b

ne-bosonble from

essentiallyhe Diract results

Physics Letters B 625 (2005) 189–195

www.elsevier.com/locate/physlet

Dark energy and right-handed neutrinos

Riccardo Barbieria,b, Lawrence J. Hallc, Steven J. Oliverc, Alessandro Strumiad

a Scuola Normale Superiore and INFN, Piazza dei Cavalieri 7, I-56126 Pisa, Italyb Theoretical Physics Division, CERN, CH-1211 Genève 23, Switzerland

c Department of Physics, University of California, Berkeley, and Theoretical Physics Group, LBNL, Berkeley, CA 94720, USAd Dipartimento di Fisica dell’Università di Pisa and INFN, Italy

Received 18 July 2005; accepted 17 August 2005

Available online 29 August 2005

Editor: G.F. Giudice

Abstract

We explore the possibility that a CP violating phase of the neutrino mass matrix is promoted to a pseudo-Goldstofield and is identified as the quintessence field for Dark Energy. By requiring that the quintessence potential be calculaa Lagrangian, and that the extreme flatness of the potential be stable under radiative corrections, we are led to anunique model. Lepton number is violated only by Majorana masses of light right-handed neutrinos, comparable to tmasses that mix right- with left-handed neutrinos. We outline the rich and constrained neutrino phenomenology thafrom this proposal. 2005 Elsevier B.V. All rights reserved.

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1. Introduction

A series of different observations and considetions [1,2] provides a strong case for a striking phnomenon: the expansion of the universe has recebegun to accelerate. Although a definitive experimwith sufficiently small systematic uncertainties is lacing, if confirmed this remarkable fact calls for an adquate explanation and, even more important, motivthe search for other correlated observable phenom

E-mail address: astrumia@mail.df.unipi.it(A. Strumia).

0370-2693/$ – see front matter 2005 Elsevier B.V. All rights reserveddoi:10.1016/j.physletb.2005.08.075

.

The accelerated expansion of the universe cobe due to a tiny Cosmological Constant (CC),Λ ≈(3 × 10−3 eV)4; tiny, but non-zero. In fact, the frustration generated by the unsuccessful attempts to sthe vacuum energy problem has led to the developmof a mild anthropic interpretation of the apparently oserved value of the CC[3]. Here we take the viewthat the search for a more fundamental understanof the cosmic acceleration remains highly motivateven if still resting on the assumption of an exacvanishing vacuum energy.

As an alternative to a CC, the accelerated expanof the universe may be due to the evolution of soscalar field, uniform or quasi-uniform in space, w

.

190 R. Barbieri et al. / Physics Letters B 625 (2005) 189–195

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the associated “Dark Energy” (DE) mostly in its ptential, usually called “quintessence”[4]. Signals re-lated to such an interpretation, although typically ovaguely determined, could be an equation of statethe associated fluid different from the one of a pCC, or the effects of the couplings of the quintessefield to the usual matter or gauge fields.

In this Letter we describe a possible microscoorigin for the quintessence field and for its potetial, guided by two general requirements and one pnomenological observation. One general requiremis that the quintessence potential should be calcble from a Lagrangian and its peculiar propertiesparticular its extreme flatness, should be stableder radiative corrections. Since the mass scale gerning this flatness is todays Hubble parameter,H0 ≈10−33 eV, this requirement is severe indeed, althouit is known to be satisfied by a Pseudo-Goldstoneson (PGB) arising from the spontaneous breakinga global symmetry near the Planck scale[5]. Our sec-ond general requirement is that the physics of DEdirectly connected to observable particle physics,that the resulting theory can be tested in the labotory. These two requirements appear to conflict—the quintessence field is coupled sufficiently stronto the standard model to give laboratory signals, tradiative corrections involving this coupling will destroy the extreme flatness of the potential. A hint opossible escape from this conundrum is providedthe phenomenological observation, already madeseveral people[6,7], of the relative closeness of thenergy scale associated with DE to the scale of ntrino masses. Thus we are led to explore the possibthat a CP violating phase of the neutrino mass mtrix is promoted to a PGB field and is identified as tquintessence field for DE.

