Neutrino Masses and Oscillations
XA9743420 IC/96/249 hep-ph/9611465
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
NEUTRINO MASSES AND OSCILLATIONS
Alexei Yu. Smirnov
INTERNATIONAL ATOMIC ENERGY AGENCY
UNITED NATIONS EDUCATIONAL. SCIENTIFIC AND CULTURAL ORGANIZATION MIRAMARE TRIESTE £ IC/96/249 hep-ph/9611465
United Nations Educational Scientific and Cultural Organization and International Atomic Energy Agency INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS
NEUTRINO MASSES AND OSCILLATIONS1
Alexei Yu. Smirnov 2 International Centre for Theoretical Physics, Trieste, Italy.
ABSTRACT
New effects related to refraction of neutrinos in different media are reviewed and implication of the effects to neutrino mass and mixing are discussed. Patterns of neutrino masses and mixing implied by existing hints/bounds are described. Recent results on neutrino mass generation are presented. They include neutrino masses in SO(IO) GUT’s and models with anomalous [/(l), generation of neutrino mass via neutrino-neutralino mixing, models of sterile neutrino.
MIRAMARE - TRIESTE November 1996
iPlenary talk given at the 28th International Conference on High Energy Physics, 25 - 31 July 1996, Warsaw, Poland. 2On leave of absence from: Institute for Nuclear Research, Russian Academy of Sci ences, 117312 Moscow, Russian Federation. E-mail: [email protected] 1 Introduction 1.3 Lower bounds on neutrino mass?
1.1 Hints Neutrinos are the only fermions for which the Standard model predicts masses. It predicts that A number of results testifies for non-zero neutrino neutrino masses are zero. This follows from the masses and mixing: content of the model, namely, from the fact that in the model there is • Solar neutrino spectroscopy. • no right-handed neutrino components, • Results on atmospheric neutrinos. • Large scale structure of the Universe. (Its • no Higgs triplets which can give the Majo- formation may imply some amount of the rana mass for the left-handed neutrinos. hot dark matter (HDM)). The absence of the i/r gives an explanation • LSND results. of strong upper bounds on the neutrino masses. However, the absence of i/r looks rather anes • Hydrogen ionization in the Universe. thetic. The Standard Model is not the end of the • Peculiar velocities of pulsars. story and we know that at least there is the grav • Excess of events in tritium spectrum. ity. Both above items can be questioned by grav ity: First four items are reviewed by Y. Suzuki \ 1. One point is related to a consistency of the fifth item was discussed in 2, and the last two the theory. It is argued 29 that invariant (Pauli- will be presented in sect. 2 . Villars) regularization in the case of local Lorentz invariance requires an existence of 16 spinors, i.e. 1.2 Upper bounds an additional spinor with properties of i/r . Once i>r exists there is no reason not to introduce the The majority of results give just upper bounds Dirac mass term for neutrinos. on neutrino masses and mixing. The most strong 2. It is believed that gravity breaks global bounds relevant for the discussion come from: quantum numbers. In the SM the lepton number - reactor oscillation experiments BUGEY3, Kras is global and therefore one expects its violation by noyarsk 4 (t>e - vx oscillations ); gravity. The effect of violation may be parameter - meson factory oscillation experiments (KAR ized in the form of the nonrenormalizable operator MEN5, LSND6 ); in the effective Lagrangian 30: - accelerator experiments E531 —h/t ) 7 and E776 K - vT )8; - direct kinematic searches of neutrino mass: tri wPLLHH ’ tium experiment in Troitzk on 9, (see also Maiz experiment 10), PSI experiment on the mass of where Mp is the Planck mass, L is the lepton dou Vft u , LEP ALEPH result on vT 12 (see also OPAL blet, H is the Higgs doublet. The operator (1) result 13; new possibility to measure the mass of leads to the mass of neutrino i/T has been suggested in H); m„p 10"5eV . (2) - searches for the neutrinoless double beta de Mp GpMp cay in Heidelberg-Moscow experiment15, (see also IGEX16); Here i) is the renormalization group factor and Gp - supernova 1987A data 17,18,19 ; dynamics of su is the Fermi constant. In fact, the interaction (1) pernovas 17 ; allows to overcome the problem in the second item: - nucleosyntesis in supernovas 20; The product HH plays the role of the effective - primordial Nucleosyntesis 21,22,23; Higgs triplet. - cosmology 24; The value (2) can be considered as the lower - structure formation in the Universe 25,26,27,28 . bound on neutrino mass. Indeed, Mp is the These results give important restrictions on a biggest mass scale we have in the theory. If possible pattern of neutrino masses and mixing. some new interactions exist below this scale at
1 M < Mp, these interactions can generate the op 2 Refraction and neutrino masses erator (1) with Mp being substituted by M. The corresponding neutrino mass, t)(H)2/M , is bigger There are several new results on neutrino refrac than rrivp. Inverting the point, one can say that tion and propagation in media which have impor the observation of mass m„ > mvp will testify for tant implications to the neutrino mass problem. new physics below the Planck scale: 2.1 Effective potentials 1 M (3) In transparent medium neutrinos undergo essen Gpm „ tially elastic forward scattering. The effect of the scattering is described by Note that the physical scale (the scale of new par Gf ticle masses, or condensates) can be even much H = UY[\ - ~f'y)v(4’e\e’rp(gv +gAls)e\ipe) , smaller than the one estimated from (3). In par V2 ticular, M can be a combination of other mass pa (5) rameters M', m3/2 which are much smaller than where ijje is the wave function of medium. (We took into account the interactions with electrons M itself: e.g. M = (M')2/m3/2- only.) For ultrarelativistic neutrinos expression Recently a phenomenological lower bound on (5) can be reduced to m„ has been suggested 31. The exchange of mass less neutrinos leads to the long range neutrino H = V2GF V v'v , (6 ) forces. In particular, two body potential due to the exchange of the i>v - pair gives 32 : where V is the effective potential. Let us summa rize the results on the potentials for some cases:
