PHYSICAL REVIEW D, VOLUME 65, 033003

Neutrino masses from operator mixing

Josep F. Oliver and Arcadi SantamariaDepartament de Fı´sica Teo`rica and IFIC, Universitat de Vale`ncia–CSIC, Dr. Moliner 50, E-46100 Burjassot (Vale`ncia), Spain

~Received 3 August 2001; published 7 January 2002!

We show that in theories that reduce, at the Fermi scale, to an extension of the standard model with twodoublets, there can be additional dimension five operators giving rise to neutrino masses. In particular thereexists a singlet operator which cannot generate neutrino masses at the tree level, but generates them throughoperator mixing. Under the assumption that only this operator appears at the tree level we calculate theneutrino mass matrix. It has the Zee mass matrix structure and leads naturally to bimaximal mixing. However,the maximal mixing prediction for solar neutrinos is very sharp even when higher order corrections areconsidered. To allow for deviations from maximal mixing a fine-tuning is needed in the neutrino mass matrixparameters. This fine-tuning relates the departure from maximal mixing in solar neutrino oscillations with theneutrinoless double beta decay rate.

DOI: 10.1103/PhysRevD.65.033003 PACS number~s!: 14.60.Pq, 12.60.Fr, 14.60.St, 14.80.Cp

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The simplest model for neutrino masses is based onseesaw mechanism@1–3#. In the seesaw mechanism the stadard model~SM! is enlarged with singlet right-handed netrinos. Then, a Dirac mass termMD mixing left-handed andright-handed neutrinos is possible. In addition, since righanded neutrinos do not carry any gauge charge, theyhave a Majorana massMR without compromising the gaugsymmetry. If the right-handed Majorana mass term is vlarge, as expected for a singlet mass term, very light Marana neutrino masses for left-handed neutrinos are obtathrough the diagonalization of the full mass matrix of neutfermions,mn5MD

2 /MR , thus justifying the small size of theneutrino masses. Since the Dirac mass term is proportionthe standard Higgs vacuum expectation value, this mecnism provides masses which aremn'LF

2/L, LF being theFermi scale andL the lepton number breaking scale. Thtype of behavior is much more general and, in fact, maneutrino mass models can be cast into this form. This canunderstood in the following way: if the SM is just the lowenergy effective manifestation of some underlying theothe effects of new physics can be represented by a seriegauge-invariant operators containing the SM fields whigher dimension operators suppressed by powers ofscale of new physics@4–6#. At low energies, the most relevant operators will be those with the lowest dimensinamely, dimension five operators. One can easily seethere is only one gauge-invariant operator of dimensionone can build with the field content of the standard model@5#

Lseesaw521

4

1

L~ lDFtW l !~ w̃†tWw!, ~1!

where l is the standard left-handed doublet of leptonsl̃

5 i t2l c, l c5C l̄ T (C is the charge conjugation operator!, w isthe Higgs doublet andw̃5 i t2w* , tW are the Pauli matrices inSU(2) space,F is a complex symmetric matrix in flavospace@SU(2) and flavor indices have been suppressed# andL is a scale related to the scale of new physics. It is clearthis Lagrangian does not conserve generational lepton nbers, but in addition it does not conserve the total lep

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number, which is violated in two units. In the SM, leptonumber is conserved because the requirement of renormability and the small particle content~no right-handed neu-trinos, no triplet scalars, etc.! but as long as the spectrumenlarged there is no strong reason for lepton number convation. Operator~1! makes this statement explicit. Thereforthis operator will be generated in any extension of the Sthat does not conserve lepton number.