2. The model

To implement this idea we introduce right-handneutrinos,Ni , at least two but most likely threeand as many complex scalars,φij , as there are independent Lorentz-invariant neutrino bilinearsNiNj .These scalars have Yukawa couplings to theNi (φij =φji, λij = λji)

(1)L NY = 1

2

∑λij φijNiNj ,

ij

which are invariant under independent phase transmations of eachNi field, say U(1)3 for concretenessU(1)3 is a subgroup of the U(1)6 which transformseach of theφij -fields by an independent phase. Tcrucial assumption is that, in the limit of vanishinλij , the full Lagrangian is invariant under this globU(1)6, which is spontaneously broken by the vacuexpectation values〈φij 〉 ≡ fij .

In the absence of any other coupling of theNi

fields, this model has three massless Goldstone boand three PGBs,Gij . The effective potential for thecombinations ofGij that are PGBs arises at one loand is given by

(2)V1 ≈ 1

32π2Tr

[MM†MM† ln

Λ2

MM†

],

where Mij = λijfij eiGij /fij is the field-dependen

right-handed neutrino mass matrix andΛ is a cut-off,to be specified later. Note the irrelevance inV of anyquadratic term inMij , however generated, since tonly such term invariant under U(1)3 is also U(1)6-invariant, and thereforeGij independent. A typicaterm inV1 contributing to the potential of a PGB fieldG, has the form

(3)V (G) = µ4 cos(G/f ),

whereµ4 = O(M4) arises as a productMijM∗jkMkl ×

M∗li , andf is an appropriate function of the symmet

breaking parametersfij . It is well known that, withµ ≈ 3×10−3 eV andf of orderMPl, G is a consistencandidate for the quintessence field[5,6,8]. However,the signals we wish to stress are not those that cfrom the form of the potential(3), but rather are due tthe required form for the underlying neutrino secto

Two natural and important questions arise at tpoint. Could we interpret theNi as the left-handedneutrinos entering the usual left-handed lepton dbletsLi? What other couplings can complete constently the neutrino sector? To answer the first queswe should first transform Eq.(1) into a gauge-invarianinteraction involving theLi and the Higgs doubleth,

(4)L LY = 1

2

∑ij

λLijφij

h2

M2L

LiLj ,

where gauge indices are left understood andML isan energy scale introduced to giveL L

Y the correct di-mensions. Indeed, if we now replace the Higgs fi

R. Barbieri et al. / Physics Letters B 625 (2005) 189–195 191

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with its vacuum expectation value, we would be lto the same contribution as in(2) to the PGB potential, except thatMij would now be the left-handeneutrino mass matrix. Apparently, this is the minimtheory, with the DE field directly related to the phasof the 3× 3 neutrino mixing matrix. It is straightforward to see, however, that radiative corrections abthe weak scale with internal Higgs fields destroy radtive stability. This is an important conclusion. Withoconsiderable complications of the Higgs sector[9], theintroduction of light right-handed neutrinos appearsa necessity. A similar conclusion applies to the cwith φij fields coupled to theLiNj bilinear.

We can now answer the second question staabove. A consistent completion of the Yukawa Lgrangian in the lepton sector has the form

LY =∑ij

λEijhLie

cj +

∑ij

λDij hLiNj

(5)+ 1

2

∑ij

λijφijNiNj ,

involving the right-handed charged leptons,eci . One

recognizes the usual Dirac neutrino mass matrix, pportional toλD , and one notices the absence of a(gauge-invariant) Majorana mass term for the lehanded neutrinos of the formλM

ij h2LiLj/ML. Sucha term, in fact, would explicitly break lepton numband, in conjunction with the Dirac mass matrix, wouallow terms in the PGB potential linear inMij , thusalso destroying radiative stability. In fact, radiative sbility requires that all non-renormalizable operatoconserve both U(1)6 on theφij and an overall lepton number, U(1)L. In the presence of non-zeroλij ,as well as genericλE,D couplings, the U(1)6 symme-try is explicitly broken to U(1)L; it is convenient tolabel the theory by “U(1)6 → U(1)L”.

