A potential test of the CP properties and Majorana nature of neutrinos

Nuclear Physics B 566 Ž2000. 92–102 www.elsevier.nlrlocaternpe

A potential test of the CP properties and Majorana nature of neutrinos S. Pastor a , J. Segura b, V.B. Semikoz c , J.W.F. Valle

d

a

SISSA – ISAS and INFN, Sezione di Trieste, Via Beirut 2-4,I-34013 Trieste, Italy Instituto de Bioingenierıa Edificio La Galia, 03206 Elche, Alicante, Spain ´ UniÕersidad Miguel Hernandez, ´ c Institute of Terrestrial Magnetism, the Ionosphere and Radio, WaÕe Propagation of the Russian Academy of Sciences, Izmiran, Troitsk, Moscow region, 142092, Russia d Instituto de Fısica Corpuscular - C.S.I.C., Departament de Fısica Teorica, UniÕersitat de Valencia, 46100 ´ ´ ` ` Burjassot, Valencia, Spain b

Received 20 May 1999; accepted 27 October 1999

Abstract The scattering of solar neutrinos on electrons may reveal their CP properties, which are particularly sensitive to their Majorana nature. The cross section is sensitive to the neutrino dipole moments through an interference of electromagnetic and weak amplitudes. We show how future solar neutrino experiments with good angular resolution and low energy threshold, such as Hellaz, can be sensitive to the resulting azimuthal asymmetries in event number, and could therefore provide valuable information on the CP properties and the nature of the neutrinos, provided the solar magnetic field direction is fixed. q 2000 Elsevier Science B.V. All rights reserved.

1. Introduction It has long been realized that, on general grounds, gauge theories generally predict that, if neutrinos are massive, they should be Majorana particles, unless protecting symmetries are imposed or arise accidentally w1x. Even though lepton-number-violating processes such as neutrinoless double beta decay are intrinsically related to the Majorana nature of neutrinos in a gauge theory w2x, their search as so far yielded only negative results w3x. It has also been shown in the early 80’s that gauge theories with Majorana neutrinos contain additional CP-violating phases without analogue in the quark sector w1x. Although these are genuine physical parameters of the theory, as they show up in D L s 2 neutrino oscillations w4x, their effects are also suppressed by the smallness of E-mail addresses: [email protected] ŽS. Pastor., [email protected] ŽJ. Segura., [email protected] ŽV.B. Semikoz., [email protected] ŽJ.W.F. Valle.. 0550-3213r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 5 5 0 - 3 2 1 3 Ž 9 9 . 0 0 6 8 1 - 1

S. Pastor et al.r Nuclear Physics B 566 (2000) 92–102

93

neutrino masses, relative to the typical neutrino energies available at accelerator and even reactor experiments. Among the non-standard properties of neutrinos the electromagnetic dipole moments w5–13x play an important conceptual role, since they can potentially signal the Majorana nature of neutrinos. Neutrino transition electromagnetic moments w5–7x are especially interesting because their effects can be resonantly enhanced in matter w14,15x and provide an attractive solution of the solar neutrino problem w16x without running in conflict with astrophysics w17x. For pure left-handed neutrinos the weak interaction amplitude on electrons does not interfere with that of the electromagnetic interaction, since the weak interaction preserves neutrino helicity while the electromagnetic does not. As a result the cross section depends quadratically on the neutrino electromagnetic form factors. However, if there exists a process capable of converting part of the initially fully polarized neutrinos, then an interference term arises proportional to the neutrino electromagnetic form factors, as pointed out e.g. in Ref. w18x. This term depends on the angle between the component of the neutrino spin transverse to its momentum and the momentum of the outgoing recoil electron. Therefore the number of events measured in an experiment exhibits an asymmetry with respect to the above defined angle. The asymmetry will not show up in terrestrial experiments even with stronger magnetic fields, since only in the Sun the neutrino depolarization would be resonant and only in the solar convective zone one will find a magnetic field extended over such a region Žabout a third of the solar radius wide.. At earth-bound laboratory experiments the helicity-flip could be caused only by the presence of a neutrino mass and is therefore small w19x, in a way analogous to the case of neutrino D L s 2 lepton-number-violating neutrino oscillations w4x. Exotic couplings to scalars might change this feature, but there are relatively strong limits. In contrast for a relatively modest large-scale solar magnetic field in the convective region B H ; 10 4 G and a neutrino magnetic moment of the order 10y1 1m B , where m B is the Bohr magneton, one has mn B H L ; 1 since L ; Lconv , 2 = 10 10 cm is the width of the convective zone. Such a spin-flip process may depolarize the solar neutrino flux at a level where neutrino electromagnetic properties may reveal the Majorana nature of neutrinos Žor alternatively, the solar magnetic field structure.. In this paper we show that the resonant enhancement of neutrino conversions induced by Majorana transition moments can provide valuable hints on the true nature of neutrinos and their CP properties, in a way which is not suppressed by the small neutrino mass. Our proposed test requires the careful investigation of neutrino-electron scattering for neutrinos from the Sun at future solar neutrino experiments with good angular resolution and low energy threshold. One such proposed experiment is Hellaz w20,21x. For completeness and pedagogy we also include a discussion of the Dirac-type magnetic moment or electric dipole moment w22,23x.

