied by the atmo- tional effects. Furthermore, the lightness of neutrinos spheric neutrino results is given by (compared to charged fermions) is again unexplained. 11Շ⌳ Շ 15 Another possibility is that neutrino masses are gener- 10 NP 10 GeV. (151) ated by mixing with singlet fermions but the mass scale The estimates (150) and (151) are very exciting. First, of these fermions is not high. Here again the lightness of the upper bound on the scale of the new physics is well neutrino masses remains a puzzle. The best known ex- below the Planck scale. This means that there is a new ample of such a scenario is the framework of supersym- scale in Nature which is intermediate between the two ϳ 19 metry without R parity. known scales, the Planck scale, mPl 10 GeV, and the Let us emphasize that the seesaw mechanism or, more electroweak breaking scale, vϳ102 GeV. generally, the extension of the standard model with non- It is amusing to note in this regard that the solar neu- renormalizable terms, is the simplest explanation of neutrino problem does not necessarily imply such a new trino masses. Models in which neutrino masses are pro- scale. If its solution is related to vacuum oscillations with ⌬ 2 ϳ Ϫ10 2 ⌳ ϳ duced by new physics at low energy imply a much more m21 10 eV , it can be explained by NP mPl . dramatic modification of the standard model. Further- However (see Sec. IV.D.4), the favored explanation for more, the existence of seesaw masses is an unavoidable the solar neutrino deficit is the large-mixing-angle solu- prediction of various extensions of the standard model. tion, which again points towards a new physics scale in In contrast, many (but not all) of the low-energy mecha- the range of Eq. (151). ⌳ ϳ 15 nisms are introduced for the specific purpose of gener- Second, the scale NP 10 GeV is intriguingly close ating neutrino masses. to the scale of gauge coupling unification. We shall say In this and in the next section, where we discuss the more about this fact when we discuss GUT’s. implications of the experimental data for theories beyond the standard model, we choose to focus on models C. Implications for flavor physics that explain the atmospheric and solar neutrino data 1. The flavor parameters through mixing among three active neutrinos. In other words, we assume three-neutrino mixing with the oscil- Flavor physics refers to interactions that are not uni- lation parameters derived in Sec. VII. We do not review versal in generation space. In the standard-model inter-

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 389 action basis, this is the physics of the Yukawa couplings. can be read from the results of the global analysis in Eq. In the mass basis, this term refers to fermion masses and (147): First, the mixing angles that are relevant to atmo- mixing parameters. There are two main reasons for the spheric neutrinos, interest in flavor physics: ͉U␮ U* ͉ϳ0.4– 0.5; (154) (1) The standard model has 13 flavor parameters. 3 ␶3 These are the nine charged-fermion masses or, equiva- second, the mixing angles that are relevant to solar neu- ϭ & lently, Yukawa couplings, Yi mi /v/ , trinos, ϳ ϳ Ϫ2 ϳ Ϫ5 ͉ ͉ϳ Yt 1, Yc 10 , Yu 10 , Ue1Ue*2 0.35– 0.5; (155) ϳ Ϫ2 ϳ Ϫ3 ϳ Ϫ4 Yb 10 , Ys 10 , Yd 10 , third, the ratio between the respective mass-squared differences, ϳ Ϫ2 ϳ Ϫ3 ϳ Ϫ6 Y␶ 10 , Y␮ 10 , Ye 10 , (152) ⌬ 2 Ϫ1 Ϫ3 m21 10 – 10 large mixing angle and the four CKM parameters, ϳͭ (156) ͉⌬m2 ͉ 10Ϫ4 – 10Ϫ5 low mass. ͉ ͉ϳ ͉ ͉ϳ ͉ ͉ϳ ␦ ϳ 32 Vus 0.2, Vcb 0.04, Vub 0.004, sin KM 1. (153) In addition, the upper bound on the third mixing angle from the CHOOZ experiments often plays a significant One can easily see that the flavor parameters are hierar- role in flavor model building: chical (that is, they have very different magnitudes from ͉ ͉Շ each other), and all but two (the top Yukawa and the Ue3 0.24. (157) CP-violating phase) are small. The unexplained smallness and hierarchy pose the flavor puzzle of the standard 2. Special features of the neutrino flavor parameters model. Its solution may direct us to physics beyond the There are several features in the numerical values of standard model. these parameters that have drawn much attention and (2) The smallness of flavor-changing neutral-current have driven numerous investigations: ⌬ ␮→ ␥ (FCNC) processes, such as mK and e , is ex- (i) Large mixing and strong hierarchy: The mixing plained within the standard model by the Glashow- angle that is relevant to the atmospheric neutrino prob- Iliopoulos-Maiani (GIM) mechanism. In many exten- lem, U␮3 , is large, of order one. On the other hand, the sions of the standard model there is, generically, no such ⌬ 2 ͉⌬ 2 ͉ ratio of mass-squared differences m21/ m32 is small. mechanism. In some cases, experimental bounds on If there is no degeneracy in the neutrino sector then the flavor-changing processes are violated. The solution of small ratio of mass-squared differences implies a small Ӷ such flavor problems leads to refinement of the models. ratio between the masses themselves, m2 /m3 1. (In the In other cases, the model predictions are within present case of inverted hierarchy, the implied hierarchy is Ӷ bounds but well above the standard-model range. We m3 /m2 1.) It is difficult to explain in a natural way a can hope, then, to probe this new physics through future situation where, in the 2–3-generation sector, there is measurements of flavor-changing processes. large mixing but the corresponding masses are hierarchi- Many mechanisms have been proposed in response to cal. Below we discuss in detail the difficulties and vari- either or both of these flavor aspects. For example, ap- ous possible solutions. proximate horizontal symmetries, broken by a small pa- (ii) If the large-mixing-angle solution to the solar neu- rameter, can lead to selection rules that both explain the trino problem holds, then the data can be interpreted in hierarchy of the Yukawa couplings and suppress flavor- a very different way. In this case, the two measured mix- ⌬ 2 ͉⌬ 2 ͉ changing couplings in sectors of new physics. ing angles are of order one. Moreover, m21/ m32 Ϫ In the extension of the standard model with three ac- ϳ10 2 means that the ratio between the masses them- tive neutrinos that have Majorana-type masses, there selves (which, for fermions, are after all the fundamental ϳ Ϫ1 are nine new flavor parameters in addition to those of parameters) is not very small, m2 /m3 10 . Such a the standard model with massless neutrinos. These are value could easily be an accidentally small number, with- three neutrino masses, three lepton mixing angles, and out any parametric suppression. If this is the correct way three phases in the mixing matrix. The counting of pa- of reading the data, the measured neutrino parameters rameters is explained in Sec. II.D. Of these, four are may actually reflect the absence of any hierarchical determined from existing measurements of solar and at- structure in the neutrino mass matrices [Hall, Mu- mospheric neutrino fluxes: two mass-squared differences rayama, and Weiner (2000); Berger and Siyeon (2001); and two mixing angles. This adds significantly to the in- Haba and Murayama (2001); Hirsch and King (2001)]. put data on flavor physics and provides an opportunity Obviously, this interpretation is plausible only if the to test and refine flavor models. large-mixing-angle solution to the solar neutrino prob- If neutrino masses arise from effective terms of the lem holds, and will be excluded if the small-mixing-angle form of Eq. (13), then the overall scale of neutrino or low-mass-ratio solutions hold. Another important test ⌳ ͉ ͉ masses is related to the scale NP and, in most cases, of this idea will be provided by a measurement of Ue3 . does not tell us anything about flavor physics. More sig- If indeed the entries in M␯ have random values of the nificant information for flavor models can be written in same order, all three mixing angles are expected to be of ͉ ͉ϳ Ϫ1 terms of three dimensionless parameters whose values order one. If experiments measure Ue3 10 , that is,

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 390 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing close to the present CHOOZ bound, it can be again bimaximal mixing is sometimes used to denote the argued that its smallness is accidental. The stronger the case of two large, rather than maximal, mixing upper bound on this angle becomes, the more difficult it angles.) will be to maintain this view. (iii) A special case of large mixing is that of maximal 3. Large mixing and strong hierarchy mixing. In a two-generation case, with a single mixing In this subsection, we focus on the large-mixing– 2 ␪ϭ angle, maximal mixing is defined as sin 2 1. In the strong-hierarchy problem. We explain the challenge to three-generation case, what we mean by maximal mixing theory from this feature and present various solutions is that the disappearance probability is equivalent to that have been proposed to respond to this challenge. that for maximal two-neutrino mixing at the relevant A large mixing angle by itself should not be a surprise. mass scale (Barger et al., 1998b). Maximal atmospheric After all, the naive guess would be that all dimensionless ͉ ͉2 Ϫ͉ ͉2 ϭ neutrino mixing corresponds to 4 U␮3 (1 U␮3 ) 1, couplings [such as the Zij couplings of Eq. (13)] are which leads to naturally of order one and consequently all mixing 2 angles are of order one. However, the quark mixing ͉U␮ ͉ ϭ1/2. (158) 3 angles are small, and this situation has led to a prejudice Maximal solar neutrino mixing corresponds to that the lepton mixing angles would also be so. Second, ͉ ͉2ϭ 4 Ue1Ue2 1, which leads to and more importantly, flavor models have a built-in mechanism for naturally inducing smallness and hierar- ͉U ͉2ϭ͉U ͉2ϭ1/2, ͉U ͉2ϭ0. (159) e1 e2 e3 chy in the quark parameters (and perhaps also in As can be seen in Sec. VII, the present data are consis- charged-lepton masses). For example, a mechanism that tent with near-maximal atmospheric neutrino mixing has been intensively studied in the literature is that of and large but not maximal solar neutrino mixing. The selection rules due to an approximate symmetry. Within possibility that both mixings are near maximal is, how- such frameworks, numbers of order one are as difficult ever, not entirely excluded. This scenario, in which both (or as easy) to account for as small numbers: one can atmospheric and solar neutrino mixing are near maxi- assign charges in such a way that small flavor parameters mal, means that the structure of the leptonic mixing ma- correspond to terms in the Lagrangian that carry trix is given to a good approximation by charges different from zero, while the order-one parameters correspond to terms that carry no charge. 1 1 The combination of a large mixing angle and strong Ϫ 0 & & hierarchy is, however, a true puzzle. To understand the difficulty in this combination, let us assume that both the 1 1 1 mixing and the hierarchy can be understood from the Uϭ Ϫ . (160) ϫ 2 2 & relevant 2 2 block in the mass matrix for light (Majo- ͩ ͪ rana) neutrinos. The generic form of this block is 1 1 1 v2 AB ϭ ͩ ͪ 2 2 & M␯ ⌳ . (161) NP BC The case of Eq. (160) is commonly called bimaximal mixing. We should like to make the following comments, This matrix would lead to a large mixing angle if regarding maximal or bimaximal mixing: ͉B͉տ͉CϪA͉, (162) (a) Theoretically, it is not difficult to construct models and to a strong mass hierarchy if (for simplicity we as- that explain near-maximal solar neutrino mixing in sume real entries) a natural way. We shall encounter some examples ͉ACϪB2͉Ӷmax͑A2,B2,C2͒. (163) below. Experimentally, it may be difficult to make a convincing case for near-maximal (rather than just If we examine the two-neutrino sector alone, these con- order-one) solar neutrino mixing (Gonzalez- ditions mean fine-tuning. Hence the challenge is to find Garcia, Penã-Garay, Nir, and Smirnov, 2001). models in which the hierarchy is naturally induced. (b) It is highly nontrivial to construct models that ex- To make the problem even sharper, let us explicitly plain near-maximal atmospheric neutrino mixing in consider models with a horizontal U(1)H symmetry a natural way. The reason is that near-maximal (Froggatt and Nielsen, 1979). A horizontal symmetry is mixing is often related to quasidegeneracy. So one in which different generations may carry different when solar neutrino data are also taken into ac- charges, in contrast to the standard-model gauge group. count, approximate degeneracy among all three We assume that the symmetry is broken by a small pa- neutrinos is required, and models of three quaside- rameter ␭. To be concrete, we take ␭ϭ0.2, close to the generate neutrinos are, in general, not easy to con- value of the Cabibbo angle. One can derive selection struct in a natural way. rules by attributing charge to the breaking parameter. (c) The case of bimaximal mixing is also very challeng- Our normalization is such that this charge is H(␭) ing for theory. Many of the attempts in the litera- ϭϪ1. While coefficients of order one are not deter- ture involve fine-tuning. (Alternatively, the term mined in this framework, one can derive the parametric

