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Relaxing nucleosynthesis bounds on sterile neutrinos
Physics Letters B 275 (1992) 112-118 North-Holland
PHYSICS LETTERS B
Relaxing nucleosynthesis bounds on sterile neutrinos K.S. Babu
~.2
and I.Z. Rothstein 3
Department qf Physics and Astronomy, University of Mao,land, College Park, MD 20742, USA Received 22 July 1991
We have analyzed the constraints arising from nucleosynthesison possibleoscillation between a doublet neutrino (u,,,uu or u~) and a sterile neutrino (us) in the presence of their couplings to a majoron. The bounds on the mixing angle and mass-squared difference derived previouslybased on the standard model interactions are found to be weakened by several orders of magnitude. For the case of a 17 keV pseudo-Diracu~,the constraint on the singlet-doublet mass splitting is relaxed to lfro 21~
1. Introduction Cosmological considerations are known to be powerful in constraining the properties of light particles. For example, limits have been placed on the n u m b e r of effective neutrino species which are in equilibrium with the primeval plasma during the epoch of nucleosynthesis. Recent calculations [ 1 ] which combine the improved measurement of neutron lifetime with estimates of primordial helium, deuterium and lithium abundance and baryon to photon ratio quote N,~< 3.3. Any extension of the standard model which introduces additional light particles must ensure that this constraint is satisfied. In this article, we are interested in the bounds on possible oscillation between a doublet neutrino (v,,, u~, or ~,~) and a "sterile" (i.e., not weakly interacting) neutrino (v~) that can be derived based on nucleosynthesis considerations. Since u~ does not interact weakly, it is expected to go out of thermal equilibrium from the plasma at temperatures well above the nucleosynthesis era. Even so, oscillation between a doublet neutrino and the sterile species may bring the sterile state back into thermal equilibrium, which would violate the b o u n d on N~. D e m a n d i n g this not e¢ Work supported by a grant from the National Science Foundation. Address after 1 September 1991: Bartol Research Institute, University of Delaware, Newark, DE 19716, USA. 2 Bitnetaddress: babu ~,umdhep. Bitnet address: rothstem~umdhep. 1 12
to occur during nucleosynthesis places constraints on the oscillation parameters. The oscillation between uc and a sterile species u~, has received much attention in connection with the possible resolution of the solar neutrino puzzle via the MSW mechanism. Earlier calculations [2] ignored the effects of coherent forward scattering off the plasma and led to rather stringent limits on the oscillation parameters. It was later realized that inclusion of the coherent matter effects [ 3 ] is important, which led to considerable relaxation of these constraints [4,5]. In particular, the entire parameter space relevant for the MSW mechanism for solar neutrinos was shown to be allowed. Nevertheless, these constraints are still more stringent than the ones that can be inferred from direct oscillation experiments. Their validity would mean that the prospects of detecting such oscillations in terrestrial experiments are bleak. We have explored the possibility of relaxing these constraints further in the case where the neutrinos have non-negligible couplings to a majoron. Our primary motivation for this study was the recent reports o f a 17 keV neutrino in/~ decay experiments [6 ], although our conclusions are more general. Limits from neutrinoless double/~ decay require the 17 keV neutrino to bc almost a Dirac particle. If the solar neutrino puzzle is resolved via the MSW resonant oscillation between u,, and ut,, the simplest explanation of the fl decay observations is that v~ pairs up with a sterile neutrino to form a pseudo-Dirac particle. Such
0370-2693/92/$ 05.00 © 1992 ElsevierScience Publishers B.V. All rights reserved.
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PHYSICS LETTERS B
a scenario is consistent with all known laboratory data. Since the 17 keV mass lies outside o f the cosmologically allowed range for stable particles, v~ should decay (or annihilate) fast enough into invisible channels so as to be consistent with the mass density constraints. This can be satisfied ifv~ decays into a light neutrino and a majoron [ 7 ]. Recently, Glashow has proposed a "singular seesaw" schemes [8] where the MSW oscillation between v,, and v~, and a pseudo-Dirac v~ arises rather naturally. Since the singularity of the heavy Majorana mass matrix is a signal of horizontal symmetry, the coupling of the neutrinos to the majoron can be rather large in this class of models, leading to a shortlived u~ ~. The coupling of the majoron to the neutrino in several models based on this scheme [ 9-12 ] goes as r n J M . If M, the scale o f B - L violation, is of the same order as the electro-weak symmetry breaking scale, this coupling is in the range I 0 - 7 - 1 0 6. Models based on the radiative mass generation scheme [ 12,13] can also yield relatively large majoron coupling. For example, in the model of ref. [ 13 ], the diagonal majoron coupling is predicted to be 1 0 - 7 - 1 0 -6"
The main result of this paper is that the standard model constraints on the oscillation parameters are relaxed by several orders of magnitude in the presence of a moderately large majoron coupling. Specializing to the case of pseudo-Dirac v~, we first ensure that the off-diagonal v~v~,Z and v~v,,z couplings (Z is the majoron) are not strong enough to keep the decays v~ ~ v,,z, v,,Z in equilibrium during nucleosynthesis. The bound on the singlet-doublet masssplitting is then found to be 18rn 21 ~
23 January 1992
In our notation, 0~<0~45 °, m2 is the mass eigenvalue corresponding to the state that is predominantly sterile, and m l corresponds to the doublet state. We allow for 8 m 2 to be either positive or negative. Assuming that the sterile neutrino (v~) has decoupled from the thermal bath at high enough temperatures, we must make sure that they are not brought back into thermal equilibrium through oscillations with a doublet neutrino Vd ( d = e , It or ;), which is in equilibrium. The probability in vacuum for finding the neutrino in a singlet state at time t given that it was in a doublet state at t = 0 is P ~ u ~ = sin220 sin2(½At) ,
( 1)
where 3 = S m 2 / 2 E . If due to lack of spatial overlap of the wave packets, or due to non-coherent scattering, the phase information is lost, then the oscillating term in eq. ( 1 ) will average to ½. In the early universe, we must modify this probability to take into account the coherent forward scattering off the plasma. The evolution of the 2 × 2 system is governed by the hamiltonian equation
id5 v, c - m y -I- S-tlq 5 2E
- + Vd cs3
csA s-my+c 2E
Pd) , Ps
m~ + V~
(2) where c = c o s 0, s = s i n 0, Vaand V~are the doublet and the singlet neutrino self-energies resulting from the plasma effects. In the standard model, since v~ is sterile, V~=0 and Vjis given by (to second order in GF)
[3]
2. Nucleosynthesis constraints in the standard model Let us briefly review the constraints that can be derived on the oscillation parameters 8m 2 = m 2 - m and sin 20 based on the standard model interactions. From the point of view of galaxy formation, such a short lifetime is preferable, since the decay products of a long-lived v~ would radiation-dominate the universe inhibiting the growth of structures.
