Neutrino physics - WPusers.df.uba.ar/sgil/physics_paper_doc/papers_phys/cosmo/neutrino... ·...

Neutrino physicsWick C. HaxtonInstitute for Nuclear Theory, Box 351550, and Department of Physics, Box 351560,University of Washington, Seattle, Washington 98195

Barry R. HolsteinInstitut fur Kernphysik, Forschungszentrum Ju¨lich, D-52425 Ju¨lich, Germany and Department of Physicsand Astronomy, University of Massachusetts, Amherst, Massachusetts 01003

~Received 19 June 1998; accepted 29 July 1999!

The basic concepts of neutrino physics are presented at a level appropriate for integration intoelementary courses on quantum mechanics and/or modern physics. ©2000 American Association of

Physics Teachers.

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I. INTRODUCTION

The neutrino has been in the news recently, with repothat the SuperKamiokande collaboration—which operate50 000-ton detector of ultrapure water isolated deep witthe Japanese mine Kamiokande—has found evidencenonzero neutrino mass.1 The neutrino, a ghostly particlewhich can easily pass through the entire earth without inacting, has long fascinated both the professional physand the layman, as this poem from writer John Updike2 at-tests:

Neutrinos, they are very smallThey have no charge and have no massAnd do not interact at all.The earth is just a silly ballTo them, through which they simply pass,Like dustmaids down a drafty hallOr photons through a sheet of glass.They snub the most exquisite gas,Ignore the most substantial wall,Cold-shoulder steel and sounding brass,Insult the stallion in his stall,And, scorning barriers of class,Infiltrate you and me! Like tallAnd painless guillotines, they fallDown through our heads into the grass.At night, they enter at NepalAnd pierce the lover and his lassFrom underneath the bed—you callIt wonderful; I call it crass.

We present this pedagogical discussion of basic neutphysics in the hope that aspects of this topical and fasciing subject can be integrated into introductory courses, pviding a timely link between classroom physics and scienews in the popular press. In this way an instructor mayable to build on student curiosity in order to enrich the criculum with some unusual new physics. In this spirit wpresent in the following some of the basic physics underlymassive neutrinos and neutrino mixing, as well as otproperties of neutrinos relevant to both terrestrial expments and astrophysics.

II. NEUTRINOS: HISTORY

We begin with a bit of history—an interesting and modetailed discussion can be found in Laurie Brown’s articlethe September 1978 issue of Physics Today.3 Nuclear betadecay is a form of radioactivity wherein a parent nucle

15 Am. J. Phys.68 ~1!, January 2000

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decays to a daughter with the same atomic mass, buatomic number changed by one unit, with the missing chacarried off by an electron or positron

~A,Z!→~A,Z61!1e7. ~1!

This is quite literally nuclear transmutation of the type thfascinated alchemists of an earlier age. Beta decay wasrecognized because of naturally occurring radioactivitytoday has been studied for a vast range of parent isotoOne example of considerable modern interest is the decaa free neutron into a proton and electron, with a half lifeabout 10 min. Another is the decay of a bound neutrontritium to produce an electron and3He with a half life of12.26 years: The effects of the nuclear binding in changthe energy released in the decay are responsible for the gincrease in the half life. At the end of the 1920’s the extence of beta decay was well established. However, the strum of the emitted electrons was puzzling. If beta decoccurs from rest into a two-body final state as given in E~1!, momentum conservation would require the momentathe emitted electron and recoiling nucleus to be equalopposite. Energy conservation would then fix the outgoelectron energy which, because the nucleus is heavy andrecoils with a negligible velocity, is nearly equal to the dference of the parent and daughter nuclear masses~known asthe reaction energy release orQ value!

Q.M ~A,Z!2M ~A,Z61!. ~2!

As theQ-value in the beta decay of tritium is 18.6 keV, onwould expect a monochromatic spectrum with all emittelectrons having this energy. Instead experimentalists foa continuous spectrum of electron energies ranging fromrest massme to the Q value, peaking at an energy abohalfway in between, as shown in Fig. 1. Various explantions were considered—Niels Bohr even proposed the pobility that energy conservation was no longer exact in susubatomic processes, and rather preserved only in a stacal sense, somewhat in analogy with the second law of thmodynamics! However, in a letter dated December, 19Pauli suggested an alternative explanation—that an unserved light neutral particle~called by him the ‘‘neutron’’ orneutral one but later renamed by Fermi the ‘‘neutrino’’little neutral one! accompanied the outgoing electron acarried off the missing energy that was required to satienergy conservation. Pauli offered this explanation ten

15© 2000 American Association of Physics Teachers

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tively as a ‘‘desperate remedy’’ to solve the energy probleAlthough he publicized it in various talks over the next thryears, no publication occurred until his contribution to tSeventh Solvay Conference in October 1933.4 He also pro-posed~correctly! that the neutrino was a particle carryinspin 1/2 in order to satisfy angular momentum conservaand statistics. Fermi was present at a number of Pauli’ssentations and discussed the neutrino with him on thesecasions. In 1934 he published his insightful model for tbeta decay process and indeed for weak interactionsgeneral.5 He described beta decay in analogy with Diracsuccessful model of the electromagnetic interaction, whetwo charged particles interact via the exchange of a~virtual!photon that is produced and then absorbed by the elecmagnetic currents associated with the particles@cf. Fig. 2~a!#.Fermi represented the weak interaction in terms of the pruct of weak ‘‘currents,’’ one connecting the initial and finnucleon and the other connecting the final state electpositron and Pauli’s neutrino@cf. Fig. 2~b!#. In electromag-netism the virtual photon connects the two currents attinct points in space–time: Indeed the masslessness ophoton is the reason for the long-range Coulomb force. Inweak interaction theory, however, Fermi connected the crents at thesamespace–time point, in effect assuming ththe weak interaction is very short ranged. The strength ofinteraction was determined by an overall coupling strenGF ,

Hw5GF

&cp

†Omcnce†Omcn1h.c. ~3!

where h.c. denotes the Hermitian conjugate of the first teon the right. As written above, one of the weak currentsassociated with the conversion of a neutron into a protand the other with the production out of the vacuum ofelectron and antineutrino, which carry off almost all of treleased energy. The electron spectrum predicted byweak Hamiltonian can be readily calculated by using Fermgolden rule for the differential decay rate, yielding in thno-nuclear-recoil approximation

Fig. 1. Theb energy spectrum for decay into a heavy daughter nucleelectron, and neutrino is compared to the monoenergetic spectrum for dinto a daughter nucleus and electron, only. The spectrum is idealized:tortions due to the Coulomb interaction between the electron and daunucleus have been neglected.

16 Am. J. Phys., Vol. 68, No. 1, January 2000

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d3pn

~2p!3 2pd~Q2Ee2En!uMwu2

;S GF

&D 2

~4p!2

~2p!5 dEeEepeA~Q2Ee!22mn

2

3~Q2Ee!uMwu2, ~4!

whereMw is the nuclear matrix element. Takingmn50 andassuminguMwu2;constant we find

dG

dEe;peEe~Q2Ee!

2, ~5!

wherepe andEe are the momentum and energy of the eletron. When this simple formula was corrected for the distoing effects of the daughter nucleus Coulomb field onoutgoing electron wave function, an excellent fit was otained to several experimentally measured spectra. Thisan important confirmation of Fermi’s theory, and thusPauli’s postulate of the neutrino. Yet it would take anothtwo decades to detect this elusive particle directly.

The reason behind the difficulty of direct detection of tneutrino can be seen in the size of the weak couplingGF .From the 887-s lifetime of the neutron one finds thatGn

5\/tn;7310228GeV, while Fermi’s theory gives

Gn;UGF

&U2E d3pe

~2p!3

d3pn

~2p!3

32pd~mn2mp2Ee2En!uMwu2

5GF

2

2

~4p!2

~2p!5 Eme

mn2mpdEeEepe~mn2mp2Ee!

2uMwu2

.4.59310219GeV5GF2 uMwu2. ~6!

As uMwu2;6, one findsGF;1025 GeV22. In order to under-stand why this interaction is called ‘‘weak,’’ we note that thcontemporary picture is thatGF can be understood in anaogy with the electromagnetic interaction as the result ofexchange of a virtual but very massive and charg‘‘photon’’—the W boson—so thatGF;e2/MW

2 with MW

;80 GeV. By the uncertainty principle, however, a virtuparticle with such a heavy mass can propagate a distaDx;cDt;\c/MW;0.002 fm, i.e., a very small fraction othe nucleon radius. This exceedingly small interactirange—which allows point particles, such as the neutrand the quarks that are the underlying constituents onucleon, to interact only if they fortuitously pass very cloto one another—is the reason that the weak process in2~b! is much less probable than the electromagnetic procof Fig. 2~a!.

Fermi’s theory is a relativistic quantum field theorwherein a given field operatorcq represents both an annihlation operator for generic particleq as well as a creationoperator for the corresponding antiparticleq. Thus Eq.~3!contains not only the interaction for beta decay—n→pe2nandp→ne1n—but also the following.

~i! Electron capture, where an atomic electron orbitingnucleus interacts with one of the nuclear protons, converit to a neutron and producing an outgoing neutrino,e21p

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16W. C. Haxton and B. R. Holstein

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→n1n. ~An analogous process for positrons,e11n→p1 n, is important in the hot plasmas encountered in thebang and in explosive stellar environments.!

~ii ! The charged current neutrino reactionsn1n→p1e2

andn1p→n1e1 @which are the inverses of the reactions~i!, but are often referred to as ‘‘inverse beta decay’’#.

~iii ! The exotic resonant reactionsn1e21p→n and n1e11n→p, the true inverse reactions of beta decay. Tfirst can occur in an atom; both can take place in astrophcal plasmas.

Reaction~ii ! is the one relevant for neutrino detection,the produced electron/positron and nuclear transformaare signals for a neutrino interaction. Such a process is cacterized by the scattering cross sections which, when mul-tiplied by the incident particle flux and the number of sctering targets and integrated over solid angles, yieldsnumber of scattering events per unit time. The cross sechas the dimensions of area. In the approximation thatneutron is much heavier than the positron

s;UGF

&U2E d3pe

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;GF

2

2ppeEeuMwu2. ~7!

