Solar neutrinos in four dimensions and beyond

ELSEVIER

Nuclear Physics B (Proc. Suppl.) 81 (2000) 102-112

PROCEEDINGS SUPPLEMENTS www.elsevier.nl/locate/npe

Solar Neutrinos in Four Dimensions and Beyond A. Yu. Smirnov a aThe Abdus Salam International Center for Theoretical Physics, 1-34100, Trieste, Italy Recent data give some indications in favor of the Large Mixing Angle (LMA) MSW-solution of the solar neutrino problem. Signatures of this solution are considered in details and confronted with experimental results. Recent theoretical developments - theories with low fundamental scale of quantum gravity and large extra dimensions give new "bulk - brahe" mechanism of generation of neutrino mass. In this framework new solution of the solar neutrino problem is possible via resonance conversion of the electron neutrino in the bulk states. The solution implies that one of extra dimensions has the size about 0.1 ram.

1. I n t r o d u c t i o n

characterize it by the winter-summer asymmetry:

High statistics SuperKamiokande data on solar neutrinos [1] opens a possibility to search for signatures of various oscillation solutions of the solar neutrino problem. This means that we have indeed started crucial checks of theory. Among these signatures are:

W-S Aw/s = 2 ~ - ~ S '

(i) Distortion of the recoil electron spectrum

R(E) =_ N(E)°bs/N(E) ssM. Some distortion (deviation of R(E) from a constant) has already been observed. The main feature of the data is that below electron energy 13.5 MeV they are in agreement with undistorted spectrum. Above 13.5 MeV there is about 3 a excess whose interpretation is still unclear: It can be just statistical fluctuation, or it can be due to large flux of the hep-neutrinos, or due to effect of the vacuum oscillations with relatively large A m 2. It will be important to see the result on the spectrum in the lowest bin 5.0 - 5.5 MeV. (ii) The night-day asymmetry

N-D

N

AN~D _ 2 N +------~~ ~ - 1,

(1)

where N (D) is the nighttime (daytime) signal integrated over energies above 6.5 MeV and averaged over the year. The asymmetry is observed at the level 6% which differs from zero by 2 a afer 825 days of observations [2]. (iii) The zenith angle distribution of events averaged over the year: F'(O). (iv) Seasonal variations of the signal. One can

(2)

where W and S are the signals averaged over winter time (November 15 - February 15) and summer time (May 15 - August 15) respectively. Important criteria can be obtained from the dependence of the asymmetry o n the energy threshold. In fact, the SK sees strong (but statistically insignificant) enhancement of the variations with increase of the threshold. Being confirmed, it will allow to identify the solution. The main conclusion from analysis of present situation is that different datasets favor different solutions of the problem. In particular, the N / D asymmetry and the zenith angle dependence indicate Large Mixing Angle (LMA) MSW solution. Seasonal variations and the spectrum distortion testify for the Vacuum Oscillation (VO) solution, whereas the total rates of signals prefer Small Mixing Angle (SMA) MSW solution. The significance of all these indications is about 2a, and clearly, more data is needed to make definite conclusion. In this paper we will consider two aspects of the solar neutrino physics: 1). Signatures of the LMA MSW solution which could be a correct solution of the solar neutrino problem. 2). New solution of the solar neutrino problem

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103

A. Yu. Smirnov/Nuclear Physics B (Proc. Suppl.) 81 (2000) 102-112

based on l/-~ l/bulk conversion. 2. In F o u r D i m e n s i o n s

Recent experimental data contain some indications in favor of the LMA solution [3] (see fig. !

1): 1)• Solution predicts flat distortion of the spectrum with survival probability determined by vacuum mixing: P = sin 2 0. For A m 2 > 5.10 -5 eV 2 the turn up of spectrum is expected at low energies due to effect of the adiabatic edge of the suppression pit. For Am 2 < 3 . 1 0 -5 eV 2 small positive slope appears due to the Earth regeneration effect• The excess of events in the high energy part of the spectrum, if confirmed, can be explained by the h e p - neutrino contribution [4] see fig. 2. The Earth matter regeneration leads to several related effects• 2). The Night-Day asymmetry. The asymmetry increases with decrease of Am2: A N / D ~ 0•22

10-SeV 2 Am 2 ,

10~

I

o.o-

,

0.060

/.

