3-Neutrino mass spectrum from combining seesaw and radiative neutrino mass mechanisms

3 August 2000

Physics Letters B 486 Ž2000. 385–390 www.elsevier.nlrlocaternpe

3-Neutrino mass spectrum from combining seesaw and radiative neutrino mass mechanisms W. Grimus a , H. Neufeld a,b a

b

Institut fur ¨ Theoretische Physik, UniÕersitat ¨ Wien, Boltzmanngasse 5, A–1090 Wien, Austria Departament de Fısica Teorica, IFIC, UniÕersitat de Valencia – CSIC, Apt. Correus 2085, E–46071 Valencia, Spain ´ ` ` ` Received 29 March 2000; received in revised form 14 June 2000; accepted 20 June 2000 Editor: R. Gatto

Abstract We extend the Standard Model by adding a second Higgs doublet and a right-handed neutrino singlet with a heavy Majorana mass term. In this model, there are one heavy and three light Majorana neutrinos with a mass hierarchy m 3 4 m 2 4 m1 such that that only m 3 is non-zero at the tree level and light because of the seesaw mechanism, m 2 is generated at the one-loop and m1 at the two-loop level. We show that the atmospheric neutrino oscillations and large mixing MSW solar neutrino transitions with D m2atm , m23 and D m2solar , m22 , respectively, are naturally accommodated in this model without employing any symmetry. q 2000 Elsevier Science B.V. All rights reserved.

1. Introduction At present, neutrino oscillations w1,2x play a central role in neutrino physics. Recent measurements of the atmospheric neutrino flux show convincing evidence for neutrino oscillations w3x with a masssquared difference D m 2atm ; 10y3 % 10y2 eV 2 . It is also likely that the solar neutrino deficit finds an explanation in terms of neutrino oscillations w4x, either by the MSW effect w5x with D m2solar ; 10y5 eV 2 or by vacuum oscillations with D m 2solar ; 10y1 0 eV 2 . For recent reviews about neutrino oscillations see, e.g., Ref. w6x. Confining ourselves to 3-neutrino oscillations and thus ignoring the LSND result w7x, neutrino flavour

E-mail address: [email protected] ŽW. Grimus..

mixing w8x is described by a 3 = 3 unitary mixing matrix U defined via 3

na L s

Ý Ua j n j L

with

a s e, m ,t ,

Ž 1.1 .

js1

where na L and n j L are the left-handed components of the neutrino flavour and mass eigenfields, respectively. Then, the solar and atmospheric neutrino mixing angles are given by sin2 2 usolar s

4 < Ue1 < 2 < Ue2 < 2

,

Ž 1.2 .

sin2 2 uatm s 4 < Um 3 < 2 1 y < Um 3 < 2 ,

Ž 1.3 .

Ž
ž

2

/

respectively. For the small-mixing MSW solution of the solar neutrino problem, sin2 2 usolar is of order 5 = 10y3 , whereas for the vacuum oscillation and