From the above arguments, the theory of(5) is es-sentially unique. The only fermion bilinear thatφij cancouple to isNiNj , and theh2LiLj operator must beabsent. We conclude that there must be two or mlight right-handed neutrinos, with typical entriestheir Majorana mass matrix of order 3× 10−3 eV—broadly comparable to the entries in the Dirac mtrix. It is remarkable that, in promoting a CP violatinphase of the neutrino sector to a field, the mass offield can be protected to the level ofH0 ≈ 10−33 eV.

The key is to ensure that the leading radiative corrtion tom2

G is of orderm4ν/M

2Pl.

It is the Dirac mass term in(5) that allows us tocall Ni the right-handed neutrinos. One may wonwhether this term introduces significant new corrtions to the PGB potential. In fact it does at two-loorder, giving a term in the PGB potential

(6)V2 ≈(

1

16π2

)2

Tr(MM†λDλD†)Λ2.

This leads us to consider a supersymmetric extenof the model with spartner masses at the Fermi scin which case a typical sneutrino mass cuts off bthe quadratic divergence of this two-loop potential athe logarithmic divergence of the one-loop term in(2).Up to loop factors, the resulting two-loop contributialso becomes of the relevant order of magnitudea quintessence potential.1 Note that with the completLY all the would-be-Goldstones associated withbreaking of U(1)6 acquire a mass from the PGB ptential, except for the linear combination related tooverall lepton number.

We must be clear that we have not explained wthe neutrinos are light—quite the reverse, sincehave introduced extremely small parameters in thλ

andλD matrices. The PGB masses are small becathey are proportional to powers of these smallplicit symmetry breaking parameters. Neglectingvor labels and emphasizing only the very smallrameters, the interactions of(5) can be rewritten ahLec + εD hLN + ε2

M φNN , where εD,M are nowthe small parameters. These parameters may nopromoted to fields, with the lightness of the neunos explained in terms of a small vacuum value, sithese fields would lead to disastrous radiative cortions to the PGB potential. Nevertheless, it is intesting that bothεD andεM should take on values oorder 10−13–10−15 for acceptable neutrino masses aDE—an approximate symmetry acts on theN fields.In higher-dimensional theories, a smallε could resultif N propagate in a bulk and have a small, expontially suppressed wavefunction at the location of

1 In contrast to the case of mass varying neutrinos[7], the effec-tive contribution to the quintessence potential from the cosmologneutrino density is numerically irrelevant. We can also ignoresmall variation of the neutrino masses induced by their dependon the dynamical PGB fields.

192 R. Barbieri et al. / Physics Letters B 625 (2005) 189–195

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φ,L and h fields. Even in this case, any paramethat sets the geometrical configuration must not crespond to a light field in the low energy effectifour-dimensional theory.

Two variations in the theory are possible: bystricting the form of the couplings or the numberφ fields in (5), alternative symmetry breaking paterns emerge, yielding PGB potentials different froV1 + V2 of the generic case. If the entire theopossesses an exact U(1)3

i symmetry, with one U(1)

for each lepton generation,λE,D become diagonaand we obtain the “U(1)6 → U(1)3

i ” variation. SinceTr(MM†λDλD†) is now independent of the three PGfields, the potential for the PGBs is given purelyV1 of (2). As this potential has only a logarithmdivergence, the quintessence potential is stable todiative corrections whether or not superpartners arthe weak scale. In this variation, lepton flavor mixiarises entirely from spontaneous breaking. Finallythe initial symmetry is restricted to U(1)3

i , so that thetheory possesses only threeφ fields,φii , we obtain the“U(1)3

i → U(1)L” variation. The potential for the 2PGBs occurs at 3 loops:

(7)V3 ≈(

1

16π2

)3

Tr(MλD†λDM†λDλD†)Λ2.