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 391 suppression of all couplings and consequently have an metrically suppressed (Binetruy et al., 1996; Elwood order-of-magnitude estimate of the physical parameters. et al., 1998; Irges et al., 1998; Vissani, 1998, 2001b; Ellis Take, for example, the case in which the H charges of et al., 1999; Sato and Yanagida, 2000). We should like to the left-handed lepton doublets Li are positive. The emphasize that, if the large-mixing-angle solution of the parametric suppression of an entry in the neutrino mass solar neutrino problem is correct, this is not an unlikely matrix is determined by the charge that it carries, solution. While oscillation experiments give the ratio of ϩ ϩ ␾ ϳ␭H(Li) H(Lj) 2H( ) (M␯)ij . The parametric suppres- masses-squared, the relevant quantity for theories is the sion of the mixing angles and mass ratios can then be ratio of masses. For the large-mixing-angle solution, ϳ estimated (see, for example, Leurer et al., 1993), m2 /m3 0.1, which is not a particularly small number Ϫ and could easily arise from a combination of order-one ͉U ͉ϳ␭H(Li) H(Lj), ij terms. Ϫ ͑␯ ͒ ͑␯ ͒ϳ␭2[H(Li) H(Lj)] ͑ р ͒ m i /m j i j . (164) (ii) Several sources for neutrino masses: the leading contribution to the neutrino mass matrix, which is re- If the various generations of left-handed fields (quarks sponsible for the order-one mixing, is rank one. The or leptons) carry different H charges, then, in general, lighter generation masses come from a different, sub- the (quark or lepton) mixing angles are suppressed. For ␪ ϳ␭ dominant source. This is, for example, the generic situ- example, sin C is naturally induced if the charges of the first two quark-doublet generations are chosen ap- ation in supersymmetric models without R parity (see, Ϫ ϭ for example, Banks et al., 1995; Borzumati et al., 1996; propriately: H(Q1) H(Q2) 1. A mixing angle of order one can be naturally induced if the charges of the Davidson and Losada, 2000). There, tree-level mixing corresponding lepton-doublet fields are equal to each with neutralinos can lead to large mixing, but gives a other. Equation (164) shows, however, that in this class mass to only one neutrino generation, while the lighter of models, independent of the charge assignments (as masses arise at the loop level. Another realization of this long as they are all positive), we have (Grossman and principle would be the scenario of a seesaw mechanism Nir, 1995) with a single right-handed neutrino (Davidson and King, 1998; King, 1998, 1999, 2000; de Gouvea and Valle, m͑␯ ͒/m͑␯ ͒ϳ͉U ͉2. (165) i j ij 2001). For another example, within supersymmetric Hence, for order-one mixing, there is no parametric sup- theories, see Borzumati and Nomura (2001). pression of the corresponding neutrino mass ratio and (iii) Large mixing from the charged-lepton sector: it is no hierarchy induced. possible that the neutrino mass matrix is hierarchical There is another possibility for inducing large lepton and nearly diagonal in the 2–3 sector, and the large mix- mixing which is unique to the case of Majorana neutri- ing is coming from the diagonalization of the charged- nos. Here one assigns, for example, opposite charges, lepton mass matrix. A variety of models have been con- ϭϪ H(Li) H(Lj), to the relevant lepton-doublet fields. structed that give this structure of mass matrices, based The selection rules lead to the following structure of the on discrete horizontal symmetries (Grossman et al., mass matrix: 1998; Tanimoto, 1999b), holomorphic zeros (Grossman ͉ ͉ v2 A␭2 H(Li) B et al., 1998) and U(1)H with two breaking parameters ϳ ͩ ͪ ϭO͑ ͒ (Nir and Shadmi, 1999). For example, Grossman et al. M␯ ͉ ͉ , A,B,C 1 . ⌳ ␭2 H(Li) NP BC (1998) work in the supersymmetric framework with a (166) horizontal U(1)ϫU(1) symmetry. By an appropriate [In a supersymmetric framework, the combination of su- choice of horizontal charges, they obtain the following persymmetry and the horizontal symmetry (Leurer structure of lepton mass matrices (arbitrary coefficients et al., 1993, 1994) leads to a situation in which either A of order one are omitted in the various entries): or C vanish.] This mass matrix has a pseudo-Dirac struc- ␭2 ␭␭ ture and it leads to near-maximal mixing and near- ͗␾ ͘2 u ␭ degenerate masses [see, for example, Joshipura and Rin- M␯ϳ ͩ 00ͪ , ⌳ NP dani (2000b); Nir (2000)]: ␭ 01 ⌬ 2 ͉ ͉ m ͉ ͉ ␭8 ␭6 ␭4 2 ␪ϭ ϪO͑␭2 H(Li) ͒ ϭO͑␭2 H(Li) ͒ sin 1/2 , 2 . m ϳ ␾ ␭7 ␭5 ␭3 MᐉϮ ͗ dͩ͘ ͪ , (168) (167) ␭7 ␭5 ␭3 Of course, a pseudo-Dirac structure is inconsistent with mass hierarchy. where the zero entries are a consequence of holomor- Here are a few mechanisms that have been suggested phy. These matrices lead to in the literature to induce strong hierarchy simulta- 2 2 3 ⌬m /͉⌬m ͉ϳ␭ , ͉U␮ ͉ϳ1, (169) neously with large mixing angle: 21 32 3 (i) Accidental hierarchy: the mass matrix (161) has where the dominant contribution to the mixing angle A,B,CϭO(1), and Eq. (163) is accidentally fulfilled. In comes from diagonalization of MᐉϮ. The solar neutrino the context of Abelian horizontal symmetries, this parameters of this model correspond to the large- means that the mass ratio is numerically but not para- mixing-angle solution.

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 392 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

(iv) Large mixing from the seesaw mechanism: it is ⑀Ϫ 11 ͗␾ ͘2 possible that sterile neutrinos play a significant role in u ⑀ ⑀ M␯ϳ ͩ 1 ϩ ϩ ͪ , the flavor parameters of light, dominantly active neutri- ⌳ NP nos. In other words, an effective mass matrix of the form 1 ⑀ϩ ⑀ϩ (161) with the condition (163) could be a result of selec- ␭ ␭ ⑀ ␭ ⑀ tion rules applied in the full high-energy theory to a e ␮ Ϫ ␶ Ϫ ϳ ␾ ␭ ⑀ ␭ ␭ mass matrix that included both active and sterile neutri- Mᐉ ͗ dͩ͘ e ϩ ␮ ␶ ͪ , (173) nos and the heavy, dominantly sterile ones integrated ␭ ⑀ϩ ␭␮ ␭␶ out (Altarelli and Feruglio, 1998; Barbieri et al., 1998; e ␭ Eyal and Nir, 1999). It is interesting to note that in a where the i’s allow for generic approximate symmetry large class of such models, the induced hierarchy would that acts on the SU(2)-singlet charged leptons. The re- be too strong for the large-mixing-angle and small- sulting neutrino masses are as follows: mixing-angle solutions, unless at least three sterile neu- m ϭm͕1ϮO͓max͑⑀ϩ ,⑀Ϫ͔͖͒, m ϭmO͑⑀ϩ͒. trinos played a role in determining the low-energy pa- 1,2 3 (174) rameters (Nir and Shadmi, 1999). An interesting example of a model of this type is presented by Altarelli Note that the quasidegenerate pair of mass eigenstates and Feruglio (1998). They consider a horizontal U(1) involved in the solar neutrino anomaly is heavier than symmetry broken by two parameters of opposite the third, separated state. For the mixing angles, one charges. By an appropriate choice of horizontal charges finds for the lepton fields, they obtain the following neutrino ͉ ͉ϭO͑ ͒ ϭO͓ ͑⑀ ⑀ ͔͒ U␮3 1 , Ue3 max ϩ , Ϫ , Dirac mass matrix MD and Majorana mass matrix for ͉ ͉2ϭ ϪO͓ ͑⑀ ⑀ ͔͒ the sterile neutrinos MN : Ue2 1/2 max ϩ , Ϫ . (175) ␭3 ␭␭2 The overall picture is that, somewhat surprisingly, the ϳ ␾ ␭␭Ј lepton parameters (154)–(156) are not easy to account MD ͗ uͩ͘ 1 ͪ , for with Abelian flavor symmetries. The simplest and ␭␭Ј 1 most predictive models have difficulties in accommodat- ing the large 2Ϫ3 mixing together with the strong 2Ϫ3 ␭2 1 ␭ hierarchy. One can find more complicated models that ϳ⌳ ␭Ј2 ␭Ј MN NPͩ 1 ͪ . (170) naturally induce these parameters, but often at the cost ␭␭Ј 1 of losing predictive power. In particular, it may be the case that, specifically for neutrinos, one cannot ignore Integrating out of the three heavy neutrinos, the follow- the existence of heavy degrees of freedom (sterile neu- ing Majorana mass matrix for the light neutrinos is ob- trinos), well beyond the reach of direct experimental tained: production, that affect the flavor parameters of the low- ␭4 ␭2 ␭2 energy effective theory. If true, this situation would ͗␾ ͘2 u ␭2 mean that measuring the low-energy neutrino param- M␯ϳ ͩ ABͪ , A,B,CϭO͑1͒, (171) ⌳ eters cannot by itself make a convincing case for the NP ␭2 BC idea of Abelian flavor symmetries. (An exception are Ϫ Ϫ ͉ Ϫ 2͉ϭO͑␭␭Ј͒ models of approximate Le L␮ L␶ symmetry.) AC B . (172) Similar difficulties are encountered in the framework If one were to apply the selection rules directly to M␯ , of non-Abelian symmetries. Again, the simplest models the generic structure of Eq. (171) would be reproduced, do not work and have to be extended in rather compli- but not the relation between the order-one coefficients cated ways (Barbieri, Hall, Kane, and Ross, 1999; Bar- in Eq. (172). bieri, Hall, and Strumia, 1999). In some cases, the non- (v) A three-generation mechanism: approximate Le Abelian symmetries can give testable (almost) exact ϪL␮ϪL␶ . One of the more interesting frameworks that relations between masses and mixing angles. For ex- produces all the observed features of neutrino flavor pa- ample, the model of Barbieri, Creminelli, and Ro- ␪ ϭͱ ␪ rameters is intrinsically a three-generation framework. manino (1999) predicts sin 12 me /m␮ cos 23 and Ϫ Ϫ ␪ ϭͱ ␪ One applies an approximate Le L␮ L␶ symmetry to sin 13 me /m␮ sin 23 . If it turns out that all three the mass matrices of light, active neutrinos and of light neutrinos are quasidegenerate, non-Abelian sym- charged leptons.3 For the most general case, the symme- metries will become an unavoidable ingredient in flavor try is broken by small parameters, ⑀ϩ and ⑀Ϫ , of charges model building, but the task of constructing realistic ϩ2 and Ϫ2, respectively. The lepton mass matrices have models will be very challenging.4 Radiative corrections the following form:

