On the oscillations of solar neutrinos in the sun

Volume 214, number 1

PHYSICS LETTERS B

10 November 1988

ON THE OSCILLATIONS OF SOLAR NEUTRINOS IN THE SUN S.T. P E T C O V DphPE, CEN-Saclay, 1-91191 Gif sur Yvette, France and Institute of Nuclear Research and Nuclear Energy, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria Received 12 August 1988

Using the exact solution of the system of evolution equations describing two-neutrino oscillations in matter with exponentially varying density we derive simple analytic expressions for the oscillating terms in the probability of solar neutrino survival in the sun. The regions of values of the parameters, characterizing the two-neutrino oscillations, for which the oscillating terms can be non-negligible are determined.

Matter-enhanced solar neutrino oscillations [ 1 ] and the possible solutions of the solar neutrino problem they imply [ 1,2 ] have been extensively studied in the last two years a~. In most o f the studies the terms in the probability o f solar neutrino survival in the sun P o (v~-,v~; t, to) (to is the time of electron-neutrino production in the sun, t >/to) which oscillate with the change o f the neutrino momentum, variation o f the density (or electron n u m b e r density Ne) in the point o f neutrino production etc., are neglected [ 1-3 ]. It is either assumed [2] that these terms will vanish as a result o f the averaging over the uncertainty in the neutrino m o m e n t u m , the position o f the solar neutrino detector etc., or an averaging procedure which practically averages them to zero is exploited [ 1,3 ]. The latter [ 1 ] corresponds effectively to averaging over an uncertainty in the position o f the solar neutrino detector equal for neutrinos with m o m e n t u m p to one oscillation length in v a c u u m Lv, Lv = 4 r t p / A m 2.

( 1)

Here p = If I and Am 2 = m 2 _ m 2, where ml,2 ( m 2 > m~ ) are the masses o f the neutrinos vLz with definite mass in vacuum. Such an averaging does not correspond to any of the possible physical uncertainties associated with the measurements of the flux and the spectrum o f solar neutrinos. In this note we study the oscillating terms present in the probability P o (Ve--'Ve; t, to) when the solar neutrinos take part in two-neutrino oscillations. We derive exact and simple approximate analytic ,expressions for the indicated terms using the exponential approximation for the matter (electron number) density distribution in the sun. According to the SSM [ 5,6 ], this approximation is rather accurate except in two regions located in the central part and close to the surface o f the sun. The system of evolution equations describing two-neutrino oscillations in matter [ 7 - 9 ] with exponentially varying density can be solved exactly [ 10,11 ] and our results are based on the solutions obtained in ref. [ l0 ]. We determine also the regions o f values o f the parameters, characterizing the two-neutrino oscillations, for which the oscillating terms can give non-negligible contributions in the probability of solar neutrino survival in the sun. We shall assume for definiteness that ve can oscillate (transform) into another flavour neutrino, say v~, and that the v e ~ v , oscillations can undergo a resonance amplification in matter [ 1,8,3,4], i.e., Am2 cos 2 0 > 0, where ~ See e.g. the review articles in refs. [ 3,4 ], where extensive lists of references can be found. 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )

139

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O is the neutrino mixing angle in v a c u u m ~2 (Am 2 > 0, 0 ~ O • n / 4 ) . The electron number density at which the resonance can take place N~-es is given by [ 1,3] N~es = AmZ(cos 20)/2pv/2Gv.

(2)

Numerical studies [ 1-3 ] have shown that the effects of solar matter on the solar neutrino oscillations can be substantial for tg220>~ 10 -a. Consider first the oscillations of solar neutrinos born in the closer (with respect to the earth) hemisphere o f the sun. According to the SSM [ 5,6 ], the Are distribution is spherically symmetric and Are changes approximately exponentially along the neutrino path in the sun:

