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Neutrino decay in matter
Volume 199, number 2
PHYSICS LETTERS B
17 December 1987
NEUTRINO DECAY IN MATTER Z.G. B E R E Z H I A N I l a n d M.I. V Y S O T S K Y Institute of Theoretical and Experimental Physics, 117259 Mosco~t USSR
Received 3 September 1987
We demonstrate that matter can induce decay of the neutrino mto the antineutrino and a light scalar particle (majoron): v~ 9+ct, or vice versa, 9--,v+ ct. Diagonal as well as non-diagonal transitions on neutrino types are possible. The decay probabilities depend on the neutrino energies in an unusual way. Applications for ( 1) accelerator neutrinos passing through the earth, (2) solar neutrinos, (3) neutrino emission accompanied by the gravitational collapse of the stellar core, are discussed.
It is well known that the oscillations o f neutrinos in m a t t e r could be different from v a c u u m oscillations [ 1,2]. We shall d e m o n s t r a t e that the coherent interaction o f neutrinos with m a t t e r could be i m p o r t a n t for their decay also. In particular, a transition o f the left-handed neutrino into a right-handed one (right neutrino in the case o f the Dirac neutrino or a n t i n e u t r i n o in the case o f the M a j o r a n a neutrino) with the emission o f massless (or sufficiently light) scalar particle can become energetically allowed, even in the case of strictly masslcss neutrinos. Coherent scattering could be taken into account directly in the " c u r r e n t × c u r r e n t " lagrangian which describes elastic neutrino scattering in terms o f neutral currents ~. In the rest frame o f m a t t e r only the time components o f the electron, proton and neutron vector currents r e m a i n nonzero after averaging over the matter, e.g. (ey~,(l + ~ s ) e ) = ~o,ne. Consequently neutrino p r o p a g a t i o n in the m e d i u m is described by the Dirac equation which is analogous to that describing electron m o t i o n in an external field e A u = (p,O). Therefore, for the left-handed neutrino and C P conjugated right-handed antineutrino, ?R = CVL, energy levels we have correspondingly: E = x / p 2 +m~2+_p. Here p and m are neutrino m o m e n t u m and mass a n d P = P e - ½ P n for v~ and P = - ½Pn for v, and v,, where p~,, = , , f 2 G F ne,. a n d n~ and nn are electron and neutron densities, correspondingly. So if there exists an interaction which changes neutrino chirality - for example, radiation o f a massless or very light scalar particle g) - the decay VL--'?R +q) is induced in m a t t e r (or vice versa, depending on the neutrino type and the relative neutron density o f m a t t e r ) . We will also d e m o n s t r a t e that even if this scalar particle interacts diagonally with a neutrino in v a c u u m flavour-nondiagonal transitions would be induced in matter. Let us illustrate the above statements within the triplet m a j o r o n model [3]. We briefly r e m i n d this model. A triplet o f scalar fields 0 = ( 0 + +,0+,~ °) is i n t r o d u c e d in the standard S U ( 2 ) L ® U ( 1 ) model o f electroweak interactions without right-handed neutrino field components. The field ~ has Yukawa couplings with the doublets ~e:(2~)L,(~ '') k (for the sake o f simplicity we shall discuss the case o f two generations). If the lagrangian does not contain couplings of the type H v e H ~ (where H is the Higgs doublet) then the global U(1 ) s y m m e t r y could be spontaneously broken by a non-zero v a c u u m average u o f the neutral c o m p o n e n t ~o o f the triplet field. In this way the neutrino acquires the M a j o r a n a masses. Simultaneously a massless G o l d s t o n e boson ( m a j o r o n ) ct= ( 1 / x / 2 ) I m ~o appears in the particle spectrum. A light Higgs boson q0 = ( 1 / v ~ ) Re 0 ° - u with a mass ,,/~u appears also (2 is an interaction constant o f the field 0) ~2. The lagrangian of the 0 ° interactions has the following form: Institute of Physics, Academy of Sciences GSSR, 380077 Tbilisi, USSR. ~J The charged ( vL.e) current interactions must also be Fierz-transformed into the form (G~:/\.2) v~,y~ ( 1 + 75 ) v,, ~'/p ( 1 + 75 ) e. ~2 The energy loss of red giants due to majoron emission puts the constraint: u < 100 keV [4]. The fields 0+ + and 0 + must be heavy, M~ Mw. 281
Volume 199, number 2
PHYSICS LETTERS B
+h.c. LP= (vev.)~/~(u + iet + ~0) (ge)R v.
