3-neutrino mass spectrum from combining seesaw and radiative neutrino mass mechanisms
TRANSCRIPT
3 August 2000
Ž .Physics Letters B 486 2000 385–390www.elsevier.nlrlocaternpe
3-Neutrino mass spectrum from combining seesaw and radiativeneutrino mass mechanisms
W. Grimus a, H. Neufeld a,b
a Institut fur Theoretische Physik, UniÕersitat Wien, Boltzmanngasse 5, A–1090 Wien, Austria¨ ¨b 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 2000Editor: R. Gatto
Abstract
We extend the Standard Model by adding a second Higgs doublet and a right-handed neutrino singlet with a heavyMajorana mass term. In this model, there are one heavy and three light Majorana neutrinos with a mass hierarchym 4m 4m such that that only m is non-zero at the tree level and light because of the seesaw mechanism, m is3 2 1 3 2
generated at the one-loop and m at the two-loop level. We show that the atmospheric neutrino oscillations and large mixing1
MSW solar neutrino transitions with Dm2 ,m2 and Dm2 ,m2 , respectively, are naturally accommodated in this modelatm 3 solar 2
without employing any symmetry. q 2000 Elsevier Science B.V. All rights reserved.
1. Introduction
w xAt present, neutrino oscillations 1,2 play a cen-tral role in neutrino physics. Recent measurements ofthe atmospheric neutrino flux show convincing evi-
w xdence for neutrino oscillations 3 with a mass-squared difference Dm2 ;10y3 %10y2 eV 2. It isatm
also likely that the solar neutrino deficit finds anw xexplanation in terms of neutrino oscillations 4 ,
w x 2 y5either by the MSW effect 5 with Dm ;10solar
eV 2 or by vacuum oscillations with Dm2 ;10y10solar
eV 2. For recent reviews about neutrino oscillationsw xsee, e.g., Ref. 6 .
Confining ourselves to 3-neutrino oscillations andw xthus ignoring the LSND result 7 , neutrino flavour
Ž .E-mail address: [email protected] W. Grimus .
w xmixing 8 is described by a 3=3 unitary mixingmatrix U defined via
3
n s U n with ase,m ,t , 1.1Ž .Ýa L a j jLjs1
where n and n are the left-handed componentsa L jL
of the neutrino flavour and mass eigenfields, respec-tively. Then, the solar and atmospheric neutrinomixing angles are given by
< < 2 < < 24 U Ue1 e22sin 2u s , 1.2Ž .solar 22 2< < < <U q UŽ .e1 e2
2 < < 2 < < 2sin 2u s4 U 1y U , 1.3Ž .ž /atm m3 m3
respectively. For the small-mixing MSW solution ofthe solar neutrino problem, sin2 2u is of ordersolar
5=10y3, whereas for the vacuum oscillation and
0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 00 00769-3
( )W. Grimus, H. NeufeldrPhysics Letters B 486 2000 385–390386
large-mixing MSW solutions this quantity is of orderw xone 4 . Future experimental data will hopefully al-
low to discriminate between the different possiblesolutions. On the other hand, for the atmosphericneutrino oscillations the results of the Super-Kamiokande experiment give best fit valuessin2 2u s0.99%1 and sin2 2u R0.84 at 90%atm atm
w xCL 9 .The above-mentioned values of the oscillation
parameters pose considerable problems for modelbuilders in addition to the problem of explaining thesmallness of neutrino masses. From now on weconcentrate on Majorana neutrinos. There is a vastliterature on models of 3-neutrino masses and mixingŽ w x w xsee, e.g., the reviews 10–12 and also Ref. 13 and
.citations therein . One possibility to explain thesmallness of the neutrino masses is the see-saw