0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 7 6 9 - 3

386

W. Grimus, H. Neufeldr Physics Letters B 486 (2000) 385–390

large-mixing MSW solutions this quantity is of order one w4x. Future experimental data will hopefully allow to discriminate between the different possible solutions. On the other hand, for the atmospheric neutrino oscillations the results of the SuperKamiokande experiment give best fit values sin2 2 uatm s 0.99 % 1 and sin2 2 uatm R 0.84 at 90% CL w9x. The above-mentioned values of the oscillation parameters pose considerable problems for model builders in addition to the problem of explaining the smallness of neutrino masses. From now on we concentrate on Majorana neutrinos. There is a vast literature on models of 3-neutrino masses and mixing Žsee, e.g., the reviews w10–12x and also Ref. w13x and citations therein.. One possibility to explain the smallness of the neutrino masses is the see-saw mechanism w10,14,15x. The other two mechanisms are obtained by extensions of the Standard Model ŽSM. in the Higgs sector w16x without adding any leptonic multiplets: The first one needs an extension by a Higgs triplet w17x and leads to neutrino masses at the tree level. The smallness of the neutrino masses is explained by the small triplet vacuum expectation value ŽVEV. which is achieved by a large mass scale in the Higgs potential Žtype II seesaw. w18x. The other possibility is given by purely radiative neutrino masses with the generic examples of the Zee model w19x Žone-loop masses. and the Babu model w20x Žtwo-loop masses.. Examples of these types can be found, e.g., in Refs. w21,22x. In this paper we will discuss a model which combines the standard see-saw mechanism with radiative neutrino mass generation. In this framework, no other Higgs multiplets apart from scalar doublets are needed. The most general version of such a scenario with n L lepton doublets and charged lepton singlets, n R right-handed neutrino singlets and n H Higgs doublets has been discussed in Ref. w23x. Here we confine ourselves to the most economic case describing a viable 3-neutrino mass spectrum, namely n R s 1 and n H s 2. As was shown in Ref. w23x Žsee also Ref. w24x., this case leads to a heavy and a light neutrino at the tree level according to the see-saw mechanism, and to one light neutrino mass at the one-loop and the two-loop level, respectively. In the following we will demonstrate that this model is capable of generating a hierarchical mass spectrum

fitting well with the mass-squared differences derived from the solar MSW effect and atmospheric neutrino data and that it naturally accommodates large mixing angles corresponding to both masssquared differences. The fact that tree level and loop neutrino masses appear in our model has an analogy with the models combining the Higgs triplet mechanism with radiative neutrino masses w13,25,26x Žfor a SUSY model combining tree level and 1-loop neutrino masses see Ref. w27x..

2. The model We discuss 3-neutrino oscillations in the framework of an extension of the SM, where a second Higgs doublet ŽFa , a s 1,2. and a right-handed neutrino singlet n R are present in addition to the SM multiplets w23x. Thus the Yukawa interaction of leptons and scalar fields is given by 2

LY s yL

Ý

ž LG F a

as1

a

l R q L DaF˜a n R / q h.c.

Ž 2.1 .

with F˜a s i s 2Fa) . Ga and Da are 3 = 3 and 3 = 1 matrices, respectively. The singlet field n R permits the construction of an explicit Majorana mass term LM s 12 MR n RT Cy1n R q h.c. ,

Ž 2.2 .

where we assume MR ) 0 without loss of generality. In this 2-Higgs doublet model, spontaneous symmetry breaking of the SM gauge group is achieved by the VEVs ²Fa : 0 s

ž

0 Õar'2

/

,

Ž 2.3 .

which satisfy the condition

(

Õ ' < Õ 1 < 2 q < Õ 2 < 2 , 246 GeV .

Ž 2.4 .

The VEVs Õ 1,2 generate the tree level mass matrix Mls

1

2

Õa Ga '2 aÝ s1

Ž 2.5 .

for the charged leptons diagonalized by ULl †M l URl s Mˆ l

Ž 2.6 .

W. Grimus, H. Neufeldr Physics Letters B 486 (2000) 385–390

with unitary matrices ULl , URl and with a diagonal, positive Mˆ l . The most general Majorana neutrino mass term in the model presented here has the form 1 2

v LT Cy1 Mn

v L q h.c. with

vL s

nL

ž / Ž nR .

c

.

The two non-vanishing mass eigenvalues are given by

m3 s

Ž 2.7 . The left-handed field vector v L has four entries according to the three active neutrino fields plus the right-handed singlet. The symmetric Majorana mass matrix Mn is diagonalized by UnT Mn Un s diag Ž m1 ,m 2 ,m 3 ,m 4 .

387

m4 s

( (

MR2

q m2D y

4 MR2

q m2D q

4

MR 2 MR 2

,

m2D MR

, MR ,

and

Ž 3.7 .

where the approximate relations refer to the limit m D < MR .