For incident neutrino energies that are large compared toelectron rest mass this becomess;GF

2En2uMwu2/2p

;10244cm2 for En;1 MeV. If we consider a single neutrino passing through a slab of material having a target dsity r ~e.g.,;1023atoms/cm3 for typical materials!, it wouldtravel a distance

Dx;1/~rs!;1021cm ~8!

before interacting, a distance equivalent to 100 billion earadii! This is indeed aweak interaction! The only way tocircumvent this problem is to have lots of low energy netrinos. The original plan of a Los Alamos team led by FrReines and Clyde Cowan was to use a fission bomb toduce the needed neutrinos. Later, however, they decideuse a nuclear reactor, which produces large numbers oftineutrinos. Working at the Savannah River reactor in SoCarolina, which has a neutrino flux of about 1013 per squarecentimeter per second, they designed a detector consistin

Fig. 2. Schematic representation of~a! the electron–proton electromagnetinteraction arising from the exchange of a virtual photon between the etromagnetic currents generated by these particles, and~b! the correspondingcharge-current weak interaction arising from the local product ofneutrino–electron and neutron–proton weak currents.


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two plastic tanks each filled with 200l of water, in whichwas dissolved cadmium chloride. The protons in the waprovided the target for the reactionn1p→n1e1, while thecadmium has a large cross section for neutron capture.tanks were sandwiched between three scintillation detectThe group looked for a signal consisting of gamma rays frthe annihilation of the emitted positron on an electron flowed closely~within a few microseconds! by gammas fromthe de-excitation of the cadium nucleus that had capturedneutron. The observed signal was correlated with the reabeing in operation. With this evidence they announced1956 that the neutrino had been detected,6 almost 25 yearsafter Pauli’s original suggestion. In 1996 Reines wawarded the Nobel Prize for this discovery.~Clyde Cowanhad died years earlier.!

However, this was not the end of the story, but only tbeginning. Indeed within seven years there was anoNobel-Prize-winning neutrino discovery: LedermaSchwartz, and Steinberger demonstrated that there was mthan one type of neutrino.7 In order to explain their discoverywe first provide a bit of theoretical background. As prevously discussed, the processes associated with Fermi’sture of beta decay are called ‘‘weak,’’ characterized by raor cross sections nearly 20 orders of magnitude smaller tthose involving strongly interacting particles, such ascross sections for scattering one nucleon off another. Pticles, such as neutrinos, that do not participate in strointeractions are called ‘‘leptons.’’ Thus the electron is alsolepton. ~Of course, as the electron carries a charge, itboth electromagnetic and weak interactions, while welieve neutrinos react only weakly.! In the 1930’s anothercharged particle was found that does not interact stronglthe muon. Except for the fact that it is about 200 timheavier, the muon’s behavior is remarkably similar to thatthe electron.8 The muon, however, is unstable and decays;1026 s into an electron and two neutral unseen particthat we now know are a neutrino and an antineutrino. Ttime scale is appropriate for a weak interaction, as canseen from the estimate

Gm51

tm;UGF

&U2E d3pe

~2p!3

d3pn

~2p!3

d3pn

~2p!3 ~2p!4

3d4~pm2pe2pn2pn !

;GF

2mm5

1636p3 , ~9!

i.e., tm;1025 s.~The numerical coefficient will change in a more care

calculation that takes into account the operatorOm .! Muondecay fits easily into Fermi’s interaction provided the leptcurrent is generalized to

Jmlep5ce

†Omcn1cm† Omcn . ~10!

Then the product of the lepton current with its Hermitiaconjugate yields an interaction responsible for muon dec

In order to understand the experiment of Ref. 7, it is useto go one step further and introduce modern quark notatIn the quark model the neutron is a composite object coprised of a pair ofd quarks and a singleu quark, while theproton consists of a pair ofu quarks and a singled quark.

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The weak current connecting the proton and neutron canbe replaced by a corresponding current connecting au quarkwith a d quark,

Jmhad5cn

†Omcp→cd†Omcu . ~11!

As the field operatorcd† can both create ad quark and de-

stroy the corresponding antiparticle (d), this same curren

describes the quark component of the process where adusystem~i.e., ap1 meson! decays to a muon and a neutrin

p1→m11n, ~12!

which is the dominant decay mode of the charged pion.9 Theexperimenters of Ref. 7 collided neutrinos from such decwith neutrons in an attempt to produce electronsandmuons,as predicted by the current of Eq.~10!. But they found onlymuons,not electrons. The explanation for this result is thneutrinos come intwo distinct species, an electron typene

and a muon typenm , with the weak current coupling electrons only tone and muons only tonm ,

Jmlep5ce

†Omcne1cm

† Omcnm. ~13!

The neutrino produced in pion decay thus must be anm andof the wrong type, or ‘‘flavor,’’ to produce an electron. I1977 a third charged lepton, thet, was discovered and another term has now been added to this equation—the cpling of the t to its neutrino, thent . Measurements of thedecay width of the neutralZ boson10 and astrophysical arguments based on helium abundance in the universe11 suggestthat this may exhaust the set of lepton–neutrino pairs: thappear to be no more light neutrinos beyond thent .

The modern picture of the weak interaction consistsonly of three doublets of charged lepton–neutrino pairsalso of three doublets~often called ‘‘generations’’! of charge2/3, charge21/3 quarks—(u,d),(c,s),(t,b). The chargedweak current then can be written as the sum of six sepacurrents connecting such quark and lepton doublets

Jm5Jmhad1Jm

lep5~cd†cs

†cb†!UKMOmS cu

cc

c t

D1~ce

†cm† ct

†!OmS cne

cnm

cnt

D . ~14!

Low-energy weak interactions are then described by anfective current–current interaction with a single overall copling GF ,

Hw5GF

&Jm

† Jm. ~15!

Such a contact interaction is a good approximation atenergies to a more complete theory described in terms oexchange of a heavy charged W boson, as we mentioearlier. HereUKM is a general unitary 333 matrix, which isnot needed in the case of the lepton current due to thesumption in the standard model that the three neutrinosdegenerate. Consequently Fermi’s weak interaction, inmodern guise, contains an enormous range of physicalcesses. In addition, in 1972 a different kind of weak intertion was found, wherein aneutralcurrent~which is diagonal


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Jm5Jmhad1Jm

lep5~cu†cc

†c t†!Om

~u,c,t !S cm

cc

c t

D1~cd

†cs†cb

†!Om~d,s,b!S cd

cs

cb

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†cm† ct

†!Om~e,m,t!S ce

cm

ct

D1~cne

† cnm

† cnt

† !Om~ne ,nm ,nt!S cne

cnm

cnt

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with

Hw;GF

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† Jm. ~17!

In this case the interaction arises from the exchange oheavy neutral particle—theZ boson with mass mZ

;91 GeV—and can again be taken to be of contact formlow-energy reactions.

Before leaving this historical journey it is useful to remaon one additional feature of the weak interaction importanmodern studies—the handedness. We have noted thaDirac fields c† can both create a particle and destroy tcorresponding antiparticle. Quarks and leptons are also s1/2 objects, coming in two magnetic spin states. Thus iconvenient to ‘‘package’’ four degrees of freedom~spin up/spin down and particle/antiparticle! in four-component Diracfields. The operatorsOm coupling these fields to form thecharged weak current are then 434 matrices. As this stillleaves quite a number of possibilities open forOm , carefulexperiments were done to determine which operatorsscribe nature. The results indicated an equal mixture of pand axial vector structures

ca†Omcb[ca

†g0gm~12g5!cb . ~18!

The Dirac matrix

12g55S 1 21

21 1 D ~19!

is called a ‘‘chirality’’ operator and, for particles of zermass, projects out only ‘‘left-handed’’ particles, i.e., thowhose spins are aligned opposite to their momenta. In oto see this, we introduce the wave equation for a Diparticle/antiparticle of massm propagating in free space

~ i ]” 2m!c~x!50, ~20!

where ]”5]0g01•g. We use the standard representatifor the 434 Dirac matrices12

g05S 1 0

0 21D , g5S 0 s

2s 0 D . ~21!

The positive energy plane wave solutions of Eq.~21! arewell-known,


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c~x!→c0~x!51

& S xs• px Dexp~2 ip•x!, m→0, ~23!

so that

~12g5!c0~x!5H c0~x! s• px52x

0 s• px5x~24!

as claimed. This result is important, as will be discussedSec. III, because the neutrino is either massless orextremelylight. Therefore, since the neutrino interacts only via tweak interactions which involve the chirality operator2g5 , all neutrinos must be left-handed!13 Similarly, it iseasy to see that all antineutrinos must be right-handed.

Another way of stating this result is to say that, althouDirac spinors arefour-component objects, those describinzero mass neutrinos involve only two of the four compnents. This ‘‘two-component neutrino’’ theory has betested in a direct measurement of neutrino helicity inreaction14

152Eu~02!1e2→152Sm* ~12!1ne→152Sm~01!1g1ne .~25!

The clever idea behind this scheme is that one can sethoseg’s from the decay of the Sm excited state which traoppositely to the direction of the electron-capturene’s ~i.e.,in the direction of the nuclear recoil! by having them reso-nantly scatter from a Sm target. By angular momentum cservation, the helicity of the downward-goingg is the sameas that of the upward-travelingne . The results of the experiment convincingly showed that neutrinos emitted in betacay have a definite helicity.

It is interesting to note that the chirality structure of tweak current also explains why the decay of the charpion proceeds predominantly viap1→m11nm rather thanby the modep1→e11ne , which is strongly favored byphase space. The point is that if the positron were massit too would be described by a two-component theory aany such particle coupled to the weak interaction would hto be purely right-handed. Then, as diagrammed in Figthe decay of a pion into a positron and a neutrino mustforbidden because angular momentum conservation prohthe coupling of a right-handed positron and left-handed ntrino to a spinless system. Of course, in the real worldpositron is light but not massless. Thus the positron deca

Fig. 3. Schematic representation of a pion at rest decaying to a maspositron and a neutrino. Such a decay is forbidden by angular momenconservation.


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the charged pion is not completely forbidden but rathhighly suppressed compared to its muonic counterpart,

Re[G~p1→e11ne!

G~p1→m11nm!5S me

mmD 2S mp

2 2me2

mp2 2mm

2 D 2

51.2331024,

~26!

which is confirmed by experiment15

Reexpt5~1.23060.004!31024. ~27!

It is interesting to note one final point about neutrinwhich has a close connection to helicity. The reader whave noticed that we have throughout distinguished thene

produced when a proton beta decays in a nucleus from thneproduced in neutron beta decay. The concept of a distantiparticle is certainly clear for charged leptons like telectron, as its antiparticle—the positron—carries the opsite charge. More generally, particle–antiparticle conjugatreverses the signs ofall of a particle’s additively conservedquantum numbers. The neutrino is immediately seen toquite interesting then, as it lacks a charge, magnetic momor other measured quantum number that would necessreverse under such an operation—it is unique amongleptons and quarks in that the existence of adistinctantipar-ticle is an open question.