w xmechanism 10,14,15 . The other two mechanismsare obtained by extensions of the Standard ModelŽ . w xSM in the Higgs sector 16 without adding anyleptonic multiplets: The first one needs an extension
w xby a Higgs triplet 17 and leads to neutrino massesat the tree level. The smallness of the neutrinomasses is explained by the small triplet vacuum
Ž .expectation value VEV which is achieved by aŽlarge mass scale in the Higgs potential type II
. w xseesaw 18 . The other possibility is given by purelyradiative neutrino masses with the generic examples
w x Ž .of the Zee model 19 one-loop masses and thew x Ž .Babu model 20 two-loop masses . Examples of
w xthese types can be found, e.g., in Refs. 21,22 .In this paper we will discuss a model which
combines the standard see-saw mechanism with ra-diative neutrino mass generation. In this framework,no other Higgs multiplets apart from scalar doubletsare needed. The most general version of such ascenario with n lepton doublets and charged leptonL
singlets, n right-handed neutrino singlets and nR Hw xHiggs doublets has been discussed in Ref. 23 . Here
we confine ourselves to the most economic casedescribing a viable 3-neutrino mass spectrum, namely
w x Žn s1 and n s2. As was shown in Ref. 23 seeR Hw x.also Ref. 24 , this case leads to a heavy and a light
neutrino at the tree level according to the see-sawmechanism, and to one light neutrino mass at theone-loop and the two-loop level, respectively. In thefollowing we will demonstrate that this model iscapable of generating a hierarchical mass spectrum
fitting well with the mass-squared differences de-rived from the solar MSW effect and atmosphericneutrino data and that it naturally accommodateslarge mixing angles corresponding to both mass-squared differences. The fact that tree level and loopneutrino masses appear in our model has an analogywith the models combining the Higgs triplet mecha-
w x Žnism with radiative neutrino masses 13,25,26 for aSUSY model combining tree level and 1-loop neu-
w x.trino masses see Ref. 27 .
2. The model
We discuss 3-neutrino oscillations in the frame-work of an extension of the SM, where a second
Ž .Higgs doublet F , as1,2 and a right-handeda
neutrino singlet n are present in addition to the SMRw xmultiplets 23 . Thus the Yukawa interaction of lep-
tons and scalar fields is given by
2
˜yLL s LG F ll qLD F n qh.c. 2.1Ž .Ý ž /Y a a a a RRas1
˜ )with F s is F . G and D are 3=3 and 3=1a 2 a a a
matrices, respectively. The singlet field n permitsR
the construction of an explicit Majorana mass term
1 T y1LL s M n C n qh.c. , 2.2Ž .M R R R2
where we assume M )0 without loss of generality.R
In this 2-Higgs doublet model, spontaneous sym-metry breaking of the SM gauge group is achievedby the VEVs
0² :F s , 2.3Ž .0a 'ž /Õ r 2a
which satisfy the condition
2 2< < < <(Õ' Õ q Õ ,246 GeV . 2.4Ž .1 2
The VEVs Õ generate the tree level mass matrix1,2
21M s Õ G 2.5Ž .Ýll a a'2 as1
for the charged leptons diagonalized by
ll † ll ˆU M U sM 2.6Ž .L ll R ll
( )W. Grimus, H. NeufeldrPhysics Letters B 486 2000 385–390 387
with unitary matrices U ll, U ll and with a diagonal,L Rˆpositive M . The most general Majorana neutrinoll
mass term in the model presented here has the form
nL1 T y1v C M v qh.c. with v s .cL n L L2 ž /nŽ .R
2.7Ž .
