Big-bang nucleosynthesis constraints on light scalars and the 17 keV neutrino hypothesis

NUCLEAR

P HY S I C S B

Nuclear Physics B 387 (1992) 193—214 North-Holland

________________

Big-bang nucleosynthesis constraints on light scalars and the 17 keV neutrino hypothesis Stefano Bertolini Istituto Nazionale di Fisica Nucleare, Sez. di Trieste, c / o SISSA, Via Beirut 4, 1-34013 Trieste, Italy

Gary Steigman Department of Physics, The Ohio State University, 174 West 18th Avenue, Columbus, OH 43210, USA Received 6 April 1992 Accepted for publication 23 June 1992 In this paper we present a detailed discussion of the Constraints that the primordial abundances of light elements impose on the interactions of light scalars which are relativistic at the time of nucleosynthesis. We discuss the implications of our results for those “17 keV neutrino” models which invoke the presence of Goldstone bosons in order to provide a fast decay or annihilation mechanism for the heavy neutrinos. Although our results can be applied to a variety of cases, we use for our discussion the “invisible-majoron” scenario. Unless an unnatural tuning of the parameters is considered, we find that the scale of spontaneous breaking of lepton number is bounded to be above 19 GeV. While the possibility of a fast decay of the heavy neutrinos into majorons is allowed, we can confidently exclude the annihilation into majorons as a means of depleting the cosmological abundance of stable 17 keV neutrinos.

1. Introduction In recent years the particle-physics—cosmology connection has provided a valuable tool for studying both high-energy physics and the universe. Astronomical observations, from stars to the universe at high red-shift, have yielded useful constraints on particle-physics model building, while accelerator data have constrained models for the structure, evolution and matter content of the universe. Primordial nucleosynthesis occupies a unique position in this symbiotic relationship, offering a quantitative probe of early-universe physics accessible to observations. Nearly 15 years ago, Steigman, Schramm and Gunn [11pointed out that the observed abundances of the light elements helium in particular provided a significant limit to the number of species of light (~1 MeV) neutrinos (or “equivalent” particles). Using the best current nuclear-physics data and astronomical observations, Walker et al. (WSSOK) [21have derived the most stringent bound to date from big-bang nucleosynthesis (BBN), —

~ 0550-3213/92/$05.OO © 1992

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Elsevier Science Publishers By. All rights reserved

—

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This cosmological bound is in excellent agreement with the beautiful LEP data on Z° decays [3], N~m~=3.OO±O.lO,(95%c.l.).

(2)

The consistency between the accelerator results and the cosmological inference from astronomical observations provides support for the utility of the particlephysics—cosmology connection. The accelerator and cosmological results are, however, complementary in that the measurement of the width of the Z° proves even massive particles (< ~m~) which, however, must couple to the Z° with a strength comparable to weak, while BBN is sensitive to all even superweakly interacting particles, provided they are light (~1 MeV). For example, while BBN would be unaffected by 1 GeV particles, it could “see” very light particles which couple too weakly to be found at LEP. Thus, although new heavy particles may be discovered at current or projected accelerators, BBN offers the most promise for revealing the presence of light particles which are “invisible” on the weak scale. Recently, interest in light, weakly coupled scalars has been stimulated by model building attempts to accommodate a possible 17 keV neutrino (for a comprehensive review and a list of references see ref. [4]). Since this mass falls into the cosmologically forbidden window for stable neutrinos (from about 40 eV to a few GeV), one of the challenges that theorists have to face is to provide scenarios for a fast enough annihilation and/or decay of the “heavy” neutrino into lighter ones or other “invisible” exotic particles. Since flavour-changing standard electroweak interactions are too weak for this purpose, an extension of the standard electroweak scenario is called for. Most of the models proposed so far invoke the presence of a physical Goldstone boson (residual of the spontaneous breaking of some global symmetry), to which the 17 keV neutrino decays or annihilates. In the majority of the cases, the global symmetry is identified with lepton number and the resulting Goldstone boson is the “majoron” (originally introduced in ref. [5]). In other cases a global family symmetry is imposed on the lepton sector, and the corresponding Goldstone boson is called the “familon” [61. In all cases, to satisfy the LEP constraint, the massless scalar must be made sufficiently “invisible”. This is most simply achieved by requiring its main component to be a weak singlet (an exception is provided by the hyperchargeless triplet majoron of ref. [7]). In this paper we will study the constraints that BBN imposes on the coupling strength of the new light fields. Since the massless Goldstone boson belongs to a complex field, it generally appears together with an additional neutral scalar whose mass is naturally of the order of the breaking scale of the global symmetry considered [81. As we will see, the presence of this massive “companion” plays a crucial role in the analysis of the BBN constraint. Although we will direct our discussion to the case of the singlet majoron, due to the universality of the form of the couplings of Goldstone bosons, our results can be cast in a model-independent —

‘~

—

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195

form and can easily be applied to different scenarios. We will discuss some explicit examples in what follows. Our main conclusion is that meeting the BBN constraint of eq. (1) requires, for a natural range of the parameters, the new spontaneousbreaking scale (SBS) to be above 19 0eV. By allowing a tuning of at least 8 orders of magnitude between the SBS and the mass of the neutral scalar partner of the majoron, when lighter than 1 0eV, the SBS bound may vary between 1 GeV and 100 TeV depending on the value of the scalar mass. In all cases, annihilation into majorons is ruled out as a mean of depleting the cosmological density of 17 keV neutrinos. In the same scenario one may exclude the existence of any heavier stable neutrino up to a mass of 550 MeV. The paper is organized as follows. In sect. 2 we briefly review the physics behind the BBN bound to N~and compare the sensitivity of LEP to that of BBN in bounding the couplings of “new” light particles. In sect. 3 we use the BBN constraints to study the consistency of models for the 17 keV neutrino which invoke the presence of an “invisible” majoron (and its massive neutral scalar companion). A summary of the results and some final considerations are then presented in sect. 4.

