Lepton mixing and neutrino oscillations

LEPTON MIXING AND NEUTRINO OSCILLATIONS

S.M. BILENKY and B. PONTECORVO Joint Institute for Nuclear Research, Dubna, USSR

I

NORTH-HOLLAND PUBLISHING COMPANY — AMSTERDAM

PHYSICS REPORTS (Section C of Physics Letters)41, No, 4(1978)225-261. NORTH-HOLLAND PUBLISHING COMPANY

LEPTON MIXING AND NEUTRINO OSCILLATIONS

S.M. BILENKY and B. PONTECORVO Joint institute for Nuclear Research. Dubna, USSR

Received 27 June 1977

Contents:

1. Introduction 2. Weak interaction theories with neutrino mixing 2.1. Electron and muon lepton numbers 2.2. Majorana neutrinos 2.3. Dirac neutrinos 2.4. Comparison of theories with neutrino mixing 3. Neutrino oscillations 4. Possible experiments in which the hypothesis of neutrino mixing might be tested 4.1. General considerations 2 by existing 4.2. Limits imposed on the parameter M data 4.3. Possible experiments 5. Solar and cosmic neutrinos 5.1. The Brookhaven experiment 5.2. The experimental results 5.3. Conclusions

227 229 229 231 234 236 237 239 239 241 242 245 245 245 247

5.4. The solar neutrino “puzzle” and neutrino oscillations 5.5. Oscillations and future solar neutrino experiments 5.6. Solar neutrinos, time intensity variations and oscillation lengths 5.7, A question about coherence of neutrino beams 5.8. Oscillations and cosmic neutrinos 6. Oscillations in the general case of N neutrino types 6.1. Left-handed fields 6.2. The general case of left-handed and right-handed neutrino fields 6.3. Concluding remarks 7. The ji ~- e~’decay and neutrino oscillations 7.!. The case when there are only neutrinos 7.2. Heavy leptons 7.3. The ~i —~ ey decay and neutrino oscillations 8. Conclusions References

Single orders for this issue PHYSICS REPORT (Section C of PHYSICS LETTERS) 41, No. 4 (1978) 225—261. Copies of this issue may be obtained at the price given below. All orders should be sent directly to the Publisher. Orders must be accompanied by check. Single issue price Dfl. 18.00, postage included.

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SM. Bilenky and B. Pontecorvo, Lepton mixing and neutrino oscillations

227

Abstract: The present article is a review of phenomena connected with neutrino oscillations. Mixing of two neutrinos (Majorana as well as Dirac) with masses m 1 and m2 is considered 2. Possibleinexperiments detail. It is shown designed thatto the reveal hypothesis neutrinoofoscillations lepton mixing at reactor, is not inmeson contradiction factory and withhigh the existingaccelerator energy data if m~facilities — m~ ~are 1 (eV) considered. In such experiments oscillation might be found if m~— m~ ~ 0.01 (eV)2. The possibilities of searching for oscillations by experiments on cosmic ray neutrinos and especially on solar neutrinos are discussed in detail. The last experiments have an incredible high sensitivity from the point of view of testing the lepton mixing hypothesis (oscillation effects might be observable if — m~ ~ i0—’~(eV)2). The “solar neutrino puzzle” is also discussed from the point of view of lepton mixing. Neutrino oscillations are considered then in the case where in nature there exist N ~ 2 neutrino types. In conclusion the case of heavy lepton mixing is considered. It is shown that in a concrete scheme with right-handed currents, the probabilities of sOch processes as p -+ ey, p —~ 3e etc. can be close to existing experimental upper limits, provided the heavy lepton masses are of an order ofa few GeV, whereas the probabilities ofthe above processes are entirely negligible if only neutrinos are mixed.

1. Introduction In the present review paper the problem of neutrino oscillations is dealt with. Qualitatively neutrino oscillations were discussed a long time ago [1], the motivation for their possible existence being an analogy with the well known oscillations of neutral kaons. Recently there have appeared many articles in which neutrino oscillations have been considered from different positions. We shall report on this work in some detail. To start with, we will briefly discuss the main results of neutrino oscillation theories. Usually, in agreement with existing data, it is assumed that the electron and the muon lepton numbers are strictly conserved. Of course, if this is the case, neutrino oscillations cannot take place. Oscillations may arise if, in addition to the usual weak interaction, an interaction which does not conserve lepton numbers is also taking place. In such a case the neutrino* masses are different from zero, and the state vectors of the ordinary electron and muon neutrinos Ve and vp are superpositions of the state vectors of neutrinos v 1 and v2 with definite masses m1 and m2. If a beam of, say, muon neutrinos if produced in a Weak process, at a certain distance from the production place the beam will be a coherent superposition of v,~and Ve (that is, there arise oscillations V~~:±Ve). Such a situation is analogous to the one we are familiar with in the case of neutral kaon oscillations. The strong interaction of kaons is analogous to the weak interaction of neutrinos, the weak interaction which does not conserve strangeness in the kaon case is analogous to a new (superweak?) interaction which does not conserve lepton numbers; Ve and v~are analogous to K°and K° (these particles are not described by stationary states), while v1 and v2 are analogous to K~and K~(these particles have definite masses and are consequently described by stationary states). Let us note also the following differences between neutral kaon oscillations and possible neutrino oscillations. 1) The mass difference MKL MKSl is much smaller than the kaon mass, whereas the difference of the neutrino masses m1 and m2 may be comparable with their masses. 2) Neutral kaons are unstable particles, whereas the instability of the heaviest neutrino may be neglected. —

* Hereafter the neutral leptons with masses, say, smaller than the electron mass will be called neutrinos, while the other hypothetical neutral leptons will be simply called heavy neutral leptons.

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SM. Bilenky and B. Pontecorro, Lepton mixing and neutrino oscillations

3) While neutral kaon oscillations have the maximum amplitude, this might not be the case in neutrino oscillations. 4) Kaons are bosons, neutrinos are fermions, a fact which has the consequence that kaon oscillations are possible with only two kaon states, while neutrino oscillations may take place only if there are (at least) four neutrino states. A quantitative theory of neutrino oscillations was first given in ref. [2], where in the hamiltonian, which does not conserve lepton numbers, two-component neutrino states were used as base states (four states in all for two types of neutrinos). In such a theory the particles with definite masses are Majorana neutrinos. More recently, work on neutrino oscillations has been closely connected with the unified theory of the weak and electromagnetic interactions of Weinberg—Salam [3]. In references [4, 5] the assumption was made that the Ve and vp field operators in the charged lepton current are othogonal superposition of fields of Dirac neutrinos with definite masses, Thus, in addition to the usual hypothesis about a lepton—quark analogy, a supplementary assumption was made: leptons are mixed, just as quarks are. In this scheme oscillations Ve ~t vp arise [5, 6]. In the theory proposed in ref. [2] leptons are mixed as well, but the Ve and v0 field operators here are orthogonal superpositions of massive Majorana neutrino fields. In both theories possible oscillations v~~ v~, ~ 13,, are described by identical expressions, in which twomasses parameters are present, the mixing 2 = m~ of the neutrino squared. angle 0 and the difference M which have been already performed, a special search for neutrino In the neutrino experiments oscillations was not made. Nevertheless from the results of such experiments one can estimate the upper limit of the parameter M2, if definite assumptions are made about the mixing angle. For example, if the mixing is maximum (0 = ~t), M2 must be smaller than I (eV)2. At the present time a number of experiments aimed to reveal oscillations are being performed or planned on neutrinos from reactors, meson factories and high energy accelerators. Such experiments will test the hypothesis of lepton mixing if M2 ~10”2(eV)2. If M2 ~ lO”2(eV)2 neutrino oscillations can be discovered [1] only in cosmic neutrino experiments and especially in solar neutrino experiments. In schemes with orthogonal lepton mixing the neutral current of the Weinberg—Salam theory contains only terms diagonal in the lepton fields. Asymmetrical neutral currents, however, effectively arise in higher order perturbation theory. The situation is entirely analogous to the case of hadron currents. Similarly to such processes as K~ ~i”~i, K’~ ~ etc., in a theory with lepton mixing there arise, in principle, processes of the type j.t ey, ~i 3e etc., in which lepton numbers are not conserved. However, calculations based on theories wherein Majorana as well as Dirac neutrinos are mixed, show that the probabilities of such processes are smaller than the experimental upper limits by tens of magnitude orders. Thus a search for neutrino oscillations is the only way to test such theories. The high sensitivity of neutrino oscillation experiments is due to the interference character of oscillation phenomena. From such experiments one can get information on matrix elements of the interaction non-conserving lepton numbers, whereas from experiments where either probabilities of decay processes or reaction cross sections are measured one can get information on matrix elements squared. In the references [7—9]theories were considered where any number N ~ 2 of either Majorana or Dirac neutrinos with different masses are mixed. In this case also an experimental search for oscillations is the most sensitive method of testing the lepton mixing hypothesis. Let us suppose that neutrinos, say, of the Ve type are generated as a result of some weak process. In the general —

—*

—÷

—*

SM. Bilenky and B. Pontecorvo, Lepton mixing and neutrino oscillations

229

case where there exist N ? 2 types of neutrinos, the minimum averaged intensity of va’s at a sufficiently large distance from the place of production may be as low as 1/N of the Ve intensity which would have been expected under the assumption that there is no mixing. When the energy of original Ve’S (in the previous example) is smaller than the threshold for muon production, which is the case of solar and reactor neutrinos, the v,,, which arise from oscillations are “sterile”. As already stated, if in nature the only neutral leptons are neutrinos, the probabilities of such processes as ~i e~,~z 3e etc., in schemes with lepton mixing turn out to be incomparably smaller than the corresponding experimental upper limits. The situation entirely changes, if there exist heavy leptons with masses of the order of a few GeV and if the heavy lepton fields are mixed. In such a case the probabilities of the ~t ey, j~ 3e, etc. processes might be close to the measured upper limits. Thus experimental searches for processes forbidden by the lepton number conservation law would be extremely important even if performed at a level close to the measured upper limits of probabilities. Below a plan is given of the present review. In the second section there are considered in detail various weak interaction theories, where the fields of two types of neutrinos with finite masses are mixed. In the same section there are presented also the most recent and relevant experimental data on lepton number conservation. In the third section neutrino oscillations are discussed from a theoretical point of view. In the subsequent, fourth section, possible search experiments on neutrino oscillations are considered. The fifth section is dedicated to the problem of cosmic neutrino, especially solar neutrino, oscillations. The general case of N ?~2 types of neutrinos is briefly considered in the sixth section. In the seventh section the decay process 4u ey is discussed . . .

—~

—~

—~

—*

—k

in a scheme with lepton mixing. Finally in the eighth concluding section a summarising account is given of the lepton mixing “ideology”.