2. Neutrino electromagnetic properties The most general effective interaction Lagrangian describing the electromagnetic properties of Majorana neutrinos has been first given in Refs. w5–7x in terms of the fundamental two-component spinors. The connection with conventional four-component

S. Pastor et al.r Nuclear Physics B 566 (2000) 92–102

94

description can be found in Ref. w1x. Other equivalent presentations are given in w11–13x and the corresponding matrix element between one-particle neutrino states for a real Ž q 2 s 0. photon, can be written as w17x X

² pX , sX , j < Leff < p, s,i : s u sj Ž pX . Gl i j Ž p, pX . u is Ž p . A l Ž q . X

s u js Ž pX . i sl r q r Ž m i j q id i jg 5 . u is Ž p . A l Ž q . ,

Ž 1.

X

where i, j denote the mass labels of the neutrinos, the indices s and s specify helicities, while the u’s are the standard wave functions of the Dirac equation and q s p y pX . Here m i j and d i j are the magnetic and electric dipole moments, respectively. From the hermiticity condition for the Lagrangian one can relate w8–10x the form factors of the i j process and its inverse,

™

m i j s m )ji ,

d i j s d ji) .

Ž 2.

Note that in the diagonal i s j case, both m and d must be real according to Eq. Ž2.. For further constraints on the form factors m i j and d i j one must assume something about the neutrino nature andror invariance under the CP symmetry w5–7,11,12x. These interactions arise only from loops in a gauge theory like the Standard Model and are therefore calculable from first principles. However, in most gauge theories magnetic moments are expected to be small. For discussions see Refs. w11,12x. Majorana neutrinos can have only off-diagonal Ž i / j . form factors, called transition moments w5–7x, while if the neutrinos are Dirac particles, just as the charged leptons, both diagonal and off-diagonal moments can exist. Let us assume that the effective Lagrangian is invariant under a CP transformation, Leff s CP Leff Ž CP .y1 . A Dirac field transforms under CP as CPCi Ž x,t .Ž CP .y1 s hi C Ci ) Žyx,t ., where hi is a phase factor and C is the charge conjugation matrix Ž Cy1 s C † s C T s yC . w10–13x. If we apply this to the smn-part of the effective Lagrangian, the CP-transformed i j part will contribute to the j i process and vice versa. The result for Gl implies that

™

™

u j Ž pX . Gl i j Ž q . u i Ž p . s hi)hj u j Ž pX . Gl i j Ž q . u i Ž p . , where Gl i j is equal to Gl i j with the change g 5 factors obey the relations

m i)j mi j

sy

d i)j di j

Ž 3.

™ yg . Eq. Ž3. implies that the form 5

s hihj) .

Ž 4.

