PHYSICAL REVIEW D 66, 113003 ~2002! Lepton masses from a TeV scale in a 3-3-1 model J. C. Montero,* C. A. de S. Pires,† and V. Pleitez‡ Instituto de Fı´sica Teo´rica, Universidade Estadual Paulista, Rua Pamplona 145, 01405-900 Sa˜o Paulo, SP Brazil ~Received 2 April 2002; revised manuscript received 16 August 2002; published 18 December 2002! In this work, using the fact that in 3-3-1 models the same leptonic bilinear contributes to the masses of both charged leptons and neutrinos, we develop an effective operator mechanism to generate mass for all leptons. The effective operators have dimension five for the case of charged leptons and dimension seven for neutrinos. By adding extra scalar multiplets and imposing the discrete symmetryZ9^ Z2 we are able to generate realistic textures for the leptonic mixing matrix. This mechanism requires new physics at the TeV scale. DOI: 10.1103/PhysRevD.66.113003 PACS number~s!: 14.60.Pq, 12.60.2i, 14.60.St n no Ac in ix o s s f an o tri t ou k rd e d ea a is od sug- eu- tors en- lets: e ged to n ip- is the ext one ons oes ion- hile eu- in- ged - the rs in he ce- ix- the ave ec. on di- ular, the I. INTRODUCTION The smallness of the neutrino masses and the patter their mixing, arising from atmospheric and solar neutri data@1–3#, suggest the extension of the standard model. cording to those experimental data, the mixing involved the atmospheric neutrino oscillation is maximal and the m ing involved in the solar neutrino oscillation is large@4,5#. From the theoretical point of view, we dispose already well established ways where explanations of the smallnes those masses arise naturally. The most popular are the saw@6# and the radiative generation@7# mechanisms. Both o them require realistic extensions of the standard model extra global and/or discrete symmetries. We also can c sider effective operators which naturally lead to light neu nos. This approach has had success in accounting for neutrino puzzle in the base of the standard model with resorting to drastic fine-tuning. It was Weinberg@8#, and independently Zee and Wilcze @9#, who first pointed out in the context of the standa model that the dimension-five effective operator 1 L Lia c L jbwk (m)w l (n)~ f abmne ike j l 1 f abmn8 e i j ekl!, ~1! with L5(n l ,l )L T , yields naturally light neutrino masses. Th success of such an effective operator approach is justifie the expression of the neutrino mass it generates: Mab n 5 f ab L ^w&2, ~2! which is a seesaw relation since^w&'246 GeV andL is a large characteristic mass. In particular, in this case the r ization of such operator turns to be important once it c guide us to all the possible realizations of that mechan using only the representation content of the standard m with operators of dimension five@10#. Higher dimension op- *Email address: montero@ift.unesp.br †Email address: cpires@ift.unesp.br ‡Email address: vicente@ift.unesp.br 0556-2821/2002/66~11!/113003~6!/$20.00 66 1130 of - - f of ee- d n- - he t by l- n m el erators have already been considered@11#, and extensions of the scalar sector of the standard model have also been gested@12#. In this work we address the problem of generating n trino and charged lepton masses through effective opera in the context of 3-3-1 models@13,14#. In general, in these models, the minimal set of scalar multiplets required to g erate the fermion masses consists in three scalar trip h5(h0,h1 2 ,h2 1)T;(1,3,0), r5(r1,r0,r11)T;(1,3,1), x5(x2,x22,x0)T;(1,3,21), and a symmetric sextetS ;(1,6,0). An important feature of this model is that th same leptonic bilinearCL cCL , where CL5(n,l ,l c)T, can give a contribution to the masses of both sectors: char leptons and neutrinos. We explore this feature in order obtain realistic textures for the mixing matrix in the lepto sector. We work first in a scenario where there are only tr lets, like h, r, and x, and neutral scalar singlets. In th context, depending on the extra symmetry added to model we obtain realistic texture for the lepton masses. N we show that a possible realization of this mechanism is in which heavy scalar sextets and singlet charged lept @15,16# are