Early on, before the handedness of the weak interacwas discovered, there appeared to be a simple test ofparticle–antiparticle properties of a massive neutrino. If odefines thene as the neutrino produced when a proton decin a b1 source, then one finds thatne’s produce electrons bythe reaction

ne1n→p1e2, ~28!

but not positrons in the analogous reaction

ne1pyn1e1. ~29!

Similarly if we define thene as the particle produced in thb2 decay of the neutron decay, thenne’s produce positronsby the reaction

ne1p→n1e1 ~30!

but not electrons by the reaction

ne1nyp1e2. ~31!

Thus it would appear that thene and thene are operationallydistinct. In fact, the absence of the reactions in Eqs.~29! and~31! became apparent around 1950 from an experiment dby nature, a form of natural radioactivity known as doubbeta decay. If, for example, the reaction in Eq.~31! wereallowed, certain nuclei could undergo the second-order wdecay

~A,Z!→~A,Z11!1e21 ne→~A,Z12!12e2, ~32!

where the neutrino produced in the first decay is reabsorby the nucleus, producing a final state with two electrons ano neutrinos. The intermediate nuclear state (A,Z11) ismore massive than the parent nucleus (A,Z), so that ordi-nary first-order beta decay is energetically impossible. Bthe uncertainty principle allows one to violate energy consvation for a short time, allowing the second-order processoccur through the ‘‘virtual’’ intermediate state (A,Z11).The absence of such ‘‘neutrinoless double beta decawhich has a distinctive experimental signal because thetire energy release is carried off by the electrons, th

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seemed to show that thene and ne were indeed distinctparticles.16 This prompted the introduction of a distinguising quantum number, lepton number. Thene and electronwere assignedl e511, the ne and positronl e521. The as-sumption of an additively conserved lepton number in weinteractions then allows the reactions in Eqs.~28! and ~30!,but explains the absence of the reactions in Eqs.~29!, ~31!,and ~32!. A neutrino with a distinct antineutrino is calledDirac neutrino: A four-component field describes tparticle/antiparticle and two helicity degrees of freedom.

However the discovery of the apparent exact handednof the weak interaction invalidates this simple conclusioAll of the results are also explained by the assignments

ne→neLH , ne→ne

RH, ~33!

and a weak interaction that violates parity maximally. HeRH denotes a right-handed particle and LH a left-handone. Thus the possibility that the neutrino is its owantiparticle—a so-called Majorana neutrino—is still opeThe field describing such a neutrino would thus have otwo components, corresponding to the two helicity statesthis case a reaction like that of Eq.~32! is not forbidden byan exact additive conservation law, but rather byhelicity: theright-handed intermediate-state neutrino has the wrong hity to initiate the ne1(A,Z11)→e21(A,Z12) reaction.But, very much as in thep1→e11ne example, if the Ma-jorana neutrino has a small mass, neutrinoless doubledecay would occur, but the decay rate would be suppresby the small quantity

S mn

EnD 2

, ~34!

whereEn;50 MeV is an energy characteristic of the virtuneutrino emitted and reabsorbed in the decay. Modsearches for neutrinoless double beta have established lon half lives of;1025yr, corresponding to a Majorana neutrino mass below 1 eV.17

Given that the familiar charged leptons have only Dirmasses, it is natural to ask why neutrinos, which can htwo kinds of masses, would then be the only masslesstons in the standard model. The absence of Dirac neutmasses in the standard model follows from the need to hboth left-handed and right-handed fields in order to constsuch masses. We have noted that neutrinos interactweakly and that weak interactions involve only left-handcomponents of the fields. The standard model, being veconomical, has no right-handed neutrino fields and thusDirac neutrino masses. However, the absence of Majormasses has a more subtle explanation. One can constrleft-handed Majorana mass with the available standmodel neutrino fields, but it turns out this term is not ‘‘renomalizable,’’ i.e., it generates infinities in the theory. Opoint-like Fermib decay theory is another example of a norenormalizable theory, though it works quite well in the dmain of low-energy weak interactions. If we were to relegthe standard electroweak model to a similar status—thaan effective theory—Majorana mass terms could then betroduced. In effect, most extensions of the standard modeprecisely that, and also generally introduce new fields sas those creating right-handed neutrinos. Thus almostheorists, believing the standard model is incompletemust be extended in some such ways, also believe that


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alldu-

trinos have masses. Indeed, the puzzle is rather to expwhy these masses are so much smaller than those of chaparticles.

With this historical background out of the way, we nomove to consider aspects of the neutrino that have recebeen in the news—masses and mixings.

III. NEUTRINO MASS: DIRECT MEASUREMENTS

The issue of whether the neutrino has a nonzero masslong been one of interest. That any such mass must be scould be seen from the feature that the maximum enemeasured in the beta spectrum agreed to high precisionthe mass difference of initial and final nuclear states. Hoever, Fermi, in his seminal paper on beta decay, notedthis question could be answered more definitively by cafully studying the end point of the electron spectrum—itpossible to plot the spectrum in such a way that a nonzmass would be revealed as a distortion at the end pointgent to~perpendicular to! the energy axis, as shown in Fig. 4In the years since Fermi’s paper there has been a seriesuch measurements, with steadily increasing precision.clear that use of ab decay parent nucleus with a relativelow Q value is helpful, as a larger fraction of the total decathen resides within a given interval from the end point. Moexperimenters have selected tritium, which has an 18.6-end point. An early tritium measurement by Hamilton, Aford, and Gross found an upper limitm( ne)&250 keV.18 Afew years later, Bergkvist, by combining electrostatic amagnetic spectrometric methods, was able to reduce thesubstantially—m( ne)&60 eV.19 Then in 1980 Lubimovet al., using a high-precision toroidal spectrometer andtium in the form of the valine molecule (C5H11NO2!,claimed the first nonzero mass—14 eV&m( ne)&46 eV,20 aresult that set off a flurry of new, high precision experimenBefore discussing the results, however, we first examineof the reasons these experiments are important—the coslogical significance of a massive neutrino.

We are all aware that at the present time the universexpanding. However, what will be its ultimate fate? Willcontinue to expand forever, or will the expansion slow afinally reverse? In order to see what role neutrinos may pin answering this question, we explore their effects inexpanding homogeneous and isotropic universe. Considsmall test massm which sits on the surface of a sphericchunk of this universe having radiusR. If the mean energydensity of the universe isr, then the mass contained insidthe spherical volume is

Fig. 4. A reproduction of curves from Fermi’s paper~Ref. 5! showing theshape of theb spectrum near the end point for the cases of zero and nonneutrino masses.


er

he

uma

semm

as

p-biigoue

ahov

et

nsle

erath-

ereturethe

tem-a-

iven

theeryns

in-

s

nisni-

neu-

ent:

rgyt re-ingo the

nlyablysseu-

ibu-of

M ~R!5 43pR3r. ~35!

The potential energy of the test mass, as seen by an obsat the center of the sphere, is

U52GM ~R!m

R, ~36!

while its kinetic energy is

T51

2mv25

1

2mS dR

dt D2

. ~37!

By Hubble’s law the expansion velocity is given by

v5HR, ~38!

where H5(1/R)(dR/dt) is the Hubble constant. Althougthe size ofH is still uncertain, most values are in the rang

50 km/s/Mpc&Hexpt&100 km/s/Mpc ~39!

~with perhaps;65 km/s/Mpc being the best value!. The totalenergy of the test particle is then

Etot5T1U5 12mR2~H22 8

3prG! ~40!

and the fate of the universe depends on the sign of this nber, or equivalently on the relation of the density to a criticvalue

rc53H2

8pG;2310229 g/cm3, ~41!

i.e.,

r,rc⇒continued expansion,~42!r.rc⇒ultimate contraction.

It is not yet clear which situation describes our univerAnalysis of the dynamics of gravitationally bound systevia the virial theorem and comparison with measured lunosities yields the ‘‘visible’’ mass density

rvis.0.02rc. ~43!

However, Doppler studies of the rotation rates of spiral gaxies indicate that these systems are much more masthan their luminosities seem to suggest

r rot;20rvis . ~44!

The origin of the ‘‘dark matter’’ responsible for this discreancy is a matter of current study: There are several possities, including massive neutrinos. But regardless of the orof the dark matter, it appears that the energy density ofuniverse is within an order of magnitude of the critical valrequired for closure.

Before discussing the relevance of neutrinos to the dmatter, we consider the more familiar case of the relic pton spectrum, which has been carefully studied recentlythe COBE satellite. In the early universe, electromagninteractions such as

e11e2↔g1g, g1e2↔g1e2 ~45!

kept the photons in thermal equilibrium with the electroand nucleons. However once the universe coosufficiently—toT;1 eV at about 105 yr after the big bang—electrons and protons combined to form neutral hydrogThe absence of charged particles rendered the universe tparent to photons, which then thermally decoupled fromrest of the matter. At the time of ‘‘recombination’’—the ep


ver

-l

.si-

l-ive

li-inur

rk-

iaic

d

n.ns-e

och when neutral hydrogen is formed—the photons wcharacterized by a blackbody spectrum at a temperaTg

recomb;4000 K. Because of the subsequent expansion ofuniverse, the spectrum today has been redshifted to aperatureTg

now;2.72 K, as was established in the COBE mesurements. The energy density of these relic photons is gby Stefan’s law,

rg52E d3q

~2p!3

q

exp~q/T!2154sTg

4;4310234 g/cm3,

~46!

wheres5p2/60 is the Stefan–Boltzmann constant.An analogous relic neutrino spectrum remains from

big bang. At the high temperatures characteristic of the vearly universe, neutrinos interact frequently with electroand other matter through weak interactions such as

e11e2↔nx1 nx , e61nx↔e61nx , e61 nx↔e61 nx~47!

~where nx represents ane , nm , or nt!. They also interactwith each other via reactions like

ne1 ne↔nm1 nm . ~48!