The left-handed field vector v has four entriesL
according to the three active neutrino fields plus theright-handed singlet. The symmetric Majorana massmatrix M is diagonalized byn
U TM U sdiag m ,m ,m ,m 2.8Ž . Ž .n n n 1 2 3 4
with a unitary matrix U and m G0.n i
3. The tree-level neutrino mass matrix
Ž . Ž . Ž .From Eqs. 2.1 , 2.2 and 2.3 we obtain thetree-level version of M :n
0 M )
DŽ0.M s 3.1Ž .n †ž /M MD R
with
21)M s Õ D . 3.2Ž .ÝD a a'2 as1
Ž .The tree-level mass matrix 3.1 is diagonalized bythe unitary matrix
U Ž0.s u ,u ,u ,u 3.3Ž . Ž .n 1 2 3 4
with
uX cosq uX1,2 3u s , u s i ,1,2 3ž / ž /0 ysinq
sinq uX3u s , 3.4Ž .4 ž /cosq
where
2mD †5 5 (tan2qs , m s M s M M . 3.5Ž .D D D DMR
The uX form an orthonormal system of complex1,2,3
3-vectors with the properties
uX H M , uX sM rm . 3.6Ž .1,2 D 3 D D
The two non-vanishing mass eigenvalues are givenby
2 2M M mR R D2m s qm y , and(3 D4 2 MR
2M MR R2m s qm q ,M , 3.7Ž .(4 D R4 2
where the approximate relations refer to the limitm <M .D R
4. One-loop corrections and the neutrino massspectrum
By one-loop corrections, the form of the neutrinomass matrix is changed to
dM M )
DŽ1.M s 4.1Ž .n †ž /M MD R
with dM being a symmetric 3=3 matrix. Its ex-w xplicit form is given by 23
M M 2 ln M rMŽ .R b R b) † ) †dMs MM MM qM AAMÝ b b D D2 2 28p M yMR bb
4.2Ž .
with
21)MM s b D . 4.3Ž .Ýb a a'2 as1
In deriving this formula, terms suppressed by afactor of order M rM have been neglected. TheD R
Ž .first term in 4.2 is generated by neutral HiggsŽ .exchange. The sum in 4.2 runs over all physical
0 ) 0' Ž .neutral scalar fields F s 2 Ý Re b F ,b as1,2 a a
which are characterized by three two-dimensionalw xcomplex unit vectors b 23 with
Õ Õa bb b s . 4.4Ž .Ý a b 2Õb
( )W. Grimus, H. NeufeldrPhysics Letters B 486 2000 385–390388
Note that we do not consider corrections to M andD
M in the neutrino mass matrix. The unitary matrixRŽ . w xdiagonalizing 4.1 can be written in the form 23
U Ž1.sU Ž0.V 4.5Ž .n n
with Vy1 being of one-loop order. By an appropri-ate choice of the matrix V we obtain
ˆ Ž1. Ž1.T Ž1. Ž1.M sU M Un n n n
uXTdMuX uXT
dMuX 0 01 1 1 2
XT X XT Xu dMu u dMu 0 02 1 2 2s . 4.6Ž .0 0 m 03� 00 0 0 m4
Ž .The second term in 4.2 containing the matrix AAŽcontributions from Z exchange and contributionsfrom neutral scalar exchange other than the first term
Ž .. Ž .in Eq. 4.2 cannot contribute to 4.6 because ofŽ . Ž .3.6 . The remaining off-diagonal elements in 4.6can be removed by choosing uX orthogonal to D1 1
and D . This shows at the same time that one of the2
neutrinos remains still massless at the one-loop level.However, there is no symmetry enforcing m s01
and the lightest neutrino will in general get a mass atw x Xthe two-loop level 28 . Finally, the vector u has to2
be orthogonal to uX and uX , and its phase is fixed by1 3
the positivity of m . Defining2
ÕX†c s u D , 4.7Ž .a 2 a'2 mD
5 5 ) )the relation m s M implies Õ c qÕ c s0,D D 1 1 2 2< < 2 < < 2but the quantity c q c remains an independent1 2
parameter of our model, only restricted by ‘‘natural-ness’’, which requires that it is of order 1. FromŽ . 5 54.2 , using the Cauchy–Schwarz inequality and bs1, we obtain the upper bound
m2 ln M rM M 2Ž .D R b b 2 2< < < <m F c q c .Ž .Ý2 1 22 2 2 28p M 1yM rM ÕR b Rb
4.8Ž .Ž .Note that cancellations in Eq. 4.2 in the summation
over the physical neutral scalars do not happen ingeneral because the vectors b are connected with thediagonalizing matrix of the mass matrix of the neu-
tral scalars. The elements of these matrix are inde-2 Ž w x.pendent of the masses M see, e.g., Ref. 29 .b
From these considerations it follows that the order ofmagnitude of m can be estimated by2
1 M 2 M0 Rm ; m ln , 4.9Ž .2 32 2 M8p Õ 0
where M is a generic physical neutral scalar mass.0
Note that for M ;Õ the relation m <m comes0 2 3
solely from the numerical factor 1r8p 2 appearingin the loop integration.