17 December 1987
(1)
where OR = C~L is the antineutrino field and h = (~, ~") is the matrix of the Yukawa coupling constants. If one transforms the neutrino mass matrix into diagonal form and arrives at the states with definite masses ml and m2 (vl=cve+sv~, v 2 = - s v e + c v . , where c - c o s 0, s=sin 0, sin220= 4heJ[4he. 2 2 + ( h . - h ~ ) 2 ] ) the Yukawa couplings (1) are diagonalized simultaneously:
h12=m12/u=½[he+h.T_x/4h2 _ _, h e ) 2 ]
+(h ,
, /~=(h01 0)h2 "
(2)
We demonstrate that a and cp couple diagonally with v~ and v2. So the heaviest neutrino can not decay in vacuum. The next step is to take into account effects of matter. For the ultrarelativistic neutrino (p= IPl >> m) the evolution of states in matter after subtraction of an unessential common phase is described by the following Schr~dinger equation [ 1 ]:
dt\v2/
\v2/'
with the hamiltonian
9ff _ ( m ~/2p + c2p~ - ½p. csp~ ) -k csp~ m~/2p+s2p _½p. .
(4)
Eigenstates of this hamiltonian are v~n =CrnVl "]-SmV2 ,
V~n = --SmV 1 "3!-CmV2
(Cm=COSOm,Sm=SinOm) ,
where sin220m = sin220/[sin220 + (cos 20 _~)2] ,
(5)
and ~= (m~-m2)/2pp~. Eigenvalues of the hamiltonian (4) look like 21,2 = ½{(rn~ +m2)/2p+pe -p~ -T-x / t ( m 2 -m2)/2p+cos 20p~] 2 + sin220p2 } •
(6)
Antineutrino evolution in the same matter is described by an analogous hamiltonian with the change Pe,n~ --Pe,n (when we interchange v and 9 the zero angle scattering amplitude f(0) changes sign). Eigenstates and eigenvalues for VR are given by formulae (5), (6) with the change p~,.--,-p~,. (g~=cosffm,g~= sin 0"m). For eigenstates in matter we obtain from (1): m m {CmCmhl"~-SmSmh2 Sm SmCmh2-cms-mhl)(V~a) gmhl .~. Cmffmh2 ~n " ~O= (Vl V2 )L~cmgmhz__Sm~mhl (la+(p) +c.c.
(7)
Therefore, nondiagonal ~,2--*v~,1 as well as diagonal ~m2--*v~,2 transitions are possible. For the forthcoming statements we need an expression for the neutrino decay probabilities. We shall give a general expression from which all needed decay probabilities could easily be obtained. We shall consider the decay of the Majorana neutrino vl with momentum p and mass m~ and an interaction potential with matter Pl into a massless scalar ct and the Majorana neutrino v2 with a mass rn2 and a potential P2, which is described by the amplitude h91LV2R a. For the width in a reference system where matter is at rest we obtain: 282
Volume 199, number 2 h2
F=
PHYSICS LETTERSB
[rn2+(pl-p2)x~2+rn~](
16~
x/'p2+rn 2
rn'~ 1- [m2+(p~_p2)2+2(pl_p2)x/lJ2+m2]
17 December 1987
) 2 .
(8)
We start our analysis from the case of dense matter (or light neutrinos) p>> ( m 2 , m 2 ) / 2 E (~ << 1 ). In this case the eigenstates of the hamiltonian (4) are directly the current ones: v~ = Ve, V~"= V~. For the neutrino decay probabilities we obtain:
F~, = (h2./16g)pn,
F¢~ = ( h 2 / 1 6 g ) ( 2 p e - P n ) ,
F.~ =Fe~ = ( h 2 . / 1 6 g ) ( p e - p n ) •
(9)
where F.~ = F(9.--.v~) etc. Negative probability means that v and 9 must be interchanged (if Pn > 2pe, then decays into v). Now, let us turn to the case of a heavier neutrino (or low densities of matter) ~ >> 1. For the diagonal transitions V~L~9~R, V2Lm~V2 R~m we obtain:
FI~ = (h~/2~r)(p~ c 0 s 2 0 - ½p~) , F2~ = (h2/2rr)(p~ s i n 2 0 - ½Pn) -
(10a)
Nondiagonal transitions have the following width:
Fl~=F~2=(h~2/16rrlP, I )( rn 4l - m 2)/m 4 2, , hl~=½sin2Om(h~-h2)l
~>>1
= [(sin20)/2~](h,-h2)=h¢./~.