A Majorana neutrino is its own anti-particle. It is easy to check in this case that in Eq. Ž1. both the ij and the ji terms in the Lagrangian will contribute to the ij form factors. One finds that for mass eigenstates w10x ² pX , j < Leff < p,i : s u j Ž pX . i sl r q r Ž m i j y m ji . q i Ž d i j y d ji . g 5 u i Ž p . A l .

Ž 5.

Finally, from the hermiticity condition Ž2. one gets m i j y m ji s 2 iImŽ m i j . and d i j y d ji s 2 iImŽ d i j .. Therefore we conclude that a Majorana neutrino has no diagonal electromagnetic factors and that the transition form factors m i j and d i j are both pure imaginary, irrespective of whether or not one assumes CP invariance w5–7x. Thus Majorana neutrinos can only possess transition magnetic or electric dipole moments. Let us now check whether CP invariance restricts them. If CP is conserved, a Majorana

S. Pastor et al.r Nuclear Physics B 566 (2000) 92–102

95

Table 1 General properties of neutrino electromagnetic dipole moments Case

Hermiticity

Hermiticity q CP

Dirac is j Dirac i/ j Majorana is j Majorana i/ j

m i i and d i i real m i j s m )ji and d i j s d ji) mi i s d i i s 0 m i j and d i j pure imaginary

dii s 0 m i j and id i j relatively real – Case Žqy.: d i j s 0 Case Žqq.: m i j s 0

neutrino is a CP eigenstate, with a phase hC P s "i w5–7x. Considering the invariance of L for the Majorana case under CP one gets a condition similar to Eq. Ž3. and Eq. Ž4.. There are two physically interesting cases to consider: two neutrino species involved in Eq. Ž1. can be either both active, weakly interacting neutrinos, or one of them can be sterile. Moreover, for each of these cases, there are two possible CP-conserving cases, depending on the relative CP sign of the neutrinos involved 1. Case Žqy .: Žhi ,hj . s Ž"i,. i ., then m i j survives and d i j s 0. This is a pure magnetic transition, and includes the Dirac-type magnetic moment if one of the neutrinos is sterile. 2. Case Žqq .: Žhi ,hj . s Ž"i," i ., then m i j s 0 and d i j survives w5–7x. This is a pure electric transition. On the contrary, as emphasized by Wolfenstein w5–7x, if CP is not conserved both magnetic and electric dipole moments will contribute to the neutrino-electron scattering cross section. The general properties of Dirac and Majorana neutrino electromagnetic dipole moments are summarized in Table 1. Note that the above discussion is completely general and covers all types of Majorana transition moments, active-active and active-sterile. In particular it covers the activesterile case with zero mass splitting ŽDirac diagonal case.. In what follows we will focus mainly on active-active Majorana transition moments, as well as the Dirac diagonal case.

3. Dipole moments for flavor states We have discussed so far the restrictions upon the neutrino electromagnetic dipole moments for mass eigenstates. Since we are interested in possible interference terms between weak and electromagnetic interactions in neutrino-electron scattering, we shall present the corresponding matrix elements in terms of flavor states. For simplicity we will restrict ourselves to the two-generation case, and for definiteness the ne y na pair, where na can be an active neutrino Žfor instance nm . or a sterile neutrino ns . In this case the mixing matrix contains a CP-violating phase for the case of Majorana neutrinos w1,4x. Such phase is absent if the two neutrinos are Dirac type, since in this case it can be removed by field redefinition, as expected in analogy with the quark sector, where CP violation sets in only for three generations.

S. Pastor et al.r Nuclear Physics B 566 (2000) 92–102

96

The u Gl u matrix element can be written as in Eq. Ž1. but for flavor eigenstates ne, a , with mn en a and dn en a. Here the restrictions on m and d for Dirac mass states still apply, and in particular for the diagonal case both mn e and dn e are real. The CP-violating phase which is present in the mixing matrix of a theory with two Majorana neutrinos may be introduced as the e i b phase in the 2 = 2 mixing matrix,

ne c na s ys

se i b ce i b

ž / ž

n1 c n 2 s ys

/ž /

s c

ž

/ ž 10

0 eib

n1 n2 .