2. Weak interaction theories with neutrino mixing 2.1. Electron and muon lepton numbers In this section we shall consider the theories in which neutrino oscillations arise. They are all based on the assumption that, in addition to the usual weak interaction, there exists also an interaction which does not conserve lepton numbers. In agreement with experimental data, it is assumed that this additional interaction is weaker than the usual weak interaction. The usual weak interaction hamiltonian is (1) of which the first term has the expression (2) Here Irx

=

(VeLYL)

+ (V,,~y~p~) + j~

(3)

is the weak charged current (j~is the ~ = ~(1 constant + y~)~’ is (M the isleft-handed 5/M2hadron is the current), weak interaction the protoncompomass), nent of the operator i/i, G = 10

230

SM. Bileoki’ and B. Pontecorro. Lepton mixing and neutrino OscillatiOliS

and j~= j~(ij~) (j.~= 1, ij~= 1). The second term of expression (1) is the neutral current contribution, the structure of which here is of no relevance. The interaction (2) does conserve separately the sum of the electron Le and muon L,1 lepton numbers* —

const.

(4)

~ I~= const.

(5)

=

The values of lepton numbers for different particles are given in table 1. Corresponding antiparticles have opposite values of lepton numbers.

Table I Lepton numbers ofvarious particles r,

e

i’~

L~

I

1

0

L5

0

0

I

Hadrons, Photon

-

0

0 0

There are a few types of experiments in which lepton number conservation can be tested. In every experiment an attempt is made to discover a process in which lepton numbers are not conserved. In table 2 there is listed a number of processes which were looked for, but not discovered, and there is given the corresponding upper limit of the ratio l4’~/l4~~ of the probabilities of a forbidden process (I) and of a corresponding, but allowed process (II). In table 2 only the most recent data** are presented. They are relevant to the question of: a) possible non-conservation of the electron lepton number, b) possible non-conservation of the muon lepton number, c) possible non-conservation of both lepton numbers, their sum being conserved and d) possible non-conservation of both lepton numbers. As it is seen, existing data are in agreement with expressions (4) and (5). Let us assume that there is an interaction, weaker than the interaction (1), which does not conserve lepton numbers. As we shall see later, in such a case there appear oscillations in neutrino beams. Thus experiments in which neutrino oscillations are searched for will allow one to test the hypothesis on the existence of an interaction non-conserving lepton numbers. Let us note that the sensitivity of such experiments is much higher than the sensitivity of experiments designed to reveal the neutrinoless double beta decay, the ~i ey decay*** and similar processes. -+

*

It is well known that other formulations of the lepton number conservation law are possible. For example, the scheme [10. 11].

in which there is only one conserved lepton charge, is attractive and economical. In this scheme e and g have opposite signs of the lepton number, and neutrinos are described by one four-component Dirac spinor. ** Nore added in proof: Recently some new upper limits were published of relative probabilities for the processes p —‘ e’ + (W(p* —‘ e~’y)/W(ji~ —* e’r~+~) < 1.1 x l0~, H.P. Powel et al., SIN Newsletter 9 (1977) and W(ji~—. ey)/W(p~ -. e’~J.,)< 3.6 x 10°, P. Depommier et al.. Phys. Rev. Lett. 39 (1977) 1113) and p + S -. e + S (W(pS —. eS)/W(0 S —. all) <4 x lO’~. A. Badertsher et al.. Phys. Rev. Lett. 39 (1977) 1385). *** This statement might be erroneous, if there exist heavy leptons (see section 7).

SM. Bilenky and B. Pontecorvo, Lepton mixing and neutrino oscillations

231

Table 2 Ratio WI/WII of the probabilities of the forbidden (I) and allowed (II) processes* Process forbidden by lepton conservation (I)

WI/WH

Confidence limits

Refs.

Double beta decay 82Se —. 82Kr + e + e + ~ + y~

<0.09

68%

12, 13

b)v,~+N-.p~+...

v

14

c) p~—~e~ + y

3 <5xl0 <2.2 x 10_8

95%

6+N—’p+... p~-÷e~+ v~+V~

90%

15

< 1.9 x iO~

90%

16

<1.6 x lO_8

90%

17

<3 x iO’~

95%

18

<2.6 x 108

90%

17

a)

Neutrinoless double beta decay 82Se ‘-~ 82Kr + e + e

c) p”

-.

c) p

+ Cu

Observed process, allowed by lepton conservation (II)

e~+ e” + e —*

e +

-.

e~+ v~+ V~

p’ + Cu

...

-÷

v 6 +

c) v5+N-+e d) p *

+ Cu

-+

v6+N—*p

+...

e~+

p

...

...

+...

+ Cu -+ v~,+

...

In the literature often a parameter a is used in order to characterise qualitatively the maximum relative amplitude

of the process which is forbidden by lepton conservation. Roughly a x \i~7W~ all cases in 82Se. In this case aspeaking, ~/ii~7i~ l0 ~, theforfactor l0 listed taking into factneutrinoless that the phase space fordecay the neutrinoless double beta decay is 106 times larger than for the tableaccount 2, exceptthe for the double beta of “ordinary” double beta decay.

2.2. Majorana neutrinos The first theory of neutrino oscillations was developed in ref. [2], and was based on the two component neutrino theory. According to this theory only left-handed components of the neutrino fields VeL

=

l+~5 2 Ve,

V,,L =

1+~’5 2 v,,

(6)

and right-handed components of the antineutrino fields =

=

(VeL)C,

1 2

V~R =

V~=

‘~‘~

(VPL)C,

(7)

can appear in the hamiltonian. Here (8)

= C~e,,,

is the charge conjugated spinor. The matrix C satisfies the relations

C+C=1, CY~’C’1=

,

‘Y 5,

CT =

~C.

(9)

232

SM. Bilenky and B. Pontecorro. Lepton mixing and neutrino oscillation.s

The hamiltonian of the interaction which does not conserve lepton numbers, quadratic in the neutrino fields, has the following general form* =

maeV~RveL+ mP,,vMRv,,L + m~e(V~RveL +

VeRV,,L)

+ h.c.,

(10)

where the parameters mre, m4,, and mpe have the dimensions of a mass. The hamiltonian (10) can be written more compactly =13~,Mv~~, +VLMVR,

.*‘

(11)

where VLI

(VeL~\

c

J’

(12)

VRI

\\VPRI

\VPL/

and M

=

(mae mge~ \m,,-~ m,,-,,j

(13)

If the interaction (10) is invariant under the CP-transformation, the parameters mae, ~ are real** In such a case the interaction hamiltonian can be written as follows: 13~M(v~ + v~)+ 13LM(V~ +

VL) =

~

mpe

(14)

Here X

= VL

+

V~= (VeL \\V,,L

+ V~R~= (Xe~ + V,,R/ \X,,J

(15)

We have

X~CXX.

(16)

Thus Xe and x,, are fields of Majorana neutrinos. Clearly the appearance in the hamiltonian (14) of Majorana fields is due to the fact that the interaction we have introduced does not conserve lepton numbers. The matrix M can be easily diagonalized. We have M

=

(17)

UMOUT,

where U is an orthogonal matrix (UTU M0

*

(nii

=

1), and (18)

rn2)

It is easy to see that

t~RV,L

—

VeRV*L

=

0. Actually.

RV*L

=

—

v~C ~(l

+

y,)v,,

=

—

v~,~(l + y[)C

‘v~=

~RV,L.

As a matter of fact UpC~(x)MvL(x)UI~R’= U~c+’fx)M+(l+ y5)s(x)U~~’= ‘ti(x’)74M4(l + ~s)~,4vc(x)=~L(x)Ms~(x), where Up( is the operator of the PC-transfotmation, and x = t(—x, = M. ix0). Because of CP-invariance U~~ir(x)U~’= )l~’(x’)and we have, consequently, M~= M. Since MT = M, we have then M **

SM. Bilenky and B. Pontecorvo, Lepton mixing and neutrino oscillations

233

From (14) and (17) we get the following expression for the hamiltonian =

=

mg~q~,

~

(19)

where U~.

=

Thus

& and ~

(20) are the fields of Majorana neutrinos with masses m1 and m2, resp. From expression

(20) we obtain

x = U4i.

(21)

It follows that the VeL and V,,L fields, which are present in the hamiltonian of the ordinary weak interaction, are connected with the Majorana neutrino fields, by the relations VeL =

~

Ul~~L,

V,,L =

~

U2gt~L.

(22)

Thus, if our assumptions are right, in the usual weak interaction hamiltonian there appear orthogonal superpositions of fields of Majorana neutrinos, the masses of which m1 and m2 are not equal to zero. As we shall see, in such a case there arise neutrino oscillations. The orthogonal matrix U has the following general form cosO sinO~ (23) \~—sinO cosO) Making use of expressions ((20)—(23)), we obtain 0X2, = Sin O~j+ cos = sin and 42

(24)

0X2,

—

VeL = C05 04~1L +

sin 04~2L’

—sin °&L + cos

V,,L =

042L•

(25)

From these expressions it is clear that the angle 0 characterises the degree of mixing of the Majorana fields 4~and ~ Let us derive now the expressions relating the masses m 1, m2 and the mixing angle 0 to the values ~ m,,-,,, m~6.From (17) we find 2Om 2Om mae = cos 1 + sin 2, (26) 2Om 2Om ma,, = sin 1 + cos 2, (27) mpe

=

sinOcosO(—m1 + m2).

(28)

It follows from these relations that tg 20

=

rn1,2

=

2m~~/(m~,,mg8),

(29)

—

2+ 4m,,~

~(m~8+ mg,, ±.,,,/(mae

— m~,,)

8).

(30)

As we shall see in the third section, oscillations do take place, if 0 ~ 0 and m1 ~ m2. It follows from (27) and (28) that this happens if m~and at least one of the two parameters m~and m,,-~

234

SM. Bilenky and B. Pontecorro, Lepton mixing and neutrino oscillations

is different from zero. From (27) we see that U 0. In such a case we have the relations VeL = ~7~I~1L

+

4~2L),

=

~it

(maximum mixing), when mae

=

mp,, and

V,,L = 7~H4~1L

+

(31)

~2L)~

similar to the well known expressions which relate the K°and K°wave functions to the K1 and K2 functions. Let us remark that the mixing is also maximum when mae, m,,-,, ~ mae. Let us discuss the scheme which was considered here from a slightly different point of view. The starting point was the two-component neutrino theory. It is well known that this theory is equivalent to the Majorana theory of neutrinos with mass equal to zero. Let us proceed then from the Majorana neutrino theory. From this point of view it is clear that when we assumed the existence of the lepton charge violating interaction (10), we assumed in essence that the two Majorana neutrinos are massive, and that their fields are mixed according to expression (25). Let us note that in such a scheme Majorana neutrinos may be distinguished only by their masses and that there is no any need of introducing the notion of lepton number. We shall see that this scheme does not contradict the existing experimental data, provided the particle masses are sufficiently small. The most spectacular consequence of the scheme is the phenomenon of neutrino oscillations. 2.3. Dirac neutrinos We shall discuss now the theory put forward in refs. [4, 5]. Its basis is the analogy between quarks and leptons. In order to explain our point of view on this analogy, we will go in some detail through the developments which led to the discovery of the fourth quark. At the moment of writing it is generally recognised that charmed particles have been discovered in neutrino experiments and in e~—e experiments (see Proc. 17th Intern. Conf. on High Energy Physics, Tbilisi 1976). It is well known that the hadron Cabibbo current is given by the expression =

ULYSdL,

(32)

where d’

=

dcos0~+ ssinO~

(33)

(O~is the Cabibbo angle). In equations (32) and (33) u, d, s are, respectively, the field operators of the u-quarks (Q = T3 = S = 0), d-quarks (Q = —~,T~= —~,S = 0), and s-quarks (Q = T = 0, S = — 1). It is known that all existing old data, including the data obtained in experiments on the deep inelastic interaction of neutrinos with nucleons, are in agreement with expression (32). The hadron Cabibbo current has the same V-A structure as the lepton current (only left-handed field components are present in both currents). However, if we confine ourselves to the consideration of expression (32), we see essential differences between lepton and hadron currents. First of all we know that there are four leptons, while in the expression (32) of the hadron current there are present the fields of only three quarks. In order to remove such asymmetry, the authors of paper [19] introduced a fourth quark with a new quantum number (charm)*. ~,

*

This question is exposed in more detail in the review paper [20].