(10b)
which are suppressed compared with diagonal ones (10a). The probabilities of matter-induced transitions depend on the initial neutrino energy in an unusual way. They do not depend on energy when ~ << 1 (see (9)). When ~ >> 1 diagonal transitions are also energy independent. As for nondiagonal probabilities (10b), they are proportional to E/m. For ordinary vacuum decays F = Fore~E, where Fo is the decay probability in the rest frame of the decaying particle. Before we discuss some applications let us give the experimental limits on the majoron coupling constants: h ~ < 2 X 1 0 -3
(11)
(213o decay with majoron emission) [5] h ~ + h 2 ~ < 4 . 5 X 1 0 -5,
h 2 + h 2 " < 2 . 4 X 1 0 -4
(12,13)
(lepton decays of ~- and K-mesons) [6]. (1) The passing of the accelerator neutrinos through the earth. The 9, are heavier in matter than the v,. So, when a bunch of accelerator produced 9, pass through the earth some of them can decay into v,. The charged current gives a distinct signature of the effect: together with ~t+ from the reaction %p-~ng + , g - from the reaction v ~ n ~ p - p appears. Let us estimate the value of the effect. We take 10 g/cm 3 for the earth density, the relative number of neutrons is given by the electron number Y~ = nJ(np + nn) ~ 0.5. The region ~ >> 1 is more profitable. With the maximal value ofh~ from (10a) and (13) we obtain:
~% = 4 n / h 2 . p . ~ (3X 1012 c m ) / ( 3 X 101° cm/s) ~ 100 s.
(14)
As the diameter of the earth is of the order of 1 0 9 c m not more than 0.03 % of 9. could transform into v.. (2) Solar neutrinos. Electron neutrinos produced in the core of the sun fall on the earth and, in particular, are detected in the Davis experiment. The deficit ofv~ in the Davis experiment in comparison with the standard solar model prediction can be understood if one supposes that ve decays inside the sun: Ve~ (ge, 9~ )a. Let us make an estimate for the matter-induced decay. As the maximal allowed value for h 2 is an order of magnitude lower than h 2 then in the domain ~ << 1 neglecting p~ in comparison with pC we obtain:
F . . . . . = (h~,/16~)p~.
(15a) 283
Volume 199, number 2
PHYSICSLETTERSB
17 December 1987
When ~ >> 1, maximum neutrino mixing is profitable, 0 ~ 45 °. Then from (10a) we obtain: V .......... = (h 2./4~z)pe.
(15b)
Taking the maximally allowed value for h2. from (12) and using a density of the sun of 100 g/cm 3 at a distance of 2X 10 m cm we obtain in the case (15b): Nv/Nvo ~ 5%,
(16)
which is too small to explain the three-times deficit of ve in the Davis data but probably rather large enough to be detected in the future with new solar neutrino detectors. (3) Neutrinos f r o m the gravitational collapse o f a stellar core. In supernovae the densities and distances have such values that in spite of the limits (10)-(13 ) neutrino decays could occur. The neutrinos emitted during the collapse have a typical energy of about 10 MeV. Taking a typical density of 1012 g / c m 3 w e obtain pe~ 10 ' eV and ~ << 1. Typical distances in supernovae are R ~ 10 7 cm and decays take place if h 2> 10- 9. The relative number of electrons in supernovae, Ye, varies between 0.3 at R ~ 10 6 c m and 0.5 at R ~ 10 7 c m [ 7 ] . Without taking decays into account, approximately equal numbers of v's and 9's of different types are expected; the decays change this prediction drastically. In particular, at Y= 0.3, % and 9 e decay into v, and v~, respectively, and no antineutrinos fall on the earth (let us remind that the detectors which are under operation now are sensitive to 9e only [8]; however, the detectors under construction will be sensitive to ve as well). It is noteworthy that the effects of neutrino decay in matter can be relevant also to the case of Dirac neutrinos. Indeed, in the standard SU (2)® U (1) model with right-handed neutrino fields vR one can introduce together with the standard H an additional Higgs doublet H ' = (Ho) with Yukawa