Physics Letters B 271 (1991) 161-164 North-Holland P H YSI C S LETTER S B

Decaying dark matter and the Zee model

W. G r i m u s Institut fiir Theoretische Ph.vsik, Universitdt Wien, Boltzmanngasse 5, A-I090 Vienna, Austria

a n d

G. N a r d u l l i Dipartimento di Fisica, Universtgt di Bari, 1NFN, Sezione di Bari, Via Amendola 173, I- 70126 Bari, Italy

Received 24 August 1991

We show that within the Zee model of lepton flavour non-conservation, radiatively decaying neutrinos with a mass my ~ 28 eV and lifetimes r ~ 2 X 1023 s can exist. Thus it is possible to realize the hypothesis of decaying dark matter recently put forward by Sciama in this rather simple extension of the standard model.

One of the most interest ing schemes proposed to explain the origin o f cosmological dark mat ter is the so-called decaying dark mat ter ( D D M ) hypothesis [ I -5 ]. In this approach dark matter is basically made up of heavy neutr inos with a mass

m v H = 2 8 _ + l e V . (1)

These neutr inos would decay radia t ively into lighter neutr inos

VH ~ V L -[-7 , (2)

with a l ifetime

r = (2_+ 1 ) X 1023 s . (3)

One o f the main virtues of this hypothesis lies in its predict ivi ty. As a mat ter of fact, from its assumpt ions one can derive predic t ions for the mean densi ty of the universe ( p = 5 . 9 6 _ 0 . 2 2 g c m - 3 ) and ifp~pcri t , for the age of the universe ( 1.2 × 101° yr) and for the Hubble constant (55 _+ 1 km s - ' M p c - ~ ). Moreover , the essential point of Sciama 's idea is that the neu- tr ino decay (2) can be used not only to detect the clark mat ter but it also natural ly explains the abun- dance of ionized hydrogen in the galaxy as well as its presence in remote regions of the universe ~'. This

~ See ref. [5 ] and references therein; for previous work on sim- ilar ideas see refs. [6-8].

lat ter aspect of the D D M hypothesis is a consequence of the final photon energy Ev= 14.0_+0.5 eV in the neutr ino decay (2) above the ionizing threshold of 13.6 eV but sufficiently close to it. In this way the cosmological red shift would lead to a strong suppres- sion of the y flux.

The D D M hypothesis has been tested using the Hopkins Ultraviolet Telescope during the Astro-1 mission of the space shuttle Columbia in December 1990 with negative results [ 9 ]. However, these data do not yet falsify the scenario because there is a nar- row region in photon energy at Ev= 14.30+0.02 eV where the da ta are still compat ib le with the D D M hy- pothesis. Moreover , it is also possible that photons from the decay (2) coming from the galaxy cluster which was observed by the telescope are substantial ly absorbed by neutral hydrogen along the line of sight [ 10 ]. In view of these considerat ions and the interest in the issues involved we consider it worthwhile to have a fresh look at the D D M hypothesis and to dis- cuss an e lementary part icle model where the require- ments ( 1 ) - ( 3 ) can be realized.

It is well known that l ifet imes of the order ~ 1023 S are not compat ib le with the s tandard model with massive neutr inos [ 7,1 1 ] and, indeed, a t tempts have been made to obta in ( 1 ), (3) within supersymmetry and a superstr ing inspired E 6 model [ 12 ]. In this let-

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Volume 271, number 1,2 PHYSICS LETTERS B 14 November 1991

ter we study the problem in the framework of the Zee model [ 13 ] which realizes lepton number non-con- servation by a minimal extension of the standard model.

Let us first recapitulate the generation of neutrino masses in the Zee model. This model is obtained from the standard model by the addition of an extra scalar gauge doublet ~2 and a charged scalar singlet h-+. Thus, the most general Yukawa couplings in the lep- ton sector are given by

c~z~= _ EFI ~l IR -- ff~F2~21R -F L T C - Ifo it2 Lh +

+h.c., (4) where L, lR denote the left-handed gauge doublets and right-handed gauge singlets, respectively. Assuming three lepton families the coupling matrices F~, F2, fo are 3 × 3 matrices in flavour space, fo is antisymme- tric because of Fermi statistics. In order to generate neutrino masses lepton number must be broken. In the Zee model there is explicit breaking by the term

h+g)~-02, with 02=iz2~'~, (5)

in the Higgs potential. Note that for such a term two different Higgs doublets are necessary.