Ne(t)=N~(to) e x p [ - ( t - t o ) / r o l ,

t>_,to,

(3)

where Are(to) is the electron number density in the point o f neutrino production and ro is a positive constant the scale height. In a large region in the sun, i.e., for 0.18Ro ~ r ~ 0.85Ro, where r is the distance from the centre of the sun and R o is the solar radius ( R o - 6.96 × 106 kin) one has [ 5,6,12 ] for the neutrinos moving radially towards the surface 0.085Ro ~
(4)

for any value o f p / A m 2 if for the values o f sin220 and Ne(to) considered 4n~ >> 1,

(5)

while in the case < 1 4no, ~

(6)

one has P o ( v e ~ V e ; t, t0)=PA(Ve--*V~; t, to)

forNr~>Ne(to)(1 --tg 2 0 ) - I ,

(7a)

Po(v~--'Ve; t, to) =/~NA(V~--'V~; t, to)

for I 1 -N~(to)/Nre~l ~
(7b)

Po(Ve--'V~; t, t0)=/~P(V~-~Ve; t, to)

f o r N ~ < N ~ ( t o ) ( l + t g 2 0 ) -1 •

(7c)

Here [ 12 ] 4n~ = 4 n o

NreeS=Ne(tO)

- dNddtlr~N~eS

AmZ2p

cos2oSin220Nres=Ne(tO)

'

(8)

4no being the adiabaticity parameter ~3 when N~(to) >~N~~ [ 1,3,4], /~A( V~~Ve; t, to) = ½+ ½ COS 2 0 COS 2Ore (to)

(9)

is the average probability o f the adiabatic [ 1 ], and [ 10] /~P(Ve ~v~; t, to) = ½ + (1 _ p , ) cos 2 O c o s 2Ore(to),

(10)

~2 It is supposed that in vacuum IVe) = IVt ) COSO+ IV2) sin O, Iv~) = -- Iv t ) sin O+ Iv2) cos O, where Iv~,~) and Iv L2) are the states of the neutrinos v~~ and v t z with momentum p, and that the neutrinos v~ 2 are stable and relativistic: E~=d ~ ~-p+m ~/2p, i=1,2. ~3We are using the same notations as in refs. [ 10,12,13 ]. 140

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where Om (t) is the neutrino mixing angle in matter [ 7,1 ], 1 -No(to)/~

e~

cos 2Ore (to) = { [ 1 -N~(to)/N~e~S]Z+tg220} ~/2'

( 11 )

and p, = exp [ -:tro ( 1 - cos 2 0 ) A m 2/2p ] - exp ( - 2nro Am 2/2p) 1-exp(-27troAm2/2p)

(12)

is [ 10,12,13 ] ~4 the analog of the L a n d a u - Z e n e r probability [ 14,15 ] for the case of exponentially varying density. The expression for the probability/~NA (Ve--'Ve; t, to) was derived in ref. [ 13 ] and we shall not give it here. Expressions (4) and ( 7 a ) - (7c) were shown [ 12 ] to reproduce with high precision (typically better than few percent) the probability/~o (v~--,v~; t, to) calculated numerically using the SSM prediction [ 5,6 ] for the Ne distribution in the sun if for given N~ eS <~N~(to) [N~e~ > N~(to) ] the value ofro entering into (4) and ( 7 b ) - ( 7 c ) is chosen to coincide with the value of the ratio N e ( t ) / d N e ( t ) / d t l in the layer of the sun wherein N e = N ~ ~ [N~=N~(to) ]. It should be added that condition (6) can be realized for the solar neutrinos only for [12] s i n 2 2 0 < ( 4 6 ) × 10- 3 #5. This implies that the interval of values of N ~ in which P o (v~--,vg t, to ) is reproduced by PrqA(V~--,V~; t, to) [see eq. (7b) ] is very narrow. Being a monotonically increasing function of N ~es (almost a straight line) in this interval, PNA (Ve--'V~; t, t0) can be substituted in m a n y practical calculations with its value in the central point of the interval [N~ ~ =N~(to) ] given by [ 13 ]

/~NA(Ve-"*Ve;t, to) IN~os=ue(,o) =

½[ 1 + e x p ( - n n ' o ) ] .