couplings of the type heLvRH' + h.c. and impose the global U(1) symmetry H' ~exp(ic~)H'. The breaking of this symmetry by the vacuum average of the scalar H ' ( ( H °' > << ( H °> =250 GeV) leads to the appearance of neutrino Dirac masses and the corresponding Goldstone boson ~ (the diron - by analogy with the majoron) simultaneously appears in the particle spectrum. Then the transitions VL~VR+13 are possible in matter (or vice versa, VR-*VL+I~, depending on the neutrino type and the neutron concentration). It is obvious that due to the sterility of VR the experimental consequences of the diron scheme substantially differ from that for the case "Majorana neutrinos + majoron". Let us discuss now the case of neutrino decays in vacuum. They are possible if we incorporate in the majoron scheme also the small Dirac mass terms. Another way is the introduction of two (or more) majoron-type global U (1) symmetries acting distinctly on the different neutrino flavours. In vacuum the neutrino-type nondiagonal transitions due to couplings of the type hglV2a+h.c. (V 1 ~---CVe'Jt-SV~t,...) are possible only, e.g. v1-,92+ ct, 9,--,v2 + ct if m , > m2. The corresponding width in the rest frame of the neutrino source is F = Fom/E, where Fo = ( h2 /16n)( m 4 - m 4 ) / r n 3. The vacuum decays can explain the deficit of the solar v~ flux in the Davis experiment in two different ways. o m ,,o ~ (1) Neutrino mixing is negligible and mvo > m~,. Supposing that %,. = r,,eE,,o/ ~ 500 s for Evo ~ 5 MeV we conclude that one third of the boron neutrinos emitted in the core of the sun decay in flight from the sun to the earth. In this scenario the flux of pp neutrinos (with energies lower than 420 keV) is diminished in e '° times. This is a strong (but negative) prediction for the G a - G e detector experiments in preparation. (2) Neutrino mixing is substantial, v~=cv, +sv> and m, > m2. If the heavier neutrino v, decays into 92 with a lifetime r~, << 500 s, the component v2 = -sv~ + cv, reaches the earth only. The magnitude of the vc flux directly measures the neutrino mixing angle: NDavis/No= sin40. In conclusion, we would like to emphasize that matter-induced neutrino decays are drastically distinct from vacuum decays and v oscillations. They can take place even for strictly massless neutrinos. The experimental signatures of the decays are also very distinct: the neutrino changes chirality, which can not take place as a result of v oscillations. Numerical estimates show that in an experiment with an accelerator 9~ beam a little part could transform into v~; a flux of Ve from the sun could contain a few percent of % and 9. and the content of neutrinos and 284
Volume 199, number 2
PHYSICS LETTERS B
17 December 1987
a n t i n e u t r i n o s of different flavours coming into the earth from stellar collapse could differ widely in n u m b e r because of v decays in dense regions of the star. Finally, note that the existence of the m a j o r o n leads to the absence of relic neutrinos: they annihilate into majorons in the early stages of the evolution of the universe if the Yukawa constant h is larger than 10-5. We are grateful to S.I. Blinnikov, A.A. Gerasimov, A.Yu. Smirnov, M.Yu. Khlopov and V.A. Tsarev for useful discussions.
References [1 ] L. Wolfenstein,Phys. Rev. D 17 (1978) 2369; D 20 (1979) 2634. [2] S.P. Mikheev and A.Yu. Smirnov, Yad. Fiz. 42 (1985) 1441. [3] G.B. Gelmini and M. Roncadelli, Phys. Lett. B 99 (1981) 411. [4] H. Georgi, S.L. Glashow and S. Nussinov, Nucl. Phys. B 193 (1980) 297. [5] D.O. Caldwell et al., Phys. Rev. Lett. 54 (1985) 281. [6] V. Barger, W. Keung and S. Pakvasa, Phys. Rev. D 25 (1982) 907. [7] D.K. Nadyozhin, Astrophys. Space Sci. 49 (1977) 399; 51 (1977) 283; 53 (1978) 131; R.L. Bowers and J.R. Wilson, Astrophys. J. 263 (1982) 366; S.W. Bruenn, Astrophys. J. Suppl. 58 (1985) 771. [8] L.N. Alekseeva, in: Chastizi i Kosmologiya,I.Ya.I.AN SSSR (Moscow, 1984) p. 58.