When the Higgs scalar develops a vacuum expectavalue~VEV!, operator~1! will give rise to a neutrino Majo-rana mass matrix given by

M n5Fv2

L, ~2!

with v5^w (0)&5174 GeV, the SM Higgs VEV. If we takethe largest eigenvalue ofF to be of order 1 and use thlaboratory bound on thet-neutrino mass,mnt

,18 MeV, we

find thatL.106 GeV. Should one take the value suggestby atmospheric neutrino data,mnt

'0.06 eV, one would ob-

tain L'531014 GeV, a scale which is close to the unification scale; physics at very high-energy scales could aflow-energy physics in a measurable way through neutrmasses. However the relationship betweenL and masses onew particles can be quite different from the naive expections. For instance, it seems that the Lagrangian of Eq.~1! isgenerated by the exchange, among the leptons andHiggses, of a scalar triplet,x, with hypercharge 1. Then1/L'm/mx

2 with m the trilinear coupling of the triplet withthe Higgs boson doublets. However, this is not the only psibility. In fact, Eq. ~1! can be identically rewritten, after aSU(2) Fierz transformation, as

Lseesaw521

2

1

L~ lDw!F~ w̃†l !, ~3!

which suggests the exchange of a neutral heavy Majorfermion; thenL should be the mass of that fermion. Indeethe seesaw mechanism described above naturally implemthis possibility.

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JOSEP F. OLIVER AND ARCADI SANTAMARIA PHYSICAL REVIEW D65 033003

Given the full generality of this mechanism, which natrally relates the smallness of neutrino masses to thephysics scale, it is quite reasonable to try to fit the differneutrino mass models into this description. However, thinot always possible; on one side it can happen that neutmasses are generated only by operators with higher dimsions~for a recent analysis of the different possibilities s@7#! and on the other side models in which neutrino masare generated through radiative corrections are also diffito fit in the simplest scheme.

In this paper, we want to show that the uniqueness ofeffective seesaw mechanism can be relaxed a bit if new fiare allowed at the electroweak scale, in particular in modwith two light doublets. This opens the door for a new claof effective seesaw mechanisms in which the light neutrmasses are generated radiatively. This fact will allow uslower the lepton number breaking scale by several ordermagnitude. In addition, as we will see, this mechanism pdicts a very particular form for the neutrino mass matwhich seems well suited for explaining both atmospheric asolar neutrino data. This is not strange since a particrealization of this mechanism is the Zee model@8,9# and itsvariations, for instance, models with spontaneous symmbreaking of the lepton number by a doublet@10# or models inwhich the spontaneous breaking of the lepton number isduced by a hyperchargeless triplet@11,12#. This last class ofmodels has the interesting property that the triplet doescontribute to the invisibleZ decay width and that the leptonumber breaking VEV could be at the electroweak scale~nobounds from red giant cooling by Majoron emission!. Mod-els with hyperchargeless triplets have also been discurecently @13,14# in connection with the results of lasstandard-model fits. Variations solving the strongCP prob-lem can be found in@15,16#. All of these models predict aZee-type neutrino mass matrix~for a comprehensive reviewof extensions of the Higgs sector of the SM see@17,18#!.Recent fits to neutrino data@19–23# suggest some type obimaximal mixing which can be accommodated naturallythis type of models@24–27#. This observation has boosteagain the interest of models with Zee-type neutrino ma@28–34#.

In the same way that a variety of seesaw mechanism mels can be described as a single effective operator it woulinteresting to see if this class of models and perhaps otypes of models with two light doublets can be describedlow energies with just a few operators.

We will assume that the low-energy~Fermi scale! theoryis just the SM model, with no right-handed neutrinos, suppmented by an additional doublet. We will denote the twdoublets asw1 and w2. Then, in principle, nothing forbidsthat both doublets couple to the two types of quarks andthe leptons. However, it is well known that this, in generwill lead to neutral current flavor changing problems@35#.Therefore, for the moment we will consider that only onethe doublets,w1, couples to the fermions. This can bachieved naturally by assigning an additional consercharge to the doubletw2, lepton number, for example. Ocourse, this additional charge should be explicitly brokenthe Higgs potential in order to avoid the appearance o

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Goldstone boson once the doubletw2 acquires a VEV~mod-els in which the lepton number is broken spontaneously bdoublet @10,36,37# are excluded by the invisibleZ widthmeasured at the CERNe1e2 collider LEP!. Therefore, forthe moment we will assume the SM Yukawa couplingsleptons

LY5 l̄ YeRw11H.c. ~4!