•

i'

I

-

-"

"

I

~,| .

o.,oo

lO-S

(3)

I

0.5

/

/

0.6

0.7

0.8

0.9

1

sin~e

and the dependence of A N / D on the mixing angle is rather weak• Recent (la) data can be explained if [3] A m 2 = /3 ~ •5 +2°~ -15)

-

10-SeV 2 .

(4)

3)• The zenith angle distribution of events (averaged over the year) can lead to unique identification of the solution• The SK data show that the excess of events is not concentrated in the vertical night bin N5; the excess is observed already in the first night bin N1, so that the data indicate flat distribution• The LMA solution predicts flat distribution for Am 2 > 2 . 1 0 -5 eV 2 (fig. 3). Indeed, for these Am 2 the oscillation length in matter (being of the order of the vacuum oscillation length) is much smaller than the diameter of the Earth: lm < Ires :

l~ = 47rE sin 20 Am 2 sin 20 <
(5)

Therefore integration over the zenith angle bin leads to averaging the oscillations. For smaller Am2: Am 2 < 1.5- 10 -5 eV 2 and also smaller 0

Figure 1. The allowed region of the LMA MSW parameter space. When only the rates in the chlorine, SuperKamiokande, SAGE, and GALLEX experiments are considered, the vertical continuous lines are part of the allowed contours at 90%, 95%, and 99% C.L. The dashed contour corresponds to the allowed region at 99% C.L. when both the total rates and the Night-Day asymmetry are included• The best fit parameters, indicated by a dot in the figure, are for the rates-only fit: sin2(20) = 0.72 and Am 2 = 1.7 x 10-SeV 2. The best-fit parameters for the combined fit are sin2(20) = 0.76 and Am 2 = 2.7 × 10-6eV 2. The approximately horizontal lines show the contours for different Night-Day asymmetries (numerical values indicated). From [3].

A. Yu. Smirnov /Nuclear Physics B (Proc. Suppl.) 81 (2000)102-112

104

1

'

'

'

'

I

,,,INS20

0.8

'

'

'

'

I

'

'

'

'

I

A m S / l O -6 eVe

o.v;)

.....

e.o

0.79

............

4.0

0.79

....

2.0

a s w (L-a)

the oscillation length becomes larger, averaging is not complete and the structure appears in the zenith angle distribution. This however happens in the range of parameters where the Night-Day asymmetry is large (practically excluded by experiment).

T

+ o

3 ""~*'"

,.--...~..,~.~..-:,.-v..-:,

.'

0.4

2.8 0 +-4 v 0.2

'

5

'

"

'

i

10

,

,

,

,

I

15 Energy (MeV)

,

,

*

*

I

20

L

sinl2e

=

0.8

2.8

= 0

o+ 2.4

Q~

2.2

Figure 2. Distortion of the SB electron recoil electron spectrum for large mixing angle MSW oscillations. The figure shows the ratio, R, of the measured number of electrons to the number expected on the basis of the standard solar model. Solutions are shown with the standard model fluxes of 8B and hep neutrinos (the approximately horizontal lines) and with the standard hep flux multiplied by factors of 8, 15, and 30 (ratio increases at highest energies). The calculated ratios are shown for three different values of Am 2- and with sin2 29 = 0.79 in order to illustrate the range of behaviors that result from choosing neutrino parameters. Results similar to Fig. 2 are obtained if sin 2 20 is allowed to vary over a representative range of the allowed global LMA solutions that are consistent with the average measured event rates. The experimental points show the SuperKamiokande results for 708 days of observations, Ref. [1]. The figure is from [3].

..... .... -- --

0 ~q

m , , , I , , , I ,

0

0.2

,,

2 . 9 x 1 0 "l eve 2 . 0 x i 0"-6 eVe 1 . 6 x l O "a eVe

I , , , I , , , I , , ,

0.4

0.6

O.O

cosO.

Figure 3. The zenith angle dependence of the total event rate above 6.5 MeV in the SuperKamiokande detector. Here ON = ~ - O is the nadir angle. Also shown are the SK data from 708 days.