285
PHYSICS LETTERS B
17 December 1987
NEUTRINO DECAY IN MATTER Z.G. B E R E Z H I A N I l a n d M.I. V Y S O T S K Y Institute of Theoretical and Experimental Physics, 117259 Mosco~t USSR
Received 3 September 1987
We demonstrate that matter can induce decay of the neutrino mto the antineutrino and a light scalar particle (majoron): v~ 9+ct, or vice versa, 9--,v+ ct. Diagonal as well as non-diagonal transitions on neutrino types are possible. The decay probabilities depend on the neutrino energies in an unusual way. Applications for ( 1) accelerator neutrinos passing through the earth, (2) solar neutrinos, (3) neutrino emission accompanied by the gravitational collapse of the stellar core, are discussed.
It is well known that the oscillations o f neutrinos in m a t t e r could be different from v a c u u m oscillations [ 1,2]. We shall d e m o n s t r a t e that the coherent interaction o f neutrinos with m a t t e r could be i m p o r t a n t for their decay also. In particular, a transition o f the left-handed neutrino into a right-handed one (right neutrino in the case o f the Dirac neutrino or a n t i n e u t r i n o in the case o f the M a j o r a n a neutrino) with the emission o f massless (or sufficiently light) scalar particle can become energetically allowed, even in the case of strictly masslcss neutrinos. Coherent scattering could be taken into account directly in the " c u r r e n t × c u r r e n t " lagrangian which describes elastic neutrino scattering in terms o f neutral currents ~. In the rest frame o f m a t t e r only the time components o f the electron, proton and neutron vector currents r e m a i n nonzero after averaging over the matter, e.g. (ey~,(l + ~ s ) e ) = ~o,ne. Consequently neutrino p r o p a g a t i o n in the m e d i u m is described by the Dirac equation which is analogous to that describing electron m o t i o n in an external field e A u = (p,O). Therefore, for the left-handed neutrino and C P conjugated right-handed antineutrino, ?R = CVL, energy levels we have correspondingly: E = x / p 2 +m~2+_p. Here p and m are neutrino m o m e n t u m and mass a n d P = P e - ½ P n for v~ and P = - ½Pn for v, and v,, where p~,, = , , f 2 G F ne,. a n d n~ and nn are electron and neutron densities, correspondingly. So if there exists an interaction which changes neutrino chirality - for example, radiation o f a massless or very light scalar particle g) - the decay VL--'?R +q) is induced in m a t t e r (or vice versa, depending on the neutrino type and the relative neutron density o f m a t t e r ) . We will also d e m o n s t r a t e that even if this scalar particle interacts diagonally with a neutrino in v a c u u m flavour-nondiagonal transitions would be induced in matter. Let us illustrate the above statements within the triplet m a j o r o n model [3]. We briefly r e m i n d this model. A triplet o f scalar fields 0 = ( 0 + +,0+,~ °) is i n t r o d u c e d in the standard S U ( 2 ) L ® U ( 1 ) model o f electroweak interactions without right-handed neutrino field components. The field ~ has Yukawa couplings with the doublets ~e:(2~)L,(~ '') k (for the sake o f simplicity we shall discuss the case o f two generations). If the lagrangian does not contain couplings of the type H v e H ~ (where H is the Higgs doublet) then the global U(1 ) s y m m e t r y could be spontaneously broken by a non-zero v a c u u m average u o f the neutral c o m p o n e n t ~o o f the triplet field. In this way the neutrino acquires the M a j o r a n a masses. Simultaneously a massless G o l d s t o n e boson ( m a j o r o n ) ct= ( 1 / x / 2 ) I m ~o appears in the particle spectrum. A light Higgs boson q0 = ( 1 / v ~ ) Re 0 ° - u with a mass ,,/~u appears also (2 is an interaction constant o f the field 0) ~2. The lagrangian of the 0 ° interactions has the following form: Institute of Physics, Academy of Sciences GSSR, 380077 Tbilisi, USSR. ~J The charged ( vL.e) current interactions must also be Fierz-transformed into the form (G~:/\.2) v~,y~ ( 1 + 75 ) v,, ~'/p ( 1 + 75 ) e. ~2 The energy loss of red giants due to majoron emission puts the constraint: u < 100 keV [4]. The fields 0+ + and 0 + must be heavy, M~ Mw. 281
Volume 199, number 2
PHYSICS LETTERS B
+h.c. LP= (vev.)~/~(u + iet + ~0) (ge)R v.