5. Discussion
Let us first discuss the neutrino mass spectrum inthe light of atmospheric and solar neutrino oscilla-tions. Due to the hierarchical mass spectrum in ourmodel we have
Dm2 ,m2 and Dm2 ,m2 . 5.1Ž .atm 3 solar 2
From the atmospheric neutrino data, using the best2 w xfit value of Dm , one gets 3atm
m2D
m s ,0.06 eV . 5.2Ž .3 MR
Ž .A glance at Eq. 4.9 shows that m is only one or2
two orders of magnitude smaller than m if M3 R
represents a scale larger than the electroweak scale.Therefore, our model cannot describe the vacuumoscillation solution of the solar neutrino problem. On
w xthe other hand, with the MSW solution one has 4
m ;10y2 .5 eV 5.3Ž .2
and, therefore, m rm ;0.05, which can easily be2 3Ž .achieved with Eq. 4.9 . In principle, the unknown
Ž .mass scales m and M are fixed by Eqs. 5.2 andD RŽ . Ž Ž .5.3 see Eq. 4.9 for the analytic expression of
.m . However, due to the logarithmic dependence of2Ž .Eq. 4.9 on M and the freedom of varying theR
scalar masses, whose natural order of magnitude isgiven by the electroweak scale, the heavy Majoranamass could be anywhere between the TeV scale andthe Planck mass.
Let us therefore give a reasonable example. As-suming that m has something to do with the massD
of the tau lepton, we fix it at m s2 GeV. Conse-D
( )W. Grimus, H. NeufeldrPhysics Letters B 486 2000 385–390 389
Ž . 11quently, from Eq. 5.2 we obtain M ,0.7=10RŽ .GeV. Inserting this value into Eq. 4.9 and using
Ž .5.3 , the reasonable estimate M ;100%200 GeV0
ensues, which is consistent with the magnitude of theVEVs. This demonstrates that our model can natu-rally reproduce the mass-squared differences neededto fit the atmospheric and solar neutrino data, wherethe fit for the latter is done by the MSW effect.
Ž .Now we come to the mixing matrix 1.1 , whichis given by
UsU ll †U X with U X s uX ,uX ,uX 5.4Ž . Ž .L n n 1 2 3
Ž Ž .. Ž .for M 4m see Eq. 3.4 and neglecting V 4.5 .R D
Since the directions of the vectors D in the 3-di-1,2
mensional complex vector space determine U X, wen
will have large mixing angles in this unitary matrixas long as we do not invoke any fine-tuning of the
Ž w xelements of D . This is in contrast to Ref. 231,2Ž ll † .where we assumed that U D ;m rÕ, whereL a j ll j
. llthe m are the charged lepton masses. Also Ull j L
might have large mixing angles, but could also beclose to the unit matrix in analogy to the CKMmatrix in the quark sector. Since we do not expectany correlations between U ll and U X, it is obviousL n
that our model favours large mixing angles in theneutrino mixing matrix U.