This again gives a successful theory for quintessewith entries ofM of order 10−3 eV, but, in contrasto the general case, supersymmetry should be abgiving a cutoffΛ ≈ MPl. As the cutoff is reduced, sthe entries ofM can be made larger—with the cuoff in this variation at the weak scale the right-handneutrino masses may be raised to an MeV.

3. Constraints and neutrino spectra

Since fij ∼ MPl the PGB interactions are extremely weak, so that the main consequences oftheory are in the neutrino mass sector. With 3 righanded neutrinos, the full neutrino mass matrix is 6×6and is made of a Dirac mass matrix,mij = λD

ij 〈h〉, andof a Majorana mass matrix for the right-handed ntrinos, Mij = λijfij , both 3× 3 and roughly of thesame size (up to differences among the varioustrix elements, which can of course be significant).M very much smaller or much bigger thanm wouldin fact give the wrong size for the PGB potenti

,

Such a neutrino mass matrix is mainly constrainby oscillation experiments. However, it is also costrained by cosmology: extra sterile neutrinos coing into equilibrium, partially or totally, affects BigBang Nucleosynthesis (BBN), the Cosmic MicrowaBackground (CMB) and Large Scale Structure (LSformation.

After symmetry breaking, the neutrino masses amixings can be described in full generality by

L = g√2ν̄V γµPLeWµ + eTmEec + νTmdN

(8)+ 1

2NTUTMdUN + h.c.,

where the flavour indices are left understood,mE,md

andMd are real and diagonal matrices,V is a unitarymatrix with a single physical phase andU is a unitarymatrix with five physical phases.2 If m andM are di-agonalized bym = VDmdUD and M = UT

MMdUM ,

respectively, thenU = UMU†D . To analyze the con

straints in full generality in the entire space of pameters is complicated and goes beyond the scopthis work. In the following we try to describe the mafeatures of the allowed parameter space.

We first consider the special caseU = 1, in whichm andM may be simultaneously diagonalized. Thisa useful starting point to understand the more gensituation or, at least, to show that there are allowedgions in parameter space that fulfill all requiremenThis situation is fully realistic and we study it blow.

The constraints on the mass parameters are eadetermine because diagonalization of the 6× 6 neu-trino mass matrix decouples into 3 separate 2× 2diagonalizations, one for each (νi,Ni ) pair. We disre-gard possible degeneracies and order the eigenvaof the Dirac and Majorana mass matrices,mi andMi ,respectively, in such a way thatm3,M3 govern theatmospheric oscillation length andm2,M2 the solaroscillation length. The constraints from oscillation eperiments onM2 and M3 are shown inFig. 1. Also

2 The proof is as follows. The first three terms in the right-haside of(8) are the usual terms in the case of pure Dirac neutrinwhich can always be reduced to this form. The last term is an atrary symmetric matrix with the overall phase which is unphysibecause it can be reabsorbed by an overall lepton number tranmation.

R. Barbieri et al. / Physics Letters B 625 (2005) 189–195 193

ctionegions).

Fig. 1. Sign(�χ2)|�χ2|1/2 of the global oscillation fit (continuous (blue in web version) line/left vertical axis) and thermalized sterile fra(dotted (red in web version) line/right vertical axis). Large scale structure data exclude thermalized heavy sterile neutrinos (shaded r

ter-

ureeno.

yforeV,

r-de

c

irac-

eteudo-

tify

ssi-irac

t-ctspearhocym-

o

) isfor

re

asis

shown are the respective fractions of thermalized sile neutrinos at BBN and CMB eras,�Nν , and theregions excluded by LSS data. An analogous figcannot be drawn forM1 since we do not know thmass of the lightest active, or quasi-active neutriHowever, form1 small enough, saym1 � 10−6 eV,M1 is almost unconstrained.

