Naturally Light Sterile Neutrinos in Gauge Mediated Supersymmetry

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hep-ph/9810257

IC/98/163

WIS-98/25/Oct-DPP

Naturally Light Sterile Neutrinos in

Gauge Mediated Sup ersymmetry Breaking

a b

Gia Dvali and Yosef Nir

ICTP, Trieste, 34100, Italy

Department of Particle Physics, Weizmann Institute of Science, Rehovot 76100, Israel

Mo duli are generic in string (M) theory. In a large class of gauge-mediated

Sup ersymmetry breaking mo dels, the fermionic comp onents of such elds

havevery light masses, around the eV scale, and non-negligible mixing with

active neutrinos, of order 10 . Consequently, these fermions could play the

role of sterile neutrinos to which active neutrinos oscillate, thus a ecting

measurements of solar neutrinos or of atmospheric neutrinos. They could

also provide warm dark matter, thus a ecting structure formation. 10/98

1. Intro duction

Light sterile neutrinos are o ccasionally invoked by theorists to explain various hints of

neutrino masses which cannot b e accommo dated in a framework of only three light active

neutrinos (see e.g. refs. [1-26]). There are, however, three puzzles related to the hyp othesis

that light sterile neutrinos may play a role in various observations:

(i) The Ma jorana mass term of a sterile neutrino is not protected byany Standard Mo del

(SM) gauge symmetry and can, therefore, be arbitrarily large. The mass that is

relevant to the various exp eriments is at or b elow the eV scale.

(ii) The Dirac mass term that mixes a sterile neutrino with an active one is protected by

the electroweak breaking scale and is exp ected to b e in the range m m . To explain

e Z

any of the exp erimental results we need this term to be at or b elow the eV scale.

(iii) The two scales describ ed ab ove are in general indep endent of each other. Yet, the

mixing b etween the sterile and the active neutrino, which is given by the ratio of the

two scales, cannot be much smaller than O (10 ) and, for some purp oses, needs to

be of O (1). Then some mechanism that relates the two scales seems to be required.

Many mo dels were prop osed that give sterile neutrinos with the required features.

Most existing mo dels employ a rather ad-ho c symmetry structure (or just give an ansatz)

to induce the relevant parameters. The case for light sterile neutrinos would b ecome

much stronger if some well-motivated extension of the SM predicted their existence. We

argue that in mo dels of Gauge Mediated Sup ersymmetry Breaking (GMSB), the fermionic

comp onents of any SM singlet sup er eld N that it massless in the Sup ersymmetry

limit and, in particular, the mo duli elds, are generically exp ected to have masses and

mixing that could be relevant to various exp erimental and observational results.

2. Light Singlet Fermions in Sup ersymmetric Mo dels

We assume that the dominant source of sup ersymmetry breaking is an F term of

a chiral sup er eld S : F 6= 0. Mass terms involving arise then from the Kahler

S N

p otential and involve sup ersymmetry breaking. The leading contribution to the mass term 1

m is of the form

NN N N

F (S ) (NN)

=) m : (2:1)

m m

Pl Pl

The singlet N eld can mix with a lepton doublet eld L. The leading contribution to the

mass term m is of the form

LN L N

( ) (LN )

=) m : (2:2)

m m

Pl Pl

Here, are the two Higgs elds of the MSSM and we used the fact that the term

d;u u d

in the sup erp otential leads to F . The mass terms m and m determine the

u NN LN

two physically relevant quantities, that is the mass of , m m , and its mixing

N N NN

with active neutrinos, s m =m .