1/(2) aGF 1. Unpolarized medium at the rest: Only 7 0 component of the vector current contributes to V 0 4?r3r5 ’ (4) and its matrix element gives the density of elec trons, nc. As a result we get 33 : where a is known coefficient. Many body (four, six ... k ...) potentials contain additional factors V = y/2Gpn egv ■ (7) {Gp/r 2)2k which are extremely small for r = Rn, (radius of neutron star). However, in compact stel 2. Polarized medium at the rest. The ax lar objects like neutron stars and whitedwarfs, the ial vector current, 775 , also gives the contribution contributions of these many body interactions to which is proportional to the vector of spin 34 : the energy of the star are greatly enhanced due to combinatorial factor. V = \Z2Gpn e gv +3a2(£- (s)) (8) The contribution of t-body interactions, Wk, to the total energy is proportional to the num where k = p/p, and p is the momentum of neu ber of combinations of 6 -neutrons from the total trino, (s) is the averaged spin of electrons in number of neutrons in a star. The combinatorial medium. The second term can be rewritten as factor leads to the series parameter Wk+2/Wk ~ V2Gf5a(m+ — n_). Here n+, n_ are the concen (GpnR ns)2 ~ (1013)2, where n is the number den trations of the electrons with polarization along sity of neutrons, so that the six body contribution and against the neutrino momentum. to the energy dominates over the four body con 3. In the case of moving medium also spa tribution etc.31. It turns out that the energy due tial components of the vector current give non-zero to the eight body interaction overcomes the mass contribution: (i/>e|7 |V'e) oc v and 35 of a star. According to 31 the only way to resolve this paradox is to suggest that all neutrinos have V = \Z2Gpn egv{l — v ■ cos 6) , (9) nonzero masses: nv > 0.4 eV: Neutrino mass cuts off the forces at r > l/m„. In section 2.3 we will where 9 is the angle between the momentum of argue that there is another resolution of the para the electrons and neutrino. In the case of isotropic dox and the mass of neutrino can be zero. distribution the correction disappears. In this case
2 the non zero effect of the motion appears via the of forces at r > l/p. Similar oscillating factors correction to the propagator of the vector boson: appear for many body interactions. As a result Gf —*■ Gf(1 4- q2/rnly), where q 2 is the four mo the many body forces do not dominate in the self mentum of the intermediate boson squared 35. In energy of the star. This can resolve the energy thermal bath q 2 ~ T2 and one gets 36 paradox suggested in 31 even for massless neutri T2 nos 38. (10) Another objection to the Fischbach result is mw related to resummation of series over the t-body where A is the constant which depends on the interactions 39. The interaction with medium composition of plasma. In all cases, apart from modifies the dispersion relation for neutrinos: the thermal correction (10), V has opposite signs for neutrinos and antineutrinos. 90 = ± V , (14) and correspondingly, the propagator of neutrino: 2.2 Neutrino sea and the long range neutrino forces i (15) At low energies a medium is transparent for neu (90 - y)7° - 97 trinos and the main effect is the refraction. Re This dressed propagator is the sum of free propa fraction index equals: gator and the results of elastic forward scattering on one neutron, two neutrons .... k —>■ oo neutrons (nr - 1) = — oc ------(11) P P in medium. If the neutrino forms a closed loop, then this process is equivalent to the summation At usual conditions: E ~ 1 MeV, p = 1 g/cm 3, of 0, 2, 4, .... k body interactions due to neutrino the deviation of the refraction index from 1 is ex exchange. Therefore the energy density due to the tremely small: (nr — 1) ~ 10_2° — 10~19. How neutrino exchange can be written as ever, at very low energies this deviation can be of the order one, leading to complete inner re 9° 7° 1 1 - 7s (16) flection of neutrinos in stars37 . For neutron star 9* 2 . with p ~ 1014 g/cm 3 the complete reflection takes place for neutrinos with energies E < 50 eV. In The energy density due to the interactions, Aw, other terms, a star can be considered as a potential is the difference of to given in (16) and too - the well with the depth V. The potential has different energy density for vacuum propagator: Ato = to — signs for neutrinos and antineutrinos. Therefore, too- The total energy of a star is the integral of A to neutrinos are trapped, whereas antineutrinos are over the volume of a star. In the approximation expelled from the star. In such a way strongly of uniform medium, V = const, one can redefine degenerate sea is formed with chemical potential the integration variable in (16): 37 p ~ 1/ ~ Gpn . (12) 9o — 9o — V . (17) In neutron stars the density of neutrinos from the After the redefinition w is reduced to too, so that sea is ~ 1017 cm-3 and the total energy in the Aw — 0. Thus the energy of a star in this approx sea is very small in comparison with the mass of imation is zero. However, this proof corresponds a star. In spite of this, an existence of the sea to infinite and uniform medium. The real star has can play an important role. The degenerate sea in finite size and the distribution of neutrons is non- stars leads to the Pauli blocking of the long range uniform. In this case the redefinition of variables forces. Instead of (4) we get for two body potential (17) is impossible and non-zero self energy of the 38. star appears.