Here N~.= ( 2 D z 2 ) ~ ( 3 ) T 3 is the photon number density and L is a sum of terms each proportional to either a lepton asymmetry or baryon asymmetry which are expected to be of order 10-~o. It is worth noting that neutrino oscillations cause the lepton asymmetry to be a dynamical variable necessitating a self-consistent analysis. Such an analysis has been performed in ref. [ 14], where it was shown that the lepton asym113
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PHYSICS LETTERS B
metry is driven to a vanishingly small n u m b e r regardless of its initial value. It is then a p p r o p r i a t e to neglect this term in comparison with the G 2 T s term of eq. (3). The coefficient A is flavor-dependent,
23 January 1992
The effective mixing angle 0m in the presence o f matter is
equality as a function of temperature, and then insist that the inequality holds at this m a x i m u m temperature. The constraints derived are dependent upon the flavor of the doublet since the decoupling temperature [or equivalently c ( T ) in eq. (7) ] and the selfenergy Vj are different for v~, and v~ from that o f v,,. For v,, the constraint is [ 5 ]
sin 20m = (A/Am) sin 20,
(sin 20)418m 2 ] -%<3.6× 10 -4 eV 2
A~-55 forv,,andA~-15 forv~,orv~.
(4)
10)
with a slightly weaker constraint for u,, and v~.
where
dm=[A2+(V,-Vd)2+2(I(,-Vd)Acos20]
I/2.
(5)
We see from the above that for an a p p r o p r i a t e sign of the effect o f matter is to suppress oscillation. In addition to coherent forward scattering off the plasma, we must also consider the effects o f non-coherent scattering which will destroy the phase information, necessitating that we average over the collision time. The rate of production of the singlet neutrino is then given by
3. Effects of coupling to the majoron
A=Sm2/2E,
F~ = F . ~ ( P.~,~,) .
(6)
Here F~u is the rate o f production o f the doublet neutrino via its weak interaction,
F,,=c(T)G~T 5 ,
(7)
where c ( T ) is a temperature d e p e n d e n t coefficient which takes into account particle thresholds ( c ( T ) =0.5 for v,, below muon decoupling [ 15 ] ). In the primeval plasma during nucleosynthesis, the average oscillation time is always smaller than either the collision time or the time it takes for the wave packets to no longer have spatial overlap. Therefore we should replace the average by ½ leaving
F,~=F,a'½(sin220,~,) .
(8)
For hot relics we can determine whether or not a reaction is in equilibrium by checking if the reaction rate exceeds the Hubble expansion rate H = 1.66g 1./2( T ) T2/MH. The condition for the sterile neutrinos to be out o f equilibrium is then
F,,
Fv~Mpl sin220m H - 1.66g~./2(T) T 2 ~ <
1.
(9)
TO place constraints on the mixing angle and 8m2, one must now maximize the left-hand side o f this in114
It has long been known that a stable neutrino m the mass range 100 eV to a few GeV would overclose the universe. A neutrino in this mass range, such as the 17 keV neutrino reported in fl decay experiments, must decay (or annihilate) sufficiently rapidly into invisible channels to be consistent with cosmology. Most of the models constructed to explain the 17 keV neutrino utilize the majoron to ensure that its decay is fast enough. In this section we show that the bounds on the oscillation parameters derived previously are relaxed considerably due to new contributions to the neutrino self-energy arising from the thermal effects of the majoron. The majoron interaction we shall consider is ih
ih'
L z, = - -x//~ Xv~Iv, + - x//2 - " Zv~l. u , + h . c .