While LSS forbids M2,3 � eV, each of thesemasses could lie in the “0.3 eV window”, given b0.1 eV� M2,3 � eV. Since the observed massesatmospheric and solar oscillations are less than 0.3these values forM2,3 lead to a mini-seesaw. Altenatively, although atmospheric oscillations excluM3 m3, any value less than about 10−2 eV is al-lowed, yielding a pseudo-Dirac pair for(ν3,N3). Onthe other hand, solar oscillations forbidM2 beneaththe 0.3 eV window all the way down to∼10−9 eV.Allowed values below 10−9 eV lead to a pseudo-Dira(ν2,M2) pair. Thus, each of(ν2,M2) and(ν3,M3) ei-ther undergo a mini-seesaw or form a pseudo-Dpair. Furthermore, fromFig. 1we see that the cosmological thermalization of the sterile state is complfor the mini-seesaw case and absent for the pseDirac case (except asM3 approaches 10−2 eV, whenpartial thermalization occurs). Hence, we can identhree possibilities3

(0) �Nν ≈ 0:

(M2,M3) ≈ (10−9,10−3) eV;

3 Here 0.3 eV means anywhere in the “0.3 eV window”,M3 ≈10−3 eV means any value ofM3 less than about 10−2 eV, andM2 ≈ 10−9 eV means any value ofM2 less than about 10−9 eV.

(1) �Nν ≈ 1:

(1a) (M2,M3) ≈ (0.3,10−3) eV,

(1b) (M2,M3) ≈ (10−9,0.3

)eV.

(2) �Nν ≈ 2:

(M2,M3) ≈ (0.3,0.3) eV.

These four mass ranges correspond to the four poble ways of assigning mini-seesaw and pseudo-Dspectra to each of(ν2,M2) and (ν3,M3), as shownin Fig. 2. One extra neutrino at BBN looks compaible with standard cosmology, with systematic effetaken into account, whereas two extra neutrinos apdefinitely problematic, unless one invokes an adnon-standard cosmology, such as large lepton asmetries or a MeV-scale reheating temperature.

Although we have setU = 1, the unitary matrixUM = UD is completely undetermined by neutrinmass phenomenology—the Euler angles,θ , of UM canbe chosen to obtain the observed DE,ρDE. If M1 issufficiently small, the relevant entries ofMij = λijfij

are given by Euler angles multiplied byM2 or M3. Incases (0), (1) and (2),ρDE typically requiresθ ≈ 1,10−2 and 10−4, respectively. Perhaps the case (0most natural, and, depending on the precise valueM3, could lead to an observable deviation of�Nν

from 0.Theories withU = 1 can be constructed that a

both more natural and more predictive than theU = 1case. In particular, suppose that, in the original bfor N , we have the texture

Mij = λijfij = M1 ε12 ε13

ε12 M2 ε23ε ε M

,

13 23 3

194 R. Barbieri et al. / Physics Letters B 625 (2005) 189–195

ass of

Fig. 2. Schematic illustration of four possible neutrino mass spectra consistent with oscillation and cosmological constraints. The mN1is largely undetermined.

atal-oft-

hreesin-e

and

u-wehtwnno

at

das-r

s-ex-d

ofos-

inscil-is-

-fa-

(9)m =

m1 0 0

0 m2 0

0 0 m3

,

where theεij are either zero or sufficiently small ththe neutrino phenomenology is not significantlytered from theU = 1 case. The standard neutrinmixing angles arise from transformations on the lehanded leptons, charged or neutral. In each of the tcases described above, it is possible to introduce agle off-diagonal entry,εij , such that the appropriatPGB potential is generated,ρDE ≈ MiiM