Thus the neutrino spectrum, which contains all flavors, isthermal equilibrium with other matter at this epoch. However, once the temperature cools to about 1010K the weakreaction rates—which depend onsweaknv, where n is thelepton density andv the relative velocity, and vary aT5—can no longer keep up with the expansion rate,H;R/R;T2, which slows less rapidly. The neutrinos thedrop out of equilibrium with the charged leptons. From thpoint they are essentially decoupled from the rest of the uverse, but, of course, cool as expansion proceeds. Thetrino energy density at the present time is given by

rn5 72NnsTn

4, ~49!

where Nn53 is the number of ‘‘massless’’ (mn,T) two-component neutrino generations.21 Note, however, that thepresent neutrino and photon temperatures are differWhile above;1010K the reactione11e2↔g1g proceedsin both directions, below this temperature the photon eneis no longer sufficient to produce pairs. Thus subsequenactions proceed only to the right, resulting in a reheatprocess that raises the photon temperature with respect tneutrinos by the factor

S rg1re21re1

rgD 1/3

5S 11

4 D 1/3

. ~50!

The present-day relic neutrino temperature is thus

Tn5TgS 4

11D1/3

51.94 K. ~51!

The corresponding average neutrino energy is o;1023 eV, so that scattering cross sections are unobservsmall. But such relic neutrinos do contribute to the madensity of the universe, yielding in the case of massless ntrinos

rn5rg78Nn~ 4

11!4/3.0.7rg . ~52!

If, however, neutrinos have a nonzero mass, then a contrtion comparable to the critical density results if the sumthe masses for the three neutrino species is as little as;25


inelhinnoto

atthritceibcasouaten

i-

ag

eh

dlehs-

al

ditsr-itiin

cin7Aa

heu

icu

too

ndininrachorsureyathhie

,

dern-

ughSi

icno

heace,res-core

ginsd-lti--is

ree

dis-thee isont

tterrentcleiin-elyrgy

eaterise

eanofpseut

le.lin

eV. Thus neutrino masses easily compatible with existexperimental limits could close the universe and ultimatlead it to recollapse. In particular, since this number is witthe range found in Ref. 20 for the electron antineutrialone, confirmation of that experiment was clearly crucialcosmology.

Stimulated by the Lubimov result, several groupstempted improved versions of this experiment. One ofcriticisms of the Russian experiment was the use of a tated valine source, which introduced a substantial untainty because of binding effects and because the contrtion of molecular excited states, populated in the beta deto the energy loss could not be calculated easily. Thugroup at Los Alamos used a much simpler source, gasetritium molecules, tackling at the same time the seriosafety issues associated with handling a kilocurie of this gAfter a series of measurements with a carefully construcmagnetic spectrometer that filled an entire room, they foumne

&9.3 eV.22 An experiment at Mainz using a frozen tr

tium source reported a similar limit,mne&7.2 eV,23 and a

Livermore group using gaseous tritium and a toroidal mnetic spectrometer achieved comparable statistics.24 All ofthese experiments, however, were troubled by a puzzlingcess of events near the end point. Although this problembeen described as anegativevalue for mne

2 (!), in fact this

was too simple a characterization: Each of the groupsscribed the anomaly with a different functional form. Whistandard statistical techniques were then used to establisabove bounds onmne

2 , it is apparent that an unknown sy

tematic contributing excess events at the end point couldmask the effects of a positivemne

2 . This has tended to

weaken the community’s confidence in the stated bounOngoing experiments at Mainz and Troitsk now claim limof ;~3–5! eV and, while improved resolution and undestanding of energy loss in the target have significantly mgated the negativemne

2 problem, certain anomalies remain

the end-point region.Because of this situation, it was fortunate that a spe

event occurred that established an independent bound one mass. This was the observation of Supernova 198which was found in the southern hemisphere on Febru23rd of that year—cf. Fig. 5. SN1987A resulted from texplosion of a star in the Large Magellanic Cloud abo170 000 years ago, the light~and neutrinos! from which fi-nally reached earth in 1987. This was the first such optsupernova in our vicinity in nearly 400 years, the previooccurrence having been noted by Kepler in 1604!

A brief discussion of stellar evolution is needed in ordermake the connection between SN1987A and neutrinSmall stars like our sun live relatively quiescent lives, speing billions of years slowly burning hydrogen to heliumtheir hot dense cores, with the liberated energy maintainthe electron gas pressure that stabilizes the star against gtational collapse. A Type II supernova—the type to whiSN1987A belongs—is the last evolutionary stage of a mmassive star, in excess of 10 solar masses. Like our sun,a star begins its lifetime burning the hydrogen in its counder conditions of hydrostatic equilibrium. When the hdrogen is exhausted, the core contracts until the densitytemperature are reached where helium can ignite via3a→12C reaction. The He is then burned to exhaustion. Tpattern ~fuel exhaustion, contraction, and ignition of th


gy

-ei-r-u-y,ausss.dd

-

x-as

e-

the

so

s.

-

althe,

ry

t

als

s.-

gvi-

ech

-nde

s

ashes of the previous burning cycle! repeats several timesleading finally to the explosive burning of28Si to Fe. For aheavy star, the evolution is rapid: The star has to work harto maintain itself against its own gravity, and therefore cosumes its fuel faster. A 25 solar mass star would go throall of these cycles in about 7 My, with the final explosiveburning stage taking only a few days!

Iron is the most strongly bound nucleus in the periodtable. Thus once the Si burns to produce Fe, there isfurther source of nuclear energy with which to support tstar. So, as the last remnants of nuclear burning take plthe core is largely supported by the electron degeneracy psure. When enough ash accumulates so that the ironexceeds the Chandrasekhar limit25—a limit of about 1.4 solarmasses above which it is no longer stable—the core beto collapse. Gravity does work on the infalling matter, leaing to rapid heating and compression of the iron, and umately ‘‘boiling off’’ a’s and a few nucleons from the nuclei. At the same time, the electron chemical potentialincreasing, making electron capture on nuclei and any fprotons favorable,

e21p→ne1n. ~53!

Both the electron capture and the nuclear excitation andassociation take energy out of the electron gas, which isstar’s only source of support. This means that the collapsvery rapid. Indeed, numerical simulations find that the ircore of the star~;1.2–1.5 solar masses! collapses at abou0.6 of the free fall velocity.

In the early stages of the infall thene’s readily escape. Butneutrinos become trapped when a density of;1012g/cm3 isreached, at which point they begin to scatter off the mathrough both charged current and coherent neutral curprocesses. The neutral current neutrino scattering off nuis particularly important, as the scattering cross sectionvolves the total nuclear weak charge, which is approximatthe neutron number. This process transfers very little enebecause the mass energy of the nucleus is so much grthan the typical energy of the neutrinos. But momentumexchanged. Thus the neutrino ‘‘random walks’’ out of thstar, frequently changing directions. When the neutrino mfree path becomes sufficiently short, the ‘‘trapping time’’the neutrino begins to exceed the time scale for the collato be completed. This occurs at a density of abo

Fig. 5. Views of a region in the Large Magellanic Cloud before~right! andafter ~left! the morning of February 23, 1987: SN1987A is clearly visibCopyright Anglo-Australian Observatory. Photograph by David Ma~http://www.aao.gov.au/local/www/dfm/aat050.html!.


itynatath

he

ltse

athet

eso

an

enothroge

usum

tho

cotoo

uc

adin

thesfudfthnh

inr

anva

icand

d

me

tsofbleer-

entsunds

in-orth

inopond-tlyun-

hed

ex-ure-ntsto

yryoesdtohasow

t’’der-t ininoes,ced

ofonpfulay

lar-a-

1012g/cm3, or somewhat less than 1% of nuclear densAfter this point, the energy released by further gravitatiocollapse is trapped within the star. If we take a neutron sof 1.4 solar masses and a radius of 10 km, an estimate ofgravitational energy is

GM2

2R;2.531053 ergs. ~54!

The collapse continues until nuclear densities are reacAs nuclear matter is rather incompressible~compressionmodulus;300 MeV!, the nuclear equation of state then hathe collapse: maximum densities of 3–4 times nuclear dsity are reached, e.g., perhaps 631014g/cm3. This suddenbraking of the collapse generates a series of pressure wwhich travel outward through the iron core and collect atsonic point~where the infall velocity of the iron matches thsound speed in iron!, eventually forming a shock wave thatravels out through the mantle of the star. Theory suggthat this shock wave in combination with neutrino heatingthe matter is responsible for the ejection of the mantle—thus the spectacular optical display we saw in 1987.

Yet this visible display represents less than 1% of theergy locked inside the protoneutron star. After the core clapse and shock wave formation are completed—eventstake on the order of tens of milliseconds—the protoneutstar cools by emitting a largely invisible radiation, a hufluence of neutrinos of all flavors.~As in the big bang, thereis an approximate equipartition of energy per flavor becathe neutrinos trapped in the core are in weak equilibriwith the surrounding matter and each other.! Essentially allof the 331053ergs released in the collapse are carried byelectron, muon, and tauon neutrinos that leak out of the cover the next 10 s.

In 1987 there existed deep underground two detectorssisting of tanks of ultrapure water instrumented with photubes. They were originally constructed for the purposedetecting proton decay, e.g.,p→e11g, a process predictedby certain grand unified theories of elementary particle strture, but not yet observed. Theg ray and Cerenkov lightgenerated by the passage of the positron through the wwould be detected by the phototubes, signaling a protoncay event. One detector was located in a Morton salt moutside Cleveland, OH, while the second was on the oside of the globe in the Kamioka mine within the Japanalps. While the search for a proton decay signal provedtile, on February 23, 1987 both detectors recorded a hanof events associated with the passage of a strong bursantineutrinos through the tanks, a few of which initiated treaction ne1p→e11n. Within several hours a Canadiaastronomer observing in Chile noted a new bright star. Twas SN1987A.

How do these supernova neutrino events limit the neutrmass? If a neutrino has mass, neutrinos of different enewill travel with different velocities according to

v5pn

Apn21mn

2.12

mn2

2En2 , ~55!

rather than with the uniform velocityc. Consequently thetime of arrival of higher energy neutrinos will be earlier ththeir lower energy counterparts. The difference in arritimes is then


.lris

d.

n-

vese

tsfd

-l-atn

e

ere

n--f

-

tere-e

ere-ulof

e

is

ogy

l

dt

t;

dvv

;mn

2

En2

dEn

En, ~56!

wheret is the total time in transit from the Large MagellanCloud. The 11 events seen by the Kamiokande detectorthe 8 seen in Ohio have an energy spreaddEn;10 MeV andan average energyEn 2–3 times larger. The events arriveover a 10-s time span. Thus we find in a simple analysis

mne&EnS dt

t

En

dEnD 1/2

;10 MeVS 10 s

1013sD1/2

;10 eV. ~57!

A more careful analysis, which takes into account the tiand energy distribution, yields a similar limit—mne

&20 eV. It is remarkable that a handful of neutrino evenfrom a star that lived and died long before the adventcivilization places a bound on the neutrino mass comparato that gained from many years of high precision weak intaction studies!