4See, for example, Fritzsch and Xing, 1996, 2000; Carone and 3Examples of this approach include Barbieri et al., 1998, 1999; Sher, 1998; Fukugita, Tanimoto, and Yanagida, 1998, 1999; Frampton and Glashow, 1999; Nir and Shadmi, 1999; Cheung Barbieri, Ross, and Strumia, 1999; Tanimoto, 1999a, 2000; Tan- and Kong, 2000; Joshipura and Rindani, 2000a; Mohapatra imoto, Watari, and Yanagida, 1999; Wetterich, 1999; Wu, et al., 2000a; Nir, 2000; Shaﬁ and Tavartkiladze, 2000d. 1999a, 1999b, 1999c; Perez, 2000; Ma and Rajasekaran, 2001.

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 393 are an important issue when examining the naturalness ϳ 2 ⌳ ϳ Ϫ3 m3 mt / SO(10) 10 eV. (177) of various models that account for quasidegeneracy among neutrinos (see, for example, Casas et al., 1999, It requires only that the lightest of the three singlet fer- 2000). mion masses be a factor of 50 below the unification scale to induce neutrino masses at the scale appropriate for the atmospheric neutrino anomaly. In other words, the ⌳ ⌳ IX. IMPLICATIONS FOR MODELS OF NEW PHYSICS intriguing proximity of NP of Eq. (150) to GU of Eq. (176) finds a natural explanation in the framework of In this section we review the implications of neutrino SO(10). In this sense the seesaw mechanism of Gell- physics for various extensions of the standard model. We Mann, Ramond, and Slansky (1979) and of Yanagida do not attempt to describe all relevant models of new (1979) predicted neutrino masses at a scale that is rel- physics, but take two representative frameworks. We evant to atmospheric neutrinos. choose to focus on well-motivated models that were not ͉ ͉ϭO The third triumph, U␮3 (1), is more subtle but constructed for the special purpose of explaining the still quite impressive (Harvey, Reiss, and Ramond, 1982; neutrino parameters. Thus the neutrino parameters ei- Babu and Barr, 1996; Sato and Yanagida, 1998). Unlike ther test these frameworks or provide further guidance the previous points, which were related to SO(10) in distinguishing among different options for model GUT’s, here the consistency relates to SU(5) GUT’s. building within each framework. The two frameworks SU(5) theories relate the charged-lepton mass matrix are the following: Mᐉ to the down-quark mass matrix Md . Specifically, the ϭ T (i) GUT’s: here the overall scale of the neutrino naive SU(5) relation reads Mᐉ Md . It follows that masses and some features of the mixing provide ͉ ͉ϳ U␮3Vcb ms /mb . (178) interesting tests. ͉ ͉ϳ (ii) Large extra dimensions: in the absence of a high Given the experimental values Vcb 0.04 and ms /mb ϳ energy scale, the seesaw mechanism is not opera- 0.03 we conclude that the naive SU(5) relations pre- ͉ ͉ϳ ͉ ͉ tive in this framework. The lightness of the neu- dict U␮3 1. Of course, U␮3 is also affected by the trinos is a challenge. diagonalization of the light-neutrino mass matrix, but there is no reason to assume that this contribution can- A. Grand unified theories cels out the charged-lepton sector in such a way that ͉ ͉Ӷ U␮3 1. There are two significant facts about the gauge sym- Many of the other naive SO(10) relations fail, as do metries of the standard model and the structure of its many of the naive SU(5) relations. Specifically, SO(10) fermionic representations that motivate the idea of su- predicts vanishing CKM mixing and mass ratios such as ϭ ϭ ϭ persymmetric grand unification. First, GUT’s provide a mc /mt ms /mb . SU(5) predicts ms m␮ and md me . unification of the standard-model multiplets (for a re- It is possible, however, that these bad predictions are cent review, see Wilczek, 2001). Second, in the frame- corrected by subleading effects, while all the successful ϭ ϳ work of the supersymmetric standard model, the three predictions (particularly mb m␶ , m␮ /m␶ ms /mb, and ͉ ͉ϳ gauge couplings unify (Dimopoulos, Raby, and Wilczek, U␮3Vcb ms /mb) are retained since they depend on 1981; Iban˜ez and Ross, 1981). The unification scale is the leading contributions. This is demonstrated in a given by (see, for example, Langacker and Polonsky, number of specific GUT models.5 1995) The flavor structure of the first two neutrino genera- ⌳ ϳ3ϫ1016 GeV. (176) tions depends on both the Majorana mass matrix for the GU singlet fermions and the subdominant effects that cor- Further support for grand unification comes from the rect the flavor parameters of the first two generations of flavor sector: the masses of the bottom quark and the quarks and charged leptons. This part of GUT’s is there- ⌳ fore much more model dependent. Explicit GUT mod- tau lepton are consistent with equal masses at GU ,as predicted by SU(5) and its extensions. els have been constructed that accommodate the various The evidence for neutrino masses from atmospheric solutions of the solar neutrino problem. We shall not ⌬ 2 describe them in any detail here. neutrinos, the implied scale of matm, and the required mixing of order one should be considered as three fur- B. Extra dimensions ther triumphs of the grand unification idea. First, as mentioned above, SO(10) theories and their New ideas concerning the possibility of large addi- extensions require that there exist singlet fermions. Neu- tional dimensions and the world on a brane can lead to trino masses are then a prediction of these theories. Second, SO(10) theories naturally give the singlet fermions heavy masses at the SO(10) breaking scale and, 5 furthermore, relate the neutrino Dirac mass matrix MD See, for example, Albright, Babu, and Barr, 1998; Albright ␯ and Barr, 1998, 1999a, 1999b, 2000a, 2000b, 2001; Altarelli and to the up-quark mass matrix Mu . Specifically, the naive Feruglio, 1998, 1999a, 1999b; Berezhiani and Rossi, 1999, 2001; Dϭ SO(10) relation reads M␯ Mu . It follows that the mass Hagiwara and Okamura, 1999; Shafi and Tavartkiladze, 1999, of the heaviest among the three light neutrinos can be 2000a, 2000b, 2000c; Altarelli, Feruglio, and Masina, 2000a, estimated: 2000b; Babu, Pati, and, Wilczek, 2000; Maekawa, 2001.

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 394 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing sources of neutrino masses that are very different from case, for example, with modulinos, the Fermionic part- those we have discussed so far. For example, the small ners of the moduli fields that are generic to string theo- mass could be related to the large volume factor of the ries. The crucial point is that the Yukawa interaction extra dimensions which suppresses the coupling to bulk between bulk singlet fermions and brane active neutri- fermions (Dienes et al., 1999; Mohapatra et al., 1999, nos is highly suppressed by a large volume factor of the 2000b; Lukas et al., 2000, 2001; Mohapatra and Perez- ͱ n extra dimensions, 1/ Vn. This suppression factor re- Lorenzana, 2000; Dienes and Sarcevic, 2001; Arkani- flects the small overlap between the wave functions of Hamed et al., 2002), to the breaking of lepton number the sterile neutrino in the bulk and the active one on the on a distant brane (Arkani-Hamed et al., 2002), or to the brane. By construction, this factor provides a suppres- warp factor in the Randall-Sundrum framework (Gross- sion of the neutrino Dirac mass by the ratio v/M . Con- man and Neubert, 2000). In this section, we briefly de- Pl sequently, large extra dimensions provide a natural scribe these three mechanisms. source of very small Dirac mass terms for the active neu- The existence of large extra dimensions not only provides new ways of generating small neutrino masses but trinos. can also lead to interesting phenomenological conse- The consequences for neutrino masses and mixing de- quences. In particular, the phenomenology of matter os- pend on the details of the physics of the bulk neutrinos. cillations in the Sun can be affected (Dvali and Smirnov, The possible scenarios are clearly described by Lukas 1999; Barbieri et al., 2000; Lukas et al., 2000; Ma et al., et al. (2000). There an explicit derivation of the effective 2000; Caldwell et al., 2001a, 2001b). Other phenomeno- four-dimensional action from a five-dimensional one is logical implications can be used to constrain the param- given. The final result for the Dirac mass is as follows: eters of the extra dimensions (Faraggi and Pospelov, vYi M 1999; Das and Kong, 1999; Ioannisian and Pilaftsis, Dirϭ ϭ v * mi n Yi . (180) 2000). ͱV M MPl n * 1. Coupling to bulk fermions Here Yi is the Yukawa coupling between the lepton doublet L and a bulk singlet, M is the string scale, and It is possible that we live on a brane with three spatial i * Vn is the volume of the n-dimensional extra space. The dimensions that is embedded in a spacetime with n ad- usual Dirac mass is of order Yv, but in Eq. (180) we see ditional large spatial dimensions (Arkani-Hamed et al., explicitly the (M /M ) suppression factor in the effec- 1998). This idea has the potential of providing an under- * Pl tive four-dimensional Yukawa couplings, leading to standing of the hierarchy between the gravitational mass highly suppressed Dirac mass terms. scale MPl and the electroweak scale mZ . The hope is to solve the hierarchy problem by avoiding a fundamental Being a bulk state, the singlet fermion has a whole high energy (that is Planck or GUT) scale altogether. tower of Kaluza-Klein associated states which are all ϭ Ϫ1/2ϳ 19 coupled to the left-handed brane neutrinos. In the sim- The observed Planck scale, MPl (GN) 10 GeV, is related to the fundamental Planck scale (most likely plest scenario and for one extra dimension of radius R, the string scale) of the 4ϩn dimensional theory, M ,by the masses of all Kaluza-Klein states are determined by * the Gauss law, the scale 1/R: ϩ 2 M2 ϭMn 2V , (179) n Pl * n m2 ϭ . (181) n R2 where Vn is the volume of the n-dimensional extra space. This picture has dramatic phenomenological con- In more general scenarios there can be other bulk mass sequences for particle physics and cosmology. Such a terms and the masses Kaluza-Klein states receive addi- situation also poses an obvious problem for neutrino tional contributions. Then the lightest Kaluza-Klein physics. If there is no scale of physics as high as Eq. mass can be taken as an independent parameter, while (151), the seesaw mechanism for suppressing light neu- the mass splitting between the states is still determined trino masses cannot be implemented. Conversely, if by the scale 1/R. there are singlet neutrinos that are confined to the three- Let us denote by Mmin the lightest mass in the Kaluza- dimensional brane where the active neutrinos live, one Klein spectrum and by ⌬m the mass scale that is rel- expects their mass to be at the string scale, M , which in * evant to atmospheric or to solar neutrino oscillations. these models is much smaller than the four-dimensional Then one can distinguish three cases: MPl , perhaps as small as a few TeV, and the resulting ӷ⌬ ӷ⌬ light-neutrino masses are well above the scales of atmo- (a) 1/R m and Mmin m. The Kaluza-Klein states spheric or solar neutrinos. play no direct role in these oscillations. Their main Within this framework the first implication of the evi- effect is to give seesaw masses to the active brane dence for neutrino masses is that there had better be no neutrinos. ӷ⌬ Շ⌬ singlet fermions confined to the brane. Alternatively, the (b) 1/R m and Mmin m. The situation is equiva- model must include some special ingredients to avoid lent to conventional models containing a small ϭ ordinary seesaw masses. number of sterile states. In particular, for Mmin 0 On the other hand, it is typical in this framework that it is a possible framework for light Dirac neutrinos. there be singlet fermions in the bulk. This would be the For small but nonvanishing Mmin it may lead to the