/ž /

Ž 6.

Here c ' cos u , s ' sin u , where u denotes the leptonic mixing angle. Let us define now in Eq. Ž5. the real parameters mXi j ' ImŽ m i j . and dXi j ' ImŽ d i j .. The expression for Eq. Ž5. is then as follows: u j Ž pX . i sl r q r 2 i Ž mXi j q idXi jg 5 . u i Ž p . .

Ž 7.

If we introduce now the weak states according to Eq. Ž6., one gets two contributions to the ne – na amplitude, corresponding to i s 1, j s 2 and i s 2, j s 1. Using the hermiticity condition one has mX21 s ymX12 and dX21 s ydX12 , and one can define the electromagnetic dipole moments for flaÕor states as follows Ž k s m ,d and note that k a e s k e)a . X X k e a ' kn en a s 2 i Ž c 2 e i b q s 2 eyi b . k 12 s 2 i Ž cos b q i Ž c 2 y s 2 . sin b . k 12 , X X k a e ' kn an e s 2 i Ž c 2 eyi b q s 2 e i b . k 21 s 2 i Ž cos b y i Ž c 2 y s 2 . sin b . k 21 .

Ž 8.

We conclude then that a pair of Majorana neutrinos Žweak states. has, in general, complex dipole moments. This is a consequence of the CP phase from the mixing matrix. The particular CP-conserving cases correspond to the values b s 0,pr2. Therefore, when assuming CP invariance the electromagnetic current for neutrinos takes the forms

b s pr2 bs0

´u Ž p . is X

´ yu Ž p . i s na

X

na

lr q

lr q

r

r

Re Ž m e a . un eŽ p . q h.c.,

Ž 9.

Im Ž d e a . g 5 un eŽ p . q h.c.

Ž 10 .

The first case is the limit that we considered in our previous paper w24x.

4. Neutrino-electron scattering cross sections We consider the scattering of neutrinos on electrons when the initial flux of neutrinos is not completely polarized, i.e. there exists a mechanism that converts part of the initial left-handed electron neutrinos Žproduced in weak processes. into right-handed ones. We assume that this is a consequence of the presence of non-zero neutrino electromagnetic dipole moments. The Sun seems to be the only physical situation where such depolarization process can occur. Let us consider the scattering n Ž k 1 . q eyŽ p 1 . n Ž k 2 . q eyŽ p 2 ., in the coordinate frame where the initial electron is at rest. The four-vectors of the particles involved, taking into account conservation of momenta, are the following:

™

k1 s Ž v , k1 . ,

p 1 s Ž m e ,0 . ,

k2 s Ž v y T , k2 . ,

p2 s Ž m e q T , p2 . ,

where T is the electron recoil energy and p 12 s p 22 s m2e . From now on we consider the limit of ultra-relativistic neutrinos, i.e. k 12 s k 22 , 0 and the low-energy limit Ž v < MW ..

S. Pastor et al.r Nuclear Physics B 566 (2000) 92–102

97

There are two contributions to the scattering process: weak and electromagnetic. As in the previous section we will consider the case of two neutrino species. There are two inequivalent physical situations, namely Ži. ne – nm and Žii. ne – ns , where ns is a sterile type neutrino. Following the conventions of Ref. w25x for f s ne , nm , the corresponding weak matrix amplitudes for Dirac and Majorana neutrinos are, respectively X

MWDf s yi2'2 GF u rf Ž k 2 . g m

ž

=gm g f L

1 y g5 2

1 y g5

q gR

2

X

u rf Ž k 1 . u es Ž p 2 .

1 q g5 2

/

u es Ž p 1 . ,

X

ž

X

MWMf s i2'2 GF u rf Ž k 2 . g mg 5 u rf Ž k 1 . u es Ž p 2 . gm g f L

1 y g5 2

q gR

1 q g5 2

/

u es Ž p 1 .