We have focused on electron neutrino mass measurembecause these have achieved the greatest precision. Bohave also been established on the masses of thenm andnt ,

mnm&170 keV, mnt

&24 MeV, ~58!

from careful kinematic analyses of decays such asp1

→m11nm and t→5p1nt . The large energy releasesdecays of thep and t and limited energy resolution of detectors result in less precise neutrino mass bounds. It is wnoting that some theories of neutrino mass predict that thentwill be the heaviest neutrino, e.g., in some models neutrmasses scale as the squares of the masses of the corresing charged leptons. From this perspective the limits direcabove are not necessarily less significant as tests of thederlying particle physics than the tighter bounds establison thene .

This discussion can be succinctly summarized: Thereists no evidence for massive neutrinos from direct measments. Naturally, when increasingly precise experimecontinue to yield values consistent with zero, it is temptingassume that the number reallyis zero. Indeed, as previouslmentioned in the so-called ‘‘standard model’’ of elementaparticles, the neutrino mass is assumed to vanish. This dnot follow from any fundamental principle, however, ancould be modified accordingly if the evidence werechange, as we have noted before. That is precisely whatbeen happening over the last several years, as we will ndescribe.

IV. PROBING MASSES THROUGH NEUTRINOMIXING

Significant improvements in the above-described ‘‘direcneutrino mass measurements will probably require consiable time and effort. For this reason there is great interesmeasurements of a different type—those exploiting neutrmixing—which might be able to probe far smaller massalbeit in a less direct fashion. The essential idea can be trato the seminal paper of Pontecorvo,26 who first pointed outthat neutrino oscillations would occur if the neutrino statesdefinite mass do not coincide with the weak interactieigenstates. To understand neutrino oscillations, it is helto first consider the more familiar phenomenon of Faradrotation, the rotation around the beam direction of the poization vector of linearly polarized light as the light prop


cula-on

tiangb

rio

nuoYribata

mel

n

tnc

-

o

nde-

less,e-

eri-

ro-ntsrac-intoe’’uxa

tionni-

uld

ear-lla-os

’’ous

rser to

gates through a magnetized material. Faraday rotation ocbecause the index of refraction~and thereby the potentiaacting on the light! depends on the state of circular polariztion, i.e., the two states of definite circular polarizati

(ux6&5A12(u1x&6 i u1y&) evolve in time with distinct

phases. Thus ifux(t50)&5u1x&5A12(ux1&1ux2&), then

ux~ t !&5A1

2S ux1&expS 2 iv0

n1t D

1ux2&expS 2 iv0

n2t D D

5expS 2 iv0

2

n11n2

n1n2t D

3S u1x&cosS v0

2

n12n2

n1n2t D

2u1y&sinS v0

2

n12n2

n1n2t D D , ~59!

so that the polarization vector rotates with frequency

vpol5v0

2

n12n2

n1n2. ~60!

Although we have three neutrino families, the essenphysics of oscillations is illustrated very well by considerithe interactions of only two of these, which we choose tothe ne and nm . In this limit the situation is similar to theabove-described two-state Faraday rotation problem. Theevant part of the weak current involves the combinatene1mnm and is given by Eq.~13!. This effectively definesthene andnm : they are the neutrino weak interaction eigestates, the neutrinos accompanying the electron and mrespectively, when these particles are weakly produced.there is a second Hamiltonian, the free Hamiltonian descing the propagation of an isolated neutrino. The eigenstof this Hamiltonian are the mass eigenstates. If the two meigenstates are distinct~and thus at least one is nonzero!,then in general the eigenstates diagonalizing the mass Hatonian will not diagonalize the weak interaction. If we labthe mass eigenstates asun1& and un2&, then

un1&5cosuune&1sinuunm& with mass m1 ,~61!un2&52sinuune&1cosuunm& with mass m2 ,

whereu is a mixing angle which distinguishes the mass aweak eigenstates. Also suppose that at timet50 an electronneutrino is produced with fixed momentump,

uc~ t50!&5une&5cosuun1&2sinuun2&. ~62!

The mass eigenstates propagate with simple phases, asare the eigenstates of the free Hamiltonian. At a dista;ct from the source the neutrino state is

uc~ t !&5cosuun1&e2 iE1t2sinuun2&e

2 iE2t, ~63!

where Ei5Ap21mi2. Projecting back upon weak eigen

states, we have

^neuc~ t !&5cos2 ue2 iE1t1sin2 ue2 iE2t,

^nmuc~ t !&5cosu sinu~e2 iE1t2e2 iE2t!.~64!


rs

l

e

el-n

-n,et-

esss

il-

d

heye

Then, noting that

E1;p1m1

21m22

4p2

dm2

4p,

~65!

E2;p1m1

21m22

4p1

dm2

4p,

wheredm25m222m1

2, we find att.0 a probability

p~ t !5u^nmuc~ t !&u25sin2 2u sin2dm2t

4p~66!

that thene will have transformed into anm . This change ofneutrino identity is called a ‘‘neutrino oscillation,’’ due tthe time- or distance-dependent oscillation inp(t). This phe-nomenon is a sensitive test of neutrino masses given nogenerate neutrinos and a nonzero mixing angleu. As thestandard electroweak model describes neutrinos as massthe observation of neutrino oscillations would constitute dfinitive evidence of physics beyond the standard model.

Three sources of neutrinos have been exploited by expmentalists seeking~and apparently finding!! neutrino oscilla-tions.

A. Accelerator and reactor experiments

Such experiments exploit an accelerator or reactor to pduce a large flux of neutrinos/antineutrinos. Measuremeare then done downstream to determine whether the chater of the neutrinos has changed. Such experiments fallone of two categories. In the first—called ‘‘disappearancexperiments—one looks for deviations in the expected flof neutrinos of a definite type. For example, the flux fromreactor is overwhelmingly of thene type. Thus oscillationsinto a second flavor would lead to an unexpected reducin the flux some distance downstream, which might be motored through reactions such asne1p→e11n. Even if thereactor flux were poorly understood, a definitive signal coresult from detectors placed at different distancesL1 andL2from the reactor, as shown in Fig. 6~a!. In the absence ofoscillations the flux must fall off as 1/L2 ~apart from correc-tions due to the finite size of the source!. However, oscilla-tions superimpose an additional factor of 12p(t), leading toa modulation that clearly signals the ‘‘new physics.’’ Thsecond type of oscillation experiment, called an ‘‘appeance’’ experiment, is a search for the product of the oscition. For example, at an accelerator like the Los AlamMeson Physics Facility~LAMPF!, proton interactions in the

Fig. 6. ~a! A neutrino oscillation experiment of the ‘‘disappearancetype—a flux of reactor antineutrinos is monitored by detectors at varidistances from the source—in which deviations from a simple 1/L2 falloffare sought;~b! A neutrino oscillation of the ‘‘appearance’’ type: detectoare placed at various distances from a beam stop neutrino source in orddetect the appearance of an oscillatoryne component.


dapa

er

p,-

--on

cee

s

in

ot t

eiveed00n-nv

r-ine

th

ino

f

he

cil-s a

ichas.

o-ost

di-riesintoan-

arth.cedauseted

ablySu-thatne

inosc-eseCer-

tral

encenate

t isos-

beam stop produce large numbers ofnm’s, nm’s, andne’s,but very few ne’s: The reaction chains involvep1→m1

1nm followed by m1→e11ne1 nm , or p2→m21 nm fol-lowed bym21(A,Z)→(A,Z21)1nm . It is the importanceof this last reaction that prevents them2 from undergoing thefree decaym2→e21 ne1nm—negative muons are sloweby their interactions with nuclei in the beam stop and ctured into atomic orbitals, then rapidly cascade electromnetically into the 1s orbital. There them2 wave function hasa strong overlap with the nucleus~especially for the highZnuclei in the beam stop!, allowing them2 capture to proceedmuch more rapidly than the free decay. Thus there are vfew electron-type antineutrinos27 and a sensitive search fothe flavor oscillationnm→ ne can be done by looking for theappearance of a flux ofne’s downstream from the beam stoas shown in Fig. 6~b!. Such neutrinos can be efficiently detected via the reactionne1p→n1e1. The rate should depend on 1/L2 modulated byp(t). The results of such experiments are generally presented in terms of a two-dimensidiagram withdm2 plotted against sin2 2u. Of course, if nosignal is observed in such an experiment it does not nesarily imply that no oscillations are occurring. It could bthat the mixing angleu is very tiny @and thusp(t) too smallto be detected#, or that dm2 is too small ~so that effectsappear only at distancesL so large that the neutrino flux hafallen below detectable limits!. In order to quantify this as-sertion note that we can write the oscillation probabilityterms of path length

p~ t !.sin2 2u sin2dm2L

4En, ~67!

which shows that the maximum effect occurs for

dm2L

4En5

p

2⇒L5

2pEn

dm2 . ~68!

Thus a smallerdm2 implies a longer oscillation length, sthat a more intense neutrino source is required to combaassociated 1/L2 fall off in the flux. Experiments have ruledout a large portion of the sin2 2u2dm2 space, e.g., sin2 2u*0.01 anddm2*1.0 eV2, as shown in Fig. 7. However onexperiment—LSND at Los Alamos—has claimed a positsignal for neutrino oscillations. This experiment, situatdownstream from the LAMPF beam stop, uses a 52 0gallon tank of mineral oil and a small amount of liquid scitillator instrumented with 1220 phototubes. A neutrino evene1p→n1e1 is indicated by a combination of Cerenkoand scintillator light produced by the positron followed~aftera couple of hundred microseconds! by a 2.2-MeV gammaray from the capture of the produced neutron,n1p→d1g. Only 0.000 000 000 1% of the LAMPF neutrinos inteact in the tank. Thus the challenge is to distinguish this tsignal from the 33108 cosmic rays which pass through thtank each day. After very careful numerical simulations,LSND collaboration announced they had detected 22neevents compared to an anticipated background of 4.660.6events.28 This excess of events is consistent with neutroscillations for the values ofdm2 and sin2 2u shown in Fig.7: the narrow allowed region includes the rangesdm2

;0.2– 2 eV2 and sin2 2u;0.03– 0.003. Since publication othe first paper, the collaboration’s data set has grown to;60events. Concurrently, the competing KARMEN group of t


-g-

ry

al

s-

he

-

t

y

e

Rutherford Laboratory in England has found no such oslation signal, though the sensitivity achieved to date leavesubstantial portion of the LSND alloweddm22sin2 2u re-gion untested. An improved experiment at Fermilab, whshould have thousands of events if LSND is correct, hrecently been approved and should yield results by 2002