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 395

four-neutrino mixing scenarios. However these sce- 3. The warp factor in the Randall-Sundrum scenario narios are not particularly favored. A different setup for the extra dimensions (leading to (c) 1/RՇ⌬m and M Շ⌬m. A large number of bulk min a different explanation of the v/M hierarchy) was pro- modes can take direct part in the oscillation phe- Pl posed by Randall and Sundrum (1999). They considered nomena. This situation modifies in a very interest- one extra dimension parametrized by a coordinate y ing way the solution to the solar neutrino problem ϭr ␾, where Ϫ␲р␾рϩ␲. r is the radius of the com- that involves MSW resonance conversion into a c c pact dimension, and the points (x,␾) and (x,Ϫ␾) are sterile neutrino. Now the ␯ can oscillate into a e identified. A visible brane is located at ␾ϭ␲ and a hid- whole set of Kaluza-Klein states. However, as dis- den brane at ␾ϭ0. This setup leads to the following non- cussed in Sec. IV, solar oscillations into sterile factorizable metric: states are now strongly disfavored by the compari- Ϫ ͉␾͉ ␮ ␯ 2ϭ 2krc ␩ Ϫ 2 ␾2 son of the charged-current and neutral-current ds e ␮␯dx dx rc d . (183) fluxes at SNO. The parameter k is the bulk curvature. All dimensionful parameters in the effective theory on the visible brane In summary, large extra dimensions with sterile bulk Ϫ ␲ ⑀ϵ krc fermions provide a natural explanation for light neutrino are suppressed by the warp factor, e . With krc Ϸ ⑀ϳ Ϫ16 masses. In the simplest realization, in which there are no 12, that is, 10 , this mechanism produces physi- bulk mass terms and lepton number conservation is im- cal masses of order v from fundamental masses of order posed in the bulk-brane couplings, light Dirac masses MPl . are generated. In the general case, the light neutrinos The two mechanisms described above to generate can be either Dirac or Majorana particles, depending on small neutrino masses do not work in this scenario be- the details of the physics of the bulk neutrinos. cause the extra dimensions are small and there is no volume suppression factor available. Grossman and Neubert (2000) proposed a different mechanism for generating small masses in the Randall-Sundrum frame- 2. Lepton number breaking on a distant brane work. (For a different mechanism, see Huber and Shafi, 2001.) With appropriate choice of orbifold boundary In its simplest realization, the mechanism described conditions, it is possible to locate the zero mode of a above provides a natural way of generating light Dirac right-handed bulk neutrino on the hidden brane. If the masses for neutrinos. In the framework of extra dimen- fundamental mass scale m of the bulk fermions is larger sions one could also generate small Majorana masses by than half the curvature k of the compact dimension, the an alternative mechanism in which lepton number is wave function of the right-handed zero mode on the vis- broken on a distant brane (Arkani-Hamed et al., 2002). ible brane is power-suppressed in the ratio v/M . Cou- Imagine that lepton number is spontaneously broken at Pl pling the bulk fermions to the Higgs and lepton doublet the scale M on a different brane located a distance r * fields yields from our brane. Further assume that the information v m/k Ϫ 1/2 about this breaking is communicated to our brane by a ϳ ͩ ͪ m␯ v . (184) bulk field with a mass m. For the sake of concreteness MPl we take the case that mrӷ1 and there are two extra dimensions. The resulting Majorana mass for the active Note that the relation between the neutrino mass and neutrinos can be estimated as follows: the weak scale is generically different from the seesaw mechanism (except for the special case m/kϭ3/2). v2 Ϫmr To summarize, in the presence of large extra dimen- m␯ϳ e . (182) M sions, neutrino masses could be suppressed by the large * volume factor if the left-handed neutrinos couple to the We learn that the naive seesaw mass, of order v2/M ,is * bulk fermions (being Dirac for massless bulk fermions), suppressed by a small exponential factor. The conse- or by the distance to a brane where lepton number is quences for the neutrino spectrum depend on various broken (which generate Majorana masses). In the model-dependent features: the number of large extra di- Randall-Sundrum framework, the suppression can be in- mensions, the string scale, the distance between our duced by a power of the warp factor. The detailed struc- brane and the brane where lepton number is broken, ture of a neutrino mass hierarchy and mixing is often and the mass of the mediating bulk field. related to the parameters that describe the extra dimen- A variant of the above two mechanisms can be imple- sions. mented if the standard-model fields are confined to a thick wall (Arkani-Hamed and Schmaltz, 2000). Dirac masses are suppressed if left-handed and right-handed X. CONCLUSIONS neutrinos are located at different points in the wall and consequently there is an exponentially small overlap of Strong evidence for neutrino masses and mixing has their wave functions. Majorana masses are suppressed if been coming from various neutrino oscillation experi- lepton number is spontaneously broken by a vacuum ments in recent years. First, atmospheric neutrinos show expectation value that is localized within the wall but at a deviation from the expected ratio between the ␯␮ and ␯ ␯ some distance from the lepton doublet fields. e fluxes. Furthermore, the ␮ flux has strong zenith-

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 396 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

␪ ␪ angle dependence. The simplest interpretation of these unique. In particular, the two mixing angles atm and ᭪ ␯␮Ϫ␯␶ ⌬ 2 ⌬ 2 results is that there are oscillations with the fol- are large. As concerns the mass hierarchy, m᭪/ matm , lowing mass and mixing parameters: the situation is still ambiguous. If there is strong hierar- Ϫ Ϫ chy, it is not easy to accommodate it together with large 1.4ϫ10 3Ͻ⌬m2 Ͻ6ϫ10 3 eV2, atm mixing angles. If the hierarchy is mild enough to be con- Ͻ 2 ␪ Ͻ 0.4 tan atm 3.0. (185) sistent with just accidental suppression, and if, in addition, the third mixing angle (bounded by reactor experi- The ranges quoted here correspond to the results of the ments) is not very small, it could well be that there is no global three-neutrino analysis presented in Sec. VII [Eq. hierarchy in the neutrino parameters at all. That would (147)]. Second, the total rates of solar neutrino fluxes call for flavor frameworks that gave a different structure are smaller than the theoretical expectations. The sup- for charged and neutral fermion parameters. Other pos- pression is different in the various experiments (which sibilities, such as quasidegeneracy in masses or bimaxi- are sensitive to different energy ranges). The recently mal mixing, also call for special structure in the neutrino announced measurements of neutral-current and sector that is very different from that of the quark sec- charged-current fluxes at SNO provide a 5.3␴ signal for tor. neutrino flavor transition that is independent of the so- The mass scales involved in Eqs. (185) and (186) have lar model. The simplest interpretation of these results is implications for many other frameworks of new physics. ␯ Ϫ␯ ␯ that there are e a oscillations (where a is some com- In particular, they can help discriminate between various bination of ␯␮ and ␯␶) with the following set of mass and options in the framework of models with extra dimen- mixing parameters: sions, models of supersymmetry without R parity, left- right symmetric models, etc. ϫ Ϫ5Ͻ⌬ 2 Ͻ ϫ Ϫ4 2 2.4 10 m᭪ 2.4 10 eV , Another hint for neutrino masses comes from the 2 LSND experiment. Here, however, the signal is pres- 0.27Ͻtan ␪᭪Ͻ0.77 ͑large mixing angle͒, (186) ently observed by a single experiment, and further ex- where again the ranges quoted correspond to the results perimental testing is required. The simplest interpreta- ␯ →␯ of the global analysis given in Eq. (147). From the global tion of the LSND data is that there are e ␮ ⌬ 2 ϭO 2 2 ␪ analysis, an upper bound on a third mixing angle arises, oscillations with mLSND (1 eV ) and sin 2 LSND ϭO ⌬ 2 ӷ⌬ 2 ⌬ 2 driven mainly by the negative results of the reactor ex- (0.003). The fact that mLSND matm , m᭪ means ⌬ 2 periments in combination with the deduced matm from that the three results cannot be explained by oscillations the atmospheric neutrino data: among the three active neutrinos alone. If the LSND result is confirmed, then more dramatic sin ␪ р0.24. (187) reactor modifications of the standard model will be required. The smallness of this angle guarantees that the results of The simplest extension, that is, the addition of a light the three-neutrino analysis combining the atmospheric singlet neutrino, is not excluded but it does run into dif- and solar neutrino data will be close to the two separate ficulties related to the fact that oscillations into purely two-neutrino analyses. sterile neutrinos fit neither the atmospheric nor the solar The evidence for neutrino masses implies that the neutrino data. standard model cannot be a complete picture of Nature. The good news is that there has been a lot of progress In particular, if the standard model is only a low-energy in neutrino physics in recent years. Measurements of effective theory, very light neutrino masses are expected. both atmospheric and solar neutrino fluxes make the The scale at which the standard-model picture is no case for neutrino masses and mixing more and more longer valid is inversely proportional to the scale of neu- convincing. Many theoretical ideas are being excluded, trino masses. Specifically, while others have at last experimental guidance in their 2 15 construction. The other piece of good news is that there m␯տͱ⌬m ϳ0.05 eV⇒⌳ Շ10 GeV. (188) atm NP is a lot of additional experimental information concern- We learn that there is a scale of new physics well below ing neutrino physics to come in the near future. We are the Planck scale. guaranteed to learn much more. The scale of new physics in Eq. (188) is intriguingly close to the scale of coupling unification. Indeed, since Note added GUT’s with an SO(10) (or larger) gauge group predict neutrino masses, plausibly at the 0.1 eV scale, the atmo- In December 2002, the KamLAND Collaboration re- spheric neutrino data can be taken as additional support leased their first data on the measurement of the flux of ␯ for the GUT idea. The large mixing angle in Eq. (185) ¯ e’s from distant nuclear reactors (Eguchi et al., 2002). also finds a natural and quite generic place in GUT’s. Below we briefly present the main implications of this The measured values of the neutrino flavor param- result. eters are useful in testing various ideas to explain the flavor puzzle. Quite a few of the simplest models that KamLAND results explain the smallness and hierarchy in the quark-sector ␯ parameters fail to explain the neutrino parameters. The KamLAND have measured the flux of ¯ e’s from dis- neutrino parameters have some features that are quite tant nuclear reactors. In an exposure of 162 ton yr (145.1