4) Seasonal asymmetry. Almost constant signal during the night allows one to establish simple relation between the Night-Day asymmetry (1) and the seasonal asymmetry (2). The asymmetry appears because the nights are longer in the winter than in the summer. Let FN and FD

105

A.Yu. Smirnov/Nuclear Physics B (Proc. Suppl.) 81 (2000) 102-112

be the constant fluxes during the night and during the day. Consider the integrated flux (signal) during to = 24 hours. In winter this averaged signal equals

0.56

i

I

|

a m a / l O - ~ eV• -

?.9

.................. 4 . 0

W = F N t w + FD(to - t w ) ;

(6)

. . . . . .

2.0

-

1.8

0.6

in summer: S -= t i n t s + FD(to -- t s ) ,

(7)

where t w (ts) is the length of the night in Winter (Summer). Inserting these two expressions in to (2) and using the definition (1) we get after simple algebra: Aw/s = AN/D

t w - t8 to

Independent test of the LMA solution follows from studies of atmospheric neutrinos [6]. Indeed, the Am 2 responsible for the LMA MSW solution can give an observable effect: the uu ++ ueoscillations driven by solar Am 2 lead to the excess of e-like atmospheric neutrino events. The excess can be defined as

Ne

1,

0.46

(9)

MeY

P-~ ~ 6 . 5

J

(8)

Thus, the seasonal variations are proportional to the Night-Day asymmetry. For the SK location the time factor in (8) is about 1/6. Thus the seasonal asymmetry due to regeneration is expected to be 6 times weaker than the N/D asymmetry. For A N / D = (3 -- 9)% which corresponds to l a range of the observed N/D effect we find A w l s = (0.5 - 1.5) % [3]. see Fig. 4. Larger seasonal vatriations have been found in [5]. There are several features of the seasonal asymmetry which will allow one to distinguish LMA regeneration case from other effects [3]. (i) No seasonal variation of the day rate is expected in contrast with Vacuum Oscillation case. The same is also correct for the night rate. (ii) The regeneration asymmetry and the geometrical effect are in phase in the northern hemisphere and they are in opposite phase in the southern hemisphere which leads to cancellation. Thus, the detector in southern hemisphere should see weaker (up to factor 1/2) seasonal variations than the detector in the northern hemisphere.

~e = NO

-

Jan÷D~

I

I

Peb÷Hov Idar÷OeL l p t ÷ ~ q )

I

I Iday+Aull

Jml+Jul

Figure 4. The predicted seasonal plus eccentricity dependences of the total event rate. The figure is taken from [3].

where Ne and N ° are numbers of events with and without oscillations. Notice that for Am ~ in the range of the LMA solution the matter suppresses the depth of oscillations (effective mixing). The suppression weakens with increase of Am ~, and correspondingly, the excess increases. In the allowed region of parameters it can reach -,, 10%. There is a complementarity of searches for the N/D asymmetry of the solar neutrino signal and the excess of the e-like events in atmospheric neutrinos. According to fig. 5 (from [7]) with increase of AN~D the excess decreases and vice versa. For the central value of the present N/D asymmetry the excess is rather small: (2 - 5)%. The excess increases with [cos 01, leading to a positive up - down asymmetry. The asymmetry is very weak in the low energy part of the sub-GeV sample (p < 0.4 GeV) but it is clearly seen in the high energy part (p > 0.4 GeV) of the sample [6]. The excess decreases with increase of energy of events: it is practically absent in the multi-GeV sample. New high precision and high statistics experiments are needed to establish this effect.

106

A. Yu. Smirnov/Nuclear Physics B (Proc. Suppl.) 81 (2000) 102-112 '1

....

p ....

J ....

; ....

i ....

l''

'1

....

i ....

i ....

j

(10 - 13) °, so that the total lepton mixing can be 0z = 0~ + 0el ~ (28 - 40) ° - close to maximal mixing. T h a t is, the required large mixing can be obtained without special arrangements.

10

N/D-I(%)

~

.......... !