(13)

Although the results (4) and ( 7 a ) - (7c) have been derived for neutrinos originating in the closer half of the solar core, i.e. for which Are(to) I> Are(t), t/> to, they can be used also to describe the transitions of neutrinos born in the far hemisphere of the sun when condition (5) is fulfilled. Indeed, it is not difficult to show that if (5) is realized for a neutrino produced in the far hemisphere, N~ (t) will change adiabatically along the neutrino path in the region where N~(t)>>.N~(to). The density will first increase along the path of the neutrino up to some maximal value Nmax; then it will decrease and at a certain point reached at time t3 and located in the closer hemisphere will be equal again to N~(to) ~6. For such a neutrino one has /~o( v~-~ve; t, t o ) = P o (v~-~v~; t, t0) COS22Om(t0)+ ½sinE2Om(t0) + ½ ( 1 - - 2 P ' ) sinE2Om(to) COS2Om(to) COS20,

t-t'o>~Ro,

independently of the n u m b e r of resonance layers it crossed (two, one or zero) on its way to the surface. The same conclusion is valid for values of p / A m 2 for which N ~ - e S < N e ( t o ) ( l + t g 2 0 ) -1 or for which N~eS>N¢(to)( l - t g 2 0 ) -1 and 4no>> 1, when condition (6) takes place. Eqs. (4) and ( 7 a ) - ( 7 c ) are not sufficient to determine/~o (v~--,v~; t, to) for neutrinos crossing two resonance layers located in the region of neutrino production ( N r~s >i Ne (to) ( 1 + tg 2 0 ) - 1, N ~ < N max ) if ( 6 ) is realized and the change of N~ in the resonance layers is non-adiabatic (4no ~< 1 ). In this case one can use the exact results [ 10,11 ] for the relevant probability ~4 See also the papers quoted in ref. [ 11 ]. ~5The upper bound on sin220 of interest is [ 12 ] inversely proportional to the values of Ne and ro in the point of neutrino production. The quoted number corresponds to Ne (to)/NA = 20 g/era 3 and ro= 0.1Ro. For Ne(to)/NA = 98 g/cm 3 it is equal approximately to 10- 4 [12]. ~6 This follows from the assumed spherical symmetry of the Ne distribution in the sun and from the fact that the sun-earth distance is approximately by a factor 103 larger than the supposed radius of the region of solar neutrino production [ 5,6 ]. 141

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amplitudes derived in the exponential density approximation to calculate/~o (Ve--'ve; t, to) or expression ( 10 ) in which the factor P' is substituted by 2P' ( 1 - P ' ) [ 16 ] ~7 Let us turn next to the oscillating terms in Po (verve; t, to). The probability of the solar neutrino survival in the sun in the case of adiabatic transitions is well known and has the form [3,4] PA(Ve---~Ve, t, to)=/~A(V~--'Ve; t, to)+PXSC(Ve--'V~; t, to), where PXSC(ve~v~; t, to)=½ sin 2Osin 2Ore(t0) t

XCOS{ 2o--~-p f A m 2 [L( 1

2

1/2

Ne(t'))N~S cos220+sin220]

dt'}

(14)

is a term which oscillates with the change ofp/Arn 2, N'~es, etc., and/~A(Ve--'V~; t, to) has been defined earlier. Thus the problem is to find adequate analytic expressions for the oscillating terms when the solar neutrino transitions in the sun non-adiabatic. Assuming that N~(t) varies exponentially in the sun [eq. (3) ] and using the results ofrefs. [ 10,13] it is not difficult to obtain the following exact expression for the probability to find neutrino v~ at time t in the point of neutrino trajectory in the sun with electron number density N~(t)~ 0 if it was produced at time to in a point with density Ne (to) and N~ changed monotonically (increased or decreased) between the two point along the neutrino path: p~o( v~ ~v~; t, to) = 1 - ¼ sin220( I q~(a, c; Zo) I21 q ~ ( a - c + 1, 2 - c ; Z) 12 + [ q~(a, c; Z)121 ~ ( a - c +

1, 2 - c ; Zo)12

-2Re{exp[ --i(t--to)Am2/2p]