For the following discussion it is sufficient and convenient to consider the case

F2 = 0 , (6)

which can be achieved e.g. by the symmetry trans- formation

L ~ iL , /R--'i/R,

01--}01, 0 2 ~ - - 0 2 , h + - ' - h + (7)

Spontaneous symmetry breaking by the vacuum expectation values ( 0 ° ) o = v a / x / 2 ( a = l , 2) leads to the mass eigenstates g~+, H + , H + in the charged Higgs sector which are obtained by the unitary transformation

(i0 0v0 , × w,, w, j H j. (8)

W~l W~/kH~ /

In this relation 0 + denotes the would-be-Goldstone

boson pertaining to the charged vector boson and v is obtained by

1 v2= Iv~ 12+ 1 U 2 1 2 - - ~ G F

4M 2 -- g ~ -- (246 GeV) 2. (9)

Gv is the Fermi coupling constant, Mw the mass of the W boson and g the SU(2) gauge coupling con- stant. The unitary matrix (W;j) diagonalizes the charged scalar states orthogonal to 0 +. Its off-diago- nal elements are induced by the term ( 5 ).

It is useful to choose a basis where the charged lep- ton matrix

UI ~= ~ F , (10)

is diagonal and positive. Then, by a common phase transformation on L and IR the matrixfo can be made real without changing ~/. This is possible because fo is an antisymmetric 3 × 3 matrix. In the following we will always assume to be in such a basis where 37I is diagonal and positive and fow > 0 for ~ < ~'.

At the one-loop level the neutrinos become mas- sive with an effective Majorana mass lagrangian

ef t T -- 1 vL~C M~, VL~, + h.c. Sm =½ Y, ( l l )

The mass matrix M~, is given by [ 13,14 ]

M w = e x p ( - 2 i ~ , ) f w (m 2, - r n 2 )

(L ~ '=e , ~t, ~) ,

with ~2

exp ( - 2i~,) fw

1 1 v2 M 2 - - 8 z c z x / ~ v v , r 2WTI W2, In ~ - J o ~ , . (12)

The phase q/is chosen in such a way that f~, > 0 for ~<~' . In eq. (12) terms of order m~/M~,2 have been neglected. Ml, 342 denote the masses of H +, H~-, respectively.

Taking advantage of the hierarchy of the charged lepton masses the neutrino masses are obtained ap- proximately as [ 14-16 ]

~2 Note that in the analogous equation of ref. [ 14] a factor 4 is missing on the RHS.

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m I -~ moas in 2c~,

m2~--mo--½ml, m3~--mo+½ml, (13)

where [ 16 ]

mo=x/lm¢~12+lm,~12 2 ~-m~F, 2 2 F =

IMpel IMpel c o s a = - - , s i n ~ x = - - ,

m o m 0

]M~.] (m,~2 fe, a = - - _ (14)

mo \m~] F"

In deriving the neutrino masses (13) we have as- sumed a<< 1. Consequently, we also get rn~ << m2 -~ m3. The neutrino mass eigenstates Z are given by

VL = U~ZL. ( 15 )

In view of a<< 1 it is convenient to split up U~ into

U~ = exp (i~,) U0 U~ ,~ ,

where

go=

COS O/ COS 0~" s i n a - x/~ x/~

sin o~ sin c~ - c ° s a - x/~ x/~

1 1 0

(16)

(17)

diagonalizes the neutrino mass matrix for a = O and UI corrects for finite a. In lowest order in a (and ne- glecting a sin 2 a ) we obtain

UI-~ 1+ ~ acos 2o~ 0 . (18)

0

Furthermore, .~ =diag( i , i, 1 ) is required by the pos- itivity of the physical neutrino masses.

Eqs. ( 13 ), ( 17 ) and ( 18 ) show that in this model one has short wavelength oscillations characterized by tim22 "-~ ~m23 = m g (am } = m 2 - m 2) whose main effect is to mix Ve with v, and v~ with a mixing angle o~. With mo-~28 eV present experimental data on neutrino oscillations give [ 17,18 ] sin 2a < 0.05. We also observe from eq. (17) and from a<< 1 that a--- ½ ~r. This is to assure that the electron neutrino is predominantly made up of the light mass eigenstate

ZI and thus v,, v~ are linear combinations mainly of the heavy Majorana states X2, Z3.

Now we turn to the decay rates which are approxi- mately given by

c~m3 [Fj, 12 (_/=2, 3) F(Zj-~)0 +y)--~ 512~4

with

Fji = 167l .2 ( UTDU)ji, U= Uo U1 .