17 December 1987
(1)
where OR = C~L is the antineutrino field and h = (~, ~") is the matrix of the Yukawa coupling constants. If one transforms the neutrino mass matrix into diagonal form and arrives at the states with definite masses ml and m2 (vl=cve+sv~, v 2 = - s v e + c v . , where c - c o s 0, s=sin 0, sin220= 4heJ[4he. 2 2 + ( h . - h ~ ) 2 ] ) the Yukawa couplings (1) are diagonalized simultaneously:
h12=m12/u=½[he+h.T_x/4h2 _ _, h e ) 2 ]
+(h ,
, /~=(h01 0)h2 "
(2)
We demonstrate that a and cp couple diagonally with v~ and v2. So the heaviest neutrino can not decay in vacuum. The next step is to take into account effects of matter. For the ultrarelativistic neutrino (p= IPl >> m) the evolution of states in matter after subtraction of an unessential common phase is described by the following Schr~dinger equation [ 1 ]:
dt\v2/
\v2/'
with the hamiltonian
9ff _ ( m ~/2p + c2p~ - ½p. csp~ ) -k csp~ m~/2p+s2p _½p. .
(4)
Eigenstates of this hamiltonian are v~n =CrnVl "]-SmV2 ,
V~n = --SmV 1 "3!-CmV2
(Cm=COSOm,Sm=SinOm) ,
where sin220m = sin220/[sin220 + (cos 20 _~)2] ,
(5)
and ~= (m~-m2)/2pp~. Eigenvalues of the hamiltonian (4) look like 21,2 = ½{(rn~ +m2)/2p+pe -p~ -T-x / t ( m 2 -m2)/2p+cos 20p~] 2 + sin220p2 } •
(6)
Antineutrino evolution in the same matter is described by an analogous hamiltonian with the change Pe,n~ --Pe,n (when we interchange v and 9 the zero angle scattering amplitude f(0) changes sign). Eigenstates and eigenvalues for VR are given by formulae (5), (6) with the change p~,.--,-p~,. (g~=cosffm,g~= sin 0"m). For eigenstates in matter we obtain from (1): m m {CmCmhl"~-SmSmh2 Sm SmCmh2-cms-mhl)(V~a) gmhl .~. Cmffmh2 ~n " ~O= (Vl V2 )L~cmgmhz__Sm~mhl (la+(p) +c.c.
(7)
Therefore, nondiagonal ~,2--*v~,1 as well as diagonal ~m2--*v~,2 transitions are possible. For the forthcoming statements we need an expression for the neutrino decay probabilities. We shall give a general expression from which all needed decay probabilities could easily be obtained. We shall consider the decay of the Majorana neutrino vl with momentum p and mass m~ and an interaction potential with matter Pl into a massless scalar ct and the Majorana neutrino v2 with a mass rn2 and a potential P2, which is described by the amplitude h91LV2R a. For the width in a reference system where matter is at rest we obtain: 282
Volume 199, number 2 h2
F=
PHYSICS LETTERSB
[rn2+(pl-p2)x~2+rn~](
16~
x/'p2+rn 2
rn'~ 1- [m2+(p~_p2)2+2(pl_p2)x/lJ2+m2]
17 December 1987
) 2 .
(8)
We start our analysis from the case of dense matter (or light neutrinos) p>> ( m 2 , m 2 ) / 2 E (~ << 1 ). In this case the eigenstates of the hamiltonian (4) are directly the current ones: v~ = Ve, V~"= V~. For the neutrino decay probabilities we obtain:
F~, = (h2./16g)pn,
F¢~ = ( h 2 / 1 6 g ) ( 2 p e - P n ) ,
F.~ =Fe~ = ( h 2 . / 1 6 g ) ( p e - p n ) •
(9)
where F.~ = F(9.--.v~) etc. Negative probability means that v and 9 must be interchanged (if Pn > 2pe, then decays into v). Now, let us turn to the case of a heavier neutrino (or low densities of matter) ~ >> 1. For the diagonal transitions V~L~9~R, V2Lm~V2 R~m we obtain:
FI~ = (h~/2~r)(p~ c 0 s 2 0 - ½p~) , F2~ = (h2/2rr)(p~ s i n 2 0 - ½Pn) -
(10a)
Nondiagonal transitions have the following width:
Fl~=F~2=(h~2/16rrlP, I )( rn 4l - m 2)/m 4 2, , hl~=½sin2Om(h~-h2)l
~>>1
= [(sin20)/2~](h,-h2)=h¢./~.