3. Beyond Four Dimensions

. /

•/ '/

i 2

d2.5 ......... 5 Z5 -10

In attempts to resolve the mass hierarchy problem new approach has been suggested recently [810] which is based on existence of large (in comparison with the Planck scale) extra dimensions. Main points of the approach relevant for our discussion are the following: 1). The fundamental scale of quantum gravity, Mr, can be as low as few TeV. The observed weakness of gravity is the result of N ( >__2) new space dimensions in which gravity can propagate. The observed (reduced) Planck scale, MR = (4~GN) -z/2 = 3.4. 10 is GeV, where GN is the Newton constant, is then related to the reduced Planck scale in 4 + N dimensions, My (fundamental scale), by

i

............

i

12.5 15 17.5 20 22.5 25

Am~r (1 0 (-') eV~)

Figure 5. The excess of the e-like events as the function of solar A m 2 for two different values of sin 2 2{712 relevant for solar neutrinos and sin 2 2023 = 0.8. Also shown by crosses is the dependence of the Night-Day asymmetry on Am 2.

One more remark: the LMA solution implies the mass of the second neutrino in the range (0.4 - 1.0) • 10 -2 eV which is only one order of magnitude smaller than the mass m3 relevant for the atmospheric neutrino anomaly:

Mp = MI /-MfV ,

(11)

where VN =-- LzL2 .... LN is the volume of the extra space, and Li is the size of the i - compact dimension. If the volume has a configuration of torus, then L~ = 27rR~, where Ri (i = 1, 2, ...N) are the radii of extra dimensions, and Vlv = (27r)N R]R2...RN •

(12)

Using (11) and (12) we get the constraint on the extra dimension radii: (27r)lVR1R2...RN = MM~ r+2 •

(13)

m 2 / m 3 "~ 10 -1. All other solutions of the solar neutrino problem require stronger mass hierarchy. A weak hierarchy indicates large mixing. Indeed, the mass matrix with above ratio of the eigenstates can naturally lead to mixing angle 0~:

The phenomenological acceptance requires that N _> 2, since for N = 1 the radius would be of the solar system size. According to present measurement the distance above which the Newtonian law should not be changed is about 1 mm, and therefore

tan 0~ ~ m ~

L~ = 27rRi < 1 mm .

.

(10)

This results in {?~ = ( 1 8 - 2 5 ) °. Diagonalization of the charge lepton matrix can give {?el ~ ~/~--~ --

(14)

For N = 2 and My --- TeV one gets from (11,12) R1 --~ R2 "-, 0.1 mm which satisfies (14). Thus, in theories under consideration it is expected

107

A. Yu. Smirnov/Nuclear Physics B (Proc. SuppL) 81 (2000) 102-112

that the Newtonian 1/r law breaks down at the scales smaller than the largest extra dimension: Lmax. The experimentally most exiting possibility would be if Lmaz ~ 1 - 10 -2 mm, that is, in the range of sensitivity of proposed experiments [11]. As we will argue in this paper the same range is suggested by neutrino physics, namely, by a solution of the solar neutrino problem based on existence of new dimensions. 2. All the standard model particles must be localized on a 3-dimensional hyper-surface ('brane') [12-14] embedded in the bulk of N large extra dimensions. The same is true for any other state charged under the standard model group. That is, all the particles split in two categories: those that live on the brane, and those which exist everywhere, 'bulk modes'. Graviton belongs to the second category. Besides the graviton there can be additional neutral states propagating in the bulk. In general, the couplings between the brane, Cbrane, and the bulk ~butk modes are suppressed by a volume factor:

~

1

Wb~neWbr~,~e~'b~tk .

(15)

According to (11) the coupling constant in (15) equals My = 3 . 1 0 -x6 Mf Mp 1TeV

(16)

and it does not depend on number of extra dimensions. It was suggested [15] to use this small model-independent coupling to explain the smallness of the neutrino mass. The left handed neutrino, Pn, having weak isospin and hypercharge must reside on the brane. Thus, it can get a naturally small Dirac mass through the mixing with some bulk fermion and the latter can be interpreted as the right-handed neutrino uR: h M f HPLuR .