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 397

days), the ratio of the number of observed inverse ␤-decay events to the number of events expected without oscillations is ϭ Ϯ RKamLAND 0.611 0.094, (189)

for E¯␯ Ͼ3.4 MeV. This deficit is inconsistent with the e ␯ expected rate for massless ¯ e’s at the 99.95% confidence level. KamLAND have also presented the energy dependence of these events in the form of the prompt energy (E ӍE¯␯ ϩm Ϫm ) spectrum with 17 bins of width prompt e p n Ͼ 0.425 MeV for Eprompt 0.9 MeV. To eliminate the background from geo-neutrinos, only bins with Eprompt Ͼ2.6 MeV are used. The measured spectrum shows a clear deficit, but there is no significant signal of any energy dependence of this effect. The KamLAND results can be interpreted in terms of ␯ ¯ e oscillations with parameters shown in Fig. 37(a) (from Bahcall, Gonzalez-Garcia, and Penã-Garay, 2002d). Since KamLAND is a disappearance experiment and, furthermore, matter effects are small, the experiment is ␯ not sensitive to the flavor into which ¯ e oscillates. The allowed region has three local minima and it is separated into ‘‘islands.’’ These islands correspond to oscillations with wavelengths that are approximately tuned to the average distance between the reactors and the detector, 180 km. The local minimum in the lowest-mass island (⌬m2ϭ1.5ϫ10Ϫ5 eV2) corresponds to an approximate first maximum in the oscillation probability (minimum in the event rate). The overall best-fit point (⌬m2ϭ7.1 ϫ10Ϫ5 eV2 and tan2 ␪ϭ0.52,1.92) corresponds approximately to the second maximum in the oscillation probability. For the same ⌬m2, maximal mixing is only slightly less favored, ⌬␹2ϭ0.4. Thus with the present statistical accuracy, KamLAND cannot discriminate between a large-but-not-maximal and maximal mixing. Because there is no significant evidence of energy distortion, the allowed regions for the KamLAND analysis extend to high ⌬m2 values for which oscillations would be averaged and the event reduction would be energy independent. These solutions, however, are ruled out by the nonobservation of a deficit at shorter baselines, and in particular by the negative results of CHOOZ, which rule out solutions with ⌬m2տ1 (0.8)ϫ10Ϫ3 eV2 at 3␴ (99% C.L.). FIG. 37. The impact of KamLAND results on the allowed regions (at 90%, 95%, 99%, and 99.7% C.L.) in the ⌬ 2 2 ␪ ( m21 ,tan 12) plane. (a) A combined analysis of KamLAND and CHOOZ results. (b) A combined analysis of KamLAND Consequences for solar neutrinos and solar data (Color in online edition). The most important aspect of Fig. 37(a) is the demon- stration by the KamLAND results that antineutrinos os- The results of such an analysis are shown in Fig. 37(b) cillate with parameters that are consistent with the (from Bahcall, Gonzalez-Garcia, and Penã-Garay, large-mixing-angle solar neutrino solution. Under the 2002d). assumption that CPT is satisfied, the antineutrino mea- Comparing Fig. 37(a) with Fig. 37(b), we can see that surements by KamLAND apply directly to the neutrino the impact of the KamLAND results is to narrow down sector, and the two sets of data can be combined to ob- the allowed parameter space in a very significant way. In tain the globally allowed oscillation parameters (in fact, particular, large-mixing-angle region is the only remain- the KamLAND results already show that CPT is satis- ing allowed solution. The low-mass solution is excluded fied in the neutrino sector to a good approximation). at 4.8␴ and vacuum solutions are excluded at 4.9␴. The

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 398 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

once-favored small-mixing-angle solution is now ex- the Spanish DGICYT under Grants Nos. PB98-0693 and cluded at 6.1␴. Within the low-mixing-angle region, the FPA2001-3031, and by the ESF network 86. Y.N. was main effect of KamLAND is the reduction in the al- supported by the Israel Science Foundation founded by lowed range of ⌬m2, while the impact on the determi- the Israel Academy of Sciences and Humanities. nation of the mixing angle is marginal. Furthermore, various alternative solutions to the solar neutrino problem which predict that there should be no REFERENCES deficit in the KamLAND experiment (such as neutrino decay or new flavor-changing interactions) are now ex- Abdurashitov, J. N., et al. (SAGE Collaboration), 1999, Phys. cluded at 3.6␴. Rev. C 60, 055801. Abdurashitov, J. N., et al. (SAGE Collaboration), 2002, J. Exp. Effect on three-neutrino mixing scenarios Theor. Phys. 95, 181. Ables, E., et al. (MINOS Collaboration), 1995, FERMILAB- The ¯␯ survival probability at KamLAND in the PROPOSAL-P-875. e Achar, H., et al., 1965, Phys. Lett. 18, 196. three-neutrino framework is Achkar, B., et al. (Bugey Collaboration), 1995, Nucl. Phys. B 434, 503. ͑␯ ↔␯ ͒ϭ 4 ␪ ϩ 4 ␪ ͫ Ϫ͚ 2 ␪ P ¯ e ¯ e sin 13 cos 14 1 fi sin 2 12 Aglietta, M., et al. (Nusex Collaboration), 1989, Europhys. i Lett. 8, 611. ⌬ 2 ͓ 2͔ ͓ ͔ Agrawal, V., T. Gaisser, P. Lipari, and T. Stanev, 1996, Phys. 1.27 m eV Li km ϫ 2ͩ 21 ͪͬ Rev. D 53, 1314. sin ͓ ͔ , (190) E␯ GeV Aguilar, A., et al. (LSND Collaboration), 2001, Phys. Rev. D 64, 112007. where Li is the distance of reactor i to KamLAND, and Ahmad, Q. R., et al. (SNO Collaboration), 2001, Phys. Rev. fi is the fraction of the total neutrino flux that comes from reactor i. Equation (190) is obtained with two ap- Lett. 87, 071301. Ahmad, Q. R., et al. (SNO Collaboration), 2002a, Phys. Rev. proximations: (a) matter effects are neglected at ⌬ 2 Lett. 89, 011301. KamLAND-like baselines, and (b) for m Ahmad, Q. R., et al. (SNO Collaboration), 2002b, Phys. Rev. տ Ϫ4 2 10 eV , the KamLAND energy resolution is not Lett. 89, 011302. good enough to resolve the corresponding oscillation Ahn, S. H., et al. (K2K Collaboration), 2001, Phys. Lett. B 511, ⌬ 2 wavelength, so the oscillation phase associated to m31 178. 1 is averaged to 2 . Ahrens, L. A., et al. (BNL E734 Collaboration), 1987, Phys. ␪ Equation (190) shows that a small 13 does not signifi- Rev. D 36, 702. cantly affect the shape of the measured spectrum. On Akhmedov, E. K., 1988, Yad. Fiz. 47, 475 [Sov. J. Nucl. Phys. the other hand, the overall normalization is scaled by 47, 301 (1988)]. 4 ␪ cos 13 and this factor may introduce a non-negligible Akhmedov, E. K., 1999, Nucl. Phys. B 538, 25. effect. Within its present accuracy, however, the Akhmedov, E. K., A. Dighe, P. Lipari, and A. Yu. Smirnov, KamLAND experiment cannot provide any further sig- 1999, Nucl. Phys. B 542,3. 2 ␪ Albright, C. H., K. S. Babu, and S. M. Barr, 1998, Phys. Rev. nificant constraint on tan 13 and the range in Eq. (190) still holds (see also Fogli et al., 2002). Lett. 81, 1167. Thus the main effect of the KamLAND results on the Albright, C. H., and S. M. Barr, 1998, Phys. Rev. D 58, 013002. global three-neutrino analysis of solar, atmospheric, and Albright, C. H., and S. M. Barr, 1999a, Phys. Lett. B 452, 287. Albright, C. H., and S. M. Barr, 1999b, Phys. Lett. B 461, 218. reactor data is to modify the allowed range of ⌬m2 in 21 Albright, C. H., and S. M. Barr, 2000a, Phys. Rev. D 62, Eq. (190) to 093008. ϫ Ϫ5Ͻ⌬ 2 2Ͻ ϫ Ϫ4 Albright, C. H., and S. M. Barr, 2000b, Phys. Rev. Lett. 85, 244. 5.5 10 m21/eV 1.9 10 . (191) Albright, C. H., and S. M. Barr, 2001, Phys. Rev. D 64, 073010. ␪ The allowed range of 12 and the corresponding entries Allison, W. W. M., et al. (Soudan Collaboration), 1999, Phys. of the mixing matrix U are only marginally affected by Lett. B 449, 137. the inclusion of the KamLAND data. Altarelli, G., and F. Feruglio, 1998, J. High Energy Phys. 9811, 021. Altarelli, G., and F. Feruglio, 1999a, Phys. Lett. B 451, 388. ACKNOWLEDGMENTS Altarelli, G., and F. Feruglio, 1999b, Phys. Rep. 320, 295. Altarelli, G., F. Feruglio, and I. Masina, 2000a, Phys. Lett. B We thank M. Maltoni and C. Penã-Garay for provid- 472, 382. ing us with updates of our previous published results. Altarelli, G., F. Feruglio, and I. Masina, 2000b, J. High Energy We thank Michael Dine and Yuval Grossman for critical Phys. 0011, 040. reading of the manuscript. M.C.G.-G. thanks the Weiz- Ambrosio, M., et al. (MACRO Collaboration), 2001, Phys. mann Institute, where this review was finalized, for their Lett. B 517, 59. warm hospitality and the Albert Einstein Minerva Cen- Apollonio, M., et al. (CHOOZ Collaboration), 1999, Phys. ter for Theoretical Physics for financial support. Lett. B 466, 415. M.C.G.-G. was supported by the European Union Arkani-Hamed, N., S. Dimopoulos, and G. Dvali, 1998, Phys. HPMF-CT-2000-00516. This work was also supported by Lett. B 429, 263.