(cos 2pr + pr sin 2pr) . (13) H 47r JrD 2.3 Oscillations in Magnetized Medium Note that 1/p ~ 10-5 cm 3 medium can be calculated as the correction to self- Complete polarization can be achieved in the case energy. Two diagrams appear: (i) the loop dia of very big magnetic field and zero temperature. gram with IT-boson: v —> We —> v, where for the The polarization equals (n + — n_)/n e, where electron we should use the effective propagator in rip , ti_: are the concentrations of the electrons thermal bath, (ii) the tadpoles diagram with Z with polarization + 1 and - 1. The energy spec and electron in the loop. The electrons couple to trum of electrons in the magnetic field is quan the electromagnetic field 40. tized: In strongly degenerate gas, Ep 3> T, where Ep is the Fermi energy one gets the following.ex e{Pz,n,\) = x/Pz + rni+ I e | fl(2n + 1 - A) , pression for the effective potential in the magnetic (21) field £40'41: where A = —2sz. It consists of main Landau level, n — 0, A = 1, and pairs of the degenerate lev V = V2GFnegv + (^j3 (k-B). (18) els with opposite polarizations. Therefore the po larization effect is determined by concentration of electrons in Landau level, The correction originates from the axial vector n + — n n o current. It influences dynamics of the neutrino 2(s) (22 ) ne n e conversion. In particular, the correction modifies the resonance condition: For strongly degenerate gas: Am2 eBpp cos 29 = 0 (19) n o (23) 2 E 2rr2 ’ shifting the position of the resonance in compar where the Fermi momentum, pp, is determined by ison with the case of zero magnetic field. It also the normalization 43 influences the adiabaticity condition. eBpp 2eB\Jpp- 2eBn There are, however, wrong statements that ne (24) 2n2 2^ the magnetic term can compensate or even be big n = l ger than the first (vector current) term. It would The first term corresponds to the main Landau induce new resonances and open the possibility to level 7 i = 0, A = 1: and the second one is the re have the flavor resonances both for neutrinos and sult of summation over all other levels. The com antineutrinos in the same medium. Actually the plete polarization corresponds to 2efl > p^, when magnetic (axial) term cannot be bigger than the nmar < 1, and the sum vanishes. In this case vector one. This can be seen immediately from all electrons are in the main Landau level: ne = another approach to the problem 42. eBpp/2ir 2, from this one gets pp = 2tt2ne/eB, Indeed, the effect of the magnetic field is re and consequently, no = ne. In the limit of small duced to polarization of electrons, so that one can field: pp % (3tt2ne)1^3 and use the result (8) for the effective potential and calculate the average polarization of the electrons. For flavor oscillations the matter effect is deter (25) mined by charge current scattering on electrons for which = gv — 1 and therefore This leads to the result (18). For oscillation to sterile neutrinos, however, V = V2Gpn e (1 + 2{s) cos a) . (20) the effective y A can be bigger than gv and the level crossing phenomena induced by magnetization are Here a is the angle between the neutrino mo possible 42. mentum and the polarization of electrons and (s) = (s(S)). Obviously, the second term cannot 2.4 Neutrino mass and the peculiar velocities of be bigger than 1, so that one can get at most the pulsars compensation of the effective potential: V — 0 in the case of the complete polarization of electrons An important application of results described in in the direction against the neutrino momentum. sect. 2.3. has been found by Kusenko and Segre 44. 4 There is the long standing problem of explanation of vT emitted in different directions and therefore of the high peculiar velocities of pulsars (v ~ 500 neutrino burst knocks the star. The observed ve km/s). Non-symmetric collapse, effects in binary locities imply the polarization effect 10~3 — 10~2, systems etc., give typically smaller velocities. or according to (18) It looks quite reasonable to relate these veloc ities with neutrino burst45. The momenta of pul sars are 10-3—10~2 of the integral momentum car ried by neutrinos. Therefore, 10-3 — 10-2 asym metry (anisotropy) in neutrino emission is enough Below the ue - neutrinosphere: ne > 10u cm* 3 for the explanation of the peculiar velocities 45. which gives B ~ 1013 Gauss. From the condition The anisotropy of neutrino properties can be that the resonance should be below the ve - neu related to the magnetic field. It was suggested that trinosphere one gets a very strong magnetic field (1015 — 1016 Gauss) Am2 > 104 eV2 , or m3 > 100 eV . (26) can influence the weak processes immediately: the probability of emission of neutrino along the field The mixing angle can be rather small: from the and against the field are different. adiabatic condition it follows sin 2 26 > 10-8. According to the mechanism suggested in 44 Thus the explanation of the peculiar velocities magnetic field influences the resonance flavor con of the pulsars based on the resonance flavor con version leading to the angular asymmetry of the version implies that the mass of the heaviest (~ conversion with respect to the magnetic field. The vT ) neutrino is larger than 100 eV. To avoid the latter results in asymmetry of the neutrino prop cosmological bound on mass, the neutrino must erties.It is assumed that the resonance layer for decay (e.g. with Majoron emission). The attempts the conversion ve — i/T lies between the ve - to diminish m3 by means of a very large magnetic neutrinoshpere and uT -neutrinoshpere (the latter field (so that the polarization effect compensates is deeper than the former due to weaker interac the density) lead to very strong asymmetry ~ 1. tions of uT ). Thus the uT which appear in the res Another problem is that due to relatively high onance layer will propagate freely and ve are im temperatures very strong polarization and conse mediately absorbed . The resonance layer becomes quently, the compensation are impossible. the “neutrinosphere ” for uT . (In fact, in the pres In connection with the Kusenko-Segre pro ence of the magnetic field the neutrinosphere be posal it is interesting to note recent results on mea comes “neutrinoellipsoid ” and this is crucial for surements of the beta spectrum in tritium decay 9. the mechanism). There are two features in the spectrum: (i) Excess It is assumed that inside the protoneutron star of events near the end point, Q, of the spectrum there is a strong magnetic field of the dipole type. Q — Ee ;$ 10 eV, (peak in the differential spec Then in one hemisphere the field is directed out trum) which leads to the negative value of the m2 side the star, so that for neutrinos leaving the star in usual fit. (ii). Excess of events at lower energies (k ■ B) > 0, whereas in another hemisphere the of the electrons: Q — Ee £ 200 eV. The excess in field points towards the center of the star and this region was also