2. Big-bang nucleosynthesis and light particles The early universe is “radiation dominated” (r.d.); that is, the total energy density is dominated by the contributions from extremely relativistic (e.r.) particles. Any “light” (m < T) particle will contribute to Pe.r. which may be normalized to the contribution from photons alone: Pe.r.

TB

1

4

-~-

B

7

~

+~(~g~)

y

F

TF

y

4

(3)

y

In eq. (3), g~,is the “effective” number of degrees of freedom; ~B(F) the number of helicity states for each boson (fermion) and TB(F) the corresponding temperatures. The primordial yield of 4He is sensitive to the universal expansion rate at T~ 1 MeV and, thereby, leads to a bound on Pe.r(T 1 MeV) since a p~/~a ~ For the “standard” case of three, light, two-component neutrinos, at BBN, TBBN

=

T~,

g~.=3Xg~=6, gB=g~=2.

g~=4, (4)

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As a result, for the standard case, (5) It is convenient to express the contributions from any “new” particles in terms of the effective number of “equivalent” (light) neutrinos: g,~=~+~N~ Thus, ~

(6)

z1N~=N,,—3.

“counts” all sufficiently light new particles (fermions and/or bosons)

~

+‘(~~B)(~)

F

P

BBN

(7)

.

B

~

BBN

The contribution to ~ of any new particle is weighted by the fourth power of the ratio of its temperature to that of the photons at BBN. A particle which couples (to e v1) as strongly as do the usual neutrinos, will have TF(B) 7~at ~,

=

BBN and, therefore, will contribute with full weight. Such particles are constrained by the powerful LEP bound of eq. (1). However, a particle X which couples more weakly than the usual neutrinos may evade the LEP constraints but, still “count” in the BBN constraint. Such a particle would have to become decoupled earlier in the evolution of the universe than the usual neutrinos (TD(Pe) 2.3 MeV, TD(v~) TD(r’~) 3.3 MeV) and, due to the “heating” of the neutrinos when massive particles (~t±, ~.± ~O, ~ decay and/or annihilate, be cooler than the usual neutrinos at BBN ((TB(F)/T},)BBN < 1). How much cooler depends on the number of interacting degrees of freedom g1(T) at the temperature of decoupling. Entropy conservation leads to the relation =

T~

—

BBN

43/4 g1(TD)

8

1/3

‘

(

where henceforth TD TD(X), unless otherwise specified. In this manner, new X-particles unconstrained by LEP may still contribute, albeit with less than full strength to the BBN bound to ~ —

—

2.1. THE CASE OF ADDITIONAL LIGHT NEUTRINOS

As an example, through which we will set the basic framework for our analysis, consider the case of N “new”, two-component neutrinos which have the same but weaker than weak interactions so that, —

—

~ B

~ F

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400

100

TD

1000

(MeV)

Fig. 1. The decoupling temperature (TD) required for N additional “equivalent neutrinos” to satisfy the BBN constraint ~ ~ 0.3. Two choices for the temperature of the quark—hadron transition are shown.

From eqs. (7) and (8) it follows, then, 43/4 BBN

~

(10)

\

(7,

~

Eq. (10) may be thought of as constraining TD as a function of N/L~N~. For the BBN bound of 4N. ~ 0.3, the N versus TD relation is shown in fig. I which is, in reality, a plot of N/0.3 versus 1’D; or, equivalently, a plot of g 1(T~)versus TD, through: 43 N(TD) gJ(T~)=—~— 0.3

We can, therefore, easily scale N N’

—~

=

~

(11)

N’ for any bound on ~

by:

(~N~/0.3)N.

(12)

The two curves in fig. 1 reflect the uncertainty in the temperature of the quark—hadron transition. Two choices are shown: Tq11 150 MeV and 400 MeV. Since the least restrictive bounds follow from TqH 150 MeV, the subsequent results will refer to that choice. For N 1, one “extra” species of neutrino, it can be seen from fig. 1 that TD> 150 MeV in order that the L1N~~ 0.3 constraint is satisfied. The constraint on TD can then be translated into a bound on the interaction strength of the new neutrino. For this purpose we find it convenient to compare the v x (production) cross section with that of a standard massless ~ neutrino, which allows us to =

=

=

=

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relate TD(X) to TD(I.~T) 3.3 MeV ~. Since the decoupling temperature is found with the production/annihilation by comparing the expansion rate (H ag~~2T2) rate (F~ ~ by defining =

Fx(TD)~

(13) ‘~‘

T 0(X)

5, one finds

**

and recalling that F~a T Fx(TD)

=

g~(T~) 1/2 3.8MeV ~ 43/4 TD

(14)

Eq. (13) can then be conveniently written as n

~

E I3ij~7~jL’)TD=<0eVV>TDFX(TD), i,j,X ~e’~

(15)