(17)

Mp

symmetry breaking the interaction (17) will generate the mixing mass mo

=

hvMf

- -

Me

,

(18)

where v is the VEV of H. For NIl ~ 1 TeV and h = 1 this mass is about 5.6.10 -5 eV. 3. Being the bulk state, uR has a whole tower of the Kaluza-Klein (KK) relatives. For N extra dimensions they can be labeled by a set of N integers n l , n 2 . . . n g (which determine momenta in extra spaces): unl,n~...,~N R. Masses of these states are given by: rrtnl,n2...nN

--

n~ ' R'-'~

(19)

Notice that the masses of the KK states are determined by the radii whereas the scale below which deviations from the Newton law become appreciable is given by the size of compact dimensions (see Eq. (14)). The left handed neutrino couples with all unR with the same mixing mass (18). This mixing is possible due to the spontaneous breaking of the translational invariance in the bulk by the brane. In ref. [17] a general case has been considered with possible universal Majorana mass terms for the bulk fermions. Neutrino masses, mixings and vacuum oscillations have been studied for various sizes of mass parameters. Let us first assume that extra dimensions have the hierarchy of radii, and only one extra dimension has radius R in sub-millimeter range and therefore only one tower of corresponding KaluzaKlein modes is relevant for the low energy neutrino physics. The number and the size of other dimensions will be chosen to satisfy the constraint (13). The right handed bulk states, uiR, form with the left handed bulk components, UiL, the Dirac mass terms which originate from the quantized internal momenta in extra dimension: +oc

Here H is the Higgs doublet and h is the modeldependent Yukawa coupling. After electroweak

Trl n DnRIJnL q- b . c . ,

Trln ~-

(2o1

108

A. Yu, Smirnov /Nuclear Physics B (Proc. Suppl.) 81 (2000) 102-112

The mass-split is determined by 1 / R . According to (17, 18) the bulk states mix with usual left handed neutrino (for definiteness we will consider the electron neutrino I/eL ) by the Dirac type mass terms with universal mass parameter: mD

+~ ~

Vnltl/eL

,

n--~. - - O 0

hvM! m D -MR

,

(21)

(22)

= - , n

where (23)

v~hvM/R ~=

MR

determines mixing with the first bulk mode. The lightest state, u0, has the mass m o ~ m D and others, vn,: n

mn ~ ~

(24)

•

According to (22) the electron neutrino state can be written in terms of the mass eigenstates as

1( 1)

,

(25)

n=l

where the normalization factor N equals N 2 = 1 + ~2 E

~-2 = 1 +

~2 .

2

(27)

H i = m-A" 2E '

whereas for the electron neutrino we have

where h is the renormalized Yukawa coupling. Diagonalization of the mass matrix formed by the mass terms (20,21) in the limit of m D R << 1 gives (see Appendix) the mixing of neutrino with n th - bulk mode, ~,L: tanOn ~ - mn

Let us consider propagation of neutrinos in medium with varying density p ( r ) keeping in mind applications to solar neutrinos. The energies of bulk states do not depend on density:

(26)

n=l

From the phenomenological point of view the bulk modes (being the singlets of the SM symmetry group) can be considered as sterile neutrinos. Thus, we deal with the coupled system of the electron neutrino and infinite number of sterile neutrinos mixed. According to (25), the electron neutrino turns out to be the coherent mixture of states with increasing mass and decreasing mixing.

He ~ Y(p)

(28)

.

Therefore the level crossing scheme (the (H - p) plot which shows the dependence of the energies of levels H e , H i on density) consists of infinite number of horizontal parallel lines (27) crossed by the electron neutrino line (28). In what follows we will concentrate on the case ~ << 1, so that m D << m n for all n, and the crossings (resonances) occur in the neutrino channels. The resonance density, Pn, of He crossing with energy of n th bulk state: H n = H e ( p n ) equals according to (27, 28) m2nmg (x n 2 . p n = 2 E G F ( Y ~ - ½Yn)

(29)

For small mixing (~ << 1 ) the resonance layers for different bulk states (where the transitions, mainly, take place) are well separated: Pn+i - Pn >> A p n R

= Pn-n

,

(30)

here A p n R is the width of the n th- resonance layer. Therefore transformation in each resonance occurs independently and the interference of effects from different resonances can be neglected. In this case the survival probability ue --+ ue after crossing of k resonances is just the product of the survival probabilities in each resonance: P = P l x P2 x

....

x Pk .