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 399

Arkani-Hamed, N., S. Dimopoulos, G. Dvali, and J. March- Barbieri, R., L. J. Hall, D. Smith, A. Strumia, and N. Weiner, Russell, 2002, Phys. Rev. D 65, 024032. 1998, J. High Energy Phys. 9812, 017. Arkani-Hamed, N., and M. Schmaltz, 2000, Phys. Rev. D 61, Barbieri, R., L. J. Hall, and A. Strumia, 1999, Phys. Lett. B 033005. 445, 407. Arpesella, C., C. Broggini, and C. Cattadori, 1996, Astropart. Barbieri, R., G. G. Ross, and A. Strumia, 1999, J. High Energy Phys. 4, 333. Phys. 9910, 020. Astier, P., et al. (NOMAD Collaboration), 2001, Nucl. Phys. B Barenboim, G., L. Borissov, J. Lykken, and A. Yu. Smirnov, 611,3. 2001, J. High Energy Phys. 0210, 001. Athanassopoulos, C., et al., 1995, Phys. Rev. Lett. 75, 2650. Barger, V., D. Marfatia, and K. Whisnant, 2002, Phys. Rev. Athanassopoulos, C., et al., 1996, Phys. Rev. Lett. 77, 3082. Lett. 88, 011302. Athanassopoulos, C., et al., 1998, Phys. Rev. Lett. 81, 1774. Barger, V., D. Marfatia, K. Whisnant, and B. P. Wood, 2002, Ayres, D. S., B. Cortez, T. K. Gaisser, A. K. Mann, R. E. Phys. Lett. B 537, 179. Shrock, and L. R. Sulak, 1984, Phys. Rev. D 29, 902. Barger, V., S. Pakvasa, T. J. Weiler, and K. Whisnant, 1998a, Babu, K. S., and S. M. Barr, 1996, Phys. Lett. B 381, 202. Phys. Rev. D 58, 093016. Babu, K. S., J. C. Pati, and F. Wilczek, 2000, Nucl. Phys. B 566, Barger, V., S. Pakvasa, T. J. Weiler, and K. Whisnant, 1998b, 33. Phys. Lett. B 437, 107. Bahcall, J. N., 1989, Neutrino Astrophysics (Cambridge Uni- Barger, V., and K. Whisnant, 1988, Phys. Lett. B 209, 365. versity Press, Cambridge). Barger, V., and K. Whisnant, 1999, Phys. Rev. D 59, 093007. Bahcall, J. N., 1997, Phys. Rev. C 56, 3391. Barger, V., K. Whisnant, and R. J. N. Phillips, 1980, Phys. Rev. Bahcall, J. N., 2002a, Phys. Rev. C 65, 015802. D 22, 1636. Bahcall, J. N., 2002b, Phys. Rev. C 65, 025801. Barr, G., T. K. Gaisser, and T. Stanev, 1989, Phys. Rev. D 39, 3532. Bahcall, J. N., N. A. Bahcall, and G. Shaviv, 1968, Phys. Rev. Battistoni, G., 2001, Nucl. Phys. B, Proc. Suppl. 100, 101. Lett. 20, 1209. Battistoni, G., A. Ferrari, P. Lipari, T. Montaruli, P. R. Sala, Bahcall, J. N., S. Basu, and H. M. Pinsonneault, 1998, Phys. and T. Rancati, 2000, Astropart. Phys. 12, 315. Lett. B 433,1. Bazarko, A., et al. (MiniBooNE Collaboration), 2000, Nucl. Bahcall, J. N., and H. A. Bethe, 1990, Phys. Rev. Lett. 65, 2233. Phys. B, Proc. Suppl. 91, 210. Bahcall, J. N., and R. Davis, 1976, Science 191, 264. Becker-Szendy, R., et al. (IMB Collaboration), 1992, Phys. ˜ Bahcall, J. N., M. C. Gonzalez-Garcia, and C. Pena-Garay, Rev. D 46, 3720. 2001, J. High Energy Phys. 0108, 014. Bemporad, C., G. Gratta, and P. Vogel, 2002, Rev. Mod. Phys. Bahcall, J. N., M. C. Gonzalez-Garcia, and C. Penã-Garay, 74, 297. 2002a, J. High Energy Phys. 0204, 007. Berezhiani, Z., and A. Rossi, 1999, J. High Energy Phys. 9903, Bahcall, J. N., M. C. Gonzalez-Garcia, and C. Penã-Garay, 002. 2002b, Phys. Rev. C 66, 035802. Berezhiani, Z., and A. Rossi, 2001, Nucl. Phys. B 594, 113. Bahcall, J. N., M. C. Gonzalez-Garcia, and C. Penã-Garay, Berezinsky, V., 2001, Astropart. Phys. 17, 509. 2002c, J. High Energy Phys. 0207, 054. Berger, M. S., and K. Siyeon, 2001, Phys. Rev. D 63, 057302. Bahcall, J. N., M. C. Gonzalez-Garcia, and C. Penã-Garay, Bergmann, S., Y. Grossman, and E. Nardi, 1999, Phys. Rev. D 2002d, J. High Energy Phys. 0302, 009. 60, 093008. Bahcall, J. N., M. Kamionkowsky, and A. Sirlin, 1995, Phys. Bilenky, S. M., J. Hosek, and S. T. Petcov, 1980, Phys. Lett. B Rev. D 51, 6146. 94, 495. Bahcall, J. N., P. I. Krastev, and E. Lisi, 1997, Phys. Rev. C 55, Bilenky, S. M., D. Nicolo, and S. T. Petcov, 2001, Phys. Lett. B 494. 538, 77. Bahcall, J. N., P. I. Krastev, and A. Yu. Smirnov, 1998, Phys. Bilenky, S. M., S. Pascoli, and S. T. Petcov, 2001a, Phys. Rev. D Rev. D 58, 096016. 64, 053010. Bahcall, J. N., P. I. Krastev, and A. Yu. Smirnov, 2000, Phys. Bilenky, S. M., S. Pascoli, and S. T. Petcov, 2001b, Phys. Rev. D Rev. D 62, 093004. 64, 113003. Bahcall, J. N., and M. H. Pinsonneault, 1992, Rev. Mod. Phys. Binetruy, P., S. Lavignac, and P. Ramond, 1996, Nucl. Phys. B 64, 885. 477, 353. Bahcall, J. N., and H. M. Pinsonneault, 1995, Rev. Mod. Phys. Boehm, F., and P. Vogel, 1987, Physics of Massive Neutrinos 67, 781. (Cambridge University Press, Cambridge). Bahcall, J. N., H. M. Pinsonneault, and S. Basu, 2001, Astro- Boliev, M. M., A. V. Butkevich, A. E. Chudakov, S. P. Mikheev, phys. J. 555, 990. O. V. Suvorova, and V. N. Zakidyshev, 1999, Nucl. Phys. B, Bahcall, J. N., and R. K. Ulrich, 1988, Rev. Mod. Phys. 60, 297. Proc. Suppl. 70, 371. Baltz, A. J., and J. Weneser, 1987, Phys. Rev. D 35, 528. Bonn, J., et al., 2001, Nucl. Phys. B, Proc. Suppl. 91, 273. Baltz, A. J., and J. Weneser, 1988, Phys. Rev. D 37, 3364. Borodovsky, L., et al. (BNL E776 Collaboration), 1992, Phys. Banks, T., Y. Grossman, E. Nardi, and Y. Nir, 1995, Phys. Rev. Rev. Lett. 68, 274. D 52, 5319. Borzumati, F. M., Y. Grossman, E. Nardi, and Y. Nir, 1996, Barbieri, R., P. Creminelli, and A. Romanino, 1999, Nucl. Phys. Lett. B 384, 123. Phys. B 559, 17. Borzumati, F., and Y. Nomura, 2001, Phys. Rev. D 64, 053005. Barbieri, R., P. Creminelli, and A. Strumia, 2000, Nucl. Phys. B Bouchez, J., et al., 1986, Z. Phys. C 32, 499. 585, 28. Bugaev, E. N., and V. A. Naumov, 1989, Phys. Lett. B 232, 391. Barbieri, R., L. J. Hall, G. L. Kane, and G. G. Ross, 1999, Caldwell, D. O., R. N. Mohapatra, and S. J. Yellin, 2001a, e-print hep-ph/9901228. Phys. Rev. Lett. 87, 041601.

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 400 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