∗ijMjjM

∗ji

for some(i, j).Realistic examples, corresponding to cases (0)

(1) above, are

(0)

(m1,m2,m3) ≈ (� 10−6 eV,msun,matm

),

(M1,M2,M3) ≈ (� 10−3,10−9,10−3) eV,

andε13 ≈ 10−2 eV;(1)

(m1,m2,m3) ≈ (0,5× 10−3 eV,matm

),

(M1,M2,M3) ≈ (0,0.3,10−8–10−3) eV,

andε23 ≈ (10−4 eV

)(10−3 eV/M3

)1/2,

with otherεij taken irrelevantly small.

4. Signals and conclusions

In promoting a CP violating phase of the netrino mass matrix to the DE quintessence field,are led, essentially uniquely, to a theory with ligright-handed neutrinos, with possible spectra shoin Fig. 2. This proposal can be tested in neutriphysics.

• The three cases with differing�Nν will easily bedistinguished by precision CMB measurementsPLANCK, and perhaps at WMAP.

• The rangeM2 ∼ 0.3 eV can be completely testeby cosmology (searching for sterile neutrino mses) and by reactor experiments (searching foν̄e

disappearance at base-line∼10 m).• Similarly, M3 ∼ 0.3 eV can be tested by co

mology, and possibly by atmospheric neutrinoperiments (HyperK, MONOLITH, IceCube) anbeam experiments (MiniBoone, MINOS).

• Long-baseline experiments will probeM3 ∼10−3 eV. If M3 approaches 10−2 eV, it can bedetermined by a CMB or BBN measurement�Nν , and signals may appear in atmosphericcillations.

• Very small M1,2,3 can give MSW resonancesthe sun and in supernovæ, as well as vacuum olations of neutrinos that travel cosmological dtances.

• MiniBoone is currently testing the LSND anomaly. Constraints from other oscillation data dis

R. Barbieri et al. / Physics Letters B 625 (2005) 189–195 195

s,

aveeu-

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rthmeme-ionto

geri-tlyd

ithm-rgyand

hen:

yby

-02-SF

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As-

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98)

v.

05,

;0)

p-

4)

-

0)

vor its interpretation in terms of sterile neutrinobut do not fully exclude it.

• The detection of a 0ν2β signal would exclude thismodel, since the left-handed neutrinos do not ha direct Majorana mass and the right-handed ntrinos are light, so that effects in 0ν2β are sup-pressed by powers ofM/Q (whereQ ∼ MeV isthe energy released in 0ν2β).

The coupling of neutrinos to light PGBs is propotional to M/f and is so small that, unlike the caof late time neutrino masses[10], all neutrinos free-stream during the eV era.

We conclude by noting other areas that are woinvestigating. Since there are several PGBs, somight have masses larger than todays Hubble parater, so that they oscillate during the recent evolutof the universe with characteristic signals relatedthe associated Jeans length[11,12]. Could such a PGBgive all of the dark matter? The mass should be larthan about 10−22 eV, otherwise the uncertainty princple prevents the formation of structures at sufficiensmall scales[12]. Parameters exist that allow a unifiepicture of both dark matter and DE. In theories wsufficiently sparse textures, it may be possible to copute the magnitude of the DE and dark matter enedensities from measurements of neutrino massesmixings. Finally, the (super-)potential that gives tφij a vev at a large scale could play a role in inflatioa candidate for a such superpotential is

W =∑ij

Sij

(σijφij φ̄ij − µ2

ij

),

whereSij , φij andφ̄ij are chiral supermultiplets.

Acknowledgements

We thank D. Larson and Y. Nomura for manconversations. This work is supported in part

MIUR and by the EU under RTN contract MRTNCT-2004-503369, by DOE under contracts DE-FG90ER40542 and DE-AC03-76SF00098 and by Ngrant PHY-0098840.

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