B. Atmospheric neutrinos

When high energy cosmic rays strike the earth’s atmsphere a multitude of secondary particles is produced, mof which travel at nearly the speed of light in the samerection as the incident cosmic ray. Many of the secondaare pions and kaons, which decay as described aboveelectrons, muons, and electron and muon neutrinos andtineutrinos. These neutrinos reach and pass through the eThe fluxes are large: about 100 such cosmic ray-induneutrinos pass through each of us every second. Yet becthese particles react weakly, only one interaction is expecper human body every thousand years! Thus a considerlarger target is required for a reasonable event rate. InperKamiokande, the massive 50 000-ton water detectorreplaced the original 3000-ton Kamiokande detector, oevent occurs every 90 min. The energies of these neutr~typically 1 GeV! are sufficiently high to produce either eletrons or muons, depending on the neutrino flavor. As thcharged particles pass through the water, they produce

Fig. 7. Regions ofdm2 and sin2 2u ruled out by the KARMEN, Bugey, andCCFR null experiments fornm( nm)↔ne( ne) oscillations are shown alongwith the allowed region corresponding to the LSND result. The censhaded region is the preferred~90% confidence level! LSND solution; theentire shaded region is possible if exclusions are done at 99% confidlevel. Thus the portion of the LSND region below the KARMEN exclusioregion and to the left of the Bugey exclusion region gives the candidoscillation parameters. An exclusion region for the NOMAD experimenalso shown, though this experiment is primarily sensitive to a differentcillation channel, the appearance ofnt from nm→nt oscillations. This figurewas provided by Bill Louis.


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enkov radiation. However, the Cerenkov ring produced byenergetic electron is more diffuse than the relatively clering of a muon. This allows the experimenters to distinguelectrons from muons with about 98% accuracy. Sincecharged lepton tends to travel in the same direction asincident neutrino, the experimenters can thus deduce bothflavor and the direction of neutrinos that react in the wat

A decade ago it was already apparent that atmosphneutrino rates seen in existing detectors were anomalousing known cross sections and decay rates, theorists haddicted about twice as many muon neutrinos as electron ntrinos from cosmic ray events. For example, we mentionpreviously that ap1 decays into am1 and anm . The muonsubsequently decays to a positron plus ane andnm . Thus thenet result is ane1, ane , anm , and anm . That is, two muonneutrinos are produced, but only onene . However, most ofthe early atmospheric neutrino experiments foundelectron-to-muon ratio from neutrino reactions to be appromately unity. The very precise measurements made withperKamiokande appear to show that the ratio has this unpected value because of a deficit in muon-like events—electron event rate is about as expected. The muon deficia strong zenith angle dependence, with the largest suppsion associated with atmospheric neutrinos coming fromlow, e.g., originating on the opposite side of the earth. Sa dependence of the muon-to-electron ratio on distancesignature of neutrino oscillations, as we have noted. Tmost plausible interpretation of the SuperKamiokande da29

is that atmosphericnm’s are oscillating intont’s, which arenot observed because thent’s are too low in energy to producet’s in SuperKamiokande. The strong suppression innm flux is characteristic of maximal mixing~u;p/4!, whilethe zenith angle dependence indicates that the oscillalength is comparable to the earth’s diameter. The cospondingdm2 is ;231023 eV2. Thus this mass differencsuggests that at least one neutrino must have a mass*0.05eV. The quality of the SuperKamiokande data—the statical error on the muon-to-electron event rate is well bel10% and there is remarkable consistency between thegiga-electron-volt and multi-giga-electron-volt data sets abetween the fully and partially contained data sets—provia powerful argument that oscillations have been observBecause the zenith-angle dependence shows that thenm fluxdepends on distance, the atmospheric data provide dproof of oscillations. Thus this may be our strongest edence for massive neutrinos and for the incompletenesthe standard model.

There is another remarkable aspect of the atmosphneutrino results. If neutrinos are massive, there must be sreason that their masses are so much lighter than thosethe more familiar quarks and leptons. In fact, a lovely expnation is provided in many proposed extensions of the sdard model that again returns to the idea that neutrinosspecial because they can have both Dirac and Majormasses. The explanation is called the seesaw mechanism30 Itpredicts that the neutrino mass is

mn5mDS mD

mRD , ~69!

wheremD is a Dirac mass often equated to the mass ofcorresponding quark or charged lepton, whilemR is a veryheavy Majorana mass associated with interactions at enefar above the reach of existing accelerators. This see


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mechanism arises naturally in models with both Dirac aMajorana masses. In the case of thent we concluded fromthe atmospheric neutrino data that its mass might be; 0.05eV. A reasonable choice formD is the mass of the corresponding third generation quark, the top quark,mD

;200 GeV. It follows thatmR;1014GeV! Thus tiny neu-trino masses might be our window on physics at enormenergy scales. This large massmR is interesting becausethere is an independent argument, based on observationsthe weak, electromagnetic, and strong interactions wouldhave approximately the same strength at;1016GeV, thatsuggests a very similar value for the ‘‘grand unificatioscale.’’ This has led many in the community to hope thatpattern of neutrino masses now being discovered may hus probe the structure of the theory that lies beyond the sdard model.31

C. Solar neutrinos

The thermonuclear reactions occurring in its core mathe sun a marvelous source of neutrinos of a single flavne . The standard solar model—really, the standard modemain sequence stellar evolution—allows us to predictflux and spectrum of these neutrinos. The standard smodel makes four basic assumptions:

~1! The sun evolves in hydrostatic equilibrium, maintaiing a local balance between the gravitational force andpressure gradient. To describe this condition, one mspecify the equation of state as a function of temperatudensity, and composition.

~2! Energy is transported by radiation and convectioWhile the solar envelope is convective, radiative transpdominates in the core region where thermonuclear reacttake place. The opacity depends sensitively on the solar cposition, particularly the abundances of heavier elements

~3! Solar energy is produced by thermonuclear reactchains in which four protons are converted to4He,

4p→4He12e112ne . ~70!

The standard solar model predicts that 98% of these retions occur through thepp chain illustrated in Fig. 8, with theCNO cycle accounting for the remainder. The sun is a slreactor, characterized by a relatively low core temperatTc;1.53107 K. Thus Coulomb barriers tend to suppress t

Fig. 8. The three cycles comprising thepp chain.


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rates of reactions involving higherZ nuclei, an effect we willsee reflected in the neutrino fluxes given in the following

~4! The model is constrained to produce today’s solardius, mass, and luminosity. An important assumption ofstandard solar model is that the sun was highly convectand therefore uniform in composition, when it first enterthe main sequence. It is furthermore assumed that the suabundances of metals~nuclei heavier than He! were undis-turbed by the sun’s subsequent evolution, and thus provirecord of the initial core metallicity. The remaining parameter is the initial4He/H ratio, which is adjusted until themodel reproduces the known solar luminosity at the supresent age, 4.6 billion years.

Figure 8 shows that thepp chain is comprised of threedistinct cycles, each of which is tagged by a distinctive ntrino. The total rate ofppI1ppII1ppIII burning is gov-erned by the rate at which protons are consumed

p1p→2H1e11ne , ~71!

a reaction which produces an allowedb decay spectrum@thatis, a spectrum like that of Eq.~5!# of low-energy~0.42-MeVend point! ne’s. The ppII rate is tagged by the distinctiveneutrino lines from electron capture on7Be ~0.86 and 0.36MeV!. Finally theppIII cycle is tagged by the high-energneutrinos from theb decay of8B ~;15-MeV end point!. Thecompetition between these three cycles depends sensiton the solar core temperatureTc . Thus the original motiva-tion for measuring solar neutrinos was to determine the rtive rates of theppI, ppII, and ppIII cycles, from which thecore temperature could be deduced to an accuracy of apercent, thereby checking the standard solar model.

The neutrino flux predictions of the standard solar moare summarized in the following:32

Reaction Enmax Flux (1010/cm2/s)

p1p→2H1e11ne 0.42 5.947Be1e2→7Li1ne 0.86~90%! 4.8031021 ~72!

0.36~10%!8B→8Be* 1e11ne 14.06 5.1531024

The first experiment to test these predictions began mthan three decades ago with a detector placed a mile unground in the Homestake Gold Mine in Lead, SD. This grdepth protects the detector from all forms of cosmic radiatother than neutrinos. Ray Davis, Jr. and his collaborato33

filled the detector with 615 tons of the cleaning fluid perchroethylene (C2Cl4) in order to make use of the reaction

37Cl~ne ,e2!37Ar. ~73!

The few atoms of the noble gas37Ar produced in the tankafter a typical~;two month! exposure could be recoverequantitatively by a helium purge, then counted via the ssequent electron capture reactione2137Ar→37Cl1ne ,which has a 35-day half life. The miniscule37Ar productionrate, less than an atom every two days, has been measuran accuracy of better than 10% by patient effort. Theduced solar neutrino capture rate, 2.5660.1660.16 SNU(1 SNU510236captures/atom/s) is about 1/3 of the standsolar model prediction. As this reaction is primarily sensitito 8B ~78%! and7Be ~15%! neutrinos, one concludes that thsun is producing fewer high energy neutrinos than expec


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Similar radiochemical experiments were done by tSAGE and GALLEX collaborations34 using a different tar-get, one containing71Ga. The special properties of thinucleus include an unusually low threshold for the react71Ga(ne ,e2)71Ge, leading to a large cross section for tcapture of low-energypp neutrinos. The resulting experimental capture rates are 6661366 and 7668 SNU for theSAGE and GALLEX detectors, respectively, which cancompared to the standard solar model prediction of; 130SNU. Most important, as thepp flux is directly constrainedby the rate of hydrogen burning and thus by the obsersolar luminosity in all steady-state solar models, there iminimum theoretical value for the capture rate of 79 SNgiven standard model weak interaction physics. With tassumption, it appears that there is virtually no contributfrom ppII and ppIII cycle neutrinos.

The remaining experiments, Kamiokande II/III and the ogoing SuperKamiokande,35 exploit water Cerenkov detectorto view solar neutrinos on an event-by-event basis. Jusdescribed in our atmospheric neutrino discussion, the scaing of high energy8B neutrinos produces recoil electrons athus Cerenkov radiation that can be recorded in the surrouing phototubes. The correlation of the electron direction wthe position of the sun is crucial in separating solar neutrevents from background. After 504 days of operationsrate measured by SuperKamiokande, which under curoperating conditions is sensitive to neutrinos with energabove;6 MeV, is consistent with an8B neutrino flux of(2.4460.0520.07

10.09)3106/cm2 s. This is about half of the standard solar model prediction.