observed by the Mainz group. (k ■ B) < 0 . Since the electronic gas in the One possible explanation of this anomaly is an ex star is strongly degenerate we can use expression istence of neutrino with mass m ~ 200 eV whose (18) for the effective potential. According to (18) admixture in the electron neutrino state is char the magnetic field modifies the resonance condi acterized by probability P ~ 1 — 2 % . This is tion differently in these two hemispheres. In hemi precisely in the range implied by pulsar velocities. sphere with (k ■ B) < 0, the resonance condition As far as the first anomaly is concerned (the is satisfied at larger densities and larger temper negative m2) one possible explanation is the ta- atures; uT emitted from this hemisphere will have chionic nature of neutrinos 46 . It should be bigger energies. On the contrary, in the neutri stressed, however, that the position of the peak nosphere with (k- B) > 0 the resonance is at lower depends on the condition of the experiment: In densities and lower temperatures and neutrinos the run of the experiment in 1994 the peak was at have smaller energies. Thus the presence of the Q — Ee as 7 eV whereas in the run 1996 the peak magnetic field leads to the difference in energies is at Q — Ee ~ 11 eV. There were some changes 5 in the experiment in the run 1996, in particular, For Am2 corresponding to vT - u, and - the strength of the magnetic field was higher. The v, channels we get from (28) Tt to 14 MeV and shift of the peak indicates that it may have the Tp to 2 MeV. (i) For T > Tt the oscillation transi instrumental origin, rather than the origin in neu tions are suppressed by the T term in the poten trino properties. tial. (ii) For T ~ Tr the T-term drops and oscilla tions become possible. Due to the non linearity of 2.5 Lepton asymmetry tn the Early Universe the equations the amplitude of oscillations blows up and the asymmetry reaches (practically dur According to (10) in the Early Universe the differ ing the same epoch t ~ 10~2 sec) AL ~ 10-5. ence of the potentials for different neutrino species With a further diminishing of temperature the can be written as asymmetry may slowly increase up to 10-2 or AV = V2GFny(AL + A^~) , (27) even higher. (The mixing is chosen to be small enough, so that the concentration of sterile neu X mW J trinos, n, 2 n-,AL, is still smaller than the equi where n 7 is the photon density, AL = [ni — librium one), (iii) In the epoch T 5- T^, when nL)/ni is the leptonic asymmetry and ul, ni are transition - us could be important, the effective the concentrations of the active neutrinos and an (matter) mixing - uT is suppressed by leptonic tineutrinos. asymmetry (AL -term of the potential) produced Matter effects can be important for oscilla previously in vT oscillations. We discuss the ap tions into sterile neutrinos. Matter influences dif plication of this result in sect. 4.7. ferently the neutrino and antineutrino channels, so that transitions i/T —► is, , and PT —> P, can create 3 Pattern of neutrino masses and mixing the uT - PT asymmetry in the Universe. Since V depends on the concentration of neu 3.1 Neutrino anomalies trinos themselves, and consequently, on conversion Existing neutrino anomalies imply strongly differ probability, the task becomes non-linear. Due to ent scales of Am2 . For the solar neutrinos, the this, depending on values of parameters, a small atmospheric neutrinos and LSND we have corre original asymmetry (one can expect ALq ~ AB ~ spondingly: 10-9) can be further suppressed 47 or blow up48,49. The leptonic asymmetry influences the primordial Attiq ~ (0.3 — 1.2) ■ 10-5 eV2 , (29) nucleosyntesis. It was realized recently, that it can suppress the production of sterile neutrinos, so AmLn~(0.3-3)-10-2eV2, (30) that the concentration of these neutrinos is much smaller than the equilibrium concentration even AmLAfD-(02-2)eV^. (31) in the case of the large mixing angle and the large That is mass squared difference. A scenario suggested in 48,49 is the following. &™lsnd » Am2tm » Am!,. (32) Suppose vT mixes with v, and parameters of the system are: Am2 ~ 5 eV2 and 0TS ~ 10-4. On The mass scale which gives the desired HDM com the contrary, i/M - u, has large mixing 9^, ~ 1 and ponent of the Universe, mu dm : Am2 ~ 10“2 eV2, so that - u, oscillations can Whdm ~ (1 — 50) eV2 (33) solve the atmospheric neutrino problem. It turns out that in spite of this large mixing the concen can cover the LSND range. tration of sterile neutrinos is small. In the case of three neutrinos there is an ob Let us consider the evolution of a system with vious relation: decrease of temperature. There are two important scales determined by the equality of the T-term in Am2! + Am|2 = Am|j. (34) A V and level splitting due to mass difference: and inequality (32) cannot be satisfied. That is, fon a T2 - Am2 with three neutrinos it is impossible to reconcile all (28) the anomalies. Furthermore, an additional larger 6 scale is needed for the explanation of the pulsar velocities (26). Three different possibilities are discussed in this connection. One can LSND • suggest (stretching the data) that ATM AmLSND = ^matm I (35) Also the possibility Atuq = Am](m was dis cussed 50 Figure 1: Qualitative pattern of the neutrino masses and mixing. Boxes correspond to different mass eigenstates. • “sacrifice” at least one anomaly, e.g. the The sizes of different regions in the boxes determine fla vors ( |f/,/|2) of given eigenstates. Weakly hatched re LSND result, or atmospheric neutrinos; gions correspond to the electron flavor (admixture of ve ), strongly hatched regions depict the muon flavor and black • introduce additional neutrino states. regions present the tau flavor. Arrows connect the eigen states involved in oscillation /conversion which solve uq - In what follows we will consider examples solar, ATM - atmospheric, LSND - problems. Scenario which realize these three possibilities. shown here reproduces simultaneously i/q, ATM, LSND and HDM. There is another important mass scale: the upper bound on the effective Majorana mass of the electron neutrino which determines the rate of (ill) The probability of the LSND/KARMEN os the neutrinoless double beta decay: cillations is determined by P oc 4|f/3e|2|t/3p|2 . (38) mee = U^nii. (36) ' = 1,2,3 (iv) The scenario can supply three component HDM, if the absolute values of the masses are in Here Uei are the elements of the lepton mixing ma eV range. In this case the spectrum is degenerate: trix. Taking into account uncertainties in the nu m i to m2 to m3 to 1 eV . clear matrix element one gets from the data mee 6 (v) If neutrinos i/j are the Majorana particles, then 0.5 — 1.5 eV . Forthcoming experiments (NEMO- mee to mi ~ 1 eV is at the level of present upper III 51) will be able to strengthen the bound by experimental bound . factor 3. Note that typically ttihdm > mee. This scheme is a variant of the previously con sidered schemes with three degenerate neutrinos 3.2 Everything with three neutrinos? and an order of magnitude smaller mass splitting: It is assumed that LSND and atmospheric neu Am23 ~ 10~2 eV2(see sect. 4.4). trino scales coincide 52: One can modify the scenario assuming mass hierarchy, so that Am23 to m\. In this case m3 ~ Am223 = Am\SND = Am2tm ~ 0.2 - 0.3 eV2 , 0.5 eV, m2 to 3 • 10-3 eV and raj 7 3u HDM LSND ATM Figure 2: The same as in fig.l. Scenario without explana Figure 4: The same as in fig.l. Scenario with strongly tion of the solar neutrino deficit. degenerate neutrino spectrum. neutrinos. The generic 3(/-case is realized, (q, - ATM i/e has an unsuppressed mode of oscillations with Am23 , and - vT has both Am\2 and Am^ LSND 'ATM modes. The CHOOZ experiment 56 will put the bound on this possibility. (ii) Effective Majorana mass of the electron neu trino is mee to m0(Ue2 + HepU^), where yep is the relative CP-parity of two massive neutrinos. The bound from /?