~X

where the sum extends to all “new” particles X which are relativistic at the time of nucleosynthesis (TBBN 1 MeV), and to all production processes which involve as initial states the particles i and I, chosen among those species that are still interacting at i’D> TBBN. All terms are evaluated at the decoupling temperature TD of the particle(s) X. This temperature can be determined by inspection of fig. 1 after scaling the “new” degrees of freedom in terms of “equivalent” neutrinos (a factor has to be included for every boson field, according to eq. (7)). The coefficient f3~counts the number of particles X produced in a given process. For instance, in the case of e~e—÷ ~ f3~ 1, whereas if a pair of majorana neutrinos r’,,~ is produced, then 13~X 2. As usual, the temperature-dependent densities n, count the number of helicity states g. of the particle i (for example, in eq. (15), g~ 1, g~ 2, so that 4.). It is worth noting that when the annihilation of a pair of majorana particles (i =1) is considered then n,n~must be replaced by 4n,~(number of i-pairs). A final remark is also quite important in order to make correct use of eq. (15) 3, (F~H). Let us assume that T’~.Since for a relativistic particle naT and H a T2, it is then obvious that for a> 1, F> H when T> TD. Conversely, for T < TD, F < H; TD is therefore the temperature at which the particles X decouple from thermal equilibrium. However, for a < 1 the opposite situation

4

=

=

=

=

~p/~e

=

—

—

*

**

Slightly different values may be found in the literature. We have derived this result consistently with the definitions and procedure here outlined, considering the processes e~e —* ~ and ~ -~ The thermally averaged cross sections include a 6% suppression effect due to Pauli blocking on the final states. We normalize the equilibrium condition to
/

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199

occurs, and TD has, rather, to be interpreted as the temperature at which the particles X enter into equilibrium. In the latter case we will denote the equilibrium temperature by TEQ and require that TEQ < TBBN, independent of the nature and the number of the particles X. Since we find KU~°~V)T 0.17G~T2, the right-hand side of eq. (15) is fully determined once g~(T~) is known. Note that g~(T~)=g 1(T~)+g~ for bosons and g~(T~)=g1(T~)+ ~gx for fermions. For our example of N additional two-component neutrinos, g~(T~) gJ(T~)+ 4N, so that, using eq. (11), we can write =

43

N

3/4

(16) where TD TD(N) is obtained from2 fig. 1.1.62, Thus, sofor thatN 1 we find TD and, correspondingly, [4g~(T~)/43]” N=1: FIP(TD)<2.6X105. =

Analogously, for N

=

=

=

150 MeV, (17)

3, TD ~ 360 MeV, [4g * (TD)/43]~’2 2.47, and one obtains =

N=3:

F

6. (18) 3~(TD)<2.9X10 Notice that Fx(TD) is a measure of how much “weaker than weak” should be the interaction strength of the new particles X, in order to comply with the BBN constraint. For instance, if we were to assume naively that F~ (M~/M~)4we would obtain N=1:

M~~/M~14,

N=3:

M~/M~24.

(19)

Let us recall that in determining the decoupling temperature TD from fig. 1, we have conservatively considered the lower quark—hadron transition temperature, 1~II-I 150 MeV. A higher ~ leads, in fact, to a more severe constraint on the new interaction strength, as the second curve in fig. 1 (TqH 400 MeV) shows. Having learned how to translate the BBN constraint into a quantitative bound on the “new” interaction strength, we now turn to the interesting case of light exotic scalars. 2.2. LIGHT SCALARS: BBN VERSUS LEP

In the case of N~one-component scalars we have gx =n~and g~,=g 1 +N~. We can then write the analogue of eq. (16) as g~(T0)

43 =

4 N~ ~-~)

3/4

+N~.

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Nucleosynthesis constraints

1.31 and 1.71, where TD For N~ 1 and 2 we have [4g~(T~)/43]l/2 160 MeV, respectively. In analogy with the neutrino case we find =

N~ 1:

Fls(TD)

Ns2:

F

=

—

100 and

7.1 x

5. (21) 2s(TD)~2.3X10 Notice that eq. (21) hold if, for the dominant mode of production of X, one finds Ka~v) T” with a> 1. Otherwise, following the discussion in subsect. 2.1, when a < 1 we must evaluate eq. (15) (and therefore Fx) at T ~ 1 MeV, for any number of relativistic degrees of freedom. Let us now apply these results to some explicit cases of theories which exhibit light scalars in the particle spectrum. A typical class is represented by models in which the spontaneous breaking of some (non-anomalous) global symmetry leads to the presence of physical Goldstone bosons in the scalar sector. If we identify the global symmetry with U(1), of lepton number, the massless scalar excitation that is left after spontaneous breaking is called a “majoron”. This name was introduced by Chikashige, Mohapatra and Peccei (CMP) in ref. [51.In their model the presence of a complex singlet scalar field, transforming non-trivially under U(1)(, is responsible for the “dynamical” generation of a majorana mass entry of the right-handed neutrinos in a see-saw scheme for neutrino masses. We shall refer to this model as the singlet majoron or CMP model. Schemes in which majorana mass entries for the left-handed neutrinos are spontaneously generated without introducing right-handed components were subsequently proposed. For this to happen, one must generally require that the majoron belongs to a scalar multiplet which transforms non-trivially under the electroweak group. The first and simplest example of this class of models was proposed by Gelmini and Roncadelli (OR) in 1981 [9] via the introduction of a complex scalar triplet carrying hypercharge and lepton number. We refer to this model as the triplet majoron or OR model. More recently, a minimal Higgs extension of the SM in which the majoron belongs to an electroweak doublet has been proposed by Bertolini and Santamaria (BS) [10]. At variance with the GR model, no triplet Higgs is needed and majorana neutrino masses are generated radiatively according to the Zee scheme [11]. We shall refer to it as the doublet majoron or BS model . Although a number of variant majoron scenarios have been proposed in the last ten years (for a review see ref. [14]), for our present discussion we may confine ourselves to the abovementioned three schemes. A crucial difference between the CMP model on one hand and the GR (BS) model on the other is that in the latter class of models, in which the majoron transforms non-trivially under SU(2)L x U(1)~,there exists a direct coupling of the —

