Volume 185, number 3,4 PHYSICS LETTERS B 19 February 1987

INFLUENCE OF DENSITY FLUCTUATIONS

ON SOLAR NEUTRINO CONVERSION

Andreas SCHAFER and Steven E KOONIN w K Kellogg Radiation Laboratory, CAL TECH, Pasadena, CA 91125, USA

Received 13 November 1986

The conversion of electron-neutrinos into muon-neutrlnos within a star was recently proposed as a mechanism to ex- plmn the solar neutrino puzzle We investigate the influence of density fluctuations on the conversion rate Although the corrections can become large for statable chmces of parameters they are probably neghglble for the sun

The idea that the solar material can induce conversion of electron-neutrinos produced at the sun's center Into muon-neutrxnos was first proposed by Mlkheyev and Smlrnov [ 1 ] and subsequently discussed in much detail by Bethe and others [ 2 - 5 ] A sufficient depletmn of the electron-neutrino spectrum by this mechanism could ex- plain the result of the 37C1 experiment [6] This hypothesis at first seemed testable by comparing the results of several different neutrino experiments, each sensitive to different neutrino energies However, Kolb et al [3] pointed out that j u d i c i o u s choices of parameters can result in rather different electron-neutrmo spectra With the limited precision of any neutrino experiment It thus became extremely difficult to test the neutrino conver- sion hypotheses One should also keep In mind that there could also exist standard physics explanations for the low rate of observed neutrinos [7] In any event, as experimental tests are some time in the future, it is Important to ensure that the theoretical t reatment is complete and that there are no additional effects that might invalidate the Mlkheyev Smlrnov mechanism

We Investigate one such effect in this letter, namely the influence of macroscopic density fluctuations The sun IS known to ha*e a convective zone extenchng from approximately seventy-five percent of the solar radius out- wards Within this zone the density fluctuations are quite important , but the density itself is too small for the neu- trino conversion to work Whether there are any appreciable density fluctuations (so-caUed g-mode oscillations) at the transition density seems to be somewhat debatable [8,9], but if they exast on a spatial scale comparable to the wavelength of the neutrino oscillations at the transition point, they can influence the conversion process To assess this Influence we must calculate the conversion probabil i ty for each reahzatlon of the density, and then average this over the ensemble of density fluctuations to obtain the observed neutrino depletion

To obtain a neutrino conversion one the energy difference between the two mass elgenstates must be very small,

A E = (m 2 - m 2 ) / 2 k < ~ 10 -11 eV, (1)

where m l , m 2 are the rest masses of the two neutrino states These latter are linear combinations of the weak In- teraction eigenstates [10]

it) I) = Iv e) cos 0 -- lug) sin 0, Iv 2) = IP e) sm 0 + IP u) Cos 0 (2)

Here k is the neutrino momentum, of the order 1 - 1 0 MeV The corresponding wavelengths are macroscopic, X = 2rr/AE/> 120 km Density fluctuations on these scales can provade Fourier components that induce transitions between the two neutrino branches

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Volume 185, number 3,4 PHYSICS LETTERS B 19 February 1987

To investigate this problem we have modulated the solar electron density by

ne(r ) = n~O)(r) (1 + e sin pr) (3)

and have studied the resulting conversion probabflxty as a function of the modulat ion amphtude e and wave num- ber p More precisely, we integrated the Schrodmger equation

d ~a(t)~ =_l(x/~GFne(t)cos20 V~GFne(t)cosO smO ~(a(t)t (4)

-dt\b(t)] ~¢~GFne(t)cosO smO AE + Vr2GFne(t)sln20 ]\ b(t) ! '

where a(t), b(t) are the projections of the neutrino state on the free mass elgenstates

lu(t)) = a(t)lv 1) + b(t)lv 2) (5)

The lmtlal conditions are a(to) = cos 0, b(to) = sm 0 As the neutrinos travel practically with the speed of hght we can put r ~ t, ne(r ) ~ ne(t ) m eq (4)

With the exponential electron density of ref [3] , we cannot find an analytic solutxon to (4) and have there- fore studied it numerically Some results are shown m figs 1, 2 ff(Pe) xs the probabxhty that an electron-neutrmo produced m the solar interior emerges as an electron-neutrino It is determined by averaging over the ordinary neutrino oscillations, due to the ansatz (2) The latter are also present in free space

1)+, P(Ve, t) = [a(t)cos 0 + b( t )sm 012, ff(pe) = P(Pe ' r ) dr t large (6,7) t t

(l t is the oscdlatlon length of the usual neutrino oscdlatxons) In figs 1 ,2 /~(ve) is plot ted as a function o f p In umts of the minimal energy difference of the two neutrino

i I I I I I I I I I

0 6 AE=5 10-13 eV 8 = 0 0 3

0 ~ 4 . . . . . . . . . . .