—~,

SM. Bilenky and B. Pontecorvo, Lepton mixing and neutrino oscillations

235

A big step forward in the building of the weak interaction theory was made in the paper [21]. First, the authors assumed that, really there exist four quarks. Second, the authors assumed that, together with the Cabibbo current (32), in the charged hadron current there is present a term — (jh\ s)GIM

—

CLYSSL,

where s’

=

—

d sin O~+ s cos O~,

(35)

and c is the field operator of the c-quarks (Q = ~, T = 0, S = 0, C is usually called the standard theory. If we assume that the hadron current is given by the expression ‘h_

=

1). At present this theory

ii~

s)C + s)GIM’ then, as it is shown in [21], the neutral current arising in weak and electromagnetic interaction gauge theories a la Salam—Weinberg has no terms changing strangeness. Since the charged current changes strangeness, however, a neutral current effectively changing strangeness does arise in higher orders of perturbation theory. In the standard theory with the current (36), as is well known, there is a compensation mechanism [21] which allows one to get a consistent agreement between experimental and theoretical values of the KL and K~mass difference and of the probabilities of such processes as K~ it~v13,KL p~jf and others. Let us note that the cancellation of the diagrams inducing strangeness changing neutral currents is connected with the form of the orthogonal combinations (33) and (35), in which the fields of the d and s quarks appear in the charged current. Let us come back to the leptons. Comparing the hadron current (36) of the standard theory with the ordinary lepton current (3), we see an important difference between them: whereas orthogonal superpositions of the d and s quark fields are present in the hadron current, in the lepton current (3) the electron neutrino and muon neutrino fields appear unmixed (the charged hadron current does not conserve strangeness, while the lepton current (3) is conserving lepton numbers). In order to remove this difference let us assume [4, 5] that there exist two neutrinos (v1, v2)

is —

—~

with finite masses and that the operators orthogonal combinations* Ve =

v1 cos 0 + v2 sin 0,

v,,

—v1 sin0 + v2cos0,

—~

Ve

and v,, in the expression (3) of the lepton current are

(37) =

where v1 and v2 are the fields of Dirac neutrinos with masses in1 and m2, 0 is the mixing angle**. There are no reasons to assume that 0 is equal to the Cabibbo angle 0~.Let us note only that two values of the mixing angle are of special significance, 0 = 0 and 0 = fir. The case 0 = 0 (no mixing) corresponds to the usual theory with strict conservation of the electron and muon lepton numbers. The case 0 = +lr corresponds to maximum amplitude of oscillations (see the third section). Thus we came to a theory where there is a full analogy between lepton and quarks weak currents. * After the present review article had been written, we have been informed that already in 1963 hadron and lepton mixing was proposed by M. Nakagawa, H. Okonogi, S. Sakata and A. Tojoda, Prog. Theor. Phys. 30 (1963) 727. ** Let us note that for the lepton mixing angle the notation 0 was used both for Majorana (see section 2.2) and Dirac fields.

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SM. Bilenky and B. Pontecorvo, Lepton mixing and neutrino oscillations

In essence in this scheme there is no lepton number distinguishing the two types of neutrinos Ve and v0 the neutrinos v1 and v2 differ only in their mass values (just as the d and s quarks do). Because of expression (37), in the neutral current of the Salam—Weinberg theory asymmetrical terms (giving processes /1 —~e~,~u—~3e and others) are absent (in first order). In higher orders of perturbation theory this scheme allows the existence in principle of such processes (note the analogy with the processes KL —~2~u, K irv~and others). However, calculations [22] show that, because of a mechanism similar to the GIM mechanism [21] for hadrons, the probabilities of these processes are extremely small in the scheme with two neutrinos we are discussing (see the seventh section for details). The main difference between the theory considered above and the usual one with two conserved lepton numbers is that the first allows neutrino oscillations [5, 6]. —*

2.4. Comparison of theories with neutrino mixing We conclude this section with a few remarks, relative to the comparison of the theory with mixing of Majorana neutrinos (theory M), exposed in section 2.2, with the theory with mixing of Dirac neutrinos (theory D), exposed in section 2.3. A. In both schemes discussed in sections 2.2 and 2.3 the neutrino masses are not equal to zero and the operators of neutrino fields are present in the hamiltonian in the form of orthogonal combinations. The difference is that in the theory M the neutrinos are Majorana particles, in the theory D they are Dirac particles. The hamiltonian (10) of the theory M does not conserve the muon L~and the electron Le lepton numbers. The hamiltonian of the theory D is conserving the sum L~+ Le. As we shall see, neutrino oscillations are described by identical expressions in both theories. At a variance with the theory D, theory M allows in principle the existence of the neutrinoless double beta decay and of other processes, in which L~+ Le is not conserved. However the probabilities of such processes turn out to be very small, so that existing experimental data permit to obtain only limits [23] (in no way stringent) on the parameters of the theory. Let us note that in the scheme D neutrinos are treated on the same foot as all the other particles (charged leptons, quarks), whereas in the scheme M neutrinos occupy a special place among the other fundamental particles. In the theory M every type of neutrino is associated with two states; in the theory D there are four states for every neutrino type. In this sense in the theory M there is no analogy between leptons and quarks. In particular the theory M, as it is easy to see [2], is equivalent to the scheme [10, 11], where there is only one lepton charge (which, in our case, is not conserved strictly, see footnote in section 2.1, page 230). B. The mass term of the hamiltonian in theory D has the form =

m1’v1v1 + m2v2v2.

Let us express =

i’1

and v2 through

mee~ve+ mp~vpv,.+

(38) Ve

~°.

and vp (see the expression (37)) and substitute in (38). We get (39)

Here =

mpe(Vpve +

~e~’p),

(40)

SM. Bilenky and B. Pontecorvo, Lepton mixing and neutrino oscillations

237

and mee mpp

m1 cos2O + m2 sin20, m 20 + m 20, 1 sin 2 cos sinOcos0(—m 1 + m2).

= =

mpe =

(41)

Clearly mee and ~ are the bare masses of the electron and muon neutrinos. The hamiltonian ~*‘ does not conserve separately L~and L~,and does conserve L~+ Le. Thus we are lead to the theory D if we make the assumption that the electron and muon neutrino masses are finite and that, together with the usual weak interaction, there exists the interaction (40), which does not conserve lepton numbers. Evidently the masses m1, m2 and the mixing angle 0 are related to the values mee, ~ mpe by expressions entirely similar to equations (29, 30) tg 20

=

in1,2

= ~(mee

2m~~/(rn~~ mee) (42) 2+ 4in~e). + ~ ±~.J(mee m~,j We have been considering the simplest theories with mixing of the fields of two neutrinos with finite masses. In refs. [7—9]the mixing of N ~ 2 neutrino types has been investigated. The corresponding results will be given briefly in the fifth section. —

—

3. Neutrino oscillations The theories we have considered above lead to neutrino oscillations. Here we shall consider such a phenomenon. Let us indicate by v~>and v~>the state vectors of the electron and muon neutrinos (that is the neutrinos participating in the usual weak interaction) with momentum p and helicity equal to 1. It follows from expressions (22) and (37) that —

lvi>

=

~

UjaIVa>

(1 = e, j.i).

(43)

Here Va> (a = 1,2) is the state vector of the neutrino with mass ma, momentum p and helicity equal to —1 (we assume that p ~ m 0). The orthogonal matrix U has the form (23). The vectors v0> describe Majorana neutrinos in the theory [2] and Dirac neutrinos in the theory [4,5]. We have HIVe>

=

EalVa>,

(44)

where H is the total hamiltonian and 2. E~=Jm~+p We have also Va> =

~

U 1eIV1>.

(45)

Let us consider the behaviour of a beam of neutrinos, produced in some ordinary weak process.

238

S. 34. Bilenk and B. Poniecorro. Lepton mixing and neutrino oscillatiosis

At the initial time

0) such a beam is described by the vector v1>. At the time t the state vector of the beam is given by the expression v,>1

(t =

= e_~t~v,> =

(46)

a=~.2 U,~ei~~~nt v,~>.

We treat neutrinos as stable particles. However in the schemes we have been considering the heavier neutrino (say v1) can decay into v2 and a photon. The probability of this decay, which was calculated on the basis of the theory with neutrino mixing [22], is given by the expression 2O cos20 —*

v2 +

~

=

~

sin

Here m 0 and M~are respectively the masses of the muon and of the charged intermediate boson (we assumed for simplicity that m1 >> m2). From the above expression it is seen that even in extreme cases (m1 = 1 MeV, 0 = ~ir), the life time in the lab. system of neutrinos having any momentum is larger than the age of the Universe by many orders of magnitude. Thus the neutrino instability can be ignored*. Let us come back to the behaviour of a neutrino beam. As it is seen from expression (46), in the theories we have been considering, such a neutrino beam is not described by a stationary state, as it should in usual theories, but by a superposition of stationary states. Clearly this happens because the vector v,> is not an eigenstate of the hamiltonian H. Let us expand the state vector (46) in terms of vectors v,.>. We have ~

VI> = I’

a~,..5,,(t)v1>,

(47)

e,p

where a51~ ~,(t)

=

LJ~~

a~,2

(48)

is the probability amplitude of finding v1 at a time t after the generation of v,. We have = ry~.2

=

Clearly in the case m1 ~ m2 and U1~~ Ve ~

~,

a5~.5~(t) = aVU.%O(t)

0 (that is there arise oscillations

vu).

The probability of the “transition” v, ~ v, is given by the expression ~v1

=

~..

=

~ UI~U,CUICUI.~cos (E~ E~)t. —

(49)

It is easy to see that the probabilities w5. ~ satisfy the relation I’

w~,1

~

%,(t) = 1.