/?oi/ — decay can be satisfied by Figure 3: The same as in fig.l. Scenario with inverse fla vor/ mass hierarchy and without explanation of the solar some amount of cancellation. neutrino deficit. (iii) If m20 > 4 eV2, CHORUS/NOMAD may observe the signal of iq, - vT oscillations. (iv) Due to inverse flavor/mass hierarchy the sce lowing. nario predicts a strong resonance conversion of an (i) The atmospheric neutrino problem is solved tineutrinos in supernova: iq, , vT —>ue . The con by i/p - uT oscillations. version results in permutation of iq , (/« energy (ii) iq and vi form two components HDM. spectra which is disfavored by SN87A data 19. (iii) The probability of oscillations in In these schemes the solar neutrino data can LSND/KARMEN experiments is determined by e be explained by virtue of introduction of the ad and fj. flavors of the lightest state: ditional (sterile) neutrino states. Poc4|[/ ei|2|Z7„ 1|2 . (40) 3-4 Sacrifice LSND. Degenerate neutrinos Mixing elements Ue 1 and U^i are restricted by Solar, atmospheric and HDM problems can be BUGEY and BNL E776 experiments. solved simultaneously, if neutrinos have strongly (iv) No observable signal of - i/> oscillations degenerate mass spectrum mi to m% to m3 ~ 1 — 2 is expected in CHORUS/NOMAD experiments eV57,58,59, with Am22 = Am| = 6 -10-6 eV2and 54,55, however these experiments may discover ve - Am23 = Am2(m = 10-2 eV2(fig.4). may have vT oscillations. the form 58 (v) fSftov — decay is strongly suppressed. m = mol + Sm, (41) A modification of the scenario is suggested where I is the unit matrix, Sm 8 LSND ATM Figure 5: The same as in fig. 1. Scenario for solar and Figure 6: The same as in fig.l. Scenario without explana atmospheric neutrinos. tion of the atmospheric neutrino deficit. 6m can be generated by the standard see-saw con (i) The solar neutrino problem is solved by the tribution with Mr ~ 1013 GeV58. ue —Hv small mixing MSW solution. The effective Majorana mass mee % mo is at (ii) All three neutrinos give comparable contribu the level of upper bound from the — decay. tions to the HDM. The mass mee can be suppressed 60 if the elec (iii) i/p - vt oscillations can be in the range of sen tron flavor has large admixture in v\ and y 2 , so sitivity of the LSND/KARMEN. that the solar neutrino problem is solved by the (iv) The Majorana mass mee is at the level of up large mixing MSW solution. Now the effective per experimental bound. Majorana mass equals mee % mo( 1 — sin 2 20), and for sin 2 26 = 0.7 one gets suppression factor 0.3. Another version 66 is characterized by mi <*C However simple formula (41) does not work 60 . m2 « m3 ~ 1 eV with Am23 = Am|, . Heavy No observable signals are expected in CHO components iq and v3 are strongly mixed in RUS/NOMAD and LSND/KARMEN. i/e and Vj and the lightest state has mainly the muon flavor (inverse hierarchy). (In contrast with Another possibility (fig. 5) is to sacrifice the the scheme of sect. 3.3, the splitting between HDM assuming (if needed) that some other par heavy states explain the solar neutrino problem.) ticles (e.g. sterile neutrinos, axino etc.) are re Comments: sponsible for structure formation in the Universe. (i) The ve —ivT conversion gives large mixing In this case m3 ~ 0.1 eV and — uT oscillations MSW solution to the solar neutrino problem. explain the atmospheric neutrino deficit. Strong (ii) The mass mee can be at the experimental Vp - i/T mixing, could be related to relatively small bound, although the cancellation is possible. mass splitting between m2 and m3 which implies (iii) One expects strong P^ -+Pe conversion in the the enhancement of the mixing in the neutrino supernova, which is disfavored by SN87A data. Dirac mass matrix61 . It could be related to the (iv) The scenario supplies two components HDM see-saw enhancement mechanism62,63 endowed by and an explanation of the LSND result. the renormalization group enhancement63 or with strong mixing in charge lepton sector 64 . 3.6 “Standard" scenario 3.5 Without the atmospheric neutrino problem The scenario is characterized by strong mass hier archy mi C m2 C m3 and weak mixing (fig. 7). The schemes which can accommodate solar neu The basic features are: trinos, HDM, and the LSND result are suggested. (i) m3 — niROM, so that y 3 forms the HDM. According to 65 (fig. 6 ) all neutrinos are in the (ii) Second mass, m2, is in the range m2 = (2 — 3) eV - range, the first two neutrinos are strongly 10-3 eV , and the ue —>■ z/M resonance conversion degenerate: Am22 = Anig , whereas Amf3 = solves the solar neutrino problem. ^mIsyvD • Mixing is small: the electron flavor (iii) There is no solution of the atmospheric neu dominates in v\ , the tau flavor -in 1/3 , and the trino problem. muon flavor in the heaviest state i/3 . Remarks: (iv) The depth of P^ — Pe oscillations with Am2 % 9 3.7 More neutrino states? HDM Another way to accommodate all the anomalies is to introduce a new neutrino state which mixes with active neutrinos (see e.g. 68,69,70) ^s fol lows from LEP bound on the number of neutrino species this state should be sterile (singlet of SM symmetry group). Mixing of sterile and active neutrinos leads to oscillations and the oscillations result in the production of sterile neutrinos in the Early Universe. The presence of the sterile com ponent in the epoch t ~ 1 sec could influence the Figure 7: The same as in fig.l. The “standard scenario. Primordial Nucleosyntesis. Several comments are in order. m\ equals 4|[/3M|2|{/3e|2. Existing experimental 1. At present, a situation with bound on the bounds on these matrix elements give the upper effective number of the neutrino species, A„, is bound on this depth: < 10-3 which is too small controversial. Depending on the abundance of pri to explain the LSND result. mordial deuterium one uses in the analysis the (v) Parameters of - i/T oscillations can be in the bound ranges from Nu < 2.5 21 to 3.9 22. A cer range of sensitivity of the