0 6 ~

"~ o 4 ~ I n 0 2

I I I I I I I I I I

AE = 4 10 -12 eV 8=0 0106

O4

O2

0 I I I I I I I I I I

15 2_0

P

AE sin (2e) Fig 1 The averaged probabflaty that an electron neutrino does not convert into a muon neutrino as a function of the density fluc- tuation wave number p (for e = 0 01) The dashed line mdmates the value for e = 0

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Volume 185, number 3,4 PHYSICS LETTERS B 19 February 1987

In

004 -

0 0 3 - ,,

OOI

I , I I i i I I J

03 05 IO ~ii[

15

P

&E sln (20)

A E = 5 10 - I~eV 8 = 0 0 6 7

I I I I I I I I I

2 0 25

Fig 2 Similar to fig 1 for AE = 5 X 10 -13 eV and 0 = 0 067

states, AE sin 20 In these calculations the amplitude of the density oscillations e was chosen to be 1% The den- slty oscillations lead to an oscdlatxon of/~(Ve) around Its value for e = 0 The amphtude of these oscillations in- creases with AxE for a fixed value of &E sin 20 The latter determines the degree of adlabataclty and thus the mean value of/~(Ve) This increase is, however, l imited because A E = 10 -12 eV is about the largest energy for which the conversion mechanism still works

We also found that the perturbations m the amphtudes are, to a very good approximation, proport ional to e In fig 1 one also notes that the amplitudes are slowly changing as a function o f p This modulat ion is more obvi- ous in fig 2 where we have plot ted P(Ve) for a nearly adiabatic case For p >> AxE sin 20 the amplitudes actually vanish and one is left with the result for e = 0 Physically this is qmte clear, as wave numbers much larger than the level spacing cannot influence the Landau Zener effect Comparing fig 2 with the uppermost part of fig 1 It be- comes also clear that the relative changes m if( re) are largest for those cases in which the electron-neutrino con- verts most strongly into muon-neutrlnos

Can density oscillations significantly Influence the neutrino conversion rate9 As they lead primarily to oscilla- tions in P(ve) and not to a general shaft, the answer is very probably no However two caveats are (1) the unhkely possibility that the density fluctuations have a rather sharp wave number distribution centered around the minimal momentum difference of the two neutrino states, and 01) the nominal value of e = 0 01 we have assumed Al- though there is no direct observational Information about small scale (g-mode) oscillations deep within the sun, most astrophysicists would probably favour a much smaller value, mainly because any such oscillation should be radlatively damped Assuming, for example, e < 10 - 6 results in a neghglble mfluence on the conversion probabili- ty (Note, however, ref [8] and the papers discussed In ref [9] )

We would hke to acknowledge helpful discussion with K Llbbrecht and P Goldrelch

References

[ 1] S P Mlkheyev and A Y Sm~'nov, 10th Intern Workshop (Savonhnna, Finland, 1985) [2] H A Bethe, Phys Rev Lett 56 (1986) 1305 [3] E W Kolb, M S Turner and T P Walker, Phys Lett B 175 (1986) 478 [4] WC Haxton, Phys Rev Lett 57 (1986)1271 [5] S J Parke, Phys Rev Lett 57 (1986)1275

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Volume 185, number 3,4 PHYSICS LETTERS B 19 February 1987

[6] B T Cleveland, R Davis Jr and J K Eawley, m Weak interactions as probes of unification, AlP Conf Proc No 72 (ALP, New York, 1981) p 322

[7] J N Bahcall et a l , Rev Mod Plays 54 (1982) 767 [8] W H Press, Astrophys J 245 (1981) 286 [9] WC Haxton, Comm Nucl Part Phys 16 (1986)95

[10] L Wolfenstem, Phys Rev D 17 (1978)2369

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