(31)

Moreover, for P i we can take the asymptotic formula which describes transition with initial density being much larger than the resonance density and the final density - much smaller than the resonance density. As the first approximation we can use the Landau-Zenner formula [18]: Pn.~

1

. e-~"

E < EnR E > Enl~

(32)

A. Yu. Smirnov /Nuclear Physics B (Proc. Suppl.) 81 (2000) 102-112

where EnR is the resonance energy which corresponds to maximal (initial) density Pm~x in the region where neutrinos are produced: rn 2nrn N Enn ~ 2EGvprnax(Ye - ½Yn) ' p TI~2n s i n 2 2 0 n ~ = -2E cos20,~ dp/dr

(33) (34)

is the adiabaticity parameter and sin 2 2t~n 4~2/n 2. We can write final expression for the survival probability as P ~, e - ~ n ' f ( E ) ,

(35)

where 2~2 p '¢1 ~ ER----~ dp/d--~ '

(36)

and f ( E ) is the step-like function

f(E)={ O, E
EnR < E <~ En+I R

(37)

Since high level resonances turn on at higher energies and ~ o¢ l / E , the effect of conversion decreases with increase of the order of the resonance, n. Moreover, since in real situation the density is restricted from above and the energies of neutrinos are in certain range, only finite number of levels is relevant and the largest effect is due to the lowest mass resonance. Let us apply these results to solution of the solar neutrino problem. We choose the lowest (non-zero) bulk mass, m l -- l / R , in such a way that A m 2 = 1 / R 2 is in the range of small mixing MSW solution due to conversion to sterile neutrino ve --+ vs [19]: 1 R---~ - ( 4 - 10). 10 -6 eV 2 .

(38)

This corresponds to 1 / R = (2 - 3) 10 -3 eV or R = 0 . 0 6 - 0.10 mm .

(39)

Using mass squared difference (38) as well as maximal density and chemical composition of the sun we find from (33) E1R ~ 0.4 + 0.8 MeV .

(40)

109

Thus the pp-neutrinos (Epp < 0.42 MeV) do not undergo resonance conversion: Epp < E1R (see (35, 37)), whereas the beryllium neutrinos (EBe = 0.86 MeV) cross the first resonance. (For smaller 1 / R 2 the pp-neutrinos from the high energy part of their spectrum can cross the resonance and the flux can be partly suppressed.) The energies of the next resonances equal: E,~R = n2E1R, or explicitly: E2R = 1 . 6 - 3.2 MeV, E3R = 3.6 - 7.2 MeV, E4R = 6.4 - 12.8 MeV, EsR = 10 - 20 MeV, E6R = 14.4 - 28.8 MeV. Higher resonances (n > 6) turn on at therefore are irrelevant. The dependence of the survival probability on energy is shown in the fig. 1. The dips of the survival probability at the energies ,~ EiR are due to turning the corresponding resonances. The effects of higher resonances lead to additional suppression of the survival probability in comparison with the two neutrino case. Therefore the parameter 4~ 2 which is equivalent to sin 2 20 should be smaller. We find that 4~ ~ = ( 0 . 7 - 1.5). 10 -3

(41)

gives average suppression of the boron neutrino flux required by the SuperKamiokande results. According to (18), the value of ~ (41) determines the fundamental scale: ~Mp = 1 M f = v/-~hvR (0.35 - 0.7) TeV .

(42)

For small h the scale/klf can be large enough to satisfy various phenomenological bounds. Let us consider features of the suggested solution of the solar neutrino problem. The solution gives the fit of total rates in all experiments as good as usual 2z: flavor conversion does: the ppneutrino flux is unchanged or weakly suppressed, the beryllium neutrino flux can be strongly suppressed, whereas the boron neutrino flux is moderately suppressed and this latter suppression can be tuned by small variations of ~. Novel feature appears in distortion of the boron neutrino spectrum. As follows from fig. 1 three resonances turn on in the energy interval accessible by SuperKamiokande ( E > 5 MeV). The resonances lead to the wave-like modulation of the neutrino spectrum. (Sharp form (35, 37) is

A. Yu. Smirnoo/Nuclear PhysicsB (Proc. Suppl.) 81 (2000) 102-112

i10

1.0

/,

t

/!

i:.....1... J"

,"

a. 0.5

i!