* Let us make a remark, which seems to be instructive, although of no other relevance: if the life time of the heavier neutrino (v1) were smaller than the age of the Universe, neutrinos )~‘2)which are described by stationary states would be actually produced as a result

of the decay

~i

~2

+

~‘-

SM. Bilenky and B. Pontecorvo, Lepton mixing and neutrino oscillations

In the case p ~ E1

E2

—

=

in

1,

239

m~,which is of interest, we have

~1

—

m~

(50)

Making use of expressions (23), (49), and (50) we obtain 220(1 — cos 2irR/L), w5.5(R) = W~.5(R)= 1 ~sin w 220(1 — cos2irR/L). 55(R) = w~.5(R)= 4sin Here w~ 51(R) is the probability of finding v1~at a distance R from a source of v,, and

(51)

—

L

=

m~I

4irp/lm~

—

(52)

(53) 22O and L are connected with

is the oscillationoflength [24]. MLet that the parameters sin the parameters the theory [2]usbynote the relations sin22O = (mae — 4m~e + 4m~’

L=

(54)

4itp

(m~e+

m~)\/(mae m~~)2 + 4m~e Clearly the relations, which connect the sin22O and L with the parameters of the theory D [5—6] —

can be obtained from relations (54), after making the obvious substitution —*

~

~

—+

mpp,

~

-+

mpe.

Let us note that the main relations of this section (the expressions (47)—(49)) are valid also in the general case of N types of neutrinos. In such a case U is an orthogonal matrix (if CP invariance holds) and the summation over a must be done from 1 to N.

4. Possible experiments in which the hypothesis of neutrino mixing might be tested 4.1. General considerations In this section we shall make use of relations (51) and (52), derived for the case of two neutrino types. The general case of N types of neutrinos will be briefly considered in the sixth section. First of all let us note that the observation of effects due to neutrino mixing (which we shall call also oscillation effects) includes either one or both of the following two aspects: i) to observe the cosine term in the neutrino intensity and ii) to establish that either the constant term in w~ 1

is different from 1 or the constant term in w~1.51(R) (1’ 1) is different from zero (see relations (51) and (52)). In order to observe the cosine term, one must require that it would not vanish on averaging over the distance from the neutrino source to the detector (which means averaging over the source and detector dimensions and over the time of measurement, if the average distance sourcedetector is not constant) and over the neutrino spectrum (see formula (53)). In particular, a necessary (but not sufficient) condition for the observation of the cosine term in the neutrino intensity is that the neutrino source dimensions were smaller than the oscillation length and that the uncer-

240

SM. Bilenky and B. Pontecorvo, Lepton mixing and neutrino oscillations

tainty in the time of neutrino emission were smaller than the oscillation period. Further, as it is seen from relations (51) and (52) ~ 0 (1’

1)

and

w~~1(R)~ 1,

if the oscillation length is much larger than the distance between the neutrino source and the detector. Thus any oscillation effect might be observable only if L.~R.

(55)

Let us introduce the notation 2 = m~— m~j. M The expression (53) for the oscillation length can be rewritten as follows L

=

4icp/M2.

(56)

(57)

Substituting the numerical value of hc, we get the convenient formula L

=

2.5 1~%~’~2m.

(58)

2 = 1 (eV)2, for values of the neutrino momentum 1 MeV, 10 MeV and 1 GeV, Foroscillation example, iflength M the is, respectively, 2.5 m, 25 m, 2.5 km. As it is seen from (55) and (57), oscillation effects are observable if M2 ~ 4irp/R.

(59)

Consequently a search for neutrino oscillations at a given facility should be conducted in conditions of “minimum” values Pmin of the neutrino momentum, which can be reached in the given experiment, and of “maximum” values Rmax of the distance from the neutrino source to the detector. In order to characterise the sensitivity of various types of experiments, we introduce the parameter ~

=

41~Pmin/Rmax.

(60)

Oscillation effects can be observed in principle if M2 ~~ ~ are given for different types of neutrino sources [25].

In table 3 values of the parameter

Table 3* values of the parameter ~ the parameter which characterises the sensitivity of oscillation experiments for various neutrino sources Neutrino source

p,,,,, (Mev)

Rmax (m)

M,~,~~((eV)2)

Reactor Meson factory High energy accel. Sun

I 10 i0~ 2 x 10’

1O~ 102 10~ 1.5 x 1011

3 x 10~ 3 x 3 x 10 4 x 1012

*

Inequalities (55) and (59) are of course very qualitative. In

addition one should keep in mind that the values of Pmj, and Rm~ (and consequently of ~ have a significance which reflects our (subjective) understanding of the experimental situation.

SM. Bilenky and B. Pontecorvo, Lepton mixing and neutrino oscillations

241

One should underline again that this table should be taken only as an illustration. Let us note also that table 3 is quite conservative. For example in the paper [26] there was proposed an “oscillation” experiment, in which neutrinos produced at Fermilab would be detected in Canada at a distance of 1000 km. (Note that, in table 3, Rmax for high energy accelerators is assumed to be only 10 km.) 4.2. Limits imposed on the parameter M2 by existing data Let us discuss now the question as to whether there may be obtained limits on the parameter M2 = m~— m~ from an analysis of existing data. It should be underlined that the data which will be analysed here did not happen to be the results of deliberate experiments on neutrino oscillations. A long time ago the cross section of the reaction

I

(61) was measured [27] for reactor antineutrinos. It was found that aexp/atheor =

0.88 ±0.13,

(62)

where 5exp is the measured cross section and 5theor is the value expected according to the V-A theory for the cross section of the process (61). Let us consider, for simplicity, the case of maximum mixing (0 = +ir). For neutrinos of definite momentum we get from expression (51) =

~(1 + cos 2irR/L),

(63)

where ‘a~v~eis the intensity Of Ve’S at a distance R from the source, and ~ is the intensity which would be expected in the absence of oscillations at the same distance. In the experiment considered [27] R -~ 10 m, the effective momenta of Ve’S from a reactor are of the order of a few MeV. Taking into account the form of the ~e spectrum from reactors, we obtain the following limit M2 ~ 1 (eV)2.

(64)

In order to estimate the upper limit of the parameter M2 one can use also the results of high energy neutrino experiments. If oscillations do take place, a beam of “muon neutrinos” is in fact a mixture of v,~’sand Ve’S. It follows from (51) and (52) that the ratio of the intensities of electron and muon neutrinos is approximately 22O(1 cos 2irR/L). (65) sin The body of Gargamelle data obtained in the neutrino beam of the CERN 30 GeV accelerator has recently been analysed [18] in terms of the expression (65). The authors took into account the background of Ve’S from K~ 3decays and showed that the intensity of Ve’S which might have appeared as a result of neutrino oscillations, cannot be larger than 0.3 % of the v~,intensity (within 95 % confidence limits). In fig. 1 the upper limit of the parameter M is plotted as a function of sin 20, the upper curve corresponding to 95 % and the lower curve to 68 % confidence limits. In conclusion a few words about the only experiment [28, 29] with solar neutrinos, which has been performed. As we shall see in the next section, existing data do not exclude in any way the 2 presence of oscillations. This means that at present it is not possible to get an upper limit of M I~e;v

—

1.(R~ p)/I~.~(R, p)

~

242

5.34. Bilenkv and B. Pontecorco. Lepton mixing and neutrino oscillations

Sin 2~

Fig. I. Upper limit of the parameter 34

2for various values of the mixing angle 0 (m

=

(m~ — m~)’

neutrinos v1 and v2) [18]. The upper curve corresponds to confidence limits 95

‘~.

1 and m2 are the masses of the the lower curve to confidence limit 68

2 from the same experiment from experiment. The possibility will bethediscussed in the fifth section. of obtaining a lower limit of M 4.3. Possible experiments We proceed now to discuss the question about future experiments, which are designed to reveal neutrino oscillations and to throw light on the phenomenon. Let us remember that (see (51)) 22O(1

~ 1(R,p)

=

[1

—

~

—

cos 2icR/L)]I~,(R,p),

(66)

sin

where ~ 51(R, p) is the intensity of v1’s (1 = e, ~.t) with momentum p at a distance R from the v, source, and I~°,(R, p) is the intensity of v,’s which would be expected in the absence of oscillations. Again (see (52)) 22O(1 cos 2~rR/L)I~° 1v~ ~1(R,p) = sin 1(R, p), (1’ ~ 1; 1, 1’ = e, ~) (67) —

;

~

where ~ ~1(R,p) is the intensity of v,.’s at a distance R from the source of v,’s. One of the possible methods of looking for oscillations consists in the comparison of the averaged intensity of neutrinos of a given type with the intensity expected in the absence of oscillations, the cosine term in expressions (66) and (67) vanishing as a result of an average over the neutrino spectrum, the region in which neutrinos are produced etc. One can perform such experiments either by detecting neutrinos of the same type as those initially emitted or by detecting different neutrinos. We write for the averaged intensities vi ‘vi’;

=

(68)

~vl ; vl’~i’

vj =

dvi’

= =

—

;

vi~i

(1’

1),

(69)

where 220,

~vi

4 sin

4sin220

(70) (1’ ~ 1).

(71)

SM. Bilenky and B. Pontecorvo, Lepton mixing and neutrino oscillations

When the mixing is maximum (0

4,

(t5~, v,)min =

=

243

kir), the coefficient ö51 5~has the minimum value (72)

while the coefficient ö~,’~ 1has the maximum value (15~~v~)max=

4

(1’ ~ 1).

(73)

1t

The relation = can be tested in experiments with low energy antineutrinos from reactors. Should it turn out that the observed intensity ~ is smaller than the expected one this result would demonstrate that neutrino mixing does exist. In this case one could determine the mixing angle from expression (70). On the contrary, if the intensity ~ is equal to 1~,clearly it would not be possible to conclude that there is no neutrino mixing, because the result could be not only due to a zero value of the mixing angle but also to a large value of the oscillation length (L~R). At high energy accelerator, as well as at meson factory facilities, one can attempt to detect (at large distances between the neutrino source and the detector) electrons among neutrino induced events produced by beams of neutrinos, originating in it and K decays. In such experiments, the presence, if any, of electrons (not due to trivial causes (Ke3, ~ .)), would prove the existence of neutrino mixing, and, according to (71) would allow one to measure the mixing angle*. If electrons will not be observed, however, it is impossible in general to conclude that oscillations are 2 (see for example fig. 1). absent, and one may only obtain an upper limit for M As already stated, reactor and accelerator experiments can reveal the phenomenon of oscillations only if M2 ~ 102 (eV)2. If M2 ~ 102 (eV)2, cosmic neutrino and especially solar neutrino experiment [1] give us the only hope of observing oscillations. The solar neutrino problem will be discussed in the fifth section. In the experiments which have been discussed above it is possible, at best, to prove the neutrino mixing and to determine the mixing angle 0. If in such experiments the effect of lepton mixing will be really observed, then there arises the very important problem of getting information on the parameter M2 = lm~ m~ For this it is necessary to observe the cosine term in either the expression (66) or in (67), that is to measure the oscillation length (and consequently M2)**. Let us first discuss this problem in the case of reactor antineutrinos [25]. We shall assume that the oscillation length is considerably larger than the reactor dimensions. Since the oscillation length is a function of the neutrino momentum p. the antineutrino spectrum N(R, p) at a distance R from the neutrino source is different from the spectrum expected in the absence of oscillations. Information on the spectrum N(R, p), on one side, can be obtained by studying the spectrum of positrons emitted in the process ‘~e+ p e~+ n. On the other side, the initial (R = 0) spectrum of antineutrinos, that is the spectrum in the place where they are generated, can be determined with a reasonable accuracy from measurements of the spectrum of beta rays emitted by fission fragments at saturation. The comparison of these spectra should give us L(p). 0~~

—

I.