CHORUS/NOMAD. tain model of evolution of the deuterium is used. There is a number of attractive features for this A conservative analysis which does not rely on any scenario: It naturally follows from the see-saw model leads to Nu < 4.5 23. If N„ > 4 is admit mechanism with Dirac mass matrix mf? ~ mup ted then obviously there is no bound on oscillation and the intermediate mass scale for the Majorana parameters of the sterile neutrinos. mass matrix of the RH neutrinos: Mi ~ 1013 GeV. 2. Even if Nu < 4, strong mixing of the sterile More precisely, for the eigenstates of this matrix and active neutrinos is not excluded. The bound one gets 28 can be avoided in the presence of large enough (AZ, £ 10-5) lepton asymmetry in the Universe, M2 ~ (2-4) • 1010 GeV as it was discussed in sect. 2.5. M3 ~ (4-8) • 1012 GeV . (42) There are bounds on oscillation parameters of sterile neutrinos from SN87A observations 18. These values of masses are in agreement with “lin Thus at present it seems possible to introduce ear” hierarchy: M2/M3 to mc/me. sterile neutrinos for explanations of different neu The decays of the RH neutrinos with mass trino anomalies. 1010 — 1012 GeV can produce the lepton asym metry of the Universe which can be transformed by sphalerons into the baryon asymmetry 67 . 3.8 Rescue the standard scenario The mixing angle desired for the solution of The atmospheric neutrino deficit is the problem the Uq problem is consistent with expression for the standard scenario. To solve it one can as sume that an additional light singlet fermion exists with the mass m ~ 0.1 eV, which mixes mainly with muon neutrino, so that vp — v, oscillations explain the data 71 . In this case one arrives at the where me and mp are the masses of the electron scheme (fig. 8) 72 . Production of us singlets in and muon, is a phase and is the angle which the Early Universe can be suppressed (if needed) comes from diagonalization of the neutrino mass by generation of the lepton asymmetry 48 in the matrix. The relation between the angles and the vT — i/, and uT — u, oscillations 49. The pres masses (43) is similar to the relation in quark sec ence of large admixture of the sterile component tor. Such a possibility can be naturally realized in in i/2 influences resonance conversion of solar i/e , terms of the see-saw mechanism. and can also modify the - t/T oscillations 72 . 10 HDM 4 On the models of neutrino mass 4-1 Predicting neutrino mass The majority of attempts to predict neutrino masses are reduced to establishing relations be ATM tween quarks and leptons. Then known parame ters in quark sector are used as an input to make some conclusions on mass and mixing in lepton sector. The see-saw mechanism allows one to realize the quark - lepton symmetry most completely. To Figure 8: The same as in fig.l. Scenario with sterile neu make the predictions one should fix the Dirac mass trino for atmospheric neutrino problem. The admixtures of sterile component are shown by white regions in boxes. matrix of neutrinos, m®, as well as the Majorana mass matrix of the right-handed components, Mr . Usually the direct Majorana masses of the left- handed components are neglected. To find m®, one can the use GUT relation, e.g. mj? = mup at Mr ATM the GUT scale. For different ansatze 73 were suggested. Also minimality of the Higgs sector can be postulated 74 which allows one to get some LSND relation between structure of Mr and quark mass matrices. The pattern of masses and mixing of the light neutrinos strongly depends on the structure of Mr , so that even for fixed mf?, practically any scenario can be realized. Relations between quarks and leptons can be Figure 9: The same as in fig.l. Scenario with sterile neu based also on certain horizontal symmetries. trino for explanation of the solar neutrino deficit. The ad Recent attempts to predict neutrino masses mixtures of sterile component are shown by white regions in boxes. are based on • GUT models with SO io symmetry, • Models with anomalous t/(l) symmetry, 3.9 The safest possibility ? • SUSY Models with R-parity violation, Even without lepton asymmetry strong nucleosyn- • Models with radiative neutrino mass gener tesis bound is satisfied, if is, has the parameters of ation. the solar neutrino problem. In this scenario 57,68 (fig. 9) mi < ms <$C m2 to m3 . Remarks. Also one can introduce some ansatze for the (i) Sterile neutrino has the mass mg ~ (2 — 3) -10—3 quark and lepton matrices. eV and mixes with ve, so that the resonance con version ve —solves the solar neutrino problem; 4-2 An ansatz for large lepton mixing (ii) Masses of iq, and i/T are in the range 2 - 3 eV, It is postulated 75 that fermion mass matrices have they supply the hot component of the DM; the following structure in a certain basis (the scale (iii) tq, and vT form the pseudo Dirac neutrino is not specified) with large (maximal) mixing and the oscillations 1/^ — uT explain the atmospheric neutrino problem; Mi = aMdem + A Mdia9 (44) (iv) The — ue mixing can be strong enough to i — u, d, l, u, where explain the LSND result. (v) No effect is expected for up - uT oscillations in 1 1 1 CHORUS/NOMAD as well as in future searches Mdem = M0 1 1 1 (45) of the f3(30„ — decay. 1 1 1 11 is the democratic matrix, and symmetry is also used to get a desired structure of the mass matrices. The neutrino Dirac mass ma f S 0 0 \ trix has the following hierarchical structure of the A Mdiag = 0 p 0 (46) V 0 0 e ) elements: m33 3> m12 1S> m23 > m22, for the Ma- jorana mass matrix one gets: M22 » M13 M33, with 6,p,c 4C Mq. It is assumed that parameters (the matrices are symmetric and all other ele c, are proportional to electric charges of fermions: ments are zero). This leads via the see-saw mech anism to the pattern of the light masses with c, oc Qi . (47) m2 to m3 3> mi. Also additional sterile neu trino is introduced to explain the solar neutrino For quarks and charged leptons the first term in problem, thus the model reproduces the pattern (44) dominates leading to big mass in one gen discussed in sect. 3.9. eration and big mixing angles which diagonalize It should be stressed, however, that the pat matrices. As a result of two similar rotations for tern is theresult of ad hoc introduction of the large the upper and for the down quarks, the mixing number of new supermultiplets and special (7(1)- in quark sector is small. The situation in lepton charge prescription. In fact, these (7(1) charges sector is different. For neutrinos: c, = 0, and should be considered as new free parameters, so therefore the neutrino mass matrix is diagonal: that high predictivity becomes not so impressive. Mu = AMdiag . (48) 4-4 Neutrino-neutralino mixing The lepton mixing follows from diagonalization of the charge lepton mass matrix and since Mi to This is the low scale realization of the see-saw Md,ag, the mixing in leptonic sector is automati mechanism. The neutrino mass equals ~ cally large. In