(11)
Here, v, is the right-handed sterile neutrino, and we are interested in Vd-V, oscillation. We have also included off-diagonal majoron couplings o f v, with a doublet state denoted by v j which is assumed to be lighter than Yd. ( F o r example, v~l may be u~ and Vd' may be re, or v,,. ) This can lead to a rapid decay o f v, (and Vd if it has a Dirac mass with v,). The off-diagonal coupling h' in the models of refs. [9-11 ] is of order Oh, where 0 is the u,/-uj, mixing angle. In the model of ref. [ 13 ], h' << h. We could also include other types o f majoron couplings, e.g., v,Va2v,Z and V~CS2VdXetc. We have omitted such terms in eq. ( 11 ) since they are not present in models of 17 keV neutrino. Our analysis can be trivially extended to the case when such couplings are also present. The interactions of eq. ( 11 ) will lead to new con-
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PHYSICS LETTERS B
(a)
if it decouples from the heat bath at a t e m p e r a t u r e TD is given by
Z /
//
,
T - TDRD R •
vs
vd
Here TD, RD are the t e m p e r a t u r e and the scale factor at the time ofdecoupling, and T, R are their values at a later time o f interest. Entropy conservation then allows us to write
vs
Z, 's ~
/J / / t
vs
Vd
Fig. 1. Self-energydiagrams for (a) sterile neutrino and (b) doublet neutrino via the majoron exchange.
tributions to the neutrino self-energy arising from the exchange of majoron. We have evaluated these contributions using the real-time finite t e m p e r a t u r e field theory techniques [16]. The relevant diagrams are shown in figs. l a and lb. The new contributions to Vd and V~are
h2
cze
Vd= 87r2(¢oj )
(f4)
•
(b)
Vd
23 January 1992
pdp[nb(Tz)+nr(T,,)],
(12)
0
(\ gg,(r) ,(T.) -
-
/
T.
(15)
Using g , ( TD ) = 113 (which includes the majoron and three singlet neutrinos) and g , ( T ) = 10.75 relevant to the nucleosynthesis era, we see that the temperatures of u, and Z are smaller by a factor o f 0.46 relative to the doublet neutrino. The fermionic integral o f eqs. (12), (13) yields 1 ~ 2 T 2 and the bosonic integral yields g1 ~r2 T-2 . The potentials V,~and I(,.are then h2 1~)=96(o9d ) (T~ + 2 T ~ )
h2 V, = 9 6 ( c o , > ( T d + 2 T ~ ) .
(16)
Their difference, which enters into the oscillation probability, can now be written in terms o f the doublet t e m p e r a t u r e alone ~3: 96(C0d)
3r 2 -
--2r
,
(17)
Here n b and nf are the Bose-Einstein and F e r m i Dirac distribution functions defined at t e m p e r a t u r e s Td, T~ or Tz corresponding to the doublet, singlet and the majoron respectively. (COd) and (~o,) are the average energy o f ud and u., ((O)d) =3.15Td, (O)s) = 3.15T~). These t e m p e r a t u r e s are not all the same, since the majoron and the singlet neutrino decouple at high temperatures. (We shall assume that this happens prior to the electro-weak phase transition ~2.) The temperature o f a relativistic particle at later times
where r = [g, ( T ) / g , ( T o ) ] ~/3 ~ 0.46. Notice that the sign o f this potential difference is negative, analogous to the standard model. Its magnitude is comparable to that o f the standard model contribution for h ~ 1 0 - 6 - 1 0 -7. More importantly, it has a milder T dependence, ~c T, in contrast with oc T 5 o f e q . (3). Before we proceed to derive bounds on the oscillation parameters, it is pertinent to ask how large a value can be tolerated for the coupling h. As noted earlier, in one class of models, the off-diagonal m a j o r o n coupling h' is expected to be o f order ~ hO. The size o f the coupling h' is restricted since a large h' would bring the reactions t'~Ud,Z and Pde'*Pd,Z into equi-
1,2 The spin-flip processes will keep a 17 keV Dirac neutrino in equilibrium until T-~ 100 GeV.
1.3 We are neglecting the standard model contribution ofeq. (3), which will be justified a posteriori.
V,= 87r2(o9, )
pdp
[nb(T~)+m(T.~)].
(13)
0
115
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PHYSICS LETTERS B
librium during the nucleosynthesis era. The former process is the helicity flipping decay, while the latter is helicity conserving. To set a bound on h', we demand that both these processes remain out of equilibrium. The reaction rate for the helicity-flipping decay is given by
i i
d.x% dx, [l--nr(x,)]rtf(x2) X-2 X 2.
h2
Note that the rate is suppressed by the square of the neutrino mass due to angular m o m e n t u m conservation. The rate for the helicity-conserving decay is also suppressed by neutrino mass due to the form of the interaction in eq. (3), and is identical to that of eq. (18). (We are assuming Va and vs have a c o m m o n Dirac mass m,,. ) The integral I was evaluated numerically yielding I n 0.3. To ensure that these interactions are out of equilibrium during the time of nucleosynthesis we impose that T~
h'2m~
<<"197~7a-"') " a I ' z T ~*, t j
<~(l/r+2r-
( 1 + y 2 +__yy ~ x / ~ ) 2 / 3 x
(23)
(l-y2)2/3(~_+y)
H e r e y = c o s 20, T~, is the v, d e c o u p l i n g t e m p e r a t u r e ~ 3 . 3 M e V a n d the + sign is f o r the case o f S m 2 > 0 .
In fig. 2, we plot the m a x i m u m value of [•m 21 / h2 as a function of sin20 for the case where 8 m 2 > 0 . For maximal mixing and a neutrino mass of 17 keV the bound is 18m 2 ]/h2<~ 1.3 MeV 2. This corresponds to 18m 2 ] ~ 4 eV 2 for h = 1.8 X 1O- 6. In the model of ref. [ 13 ], since the off-diagonal coupling of Z is negligibly small, the bound can be even weaker: 18m21 20
''11 ....
I ....
I ....
I ....
I'
I .... I .... I .... I .... 0.2 o.4 o.6 0.8
I, 1
g-.