(10b)
which are suppressed compared with diagonal ones (10a). The probabilities of matter-induced transitions depend on the initial neutrino energy in an unusual way. They do not depend on energy when ~ << 1 (see (9)). When ~ >> 1 diagonal transitions are also energy independent. As for nondiagonal probabilities (10b), they are proportional to E/m. For ordinary vacuum decays F = Fore~E, where Fo is the decay probability in the rest frame of the decaying particle. Before we discuss some applications let us give the experimental limits on the majoron coupling constants: h ~ < 2 X 1 0 -3
(11)
(213o decay with majoron emission) [5] h ~ + h 2 ~ < 4 . 5 X 1 0 -5,
h 2 + h 2 " < 2 . 4 X 1 0 -4
(12,13)
(lepton decays of ~- and K-mesons) [6]. (1) The passing of the accelerator neutrinos through the earth. The 9, are heavier in matter than the v,. So, when a bunch of accelerator produced 9, pass through the earth some of them can decay into v,. The charged current gives a distinct signature of the effect: together with ~t+ from the reaction %p-~ng + , g - from the reaction v ~ n ~ p - p appears. Let us estimate the value of the effect. We take 10 g/cm 3 for the earth density, the relative number of neutrons is given by the electron number Y~ = nJ(np + nn) ~ 0.5. The region ~ >> 1 is more profitable. With the maximal value ofh~ from (10a) and (13) we obtain:
~% = 4 n / h 2 . p . ~ (3X 1012 c m ) / ( 3 X 101° cm/s) ~ 100 s.
(14)
As the diameter of the earth is of the order of 1 0 9 c m not more than 0.03 % of 9. could transform into v.. (2) Solar neutrinos. Electron neutrinos produced in the core of the sun fall on the earth and, in particular, are detected in the Davis experiment. The deficit ofv~ in the Davis experiment in comparison with the standard solar model prediction can be understood if one supposes that ve decays inside the sun: Ve~ (ge, 9~ )a. Let us make an estimate for the matter-induced decay. As the maximal allowed value for h 2 is an order of magnitude lower than h 2 then in the domain ~ << 1 neglecting p~ in comparison with pC we obtain:
F . . . . . = (h~,/16~)p~.
(15a) 283
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PHYSICSLETTERSB
17 December 1987
When ~ >> 1, maximum neutrino mixing is profitable, 0 ~ 45 °. Then from (10a) we obtain: V .......... = (h 2./4~z)pe.
(15b)
Taking the maximally allowed value for h2. from (12) and using a density of the sun of 100 g/cm 3 at a distance of 2X 10 m cm we obtain in the case (15b): Nv/Nvo ~ 5%,
(16)
which is too small to explain the three-times deficit of ve in the Davis data but probably rather large enough to be detected in the future with new solar neutrino detectors. (3) Neutrinos f r o m the gravitational collapse o f a stellar core. In supernovae the densities and distances have such values that in spite of the limits (10)-(13 ) neutrino decays could occur. The neutrinos emitted during the collapse have a typical energy of about 10 MeV. Taking a typical density of 1012 g / c m 3 w e obtain pe~ 10 ' eV and ~ << 1. Typical distances in supernovae are R ~ 10 7 cm and decays take place if h 2> 10- 9. The relative number of electrons in supernovae, Ye, varies between 0.3 at R ~ 10 6 c m and 0.5 at R ~ 10 7 c m [ 7 ] . Without taking decays into account, approximately equal numbers of v's and 9's of different types are expected; the decays change this prediction drastically. In particular, at Y= 0.3, % and 9 e decay into v, and v~, respectively, and no antineutrinos fall on the earth (let us remind that the detectors which are under operation now are sensitive to 9e only [8]; however, the detectors under construction will be sensitive to ve as well). It is noteworthy that the effects of neutrino decay in matter can be relevant also to the case of Dirac neutrinos. Indeed, in the standard SU (2)® U (1) model with right-handed neutrino fields vR one can introduce together with the standard H an additional Higgs doublet H ' = (Ho) with Yukawa couplings of