—

*

A supersymmetric scenario in which R-parity breaking gives rise to a doublet majoron was first discussed by Aulakh and Mohapatra in ref. [12] and more recently by Santamaria and Valle [13].

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majoron to the Z° boson The gauge coupling to the Z°,due to its derivative nature, actually involves the majoron (pseudoscalar) together with a scalar partner. In fact, it is worth remarking that since the majoron belongs to a complex field its presence is generally related to that of an additional neutral scalar, henceforth referred to as p. A remarkable property of this particle is that its mass is bounded from above by the scale of spontaneous breaking of lepton number [8] (m~ generally turns out to be proportional to the square of the lepton-number-breaking vacuum expectation value (vev) via a quartic coupling in the scalar potential). A sharp bound on the lepton-breaking vev (and indirectly on the p-mass) can be obtained from astrophysical arguments related to energy loss in helium-burning red-giant stars. In fact, too large a production of majorons in the interior of the star would lead, through their escape, to an excessive cooling of the core. This would prevent helium ignition and change the stellar evolution. In red-giant cores, majorons (henceforth denoted by J) are mainly produced via the process y + e e+ J. A direct majoron coupling to electrons can proceed only through the presence of a doublet-Higgs components in the physical majoron field, and it is therefore bounded from above by the standard Yukawa coupling. Since the Higgs fields responsible for quark and lepton masses do not carry lepton number, they do not mix with the majoron field if lepton number is conserved. As a consequence, if such components are present, the mixing coefficient must be proportional to the lepton-number-breaking vev, henceforth denoted by v. We come, therefore, to the conclusion that a bound on the majoron—electron coupling gjee, when present, reduces to an upper bound on v. The analysis of the red-giant constraint gives gjee < 1.4 x iO’~ [15]. This limit corresponds for the OR and BS models to a bound on v (and consequently on m~)of the order of 10 keV. Notice, however, that in the CMP scheme gjee 0 at the tree level, due to the singlet nature of the majoron field, and the astrophysical bound is not effective. In the OR and BS models the presence of the massless majoron and the light p allows for a new contribution to the “invisible” decay width of the Z°gauge boson, namely Z° Jp. The strength of this mode is proportional to the value of the hypercharge of the scalar multiplet to which J and p belong. In the OR model, F(Z° Jp) turns out to be twice a large as the Z° decay width into a pair of massless neutrinos. In the BS model F(Z°—s Jp) is four times smaller (the hypercharge of a doublet is half that of a triplet) and therefore equivalent to 0.5 F(Z° v1). Given the present LEP bound of eq. (2), both models are now ruled out. Suppose, however, that the physical majoron field is an admixture of singlet and non-singlet components. Then the contribution to the invisible Z°width depends ~.

- —~

=

-+

—~

—

*

A remarkable exception is provided by the hyperchargeless triplet-majoron model of ref. [7].

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Nucleosynthesis constraints

on the weight with which the non-singlet components appear in the mass eigenstate. The LEP bound then becomes a limit on such mixing. By writing ~z0Jp =a~0Z~(J~p),

(22)

we obtain F(Z° F(Z°

—*

Jp)

—~

v1)

2. =

(23)

2a

Note that a 0, 4, 1 corresponds to a pure singlet, doublet, triplet majoron, respectively. Assuming that m~~‘~z M~,the bound of eq. (1) then becomes =

a2<5x102,

(24)

thus excluding, as we already know, pure doublet or triplet majoron states. We now consider the constraint that arises from BBN. Under the assumptions that (i) m~< 1 MeV (the p-particle is relativistic at the time of nucleosynthesis), and that (ii) s-channel Z° exchange is the most efficient mode of production of J and p through neutrino (and electron) annihilation, we obtain —s

Jp)

=

—‘

Jp)

=

G~s —a2, l2ir

/

(25)

G~s 1.0064—a2. 24ir

(26)

These cross sections should be compared to G~s

-

o-(ee —s 1.0064-~——. Summing over three massless neutrinos, application of eq. (15) gives =

~5a2

KUe~TOUP~~~V>TD

where we used for N~ 2, TD =

<

1(nJ+n~)

(28)

4 and 13’f 2. Since (n~+ n~)/n~ 160 MeV, we finally obtain (see eq. (21))

~p/fle

=

=

a2<6X106,

(27)

=

~,

and recalling that

(29)

a bound about four orders of magnitude stronger than that obtained at LEP, albeit with different assumptions involved.

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This example shows how important the BBN constraint can be in limiting the properties of light particles which may escape detection at present high-energy colliders. In the next section we utilize this approach for the analysis of the implications of the nucleosynthesis bound on proposed scenarios for the existence of a 17 keV neutrino.