(21)
For a 17 keV neutrino, the bound is h'~< 1.8X 10 -7. If its mixing with v,, is 10%, we expect the diagonal coupling to be h~< 1.8X 10 -6. Interestingly, the maj oron couplings of the neutrino in refs. [ 9,10 ] nearly saturate but are below this upper limit. Now we can derive bounds on the oscillation parameters. The constraint that v~ is not to be brought into equilibrium (i.e., F,,/H<~ 1 ) is given in eq. (9). The effective mixing angle in matter sin 20r~, is given in eqs. (4), (5) and (17). The function F,,/H is peaked at the temperature given by 116
3r 2 )T~,2
(20)
Since the reaction rate goes like ( I / T ) , we must choose the smallest allowed value for the temperature to evaluate the rate. This is approximately 1.9 MeV, the decoupling temperature of v,,, below which weak interactions freeze-out. Choosing this value for T gives the following constraint on h':
m~h' ~<3X 10 -9 ( M e V ) .
(22)
I~m21
0
l'66gZ*/2( T) ~
× (x/cos220+ 3 +__cos 20) .
(19)
-
m,,/7"
8m2l
48
(18)
where
l=
//
Tm"~=~l/r+2rr_3r2) I h 2
Here the plus sign is for the case when 8m2>~0 and the minus sign is for 8rn2~< 0. We have neglected the temperature dependence in c(T) [see eq. (7) ], and from g, in the Hubble expansion, since they are constant over the temperature range we are considering. Using eqs. (4), ( 5 ), (9), (22) we arrive at the following constraint on 8m2:
m~
F : h '2 1927t{(3)T I ,
2
23 January 1992
....
sin2O Fig. 2. Maximum value of 18m2[/h 2 as a function of sin20 for 8m2>0.
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PHYSICS LETTERS B
~
(24)
We see that resonance can only occur i f S m 2 < 0, since Vj-V~<~O. I m p o s i n g the c o n s t r a i n t that r e s o n a n c e should o c c u r e i t h e r b e l o w the n e u t r i n o d e c o u p l i n g scale or a b o v e the Q C D phase transition, leads to cos 20 lSrn21
h2
44~(l/r+2r-3r2)T~
cos 20 18rn21
h2
or
(25)
>~(1/r+2r--3r2)T~cD.
(26)
T h e s e c o n d c o n s t r a i n t g i v e n here a s s u m e s that the sterile n e u t r i n o s are no longer in e q u i l i b r i u m b e l o w the r e s o n a n c e t e m p e r a t u r e . In fig. 3, the solid line shows the c o n s t r a i n t o f e q . ( 2 5 ) . N o t e that this constraint is m o r e stringent t h a n eq. ( 2 3 ) for small m i x ing angles. T h e c o n s t r a i n t f r o m eq. ( 2 6 ) is seen to be irrelevant. If the n e u t r i n o is a p s e u d o - D i r a c particle, t h e n
....
I ....
1,2
I ....
I ....
1.0
0.8
0.6 ....
I .... 0.2.
I .... 0.4
4. Conclusion We h a v e s h o w n that in the p r e s e n c e o f a m a j o r o n c o u p l i n g b e t w e e n a light singlet a n d d o u b l e t neutrino, the c o n s t r a i n t s f r o m n u c l e o s y n t h e s i s on the oscillation p a r a m e t e r s are relaxed by several orders o f m a g n i t u d e . F o r a 17 keV p s e u d o - D i r a c v~, the b o u n d is 18m21 ~
Acknowledgement We wish to t h a n k R a b i M o h a p a t r a for v a l u a b l e discussions. We h a v e also b e n e f i t t e d f r o m discussions with L. B e n t o and J.W.F. Valle.
References
6m 2 ~ 0
0
cos 20 ~lr/2"-~/t2 w h e r e / t is the l e f t - h a n d e d d i a g o n a l m a s s ( a s s u m e d to be s m a l l e r t h a n the D i r a c m a s s ) . F o r h -~ 1 0 - 7 - 1 0 -6 we see that the c o n s t r a i n t on # is o f o r d e r 0.1-1 eV. T h i s c o r r e s p o n d s to ]8m21 ~< 103104 eV 2 for a 17 keV v~, well b e l o w the p r e d i c t i o n s o f refs. [ 8 - 1 3 ] .
[''
g--
~
23 January 1992
I .... 0,0
I,, 0,8
sin20 Fig. 3. Maximum value of ]Sm2[/h 2 versus sin20 for 8 m 2 ~ 0 . The dotted line corresponds to the constraint from ordinary oscillation and the solid line corresponds to the case of resonant oscillation.
[ 1] K. Olive, D.N. Schramm, G. Steigman and T.P. Walker, Phys. Lett. B 236 (1990) 454; K. Olive, D.N. Schramm, G. Steigman, T.P. Walker and H.S. Kang, Ohio State preprint OSU-TA-10/90. [2] A. Dolgov, Soy. J. Nucl. Phys. 33 ( 1981 ) 700; M. Kholpov and S. Petcov, Phys. Lett. B 99 ( 1981 ) 117; B 100 (1981) 520(E); D. Fragion and M. Shepkin, Phys. Len. B 146 (1984) 46; P. Langacker, B. Sathiapalan and G. Steigman, Nucl. Phys. B266 (1986) 669; P. Langacker, Pennsylvania preprint UPR 040 IT (1989). [3] D. Notzold and G. Raffelt, Nucl. Phys. B 307 (1988) 924. [4] R. Barbieri and A. Dolgov, Phys. Len. B 237 (1990) 440. [5] K. Kainulainen, Phys. Lett. B 244 (1990) t91; M.J. Thomson and B.H.J. McKellar, Melbourne preprint UM-P-90/44 ( 1990); K. Enqvist, K. Kainulainen and J. Maalampi, Phys. Lett. B 249 (1990) 531; J.M. Cline, OSU preprint DOE/ER/01545-552. [6] J. Simpson, Phys. Rev. Lett. 54 (1985) 1891; J. Simpson and A. Hime, Phys. Rev. D 39 (1989) 1825, 1837; 117
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A. Hime and N.A. Jelley, Phys. Lett. B 257 ( 1991 ) 441; B. Suret al., Phys. Rev. Lett. 66 ( 1991 ) 2444; J. Zliman et al., Zagreb preprint 11-9/90LEI. [7] Y. Chikashige, R.N. Mohapatra and R.D. Peccei, Phys. Lett. B98 (1981) 265. [8] S.L. Glashow, Phys. Lett. B 256 ( 1991 ) 218. [ 9 ] K.S. Babu and R.N. Mohapatra, Maryland preprint 91-186 (1991). [ 10] D. Ng, North Carolina preprint IFP-403-UNC ( 1991 ).