Again,SU(2) and flavor indices have been suppressed anYis a 333 complex matrix in flavor space. However, becauno right-handed neutrinos have been introduced in the moand because onlyw1 couples to leptons, it can be chosewithout loss of generality, as diagonal with all its elemenreal and positive. As in the SM, neutrinos remain masslbecause there are no right-handed neutrinos and becauston number is conserved.

Now we will assume that this model is just the lowenergy manifestation of a more complete theory which wonly show up at higher energies. As discussed above, ifscale of new physics is high enough its effects can beparametrized by operators with higher dimensionality:

Leff5L2HSM11

LL11

1

L2L21•••. ~5!

Here L2HSM represents the renormalizable Lagrangianjust outlined, which is a minimal extension of the SM cotaining two doublets.

If the theory at the Fermi scale contains two doublets, ocan write four independent dimension five operators1

LT152

1

8

1

L~ lDT1tW l !~ w̃1

†tWw1!,

LT252

1

8

1

L~ lDT2tW l !~ w̃2

†tWw2!, ~6!

LT521

4

1

L~ lDTtW l !~ w̃2

†tWw1!, ~7!

and

LS521

4

1

L~ lDSl!~ w̃2

†w1!. ~8!

OperatorsLT1, LT2

can be excluded by the same symm

try used to forbid Yukawa couplings of the doubletw2 to thefermions. For instance, one can assign lepton numbers tow2in such a way that it does not have Yukawa couplingsfermions while the couplingsLT and LS remain allowed.

1In general one also expects higher-dimension operators wcould contribute to interesting processes asm-e conversion in nu-clei, m→eg, etc. For instance, when a charged scalar is integraout all those operators appear at one loop@38#.

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NEUTRINO MASSES FROM OPERATOR MIXING PHYSICAL REVIEW D65 033003

L(w2)522 will do the job and will forbid operatorsLT1,

LT2. Therefore, in the following we will only consider op

eratorsLT andLS .It is important to notice that the operator in Eq.~8! does

not exist with only one doublet because the singlet coupof two scalar doublets is antisymmetric@in SU(2) compo-nents it is juste i j w2iw1 j #. In addition, one can see that, duto Fermi statistics, the 333 matrix in flavor space,T, issymmetric while the singlet couplingS is a complex anti-symmetric matrix.

When the Higgs boson fields develop a charge conserVEV, operatorLT gives rise to a neutrino Majorana masHowever, the singlet operatorLS does not give any mass tthe neutrinos because the product of the two doubletshypercharge 1, and a singlet with hypercharge 1 alsocharge 1@different from the triplet combination~7! whichhas neutral components#. For this reason the singlet operatdoes not seem very interesting at first sight. However, a vinteresting situation arises when, for some reason~limitedparticle content of the full theory!, only operator~8! arises attree level.2 As commented before, it cannot give rise to netrino masses after spontaneous symmetry breaking. Hever, one expects that renormalization effects will mixoperators with the same dimensionality and the same qtum numbers. Therefore, Eqs.~8! and~7! will mix under therenormalization group and operator~7! will be generated atone loop even if it did not appear at tree level. In fact, ocan easily see, by computing the diagram in Fig. 1~and thecrossed diagram! and taking the divergent part, that the mtrix T obeys the following renormalization-group equati~RGE!:

md

dmT5

1

~4p!2~SYY†1Y* YTST!1•••. ~9!