the type heLvRH' + h.c. and impose the global U(1) symmetry H' ~exp(ic~)H'. The breaking of this symmetry by the vacuum average of the scalar H ' ( ( H °' > << ( H °> =250 GeV) leads to the appearance of neutrino Dirac masses and the corresponding Goldstone boson ~ (the diron - by analogy with the majoron) simultaneously appears in the particle spectrum. Then the transitions VL~VR+13 are possible in matter (or vice versa, VR-*VL+I~, depending on the neutrino type and the neutron concentration). It is obvious that due to the sterility of VR the experimental consequences of the diron scheme substantially differ from that for the case "Majorana neutrinos + majoron". Let us discuss now the case of neutrino decays in vacuum. They are possible if we incorporate in the majoron scheme also the small Dirac mass terms. Another way is the introduction of two (or more) majoron-type global U (1) symmetries acting distinctly on the different neutrino flavours. In vacuum the neutrino-type nondiagonal transitions due to couplings of the type hglV2a+h.c. (V 1 ~---CVe'Jt-SV~t,...) are possible only, e.g. v1-,92+ ct, 9,--,v2 + ct if m , > m2. The corresponding width in the rest frame of the neutrino source is F = Fom/E, where Fo = ( h2 /16n)( m 4 - m 4 ) / r n 3. The vacuum decays can explain the deficit of the solar v~ flux in the Davis experiment in two different ways. o m ,,o ~ (1) Neutrino mixing is negligible and mvo > m~,. Supposing that %,. = r,,eE,,o/ ~ 500 s for Evo ~ 5 MeV we conclude that one third of the boron neutrinos emitted in the core of the sun decay in flight from the sun to the earth. In this scenario the flux of pp neutrinos (with energies lower than 420 keV) is diminished in e '° times. This is a strong (but negative) prediction for the G a - G e detector experiments in preparation. (2) Neutrino mixing is substantial, v~=cv, +sv> and m, > m2. If the heavier neutrino v, decays into 92 with a lifetime r~, << 500 s, the component v2 = -sv~ + cv, reaches the earth only. The magnitude of the vc flux directly measures the neutrino mixing angle: NDavis/No= sin40. In conclusion, we would like to emphasize that matter-induced neutrino decays are drastically distinct from vacuum decays and v oscillations. They can take place even for strictly massless neutrinos. The experimental signatures of the decays are also very distinct: the neutrino changes chirality, which can not take place as a result of v oscillations. Numerical estimates show that in an experiment with an accelerator 9~ beam a little part could transform into v~; a flux of Ve from the sun could contain a few percent of % and 9. and the content of neutrinos and 284
Volume 199, number 2
PHYSICS LETTERS B
17 December 1987
a n t i n e u t r i n o s of different flavours coming into the earth from stellar collapse could differ widely in n u m b e r because of v decays in dense regions of the star. Finally, note that the existence of the m a j o r o n leads to the absence of relic neutrinos: they annihilate into majorons in the early stages of the evolution of the universe if the Yukawa constant h is larger than 10-5. We are grateful to S.I. Blinnikov, A.A. Gerasimov, A.Yu. Smirnov, M.Yu. Khlopov and V.A. Tsarev for useful discussions.
References [1 ] L. Wolfenstein,Phys. Rev. D 17 (1978) 2369; D 20 (1979) 2634. [2] S.P. Mikheev and A.Yu. Smirnov, Yad. Fiz. 42 (1985) 1441. [3] G.B. Gelmini and M. Roncadelli, Phys. Lett. B 99 (1981) 411. [4] H. Georgi, S.L. Glashow and S. Nussinov, Nucl. Phys. B 193 (1980) 297. [5] D.O. Caldwell et al., Phys. Rev. Lett. 54 (1985) 281. [6] V. Barger, W. Keung and S. Pakvasa, Phys. Rev. D 25 (1982) 907. [7] D.K. Nadyozhin, Astrophys. Space Sci. 49 (1977) 399; 51 (1977) 283; 53 (1978) 131; R.L. Bowers and J.R. Wilson, Astrophys. J. 263 (1982) 366; S.W. Bruenn, Astrophys. J. Suppl. 58 (1985) 771. [8] L.N. Alekseeva, in: Chastizi i Kosmologiya,I.Ya.I.AN SSSR (Moscow, 1984) p. 58.
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