—~

*

If in nature there exist N

>

2 types of neutrinos, neutrino mixing is described by an N x N matrix; the relation (70) should be

substituted by the relation (80), given in the sixth section. In the present paragraph we limit ourselves to the discussion of the simplest case of two neutrinos. ** If it turns out that the expected and measured average intensities are not equal in either reactor or accelerator experiments, the determination of the oscillation length at these facilities is quite real and not too difficult. In the case of solar neutrino experiments, instead, the problem of measuring the oscillation length is much more difficult than that of proving neutrino mixing (see details in the fifth section).

244

S .31. Bilenkv and B. Foiitecouio, Lept on miring ;tii~lneutrino 0scillill

Ii

One can obtain information on the oscillation length also by measuring at various distances from a reactor the antineutrino intensity, averaged over a relatively narrow part of the ~iJespectrum. Let us discuss now possible experiments at meson factory and high energy accelerator facilities. in which the cosine term in expressions (66) and (67) might be determined. Let us assume that the effective dimensions of the neutrino source are considerably smaller than the oscillation length. A good method would be to measure, at a given distance R from the source of v0’s, the ratio r(p) of the number of electrons to the number of muons in neutrino induced events. If the intensity of Ve’5 in the region where neutrinos are initially produced can be neglected, we obtain from (66) and (67) for the ratio r(p) the following expression:

r(p)

— —

1

220(1 4 sin 220(1 —

4sin

—

coscosa/p)’ a/p)

—

where a=4RIm~—in~j.

(75)

Of course it would be useful to measure also r(p) at various distances R. We list now the advantages and disadvantages of possible experiments at facilities of the meson factory type (proton energy 500—800 MeV) and of the high energy accelerator type (proton energy of a few hundreds of GeV). The advantages of meson factory* experiments are: 1) small dimensions of the neutrino production region, 2) relatively small oscillation lengths (small neutrino momenta), and 3) the appearance of electrons in neutrino induced events is practically a direct indication that oscillations take place (since there is no sufficient energy for ir°production, which gives the most dangerous background). The main disadvantage of such experiments is that it is impossible to measure directly the “expected” intensity I~ (since the original vs’s, having an energy smaller than the muon mass, are “sterile”). The advantages of high energy accelerator experiments designed to reveal oscillations are: 1) large distances between the neutrino source and the detector, 2) the possibility of measuring directly the “expected” intensity I~,and 3) the fascinating possibility that heavy charged leptons are produced in the oscillations process (if there existed new types of neutrinos, coupled to charged heavy leptons in the weak current, and if the fields of all neutrinos were mixed). The disadvantages of such experiments are: 1) relatively large oscillation lengths (large neutrino momenta) and 2) the large electron background from neutral pions. In conclusion a few words about some known concrete proposal and projects of experiments designed to reveal neutrino oscillations. Such experiments at reactor facilities are being planned at least in three laboratories: at the Moscow Kurchatov Institute of Atomic Energy [30], at the Grenoble Institute Laue-Langevin [31], and at the Irvine University of California [32]. An oscillation experiment is soon to be performed at the Brookhaven accelerator [33]. The originality of the proposal is that the 30 GeV proton accelerator will be used as a meson factory (at a proton energy of only 800 MeV and at a beam intensity of 1014 protons per second). *

We have in mind here a facility where v

5 beams ata meson factory are obtained from decaying pions in flight. and not from stopping pions. In particular neutrinos from pions of a few hundred MeV trapped in a magnetic bottle, might be of interest (see V.M. Lobashev and O.V. Sedryuk, Nucl. Inst. Meth. 136(1976)61).

SM. Bilenky and B. Pontecorvo. Lepton mixing and neutrino oscillations

245

In conclusion we would like to mention the paper [26], in which it was proposed to place the neutrino detector in Canada at a distance of 1000 km from the Fermilab accelerator. Without doubt this type of experiment is not only of interest in connection with the oscillation problem, but also because it gives the possibility of measuring directly a distance between two points on the Earth separated by enormous quantities of matter.

5. Solar and cosmic neutrinos 5.1. The Brookhaven experiment In this section we shall consider various possible cosmic neutrino oscillation experiments. Let us start with solar neutrinos. From table 3 it can be concluded that solar neutrino experiments are an extremely sensitive method of revealing neutrino oscillations. This is due to the fact that the energy of solar neutrinos is relatively small and the Sun—Earth distance is enormous. For many years Davis and collaborators [28] have been conducting an experiment with the aim of detecting solar neutrinos (see in the paper [28] the relevant references). In this experiment the Cl—Ar method [34] is used, wherein neutrinos are supposed to be detected through the reaction 37C1 e + 37Ar. (76) Ve + —*

The isotope 37Ar undergoes K-capture decay with a period of 35.1 days. The detector-target, consisting of 3.8 x l0~liters of C 37C1 atoms), is placed at a depth of 4400 m 2C14 (2.2 x l0~° H 37Ar plus the argon 20 equivalent in a goldmine in South Dakota (the He isotope tracer which is added) is first extracted from the(U.S.A.). detectorArgon through purging, then separated from the He mass and finally introduced in a small proportional counter, where practically every K-capture event is registered~Few words about the advantages of the Cl—Ar method. 1) The simplicity of extracting few 37Ar atoms from an anormous quantity of C 2C14 (let us stress here that in spite of this “simplicity” the experiment of Davis et aL is heroic indeed); 2) C2C14 is a cheap, non-inflamable, liquid; 3) The fact that K-capture events are accompanied by the release of 2.8 keY gives the possibility of using proportional counters as low background detectors [35]*. The low level of the background is obtained not only through pulse has amplitude analysis butbecause also by measuring the formionization of the pulse: 37Ar K-capture a fast rising time, the corresponding in a pulse from the the counter gas is well localized, which is not the case for the ionizion from most background processes [36]. At the present time the effective counterbackground in the Brookhaven experiment is only 0.5 counts/35 days (in the energy region 1.5—5 keV, if the proper pulse form discrimination has been made). Between 1970 and 1975 the exposure and the counting runs had a length of a few months. At the present time the exposure time is decreased to about a month. 5.2. The experimental results In fig. 2 the results of the experiment of Davis et al. are presented. The rate of production of 37Ar atoms over the detector volume is plotted against the run number in the period 1970—1975. *

High gas amplification coefficient

(~10’)

proportional counters have been first developed just for the Cl—Ar method.

246

SM. Bilenky and B. Pontecorvo, Lepton mixing and neutrino oscillations i.E 11 —

1.2

j,.

SOLAR NEUTRINO RATE (SNUI ~-STANOARD

-I

5SOLARM0(~

0’. _____________________ 31 32 1970 1971 1972 1973 1974 1975 1976 1977 YEAR 37Ar production in the Cl—Ar Brookhaven solar neutrino experiment [28, 29]. At left the scale indicates the number of Fig. 37Ar 2. atoms/day Rate of produced in the detector volume of 3.8 x 10’ liters. The unit SNU at right is iO’~37Ar atoms/37C1 atom sec.

The left ordinate axis scale is in units of 37Ar atoms/day; the right ordinate scale is in the so called solar neutrino units or SNU units (1 SNU = 10_36 37Ar atoms/sec 37C1 atom), the units in which usually theoretical results are presented. An arrow on the right side indicates the 37Ar production rate by neutrinos, calculated according to the standard solar model (see [29] and [37]). In fig. 2 also the expected cosmic ray background and the 37Ar production rate averaged over all measurements are indicated. The results of the experiment are presented in table 4. Table 4 Results of the Brookhaven solar neutrino experiment 3’Ar atoms/day 35Ar production rate averaged over the period April 1975—February 1976

0.32 ±0.08

37Ar production rate by cosmic g’s and

0.08 ±0.02

va’s

37Ar production rate ascribable to solar neutrinos

0.24 ±0.09

If we express in solar neutrino units the average 37Ar production rate given in table 4, we have 1.3 ±0.4 SNU (1970—1975), a value which must be compared with the value Jc~(E)a(E)dE=

6 ± 2

SNU,

(77)

calculated (see [29], [37]) on the basis of the standard model of the Sun. In expression (77), 4t(E) is the flux of neutrinos with energy E and a(E) is the cross section for the reaction (76).

SM. Bilenky and

B. Pontecorvo, Lepton mixing and neutrino oscillations

247

5.3. Conclusions Which conclusions can be drawn from the above results? Davis and Evans write [28]: “In view of uncertainties described above and uncertainties in background processes we regard this result (that is the already quoted rate of 37Ar production (1.3 ±0.4) SNU) as a 1o upper limit to the solar neutrino flux of 1.7 SNU”. One can only welcome such a conservative approach. Let us keep in mind, however, the wide-spread opinion held in particular by the authors of ref. [29] for number of years, that there is a “solar neutrino puzzle”, that is that the solar neutrino flux is much smaller than it should be. Our impression, which is not really in contradiction with the quoted conclusion of Davis et al., is that these authors have, apparently, observed a solar neutrino signal of the order of magnitude of the expected value, although, probably, smaller than such value. The expected 37Ar production rate by solar neutrinos was calculated [37] according to the universally recognised basic hypothesis that thermonuclear reactions of the hydrogen cycle (see table 5) are the source of energy in the Sun. It is assumed that the primordial chemical constitution in the Sun was uniform all over the Sun volume. In addition it is taken that nuclear data, relevant to the problem at issue, are known with sufficient accuracy. Table 5 Thermonuclear reactions of the hydrogen cycle p(p,e~v,)

99.75°/) It.

p(ep, v,)

0.25

j

((~He,2p)~He ((e, v )7Li(p, 4He)4He 2H(p, y)’He 4He, y)7Be 1(p Y):B -. v, + e~+ SBe* 1(

-+

4He + 4He

86 % 14 °/ 0.02%

The maximum energies of the neutrinos produced in various reactions, the corresponding neutrino fluxes at the Earth surface and the contribution of every process to the 37Ar production rate are presented in table 6. Table 6 Contribution of various reactions in the Sun to the expected “Ar production rate in the process v, + “CI —+ e + “Ar Neutrino source

Maximum neutrino energy (MeV)

p + p -. d + e~+ v, 0.42 p + e + p -. d + v~ 1.44 (monochr.) 7Be + e -. 7Li + v, 0.86 (90%), 0.38 (10%) 8B —~ SBe* ±e~+ v, 14

Neutrino flux (cm’2 sec~)

Expected 37Ar production rate (SNU)

6 x iO~° 1.5 x io~ 4.5 x iO~ 5.4 x 106

0 —0.3 —1 .