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[ 11 ] R. Foot and S.F. King, Southampton preprint SHEP 90/ 91-18 (1991). [12]A.L. Kagan, City College preprint CCNY-HEP-91/3 (1991). [ 13] L. Bento and J.W.F, Valle, Valencia preprint FTUV/91-6, IFIC/91-6 (1991). [14] K. Enqvist, K. Kainulainen and J. Maalampi, Nucl. Phys. B 349 (1991) 754. [ 15 ] L. Stodolsky, Phys. Rev. D 36 (1987) 2273. [ 16] H.A. Weldon, Phys. Rev. D 26 (1982) 2789.
PHYSICS LETTERS B
Relaxing nucleosynthesis bounds on sterile neutrinos K.S. Babu
~.2
and I.Z. Rothstein 3
Department qf Physics and Astronomy, University of Mao,land, College Park, MD 20742, USA Received 22 July 1991
We have analyzed the constraints arising from nucleosynthesison possibleoscillation between a doublet neutrino (u,,,uu or u~) and a sterile neutrino (us) in the presence of their couplings to a majoron. The bounds on the mixing angle and mass-squared difference derived previouslybased on the standard model interactions are found to be weakened by several orders of magnitude. For the case of a 17 keV pseudo-Diracu~,the constraint on the singlet-doublet mass splitting is relaxed to lfro 21~
1. Introduction Cosmological considerations are known to be powerful in constraining the properties of light particles. For example, limits have been placed on the n u m b e r of effective neutrino species which are in equilibrium with the primeval plasma during the epoch of nucleosynthesis. Recent calculations [ 1 ] which combine the improved measurement of neutron lifetime with estimates of primordial helium, deuterium and lithium abundance and baryon to photon ratio quote N,~< 3.3. Any extension of the standard model which introduces additional light particles must ensure that this constraint is satisfied. In this article, we are interested in the bounds on possible oscillation between a doublet neutrino (v,,, u~, or ~,~) and a "sterile" (i.e., not weakly interacting) neutrino (v~) that can be derived based on nucleosynthesis considerations. Since u~ does not interact weakly, it is expected to go out of thermal equilibrium from the plasma at temperatures well above the nucleosynthesis era. Even so, oscillation between a doublet neutrino and the sterile species may bring the sterile state back into thermal equilibrium, which would violate the b o u n d on N~. D e m a n d i n g this not e¢ Work supported by a grant from the National Science Foundation. Address after 1 September 1991: Bartol Research Institute, University of Delaware, Newark, DE 19716, USA. 2 Bitnetaddress: babu ~,umdhep. Bitnet address: rothstem~umdhep. 1 12
to occur during nucleosynthesis places constraints on the oscillation parameters. The oscillation between uc and a sterile species u~, has received much attention in connection with the possible resolution of the solar neutrino puzzle via the MSW mechanism. Earlier calculations [2] ignored the effects of coherent forward scattering off the plasma and led to rather stringent limits on the oscillation parameters. It was later realized that inclusion of the coherent matter effects [ 3 ] is important, which led to considerable relaxation of these constraints [4,5]. In particular, the entire parameter space relevant for the MSW mechanism for solar neutrinos was shown to be allowed. Nevertheless, these constraints are still more stringent than the ones that can be inferred from direct oscillation experiments. Their validity would mean that the prospects of detecting such oscillations in terrestrial experiments are bleak. We have explored the possibility of relaxing these constraints further in the case where the neutrinos have non-negligible couplings to a majoron. Our primary motivation for this study was the recent reports o f a 17 keV neutrino in/~ decay experiments [6 ], although our conclusions are more general. Limits from neutrinoless double/~ decay require the 17 keV neutrino to bc almost a Dirac particle. If the solar neutrino puzzle is resolved via the MSW resonant oscillation between u,, and ut,, the simplest explanation of the fl decay observations is that v~ pairs up with a sterile neutrino to form a pseudo-Dirac particle. Such
0370-2693/92/$ 05.00 © 1992 ElsevierScience Publishers B.V. All rights reserved.