Caldwell, D. O., R. N. Mohapatra, and S. J. Yellin, 2001b, Fogli, G. L., and E. Lisi, 1995a, Astropart. Phys. 3, 185. Phys. Rev. D 64, 073001. Fogli, G. L., and E. Lisi, 1995b, Phys. Rev. D 52, 2775. Carone, C. D., and M. Sher, 1998, Phys. Lett. B 420, 83. Fogli, G. L., E. Lisi, and A. Marrone, 2001, Phys. Rev. D 64, Casas, J. A., J. R. Espinosa, A. Ibarra, and I. Navarro, 1999, 093005. Nucl. Phys. B 556,3. Fogli, G. L., E. Lisi, A. Marrone, D. Montanino, and A. Casas, J. A., J. R. Espinosa, A. Ibarra, and I. Navarro, 2000, Palazzo, 2002, Phys. Rev. D 66, 053010. Nucl. Phys. B 569, 82. Fogli, G. L., E. Lisi, A. Marrone, and G. Scioscia, 1999a, Cheung, K., and O. C. Kong, 2000, Phys. Rev. D 61, 113012. e-print hep-ph/9906450 Cleveland, B. T., et al., 1998, Astrophys. J. 496, 505. Fogli, G. L., E. Lisi, A. Marrone, and G. Scioscia, 1999b, Phys. Cocco, A. G., et al. (OPERA Collaboration), 2000, Nucl. Phys. Rev. D 59, 033001. B, Proc. Suppl. 85, 125. Fogli, G. L., E. Lisi, and D. Montanino, 1994, Phys. Rev. D 49, Cowsik, R., Yash Pal, T. N. Rengarajan, and S. N. Tandon, 3626. 1965, in Proceedings of the International Conference on Cos- Fogli, G. L., E. Lisi, and D. Montanino, 1995, Astropart. Phys. mic Rays, Jaipur, 1963, edited by R. R. Daniels, Vol. 6, p. 211. 4, 177. Creminelli, P., G. Signorelli, and A. Strumia, 2001, J. High En- Fogli, G. L., E. Lisi, and D. Montanino, 1996, Phys. Rev. D 54, ergy Phys. 0105, 052. 2048. Das, A., and O. C. Kong, 1999, Phys. Lett. B 470, 149. Fogli, G. L., E. Lisi, and D. Montanino, 1998, Astropart. Phys. Davidson, S., and S. F. King, 1998, Phys. Lett. B 445, 191. 9, 119. Davidson, S., and M. Losada, 2000, J. High Energy Phys. 0005, Fogli, G. L., E. Lisi, D. Montanino, and A. Palazzo, 2000a, 021. Phys. Rev. D 62, 013002. Davis, R., Jr., D. S. Harmer, and K. C. Hoffman, 1968, Phys. Fogli, G. L., E. Lisi, D. Montanino, and A. Palazzo, 2000b, Rev. Lett. 20, 1205. Phys. Rev. D 62, 113004. Daum, K., et al. (Frejus Collaboration), 1995, Z. Phys. C 66, Fogli, G. L., E. Lisi, D. Montanino, and A. Palazzo, 2001, Phys. 417. Rev. D 64, 093007. de Bellfon, A. (for the HELLAZ Collaboration), 1999, Nucl. Fogli, G. L., E. Lisi, D. Montanino, and G. Scioscia, 1997, Phys. B, Proc. Suppl. 70, 386. Phys. Rev. D 55, 4385. de Gouvea, A., A. Friedland, and H. Murayama, 2000, Phys. Fogli, G. L., and G. Nardulli, 1979, Nucl. Phys. B 160, 116. Lett. B 490, 125. Fogli, G. L., et al., 2002, e-print hep-ph/0212127. de Gouvea, A., and J. W. Valle, 2001, Phys. Lett. B 501, 115. Foot, R., R. R. Volkas, and O. Yasuda, 1998, Phys. Rev. D 58, de Holanda, P. C., and A. Yu. Smirnov, 2002, Phys. Rev. D 66, 013006. 113005. Frampton, P. H., and S. L. Glashow, 1999, Phys. Lett. B 461, 95. de Rujula, A., M. B. Gavela, and P. Hernandez, 2001, Phys. Friedland, A., 2000, Phys. Rev. Lett. 85, 936. Rev. D 63, 033001. Friedland, A., 2001, Phys. Rev. D 64, 013008. Dienes, K. R., E. Dudas, and T. Gherghetta, 1999, Nucl. Phys. Fritzsch, H., and Z. Xing, 1996, Phys. Lett. B 372, 265. B 557, 25. Fritzsch, H., and Z. Xing, 2000, Phys. Rev. D 61, 073016. Dienes, K. R., and I. Sarcevic, 2001, Phys. Lett. B 500, 133. Froggatt, C. D., and H. B. Nielsen, 1979, Nucl. Phys. B 147, Dimopoulos, S., S. Raby, and F. Wilczek, 1981, Phys. Rev. D 277. 24, 1681. Fukuda, S., et al. (SuperKamiokande Collaboration), 2000, Dvali, G., and A. Yu. Smirnov, 1999, Nucl. Phys. B 563, 63. Phys. Rev. Lett. 85, 3999. Dydak, F., et al. (CDHS Collaboration), 1984, Phys. Lett. Fukuda, S., et al. (SuperKamiokande Collaboration), 2001, 134B, 281. Phys. Rev. Lett. 86, 5651. Dziewonski, A. M., and D. L. Anderson, 1981, Phys. Earth Fukuda, Y., et al. (Kamiokande Collaboration), 1994, Phys. Planet. Inter. 25, 297. Lett. B 335, 237. Eguchi, K., et al. (KamLAND Collaboration), 2002, e-print Fukuda, Y., et al. (Kamiokande Collaboration), 1996, Phys. hep-ex/0212021. Rev. Lett. 77, 1683. Ellis, J., G. K. Leontaris, S. Lola, and D. V. Nanopoulos, 1999, Fukuda, Y., et al. (SuperKamiokande Collaboration), 1998, Eur. Phys. J. C 9, 389. Phys. Rev. Lett. 81, 1562. Elwood, J. K., N. Irges, and P. Ramond, 1998, Phys. Rev. Lett. Fukuda, Y., et al. (SuperKamiokande Collaboration), 1999a, 81, 5064. Phys. Lett. B 467, 185. Engel, R., T. K. Gaisser, and T. Stanev, 2000, Phys. Lett. B 472, Fukuda, Y., et al. (SuperKamiokande Collaboration), 1999b, 113. Phys. Rev. Lett. 82, 5194. Ermilova, V. K., V. A. Tsarev, and V. A. Chechin, 1986, Kratk. Fukugita, M., M. Tanimoto, and T. Yanagida, 1998, Phys. Rev. Soobshch. Fiz. 5, 26. D 57, 4429. Eskut, E., et al. (CHORUS Collaboration), 2001, Phys. Lett. B Fukugita, M., M. Tanimoto, and T. Yanagida, 1999, Phys. Rev. 497,8. D 59, 113016. Eyal, G., and Y. Nir, 1999, J. High Energy Phys. 9906, 024. Gago, A. M., H. Nunokawa, and R. Zukanovich Funchal, 2001, Faı¨d, B., G. L. Fogli, E. Lisi, and D. Montanino, 1997, Phys. Phys. Rev. D 63, 013005. Rev. D 55, 1353. Gaisser, T. K., M. Honda, K. Kasahara, H. S. Lee, S. Mi- Faraggi, A. E., and M. Pospelov, 1999, Phys. Lett. B 458, 237. dorikawa, V. Naumov, and T. Stanev, 1996, Phys. Rev. D 54, Farzan, Y., O. L. Peres, and A. Yu. Smirnov, 2001, Nucl. Phys. 5578. B 612, 59. Gaisser, T. K., and T. Stanev, 1998, Phys. Rev. D 57, 1977. Fiorentini, G., V. A. Naumov, and F. L. Villante, 2001, Phys. Gaisser, T. K., T. Stanev, and G. Barr, 1988, Phys. Rev. D 38, Lett. B 510, 173. 85.

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing 401

Garzelli, M. V., and C. Giunti, 2001, J. High Energy Phys. 0112, Irges, N., S. Lavignac, and P. Ramond, 1998, Phys. Rev. D 58, 017. 035003. Gell-Mann, M., P. Ramond, and R. Slansky, 1979, in Super- Joshipura, A. S., and S. D. Rindani, 2000a, Eur. Phys. J. C 14, gravity, edited by P. van Nieuwenhuizen and D. Z. Freedman 85. (North-Holland, Amsterdam), p. 315. Joshipura, A. S., and S. D. Rindani, 2000b, Phys. Lett. B 494, Gemmeke, H., et al., 1990, Nucl. Instrum. Methods Phys. Res. 114. A 289, 490. Kajita, T., and Y. Totsuka, 2001, Rev. Mod. Phys. 73, 85. Gonzalez-Garcia, M. C., P. C. de Holanda, C. Penã-Garay, and Kayser, B., F. Gibrat-Debu, and F. Perrier, 1989, The Physics of J. W. F. Valle, 2000, Nucl. Phys. B 573,3. Massive Neutrinos, World Scientific Lecture Notes in Physics Gonzalez-Garcia, M. C., N. Fornengo, and J. W. F. Valle, 2000, No. 25 (World Scientific, Singapore), p. 1. Nucl. Phys. B 580, 58. Kim, C. W. and A. Pevsner, 1993, Neutrinos in Physics and Gonzalez-Garcia, M. C., and M. Maltoni, 2002, Eur. Phys. J C Astrophysics, Contemporary Concepts in Physics, No. 8 (Har- 26, 417. wood Academic, Chur, Switzerland). Gonzalez-Garcia, M. C., M. Maltoni, C. Penã-Garay, and J. W. King, S. F., 1998, Phys. Lett. B 439, 350. F. Valle, 2001, Phys. Rev. D 63, 033005. King, S. F., 1999, Nucl. Phys. B 562, 57. Gonzalez-Garcia, M. C., H. Nunokawa, O. L. G. Peres, T. King, S. F., 2000, Nucl. Phys. B 576, 85. Stanev, and J. W. F. Valle, 1998, Phys. Rev. D 58, 033004. Kirsten, T. (GNO Collaboration), 2002, Talk given at ␯-2002 Gonzalez-Garcia, M. C., H. Nunokawa, O. L. G. Peres, and J. conference, Munich, May 2002. W. F. Valle, 1999, Nucl. Phys. B 543,3. Klapdor-Kleingrothaus, H. V., H. Pas, and A. Yu. Smirnov, Gonzalez-Garcia, M. C., and C. Penã-Garay, 2000, Phys. Rev. 2001, Phys. Rev. D 63, 073005. D 62, 031301. Klapdor-Kleingrothaus, H. V., et al., 2001, Eur. Phys. J. A 12, Gonzalez-Garcia, M. C., and C. Penã-Garay, 2001, Phys. Rev. 147. D 63, 073013. Kobayashi, M., and T. Maskawa, 1973, Prog. Theor. Phys. 49, Gonzalez-Garcia, M. C., C. Penã-Garay, Y. Nir, and A. Yu. 652. Smirnov, 2001, Phys. Rev. D 63, 013007. Krastev, P. I., and S. T. Petcov, 1988, Phys. Lett. B 207, 64. Gonzalez-Garcia, M. C., C. Penã-Garay, and A. Yu. Smirnov, Kuo, T. K., and J. Pantaleone, 1986, Phys. Rev. Lett. 57, 1805. 2001, Phys. Rev. D 63, 113004. Kuo, T. K., and J. Pantaleone, 1989, Rev. Mod. Phys. 61, 937. Groom, D. E., et al. (Particle Data Group), 2000, Eur. Phys. J. Landau, L., 1932, Phys. Z. Sowjetunion 2, 46. C 15,1. Lande, K., et al., 1999, Nucl. Phys. B, Proc. Suppl. 77, 13. Grossman, Y., and M. Neubert, 2000, Phys. Lett. B 474, 361. Langacker, P., S. T. Petcov, G. Steigman, and S. Toshev, 1987, Grossman, Y., and Y. Nir, 1995, Nucl. Phys. B 448, 30. Nucl. Phys. B 282, 589. Grossman, Y., Y. Nir, and Y. Shadmi, 1998, J. High Energy Langacker, P., and N. Polonsky, 1995, Phys. Rev. D 52, 3081. Phys. 9810, 007. Lanou, R. E., et al., 1993, J. Low Temp. Phys. 93, 785. Haba, N., and H. Murayama, 2001, Phys. Rev. D 63, 053010. Learned, J. G., S. Pakvasa, and T. J. Weiler, 1988, Phys. Lett. B Hagiwara, K., and N. Okamura, 1999, Nucl. Phys. B 548, 60. 207, 79. Hall, L., and H. Murayama, 1998, Phys. Lett. B 436, 323. Lee, H., and Y. S. Koh, 1990, Nuovo Cimento Soc. Ital. Fis., B Hall, L., H. Murayama, and N. Weiner, 2000, Phys. Rev. Lett. 105, 883. 84, 2572. Leurer, M., Y. Nir, and N. Seiberg, 1993, Nucl. Phys. B 398, Halprin, A., 1986, Phys. Rev. D 34, 3462. 319. Hampel, W., et al. (GALLEX Collaboration), 1999, Phys. Lett. Leurer, M., Y. Nir, and N. Seiberg, 1994, Nucl. Phys. B 420, B 447, 127. 468. Harvey, J. A., D. B. Reiss, and P. Ramond, 1982, Nucl. Phys. B Lipari, P., 2000, Astropart. Phys. 14, 153. 199, 223. Lipari, P., T. K. Gaisser, and T. Stanev, 1998, Phys. Rev. D 58, Hata, N., S. Bludman, and P. Langacker, 1994, Phys. Rev. D 49, 073003. 3622. Lipari, P., and M. Lusignoli, 1998, Phys. Rev. D 58, 073005. Hata, N., and P. Langacker, 1997, Phys. Rev. D 56, 6107. Lipkin, H. J., 1999, e-print hep-ph/9901399. Haxton, W. C., 1986, Phys. Rev. Lett. 57, 1271. Lisi, E., A. Marrone, D. Montanino, A. Palazzo, and S. T. Pet- Heeger, K. M., and R. G. H. Robertson, 1996, Phys. Rev. Lett. cov, 2001, Phys. Rev. D 63, 093002. 77, 3720. Lobashev, V. M., et al., 2001, Nucl. Phys. B, Proc. Suppl. 91, Hidaka, K., M. Honda, and S. Midorikawa, 1988, Phys. Rev. 280. Lett. 61, 1537. Lohmann, W., R. Kopp, and R. Voss, 1985, CERN Yellow Re- Hirata, S. H., et al. (Kamiokande Collaboration), 1992, Phys. port EP/85-03. Lett. B 280, 146. Lukas, A., P. Ramond, A. Romanino, and G. G. Ross, 2000, Hirsch, M., and S. F. King, 2001, Phys. Lett. B 516, 103. Phys. Lett. B 495, 136. Honda, M., T. Kajita, K. Kasahara, and S. Midorikawa, 1996, Lukas, A., P. Ramond, A. Romanino, and G. G. Ross, 2001, J. Prog. Theor. Phys. Suppl. 123, 483. High Energy Phys. 0104, 010. Honda, M., T. Kajita, S. Midorikawa, and K. Kasahara, 1995, Ma, E., M. Raidal, and U. Sarkar, 2000, Phys. Rev. Lett. 85, Phys. Rev. D 52, 4985. 3769. Honda, M., K. Kasahara, K. Hidaka, and S. Midorikawa, 1990, Ma, E., and G. Rajasekaran, 2001, Phys. Rev. D 64, 113012. Phys. Lett. B 248, 193. Maekawa, N., 2001, Prog. Theor. Phys. 106, 401. Huber, S. J., and Q. Shafi, 2001, Phys. Lett. B 512, 365. Maki, Z., M. Nakagawa, and S. Sakata, 1962, Prog. Theor. Ibanez, L. E., and G. G. Ross, 1981, Phys. Lett. B 105, 439. Phys. 28, 870. Ioannisian, A., and A. Pilaftsis, 2000, Phys. Rev. D 62, 066001. Mannheim, P. D., 1988, Phys. Rev. D 37, 1935.