The antisymmetric matrix D is defined by

D =fYt('+ C~f,

where

C = d i a g ( C ~ ) ,

and

1 C~ - In M2/M~

×~-{lnM2/m2-1~22 lnM2/m~-l)M~ "

(19)

(20)

After some algebraic manipulations one arrives at

167r 2 ~j-~ ~ - m~[ +fe~(C~ - c o s 2c~ C.)

- ½sin 2o~ F( C~ +G) ] (21)

where + and - correspond to j = 2, 3 respectively. In eq. (21 ) we will neglect the term containing ½sin 2c~ in the following, given the experimental constraints on this quantity, and thus we obtain

F(Z2 --4"~1 '~-'Y) ~-F(x3 ---~1 "~- ~/) ~ ~ ' - l . ( 2 2 )

Furthermore, in view of a_~ 1 lr we will replace cos 2c~ by - 1 for the numerical calculation. Taking

m2 --~ m3 --~ m o = 28 eV, (23)

r = 2 × 1023 s and M~, M2 in the range 50-120 GeV (with IMl -M21 >i 10 GeV) we obtain the result

fe, = (1.5-5.7) × l0 -16 e g - 1 (24)

Moreover, since a~-fe,m2/mo, one gets

a = 0 . 0 6 - 0 . 2 3 , (25)

which satisfies the criterion a<< 1. Finally, from eqs. (13) and (14)

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Volume 271, number 1,2 PHYSICS LETTERS B 14 November 1991

F _ ~ 8 . 8 × 1 0 - ~ 8 e V -~ and m ~ < 0 . 3 e V ( 2 6 )

are der ived .

It is easily seen tha t these va lues o f f w and cr are

compa t ib l e wi th all p resen t expe r imen t a l data. As a

m a t t e r o f fact the mos t s t r ingent l imi t s ar ise f r o m

/.t~e?, and/a-13 un ive r sa l i ty ~3. In bo th cases one gets

b o u n d s onfo~ , which by an a p p r o p r i a t e cho ice o f the

rat io I v2/Vll (see eqs. (8 ) and ( 1 2 ) ) can peacefu l ly coexis t wi th the results ( 24 ) , (26 ) o n f ~ , .

It should be stressed that the rea l i za t ion o f the

D D M hypothes i s in the m o d e l o f Zee has s o m e im-

pact on the cosmolog ica l consequences o f Sc i ama ' s

Ansatz . The reason is tha t the neu t r i no mass dens i ty

o f the un ive r se wou ld be roughly twice as large as the

va lue o b t a i n e d by S c i a m a since in the Zee m o d e l one

has two mass ive decay ing neut r inos . Consequen t ly ,

i f the m e a n dens i ty o f the un ive r se is the cr i t ical one,

the H u b b l e cons tan t wou ld be H ~ 80 km s - ~ M p c -

which is larger than Sc i ama ' s va lue and also the age

o f the un iverse wou ld be smaller . Th is migh t be a

p r o b l e m for the present rea l iza t ion o f the D D M hy-

pothesis . We wou ld also like to m e n t i o n that there is

no so lu t ion to the solar neu t r i no p r o b l e m in this s im-

ple scheme.

We conc lude that the Zee m o d e l o f l ep ton n u m b e r

n o n - c o n s e r v a t i o n can a c c o m m o d a t e M a j o r a n a neu-

t r inos wi th masses a r o u n d 28 eV and a l i f e t ime o f

r ~ 2 × 1023 s as needed for the D D M hypothes is . Pre-

v ious es t ima tes o f the lower b o u n d for the rad ia t ive

l i f e t ime o f a 30 eV neu t r i no gave va lues an o rde r o f

magn i tude larger in the Zee mode l [ 14 ] but they were

based on add i t iona l a s s u m p t i o n s which can safely be

released. It is well k n o w n that a m o n g the mode l s that

~3 Data from ~--. 3e and the anomalous magnetic moment of the muon are less restrictive; for analyses of the phenomenologi- cal constraints on the Zee model see refs. [ 11,14,19 ].

ex tend the s t andard m o d e l by inc lud ing add i t iona l

scalars the Zee m o d e l p rov ides the smal les t l i fe t imes.

There fo re , i f the D D M hypothes i s t u rned ou t to be

correc t this m o d e l wou ld be one o f the s imples t can-

d ida tes to expla in l ep ton f l avour n o n - c o n s e r v a t i o n in

this context .

We thank D.W. Sc i ama for useful c o r r e s p o n d e n c e

and we are grateful to H. Neu fe ld for helpful

discussions.

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