Volume 275, number 1,2
PHYSICS LETTERS B
a scenario is consistent with all known laboratory data. Since the 17 keV mass lies outside o f the cosmologically allowed range for stable particles, v~ should decay (or annihilate) fast enough into invisible channels so as to be consistent with the mass density constraints. This can be satisfied ifv~ decays into a light neutrino and a majoron [ 7 ]. Recently, Glashow has proposed a "singular seesaw" schemes [8] where the MSW oscillation between v,, and v~, and a pseudo-Dirac v~ arises rather naturally. Since the singularity of the heavy Majorana mass matrix is a signal of horizontal symmetry, the coupling of the neutrinos to the majoron can be rather large in this class of models, leading to a shortlived u~ ~. The coupling of the majoron to the neutrino in several models based on this scheme [ 9-12 ] goes as r n J M . If M, the scale o f B - L violation, is of the same order as the electro-weak symmetry breaking scale, this coupling is in the range I 0 - 7 - 1 0 6. Models based on the radiative mass generation scheme [ 12,13] can also yield relatively large majoron coupling. For example, in the model of ref. [ 13 ], the diagonal majoron coupling is predicted to be 1 0 - 7 - 1 0 -6"
The main result of this paper is that the standard model constraints on the oscillation parameters are relaxed by several orders of magnitude in the presence of a moderately large majoron coupling. Specializing to the case of pseudo-Dirac v~, we first ensure that the off-diagonal v~v~,Z and v~v,,z couplings (Z is the majoron) are not strong enough to keep the decays v~ ~ v,,z, v,,Z in equilibrium during nucleosynthesis. The bound on the singlet-doublet masssplitting is then found to be 18rn 21 ~
23 January 1992
In our notation, 0~<0~45 °, m2 is the mass eigenvalue corresponding to the state that is predominantly sterile, and m l corresponds to the doublet state. We allow for 8 m 2 to be either positive or negative. Assuming that the sterile neutrino (v~) has decoupled from the thermal bath at high enough temperatures, we must make sure that they are not brought back into thermal equilibrium through oscillations with a doublet neutrino Vd ( d = e , It or ;), which is in equilibrium. The probability in vacuum for finding the neutrino in a singlet state at time t given that it was in a doublet state at t = 0 is P ~ u ~ = sin220 sin2(½At) ,
( 1)
where 3 = S m 2 / 2 E . If due to lack of spatial overlap of the wave packets, or due to non-coherent scattering, the phase information is lost, then the oscillating term in eq. ( 1 ) will average to ½. In the early universe, we must modify this probability to take into account the coherent forward scattering off the plasma. The evolution of the 2 × 2 system is governed by the hamiltonian equation
id5 v, c - m y -I- S-tlq 5 2E
- + Vd cs3
csA s-my+c 2E
Pd) , Ps
m~ + V~
(2) where c = c o s 0, s = s i n 0, Vaand V~are the doublet and the singlet neutrino self-energies resulting from the plasma effects. In the standard model, since v~ is sterile, V~=0 and Vjis given by (to second order in GF)
[3]
2. Nucleosynthesis constraints in the standard model Let us briefly review the constraints that can be derived on the oscillation parameters 8m 2 = m 2 - m and sin 20 based on the standard model interactions. From the point of view of galaxy formation, such a short lifetime is preferable, since the decay products of a long-lived v~ would radiation-dominate the universe inhibiting the growth of structures.
Here N~.= ( 2 D z 2 ) ~ ( 3 ) T 3 is the photon number density and L is a sum of terms each proportional to either a lepton asymmetry or baryon asymmetry which are expected to be of order 10-~o. It is worth noting that neutrino oscillations cause the lepton asymmetry to be a dynamical variable necessitating a self-consistent analysis. Such an analysis has been performed in ref. [ 14], where it was shown that the lepton asym113
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PHYSICS LETTERS B
metry is driven to a vanishingly small n u m b e r regardless of its initial value. It is then a p p r o p r i a t e to neglect this term in comparison with the G 2 T s term of eq. (3). The coefficient A is flavor-dependent,
23 January 1992
The effective mixing angle 0m in the presence o f matter is
equality as a function of temperature, and then insist that the inequality holds at this m a x i m u m temperature. The constraints derived are dependent upon the flavor of the doublet since the decoupling temperature [or equivalently c ( T ) in eq. (7) ] and the selfenergy Vj are different for v~, and v~ from that o f v,,. For v,, the constraint is [ 5 ]
sin 20m = (A/Am) sin 20,
(sin 20)418m 2 ] -%<3.6× 10 -4 eV 2
A~-55 forv,,andA~-15 forv~,orv~.
(4)
10)
with a slightly weaker constraint for u,, and v~.
where
dm=[A2+(V,-Vd)2+2(I(,-Vd)Acos20]
I/2.
(5)
We see from the above that for an a p p r o p r i a t e sign of the effect o f matter is to suppress oscillation. In addition to coherent forward scattering off the plasma, we must also consider the effects o f non-coherent scattering which will destroy the phase information, necessitating that we average over the collision time. The rate of production of the singlet neutrino is then given by
3. Effects of coupling to the majoron
A=Sm2/2E,
F~ = F . ~ ( P.~,~,) .
(6)
Here F~u is the rate o f production o f the doublet neutrino via its weak interaction,
F,,=c(T)G~T 5 ,
(7)
where c ( T ) is a temperature d e p e n d e n t coefficient which takes into account particle thresholds ( c ( T ) =0.5 for v,, below muon decoupling [ 15 ] ). In the primeval plasma during nucleosynthesis, the average oscillation time is always smaller than either the collision time or the time it takes for the wave packets to no longer have spatial overlap. Therefore we should replace the average by ½ leaving
F,~=F,a'½(sin220,~,) .
(8)
For hot relics we can determine whether or not a reaction is in equilibrium by checking if the reaction rate exceeds the Hubble expansion rate H = 1.66g 1./2( T ) T2/MH. The condition for the sterile neutrinos to be out o f equilibrium is then
F,,
Fv~Mpl sin220m H - 1.66g~./2(T) T 2 ~ <
1.