Here, the dots represent extra contributions to the renormization group of theT matrix which are proportional to theT

2An example of this situation is provided by the Zee modelwhich a charged singlet,h1, and an extra doublet are introduce

with couplings lD f lh1 and mh2w̃2†w1, then, for a heavyh1 one

finds, at tree level, thatS/L54m f /mh2 while the triplet operator

cannot be obtained at tree level.

FIG. 1. Diagram contributing to the mixing of the singlet antriplet operators.

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matrix itself and, therefore, cannot generate it if it did nexist at some scale. This RGE is peculiar with respect toRGE we are used to seeing for the Yukawa couplings, in bthe SM and the MSSM, in that there is a piece that doesdepend on theT matrix. In both the SM and the MSSM therare several chiral symmetries broken only by the Yukacouplings that ensure that the RGE of those couplings shotransform in a covariant way with respect to those symmtries. This ensures, in this type of theory, that fermion mascannot be generated through radiative corrections. Equa~9! is not of this type and, therefore, even if the couplingTdid not exist at the scaleL at which the operators wergenerated, it will arise through operator mixing. It is veeasy to integrate Eq.~9! by keeping only the leading logarithm @a more sophisticated integration can be performedtaking into account the running of the standard Yukawa cplings, the running of theS coupling itself, and the extracouplings we neglected in Eq.~9!, however, this effect ishigher order in the couplings and since the couplingssmall we expect a small effect#. The result is that

T~mZ!'1

~4p!2~SYY†1Y* YTST!logS mZ

L D1T~L!,

~10!

where we have identifiedmZ with the Fermi scale. We willassume thatT(L) is not generated at tree level. Of courT(L) could also pick up contributions at one loop~or higherloops!. To compute them one would need to know the detaof the full theory in which our effective theory is embeddeHowever, ifL is large enough,T(mZ) will be dominated bythe model independent logarithmic piece in Eq.~10! whichcan be computed in the effective theory. So, as a firstproximation we will assumeT(L)'0 and further on we willkeepT(L) only when the logarithmic pieces vanish.

After spontaneous symmetry breaking, if bothw1 andw2develop a VEV, operator~7! will give rise to a neutrino massmatrix for the left-handed neutrinos given by

M n'tanb1

~4p!2~SYY†1Y* YTST!logS mZ

L D v12

L, ~11!

wherev15^w1(0)&, v25^w2

(0)& are the VEV’s of the two dou-blets and tanb5v2 /v1. The seesaw structure is apparentthe last term. The other factors, however, are also importFirst, the neutrino mass matrix comes naturally proportioto the mass of the leptons squared which gives an imporsuppression since lepton Yukawa couplings are small. Sond, it contains the standard loop suppression factor 1/(4p)2

and, third, the ratio of VEV’s of the second doubletv2 to thestandard doublet VEV, tanb5v2 /v1, can give an additionasuppression factor. All together, with this mechanism oneachieve the same neutrino masses one would obtainstandard seesaw mechanism with aL which is at least sixorders of magnitude smaller. This could put the scale ofnew physics responsible for neutrino masses at the reacthe next generation of accelerators if the ratio of VEVv2 /v1 and/or the largestSi j are very small. However, per

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JOSEP F. OLIVER AND ARCADI SANTAMARIA PHYSICAL REVIEW D65 033003

haps the most interesting aspect of Eq.~11! is the structure ofthe mass matrix, inherited from the antisymmetric structof the singlet couplingS; if we choose for the Yukawa coupling of the leptons a diagonal form we find

M n'2m0S 0 Sem~xe2xm! Set~xe21!

Sem~xe2xm! 0 Smt~xm21!

Set~xe21! Smt~xm21! 0D ,

~12!

with xe5me2/mt

2 , xm5mm2 /mt

2 and

m05tanbmt

2

L~4p!2logS L

mZD .

The structure of this mass matrix is very interesting sincediagonal elements are zero and contains only three freerameters~the three elements of the antisymmetric matrixS).It is convenient to rewrite it as

M n'S 0 mem met

mem 0 mmt

met mmt 0D . ~13!