Ž 11 . and obviously zero for sterile neutrinos. Here g e L s sin2u W q 1r2, gm L s sin2u W y 1r2 and g R s sin2u W . The electromagnetic amplitudes are e rX X r r s l s M aem Ž 12 . b s 2 u b Ž k 2 . sl r q Ž m a b q id a b g 5 . u a Ž k 1 . u e Ž p 2 . g u e Ž p 1 . , q where ab denotes ne nm or ne ns , and the form factors m a b and d a b depend on whether neutrinos are Dirac or Majorana. In order to simplify the notation let us set m a ' m a a and d a ' d a a Žboth real. for the diagonal case, and m ' m e a s m )a e and d ' d e a s d a)e for the transition dipole moments, which are complex in general. We will perform the calculation of the differential cross section for neutrino–electron scattering without assuming CP invariance and in the two physical situations. It can be written as a sum of three terms, ds s dT d f

ž

ds

/ ž

ds

q

dT d f

weak

/ ž

ds

q

dT d f

em

dT d f

/

,

Ž 13 .

int

that correspond to the purely weak, the purely electromagnetic and the interference term, respectively. In the last equation f is the azimuthal angle, as defined in Fig. 1.

Fig. 1. Coordinate system conventions.

S. Pastor et al.r Nuclear Physics B 566 (2000) 92–102

98

4.1. ActiÕe–actiÕe case In this case Ž ne y nm . the purely weak term can be written in a general form as

ž

ds d f dT

/

s weak

GF2 m e

p2

Pe h Ž g e L , g R . q Pe h Ž g R , g e L . q Pm h Ž gm L , g R .

qPm h Ž g R , gm L . ,

Ž 14 .

where we have defined hŽ x, y . ' x 2 q y 2 Ž1 y Trv . 2 y xym e Trv 2 . Here PA Ž PA . is the probability of measuring nA s ne, m Ž nA s ne, m ., and from unitarity Pe q Pe q Pm q Pm s 1. The purely electromagnetic term in the presence of transition dipole moments is

ž

ds dT d f

/

a2 s em

2 m2e

ž

1

1 y

T

v

/

< m<2 q< d<2

m2B

q 2 Ž Pe q Pm y Pe y Pm .

Im Ž m d ) .

m2B

.

Ž 15 . Note that in the limit of zero mixing only ne and nm are present Ž Pe s 0 s Pm and Pe q Pm s 1., so that the last equation reduces to the form Žsee e. g. w26x.

ž

ds dT d f

/

a2 s em

2 m2e

ž

1

1 y

T

v

/

< m y id < 2

m2B

.

Ž 16 .

It will be convenient in order to calculate the interference term of the cross section to describe the flux of initial neutrinos in terms of a density matrix r which generalizes the usual to account for the case of two different flavors A, B. The neutrino part of the amplitude squared MM X † is thus calculated as follows:

Ý

u B Ž k 2 . Mu A Ž k 1 .

uAŽ k 2 . M X uB Ž k1 .

†

s Tr M rA B Ž k 1 . M X† kˆ 2 ,

Ž 17.

spins

where M X† s g 0 M X†g 0 and xˆ ' gl x l. The density matrix can be written as a function of the different neutrino probabilities and the components of the corresponding polarization vectors as follows 1 1 rA B Ž k 1 . s Ž PA q PB q Ž PB y PA . g 5 . kˆ 1 q jˆHA B kˆ 1 Ž 1 q g 5 . 2 4 1 q kˆ 1 jˆHA B Ž 1 y g 5 . , Ž 18 . 4 where the polarization four-vectors j HA B s Ž0, j HA B . are orthogonal to the neutrino BA momentum, i.e. k 1 P j HA B s 0 s k 1 P j H and < j HA B < s 2 PA PB . Note that in the case of one Dirac neutrino A s B this reproduces the result given in the Appendix E of Ref. w27x, namely 1 rA Ž k 1 . s Ž 1 y j I g 5 y jˆH g 5 . kˆ 1 , Ž 19 . 2 where j is the normalized polarization vector at the neutrino’s rest frame, with < j H < s 2 PA Ž 1 y PA . and j I s 2 PA y 1. It is important to remark that the transversal component j H of the neutrino polarization vectors are aligned along the direction of the solar magnetic field B( w28x.