~*(a, c, Zo)q~(a, c; Z ) ~ * ( a - c + 1, 2-c, Z ) ~ ( a - c + 1, 2 - c ; Zo)}). (15)

Here q~(a, c; Z~o) ) and Z l f f f ~ ( a - c + 1, 2 - c ; Z~o) ) ( c ¢ 0, _+ 1, _+2, ... ) are two linearly independent confluent hypergeometric functions [ 17 ] having the property ~8 ~ (a,, c'; 0) = 1, when a', c' ¢ 0, - 1, - 2 , ..., a', c' being arbitrary parameters Z(o) =iro v/2 GvNe(t~o)) ,

(16)

and

a= 1 +iro( Am2/2p) sin20,

c = 1 +iroAm2/2p.

(17)

Note that ( 15 ) is symmetric with respect to the interchange of the initial and final points of the neutrino path. Obviously, expression (15) represents a generalization of the expression for the probability of solar neutrino oscillations in vacuum (see e.g. ref. [ 4 ] ) and reduces to the latter in the vacuum limit N~ (tto))-~ 0. It follows from ( 15 ) and (16) that for the neutrinos originating in the closer solar hemisphere the survival probability in the sun [ N~ (t) = 0 ] is given by [ 13 ] P~P(Ve--'Ve; t, to) = 1 -- ~ sin220( I q~(a, c; Zo) 12+ I ~ ( a - c +

1, 2 - c ; Zo) 12

-2Re{exp[i(t-to)Am2/2p] ~' (a, c; Z o ) ~ ( a - c + 1, 2 - c ; Z o ) } ) .

(18)

Averaging ( 18 ) over one vacuum oscillation length [ 1 ] we get [ 13 ] using ( 1 ) ~7The accuracyof this approximation has not been studied. It is, probably, not better than (20-30)% for a certain range of values of p~ Am2 [12,13]. ~8The function qb(a, c; Z) is related to Whittaker's function M~.u(Z) [17 ]: Mx.u(Z) =exp(- ½Z)ZC/Z~(a,c; Z), where a= ½-x+/t and c= 1+ 2,u. 142

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R+Lv

P~P(v¢~v¢)=Lv t

_1 P ~ P ( v ¢ ~ v ¢ ; t ' , t o ) dt' r~Ro

= 1 -~

sin220( I ~b(a, c; Z o ) 1 2 + l ~ ( a - c +

1, 2 - c ; No)12) .

(19)

In certain cases ("large" I Zol limit) is more convenient to express P~p (Ve--'Ve; t, to) in terms of a different set of linearly independent confluent hypergeometric functions [ 17 ] ~(a, c; Z) and 7.,(c_ a, c, - Z ) : • (a, c; Z ) = [ F ( c ) / F ( c - a ) ]

exp(ina) ~U(a, c; N ) + [ F ( c ) / F ( a ) ] e x p [ i n ( a - c ) + Z ]

~ ( c - a , c; - Z ) ,

(20) where [ 17 ] ~P(a, c; Z ) -, :~ o IF( 1 - c ) / F ( a - c + 1 ) ] + Z ' - c i F ( c _ 1 ) / F ( a ) ], and F ( a ) , is the gamma function. From eq. ( 18 ) using (20) we find for the solar neutrinos born in the closer hemisphere of the sun: 3

P o ( v c ~ v G t, t o ) = / 5 ( v c - , v ~ ) + ~ P°SC(Ve--~Ve) . i=l

(21)

Here P(v~-,v~) = 1 + ( 1 _ p , ) (cos 20)F1 +F2

(22)

is the average solar neutrino survival probability in the sun [ 13 ] and P?SC(Ve--*V~) = --~/P' ( 1 - P ' ) (sin 2 0 ) F I c o s ( ~ z + q~22),

(23)

p~sc(ve ~ v e ) = - x/P' ( 1 - P ' )

(24)

cos 20{ IF3l [Nr~/N~(to)] tg 20} COS(~,2 - ~22),

P]~(v~ ~v¢) = - 1p, sin 20{ IF31 [N~e~/N~(to) ] tg 20} (cos 2 ~ 2 +cos 2 ~ 2 2 ) ,

