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Sterile neutrino dark matter in warped extra dimensions

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UMN-TH-2621/07FTPI-MINN-07/30Sterile neutrino dark matter in warped extra dimensions Kenji Kadota William I.Fine Theoretical Physics Institute,University of Minnesota,Minneapolis,MN 55455February 4,2008Abstract We consider a (long-lived)sterile neutrino dark matter scenario in a ?ve dimensional (5D)warped extra dimension model where the ?elds can live in the bulk,which is partly motivated from the absence of the absolutely stable particles in a simple Randall-Sundrum model.The dominant production of the sterile neutrino can come from the decay of the radion (the scalar ?eld representing the brane separation)around the electroweak scale.The suppressions of the 4D parameters due to the warp factor and the small wave function overlaps in the extra dimension help alleviate the exceeding ?ne-tunings typical for a sterile neutrino dark matter scenario in a 4D setup.PACS :98.80.Cq 1Introduction

We will discuss the cosmological aspects of the radion phenomenology in the Randall-Sundrum background with the ?elds in the bulk [1,2,3,4,5,6,8,9,10,11].In particular we study the decay of the radion into the sterile right-handed neutrino which can account for all the dark matter in the Universe.This is partly motivated from the absence of the discrete symmetry in a simple warped extra dimension model and consequently the lack of an absolutely stable particle which could be a promising dark matter candidate.This is in contrast to,for instance,a ?at extra dimension model which possesses the discrete symmetry corresponding to the translational invariance in the extra dimension,called the Kaluza-Klein (KK)parity,leading to the absolutely stable KK dark matter

[12,13]1.We consider a simple addition of three right-handed neutrinos to the minimal Standard Model particle contents,where one of them is sterile (i.e.very weakly coupling to the other ?elds)

with a lifetime longer than the age of the Universe(and the other two explain the atmospheric and solar neutrino data[15]for the?nite small neutrino masses).

The sterile neutrino can be either warm or cold dark matter in our scenarios.

The sterile neutrino warm dark matter in such a minimal extension of the Standard Model (sometimes referred to asνMSM[16,17,18,19]in a4D setup)would have a potential interest from the astrophysical viewpoint because it can ameliorate the shortcomings in the small(i.e. galactic)scale structures of the cold dark matter scenarios which the N-body numerical simulations apparently su?er from,such as the missing satellite problem(the cold dark matter models predict too many satellites(small dwarf galaxies about a thousandth the mass of the Milky Way)than observed)and the cusp/core problem(the cusped central density distributions rather than the observed smoother core in the dark matter haloes)[20].

One of the notable features in a warped extra dimension setup is the production mechanism of the sterile neutrinos which can come from the decay of the radion because the radion can couple to all the degrees of freedom.This would be of signi?cant interest for a warm dark matter scenario where a simple Dodelson-Widrow((non-resonant)active-sterile neutrino mixing[21,22])mechanism alone cannot account for all the dark matter of the Universe due to the con?icts from the astrophysical observations such as the Lyman-αforest(giving the sterile neutrino dark matter mass lower limit of at least～10keV)and the X-ray data(giving the upper bound of at most～8keV)[23,24,25, 26,27,28,29,30,31].

The big?ne-tunings,such as those typical for a sterile neutrino dark matter scenario in the4D setups,are relaxed in the warped extra dimension models because the4D parameters are exponen-tially suppressed naturally due to the warp factor and the small wave function overlaps in the extra dimension.We also mention that the desirable baryon asymmetry of the Universe in our simple warped extra dimension model can arise from the decay of the higher KK modes of the right-handed neutrinos[32].

2Setup

We consider the scenario where the?elds can live in the bulk of the5D Randall-Sundrum spacetime whose AdS5metric in the conformally?at coordinate is[1]

ds2= R

where z is in the interval [R,R ′]with the UV (Planck)brane and the IR (TeV)brane respectively at z =

R ′.The radion corresponds to the scalar perturbation F of the metric [5,11,33,34]2

ds 2= R

R ′ 2r (x )6

gF (Tr T MN ?3T 55g 55)(5)

can give us a coupling of the radion to the matter ?elds,where g is the determinant of the 5D metric and we use the upper-case Latin (lower-case Greek)letters to denote the 5D (4D)indices.Let us now consider a concrete example of a 5D bulk Dirac spinor consisting of two two-

component spinors ΨT =(χα,ˉψ˙α)whose standard 5D bulk action including the bulk Dirac mass

term reads [8,9,10,11]

d 5x R

R ?i ˉχˉσμ?μχ?iψˉσμ?μˉψ

+1g = R

R (e F ,e F ,e F ,e F ,1/(1+2F ))(7)

z 2(e ?2F (x,z )ημνdx μdx ν?(1+G (x,z ))2dz 2)(2)