If one combines the various experimental results andsumes that the neutrino spectra are not being distortedoscillations or other new physics, the following patternfluxes emerges:

f~pp!;0.9fSSM~pp!,

f~7Be!;0, ~74!

f~8B!;0.4fSSM~8B!,

where SSM stands for the standard solar model. In fact,preferred value off(7Be) turns out to be negative~at 2–3s!in unconstrained fits. A reduced8B neutrino flux can be pro-duced by lowering the central temperature of the sun sowhat, asf(8B);Tc

18. However, such an adjustment, eithby varying the parameters of the standard solar model oadopting some nonstandard physics, tends to pushf(7Be)/f(8B) ratio to higher values rather than the low onrequired by the above-mentioned results,

f~7Be!

f~8B!;Tc

210. ~75!

Thus the observations seem difficult to reconcile with plasible solar model variations: one observable,f(8B), requiresa cooler core while a second, the ratiof(7Be)/f(8B), re-quires a hotter one.

These arguments seem to favor a more radical solutone involving new properties of neutrinos. Originally thmost plausible such solution was neutrino oscillations oftype we discussed in connection with atmospheric neutriand LSND. For simplicity we consider the mixing of thene

with a single second flavor, which we will call thenm

~though it could equally well be thent or even a linear


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combination of thenm and nt!. This will clearly alter theexpectation for solar neutrino experiments, as the Homesand SAGE/GALLEX experiments cannot detectnm’s, whileSuperKamiokande detectsnm’s with reduced efficiency.~Thecross section fornm scattering off electrons is about 1/6 thfor ne scattering.! The ne survival probability at the earth is

pne~x!512sin2~2u!sin2S dm2c4x

4\cE D;121

2sin2~2u!, ~76!

whereu is thene2nm mixing angle.@This is just our earlierresult of Eq.~66!, with the replacementp;E and with thefactors ofc and\ reinserted.# The result on the right is appropriate if the oscillation lengthL054p\cE/dm2c4 ismuch smaller that the earth–sun distancex. In that case~fora broad spectrum such as the8B neutrinos! the oscillatoryfactor averages to 1/2. For8B neutrinos this averaging iappropriate ifdm2c4 exceeds 1029 eV2.

Such vacuum oscillations were discussed many yearsas a solution to the solar neutrino problem and remain a vinteresting possibility.36 If the mixing angle were maximal—u;p/4—such vacuum oscillations would then produce a ftor of 2 suppression in the neutrino flux, under the abostated assumptions. If the oscillation length were comparato the earth–sun separation for some neutrino source~such asthe 7Be neutrino line!, the suppression could be nearly complete. In this case, variations in the suppression due toearth’s orbital eccentricity could then result. The recentmospheric neutrino results, which strongly favor large ming angles, have renewed interest in such vacuum oscillasolutions to the solar neutrino problem. Nevertheless, onthe marvelous properties of our sun is that it can grea

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enhance oscillations even if mixing angles are quite smWe now turn to describing this effect, which is known as tMihkeyev–Smirnov–Wolfenstein mechanism.

The starting point is a slight generalization of our vacuuneutrino oscillation discussion. Previously we discussedcase where our initial neutrino had a definite flavor. Butcould have considered the somewhat more general case

un~ t50!&5ae~ t50!une&1am~ t50!unm&. ~77!

Exactly as before, we could expand this wave functionterms of the mass eigenstates, which propagate simplyfind ~this takes a bit of algebra!

id

dx S ae

amD5

1

4E S 2dm2 cos 2uv dm2 sin 2uv

dm2 sin 2uv dm2 cos 2uvD S ae

amD . ~78!

Note that the common phase has been ignored: it canabsorbed into the overall phase of the coefficientsae andam ,and thus has no consequence. We have also labeled theing angle asuv , to emphasize that it is the vacuum valuand equatedx5t, that is, setc51.

The view of neutrino oscillations changed when Mikheyand Smirnov37 showed in 1985 that the density dependenof the neutrino effective mass, a phenomenon first discusby Wolfenstein38 in 1978, could greatly enhance oscillatioprobabilities: Ane is adiabatically transformed into anm as ittraverses a critical density within the sun. It became clthat the sun was not only an excellent neutrino source,also a natural regenerator for cleverly enhancing the effeof flavor mixing.

The effects of matter alter our neutrino evolution equatin an apparently simple way,

id

dx S ae

amD5

1

4E S 2E&GFr~x!2dm2 cos 2uv dm2 sin 2uv

dm2 sin 2uv 22E&GFr~x!1dm2 cos 2uvD S ae

amD , ~79!

the

by

wherer(x) is the solar electron density. The new contribtion to the diagonal elements, 2E&GFr(x), represents theeffective contribution tomn

2 that arises from neutrinoelectron scattering. The indices of refraction of electron amuon neutrinos differ because the former scatter by charand neutral currents, while the latter have only neutral crent interactions: The sun contains electrons but no muThe difference in the forward scattering amplitudes demines the density-dependent splitting of the diagonal ements of the new matter equation.

It is helpful to rewrite Eq.~79! in a basis consisting of thelight and heavy local mass eigenstates~i.e., the states thadiagonalize the right-hand side of the equation!,

unL~x!&5cosu~x!une&2sinu~x!unm&,~80!

unH~x!&5sinu~x!une&1cosu~x!unm&.

The local mixing angle is defined by

-

dedr-s.

r--

sin 2u~x!5sin 2uv

AX2~x!1sin2 2uv

,

~81!

cos 2u~x!52X~x!

AX2~x!1sin2 2uv

,

where X(x)52&GFr(x)E/dm22cos 2uv . Thus u(x)ranges fromuv to p/2 as the densityr(x) goes from 0 to .

If we define

un~x!&5aH~x!unH~x!&1aL~x!unL~x!&, ~82!

the neutrino propagation can be rewritten in terms oflocal mass eigenstates

id

dx S aH

aLD5S l~x! ia~x!

2 ia~x! 2l~x!D S aH

aLD , ~83!

with the splitting of the local mass eigenstates determined

2l~x!5dm2

2EAX2~x!1sin2 2uv ~84!


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and with mixing of these eigenstates governed by the dengradient

a~x!5S E

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d

dxr~x!sin 2uv

X2~x!1sin2 2uv. ~85!

The above-mentioned results are quite interesting: The lmass eigenstates diagonalize the matrix if the density is cstant, that is, ifa50. In such a limit, the problem is no morcomplicated than our original vacuum oscillation case,though our mixing angle is changed because of the maeffects. But if the density is not constant, the mass eigstates in fact evolve as the density changes. This is theof the MSW effect. Note that the splitting achieves its minmum value, (dm2/2E)sin 2uv , at a critical density rc

5r(xc),

2&EGFrc5dm2 cos 2uv , ~86!

which defines the point where the diagonal elements ofmatrix in Eq.~79! cross.

Our local-mass-eigenstate form of the propagation eqtion can be trivially integrated if the splitting of the diagonelements is large compared to the off-diagonal elementsthat the effects ofa(x) can be ignored,

g~x!5Ul~x!

a~x!U

5sin2 2uv

cos 2uv

dm2

2E

1

U 1

rc

dr~x!

dx U@X~x!21sin2 2uv#3/2

sin3 2uv@1,

~87!

a condition that becomes particularly stringent nearcrossing point, whereX(x) vanishes,

Fig. 9. Schematic illustration of the MSW crossing. The dashed linesrespond to the electron–electron and muon–muon diagonal elements omn

2 matrix in the flavor basis. Their intersection defines the level-crossdensityrc . The solid lines are the trajectories of the light and heavy lomass eigenstates. If the electron neutrino is produced at high densitypropagates adiabatically, it will follow the heavy-mass trajectory, emergfrom the sun as anm .


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e

a-

so

e

gc5g~xc!5sin2 2uv

cos 2uv

dm2

2E

1

U 1

rc

dr~x!

dx Ux5xc

U @1. ~88!

The resulting adiabatic electron neutrino survivprobability,39 valid whengc@1, is

pne

adiab5 121 1

2 cos 2uv cos 2u i , ~89!

where u i5u(xi) is the local mixing angle at the densitwhere the neutrino was produced.~So if uv;0 and if thestarting solar core density is sufficiently high so thatu i

;p/2, pne

adiab;0.!

The physical picture behind this derivation is illustratedFig. 9. One makes the usual assumption that, in vacuum,ne is almost identical to the light mass eigenstate,nL(0), i.e.,m1,m2 and cosuv;1. But as the density increases, the mter effects make thene heavier than thenm , with ne

→nH(x) asr(x) becomes large. The special property of tsun is that it producesne’s at high density that then propagate to the vacuum where they are measured. The adiaapproximation tells us that if initiallyne;nH(x), the neu-trino will remain on the heavy mass trajectory provided tdensity changes slowly. That is, if the solar density gradiis sufficiently gentle, the neutrino will emerge from the sas the heavy vacuum eigenstate,;nm . This guaranteesnearly complete conversion ofne’s into nm’s, producing aflux that cannot be detected by the Homestake or SAGGALLEX detectors.

Although it goes beyond the scope of this discussion,case where the crossing is nonadiabatic can also be hanin an elegant fashion by following a procedure introducedLandau and Zener for similar atomic physics level-crossproblems. The result40 is an oscillation probability valid forall dm2/E anduv ,

pne5 1

21 12 cos 2uv cos 2u i~122e2pgc/2!. ~90!

As it must by our construction,pnereduces topne

adiab for gc

@1. When the crossing becomes nonadiabatic~e.g.,gc!1!,the neutrino ‘‘hops’’ to the light mass trajectory as it reachthe crossing point, allowing the neutrino to exit the sun ane , i.e., no conversion occurs.