(25)

e~(v~--,v~) = ½ sin 20{ IF31 [N~d~/Ne(to) ] tg 20} cos 2~22

(26)

are oscillating terms. In eqs. ( 2 2 ) - ( 2 6 ) F~, i = 1, 2, 3, are functions of AmZ/2p, O, ro and N~(to) and [ 13] F, = - I ( - Z o ) " - " T ( c - a ,

c; - No) 12+ (roe,2 IZo I - ' ) 2lZg ~U(a, c; No) I z ,

F 2 = ½ [ 1 -- I ( - - Z o ) " - ~ ( c - a ,

c; - N o ) I 2 - (ro812 IZo 1-1) ZiNg ~U(a' c; No) 121,

F3 = [Zg ~P(a, c; N o ) ] * ( - Z o ) ~ - ~ ( c - a ,

c; - N o ) ,

(27) (28) (29)

qh2 and ~22 are functions of AmZ /Zp, O, ro, N~( to) and ( t - t o ) A m Z /Zp, ~)12--~22 =

lEo I -~0~ - ~ 2 +~O-ro(AmZ/ZP)(COS 2 0 ) lnlZo I ,

~12 + rib22=2go3 q-~l -~02 + r o ( A m Z / Z p ) lnlZo I - ( t - t o ) A m Z / 2 p ,

(30) (31)

where e,2= ( A m Z / 4 p ) sin 20, ~l = a r g F ( a -

1),

~03=argF(1-c),

~02= a r g F ( a - c ) ,

(32)

~o=argF3( AmZ/Zp, O, ro, N~(to) ) .

(33)

Let us note once again that expressions ( 15 ), ( 18 ) and ( 2 2 ) - (26) are exact within the exponential approximation [ eq. ( 3 ) ] for the electron number density distribution in the sun. It follows from ( 2 3 ) - ( 26 ) and ( 30 ) (33 ) that of the four oscillating terms only P~Se (Ve~ V~) survives if we apply the averaging ( 19 ) adopted in refs. [1,31. On the basis of the exact results ( 2 3 ) - ( 2 6 ) we shall derive next simple approximate but very accurate formulae for the probabilities po~ (v~-~v~) using the asymptotic series expansions [ 17 ] in powers of Z6- ~ of the functions (Zg ~ ( a , c; No) ) and ( - Zo) ~- ~~ ( c - a, c; - No). Note that since according to the SSM, the solar 143

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neutrinos are produced in the central region of the sun wherein 20 g/cm3 ~~0.1R o, and i Z o l = x / ~ GFroNe(to)~_5.2XlO 2

ro

Ne(to)

0.1Ro 20 g / c m 3 NA '

(34)

one has IZo I >> 1. Under the conditions [ 10,12,13 ] N ¢ ( t o ) > N ~ es ,

(35)

tg22Om(to) < 1,

(36)

one obtains neglecting terms of sixth (fourth) or higher orders in Zr- ~ in the corresponding expansions in powers of Z6 t ofF~ andF2 (F3): F~ =cos 2Ore(to)- 11Zo i - 2 ( 3 + 2 0 x ) tg/2Om(t0) =COS 2Om(to)--O( ~ 10--5), N~S " I F 3 1N- -. (t tgo2) 0 = s l n 2 O m ( t o )

E(

Fz = 0 ,

(37)

( l - - x ) ( 2 + 1 lx) '~2 1+ iZolUCOS2Om(tO)}

(1 - x ) 2 --]1/2 + IZo Ii cos"~'Om(to) [1 + 3 x + 6 x 2 - 6 IZo I - 2 - 3 ( 1 - x ) z tgZ2Om(tO)]2J =sin 2Om(to) [1 + O ( ~ 10 - s ) ] ,