a sense, large mixing in lepton sec mlN/mN, where m^jv is the mixing mass term, tor is related to smallness of the neutrino mass. and mjv is the typical neutralino mass. Mixing of All three neutrinos are strongly mixed. If neutrinos and neutralinos implies violation of the 50(10) —>5(7(5) —>SM at mass scales Mp , iqo, Yukawa couplings. This can be related to 72- and i>s correspondingly. All fermions are predicted symmetry 82. The fields Lq and La (a = 1,2,3) in terms of 4 continuous parameters: v$/Mp, may have different 72-charges: e.g. 72 (Lq, 77 2) = viq/Mp, the ratio of MSSM VEV: tan/?, and uni 2, whereas R(Q,UC, Dc, La) = 0. In this case versal Yukawa coupling A. An additional (7(1) the 72-parity breaking Yukawa couplings are sup “Previously dihedral group A(75) was suggested in 77 pressed. Moreover, the p-terms can be generated 12 by nonrenormalizable interactions with new fields This contribution to neutrino mass is typically Z{. The E-symmetry is broken spontaneously by larger than the one produced by the loop-diagram the VEV of these field z*: (z,) C Mp, and the stipulated by E-parity violating Yukawa cou Hi parameters of the superpotential may have the plings. For Hi ~ H and large tan (3 (hp ~ 1) we hierarchy determined by (z,)n/Mp-i, where n is find ra„ ~ 0(10 MeV). This neutrino can be iden fixed by the E-charge of z, 82. tified with uT . It is assumed (here we will follow a discussion There are several possibilities to get much in 81) that soft SUSY breaking terms for L, are smaller mass. For small tan /?(~ 1) the Yukawa universal at, e.g., GUT scale: coupling is small and the m„ is of the order 10 eV. Also the mass can be suppressed if there is the hi V = B ■ Hi Li H 2 + mg |Zj |2 + ... (50) erarchy of Hi- Hi!P ^ 1 ■ For Hill1 ~ Mgut/Mpi : m„ ~ 10 eV even for large tan/3 80. Due to the universality one can diagonalize the h term in the superpotential, and simultaneously in Another possibility for the suppression of m„ the potential (50), by rotation Li — is a cancellation between the two terms in (53). If there is no cancellation, the neutrino mass HiLiHi —t hL'qH-z • (51) turns out to be related to the E-parity violat ing Yukawa coupling generated by rotation (51): (This rotation generates simultaneously the E- A4 to Cm„mg(/q//r) 2, where C is a known con parity violating Yukawa couplings.) There are no stant 81. Thus a certain relation between the prob terms like L’aH2 (a = 1,2,3) at GUT scale. These abilities of E-parity violating processes (due to A) terms, however, appear at the electroweak scale and neutrino mass gives signature of this mecha due to the renormalization group effect. Indeed, nism. Yukawa coupling (49) distinguishes different com In the case of three generations only 79,80 ponents of Li and this leads to different renor one neutrino acquires the mass due to neutrino- malization of terms with Lq and La in (50). The neutralino mixing. Loop corrections induced by universality turns out to be broken, and the ro E-parity violating couplings make all neutrinos tation (51) will not diagonalize the potential. We massive. In a certain region of parameters one get after rotation (51) the mixing term can explain the solar neutrino problem and sup ply HDM (i.e. reproduce the standard scenario). — x [8m2 L'0* + SB-h H2] Z' + h.c. (52) Also a simultaneous solution of the solar and at A* mospheric neutrino problems is possible 80. where 8m2 and SB describe the renormalization group effect. After electroweak symmetry break ing the mixing terms (52) (linear in Z'), together 4-5 Models with anomalous U(1)a symmetry with soft symmetry breaking masses, induce a Masses of neutrinos are generated by the see-saw VEV of “sneutrino ” (neutral component of the mechanism. A structure of the neutrino mass ma doublet in Z-) of the order: trices is determined by U(1)a charges of neutrinos 83,84,85,64 ^ relation between the neutrino and the {£) to v — x f cos /? + sin /?'] , (53) quark mass matrices is established via the charges H \ ml m2 J (rather than immediately, as in the simplest GUT here t> is the electroweak scale. The VEV of sneu theories). It is assumed that charges of neutrinos trino leads via the gauge coupling to the neutrino- coincide with charges of (electrically charged) lep gaugino mixing: m„/v ~ g(i>). In turn the see-saw tons: mechanism results in the mass OMi) = 0("&) = GW = Q, (55) Masses are generated a la Froggatt-Nielsen mech m„ to HI anism, and elements of the mass matrix appear 2 as (54) J , (56) where 6 is the VEV of singlet field with unit (7(1)a The first neutrino tq practically coincides with ve charge and M is the mass scale of new heavy scalar and has a much smaller mass: mi 14 metry which is embedded in E$ and broken at low m3/2- Then the natural scale of mass of S is scale. 3. In 92 it was suggested that v, is the mirror m3/2 ms (62) neutrino from the mirror standard model. The Mp mass of v, is generated by the see-saw mechanism in the mirror world which, however, has the elec The mixing of S with active neutrinos involves tro weak symmetry breaking scale (Hm) about two electroweak symmetry breaking. The simplest ap orders of magnitude larger than in usual world. propriate effective operator is (m3/2/Mp)LSH . It (Here Hm is the mirror Higgs doublet.) General generates the mixing mass parameter izing (1) we get ms = (Hm)2/Mp ■ The mixing of m3/2 {H) usual neutrinos with the mirror one proceeds via m ~ (63) the gravitational interactions Mp For small electron neutrino mass m„= ms the ve — vs mixing angle 6e , is of the order ——LHLmHm + h.c. , (61) Mp (H) 6€3 (64) where Lm is the mirror lepton doublet. Therefore TO3/2 the mixing angle is determined essentially by the For Mp ~ 2 x 1018 GeV and m3/2 ~ 103 GeV one ratio ofVEV: (H)/{Hm)- gets ms and m 4. The origin and properties of vs can be re lated to SUSY. A number of singlet superfields ms ~ 10-3eV , m ~ 2 - 10"4eV (65) was introduced for different purposes: to gener ate the term, to realize PQ-symmetry breaking, precisely in the range desired for a solution of the to break spontaneously the lepton number, etc. solar neutrino problem via resonance conversion String theory typically supplies a number of sin ve -+ S in the sun. Moreover, varying the param glets. Fermionic components of these superfields eters (constants of the order 1) and taking into could be identified with desired sterile neutrino. account the renormalization group effect it is easy It was shown in 93 that masses and mixing to achieve both small and large mixing solutions of va can be protected by /(-symmetry. 