3. Invisible majoron models and the 17 keV neutrino In sect. 2 we have seen that accelerator and cosmological constraints considerably limit the strength of the majoron coupling to the Z°.A way to circumvent this constraint is to require that the leading component in the physical Goldstone field is an electroweak singlet, so as to make the majoron sufficiently “invisible.” The simplest realization of this approach is the original CMP model. For the purpose of our analysis we will refer explicitly to the singlet-majoron case, although our results hold for a larger class of invisible-majoron scenarios. Borrowing some of the notation from ref. [16], the part of the lagrangian involving J, p and their interactions with the chiral neutrino fields NL,R can be written as .~(J,p,

~)

=

4(a~p)2+ 4(1

+

~

—{NRM 2NL +

4(1

+ ~)NRMI(NR)C

+

(30) where J and p belong to a scalar singlet u defined as (31) 2 2A 2coupling in the scalar and m~/vThe 3 3,x 3the latter being the quartic (~~*) potential. matrices M 1 and M2 respresent the majorana and Dirac entries in the 6 X 6 neutrino mass matrix. In a see-saw scheme for neutrino masses one assumes that the elements of M1 are much larger than those of M2, which are typically of the size of ordinary fermion mass scales. As a consequence, the diagonalisation of the neutrino mass matrix leads approximately to three heavy =

5. Bertolini, G. Steigman

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/ Nucleosynthesis constraints

majorana neutrinos, mostly singlets, and to three light states, mostly doublets, with mass matrix M~M2. Considering only the light-neutrino sector, relevant to our discussion ~, the interaction terms with J and p can then be written in terms of the three light-majorana mass eigenstates and their masses m1 as i.’,

1 —

—

~

~—p(m~i~v~

—

—X,1(m,

~

+

m1)i~t.~,) + h.c..

(32)

The mixing factors X,~depend on the detailed form of the matrices M1 and M2. In general, for M2 <
~(~j)2

~&~j2

1

m —~

*

~-~-J(iy5v).

(33)

(34)

We thank J. Cline for pointing out to us that the heavy-neutrino sector may play a relevant role in a limited region of the parameter space. The results and conclusion of the present analysis are however unaffected.

S. Bertolini, G. Steigman V

~-

T

_____

——

J.p

——

J,p

/

Nucleosynthesis constraints

,-J,p

—~---~

N

\J,p

P

(b)

(a)

205

N

(c)

Fig. 2. The neutrino annihilation processes responsible for J, p production.

Notice that these results can also be obtained by using the equations of motion and discarding vanishing contributions to the action. At any rate, the consistent use of one parametrization or the other leads to identical results, although intermediate steps and partial contributions to a physical amplitude may differ substantially in the two cases and can lead to misleading estimates. The calculation of o~(r’v JJ) from the diagrams in fig. 2 gives the following (exact) result —‘

—*

JJ)

=

%/1—4y 1 rn2 64ir v’~ (x_1)2+y2 2y(x2 1 + l—4y —

x

1+

1 ~Il—4y

~2)

1 + ~Ii —2 1—~/1—4y —

ln

(35)

in agreement with ref. [16]. In the previous equation y m~/s, x s/m~ and y F~/m~, where F.~, is the p-width. For m~~ m~,F,, is dominated by p JJ leading to T, m~/(32~v2). Thus, since rn, ~ v, one obtains y < 102. We can, therefore, neglect y2 in the factor in the curly brackets, but its presence in the denominator of eq. (35) is crucial for the case of resonant p-production (x 1). Due to the smallness of when thermally averaging the cross section we will replace, at resonance, [(x 1)2 + y2]—’ by (ir/y)8(x 1). As a final remark, we note that for s — 4m~the term in curly brackets reduces to 1 + 4(x2 1), and the cross section vanishes as ~/~L-4y. In order to implement our result in eq. (15), we must evaluate the thermally averaged cross section Ko~v>T. The analysis is conveniently done by separately considering the following kinematical regions: (a) m~>> s; (b) rn,2, s; (c) rn,2 <
~2

—

—

—

1rn~ ~oo,

(36)

—~(x —1),

(37)

——~-

64~r

V

IT —~

JJ)

‘I

1 —~

JJ)

2o0y In— —2

.

(38)

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/

5. Bertolini, G. Steigman

In case (c), however, the process vv same limit we obtain cr~(r’v—

Jp)

—

Nucleosynthesis constraints

Jp and r’v

—‘

pp are also allowed. In the

1

4o-

0y ln— —2

,

~@“—~ic) ~2u0y.

(39)

(40)

In passing, it is instructive to remark that by using the lagrangian in eqs. (30) and (32), obtained with the exponential parametrization of eq. (31), the limit (a) is recovered computing only the first two diagrams in fig. 2. The opposite situation occurs if one uses the linear parametrization. In this case, the calculation of the first two diagrams gives in fact the limit (c). This behaviour is easy to understand, by looking 2atcoupling the thirdis diagram in thetofigure (p-exchange), weitrecall that m,2/v. in the proportional s/v, whereas in theifl.p. becomes e.p. the Jp The above conclusions follow by considering that in both cases the p-propagator (s m~)~multiplies the coupling. This simple example justifies our warnings on the careful and consistent use of the two parametrizations. Corresponding to the three kinematical regions above, by thermally averaging the cross section we find the following limiting expressions: For x~ rn,2/T2>> 1 one simply obtains —

(41) For

0(1) we have V2 (O-~.]V)T~O

2~I(X~),

(42)