(38)

where x = N ~es/N¢ (to) and the result ( 37 ) for Ft and Fz has been derived in ref. [ 10 ]. Eqs. (10), (22) and (37), (38) imply that in the approximation used to derive (37), (38), IP(vo-'Ve; t, to) - P ~ P (v~--,ve; t, to) I ~ 10 -s. From ecls. ( 2 3 ) - ( 2 6 ) using (37), (38) we get for the oscillating terms p~sc(v,--,v~) = - x / P ' ( 1 - e ' ) sin 2 0 cos 2Ore (to) cos(O12 + O22),

(39)

P~SC(v,--,v~) = - x / P ' ( 1 - P ' ) cos 2 0 sin 2Ore(t0) COS(O,Z --~22) ,

(40)

P]~¢( VeoV~) = --1p, sin 2 0 cos 2Om(to)(COS 2~12 +COS 2~22),

(41)

P~"~(ve o v ¢ ) = ½ sin 2Osin 20m(to) cos 2q022.

(42)

Our further discussion will be concerned with these results. It follows from general considerations that under the conditions (35 ), (36) the quantity P' [ecl. ( 12 ) ] entering into ecls. (10) and ( 3 9 ) - ( 4 2 ) represents the probability of the transition v~(to)--.vz in the sun, where v~}2~ (to) denotes matter eigenstate neutrino [3A] in the point of neutrino production ~9 [ v ~ 2 ) ( t ) = v~2~ if Are(t) = 0 ]. Thus the limit P'--*0 (which is attained for fixed sin220 at sufficiently small values of p / A m 2) corresponds [ 3,4] to purely adiabatic transitions. In this limit P~.~.3 (VerVe)--,0, which, together with a comparison of eq. (14) with eq. (42), permits to identify P2 ~ (Ve--,Ve) as the oscillating term in the probability of the adiabatic transitions ~lO. The exact expression for this term is given, however, by eq. (14) and we shall use eq. (14) instead of P] ~ (Ve--,V~) in what follows. Performing an analysis analogous to the one made in ref. [ 12 ], which leads to the results (4) and ( 7a ) - (7c), we find for the oscillating term P~C (VerVe; t, to) in the probability of solar neutrino survival in the sun; for given sin220 and N~ (to) ~9 As can be shown, in this case ~12(22)=arg(a(v~2) (to)-~v2)) - (t-to)Am2/4p+ ½f~o[ (Am2/2p) cos 20-x/~GFN~(t' ) ]dt', where to is the time at which the neutrino reaches the surface of the sun (t t>to ) and a (v~ z) (to)-~v2) is the amplitude of the probability of the transition v~'t2) (t0)--.v2in the sun. ~o We have checked that for any fixed sin 220 and N~( to) relevant for the oscillations of solar neutrinos the regions of p/Am2 for which P~SC(Ve--~Ve)[orP~,~(v~--,v~)]and Zi= 3 ~P~o~¢(v.~v~) can give non-negligiblecontributions in P ~ (v¢--,v~;t, to) practically do not

overlap. 144

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_ osc P~C(ve~ve; t, to)--PNA(Ve ~V~; t, to)

10 November 1988

(43)

for all p / A m 2 if condition (5) is fulfilled, and P~C(Ve~V~; t, to)=P~SC(v~ve; t, to)

f o r N ~ S > N e ( t o ) ( 1 - t g 2~9) -~ ,

(44a)

P~C(Ve ~v~; t, to) =p~s~(v~__.v~; t, to)

for N~es
(44b)

when (6) takes place. In eqs. (43) and (44b) 3

pose, NA~V~__,V~,. t, to)=e~,~(V~V; t, to)+ ~ e ° ~ ( v ~ v ~ )