(

(

S. Pastor et al.r Nuclear Physics B 566 (2000) 92–102

99

For ultra-relativistic neutrinos the interference term arises only if the initial flux contains some mixture of right-handed neutrinos. Note that in this subsection we restrict to the simple situation where the initial ne convert to nm through a neutrino transition dipole moment Žzero neutrino mixing.. Then one sets Pm s Pe s 0, so that j He m s 0. This is the process that can occur in the Sun. The expression for the interference term is found to be

ž

ds dT d f

/

sy int

q

a GF

ž

4'2 p m e T

ž

Re Ž m . q Im Ž d .

mB

Re Ž d . y Im Ž m .

mB

/

/

p2 P A M Ž T , v .

kˆ 1 P Ž p 2 = A M Ž T , v . . ,

Ž 20 .

where kˆ 1 ' k 1rv and we have defined AM Ž T ,v . '

Ž g e L q gm L q 2 g R .

ž

2y

T

v

/

q Ž g e L y gm L .

T

v

em jH .

Ž 21 .

Note that Eq. Ž20. depends explicitly on the azimuthal angle f . Choosing the coordinate system as shown in Fig. 1, this dependence is like cos f or sin f , since it is easily checked that p 2 P j H sN p 2 N sin u N j H N cos f s

(

ž

2 m eT 1 y

T Tmax

(

kˆ 1 P Ž p 2 = j H . s yN p 2 N sin u N j H N sin f s y

/

N j H N cos f ,

ž

2 m eT 1 y

T Tmax

/

N j H N sin f ,

Ž 22 . where Tmax s 2 v 2rŽ m e q 2 v . is the maximum electron recoil energy. 4.2. ActiÕe–sterile case In this case the three terms of the differential cross section in Eq. Ž13. are different with respect to the active-active case, since sterile neutrinos do not have weak interactions. For instance the purely weak term will consist only of the electron neutrino contribution in Eq. Ž14., while the purely electromagnetic term in the presence of Dirac-type dipole moments is the well-known result

ž

ds dT d f

/

a2 s em

2 m2e

ž

1

1 y

T

v

m2e q d e2

/

m2B

.

Ž 23 .

Finally the interference term in the presence of active–sterile dipole moments is

ž

ds dT d f

/

sy int

q

a GF 2'2 p m e T

ž

ž

Re Ž m . q Im Ž d .

Re Ž d . y Im Ž m .

mB

mB

/

/

p2 P A S Ž T , v .

kˆ 1 P Ž p 2 = A S Ž T , v . . ,

Ž 24 .

S. Pastor et al.r Nuclear Physics B 566 (2000) 92–102

100

where

ž

AS Ž T , v . ' ge L q gR 1 y

T

es jH .

/

v

Ž 25 .

In the limit when the active-sterile pair form a Dirac neutrino Eq. Ž24. reduces to

ž

ds dT d f

/

sy int

q

a GF

me

ž /

2'2 p m e T de

ž / mB

mB

p2 P A D Ž T , v .

kˆ 1 P Ž p 2 = A D Ž T , v . . ,

Ž 26 .

where

ž

A D Ž T , v . ' ge L q gR 1 y

T

v

/

e jH .

Ž 27 .

The first term in this result was obtained in Ref. w18x, while the second CP-violating term was given in Ref. w29x Žsee their Eq. Ž9c...