6+2

The threshold of the reaction Ve + 37C1 e + 37Ar is 0.81 MeV, a value considerably above the maximum energy of neutrinos emitted in the reaction p + p d + e + Ve, which is the main thermonuclear reaction in the Sun*. As is seen in table 6, the main contribution to the 37Ar —*

-+

*

We listed above the considerable advantages of the Cl—Ar method; the high threshold of the v, + “Cl

is clearly a definite disadvantage ofthis method.

-‘*

e

+ ‘7Ar reaction

248

SM. Bilenkv and B. Ponlecorvo. Lepton mixing and neutrino oscillations

production rate is due to high energy neutrinos from the 8B decay (this happens because of the rapid increase with neutrino energy of the relevant cross section). The expected neutrino flux from the 8B decay is a very small fraction of the total solar neutrino flux and is strongly depending upon the model parameters. In particular, its value is extremely sensitive to the temperature in the central region of the Sun. In spite of these circumstances, the general opinion (see for example reference [29]) is that in the numerical value 6 ±2 SNU of the calculated rate of 37Ar production by solar neutrinos the indicated error is not seriously overestimated. 5.4. The solar neutrino “puzzle” and neutrino oscillations Let us remember that if there are neutrino oscillations, the solar neutrino flux in the case of two neutrino types may be as small as 4 of the flux expected in the absence of oscillations [1] (see expressions (66) and (68) for maximum mixing 0 = fir). If the mixing angle is different from *n, the neutrino flux may have any value in the interval from one to 4 of the “expected” flux. In the case of N ~ 2 neutrino types, as it will be shown in the sixth section, the solar neutrino flux may be as small as 1/N of the value expected in the absence of oscillations. If the “solar neutrino puzzle” really exists, that is if the signal in the experiment of Davis et al. is really smaller than the calculated one and if the calculations at issue are reliable, then the puzzle solution based on neutrino mixing seems to us much more natural than any other solution which has been proposed either in terms of elementary particle physics or astrophysics. These suggestions are listed in ref. [29], wherein the reader may find the relevant literature. They include the hypothesis that neutrinos are unstable particles, so that they are lost on their way from the Sun to the Earth [38], as well as the following exotical astrophysical assumptions: the energy in the Sun does not arise from thermonuclear reactions; inside the Sun there is a black hole; the Sun is in a transient state and its luminosity, due to the extremely slow process of photon diffusion from the central region to the surface, is much larger than the “internal luminosity”, information on which is obtained almost right away from the solar neutrino experiment; the calculations based on the Sun uniformity of composition are wrong, the internal and external regions having entirely different compositions, since some time in the past, as it were, the Sun acquired a considerable part of its mass from outside; etc. From the elementary particle physics point of view, lepton mixing is a reasonably likely and quite attractive hypothesis. Thus if we believe that the solar neutrino flux is really too “small”, we have in our hands an explanation reasonable, not exotic and in agreement with modern thinking about quark--lepton analogies: neutrino mixing and its corollary, the neutrino oscillation phenomenon. Let us note also that the solution of the “neutrino puzzle” in terms of neutrino oscillations was not invented ad hoc, for the sake c~fexplaining the result of the experiment of Davis et al., as it was the case for all the other explanations of the puzzle. In conclusion let us remark that if lepton mixing were the correct solution of the neutrino puzzle (if any), this would imply that: 1) the degree of neutrino mixing is considerable (0 2) the oscillation length L(~)= 4ir~/~ m~ m~ is smaller than the Sun—Earth distance, a circumstance requiring that m~ m~j~ 10_it (eV)2 ~-‘

—

—

*

* This limit is obtained by taking into account that the effective neutrino momentum, in the experiment of Davis et al. is about 10 MeV.

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249

5.5. Oscillations andfuture solar neutrino experiments In order to test the neutrino mixing hypothesis by measuring the average neutrino intensity, one must determine the coefficient ö,~.~ of expression (68). Clearly for this purpose one should know with sufficient accuracy the value 7~of the intensity expected in the absence of oscillations. As it has been already stated, the uncertainty affecting the calculated intensity is quite large in the case of the Cl—Ar method (see (77)) and, in addition, can be seriously underestimated (because of the strong dependence of the expected intensity upon the model parameters). The expected intensity 1~could be estimated in a reliable way [37], were it possible to detect a large fraction of the solar neutrinos, namely the low energy (E <0.4 MeY) neutrinos emitted in the reaction p + p d + e~+ v~.As a matter of fact, the total flux ofsolar neutrinos I,,, is cdnnected with the luminosity L0 by the relation 2Q, I~.= 2L®/4nR where R is the Sun—Earth distance and Q ~ 25 MeV. This relation is easy to understand and simply expresses the fact that the final result of a thermonuclear cycle in the Sun is the transformation of 4 protons in a 4He nucleus, 2e~and 2v~with an energy release Q: 4p—+4He+2e~+2Ve+Q. Thus the problem of measuring the coefficient ~ (see (68)) could be solved if a new type of neutrino detector, capable of detecting neutrinos from the reaction p + p d + e + Ve were operative. Such detector is being designed at the present time; it is based on the radiochemical Ga—Ge method. The reaction* Ve + 71Ga 71Ge + e. has a threshold of only 230 keY. The decay period of 71Ge (K-capture) is about 11 days. The chemical problem of extracting a few 71Ge atoms from a large quantity of a gallium compound, as well as the problem of counting the 71Ge K-capture events are now being solved [28, 39]. The realization of a full scale detector requires several tens of tons of Ga, which is more than the today annual production. Nevertheless, this experiment, apparently, will be made. In conclusion let us make two remarks. 1) Different methods of detecting solar neutrinos have been proposed. We shall not report on such work and simply quote the relevant literature [40]. 2) At the present time at the Neutrino Observatory of the Institute for Nuclear Research, Academy of Sciences, of the USSR, Baxan Yalley, Northern Caucasus, an experiment is being prepared to detect solar neutrinos by the Cl—Ar method [41]. The quantity of C 2C14 which will be used in such experiment is about five times larger than in the experiment of Davis et al. i5~

—*

+

—~

5.6. Solar neutrinos, time intensity variations and oscillation lengths We have already stressed that a possible discovery either at reactor or at accelerator facilities that öV; v~ 1 and/or 5, 0 (see (68) and (69)), i.e. that there takes place neutrino mixing, would imply that at the same facility it would certainly be possible to perform experiments of comparable difficulty, the aim of which is to measure the oscillation length and consequently the difference in the neutrino squared masses. For solar neutrino experiments the situation is entirely different, the problem of determining the oscillation length being incomparably more difficult than that of establishing neutrino mixing, if any, from a measurement of ö,.,. *

The use of this reaction for the detection ofsolar neutrinos was proposed by V.A. Kuzmin, Zh. Eksp. Theor. Fiz. 49 (1965) 1532.

250

SM. Bilenkv and B. Pontecorro. Lepton mixing and neutrino o,sc’illation.s

In this paragraph we discuss the possibilities of performing solar neutrino experiments capable of yielding information on the oscillation length. To start with, let us assume for simplicity that the detector is sensitive only to neutrinos of momentum p. The Ve intensity then is given by expression (66), if we are confining ourselves to the discussion of the simplest case of two neutrino types. If the oscillation length L(p) is smaller than the dimension r l0~km of the Sun region wherein effectively neutrinos are produced, the cosine term in the neutrino intensity is vanishing and at the Earth surface only the mixing effects discussed above (s,., 1) can be observed. Under different conditions, as it was pointed out by Pomeranchuk, there may arise time variations of the solar neutrino intensity, which are connected with the Sun—Earth distance R(t) not being constant in time. In principle the variations will be present if the oscillation length L(p) is larger than r and if changes in the Sun—Earth distance comparable to L(p) do take place, that is if r < L(p) ~ AR, where AR 5 x 106 km is the maximum change in R (equal to the difference in the semi-axis of the Earth orbit). The observation of the time variations is an extremely difficult task, because even the maximum relative change in the Sun—Earth distance (AR/R ~ 0.03) is in most situations much smaller than the relative momentum spread for neutrinos which are effectively detected (the detector response only to neutrinos of a definite momentum being an idealization). Thus, except for special circumstances, the cosine term will vanish upon averaging over the neutrino momentum. Is there any possibility of observing actually time variations of the solar neutrino intensity? It is clear from above that one must rely upon the nuclear reactions inside the Sun which do produce “neutrino lines”, that is monoenergetic neutrinos (see table 6). The problem of observing time variations would be relatively simplified if there existed the possibility of detecting exclusively or almost exclusively neutrino lines from such reactions~.Let us note here that the intensity variations of neutrinos with definite momentum p would have a period of the order of hundred days if L(p) AR 5 x 106 km and of the order of a fraction of a month if L(p) ~ 5 x l0~km. Period variations smaller than, say, 10 days, should not be considered even from an abstract point of view, for the following reasons. Neutrinos emitted in such reactions as e + 7Be v~+ 7Li and p + e + p d + are not strictly monoenergetic lines, since the thermal energy of the particles kT is 1 keY. The minimum momentum spread Ap/p is about equal to i0~ at 1 MeV neutrino momentum. Then for intensity variations with periods of the order of one day, the relevant relative change in the Sun—Earth distance is smaller than the minimum spread Ap/p, so that the cosine term is vanishing when averaging over the neutrino momentum. To summarize, there is in principle the possibility of measuring oscillation lengths of I MeV neutrinos, if they are in the 5 x l0~km—S x 106 km range, by investigating time variations of the solar neutrino intensity with periods in the range from a fraction of a month to hundreds of days (this would correspond (see eq. (58)) to an M2 m~ m~range of values from 0.5 x 10_8 (eV)2 to 0.5 x iO~(eV)2). The possibility of the exclusive detection of neutrino lines is at present quite remote from an experimental point of view. Let us then come back to the Cl—Ar method. In table 6 one can see that about 15 % of the expected solar neutrino signal in the detector is due to 0.86 MeV neutrinos emitted in the 7Be + e 7Li + reaction. This means that at much improved accuracies the —*

—~

Ve

—

—+

Ve

* There is a remote, but possible radiochemical method for detecting almost exclusively neutrinos from the reaction p -~-c -e p -.d + v, in the Sun. The method would make use of the process s’~+ 7Li -. e + 7Be (see. V.A. Kuzmin and G.T. Zatsepin [40] and J. Bahcall [40]).