0

0.1

,,"

"'4 .....

I

1.0

I0

E, M e V

requires larger original boron neutrino flux. The SNO experiment [20] will have better sensitivity to distortion of the spectrum. The slope of distortion of the neutrino spectrum is substantially smaller than in the case of conversion to one sterile neutrino (see fig. 1). In view of smearing effects due to integration over neutrino and electron energies (due to finite energy resolution) we can approximate the steplike function ] ( E ~ i n (37) by smooth function .law(E) ~ "x /~E / ~E l n . Then the smeared survival probability in the high energy range can be written as P ~ e-

Figure 6. The survival probability as the function of neutrino energy for the electron neutrino conversion to the bulk states in the Sun (solid line), 4~ 2 = 10 -3. Dot-dashed line shows the survival probability of the two neutrino conversion for the equivalent mixing sin 2 28 = 10 -3. Dashed line corresponds to the survival probability of the two neutrino conversion for sin s 28 = 4 - 10 -3 which gives good fit of the total rates in all experiments.

smeared due to integration over the production region.) The observation of such a regular wave structure with E o¢ n 2 would be an evidence of the extra dimensions. However, in practice this will be very difficult to realize. The SuperKamiokande experiment measures the energy spectrum of the recoil electrons from the reaction ~ e - ~ e [1]. Integrations over the neutrino energy as well as the electron energy of the survival probability folded with the neutrino cross-section and the electron energy resolution function lead to strong smearing of the distortion in the recoil spectrum. As the result, the electron energy spectrum will have just small positive slope (the larger the energy the weaker suppression) with very weak ( < 2 - 3 %) modulations. It is impossible to observe such a modulations with present statistics. Notice that relative modulations become stronger if mixing, ~, is larger than 10 -a and therefore suppression is stronger. This, however,

,z¢

,

(43)

where v ~ o = r~2 P/(R2dp/drv/-E-~). In the case of two large dimensions with R1 ~ R2 ~ 0.02 - 0.03 mm (see sect. 5) the number of bulk states, and therefore the number of relevant resonances increases quadratically: n 2. (Here n ~ 5 - 6 is the number of resonances in the energy range of solar neutrinos in the one dimension case.) Correspondingly, the approximating function lapp(E) will be proportional to E. As the result, hi • f ( E ) = const and the smeared survival probability will not depend on energy. In this case P(E) ~ const for E > E1R and there is no distortion of the recoil electron spectrum. For two different radii: R2 < R1 one can get any intermediate behaviour of the probability between that in (43) and P = const. Common signature of both standard v~ - v~ conversion and conversion to the bulk modes is the suppression of the neutral current (NC) interactions. The two can be, however, distinguished using the following fact. In the case of the ve - Vs conversion there is certain correlation between suppression of the NC interactions and distortion of the spectrum. The weaker distortion the weaker suppression of the NC interactions and vice versa. In the case of ve - Vbulk conversion a weak distortion can be accompanied by significant suppression of the NC events. This can be tested in the SNO experiment. No significant Day-Night asymmetry is expected due to smallness of mixing angle. Thus, the smeared distortion of the energy spectrum (for E > 5 MeV ) is weak or absent

A. Yu. Smirnov/Nuclear Physics B (Proc. Suppl.) 81 (2000) 102-112

in the case of ve - Vbutk conversion. However, in contrast to other energy independent solutions here pp-neutrino flux may not be suppressed, or the energy spectrum of pp- neutrinos can be significantly distorted. The suggested solution of the solar neutrino problem implies that the radius of at least one extra dimension is in the range 0.06 - 0.10 mm, that is, in the range of sensitivity of proposed gravitational measurement. The fundamental scale should be about h M f ~ 1 TeV. This mass satisfies the fixed overall volume condition provided additional large extra dimensions exist. In the case of one additional extra dimension its radius should be 1/R' = 1.3- 10-3(MI/1TeV) 4 eV. For h ,~ 1 one has M! --, 1 TeV, and 1/R' ,,~ 2- 10-3 eV, so that the second extra dimension will influence the solar neutrino data too. If h << 1, the fundamental scale can be much larger than 1 TeV, and R' can be much smaller than 1 mm. For h = 0.1 and M l = 10 TeV we get 1/R' ,,, 10 eV. For two additional dimensions the common radius equals 1/R' ,,, 3 . 1 0 -2 GeV, if h = 1 and M f = 10 TeV. For large fundamental scales (M/ > 10 - 20 TeV), direct laboratory searches at high energies will be practically impossible and neutrinos can give unique (complementary to gravitational measurements) opportunity to probe the effects of large extra dimensions.