Ž 2.8 .

with a unitary matrix Un and m i G 0. 4. One-loop corrections and the neutrino mass spectrum 3. The tree-level neutrino mass matrix From Eqs. Ž2.1., Ž2.2. and Ž2.3. we obtain the tree-level version of Mn : MnŽ0. s

ž

0

MD)

MD†

MR

/

Ž 3.1 .

with MD s

Õa)Da . '2 aÝ s1

Ž u1 ,u 2 ,u 3 ,u 4 .

uX u1,2 s 1,2 , 0

ž

/

MR

,

Ž 3.4 .

(

m D s 5 MD 5 s MD† MD .

Ž 3.5 .

The uX1,2,3 form an orthonormal system of complex 3-vectors with the properties uX1,2

H MD ,

dMs

/

Ž 4.1 .

MR 8p 2

Ý

Mb)

Mb2 ln Ž MRrMb .

b

MR2 y Mb2

Mb† q MD) AMD†

Ž 4.2 .

Mb s

where 2 mD

MR

1

/

sinq uX3 , cosq

tan2q s

MD†

with

cosq uX3 u3 s i , ysinq

ž /

ž

MD)

Ž 3.3 .

with

u4 s

ž

dM

Ž 3.2 .

The tree-level mass matrix Ž3.1. is diagonalized by the unitary matrix UnŽ0. s

MnŽ1. s

with d M being a symmetric 3 = 3 matrix. Its explicit form is given by w23x

2

1

By one-loop corrections, the form of the neutrino mass matrix is changed to

uX3 s MD rm D

.

Ž 3.6 .

2

ba)Da . '2 aÝ s1

Ž 4.3 .

In deriving this formula, terms suppressed by a factor of order MD rMR have been neglected. The first term in Ž4.2. is generated by neutral Higgs exchange. The sum in Ž4.2. runs over all physical neutral scalar fields F b0 s '2 Ý a s1,2 ReŽ ba)Fa0 ., which are characterized by three two-dimensional complex unit vectors b w23x with

Ý ba bb s b

Õa Õb Õ2

.

Ž 4.4 .

W. Grimus, H. Neufeldr Physics Letters B 486 (2000) 385–390

388

Note that we do not consider corrections to MD and MR in the neutrino mass matrix. The unitary matrix diagonalizing Ž4.1. can be written in the form w23x UnŽ1. s UnŽ0. V

Ž 4.5 .

with V y 1 being of one-loop order. By an appropriate choice of the matrix V we obtain



X uXT 1 d Mu1

X uXT 1 d Mu 2

0

X uXT 2 d Mu1

X uXT 2 d Mu 2

0 0

0 0

0 m3 0

0

0 . 0 m4

Ž 4.6 .

the relation m D s 5 MD 5 implies Õ 1) c1 q Õ 2) c 2 s 0, but the quantity < c1 < 2 q < c 2 < 2 remains an independent parameter of our model, only restricted by ‘‘naturalness’’, which requires that it is of order 1. From Ž4.2., using the Cauchy–Schwarz inequality and 5 b 5 s 1, we obtain the upper bound m2D 8p 2 MR

ln Ž MRrMb . Mb2

Ý 1 y M 2rM 2 b

1 8p

2

m3

M02 Õ

2

ln

MR M0

,

Ž 4.9 .

0

The second term in Ž4.2. containing the matrix A Žcontributions from Z exchange and contributions from neutral scalar exchange other than the first term in Eq. Ž4.2.. cannot contribute to Ž4.6. because of Ž3.6.. The remaining off-diagonal elements in Ž4.6. can be removed by choosing uX1 orthogonal to D1 and D2 . This shows at the same time that one of the neutrinos remains still massless at the one-loop level. However, there is no symmetry enforcing m1 s 0 and the lightest neutrino will in general get a mass at the two-loop level w28x. Finally, the vector uX2 has to be orthogonal to uX1 and uX3 , and its phase is fixed by the positivity of m 2 . Defining Õ X† ca s Ž 4.7 . '2 m D u 2 Da ,

m2 F

m2 ;

where M0 is a generic physical neutral scalar mass. Note that for M0 ; Õ the relation m 2 < m 3 comes solely from the numerical factor 1r8p 2 appearing in the loop integration.