5. Test of CP conservation at Hellaz We propose to measure solar neutrino-electron scattering in upcoming experiments that will be capable of measuring directionality of the outgoing ey Žlike Hellaz.. The relevant observable is the azimuthal distribution of events, namely dN df

s Ne Ý F 0 i i

Tmax

HT

Th

dT

vmax

Hv

min Ž T .

d v li Ž v . e Ž v .

ds dT d f

Ž v ,T . ,

Ž 28 .

where d srdT d f is the complete differential cross section of Eq. Ž13., e Ž v . is the efficiency of the detector and Ne is the number of electrons in the fiducial volume of the detector. The sum in the above equation is done over the solar neutrino spectrum, where i corresponds to the different reactions i s pp, 7 Be, pep, 8 B . . . , characterized by a differential spectrum l i Ž v . and an integral flux F 0 i . In Section 4 we found the expressions for the differential cross section. The azimuthal distribution of the number of events can be written in a general form as dN df

s n weak q n em q n int cos Ž f q d . ,

Ž 29 .

where n weak Ž n em . accounts for the weak Želectromagnetic. contributions, while n int is the interference term. The dependence of the last term on the azimuthal angle f is parametrized with d . Thus a pure cos f Žysin f . dependence corresponds to d s 0 Ž d s pr2.. We can define the differential azimuthal asymmetry as dN d f

dA s df

f

X

dN d f

f X ydN f

X

df

qdN d f

f Xq p X

f qp

s

n int n weak q n em

cos Ž f X q d . ,

Ž 30 .

S. Pastor et al.r Nuclear Physics B 566 (2000) 92–102

101

where f Ž f X . is measured with respect to the direction of the magnetic field B( , which we will assume to be along the positive x-axis Žsee Fig. 1.. By integrating over f one can also define an asymmetry A as f Xq p X

AŽ f . s

Hf

X

Hf

X

dN

df X f q p dN df

df y df q

f Xq2 p

dN

Hf qp

df X f q2 p dN X

Hf qp X

df

df s yAsin Ž f X q d . ,

Ž 31 .

df

where A ' 2 n intrp Ž n weak q n em . is the maximum integrated asymmetry measurable by the experiment, which is manifestly positive. In our previous paper w24x we calculated the expected values of A for pp solar neutrinos at Hellaz, for different choices of the survival probability of ne’s, in the CP-conserving case of Eq. Ž9.. Let now discuss how the measurement of the azimuthal asymmetry could be carried out considering that B( is constant over a given period of time and its direction is known. One should collect events in every f-bin, where f is defined with respect to the positive x-axis and then take for different f X s the ratio AŽ f . which should show a sin f dependence with a maximum equal to A. This will allow us to identify the value of d . Note that if we were able to find from the measurements that d / 0 beyond experimental uncertainties, then this would lead to the conclusion that CP is not conserÕed in the electromagnetic interactions of neutrinos if we consider Dirac diagonal or Majorana transition dipole moments. In the Dirac transition case the CP phases hi of the neutrinos can be chosen so as to have CP conservation for any value of d . However, Dirac dipole moments do not seem to be favored by theoretical models nor by the existent astrophysical and cosmological constraints w17x.

6. Discussion We have shown that the scattering of solar neutrinos on electrons may reveal their CP properties, which are particularly sensitive to their Majorana nature, due to the interference of electromagnetic and weak amplitudes. We showed how future solar neutrino experiments with good angular resolution and low energy threshold can be sensitive to the resulting azimuthal asymmetries in event number, and could therefore provide valuable information on the CP properties and the nature of the neutrinos, provided the solar magnetic field direction is fixed. Hellaz will be the first experiment which is potentially sensitive to azimuthal asymmetries since the directionality of the outgoing ey can be measured. The angular resolution is expected to be Du ; Df ; 30 mrad ; 28, substantially better than that of Super-Kamiokande. Notice also that the width of the Cerenkov cone defined by the angle u is very narrow for high-energy boron neutrinos, as one can see from Eq. Ž22.. In contrast, for pp neutrino energies accessible at Hellaz ŽTmax , 0.26 MeV, Tt h , 0.1 MeV. we estimate that u can be as large as 488. It is important to emphasize here that, while the existence of an asymmetry in event number can be ascribed to a non-zero neutrino electromagnetic dipole moment, one can not infer any information on the specific issue of CP conservation in the neutrino sector and the nature of neutrinos without an accurate knowledge of the direction of the solar

S. Pastor et al.r Nuclear Physics B 566 (2000) 92–102

102

magnetic field. Such a knowledge is indeed possible except at minimal solar activity periods, when the toroidal magnetic field vanishes.