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251

usual Cl—Ar method might allow the observation of time variations of the intensities, due to the cosine term. As already noted, the relevant percentage change in the Sun—Earth distance is usually considerably smaller than the relative typical differences in the solar neutrino momenta from various nuclear reactions. Thus, it is desirable to utilize this fact in attempts to obtain information about oscillation lengths*. This suggests that solar neutrino detectors having different thresholds should be used; it is not excluded that an analysis of the body of data from various experiments will allow valuable information to be obtained on the question at issue. To conclude this paragraph we shall make the following remarks. We have already stated that the cosine term in the solar neutrino intensity (see eq. (66)), as a rule, is vanishing, so that the average Ve intensity (for two neutrino types) can be found in the interval from 4 to 1 of the expected intensity. In the general case of N ~ 2 neutrino types the average Ve intensity can change from 1/N of the “expected” intensity 1~to 1~. Let us note here that even in the case of two neutrino types the Ve intensity may turn out to be considerable smaller than the expected one [2]. Actually let us suppose that the oscillation length, corresponding to an effective neutrino momentum p, is comparable with the Sun—Earth distance. In that case the cosine term in expression (66) of the neutrino intensity may “survive” on averaging over the momentum. Thus, there arises one more possibility (which is, true, connected with a rather special value of the oscillation length) of accounting for a low value of the solar neutrino signal in the Cl—Ar experiment. 5.7. A question about coherence of neutrino beams Let us discuss the condition, considered in ref. [42], which is required for the coherence of neutrino beams. According to [42] the packet dimensions of a solar neutrino beam may be quite small. This effect, due to nuclear collisions in the Sun, is similar to the well known increase of the natural width of spectral lines. According to ref. [42] the dimension of a solar neutrino packet is d

‘-~

10_6 cm.

(78)

Clearly coherence is maintained provided that RA/3 ~ d,

(79)

where R is the Sun—Earth distance and 2= 2ir/pL = m~— m~I/2p is the velocity difference of neutrinos of momentum p and masses m 1 and m2. Making use of the estimate (78) the condition (79) for 1 MeV neutrino can be rewritten as 4R. (80) L ~ 10’ For example, if m~— m~ 1 (eV)2, then L 2.5 m and the condition (80) is certainly not fulfilled (this is just the case considered in ref. [42]). We often have underlined, however, that a necessary (although not sufficient) condition for the *

See also B. Pontecorvo, Uspekhi Fiz. Nauk 104 (1971) 3, where a proposal is made to utilize the considerable spread i.~p/pof

neutrinos from 8B beta decay.

252

S,JI.1. Bilenk~’and B. Pont ecorio, Lepton mixing and neutrino oscillations

“survival” of the cosine term is that the effective source dimension r must be smaller than the oscillation length r~
(81)

Since r iO~R, the inequality (81) is comparable, in essence, with the inequality (80). Thus we conclude that the condition of coherence (79) does not impose any conditions supplementary to the condition (81). 5.8. Oscillations and cosmic neutrinos The phenomena of neutrino oscillations, if it does take place, could be of importance in cosmic ray neutrino* experiments. Let us give a few examples. 1) At the underground Neutrino Observatory of the Institute for Nuclear Research Academy of Sciences of the USSR an experiment is being prepared [43], in which there will be detected high energy muon neutrinos emitted by mesons, which are produced in collisions of cosmic ray protons with nitrogen and oxygen nuclei in the atmosphere. The energy spectra and other properties of those neutrinos have been calculated and the results are given in ref. [44]. High energy muons produced by v,~’sinteracting with nuclei in the Earth will be detected by 8 hodoscope plane systems (every one of which has an area of 1500 m2) of organic scintillators. The scintillator systems are in coincidence, the logic giving information on the muon trajectory and also establishing whether the detected muon has come either from “above” or from “below” (in the last case it is produced by a muon neutrino impinging upon the Earth opposite face and passing through the Earth). The average neutrino momentum in such experiments is 5—10 GeV, and the distance from the neutrino source to the detector is R ~ iO~km for neutrinos coming from the Earth opposite face. Making use of formula (66) it is possible to test the neutrino mixing hypothesis by comparing the measured and “expected” vu intensities. The sensitivity of those experiments for testing neutrino mixing is, in principle, quite high [43], the value of M~ 1~ (see the definition 2. Thus, these experiments have a sensitivity intermediate between (60)) that being M~,,10 iO~ (eV) of the experiments wherein artificial (reactor, accelerator) neutrinos are used and that of the investigations wherein solar neutrinos are used. However, the statistical accuracy which can be attained (~100 events/year) is quite low, and, in addition, the intensity, spectrum etc. calculations are relatively inaccurate. 2) At present a project for the investigations of cosmic neutrinos with the help of an enormous detector (> I O” tons of ocean water) is being widely discussed (project DUMAND, which is an acronym for Deep Underground Muon and Neutrino Detector [45—46]).The detector is supposed to be located at an ocean depth of about 5 km. The high luminosity of this detector allows oscillation investigations to be made. 3) If there exist more than two neutrino types with mixing of all neutrinos, cosmic neutrino oscillations may result in the appearance of new type neutrinos, the field of which may be present in the weak interaction hamiltonian together with heavy charged lepton fields. Such a mechanism could explain [47] the contradiction between the so called Kolar Goidmine events [48] and the high energy neutrino Batavia data, if the oscillation length is about 100 km (a typical distance *

The importance ofcosmic ray neutrinos investigations as a tool inelementary particle physics was first underlined by MA. Markov

and TM. Zheleznych, NucI. Phys. 27(1911)385.

SM. Bilenky and B. Pontecorvo, Lepton mixing and neutrino oscillations

253

between the production and detection locations for high energy neutrinos produced in the atmosphere and detected underground).

6. Oscillations in the general case of N neutrino types 6.1. Left-handed fields Until now we have been considering neutrino oscillations in the case of two neutrino types. It is of interest to consider also the general case of an arbitrary number N ? 2 of neutrino types. This study was presented in refs. [7—9],and we shall briefly expose below the content of these papers. The relation (22) of the theory M [2] with two Majorana neutrinos can be generalized as follows for the case of N neutrinos: U,C~,,L.

V,L =

(82)

Here v, (1 = e, ~.i, M...) is the field operator of the neutrino participating in the usual weak interaction (electron, muon and others), 4~(a = 1, . .. N) is the field operator of the Majorana neutrino of mass me, U,~is an orthogonal (if there is CP-invariance) N x N matrix. Similarly, instead of the relations (37) of the theory with 2 Dirac neutrinos [4, 5] we have [9~ VIL =

~ U1eVeL,

(83)

where ye (a = 1,. N) is the field operator of the Dirac neutrino of mass me. Let v,’s be produced as a result of a weak process. The probability of finding a neutrino v, at a distance R from the place of production of the v1 neutrino turns out to be . .

w~p1(R)

=

~ ~

+ ~ U1aU1’eUig’Ui’e cos 2rrR/Lag’,

(84)

o~c’

where Lgc~=

4~p/Im~ — m~’I.

(85)

If the oscillation lengths are smaller than R and if the average values of cos (2ltR/L,a’) do vanish in expression (84), one finds that the average v, intensity I~,.~and the intensity expected in the absence of oscillations I~°~ are connected by the expression 1vg ; V1 = vlt~l, (86) ;

where vi =

e~1

~

(87)

Making use of the orthogonality of the matrix U it is not difficult to show that ~

vi)min =

1/N.

(88)

254

sM. Bilenkv

and

B.

Pont ecorro. Lepton mixing and neutrino oscillations

The minimum value of ~ ,~takes place when (89) U~= U,22 = ... = U,2~= I/N. Thus, in the case of N neutrino types, the solar neutrino intensity as a consequence of oscillations may be as low as 1/N of the intensity expected in the absence of oscillations [7—9]. From expression (84) it follows also that the average v, intensity I~..~,, 1!’ 1: 1, 1’ = e, ji. M . is related to the v, intensity 1~by the relation =

or 1,

(90)

vj~i~

where vi

If the ~

~

=

(1’

1).

value is equal to its minimum v,)max

=

~

(91) I/N,

we have, making use of expression (89)

=

(92)

(1’ ~ 1)

Thus, if the average probability of “transition” v, v, (for a given 1) is equal to 1/N, the “transition” probabilities v, v1. (1’ I) are also equal to I/N. Let us note that for the N = 2 case, if —+

—#

—

2

0 is equal to ~ir and =

If N [42]

4.

(93)

2 and if the coefficient O~ ~ is equal to the minimum value I/N, then in the general case we have for I’ 1: >

> I/N. (94) As a matter of fact, the elements of the orthogonal matrix U cannot satisfy the conditions (89) simultaneously for different values 1, if N ~ (n is an integer number)~.

6.2. The general case of left-handed and right-handed neutrino fields Let us assume that in the lepton-number violating hamiltonian there appear both left-handed and right-handed components of neutrino fields and that the bare neutrino masses are different from zero. Then the eigenstates of the total hamiltonian are the states of 2N Majorana neutrinos [8] (in the weak interaction hamiltonian there are present N types of four-component neutrino). 7eL’ i.’e ~ VPL, ... (~cL, In such a case in a Ve beam there arise oscillations of the type Ve # ~ ~‘e~ are antineutrinos with negative helicity). The minimum value of the parameter 5.,,.~in this theory is (Ove;ve)min

=

1/2N.

(95)

* For example, when N = 3, the relation (89) will hold, if the square of every element of one of the lines is equal to 1/3. Clearly the orthogonality ofthe matrix U implies that the elements ofanyone ofthe two remaining lines cannot satisfy this condition.

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255

Let us note that such a scheme would become attractive, if right-handed currents, as well as left-handed ones, could be present in the weak interaction hamiltonian. This possibility is being widely discussed in the literature (see, for example, refs. [49—50]). 6.3. Concluding remarks

1) In table 7 the values (ö,~v)min are given for all the schemes of neutrino mixing we have been discussing. Table 7 Schemes of neutrino mixing and oscillations in v, beams Number of neutrino types

Number of neutrino states

1. Two (v,, v

5)

Bare neutrino masses

4

0

Particles with definite masses

Oscillations in v, beams

(&s..v,)min

Ref.

2 Majorana neutrinos

v~~ v,~

1/2

2

2.

-“-

8

+0

2 Dirac neutrinos

v, e~

1/2

5,6

3.

-“-

8

+0

4 Majorana neutrinos

v, ~ v~, v, ~ ~L’ ~o ~

1/4

8

N Majorana

v, ~ v~,

i/N

7

Ta.

2N

N>2(V,,56,VM,...)

0

neutrinos 2a.

3a.

4N

-“-

4N

-“-

+0

+0

N Dirac

~6 VM,...

v, ~ vp, ~ VM,.

1/N

7,9

neutrinos 2N Majorana neutrinos

v~~z2v6, v~~ VM,... v, ~~L’ V, ~*VUL, V~#VML,...

l/2N

8

2) In recent SLAC experiments [51] rather a convincing evidence has been obtained for the existence of a heavy charged lepton with mass 1.9 GeV. In connection with such a discovery the possibility that there exist N > 2 neutrino types seems now quite natural (in analogy with the known leptons, a new type of neutrino might correspond to every new charged lepton). 3) All the theories which have been considered above are based on the assumptions that the neutrino masses m1, m2,.. . are different from zero and that the neutrino fields appear in the weak interaction hamiltonian in a mixed form. Clearly from the experiments in which there were found upper limits for the electron and muon neutrino masses one can find also limits [5,42] for the masses m1, m2, The “best” upper3Hlimit the mass of the electron neutrino has been found and for is [52] by measuring the beta spectrum of m~< 35 eV. (96) ...