(45)

i=1

Obviously, eqs. (43) and (44a), (44b) are the counterparts of the eqs. (4) and ( 7 a ) - ( 7 c ) for the oscillating terms in the full probability Po(ve~v~; t, to) = / ~ e ( v ~ V e ; t, to) + P ~ ( v ~ v ~ ;

t, to) •

If follows from (44a) and (44b) that we lack an analytic expression for P ~ (v~--,v~; t, to) only for the case of non-adiabatic transitions occuring for values of N~e~ (or p / A m 2) from the interval I 1 -N~ ( t o ) / N ~ I <~tg 2 0 when 4no' <~ 1. As was indicated earlier, for the solar neutrino transitions of interest this interval is very narrow [sin22~9< ( 4 - 6 ) X 1 0 - 3 ] . For this reason the knowledge of the form of the oscillating term in it will not be essential for our conclusions. We shall next find the ranges of values of sin220 and p / A m 2 for which the oscillating terms po~>, i = 1, 2, 3 in Po (VerVe; t, t0) are non-negligible in comparison with the average probability/~o (v~--.v; t, to), i.e., for which Ie~°~'~ I//~o ( v e r y ; t, to) >/0.05. Using this criterion, eqs. ( 3 9 ) - ( 4 5 ) , (14), (4), ( 7 a ) - ( 7 c ) and ( 9 ) - ( 1 2 ) , the SSM predictions [5,6] for Ne ( t ) / I d N d d t l and the values of N~ in the region of neutrino production, it is not difficult to convince oneself that (i) all oscillating terms are negligible for sin220< (6-8) X 10-4; (ii) the term P ~ (v~--,v~) is negligible for all values of sin22~9 and p / A m 2 of interest; (iii) P ~ (VerVe) can be non-negligible [13] only for sin22~9<~ 10 -2 and when both/~o(v~--.v~; t, to) and I P ~ ~c ( V e " ' ~ V e ) I are smaller than 10-2; (iv) P~C (v~--,v~) is negligible for sin220< 6 X 10 -3, while for values o f p / A m 2 for which P ~ (ve--,v~) cannot be neglected (according to the criterion chosen by us) IPX~(v¢--.v¢)I < 5 x 10 -2 if sin220< 10 -2, and for 10-2~> N~(to), which is realized for the solar neutrinos for p / A m 2 << 6 × 104 MeV/eV 2, P ~ (VerVe) reduces to the vacuum oscillating term which averages to zero for the indicated values of p/Am2; (v) If sin220< 10 -2, I P~(Ve~V~) I < 5 × 10 -2 for the values ofp/Am 2 for which P ~ ( v ~ v ¢ ) is non-negligible. To summarize, oscillations with sufficiently large amplitudes exceeding 5 X 10- 2 can be generated only by the terms P~e (verve) [eq. (41) ] and P~S~( v ~ v ~ ) [eq. (14) ] for sin'~20~>0.01. As can be shown, p~c ( v ~ v ~ ) and P ~ (V~--'Ve) can be non-negligible in the indicated interval of values of sin220 for p / A m 2 > 2 X 106 MeV/ eV 2 and for p/Am2< 2 × 106 MeV/eV 2, respectively. In the limit Am 2/2p--,0, which corresponds to extreme non-adiabatic transitions, p~c (v~--,v~; t, to)--,0 while P?~ (v~--.v~) converges to the vacuum oscillating term. Although eqs. (39 ) - ( 4 5 ) have been derived for scalar neutrinos originating in the closer solar hemisphere, for the values of sin220 for which which P .o. . .tve ~ v~,. t, to) can be non-negligible our results permit to find the oscillating terms in P e ( v ~ v e ; t, to) for the neutrinos produced in the far solar hemisphere. In conclusion, we have derived simple analytic expressions for the oscillating terms in the probability of solar neutrino survival in the the sun when two neutrinos take part in the oscillations, and have determined the regions of values of the oscillating parameters for which these terms can be non-negligible. The effect of averaging over the uncertainty in the solar neutrino energy, in the position of the solar neutrino detector etc., on the oscillating terms will be studied in a forthcoming publication. 145

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10 November 1988

It is a p l e a s a n t o b l i g a t i o n to t h a n k P. K r a s t e v for help in the n u m e r i c a l calculations m a d e for the final part o f the p r e s e n t paper. I w o u l d like also to a c k n o w l e d g e w i t h gratefulness the k i n d h o s p i t a l i t y o f the m e m b e r s o f D P h P E - C E N - S a c l a y , w h e r e this w o r k has b e e n c o m p l e t e d .

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