Plugging this parameterization ansatz into the Einstein equations reveals G and ?can be expressed in terms of F (in particular G =2F up to the linear order).Therefore we shall use only a single scalar mode F to describe the scalar perturbations in our discussions.

the linear order couplings of the radion to a fermion for the above action can be obtained from3

d5xF R z(ψχ+ˉχˉψ) (8)

We are mainly interested in the radion couplings to the lightest4D modes in the KK eigenstate decompositions

χ= n g n(z)χn(x),ψ= n f n(z)ˉψn(x)(9) where the4D components satisfy the4D equations of motion with the mass m n for each n th mode ?iˉσμ?μχn+m nˉψn=0,?iσμ?μˉψn+m nχn=0(10)

In particular,for n=0,there are massless zero modes of form,with the standard Dirichlet boundary conditionsψ|z=R,R′=0,

g0=A z1?2c√R′(11)

where the normalization constants are obtained by requiring the canonically normalized4D kinetic terms.Eq.(11)shows the fermion zero modes are localized around the Planck(TeV)brane for the bulk mass parameter c>1/2(c<1/2).The radion,on the other hand,is always localized to the TeV brane as can be seen by Eq(4).This means that,from the AdS/CFT correspondence [36],the radion is a composite state which shows up when the conformal symmetry is broken at the temperature around the TeV scale.We implicitly assume the con?nement of the composites when we discuss the radion decay around the electroweak scale temperature.

3Sterile neutrino mass

To obtain the abundance of the sterile neutrinos produced by the decay of the radion,we need to know the sterile neutrino mass in addition to the couping strength to the radion outlined in the last section.

We assume,for simplicity,three gauge-singlet right-handed neutrinos in addition to the minimal Standard Model particles,and consider the possibility for one of them,denoted as N,to be the sterile neutrino dark matter taking account of all the dark matter in the Universe.For it to be sterile,we assume the4D Dirac Yukawa coupling of N to the Higgs and left-handed neutrino is negligibly small(consequently its Dirac mass is also negligible compared with its Majorana mass) which could be justi?ed by the suppression due to the warp factor and small wave function overlaps

in the extra dimension(more quantitative discussions on how small it needs to be will be given in§6).Because no symmetry prohibits the Majorana mass terms for the gauge singlet N,we can consider the lightest eigenmass of N in existence of the brane localized Majorana mass term which dominates over the Dirac mass contributions.

We brie?y review here,for the illustration purpose,the Majorana mass term con?ned on the Planck brane(we choose the basis where the Majorana mass is real)4[8,9,10,37,38]

d5x√2m M(N R N R+h.c.),m M=d Mδ(z?R)(12) and the boundary conditions are accordingly modi?ed to

N L(z=R)=d M N R(z=R),N L(z=R′)=0(13) where d M is a dimensionless constant and

N(x,z)= N L(x,z)ˉN R(x,z) ,N L= n f(n)L(z)N(n)L(x),ˉN R= n f(n)R(z)ˉN(n)R(x)(14)

We can however simplify our analysis by calculating the4D mass eigenvalues/states via the di-agonalization of the mass matrix in the basis obtained without the boundary Majorana mass

(N(0)

R ,N(1)

,N(1)

,...).In this approximation,the symbols f(n)

R,L

,N(n)

R,L

are used for the states obtained

without the Majorana mass terms rather than the mass eigenstates,and the lightest eigenstate

dominantly consists of N(0)

R while N(0)

is decoupled from the low-energy theory as the result of the

boundary conditions.The approximate mass spectrum up to the n th KK level can be calculated by truncating at the n th level mass matrix when the mixing contributions from the higher modes are

small[37,39].We are interested in the lightest4D eigenstate(because of the boundary Majorana

mass,the lightest mode now obtains a non-vanishing eigenmass m N),and we omit the upper mode indices in the following unless stated otherwise.For instance,the lightest4D eigenmass for the

range of our interest m N?1/z can be obtained,up to the leading order,by substituting the wave functions of Eq.(11)into the brane localized Majorana mass term of Eq.(12)as(note our use of

c is such that c<1/2(c>1/2)for the localization to the TeV(Planck)brane)5

m N≈1R′ 1?2c(1?2c)for c<1/2(15)