On the other hand, there is a restriction on theelement U from the results of the Super-Kamio-e3
kande atmospheric neutrino experiment and thew x Ž .CHOOZ result 30 absence of n disappearance ,e
w x < < 2which is approximately given by 31 U Q0.1.e3
Furthermore, the Super-Kamiokande results imply2 Ž .that sin 2u is close to 1 see introduction . Theseatm
restrictions find no explanation in our model, but, aswe want to argue, not much tuning of the elements
Ž ll † X .U s U u is needed to satisfy them. If we takea3 L 3 a< < < < < <the ratios U : U : U s1:2:2 as an example wee3 m3 t 3
< < 2 Ž .find U s 1r9 , 0.11 and Eq. 1.3 givese3
sin2 2u s 80r81 , 0.99. To show that theatm
favourable outcome for sin2 2u does not dependatm< < < <on having U , U , let us consider now 1:3:2 form3 t 3
< < < < 2the elements U . Then we obtain U s1r14,a3 e3
0.07 and sin2 2u s45r49,0.92. Thus not muchatm
fine-tuning is necessary to meet the restrictions on< < 2 2 w xU and sin 2u 10,15,32 . Obviously, our modele3 atm
would be in trouble if it turned out that the atmo-spheric and solar neutrino oscillations decouple with
Ž .high accuracy U ™0 or atmospheric mixing ise3
Õery close to maximal.Some tuning of U is also required to reproducee2
the large-mixing MSW solution of the solar neutrinoproblem. As an illustration of the amount of the
w xtuning, looking at Fig. 8 of Ref. 4 , Gonzalez-GarciaŽ . < <et al. combined analysis , we estimate 0.44Q Ue2
Q0.64 from the 90% CL area in the limit U ™0.e3
Since there is no lepton number conservation inthe present model, lepton flavour changing processesare allowed. The branching ratios of the decaysm"™e"g and m"
™e"eqey have the most strin-w xgent bounds 33 . With our assumption on the size of
the Yukawa couplings D , the contribution of thea
w x " "charged Higgs loop 2 to m ™e g leads to alower bound of about 100 m for the charged HiggsD
mass. The decay m"™e"eqey, proceeding through
neutral Higgs scalars at the tree level, restricts onlysome of the elements of the Yukawa coupling matri-ces G , but not those of the D couplings relevant ina a
the neutrino sector. It is well known that the effec-Ž .tive Majorana mass relevant in bb decay is0n
suppressed to a level below 10y2 eV in the 3-neu-w xtrino mass hierarchy 34 , which is considerably
smaller than the best present upper bound of 0.2 eVw x35 .
In summary, we have discussed an extension ofthe Standard Model with a second Higgs doublet anda neutrino singlet with a Majorana mass being sev-eral orders of magnitude larger than the electroweakscale. We have shown that this model yields ahierarchical mass spectrum m 4m 4m of the3 2 1
three light neutrinos by combining the virtues ofŽ .seesaw m and radiative neutrino mass generation3
Ž .m /0 and m s0 at the one-loop level , and that2 1
it is able to accommodate easily the large mixingangle MSW solution of the solar neutrino problemand the n ™n solution of the atmospheric neutrinom t
anomaly. We want to stress that by virtue of Eq.Ž . 2 24.9 the order of magnitude of Dm rDm issolar atm
fixed apart from a logarithmic dependence on M .RŽ .This logarithmic factor in 4.9 varies roughly be-
tween 2 and 40 for M between 1 TeV and theR
Planck scale and M ;100 GeV. By construction,0
the neutrino sector of our model is very differentfrom the charged lepton sector. The model offers noexplanation for the mass spectrum of the chargedleptons. We want to stress that the scalar sector of
( )W. Grimus, H. NeufeldrPhysics Letters B 486 2000 385–390390
the model is exceedingly simple and that – apartfrom the Standard Model gauge group – no symme-
< < 2try is involved. The moderate smallness of Ue3
and closeness of sin2 2u to 1 is controlled by theatm
ratios of the elements of the third column of themixing matrix U. We have argued that the ratios of< < Ž . < < 2U ase,m,t required to give U Q0.1 anda3 e3
sin2 2u R0.84 are quite moderate, with 1:2:2 be-atm
ing a good example. An similar moderate tuning isnecessary for the reproduction of the large mixingangle MSW solution. Suitable ratios of the elementsof the mixing matrix have to be assumed in themodel presented here, but might eventually find anexplanation by embedding it in a larger theory rele-vant at the scale M .R
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