LN LN NN

Note that the contribution from F to m is crucial for to be relevant to neu-

LN N

trino physics. The reason is that m breaks b oth sup ersymmetry and the electroweak

symmetry. Without F -terms of SU (2) non-singlets, there would b e a separate suppression

factor for each of the two breakings, making m to o small for our purp oses. Explicitly,

S u

and conse- if the only F term to play a role were F , then we would get m

S LN

quently s 10 , indep endent of the mechanism that mediates sup ersymmetry

breaking. Such mixing is to o small to a ect any neutrino exp eriment. In contrast, the

contribution to m from F leads toavalue for s that is mo del dep endent and that

LN LN

can be sizable. Assuming that is of the order of the electroweak breaking scale, we get

10 eV : (2:3) m

The scale of F (and, consequently, the values of m and s ) dep ends on the

S N LN

mechanism that communicates SUSY breaking to the observable sector. In sup ergravity

mo dels, where F m m , we get

S Z Pl

2 16

m m 10 GeV ; s 10 : (2:4)

N Z LN

We implicitly assume here that the mass and mixing of are describ ed e ectively by a

2 2 matrix, and that m m .

LL NN

Then is practically decoupled from the observable sector and do es not have any ob-

servable signatures.

2 2

= , In GMSB mo dels [27-29] wehave a more interesting situation. There, F Cm

where C 1 dep ends on the details of the mo del (for a review, see [30]). We now get

2 2 4

m 0:1 eV C; s : (2:5)

N LN

m C C

The mass scale for is not far from those relevant to galaxy formation ( 10 eV ),

atmospheric neutrinos ( 0:1 eV ) and solar neutrinos ( 10 eV ). The mixing is small

but non-negligible. We conclude then that in GMSB mo dels, the fermionic elds in the

mo duli can, in principle, play the role of sterile neutrinos that are relevant to various

observations.

We emphasize that eq. (2.5) gives only naive order of magnitude estimates. Each of

its relations might b e somewhat mo di ed by unknown co ecients, exp ected to b e of O (1).

Furthermore, there might be other ingredients in the mo del that a ect even the order of

magnitude estimates. In the next section we show how simple variations within our basic

framework might bring the mass and the mixing of closer to those required to explain

the various exp erimental results.

3. Solar and Atmospheric Neutrinos

Simple variations on the naive estimates given ab ove could make the sterile neutrino

parameters consistent with solutions to the solar neutrino problem [32] or to the atmo-

spheric neutrino problem [33].

Let us consider rst the p ossibility that the relevant sup er elds N transform under

some approximate symmetry. This could b e a horizontal symmetry invoked to explain the

smallness and hierarchy in the avor parameters. Take, for example, a U (1) symmetry

A sup ergravity scenario where the fermionic elds in the mo duli play the role of sterile

neutrions was prop osed in ref. [12]. This was done, however, with a sp ecial ansatz for the

sup ersymmetry breaking mass terms.

Neutrino masses in the GMSB framework were recently discussed in ref. [31]. Their mo del,

however, has no sterile neutrinos and involves R parity violation. 3

broken by a small parameter , to which we attribute charge 1. Take N and L to carry

charges p and q , resp ectively, under the symmetry. Then, (2.5) is mo di ed:

p+q 2 2p 2

m Cm

2p p+q 5

Z Z

C 0:1 eV ; m 10 eV ; m

LN NN

m m

Pl Pl

(3:1)

2 4

s :

pq pq

C C

10 eV , so that is unlikely (in this To get s = O (1), we would need m

N LN NN

simple scenario) to play a role in the atmospheric neutrino anomaly or in the large angle

MSW solution to the solar neutrino problem. On the other hand, two relevant sets of

parameters can be easily pro duced by the approximate symmetry:

(I) Take C 1, 0:1, and q =0:

3 6 3

m 10 eV ; m 10 eV ; s 10 : (3:2)

NN LN LN

This is not far from the small angle MSW solution to the solar neutrino problem. (The mix-

ing angle is somewhat small but, as mentioned ab ove, could be mo di ed by the unknown

co ecients of O (1).)

p 2

(I I) Take C 1, 10 , and q = p:

5 5

m 10 eV ; m 10 eV ; s 1: (3:3)

NN LN LN

This set of parameters is appropriate for the vacuum oscillation solution to the solar

neutrino problem.

Another variation on the naive estimates arises if the relevant heavy scale (call it m

for New Physics) in the nonrenormalizable terms is lower than m . Then b oth m and

Pl NN

m will b e enhnaced compared to (2.1) and (2.2). A particularly intriguing option is that

the string scale identi es with the scale of gauge uni cation [34], that is m 10 GeV .