The leading-order radion decay channel into the zero mode sterile neutrinos(approximated by N(0)

(x))can be thus read o?from Eq.(8)as6

d4xa0m N R′ 2 R

R′

d M R

+3Hn N=C col(20) which can be rewritten in terms of the so-called yield parameter Y≡n/s

HT s 1+T dT (21)

In the above,n N is the number density for the sterile neutrino,s is the entropy density and H is the Hubble constant given by,in terms of the radiation energyρR,

H2=8πG

g?T4,s=

2π2

6We here note that the radion couples only to the bulk terms and the brane-localized Majorana mass terms do not a?ect the radion couplings[5].At?rst sight,one may naively expect that the Majorana mass terms of Eq.(12) could lead to the radion coupling of

d4xe?4F1

2 d4x

The total numbers of the e?ectively massless degrees of freedom are given by

g?(T)= B g B T B8 F g F T F T 3+7T 3(23) where g B(F)is the number of helicity states for each boson(fermion)with their corresponding

temperature T B(F).In deriving Eq.(21),we assumed the constant entropy

and used Eq.(22).

We note that the thermalization of the radion can be justi?ed from the interactions involving the gauge couplings.For instance,its coupling to the gauge?elds in4D[4,5]can be obtained in the same manner as those outlined in§2

Rg24

4Λr Fμν(x)Fμν(x)～

2π2 m r T ,γ= m r E r K2(m r/T)(25)

where K n is the modi?ed Bessel function of the second kind of order n(as expected,n r∝g?(m r T)3/2exp(?m r/T),γ～1for T?m r and n r∝g?T3,γ～m/T for T?m r)[40,41].

Then Y can be obtained by integrating the Boltzmann equation

Y(x)=aλ2

M p

8There is an uncertainty for when the conformal symmetry breaking occurs,even though changing the lower bound to m r/μT eV=0.01changes the asymptotic value less than1%and m r/μT eV=1.0just slightly changes the asymptotic value to y(x?1)～4.5.

In the above integration,the time dependence of g s was assumed to be small.This assumption would be reasonable during the decay of the radion which dominantly occurs around T ～m r (the electroweak scale).The most of the higher KK states which could possibly have a non-negligible e?ect on dg s (T )/dT are not energetically accessible around the conformal symmetry breaking energy scale when the composite particles appear.More detailed studies

the

physics

around

the

phase

transition period will be presented in our future work 9.Matching m N Y to the current matter density value ρm /s ～0.4×10?9GeV [43]gives an estimation of

λ2～0.3×10?20 1MeV 100GeV

(27)5Examples

Let us now discuss a few concrete examples for the illustration purpose.For de?niteness,we set 1/R to be the Planck scale and use the values for the radion mass and Majorana mass parameter as m r =300GeV and d M =1.There then remain only two free parameters which are relevant for the dark mater abundance constraint Eq.(27):Λr (≡

√m N p N

For instance,in our simple numerical estimations,we neglected the e?ects of the the thermal mass corrections whose signi?cance would depend on the nature of the electroweak and conformal symmetry breaking phase transitions

[42].

average momentum produced from the thermal radion decay has the average momentum p N ≈2.45T[19].Taking account of the additional redshifting of p N by the decrease of the e?ective degrees of freedom(g(T～TeV)/g(T?MeV))?1/3,our scenario has p N T?1MeV≈0.76T.We then?nd the Lyman-αconstraints assuming the the active-sterile neutrino mixing mechanism for the sterile neutrino production[23]

m N 10～28[keV](29) correspond to10,in our radion decay production scenario,

m N 3～8[keV](30) The values realizable in our scenarios,such as those in our concrete examples,hence can be indeed consistent with the Lyman-αforest data,and they can also be consistent with another independent probe using the QSO gravitational lensing which gives the constraint m N 10keV[47].