0IT

where 1(x~) dx exp[ —x (x,~/4x)]. Numerically, i(x~)obtains a maximum value 1(2.4) 1.16, falls off exponentially for x,,>> 1 and scales as x~ for <<1. Finally, in the limit x,, ~z 1 (rn~4z T) we find =

—

rn2 T2 T7~0.208o-o-~3ln—~—0.522

(43)

KJ,~’V)T=2KU~PV)T,

(44)

rn2 Ko~f’V)T~O.2O8 o-o-~,

(45)

We can now apply these results to the analysis of the BBN constraint. The first important remark, from inspection of eqs. (41)—(45), is that for x,, ~ 3.4 (1,, is

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the point on the resonance at which TxI’(T)/1(T)~ —x~I’(x~)/1(x~)= 1) the thermally averaged cross section behaves as T~with a < 1, whereas for x,, > xp a> 1. Recalling our discussion in subsect. 2.1, for x,~,<~ we must require that the majorons enter into equilibrium at a temperature ‘~q < 1 MeV, i.e. after BBN has begun. On the other hand, for x~> ~,, we require that J decouples at a temperature TD> 100 MeV. With this in mind we begin by analysing the case rnp>> 100 MeV, where the only additional relativistic degree of freedom at nucleosynthesis is the majoron. Applying eq. (15) one obtains —

—

(46)

< (O-C~”V)TDF!S(TD).

4KO~,],’V)TD

In order to understand the factor 4 we should recall that for a majorana neutrino g~ 2, and therefore we have n~/n~ 4 and flp/fle 1. In addition, n~ñ,,must be replaced by 4n~(number of pairs). Together with f3~, 2, this gives the factor 4. Notice that when considering the case of a pseudo-Dirac 17 keV neutrino, the presence of two (almost degenerate) majorana neutrinos cancels the factor 4 in the square of the densities and gives correctly the result expected for a Dirac neutrino of the same mass. Implementing in eq. (46) the results of sects. 2 and 3, one readily finds =

=

=

=

v> 6.06~/l7keV T~ 0eV

where obtain

TMeV

is the temperature in units of MeV. For TD

(47)

=

100 MeV we therefore

v> 19.2 0eV.

(48)

In the case of a pseudo-Dirac 17 keV neutrino the bound is increased by 2 1/4 to > 22.8 GeV. In the limit mp 1 MeV our master equation reads V

<<

4{Ko~V>+

4Ko~v) +

(O;f~PV)}T~q <


(49)

2s(Teq).

The relative factor 4 among the cross sections arises from n~+ n,, 2: 2n~.In terms of the neutrino mass and the temperature we obtain for m~~ T =

1 T v > 0.75 T~J~IJ [ln__]

1/2

m (17 keY) 0eV

=

(50)

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‘~

/IQOMV

io~~’ io~ 1o~10’ 10’ 10’ 100 mp

10’

102 10’

~

10’

(Mev)

Fig. 3. The v—mp plane. The region allowed by the BBN constraint ~N, o~0.3 is above the solid curve. Above the dash-dotted curve the J, p are never in equilibrium during the early evolution of the

universe.

By requiring ‘~q< 1 MeV we finally have V>1.O7GeV

(51)

Once again, in the case of a pseudo-Dirac 17 keV neutrino, one obtains V > 21/4 x 1.07 GeV 1.27 0eV. In the intermediate region 0(1 MeV)
~‘.

V>

/1(x) 115~/

V

“

TMeV

____

TeV.

“

(52)

l7keV)

Notice that much stronger bounds are obtained in this case. This fact depends on two factors: (i) the resonant enhancement of the cross section, and (ii) the quadratic dependence of the cross section on For a pseudo-Dirac 17 keY neutrino, the resonant bounds have to be scaled upward by a factor f~. All the results for one 17 keV majorana neutrino are summarized in fig. 3. The solid line indicates, as a function of m~,the lowest value of allowed by the BBN constraint L1N~~ 0.3. The dashed line on the left shows how the limit changes by requiring Teq <0.1 MeV. We have to remark that in drawing the convolution of V.

V

*

The possibility that the lepton-breaking scale is below TBHN is forbidden by laboratory limits on leptonic kaon decays which involve a majoron in the final state [17]. For a 17 keV neutrino one finds v >1 MeV [18].

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the resonances for T in the 1—100 MeV range, a constant value of g4(100 MeV) has been used. Since g~(1MeV)/g4(100 MeV) 2 and Fx ~ this approximation underestimates the bound on by 20% on resonance, and by 5% in the limit (c). The dot-dashed line on the right extrapolates the convolution of the resonances for T> 100 MeV, and separates the region in the mr—v plane where majorons never enter thermal equilibrium (above the line) from the region in which they have entered and leftwhere thermal at temperatures higher than 100 MeV. 2 A, A is equilibrium a quartic coupling in the scalar potential, considerSince rn,2/V ing values of rn,, amounts to assuming very small values for this scalar V

—‘

~‘4Z V

coupling. Since A 0 does not increase the symmetry of the lagrangian, values of A smaller than, say, 10_2_10_3 should be considered unnatural (unstable to quantum corrections). As a consequence, from the results shown in fig. 3 it is obvious that only the region in the rn,,—~ plane corresponding to rn,, > few 100 MeV can be considered natural. With this in mind, our results can be summarized as follows: unless a fine tuning in ma/v of at least 8 orders of magnitude is accepted, the BBN constraint corresponding to z1 N~< 0.3 implies, in the invisible majoron scenario, a lower bound on the scale of lepton-number breaking of about 19 0eV (23 0eV for a pseudo-Dirac 17 keV neutrino). In the next section we will discuss the implications of these results for specific invisible majoron scenarios which have been (re)considered recently in the literature. =