MˆnŽ1. s UnŽ1.T MnŽ1.UnŽ1.

s

tral scalars. The elements of these matrix are independent of the masses Mb2 Žsee, e.g., Ref. w29x.. From these considerations it follows that the order of magnitude of m 2 can be estimated by

b

R

Õ2

Ž < c1 < 2 q < c 2 < 2 . . Ž 4.8 .

Note that cancellations in Eq. Ž4.2. in the summation over the physical neutral scalars do not happen in general because the vectors b are connected with the diagonalizing matrix of the mass matrix of the neu-

5. Discussion Let us first discuss the neutrino mass spectrum in the light of atmospheric and solar neutrino oscillations. Due to the hierarchical mass spectrum in our model we have

D m2atm , m23 and

D m2solar , m22 .

Ž 5.1 .

From the atmospheric neutrino data, using the best fit value of D m2atm , one gets w3x m3 s

m2D MR

, 0.06 eV .

Ž 5.2 .

A glance at Eq. Ž4.9. shows that m 2 is only one or two orders of magnitude smaller than m 3 if MR represents a scale larger than the electroweak scale. Therefore, our model cannot describe the vacuum oscillation solution of the solar neutrino problem. On the other hand, with the MSW solution one has w4x m 2 ; 10y2 .5 eV

Ž 5.3 .

and, therefore, m 2rm 3 ; 0.05, which can easily be achieved with Eq. Ž4.9.. In principle, the unknown mass scales m D and MR are fixed by Eqs. Ž5.2. and Ž5.3. Žsee Eq. Ž4.9. for the analytic expression of m 2 .. However, due to the logarithmic dependence of Eq. Ž4.9. on MR and the freedom of varying the scalar masses, whose natural order of magnitude is given by the electroweak scale, the heavy Majorana mass could be anywhere between the TeV scale and the Planck mass. Let us therefore give a reasonable example. Assuming that m D has something to do with the mass of the tau lepton, we fix it at m D s 2 GeV. Conse-

W. Grimus, H. Neufeldr Physics Letters B 486 (2000) 385–390

quently, from Eq. Ž5.2. we obtain MR , 0.7 = 10 11 GeV. Inserting this value into Eq. Ž4.9. and using Ž5.3., the reasonable estimate M0 ; 100 % 200 GeV ensues, which is consistent with the magnitude of the VEVs. This demonstrates that our model can naturally reproduce the mass-squared differences needed to fit the atmospheric and solar neutrino data, where the fit for the latter is done by the MSW effect. Now we come to the mixing matrix Ž1.1., which is given by U s ULl †UnX

with

UnX s Ž uX1 ,uX2 ,uX3 .

Ž 5.4 .

for MR 4 m D Žsee Eq. Ž3.4.. and neglecting V Ž4.5.. Since the directions of the vectors D1,2 in the 3-dimensional complex vector space determine UnX , we will have large mixing angles in this unitary matrix as long as we do not invoke any fine-tuning of the elements of D1,2 . ŽThis is in contrast to Ref. w23x where we assumed that ŽULl †Da . j ; m l jrÕ, where the m l j are the charged lepton masses.. Also ULl might have large mixing angles, but could also be close to the unit matrix in analogy to the CKM matrix in the quark sector. Since we do not expect any correlations between ULl and UnX , it is obvious that our model favours large mixing angles in the neutrino mixing matrix U. On the other hand, there is a restriction on the element Ue3 from the results of the Super-Kamiokande atmospheric neutrino experiment and the CHOOZ result w30x Žabsence of ne disappearance., which is approximately given by w31x < Ue3 < 2 Q 0.1. Furthermore, the Super-Kamiokande results imply that sin2 2 uatm is close to 1 Žsee introduction.. These restrictions find no explanation in our model, but, as we want to argue, not much tuning of the elements Ua3 s ŽULl † uX3 .a is needed to satisfy them. If we take the ratios < Ue3 <: < Um 3 <: < Ut 3 < s 1:2:2 as an example we find < Ue3 < 2 s 1r9 , 0.11 and Eq. Ž1.3. gives sin 2 2 uatm s 80r81 , 0.99. To show that the favourable outcome for sin2 2 uatm does not depend on having < Um 3 < , < Ut 3 <, let us consider now 1:3:2 for the elements < Ua3 <. Then we obtain < Ue3 < 2 s 1r14 , 0.07 and sin2 2 uatm s 45r49, 0.92. Thus not much fine-tuning is necessary to meet the restrictions on < Ue3 < 2 and sin2 2 uatm w10,15,32x. Obviously, our model would be in trouble if it turned out that the atmospheric and solar neutrino oscillations decouple with