The upper limit for the mass of the muon neutrino has been found [53] in the investigation of the K~ 3decay and is m5 <0.65 MeY.

(97)

256

5.

Bilenki.’ and B. Pontecort’o. i.eptoil in ring aiiil neutrino o.scilhitioii.s

‘ii.

Let us consider the case of two neutrino types with neutrino mixing and let us assume that the mixing angle is close to ~7r. From (95) we have 2. ~ < (35 eV) Taking into account that in~ ,n~j~ I (eV)2 (see section 4.2, eq. (64)) we have —

m 1,rn2 <35eV.

7. The p

—~

(98)

ey decay and neutrino oscillations

7.1. The case when there are only neutrinos In the theories considered above, in addition to neutrino oscillations, there must take place processes of the type 1u ey, 1u 3e, etc., which are absolutely forbidden in the ordinary theory. These processes arise in high order perturbation theory (see diagrams in fig. 3). —*

—+

/

~,v2 Fig. 3. Diagrams ofthe process p

—~

~

e~’with virtual neutrinos

i’~

and

‘2.

The ratio Ru of the ~ ey rate to the ~ e~v5~ rate has been first calculated according to the Weinberg—Salam theory in ref. [22] and is given by the following expression: —+

=

r’~

FCU

—+

=

~

e

VeVu)

—~

~ 32 it

\

(99)

M~ j

where M~is the mass of the intermediate charged boson (M~~r 37 GeV). It is easy to verify that even in the case where the values 35 eV and 0.65 MeY (the upper limits of the Ve and v~masses) are taken for m1 and m2, the value Ru turns out to be smaller by many orders of magnitude than the experimental upper limit. For this case we have: Ru

<3

x

(100)

10-26,

whereas the experiment [54] gives R~XP <2.2

x

(101)

l0~8.

2 given by the oscillation If the mixing angle is not small, we must use the limit m~ ~ I (eV) ideology (see relation (64)). In this case the ratio R~is even smaller than the value (100) by many orders of magnitude. It can be shown as well that in the general case of N types of finite mass neutrinos with neutrino mixing the ratio Ru is many orders of magnitude smaller than the experi—

SM. Bilenky and B. Pontecorvo, Lepton mixing and neutrino oscillations

257

mental limit (101). Thus, if neutrinos are the only existing neutral leptons, there is only one method of revealing lepton mixing: neutrino oscillations*. 7.2. Heavy leptons The situation might change [55] radically**, if in nature there exist heavy leptons. (It should be noted that mixing in a scheme with heavy leptons was discussed in ref. [57].) Let us consider the decay ~ ey in a scheme with right-handed currents. As an example, let us assume that together with the left-handed doublets of the standard theory —*

(vu~

(Ve~

(102)

\eJL there are right-handed doublets (N~~

(N~~

(103)

,

\P’JR

where eR = Ne =

~{l

—

y5)e

etc., and

N1 cos 0’ + N2 sin 0’,

(104)

Nu= —N1sinO’ + N2cosO’. Here N1, N2 are the field operators of heavy neutral leptons with masses M1 and M2 (M1, M2 > MK, MK being the kaon mass) and 0’ is the mixing angle. Clearly the charged lepton current has the form

j~=j~+j~,

(105)

where the first term is given by the expression (37) and .1r~ = NeRY~R+

(106)

NuRYS/AR.

Together with the diagram of fig. 3 supplementary diagrams arise (fig. 4) with virtual heavy neutral leptons N1 and N2. Neglecting the negligible contribution of fig. 3 diagrams, we find the following expression: Ru

=

3~(MiM2)i20~201

(107) 2/Mw

From (101) and (107) and for the case of maximum mixing we get IM~— M~“ *

<

1.4 x 10

‘.

The physical reason why the oscillation method is so sensitive is that it allows process amplitudes to be measured, instead of

amplitudes squared (as is the case for usual methods, whereby process probabilities are measured). A good illustration of the sensitivity ofoscillation methods is the measurement ofthe mK~— mKS mass difference, the only second order weak effect, which has been measured with high accuracy in relatively simple experiments. ** After the main content of our paper [55] had been already written for the present review paper, we became acquainted with a number of similar papers, in which the processes g —~ey, j~ ~ 3e etc. have been considered on the basis of heavy lepton mixing. we list only [56] the references which are known to us (April 10, 1977) while in the present paragraph an exposition of paper [55] is given as an example of a possible model.

258

SM. Bilenky and B. Pontecorvo. Lepton mixing and neutrino o.scillarion.s

~w2

~‘

Fig. 4. Diagrams ofthe process p

—+

~

e~’with virtual neutral heavy leptons

N1

and

N2.

If M~= 60 GeV, the value M~ M~1/2 is smaller than 8.5 GeV. From this it also follows that M2~< 8.5 GeV*. 2equal to 1, 2, 3, 4 GeV, one obtains for Ru values equal to Assuming values 4.2 x l0 12 6.7 xof M~ M~’ 10_li, ~ x 10_b, 1.1 x iO”~,respectively. Thus, the p ey decay probability could turn out to be relatively close to its upper experimental limit, if there exist neutral leptons with masses of the order of a few GeV and if mixing is taking place. Let us note that the decay p ey could be revealed in the spectrometer ARES [58], if Ru ~ 10_li. —

—

—

—p

—+

7.3. The p

—s

ey decay and neutrino oscillations

Below we discuss the relation between the phenomenon of neutrino oscillations and of the p ey decay. The observation of such effects would show that lepton mixing takes place indeed. In such a general sense the observation of either of these phenomena would make the existence of the other more likely, in particular, the neutrino masses would be probably finite. Apart from this general connection, it should be emphasized, however, that neutrino oscillations and processes as the p ey decay etc. might well be entirely unconnected. First, one can think of the case in which neutrino oscillations might be observable, but the process p ey is in fact unobservable; this is just the situation we have been discussing in the six preceeeding sections: in Nature the only neutral leptons are neutrinos. Second, one can imagine a situation in which the p ey decay is perfectly observable (let us say that there are heavy neutral leptons of sufficiently large masses) but neutrino oscillations are unobservable, for example, because of a small mixing angle and/or a small difference in the neutrino masses. Third, one can imagine a state of affairs, in which the p ey decay probability is relatively high (for example, because there exist heavy charged leptons and asymmetrical neutral currents are present in the hamiltonian), but neutrino oscillations may be completely absent (not only unobservable in practicle), since the p ey process has no relation whatever to the oscillations. —~

-~

—*

—*

-+

—*

8. Conclusions 1) The physical problems considered here are based on the fundamental assumption of lepton mixing. The main consequences of this hypothesis depend upon how many leptons exist in nature * Here the lepton-.quark analogy is quite striking: the upper limit of the decay p -. e~’probability gives us an upper limit for the mass difference of heavy leptons just as a few years ago the data on rare decay processes of kaons (and the mass difference of K~,and K,) allowed one to find an upper limit for the mass of the charmed quark.

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and what kind. In the case of only four leptons (muon, electron, ~u’ va), the main observable effect of the mixing hypothesis is the existence of neutrino oscillations. The theoretical rates of such processes as p ey, p 3e, etc. are finite but turn out to be by many orders of magnitude smaller than the rates which can be actually measured. Neutrino oscillations can be revealed by finding out that the intensity of a definite type of neutrinos is smaller than its expected value and/or by actually observing a cosine term in the neutrino intensity as a function of either the source-detector distance or the neutrino momentum. 2) Neutrino mixing implies that neutrino masses are finite. Let us note that finite values of neutrino masses do not contradict any known principle (similar to gauge invariance, according to which the photon mass must be zero). In this sense the question about the finite values of neutrino masses is completely open (and not only in the “trivial” sense that the experimental upper limits of the vp and Ve masses are relatively large). 3) The question about the number of existing leptons is open, as well. If the number of neutrino types is larger than two and if all the neutrino fields are mixed, oscillations among all types of neutrinos become possible. The rates of such processes as p —~ey, p 3e, etc. remain unobservable, as before. 4) The situation is changing radically if there exist heavy leptons, the fields of which are present in the hamiltonian in a mixed form: if the heavy lepton masses are of the order of a few GeY, the rates of the p —~ e~’,p 3e, etc. processes may be close to their experimentally determined upper limits. In the general case there is no definite connection between the probabilities of these processes and the neutrino oscillation parameters. 5) We saw that there are two types of theories, one with Majorana neutrinos, the other with Dirac neutrinos. What makes Majorana fields attractive is their “economy” (in the case of two neutrino types, in all there are only four states). Dirac neutrinos are described just as all the other fundamental fermions (leptons and quarks). This is very nice from the point of view of quark—lepton analogies. The picture of oscillations in both theories is identical and practically it will be impossible to select one of them on the basis of experimental and theoretical arguments only, without using arguments of aesthetical character. 6) At a first impression, the very notion of lepton charge is lost as soon as there is neutrino mixing. However, orthogonal mixing in essence is equivalent to the conservation of lepton charges, provided the neutrino masses are sufficiently small (as they are). In the first order of the weak interaction constant the amplitudes of processes (such as p —~e~’,p —. 3e, etc.), due to asymmetrical neutral currents, are equal to zero. In high order, however, these processes do take place, their amplitude being proportional to the difference in the squared neutrino masses. Thus, it is just the small value of neutrino masses which ensures an effective conservation of lepton numbers and we have already seen that the processes p —~ey, p —~ 3e, etc. might be observable if heavy leptons with masses of a few GeY were existing. Here the lepton—quark analogy is appearent through the striking similarity of the quark GIM mechanism and of the orthogonal lepton mixing. 7) Until now neutrino oscillations have not been observed and experimental searches for them are only being planned. An analysis, based on the oscillation ideology of all the neutrino experiments already performed at reactor and accelerator facilities, nevertheless, allows one to set an upper limit for the difference in the squared neutrino masses: m~— m~ ~ 1 (eY)2 for maximum mixing. On the other hand, in the only solar neutrino experiment which has been performed, the signal turned out to be weaker than it was expected (this neutrino “shortage” is the so-called solar “neutrino” puzzle). The explanation of such puzzle (if any) in terms of neutrino oscillations —~

-+

—+

—*

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seems to be much more natural than other quite exotic solutions of it which have been suggested. Let us remark incidentally that solar neutrino experiments will allow oscillation effects to be seen if m~ m~~ 10_12 (eV)2 (for maximum mixing). 8) Clearly it is of the utmost importance to perform search experiments for neutrino oscillations and for processes such as p ey, p 3e, p~+ A e~+ ... etc. at any level, improving on the existing experimental limits. —

—*

—~

—*

In conclusion we wish to express our gratitude to A. Chudakov, S. Korenchenko, G. Mice!macher, L. Okun, S. Petcov, M. Podgoretsky, A. Pomansky, G. Zatsespin for fruitful discussions.

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