(9)
TO place constraints on the mixing angle and 8m2, one must now maximize the left-hand side o f this in114
It has long been known that a stable neutrino m the mass range 100 eV to a few GeV would overclose the universe. A neutrino in this mass range, such as the 17 keV neutrino reported in fl decay experiments, must decay (or annihilate) sufficiently rapidly into invisible channels to be consistent with cosmology. Most of the models constructed to explain the 17 keV neutrino utilize the majoron to ensure that its decay is fast enough. In this section we show that the bounds on the oscillation parameters derived previously are relaxed considerably due to new contributions to the neutrino self-energy arising from the thermal effects of the majoron. The majoron interaction we shall consider is ih
ih'
L z, = - -x//~ Xv~Iv, + - x//2 - " Zv~l. u , + h . c .
(11)
Here, v, is the right-handed sterile neutrino, and we are interested in Vd-V, oscillation. We have also included off-diagonal majoron couplings o f v, with a doublet state denoted by v j which is assumed to be lighter than Yd. ( F o r example, v~l may be u~ and Vd' may be re, or v,,. ) This can lead to a rapid decay o f v, (and Vd if it has a Dirac mass with v,). The off-diagonal coupling h' in the models of refs. [9-11 ] is of order Oh, where 0 is the u,/-uj, mixing angle. In the model of ref. [ 13 ], h' << h. We could also include other types o f majoron couplings, e.g., v,Va2v,Z and V~CS2VdXetc. We have omitted such terms in eq. ( 11 ) since they are not present in models of 17 keV neutrino. Our analysis can be trivially extended to the case when such couplings are also present. The interactions of eq. ( 11 ) will lead to new con-
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PHYSICS LETTERS B
(a)
if it decouples from the heat bath at a t e m p e r a t u r e TD is given by
Z /
//
,
T - TDRD R •
vs
vd
Here TD, RD are the t e m p e r a t u r e and the scale factor at the time ofdecoupling, and T, R are their values at a later time o f interest. Entropy conservation then allows us to write
vs
Z, 's ~
/J / / t
vs
Vd
Fig. 1. Self-energydiagrams for (a) sterile neutrino and (b) doublet neutrino via the majoron exchange.
tributions to the neutrino self-energy arising from the exchange of majoron. We have evaluated these contributions using the real-time finite t e m p e r a t u r e field theory techniques [16]. The relevant diagrams are shown in figs. l a and lb. The new contributions to Vd and V~are
h2
cze
Vd= 87r2(¢oj )
(f4)
•
(b)
Vd
23 January 1992
pdp[nb(Tz)+nr(T,,)],
(12)
0
(\ gg,(r) ,(T.) -
-
/
T.
(15)
Using g , ( TD ) = 113 (which includes the majoron and three singlet neutrinos) and g , ( T ) = 10.75 relevant to the nucleosynthesis era, we see that the temperatures of u, and Z are smaller by a factor o f 0.46 relative to the doublet neutrino. The fermionic integral o f eqs. (12), (13) yields 1 ~ 2 T 2 and the bosonic integral yields g1 ~r2 T-2 . The potentials V,~and I(,.are then h2 1~)=96(o9d ) (T~ + 2 T ~ )
h2 V, = 9 6 ( c o , > ( T d + 2 T ~ ) .
(16)
Their difference, which enters into the oscillation probability, can now be written in terms o f the doublet t e m p e r a t u r e alone ~3: 96(C0d)
3r 2 -
--2r
,
(17)
Here n b and nf are the Bose-Einstein and F e r m i Dirac distribution functions defined at t e m p e r a t u r e s Td, T~ or Tz corresponding to the doublet, singlet and the majoron respectively. (COd) and (~o,) are the average energy o f ud and u., ((O)d) =3.15Td, (O)s) = 3.15T~). These t e m p e r a t u r e s are not all the same, since the majoron and the singlet neutrino decouple at high temperatures. (We shall assume that this happens prior to the electro-weak phase transition ~2.) The temperature o f a relativistic particle at later times
where r = [g, ( T ) / g , ( T o ) ] ~/3 ~ 0.46. Notice that the sign o f this potential difference is negative, analogous to the standard model. Its magnitude is comparable to that o f the standard model contribution for h ~ 1 0 - 6 - 1 0 -7. More importantly, it has a milder T dependence, ~c T, in contrast with oc T 5 o f e q . (3). Before we proceed to derive bounds on the oscillation parameters, it is pertinent to ask how large a value can be tolerated for the coupling h. As noted earlier, in one class of models, the off-diagonal m a j o r o n coupling h' is expected to be o f order ~ hO. The size o f the coupling h' is restricted since a large h' would bring the reactions t'~Ud,Z and Pde'*Pd,Z into equi-
1,2 The spin-flip processes will keep a 17 keV Dirac neutrino in equilibrium until T-~ 100 GeV.
1.3 We are neglecting the standard model contribution ofeq. (3), which will be justified a posteriori.
V,= 87r2(o9, )
pdp
[nb(T~)+m(T.~)].
(13)
0
115
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PHYSICS LETTERS B
librium during the nucleosynthesis era. The former process is the helicity flipping decay, while the latter is helicity conserving. To set a bound on h', we demand that both these processes remain out of equilibrium. The reaction rate for the helicity-flipping decay is given by
i i
d.x% dx, [l--nr(x,)]rtf(x2) X-2 X 2.
h2
Note that the rate is suppressed by the square of the neutrino mass due to angular m o m e n t u m conservation. The rate for the helicity-conserving decay is also suppressed by neutrino mass due to the form of the interaction in eq. (3), and is identical to that of eq. (18). (We are assuming Va and vs have a c o m m o n Dirac mass m,,. ) The integral I was evaluated numerically yielding I n 0.3. To ensure that these interactions are out of equilibrium during the time of nucleosynthesis we impose that T~
h'2m~
<<"197~7a-"') " a I ' z T ~*, t j
<~(l/r+2r-
( 1 + y 2 +__yy ~ x / ~ ) 2 / 3 x
(23)
(l-y2)2/3(~_+y)
H e r e y = c o s 20, T~, is the v, d e c o u p l i n g t e m p e r a t u r e ~ 3 . 3 M e V a n d the + sign is f o r the case o f S m 2 > 0 .