389

™

high accuracy ŽUe3 0. or atmospheric mixing is Õery close to maximal. Some tuning of Ue2 is also required to reproduce the large-mixing MSW solution of the solar neutrino problem. As an illustration of the amount of the tuning, looking at Fig. 8 of Ref. w4x, Gonzalez-Garcia et al. Žcombined analysis., we estimate 0.44 Q < Ue2 < Q 0.64 from the 90% CL area in the limit Ue3 0. Since there is no lepton number conservation in the present model, lepton flavour changing processes are allowed. The branching ratios of the decays m " e "g and m " e "eqey have the most stringent bounds w33x. With our assumption on the size of the Yukawa couplings Da , the contribution of the charged Higgs loop w2x to m " e " g leads to a lower bound of about 100 m D for the charged Higgs mass. The decay m " e "eqey , proceeding through neutral Higgs scalars at the tree level, restricts only some of the elements of the Yukawa coupling matrices Ga , but not those of the Da couplings relevant in the neutrino sector. It is well known that the effective Majorana mass relevant in Ž bb . 0 n decay is suppressed to a level below 10y2 eV in the 3-neutrino mass hierarchy w34x, which is considerably smaller than the best present upper bound of 0.2 eV w35x. In summary, we have discussed an extension of the Standard Model with a second Higgs doublet and a neutrino singlet with a Majorana mass being several orders of magnitude larger than the electroweak scale. We have shown that this model yields a hierarchical mass spectrum m 3 4 m 2 4 m1 of the three light neutrinos by combining the virtues of seesaw Ž m 3 . and radiative neutrino mass generation Ž m 2 / 0 and m1 s 0 at the one-loop level., and that it is able to accommodate easily the large mixing angle MSW solution of the solar neutrino problem and the nm nt solution of the atmospheric neutrino anomaly. We want to stress that by virtue of Eq. Ž4.9. the order of magnitude of D m2solarrD m 2atm is fixed apart from a logarithmic dependence on MR . This logarithmic factor in Ž4.9. varies roughly between 2 and 40 for MR between 1 TeV and the Planck scale and M0 ; 100 GeV. By construction, the neutrino sector of our model is very different from the charged lepton sector. The model offers no explanation for the mass spectrum of the charged leptons. We want to stress that the scalar sector of

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W. Grimus, H. Neufeldr Physics Letters B 486 (2000) 385–390

the model is exceedingly simple and that – apart from the Standard Model gauge group – no symmetry is involved. The moderate smallness of < Ue3 < 2 and closeness of sin2 2 uatm to 1 is controlled by the ratios of the elements of the third column of the mixing matrix U. We have argued that the ratios of < Ua3 < Ž a s e, m ,t . required to give < Ue3 < 2 Q 0.1 and sin2 2 uatm R 0.84 are quite moderate, with 1:2:2 being a good example. An similar moderate tuning is necessary for the reproduction of the large mixing angle MSW solution. Suitable ratios of the elements of the mixing matrix have to be assumed in the model presented here, but might eventually find an explanation by embedding it in a larger theory relevant at the scale MR .

w12x w13x w14x

w15x w16x w17x w18x

w19x

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