This form of mass matrix has been considered recentlyorder to fit both atmospheric and solar neutrino data@24–27#. Let us review some of the results; in order to explasolar neutrino data in terms of oscillations one needs a msquared difference, Ds which is 10211 eV2<Ds<1025 eV2. A global analysis including the results of SNsuggests mixings close to maximal. On the other handorder to explain atmospheric neutrino data one needs a msquared difference,Da'331023 eV2 and also a very largemixing. The particular structure of the obtained mass ma~it is traceless! implies that the sum of the eigenvalues is zea fact which constrains the possible solutions. An analysithe different possibilities in terms of this mass matrix hbeen carried out in@24–27# where it has been shown thaonly the case withmem;met andmmt!mem ,met is accept-able. This naturally predicts maximal mixing for solar osclations which, after SNO, seems to be the only viable pobility. In fact the Zee mass matrix predicts, in this case@28#,

sin2 2us5121

16S Ds

DaD 2

. ~14!

This is a very strong prediction which is compatible with tlow probability, low mass~LOW! and the vacuum~VAC!solutions of the solar neutrino problem. The large mixiangle~LMA ! solution, which right now seems to be the prferred solution, would be marginally compatible with thprediction. However, it is important to notice that in genewe expect corrections to the Zee mass formula. By requirthat only one doublet couples to fermions we have takenmost restrictive~and predictive! possibility. By relaxing thisassumption one can also perfectly fit the LMA region in soneutrino parameters@29#. But even in the restrictive case iwhich only one doublet couples to fermions one can a

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take into account subdominant contributions. In fact, on geral grounds we expect modifications to the particular foof the mass matrix with all the elements in the diagonvanishing. There is no symmetry that enforces this structtherefore one expects that at some point this structurereceive corrections. This happens, for instance, in themodel, where diagonal entries are generated at two lo@31#. For our purposes we can include all those correctionthe initial contributions at the scaleL, that is, by consideringa nonvanishingT(L) which we could take as a general symmetric matrix. Given the bimaximal mixing required by thdata it is natural to parametrize the neutrino mass matrixfollows:

M n'm03 S 01

A2

1

A2

1

A20 0

1

A20 0

D 1eS gee gem get

gem gmm gmt

get gmt gtt

D 4 ,

(15)

where, without loss of generality we can takeget52gemand choose one of theg ’s equal to one, defining in this waythe normalization ofe. We will assume that the first termgenerated by running and containing the logarithmichancement, is the dominant one and gives the scale of aspheric neutrinos. The term proportional toe is subdominantand cannot be computed in the effective theory. Its explform depends on the details of the underlying theory ancould contain one-loop contributions not enhanced by lalogarithms and/or higher loop contributions, dependingthe underlying theory. For simplicity, we will also take all thg ’s as real. Notice that the mechanism suggested in thisper only gives a leading-order contribution with zeros in tdiagonal. The extra structure assumed in the leading mterm (mem'met and mmt'0) must be imposed with somadditional symmetries which, however, are not unnaturathis type of model@8#. The modifications introduced by thterm proportional toe can be qualitatively very importantOne of the most interesting predictions of the leading-ormass matrix is that there is no neutrinoless double beta de~NDBD!; since the NDBD amplitude is proportional t(M n)ee it is obvious that a mass matrix with zeros in thdiagonal forbids NDBD. However any correction to thleading-order mass matrix introducing diagonal entries wlead to NDBD. On the other hand, corrections to the leadiorder mass matrix are necessary in order to accommosolar neutrino mass differences and small departures fmaximal mixing in solar and/or atmospheric neutrinos. Itimportant to check how these corrections could modifysharp predictions of the model.

By diagonalizing the mass matrix at second order ine oneeasily obtains that

Da'm02 , ~16!