Acknowledgements The authors thank Thomas Ypsilantis for fruitful discussions on the Hellaz experiment. This work has been supported by DGICYT under Grants PB95-1077, by the TMR network grant ERBFMRXCT960090 and by INTAS grant 96-0659 of the European Union. V.S. acknowledges also the support of Generalitat Valenciana and RFBR grant 97-02-16501.

References w1x w2x w3x w4x w5x w6x w7x w8x w9x w10x w11x w12x w13x w14x w15x w16x

w17x w18x w19x w20x w21x w22x w23x w24x w25x w26x w27x w28x w29x

J. Schechter, J.W.F. Valle, Phys. Rev. D 22 Ž1980. 2227. J. Schechter, J.W.F. Valle, Phys. Rev. D 25 Ž1982. 2951. L. Baudis, Phys. Rev. Lett. 83 Ž1999. 41. J. Schechter, J.W.F. Valle, Phys. Rev. D 23 Ž1981. 1666. J. Schechter, J.W.F. Valle, Phys. Rev. D 24 Ž1981. 1883. J. Schechter, J.W.F. Valle, Phys. Rev. D 25 Ž1982. 283, E. L. Wolfenstein, Phys. Lett. B 107 Ž1981. 77. R. Shrock, Nucl. Phys. B 206 Ž1982. 359. B. Kayser, Phys. Rev. D 26 Ž1982. 1662. J.F. Nieves, Phys. Rev. D 26 Ž1982. 3152. For reviews and discussion on various models see, for instance R.N. Mohapatra, P.B. Pal, Massive Neutrinos in Physics and Astrophysics ŽWorld Scientific, Singapore, 1991.. J.W.F. Valle, Prog. Part. Nucl. Phys. 26 Ž1991. 91. C.W. Kim, A. Pevsner, Neutrinos in Physics and Astrophysics ŽHarwood, 1993. C.S. Lim, W. Marciano, Phys. Rev. D 37 Ž1988. 1368. E.Kh. Akhmedov, Phys. Lett. B 213 Ž1988. 64. E.Kh. Akhmedov, The neutrino magnetic moment and time variations of the solar neutrino flux, Invited talk given at the 4th Int. Solar Neutrino Conf., Heidelberg, Germany, April 1997, preprint ICr97r49, hep-phr9705451. G. Raffelt, Stars as Laboratories for Fundamental Physics, ŽUniv. of Chicago Press, Chicago–London, 1996.. R. Barbieri, G. Fiorentini, Nucl. Phys. B 304 Ž1988. 909. W. Grimus, P. Stockinger, Phys. Rev. D 57 Ž1998. 1762. F. Arzarello et al., Preprint CERN-LAAr94-19, College de France LPCr94-28 Ž1994.. J. Seguinot et al., Preprint LPC 95 08, College De France, Laboratoire de Physique Corpusculaire Ž1995. M.B. Voloshin, M.I. Vysotsky, Sov. J. Nucl. Phys. 44 Ž1986. 845. M.B. Voloshin, M.I. Vysotsky, L.B. Okun, Sov. Phys. JETP 64 Ž1986. 446. S. Pastor, J. Segura, V.B. Semikoz, J.W.F. Valle, Phys. Rev. D 59 Ž1999. 013004. F. Mandl, G. Shaw, Quantum Field Theory ŽWiley, New York, 1984.. G. Raffelt, Phys. Rev. D 39 Ž1989. 2066. S.M. Bilenky, Introduction to Feynman Diagramas and Electroweak Interactions Physics ŽFrontieres, ` 1994.. V.B. Semikoz, Nucl. Phys. B 498 Ž1997. 39. P. Stockinger, W. Grimus, Phys. Lett. B 327 Ž1994. 327.