5. An to the problem. other possibility is that vs is the would-be Nambu- Fermion S can also mix with the other neu Goldstone fermion 94: the superpartner of the trino species. If the coupling of S with fermion Nambu-Goldstone boson which appears as the re generations is universal; i.e. m, are the same (or of sult of spontaneous violation of some (7(1) global the same order) for all generations, then 5-mixing symmetry like Peccei-Quinn symmetry or lepton with Vn and vT are naturally suppressed as the number symmetry etc. (i.e. vs is the axino, or ma- mixing angles behave as 0; ~ m/m,-. For instance, jorino ....). A general problem is that SUSY break taking m2 ~ 10“1 eV and m3 ~ 1 eV we get ing generates typically the mass of v, of the order sin 2 20S)i ~ 10~5 and sin 2 26 sT ~ 10-7 . Thus the of the gravitino mass and further suppression is lightest neutrino has naturally the biggest mixing needed. One can use here the ideas of non-scale with v, . supergravity, or possibly, gauge mediated SUSY The desired properties of 5 could be realized breaking. for some fields in a hidden sector, and probably 6 . Sterile neutrino as modulino? Suppose that for fermionic components of some moduli field ^5 . there is a singlet S = vs which is massless in the supersymmetric limit and couples with observable 5 Conclusion sector via the gravitational interactions. The mass and effective interactions are induced when super- 1. New effects of the neutrino refraction in me symmetry is broken. 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Lett. 70 (1993) 2845; did not have time to submit as I was working on 75. H. Fritzsch and Zhi-zhong Xing, These Pro molecular genetics. However, I did send him a ceedings pa07-094. paper written earlier this year which raises worries 76. K. C. Chou and Y.-L. Wu, OHSTPY-HEP- about 3 things- Errors, the Sun ’s luminosity and T-96-008, paper pa09-001. motion inside the Sun. 77. D. Kaplan and M. Schmaltz, Phys. Rev. 1. ERRORS: We seem to agree there is a D49 (1994) 3741; M. Schmaltz Phys. Rev. problem with the very different errors of different D 52 (1995) 1643. SSM’s. 78. L. J. Hall and M. Suzuki, Nucl. Phys. 2. LUMINOSITY: What we measure is what B231 (1984) 419. we see on the surface of the Sun over the last few 79. N. Polonsky, hep-ph/9607350. years. But what we need is an average over the last 80. R. Hempfling, MPI-PhT/95-59. few million years as the time for thermal informa 81. A. Yu. Smirnov and F. Vissani, Nucl. Phys. tion to travel from the core to the surface is be 460 (1996) 37. tween one and ten million years (Douglas Gough ’s 82. H.-P. Nilles and N. Polonsky, TUM-HEP- estimate). The latest satellite measurements show 245/96. the luminosity follows the sunspot cycle. If we 83. H. Dreiner et al., Nucl. Phys. B436 (1995) were living near the year 1700, the luminosity 461. would have been quite different as the earth’s tem 84. E. Papageorgieu, Z. Phys., C65 (1995) 135; perature was much lower - in London people had Phys. Lett. B343 (1995) 263. fairs with bonfires on the ice on the Thames - and 85. P. Binetruy, S Lavignac, P. Ramond, hep- there were no sunspots between 1650 and 1710. 18 Similarly, there were few sunspots about 1400 1. At present the solar neutrino problem can be when there was another cold spell whereas near formulated practically without reference to a spe 1200, there was a hot period with extra sunspots. cific standard solar model. The problem can be In other words, the surface of the Sun changes formulated as a discrepancy between different ex in ways not included in the SSM which does not perimental results, and in this connection a more consider sunspots nor variation of the apparent relevant question is how reliable are the experi luminosity. Going back further, for many million mental results. years, the sea level was much higher indicating 2. Possible variations of the Solar luminosity that the luminosity was much greater. For ex are certainly much smaller than those which could ample, when the dinosaurs were extinguished, the correspond to the depletion of the Gallium result sea level was consistently about 200 metres higher by factor 2. than now and half of the present land surface was 3. W. Haxton has suggested an unusual mix under water. We do not have a good measurement ing of elements. It involves a fast filamental flow of the luminosity over a suitably long time period of matter from the layers with maximal concen and hence the error on the luminosity should be tration of 3He downward, and slow restoring flow greatly increased. upward. This leads to the enhancement of the pp-I 3. INTERNAL MOTION: There are three branch and therefore suppression of the 7Be neu pieces of evidence. Initially the Sun was a T Tauri trino flux. However, (i) the suppression achieved star - very bright and rotating quickly. Standard by mixing is not enough for a good description Solar Models cannot slow this rotation to zero, so of data, (ii) In fact, no consistent solar model one expects a differential rotation even to the core has been elaborated which incorporates the mix of the Sun. This is supported by helioseismological ing, and it is unclear what is a feedback of the measurements which show that the rotation at the mixing on other observable characteristics of the poles and at the equator is different down to 0.2 Sun. (iii) It has been shown by Bahcall and col of the Sun ’s radius. Helium-3 has an unusual dis laborators, that the mixing suggested by Haxton tribution being sharply peaked at a radius of 0.3. strongly contradicts the helioseismology data. Calculations by Wick Haxton have shown that a Results obtained by S. Vauclair et al. show motion of only 700 metres per year, is enough to that the solution of the Lithium-7 problem has cause this Helium-3 to move and to be burnt thus no serious impact on solar neutrino fluxes if one changing the temperature of the Sun ’s core ap takes into account the helioseismological data. preciably. Lithium-7 has a measured abundance The rotation-induced mixing of elements below which is less than one hundredth of that predicted the connective zone is introduced; the agreement by the SSM. Also, looking at other stars, the Boes- with the helioseismological data imply that mix gaard dip is not explained by the SSM. Sylvie Vau- ing should terminate below R < 0.4R©. This clair et al. have explained this by meridional mo is enough to solve the lithium problem, but this tion inside the Sun. However they cut the motion is not enough to change appreciably the neutrino at a radius higher than 0.3 and hence do not allow fluxes which are generated in deeper layers. any Helium-3 movement, and so find little change In the paper presented at Rencontres de Blois in the neutrino flux. Without this cut which seems Vauclair et al., added some mixing in the central in contradiction to the helioseismological results regions (R ~ (0.1 — 0.2) R©) to have a better de which show effects down to at least 0.2 radius, the scription of the helioseismology data. It turns out neutrino flux would have been changed. that the mixing should be weak and its effect on neutrino fluxes is of the order of 20% only. A. Yu. S. : You said so many things that I forget what they were. D. R. 0. Morrison: Errors, Luminosity, Internal motion. A. Yu. S.: 19