4. Conclusion The LMA MSW solution predicts flat distortion of the recoil electon energy spectrum in the SuperKamiokande as well as SNO experiments. It predicts the Day-Night asymmetry and flat zenith angle distribution of events during night as is hinted by present data. The seasonal variations of the signal are smaller then the geometrical (eccentricity) effect in the allowed range of neutrino parameters. Moreover, there is a linear correlation between the D/N asymmetry and the seasonal variations. Further studies of these effects will allow us to identify (or exclude) this solution.

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The resonance conversion of the electron neutrino to the bulk states can solve the solar neutrino problem. Properties of this solution are similar to those due to conversion to sterile neutrino. The important difference is that significant suppression of the boron neutrino flux can be accompanied by weak distortion of the energy spectrum. Moreover, weak modulation of the boron neutrino spectrum is expected due to conversion to several KK-states. REFERENCES 1. K. Inoue, VIII Int. Workshop on Neutrino Telescopes, Venice Feb. 1999; Y. Fukuda et al., (Super-Kamiokande Collaboration), Phys. Rev. Lett. 82 (1999) 2430. 2. T. Kajita, "Beyond the Desert", Castle Ringberg, Tegernsee, Germany, June 6 - 12 (1999). 3. J.N. Bahcall, P. I. Krastev and A. Yu. Smirnov, hep-ph/9905220. 4. R. Escribano et al., Phys. Lett. B444 (1998) 397; J. N. Bahcall and P. I. Krastev Phys. Lett. B436 (1998) 243. 5. By P.C. de Holanda, C. Pena-Garay, M.C. Gonzalez-Garcia, J.W.F. Valle, Phys. Rev. D60 093010, 1999, hep-ph/9903473. 6. O.L.G. Peres, A. Yu. Smirnov, Phys. Lett. B456 (1999) 204; hep-ph/9902312. 7. O . L . G . Peres and A. Yu. Smirnov, in preparation. 8. N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. B429 (1998) 263, hepph/9803315. 9. I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Lett. B436 (1998) 257, hep-ph/9804398. 10. N. Arkani-Hamed, S. Dimopoulos and G. Dvali, Phys. Rev. D59 (1999) 086004 hepph/9807344. 11. J. C. Price, in proc. Int. Symp. on Eperimental Gravitational Physics, ed. P. F. Michelson, Guangzhou, China (World Scientific, Singapore 1988); J. C. Price et. al., NSF proposal 1996; A. Kapitulnik and T. Kenny, NSF proposal, 1997; J. C. Long, H. W. Chan and J. C. Price, hep-ph/9805217.

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12. V.A. Rubakov and M.E. Shaposhnikov, Phys. Lett. B125 (1983) 136; see also, A. Barnaveli and 0. Kancheli, Sov. J. Nucl. Phys. 51 (1990) 573. 13. G. Dvali and M. Shifman, Nucl. Phys. B504 (1997) 127. 14. G. Dvali and M. Shifman, Phys. Lett. B396 (1997) 64. 15. N. Arkani-Hamed, S. Dimopoulos, G. Dvali and J. March-Russell, Talk given at SUSY 99 by SD.; hep-ph/9811448. 16. A. E. Faraggi and M. Pospelov, hepphf9901299. 17. K.R. Dienes, E. Dudas and T. Gherghetta, hep-phJ9811428. 18. S. J. Parke, Phys. Rev. Lett., 57 (1986) 1275; W. C. Ha&on, Phys. Rev. Lett. 57, (1986) 1271. 19. see e. g., J.N. Bahcall, P.I. Krastev, A. Yu. Smirnov Phys. Rev. D58 096016 (1998). 20. G. T. Evan et al, SNO Collaboration, Report SNO-87-12 (1987).