In fig. 2, we plot the m a x i m u m value of [•m 21 / h2 as a function of sin20 for the case where 8 m 2 > 0 . For maximal mixing and a neutrino mass of 17 keV the bound is 18m 2 ]/h2<~ 1.3 MeV 2. This corresponds to 18m 2 ] ~ 4 eV 2 for h = 1.8 X 1O- 6. In the model of ref. [ 13 ], since the off-diagonal coupling of Z is negligibly small, the bound can be even weaker: 18m21 20
''11 ....
I ....
I ....
I ....
I'
I .... I .... I .... I .... 0.2 o.4 o.6 0.8
I, 1
g-.
(21)
For a 17 keV neutrino, the bound is h'~< 1.8X 10 -7. If its mixing with v,, is 10%, we expect the diagonal coupling to be h~< 1.8X 10 -6. Interestingly, the maj oron couplings of the neutrino in refs. [ 9,10 ] nearly saturate but are below this upper limit. Now we can derive bounds on the oscillation parameters. The constraint that v~ is not to be brought into equilibrium (i.e., F,,/H<~ 1 ) is given in eq. (9). The effective mixing angle in matter sin 20r~, is given in eqs. (4), (5) and (17). The function F,,/H is peaked at the temperature given by 116
3r 2 )T~,2
(20)
Since the reaction rate goes like ( I / T ) , we must choose the smallest allowed value for the temperature to evaluate the rate. This is approximately 1.9 MeV, the decoupling temperature of v,,, below which weak interactions freeze-out. Choosing this value for T gives the following constraint on h':
m~h' ~<3X 10 -9 ( M e V ) .
(22)
I~m21
0
l'66gZ*/2( T) ~
× (x/cos220+ 3 +__cos 20) .
(19)
-
m,,/7"
8m2l
48
(18)
where
l=
//
Tm"~=~l/r+2rr_3r2) I h 2
Here the plus sign is for the case when 8m2>~0 and the minus sign is for 8rn2~< 0. We have neglected the temperature dependence in c(T) [see eq. (7) ], and from g, in the Hubble expansion, since they are constant over the temperature range we are considering. Using eqs. (4), ( 5 ), (9), (22) we arrive at the following constraint on 8m2:
m~
F : h '2 1927t{(3)T I ,
2
23 January 1992
....
sin2O Fig. 2. Maximum value of 18m2[/h 2 as a function of sin20 for 8m2>0.
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PHYSICS LETTERS B
~
(24)
We see that resonance can only occur i f S m 2 < 0, since Vj-V~<~O. I m p o s i n g the c o n s t r a i n t that r e s o n a n c e should o c c u r e i t h e r b e l o w the n e u t r i n o d e c o u p l i n g scale or a b o v e the Q C D phase transition, leads to cos 20 lSrn21
h2
44~(l/r+2r-3r2)T~
cos 20 18rn21
h2
or
(25)
>~(1/r+2r--3r2)T~cD.
(26)
T h e s e c o n d c o n s t r a i n t g i v e n here a s s u m e s that the sterile n e u t r i n o s are no longer in e q u i l i b r i u m b e l o w the r e s o n a n c e t e m p e r a t u r e . In fig. 3, the solid line shows the c o n s t r a i n t o f e q . ( 2 5 ) . N o t e that this constraint is m o r e stringent t h a n eq. ( 2 3 ) for small m i x ing angles. T h e c o n s t r a i n t f r o m eq. ( 2 6 ) is seen to be irrelevant. If the n e u t r i n o is a p s e u d o - D i r a c particle, t h e n
....
I ....
1,2
I ....
I ....
1.0
0.8
0.6 ....
I .... 0.2.
I .... 0.4
4. Conclusion We h a v e s h o w n that in the p r e s e n c e o f a m a j o r o n c o u p l i n g b e t w e e n a light singlet a n d d o u b l e t neutrino, the c o n s t r a i n t s f r o m n u c l e o s y n t h e s i s on the oscillation p a r a m e t e r s are relaxed by several orders o f m a g n i t u d e . F o r a 17 keV p s e u d o - D i r a c v~, the b o u n d is 18m21 ~
Acknowledgement We wish to t h a n k R a b i M o h a p a t r a for v a l u a b l e discussions. We h a v e also b e n e f i t t e d f r o m discussions with L. B e n t o and J.W.F. Valle.
References
6m 2 ~ 0
0
cos 20 ~lr/2"-~/t2 w h e r e / t is the l e f t - h a n d e d d i a g o n a l m a s s ( a s s u m e d to be s m a l l e r t h a n the D i r a c m a s s ) . F o r h -~ 1 0 - 7 - 1 0 -6 we see that the c o n s t r a i n t on # is o f o r d e r 0.1-1 eV. T h i s c o r r e s p o n d s to ]8m21 ~< 103104 eV 2 for a 17 keV v~, well b e l o w the p r e d i c t i o n s o f refs. [ 8 - 1 3 ] .
[''
g--
~
23 January 1992
I .... 0,0
I,, 0,8
sin20 Fig. 3. Maximum value of ]Sm2[/h 2 versus sin20 for 8 m 2 ~ 0 . The dotted line corresponds to the constraint from ordinary oscillation and the solid line corresponds to the case of resonant oscillation.
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