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NEUTRINO MASSES FROM OPERATOR MIXING PHYSICAL REVIEW D65 033003

sin2 2ua'121

2@16gem1~gmm2gtt!

2#e2, ~17!

Ds'm02~2gmt12gee1gmm1gtt!e, ~18!

sin2 2us'121

16S Ds

DaD 2

11

2@gee~2gmt1gmm1gtt!

2~gmm2gtt!2#e2. ~19!

SinceDs /Da is known to be small, the natural predictionthe model is just maximal mixing for the two angles tovery good degree of precision. Departures from maximmixing are allowed naturally for the atmospheric mixinangle sinceDs does not depend ongem which controls theatmospheric mixing~for gmm5gtt); therefore it can be madlarge without any conflict with solar neutrino data. Howevto accommodate deviations from maximal mixing in the slar neutrino parameters is more delicate. One needs to mDs small while keeping the last term in Eq.~19! relativelylarge. This is clearly unnatural. For instance, in the casewhich gmm5gtt50 one obtainsDs /Da'2(gmt1gee) whilesin2 2us'12(Ds/Da)

2/161gmtgee. Therefore, to obtain avery smallDs /Da while keeping a sizeable contribution tsin2 2us one would need to fine-tune the couplings in sucway thatgee'2gmt . Although it is not natural, this possibility might be interesting because perhaps it is the only wto accommodate LMA within this scheme and because if irealized in nature it will link the deviation from maximamixing in solar neutrino oscillations with neutrinoless doubbeta decay; as commented before, the NDBD amplitudproportional to the so-called effective neutrino mass^mn&[(Uei

2 mn i5(M n)ee.In this work we have shown that, in models that contain

their low-energy~Fermi scale! effective theory two Higgsboson doublets, there are four independent dimensiongauge-invariant operators which violate lepton numbThree of them, which couple leptons to doublets in the tripchannel, generate masses at tree level when the doublequire VEV’s. The other operator, which couples leptons adoublets in the singlet channel, cannot generate masse

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tree level. However, loop corrections mix all operators unthe renormalization group and, therefore, the singlet operalso gives rise to neutrino masses when the doublets acqVEV’s.

Under the assumption that only the singlet operatorgenerated at tree level, at the scale of new physics, onecompute the induced neutrino masses at one loop. Neutmasses are suppressed by several factors, the loop factomasses of charged leptons, and the ratio of the two doubVEV’s, therefore allowing for a new physics scale seveorders of magnitude lower than the one needed in tree-lemechanisms for neutrino masses.

Because of the structure of the singlet effective operawhich necessitates antisymmetric couplings in flavor, thetained mass has the structure of the Zee mass matrixzero entries in the diagonal. This structure naturally accomodates bimaximal mixing and is well suited for explaininboth solar and atmospheric neutrino data. However, insimplest scheme it is very difficult to accommodate the LMsolution for solar neutrinos because it gives a sharp pretion for sin2 2us51 once one takes into account the madifferences needed for solar neutrinos.

We have investigated the possibility of accommodatsin2 2usÞ1 in this scheme by considering nonleading conbutions to the mass matrix since, on general grounds,expects to generate nonzero entries for the diagonal elemof the mass matrix. We found that the prediction sin2 2us51 is quite stable and, only by fine-tuning the parametersthe mass matrix, is it possible to accommodate at the stime sin2 2usÞ1 and Ds /Da!1. This fine-tuning, eventhough unnatural, has the interesting property that relatesrate of neutrinoless double beta decay to the departure fmaximal mixing in solar neutrino oscillations.

J.F.O. and A.S. are indebted, for their hospitality, to tInternational School for Advanced Studies, SISSA~Trieste!and the CERN TH division, respectively, where part of thwork has been done. This work has been funded by CICunder the Grant AEN-99-0692, by DGESIC under the GrPB97-1261, and by the DGEUI of the Generalitat Valenciaunder the Grant GV98-01-80.

ds

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