This leads to our third example:

16 p 2

(III) Take m 10 GeV , C 1, 10 , and q = p:

3 3

m 10 eV ; m 10 eV ; s 1: (3:4)

NN LN LN

These parameters give the large angle MSW solution to the solar neutrino problem. 4

Either a surprisingly small m or a surprisingly large may make relevantto

NP d N

the atmospheric neutrino problem. First, an even lower cut-o scale, m 10 GeV ,

would give m = O (0:1 eV ). However, there is no particularly attractive scenario that

requires such a scale for m . Second, a large [35] could also lead to a large m .

NP LN

Note, however, that in order to prevent a negative mass-squared for the stop, one needs

. In this scenario we have then an interesting relation between the stop sector m

and the neutrino sector: the mixing b etween the sterile and the active neutrinos is b ounded

by m =m .

14 p 2 4

(IV) Take m 10 GeV , C 1, 10 , and q = p, or F , C 10 ,

NP S

p 2

10 , and q = p. Then

1 1

m 10 eV ; m 10 eV ; s 1; (3:5)

NN LN LN

which can solve the atmospheric neutrino problem.

4. Nucleosynthesis and Galaxy Formation

The numb er of light neutrinos (m 1 MeV ) that were in equilibrium at the neutrino

decoupling temp erature (T a few MeV ), N , cannot be much larger than three.

dec

Otherwise, the consistency between the predictions of the standard mo del of Big Bang

Nucleosynthesis (BBN) and the observed abundance of primordial light elements will be

lost. A sterile neutrino that mixes with the active ones contributes to N b ecause neutrino

oscillations can bring it into equilibrium ab ove T . Consequently, one can use the BBN

dec

constraints to exclude regions in the m sin 2 plane [36-41]. (Here is the light

4i 4

2 2

mass eigenstate with a dominant comp onent.) In our framework, m = O (m ) and

sin 2 = O (4s ). The calculation of the b ounds is quite complicated. An approximate

2 6 2

analytical constraint is given in [38], m sin 2 5 10 eV , corresp onding to

N 3:4. This leads to

2 4

m s 5 10 eV : (4:1)

2 9

The naive estimates of eq. (2.5) yield m s 10 eV =C , whichiswell b elow the b ound

(4.1). Note, however, that (4.1) is very sensitive to s . Taking into account the various

LN 5

variations discussed in the previous section, we nd that (4.1) leads to

5 10 : (4:2)

C m

This b ound is ful lled for the small angle MSW (3.2) and the vacuum oscillation (3.3)

solutions of the solar neutrino problem but (as is well known) violated if plays a role

in the large angle MSW solution of the solar neutrino problem (3.4) or, in particular, in

the atmospheric neutrino anomaly (3.5).

A sterile neutrino could also provide a signi cant warm dark matter comp onent (see

e.g. [45]). To have = O (1) requires

< <

1 keV: (4:3) m s 0:1 eV ; 10 eV m

N LN N

A particulary plausible framework where (4.3) is realized is that of GMSB mo dels with

C 10 , which gives:

12 2

F 10 GeV =) m 0:1 eV ; m 1 keV: (4:4)

S LN N

5. Intermediate Scale Gauge Mediated Sup ersymmetry Breaking

All the examples that we discussed ab ove require that the scale of sup ersymmetry

4 6

breaking is low, F 10 10 GeV . We now discuss another class of GMSB mo dels,

where can play the role of a sterile neutrino if the sup ersymmetry breaking scale is

6 9

F 10 10 GeV . much higher,

Consider the case where S , which is resp onsible for Sup ersymmetry breaking, trans-

forms under some U (1) symmetry [46]. Assume that N is neutral under this symmetry.