4. Discussion and conclusions

Among the questions that the existence of a 17 keY neutrino raises is the problem of its relic density. The question is how to deplete its abundance consistently with the limits on the mass density in our present universe without affecting the black-body spectrum of the background radiation. A 17 keY neutrino cannot be stable, but at the same time its decay products cannot be too visible either. Since decay and/or annihilation mechanisms mediated by the known electroweak forces turn out to be unsuitable for this purpose, one is lead to devise new scenarios in which new interactions and exotic particles are involved. One possibility is, in fact, invoking the presence of massless physical scalars, residue of the breakdown of some global symmetry, which are sufficiently coupled to neutrinos but, which interact very weakly with ordinary matter. The invisible-majoron scenario is an attractive possibility, since it is also directly relevant for the problem of neutrino mass generation. There are two possible ways of diluting the primordial abundance of neutrinos: through decay or through annihilation. In the first case one needs lepton-flavourviolating interactions between majorons and neutrinos. This depends strictly on the

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/ Nucleosynthesis constraints

relation between the neutrino mass matrix and the Yukawa interactions responsible for its generation. If they are merely proportional, no off-diagonal majoron— neutrino interactions arise and, as a consequence, a tree-level neutrino decay involving the majoron is forbidden. In a singlet-majoron scheme with the see-saw mechanism for neutrino masses, the Dirac entries of the neutrino mass matrix have nothing to do with the presence of the majoron which is coupled solely to the right-handed neutrinos. Within this scenario one may therefore expect, after diagonalization of the mass matrix, the presence of lepton-flavour-violating couplings between neutrinos and majoron. In order to accommodate the Simpson 17 keY neutrino, a simple modification of the singlet-majoron hypothesis has been first proposed by Glashow [191. The model addresses specifically the second major issue related to the existence of a majorana 17 keY neutrino having a 10% mixing with the electron neutrino, namely its too large contribution to neutrinoless double-a decay (for a review on this and related topics see ref. [4]). Glashow observes that the presence of a zero eigenvalue in the majorana mass matrix of the right-handed neutrinos is enough to ensure the presence of two degenerate majorana 17 keV neutrinos with opposite CP eigenvalues. Their contributions to neutrino-less double-13 decay cancel, thus solving the problem. At the other end of the spectrum, a typical example of an invisible majoron scenario in which the decay mechanism is absent at tree level is given by the triplet—singlet majoron model of Choi and Santamaria [181.This model is a simple modification of the triplet majoron model of Gelmini and Roncadelli. In ref. [18] a singlet component for the majoron field is added to the triplet in order to weaken its electroweak interactions. At variance with the Glashow proposal, the majoron couplings to neutrinos are flavor diagonal, and no right-handed neutrino components are required. Although the structure of the model differs substantially from that of the singlet majoron with see-saw, it is easy to check that the various J- and p-couplings to neutrinos given in ref. [18] coincide with those given in eqs. (30)—(32). As already mentioned, this is a consequence of the universal character of Goldstone boson couplings. Although a plethora of different models for the 17 keY neutrino have appeared in the last year in the literature (see for instance ref. [41)for our discussion it is enough to consider the two abovementioned models. They are representative of two classes of scenarios in which the relic abundance of the 17 keV neutrino is depleted via (i) decay involving, as final products, light neutrinos and majorons (class I) or (ii) annihilation into two majorons (class II). For what concerns class I models, our results have little to say. An estimate of the neutrino lifetime in the modified singlet-majoron scheme as reported in ref. [191gives T

16ITM~ V Y~ 17 keY ~1.5 x 2 4ITg2 300eVj I grn, rn, 5

106

s.

(53)

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In obtaining eq. (53) we used the relation M1 gv, where is the singlet vev and g the Yukawa coupling of the scalar singlet to right-handed neutrinos. Since galaxy formation, which provides the most stringent constraint, requires for a pseudo-Dirae 17 keY neutrino ~ ~ 10~s [20], we eq. (53) problem arises as 2(V/30 0eV)4 ~ 0(1). Thissee is from consistent withthat ournoresult v> 23 GeY, long as 4ITg obtained in the most natural region of the mr—V plane. Let us also recall that =

V

demanding only that the present relic abundance is consistent with the closure density of the universe requires ~ 1011 s, a much weaker constraint, albeit less model dependent. Instead, our results are quite relevant for class II models where annihilation into majorons is required in order to solve the relic-density problem for the 17 keV neutrino. In a more general context, annihilation of massive neutrinos into majorons has recently been reconsidered by Carlson and Hall [21] in connection to the dark-matter problem. The authors of ref. [21] consider the singlet CMP scenario and show that neutrinos with mass in the range 1 keY—35 MeY may provide the missing matter in our present universe if they have a large enough cross section for annihilation into majorons. They show that this result is achieved for lepton-number-breaking scales of 0(0eV). Neglecting factors of order unity they find T~

V~ieV

m,,

(54)