The other astrophysical constraints,such as the observation of di?use photon backgrounds [24,25,27,29,48],are related to the neutrino Yukawa coupling and consequently the mixing of the active and sterile neutrinos.Our scenarios can satisfy these astrophysical and cosmological constraints without a?ecting the dark matter abundance,simply because the dominant production of the sterile neutrinos in our scenarios can come from the radion decay and is unrelated to the active-sterile neutrino mixing[21,22,26,28,49](also see Refs.[19,50]for other production mechanisms).In fact,the desirable dark matter abundance is possible even with the vanishing neutrino Yukawa coupling for the sterile neutrino in our scenarios.

6Discussion and conclusion

We showed that the sterile neutrino can be an interesting long-lived dark matter candidate in a simple warped extra dimension model which in general does not necessarily possess a discrete symmetry.

Although we assumed the(4D)neutrino Yukawa couplingλ4D for the sterile neutrino is negligibly small in our discussions,it may be still of possible interest to check how small it has to be.Let us for this purpose outline the constraints on the mixing angle sinθ～λ4D H /m N where H is the Higgs vev.

There arises a constraint from the requirement of the negligible sterile neutrino production by the active-sterile neutrino mixings[21,22,26].The?tting function for the sterile neutrino abundance produced from the(non-resonant)active-sterile neutrino mixing is11

h2～0.3 sin22θ100keV 2(31)

mix

Another constraint comes from the long enough lifetime[26,29,51]by considering the decay into an active neutrino and two leptons N→ν+l+ˉl via Z boson exchange

τ～1015sec× MeV sin22θ (32)

which should be at least of order the age of the Universe～4×1017s.12There also exists the constraint from the di?use photon backgrounds[29,30]due to the radiative decay into an active neutrino and a photon N→νγ.The emitted monoenergetic photon has a narrow decay line whose width is determined by the Doppler broadening for a potential astrophysical probe.For instance, for m N 20keV,the X-ray observatories such as Chandra and XMM-Newton[24,25,27]can give a severer constraint than the lifetime constraint.The recent analysis of the XMM-Newton observation of the Andromeda galaxy[24],for example,indicates the mixing angle should be sin2(2θ) 10?11 for one of our examples m N～17keV.Even though the more massive cold dark mass cases leading to theγ-ray backgrounds still have less precise data for detecting the dark matter decay lines,the recent analysis using the high-resolution spectrometer SPI set up on INTEGRAL satellite[58,59] can constrain the dark matter lifetime for40keV m N 14MeV and it constrains the mixing angle to be less than sin2(2θ) 10?24for m N～5MeV.13

We hence see that our two examples m N=17keV for a Planck brane localized Majorana mass and m N=5MeV for a TeV brane localized Majorana mass in§5require the4D neutrino Yukawa couplingsλ4D which can be obtained after integrating the wave functions over the?fth dimension to be respectivelyλ4D 10?13andλ4D 10?17(we took the Higgs vev H =246GeV).The free parameters in our model to determineλ4D are5D neutrino Yukawa couplingλ5D and the bulk mass parameters c L for the left-handed neutrino(assuming the other parameter c N relevant forλ4D is already?xed from the dark matter abundance constraint as in§5).It is easy to obtain such a small λ4D by localizing the left-handed neutrino toward the Planck brane(i.e.by letting c L>1/2)due to the exponential suppressions of the wave function overlaps in the?fth dimension.We however found that c L is essentially constrained to be1/2

The lightest mass eigenvalue for the light left-handed neutrino in the same generation as that of the sterile(dark matter)neutrino is much smaller than O(10?2)eV due to the negligible active-sterile neutrino mixing in our model,so that the atmospheric and solar neutrino data[15]should be explained by the other two remaining right-handed neutrinos in our simple scenarios.We also mention that the desirable baryon asymmetry of the Universe can be induced in our model from the decay of the higher KK modes of the right-handed neutrinos[32].

While most of the radion phenomenology and in particular the cosmology relevant for the ra-dion has been studied for the?elds con?ned on the brane,our study would hopefully open up an interesting possibility for the further investigations of both astrophysics and particle physics model building in a warped extra dimension with the?elds in the bulk.

Acknowledgments

The author thanks T.Gherghetta,K.Olive,M.Peloso,Y.Qian,M.Shaposhnikov and M.Voloshin for the useful discussions.He especially thanks T.Gherghetta for the early stage of collaboration and the continuous encouragement.This work was supported by DOE grant DE-FG02-94ER-40823. References

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