Then the contribution (2.1) to m is forbidden. Instead, the leading contribution is of

the form

y 2

(S ) (S ) (NN) S

=) m : (5:1)

2 2

m m

Pl Pl

The constraint (4.1) could b e evaded if there had b een a large lepton asymmetry in the early

Universe [42-44]. The constraint could also be relaxed if the b ound on N is weaker than the

one we quoted. 6

Here is the dimensionful parameter that sets the scale for the masses of the MSSM

): particles in GMSB mo dels (m

4 6

10 10 GeV : (5:2)

Assuming that , and L are also neutral under the U (1) symmetry (or that they carry

d u

appropriate charges), the estimate of m in eq. (2.2) remains valid.

Unlike our discussion ab ove, where F led us to exp ect that m >m ,we

S u NN LN

now have to distinguish between two cases, dep ending on the value of

: (5:3) C =

6 8

10 GeV 10 GeV

m , we get, The p oint is that m =m C . For m

LN NN LN NN

0 5 0 0

m C 10 eV ; s 1=C ; (C 1): (5:4)

N LN

But for m m (and assuming, as b efore, that m m ) the situation is dras-

NN LN LL NN

tically di erent: the active and the sterile neutrino form a pseudo-Dirac neutrino of mass

m and small splitting m :

LN NN

m m

N L

5 0

m ' m 10 eV ; C ;

N L

m + m

N L

(5:5)

s ' 2=2; (C 1):

Again, could play a role in the various neutrino exp eriments. Here are a few

examples:

0 2

(I) C 10 corresp onds to the small angle MSW solution to the solar neutrino problem.

(I I) C 1 corresp onds to the vacuum oscillation solution to the solar neutrino problem.

16 0 2

(III) m 10 GeV and C 10 corresp ond to the large angle MSW solution to the

solar neutrino problem.

4 14 0

10 can explain the atmospheric neutrino results. (IV) m 10 GeV and C

Note that, in order that will be relevant to our purp oses, we need C that is not

much larger than 1. This is imp ortant since a to o large C leads to phenomenological

problems: 7

a. For S 10 GeV , the non-universal sup ergravity contributions to sfermion masses

b ecome comparable to the universal gauge-mediated contributions. Consequently, the

sup ersymmetric avor problem is no longer solved [47].

b. For S 10 GeV , the S -scalar decays late and its hadronic decay pro ducts over-

pro duce light nuclei. Consequently, the successful predictions of the standard BBN

no longer hold [46].

Wewould like to emphasize two attractive p oints ab out sterile neutrinos in the frame-

work of intermediate-scale GMSB mo dels discussed in this section (compared to the GMSB

mo dels of section 2). First, for mo dels with m F =m to give 's that are relevant

NN S Pl N

12 2

10 GeV ). Direct exp erimental to neutrino physics, a rather low F is required (F

S S

searches for diphoton events with large missing transverse energy [48-50] exclude large re-

gions in the parameter space where the scale of F is low. Second, in many GMSB mo dels

of direct gauge mediation [30] the scale of F cannot be low.

6. Conclusions

If low-energy sup ersymmetry is a result of a high-energy string theory, then we exp ect

quite generically that there exist singlet elds N that are massless in the sup ersymmetric

limit (mo duli). Our main p oint is very simple: for two large classes of gauge mediated

sup ersymmetry breaking mo dels, the sup ersymmtry-breaking masses of the fermionic elds

is around the eV scale and their mixing with active neutrinos is non-negligible. (In the

8 12 2 y

rst class, sup ersymmetry is broken at a scale F 10 10 GeV and a term S NN

13 17 2 y

in the Kahler p otential is allowed. In the second class, F 10 10 GeV and S NN

is forbidden.) Consequently, such elds could play the role of sterile neutrinos to which

( ) oscillate, thus solving the solar (atmospheric) neutrino problem. They could also

provide a warm comp onent to the dark matter, thus a ecting galaxy formation.

Acknowledgements

We thank Yael Shadmi, Alexei Smirnov and Eli Waxman for helpful discussions. Y.N. is

supp orted in part by the United States - Israel Binational Science Foundation (BSF), by

the Israel Science Foundation and by the Minerva Foundation (Munich). 8

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