MeV~~

Using (1 ~ 1 and bounding the age of the universe to be t 0 ~ 10~° years, constrains 2 ~ 0.43. Inserting this value in eq. the Hubble parameter (h ~ 0.65) so that Qh (54), for a 17 keV neutrino one finds V 100 MeV, which is excluded by our bounds. On the other hand, if we keep this scenario and assume ~ 19 0eV, we obtain rn,, ~ 550 MeV, thus excluding even as a possible dark matter candidate in this scenario. In class II models a strong constraint on the lepton-breaking vev comes from the observed diffuse photon background as pointed out by Choi and Santamaria [18]. They note that, considering the electroweak-induced radiative decay r’ — r”-y for 17 keV neutrinos, an X-ray peak at E 8.5 keY, coming from halo neutrinos in our galaxy, should be observed. Comparison with the recent measurements give a bound on the present-day energy density of 17 keV neutrinos equal to 0.1 times the critical mass density. In turn, this translates into a bound ~ 56 MeV [18] on the lepton-number-breaking scale which is clearly ruled out by the results of our present analysis. As a consequence, we can confidently exclude all those scenarios in which annihilation into majorons is invoked in order to deplete the cosmological abundance of stable 17 keY neutrinos. We can state the conclusions of our analysis in a different way by parameterizing the diagonal majoron-neutrino coupling as g~~~J(iy 5~). In this form our bound of ~ 19 0eV for a 17 keV majorana neutrino translates into V

V

V

~

~ 0.5 x 10—6.

(55)

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Nucleosynthesis constraints

Much stronger bounds are obtained if rn~is in the neighborhood of the critical temperature interval of 1—100 MeV. However, this would require the presence of a tiny coupling in the scalar potential A 0(m,2/V2) < 10-16, which would be highly unstable to quantum corrections. Our results are complementary to those obtained from the study of the constraints coming from the observed t”e pulse from supernova SN1987A. Based on the latter, the authors of ref. [16] exclude, for a 17 keV neutrino, the region 3 <8 GeV. We also want to mention that Babu and Rothstein [22], analyzing the constraints arising from nucleosynthesis and oscillations between doublet and sterile neutrinos in a background majoron bath, find a limit of the off-diagonal majoron—neutrino coupling ~ ~ 2 X i0~. Assuming then a 10% mixing with the light (n’) neutrino suggests a bound on the diagonal coupling ~ ~ 2 x 10-6, comparable to but slightly larger than our bound in eq. —‘


(55). Before closing this section let us recall that our constraint on of eq. (55) depends on our choice of the decoupling temperature, TD> 100 MeV, which, in turn, depends on the constraint z.1N~~ 0.3. Clearly, for L1N~>4 0.6, the presence at nucleosynthesis of one additional relativistic scalar would be allowed, and our constraint, at least for rn,, 1 MeV, will disappear completely. How uncertain then is our adopted bound of ~in~ ~ 0.3? This bound was derived by the authors of ref. [2] using the lower bound to the nucleon abundance of ~> 2.8 (to satisfy the observational constraints imposed by D and 3He), a lower bound to the neutron lifetime of r,, ~ 882 s and, an upper bound to the primordial helium mass fraction of Y~,< 0.240. Recent, high-precision measurements of the neutron lifetime have reduced its uncertainty and Dubbers [23] compilation leads to a 95% c.l. lower bound of r,, > 885 s. With all else as in ref. [2], this would decrease our bound to ~ < 4. How certain then is the adopted upper bound to from the data? Recently, Olive, Steigman and Walker [24], using the data of Pagel [25] made several independent fits to the data and inferred that the range in maximum primordial abundances consistent with the data was: 0.237 < yrnax ~ 0.243. Thus, it is not unreasonable to adopt yrnax 0.240 ±0.003. In this case we infer: ~ 4± 4 ~4. If we add the uncertainties in quadrature, 4NPIr~~X~ 0.35. Even for ~ ~ 0.5 we significantly constrain majoron models for the 17 keV neutrino. In fact, allowing for iiN~~ 0.5 implies, for n~ 1, a decoupling temperature TD 20 MeV (compare fig. 3.5 in ref. [26] with our fig. 2). From eq. (49) one sees that our bound of V ~ 19 0eV is only relaxed to ~ 13 0eV, and all our earlier conclusions remain unaffected. In conclusion we may summarize our results as follows. The limits on the presence of additional light excitations at the time of nucleosynthesis impose strong constraints on the interaction strength of such exotic particles. In particular, those 17 keV neutrino scenarios in which the stable heavy neutrinos annihilate into invisible majorons are excluded with high confidence. Those models in which the 17 keV neutrinos are unstable and decay into majorons are allowed, provided that V

=

=

=

t~

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213

the lepton number breaking scale is above 19 GeV (23 0eV for a pseudo-Dirac neutrino). In the same 17 keV neutrino scenario a stable heavier neutrino may solve the missing-mass problem as long as its mass is larger than 550 MeV. This excludes either j.t- or T-neutrinos as possible candidates.

One of the authors (S.B.) is grateful to the Theory Division at CERN, for the kind hospitality during the summer of 1991 when a preliminary part of this work was carried out, and to the Theory Group at the Max Planck Institute in Munich, where most of this analysis has taken place. G.S. acknowledges the support of the Humboldt Stiftung and the hospitality of the Max-Planck-Institute für Physik (Munich). We have benefitted from helpful discussions with J. Cline, A. Duncan, 0. Ogivetsky, K. Olive, 0. Raffelt, A. Santamaria and T. Walker. This work is supported at OSU by the DOE (DE-FGO2-91ER 40690).

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