Rev. Mod. Phys., Vol. 75, No. 2, April 2003 402 M. C. Gonzalez-Garcia and Yosef Nir: Neutrino masses and mixing

Maris, M., and S. T. Petcov, 1997, Phys. Rev. D 56, 7444. Romosan, C., et al. (CCFR Collaboration), 1997, Phys. Rev. Marquet, C., et al. (NEMO Collaboration), 2000, Nucl. Phys. Lett. 78, 2912. B, Proc. Suppl. 87, 298. Sato, J., and T. Yanagida, 1998, Phys. Lett. B 430, 127. McDonald, A. B., et al. (SNO Collaboration), 2000, Nucl. Sato, J., and T. Yanagida, 2000, Phys. Lett. B 493, 356. Phys. B, Proc. Suppl. 91, 21. Schechter, J., and J. W. F. Valle, 1980a, Phys. Rev. D 21, 309. McFarland, K. S., et al. (CCFR Collaboration), 1995, Phys. Schechter, J., and J. W. F. Valle, 1980b, Phys. Rev. D 22, 2227. Rev. Lett. 75, 3993. Schechter, J., and J. W. F. Valle, 1981, Phys. Rev. D 24, 1883. Mikheyev, S. P., and A. Yu. Smirnov, 1985, Yad. Fiz. 42, 1441 Shafi, Q., and Z. Tavartkiladze, 1999, Phys. Lett. B 451, 129. [Sov. J. Nucl. Phys. 42, 913 (1985)]. Shafi, Q., and Z. Tavartkiladze, 2000a, Phys. Lett. B 473, 272. Mikheyev, S. P., and A. Yu. Smirnov, 1987, Sov. Phys. Usp. 30, Shafi, Q., and Z. Tavartkiladze, 2000b, Phys. Lett. B 487, 145. 759. Shafi, Q., and Z. Tavartkiladze, 2000c, Nucl. Phys. B 573, 40. Mohapatra, R. N., S. Nandi, and A. Perez-Lorenzana, 1999, Shafi, Q., and Z. Tavartkiladze, 2000d, Phys. Lett. B 482, 145. Phys. Lett. B 466, 115. Shi, X., and D. N. Schramm, 1992, Phys. Lett. B 283, 305. Mohapatra, R. N., and A. Perez-Lorenzana, 2000, Nucl. Phys. Shibuya, H., et al., 1997, CERN-SPSC-97-24. B 576, 466. Shrock, R., 1980, Phys. Lett. B 96, 159. Mohapatra, R. N., A. Perez-Lorenzana, and C. A. de Sousa Simpson, J. A., 1983, Annu. Rev. Nucl. Part. Sci. 33, 323. Pires, 2000a, Phys. Lett. B 474, 355. Smith, C. L., 1972, Phys. Rep. 3, 261. Mohapatra, R. N., A. Perez-Lorenzana, and C. A. de Sousa Smy, M., 2002, Talk given at ␯-2002 conference, Munich, May Pires, 2000b, Phys. Lett. B 491, 143. 2002. Mohapatra, R. N., and G. Senjanovic, 1980, Phys. Rev. Lett. Tanimoto, M., 1999a, Phys. Rev. D 59, 017304. 44, 912. Tanimoto, M., 1999b, Phys. Lett. B 456, 220. Murayama, H., and T. Yanagida, 2001, Phys. Lett. B 520, 263. Tanimoto, M., 2000, Phys. Lett. B 483, 417. Nakahata, M., et al., 1986, J. Phys. Soc. Jpn. 55, 3786. Tanimoto, M. T., T. Watari, and T. Yanagida, 1999, Phys. Lett. Nakamura, S., T. Sato, V. Gudkov, and K. Kubodera, 2001, B 461, 345. Phys. Rev. C 63, 034617. Teshima, T., and T. Sakai, 1999, Prog. Theor. Phys. 101, 147. Naples, D., et al. (CCFR Collaboration), 1999, Phys. Rev. D Toshito, T., et al., 2001, e-print hep-ex/0105023. 59, 031101. Tserkovnyak, Y., R. Komar, C. Nally, and C. Waltham, 2003, Nir, Y., 2000, J. High Energy Phys. 0006, 039. Astropart. Phys. 18, 449. Nir, Y., and Y. Shadmi, 1999, J. High Energy Phys. 9905, 023. Turck-Chieze, S., S. Cahen, M. Casse, and C. Doom, 1988, Nishikawa, K., et al., 1997, KEK-PS proposal, Nucl. Phys. B, Astrophys. J. 335, 415. Proc. Suppl. 59, 289. Ushida, N., et al. (BNL E531 Collaboration), 1986, Phys. Rev. Nishikawa, K., et al. (K2K Collaboration), 2001, J. High En- Lett. 57, 2897. ergy Phys., Conference Proceedings, PRHEP-hep2001/298. Valuev, V., et al. (NOMAD Collaboration), 2001, J. High En- Oberauer, L., 1999, Nucl. Phys. B, Proc. Suppl. 77, 48. ergy Phys., Conference Proceedings, PRHEP-hep2001/190. Ortiz, C. E., A. Garcia, R. A. Waltz, M. Bhattacharya, and A. Vidyakin, G. S., et al., 1994, JETP Lett. 59, 390. K. Komives, 2000, Phys. Rev. Lett. 85, 2909. Villante, F. L., G. Fiorentini, and E. Lisi, 1999, Phys. Rev. D 59, Osborne, J. L., S. S. Said, and A. W. Wolfendale, 1965, Proc. 013006. Phys. Soc. London 86, 93. Vissani, F., 1999, J. High Energy Phys. 9906, 022. Pakvasa, S., and J. Pantaleone, 1990, Phys. Rev. Lett. 65, 2479. Vissani, F., 1998, J. High Energy Phys. 9811, 025. Pantaleone, J., 1990, Phys. Lett. B 251, 618. Vissani, F., 2001a, Nucl. Phys. B, Proc. Suppl. 100, 273. Parke, S. J., 1986, Phys. Rev. Lett. 57, 1275. Vissani, F., 2001b, Phys. Lett. B 508, 79. Parke, S. J., 1995, Phys. Rev. Lett. 74, 839. Vissani, F., and A. Yu. Smirnov, 1998, Phys. Lett. B 432, 376. Perez, G., 2000, J. High Energy Phys. 0012, 027. Wetterich, C., 1999, Phys. Lett. B 451, 397. Petcov, S. T., 1987, Phys. Lett. B 191, 299. Wilczek, F., 2001, Int. J. Mod. Phys. A 16, 1653. Petcov, S. T., 1988a, Phys. Lett. B 200, 373. Wojcicki, S. G., 2001, Nucl. Phys. B, Proc. Suppl. 91, 216. Petcov, S. T., 1988b, Phys. Lett. B 214, 139. Wolf, J., et al. (KARMEN Collaboration), 2001, J. High En- Petcov, S. T., 1998, Phys. Lett. B 434, 321. ergy Phys., Conference Proceedings, PRHEP-hep2001/301. Petcov, S. T., and J. Rich, 1989, Phys. Lett. B 224, 426. Wolfenstein, L., 1978, Phys. Rev. D 17, 2369. Piepke, A., et al., 2001, Nucl. Phys. B, Proc. Suppl. 91, 99. Wolfenstein, L., 1981, Phys. Lett. 107B, 77. Pontecorvo, B., 1957, Zh. Eksp. Teor. Fiz. 33, 549 [Sov. Phys. Wu, Y., 1999a, Phys. Rev. D 60, 073010. JETP 6, 429 (1958)]. Wu, Y., 1999b, Eur. Phys. J. C 10, 491. Raghavan, R. S., 1997, Phys. Rev. Lett. 78, 3618. Wu, Y., 1999c, Int. J. Mod. Phys. A 14, 4313. Ramond, P., 1979, preprint CALT-68-709, Invited talk given at Yanagida, T., 1979, in Proceedings of Workshop on Unified Sanibel Symposium, hep-ph/9809459. Theory and Baryon Number in the Universe, edited by O. Ramond, P., 1999, Journeys Beyond The Standard Model (Per- Sawada and A. Sugamoto, KEK Report KEK-97-18, p. 315. seus, Reading, MA). Yasuda, O., 1998, Phys. Rev. D 58, 091301. Randall, L., and R. Sundrum, 1999, Phys. Rev. Lett. 83, 3370. Zacek, G., et al., 1986, Phys. Rev. D 34, 2621. Reines, F., M. F. Crouch, T. L. Jenkins, W. R. Kropp, H. S. Zatsepin, G. T., and V. A. Kuzmin, 1962, Sov. J. Nucl. Phys. 14, Gurr, G. R. Smith, J. P. F. Sellschop, and B. Meyer, 1965, 1294. Phys. Rev. Lett. 15, 429. Zener, C., 1932, Proc. R. Soc. London, Ser. A 137, 696.

Rev. Mod. Phys., Vol. 75, No. 2, April 2003