Thus there are two conditions for strong conversionsolar neutrinos: there must be a level crossing@that is, thesolar core density must be sufficient to renderne;nH(xi)when it is first produced# and the crossing must be adiabatThe first condition requires thatdm2/E not be too large, andthe secondgc*1. The combination of these two constrainillustrated in Fig. 10, defines a triangle of interesting paraeters in the dm2/E2sin2 2uv plane, as Mikheyev andSmirnov found by numerical integration. A remarkable feture of this triangle is that strongne→nm conversion canoccur for very small mixing angles (sin2 2u;1023), unlikethe vacuum case. One can envision superimposing on Figthe spectrum of solar neutrinos, plotted as a functiondm2/E for some choice ofdm2. Since Davis seessomesolarneutrinos, the solutions must correspond to the boundariethe triangle in Fig. 10. The horizontal boundary indicatesmaximumdm2/E for which the sun’s central density is suficient to cause a level crossing. If a spectrum propestraddles this boundary, we obtain a result consistent w

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the Homestake experiment in which low energy neutrin~large 1/E! lie above the level-crossing boundary~and thusremainne’s!, but the high-energy neutrinos~small 1/E! fallwithin the unshaded region where strong conversion taplace. Thus such a solution would mimic nonstandard smodels in that only the8B neutrino flux would be stronglysuppressed. The diagonal boundary separates the adiaand nonadiabatic regions. If the spectrum straddlesboundary, we obtain a second solution in which low eneneutrinos lie within the conversion region, but the higenergy neutrinos~small 1/E! lie below the conversion regionand are characterized byg!1 at the crossing density.~Ofcourse, the boundary is not a sharp one, but is characteby the Landau–Zener exponential.! Such a nonadiabatic solution is quite distinctive as the flux ofpp neutrinos, which isstrongly constrained in the standard solar model and insteady-state nonstandard model by the solar luminoswould now be sharply reduced. Finally, one can imag‘‘hybrid’’ solutions where the spectrum straddles both tlevel-crossing~horizontal! boundary and the adiabaticity~di-agonal! boundary for smallu, thereby reducing the7Be neu-trino flux more than either thepp or 8B fluxes.

Remarkably, this last possibility seems quite consistwith the experiments we have discussed. In fact, a neperfect fit to the data results from choosingdm2;531026 eV2 and sin2 2uv;0.006. As thedm2 is quite differ-ent from that found in the atmospheric neutrino results,mixing seen in solar neutrinos is distinct from that seenatmospheric neutrinos, consistent with our attribution offormer tone→nm oscillations and the latter tonm→nt .

The argument that the solar neutrino problem must beto neutrino oscillations is quite strong, but not as compell

Fig. 10. MSW conversion for a neutrino produced at the sun’s center.upper shaded region indicates thosedm2/E where the vacuum mass splittinis too great to be overcome by the solar density. Thus no level crosoccurs. The lower shaded region defines the region where the level crois nonadiabatic~gc less than unity!. The unshaded region correspondsadiabatic level crossings where strongne→nm will occur.


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as in the case of atmospheric neutrinos. Our conclusionslow from combining the results of several experiments, anot from direct observation of new physics, such as thenith angle dependence in the atmospheric results. Forreason there is great interest in a new experiment now breadied in the Creighton nickel mine in Sudbury, Ontar6800 ft below the surface. The Sudbury Neutrino Obsertory ~SNO!41 has a central acrylic vessel filled with 1 kt overy pure~99.92%! heavy water, surrounded by a shield7.5 kt of ordinary water. SNO can detect neutrinos throuthe charged current reaction

ne1d→p1p1e2, ~91!

as well as through the neutral current reaction

nx~ nx!1d→nx~ nx!1p1n. ~92!

Thus SNO offers the exciting possibility of comparing thsolar flux inne’s with that in all flavors, thereby providing adefinitive test of flavor oscillations. The electrons from reation ~91! will be detected by the Cerenkov light they geneate: SNO’s central vessel is surrounded by 9800 phototuThe neutrons produced in the neutral current reaction candetected using either the~n,g! reaction on salt dissolved inthe heavy water or proportional counters exploiting t3He(n,p)3H reaction. SNO began taking data in summ1999.

V. CONCLUSIONS

In this article we have summarized some of the basic idof neutrino physics. Neutrinos come in three speciesflavors—ne ,nm ,nt—with their corresponding antiparticlesWe have discussed the unresolved problem of the naturthese antiparticles: Aren’s and n ’s distinguished by someadditive quantum number~Dirac neutrinos!, or do they in-stead correspond to the two projections of opposite handness of the same state~Majorana neutrinos!? We revieweddirect mass measurements, which have not yet yieldeddence for nonzero masses, and the reasons that the minstandard model cannot accommodate massive neutrinoswe have seen strong though indirect evidence for masneutrinos in three classes of neutrino oscillation experimeIn an accelerator experiment—LSND—an unexpected flof electron antineutrinos has been attributed tonm→ ne os-cillations. Over the past 15 years a series of experimentsbeen carried out on the neutrinos produced when high encosmic rays interact in the upper atmosphere. The increaevidence of an anomaly has culminated with the Supermiokande measurements that confirm a deficit in the mneutrino flux, and find a zenith angle dependence direindicating oscillations. When constraints from reactor aaccelerator experiments are taken into account, the expltion for the atmospheric neutrino anomaly appears to benm

→nt oscillations corresponding to a nearly maximal mixinangle. Finally, we discussed the evidence for a deficit influx of solar electron neutrinos and the difficulty in attribuing this deficit to uncertainties in the solar model. Again thypothesis of neutrino oscillations accounts for the obsertions, with one attractive possibility being ane→nm oscilla-tion enhanced by matter effects within the sun. One popuchoice for the mass differences and mixing angles neeto account for these observations is shown in Fig. 11. Tpattern that emerges is not, unfortunately, compatiwith the simplest scenario of three mixed neutrinos: th

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distinct dm2 values are required, while a three-neutrino snario provides only two. This could mean that one~or more!of the experiments we have discussed has been misipreted. Alternatively, it could indicate the existence of adtional light neutrinos that have so far avoided detectioncause they are sterile, lacking the usual standard minteractions. In any case, it is clear that the possibilities sumarized in Fig. 11 will require much more study in a negeneration of heroic neutrino experiments. We have notwo efforts that may provide important results in the not tdistant future: SNO will soon tell us whether there existsnm or nt component in the solar neutrino flux, while a neFermilab experiment will test the LSND conclusions. Thperhaps the picture will become far clearer in the next fyears. In the meantime, neutrinos will remain in the newWe have tried to illustrate how the many profound issuconnected with neutrinos can be readily appreciated wonly simple quantum mechanics. It is our hope, therefothat the developing discoveries in neutrino physics—whmay show the way to the next standard model of partphysics—will be accessible to and followed by beginnistudents of physics. We hope the material presented hwhich is suitable for beginning courses in quantum mechics and modern physics, will help make this possible.

ACKNOWLEDGMENTS

This work is supported in part by the National ScienFoundation ~BRH! and by the Department of Energ~WCH!. In addition BRH would like to acknowledge thsupport of the Alexander von Humboldt Foundation andhospitality of Forschungszentrum Ju¨lich. We also thank BillLouis and Vern Sandberg for help with Figs. 7 and 11.

1See, e.g., ‘‘A Massive Discovery,’’ Sci. Am.279, 18–20~August 1998!.2From Telephone Poles and Other Poems, by John Updike Copyright ©1963. Reprinted by permission of Alfred A. Knopf, Inc.

Fig. 11. Regions ofdm2 and sin2 2u that account for the solar neutrinoSuperKamiokande atmospheric neutrino, and LSND results. The LSNDgion is the portion from Fig. 7 that is allowed by the various null expements. In addition to the two solar neutrino solutions shown, there is a ththe vacuum neutrino oscillation solution discussed in the text, residinmuch smallerdm2. Note that three distinctdm2 are required to fit the data


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3L. M. Brown, ‘‘The Idea of the Neutrino,’’ Phys. Today31, 23–28~Sep-tember 1978!; also see ‘‘Celebrating the Neutrino,’’ Los Alamos Sci.25,1–191~1997!.

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8Often quoted is Rabi’s remark about the discovery of the muon—‘‘Wordered that?’’

9Of course, decay of thep1 to an electron and neutrino is also possible bis highly suppressed for helicity reasons, as we shall subsequently dis

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11G. Steigman, D. Schramm, and J. Gunn, ‘‘Cosmological Limits toNumber of Massive Leptons,’’ Phys. Lett.B66, 202–204~1978!; J. Yanget al., ‘‘Primeval Nucleosynthesis: A Critical Comparison of Theory anObservation,’’ Ap. J.281, 493–511~1984!.

12J. D. Bjorken and S. D. Drell,Relativistic Quantum Mechanics~McGraw–Hill, New York, 1965!.

13In principle there could be right-handed neutrinos present, but they cnot interact and hence we would not be aware of their presence.

14M. Goldhaber, L. Grodzins, and A. Sunyar, ‘‘Helicity of Neutrinos,Phys. Rev.109, 1015–1017~1958!.

15Particle Data Group Phys. Rev. D54, 321 ~1996!.16W. C. Haxton and G. J. Stephenson, Jr., ‘‘Double Beta Decay,’’ Pr

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21The constant in front of the neutrino expression differs from the corsponding photon number because of the use of Fermi rather thanstatistics

*0`dz@z3/(ez21)#5p4/15→*0

`dz@z3/(ez11)#57p4/120. ~93!22R. G. H. Robertsonet al., ‘‘Limit on Electron Antineutrino Mass from

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23Ch. Weinheimeret al., ‘‘Improved Limit on the Electron-AntineutrinoRest Mass for Tritiumb-Decay,’’ Phys. Lett. B300, 210–216~1993!.

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26B. Pontecorvo, ‘‘Neutrino Experiments and the Problem of Conservaof Leptonic Charge,’’ Sov. Phys. JETP26, 984–988~1968!.

27The ne’s come from both the fraction ofm2’s that decay viam2→e2

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28C. Athanassopouloset al., ‘‘Candidate Events in a Search fornm→ ne

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29Y. Fukudaet al., ‘‘Measurement of a Small Atmosphericnm /ne Ratio,’’Phys. Rev. Lett.81, 1562–1565~1998!.

30M. Gell-Mann, P. Ramond, and R. Slansky, inSupergravity, edited by P.

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MILLICURIES IN THE MAIL

Otto Stern had extolled the future of Ernest Lawrence’s cyclotron to us in previous years. In1935, when we were in Ann Arbor, Fermi and I had corresponded with Lawrence. At that time, Ido not remember for what reason, he offered Fermi a millicurie of radiosodium. Doubting Ber-keley’s radioactivity measurements, Fermi replied suggesting that Lawrence had perhaps made amistake and actually meant a microcurie, a thousand times less. In answer, he received a lettercontaining a millicurie of radiosodium. We were dumbfounded. By then I was sure I wanted to goto see the cyclotron.

Emilio Segre, A Mind Always in Motion—The Autobiography of Emilio Segre` ~University of California Press, Berkeley,1993!, p. 112.


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