added to the model. The scalar sextet which d not gain a vacuum expectation value induces a dimens five operator that give mass for the charged leptons; w the dimension-seven operator, which gives mass to the n trinos, arises from the interactions of neutrinos with the s glet charged leptons and a mixture among the singly char scalar bosons. The outline of this work is as follows. In Sec. II we de velop a formalism to address the issue of generating charged lepton and neutrino masses by effective operato the context of a 3-3-1 model. Next, in Sec. III we use t formalism previously developed to generate realistic s narios to accommodate lepton masses with appropriate m ing in the lepton sector. In Sec. IV we suggest what are main ingredients that a more fundamental theory has to h in order to realize such effective operators. We reserve S V for our conclusions. II. THE MECHANISM In this section we build effective operators of dimensi five for the generation of the charged lepton masses and mension seven for the case of neutrino masses. In partic we show that with a dimension-seven operator we attain ©2002 The American Physical Society03-1 ce th es si a et o d r tr re er a b , ed , odel der ber The alar V ber, r y we we n- - ed e - o J. C. MONTERO, C. A. de S. PIRES, AND V. PLEITEZ PHYSICAL REVIEW D66, 113003 ~2002! desired order of the neutrino masses required by the re experiments, i.e., at the eV scale. It is interesting to note the energy scale required to obtain those neutrino mass of the order of 5 TeV. In the 3-3-1 models of Refs.@13–15# all leptons transform as triplets under the electroweak gauge symmetry: CaL5S na l a l a c D L ;~1,3,0!, a5e,m,t. ~3! Below we generate the charged lepton masses by u the two scalar tripletsr and x, mentioned in the previous section, and the neutrino masses using the tripleth and a neutral scalar singletf. In order to generate a realistic texture of the relevant m trices in the lepton sector we will impose appropriate discr or global symmetries~see the next section!. A. Charged lepton masses In the version without the scalar sextet we still dispose the scalar tripletsr and x in order to generate the charge lepton masses by dimension-five effective operators. Acco ing to the transformation properties under the symme SU(3)C^ SU(3)L ^ U(1)N of the last two scalar triplets we can form withCaL c CbL the following, effective dimension- five operator: L5 f ab L CaL c CbLx* r* 1H.c. 5 f ab L $naL c @nbLx1r21 l bLx1r0* 1~ l c!bLx1r22# 1~ l aL!c@nbLx11r21 l bLx11r0* 1 l bLx11r22# 1 l aR@nbLx0* r21 l bLx0* r0* 1~ l c!bLx0* r22#% 1H.c. ~4! After the neutral componentsx0 and r0 develop their respective vacuum expectation values~VEVs!, ^x& and^r&, the effective operator above generates the following exp sion for the charged lepton mass matrix: Mab l 5 f ab L ^r&^x&, ~5! with a,b5e,m,t. Let us discuss the values of the paramet present in the expression above. None of them has alre been fixed by the model. We just expect that they can found in some range of values. For example,^x& must be in the range 300 GeV,^x&,4 TeV @17,18#. The constraint on ^r& comes from the mass of the gauge bosonsW6 and Z0 i.e., ^r&21^h&25(246)2 GeV2. Assuming, as an illustration the following set of values: ^h&.22 GeV, ^r&.245 GeV, ~6! ^x&.1 TeV, and L.5 TeV, 11300 nt at is ng - e f d- y s- s dy e the charged lepton mass matrix takes the form Mab l .49f ab GeV. ~7! If f ab is a diagonal matrix, for obtaining the correct charg lepton masses we needf ee;31025 for the electron mass f mm;231023 for the muon mass, andf tt;3.631022 for the tau mass. We recall that in the case of the standard m we have 1026, 1023, and 1022, respectively. B. Neutrino mass In order to generate neutrino masses we will consi only effective operators that conserve the total lepton num and also some other global or/and discrete symmetries. simplest way to obtain such operator is by adding an sc singlet f;(1,1,0), coming from new physics at the Te scale, carrying the total lepton number,L5Le1Lm1Lt , with the following assignmentL(f)521, and forming with h andCaL the dimension-seven effective operator: L5 f ab8 L3 CaL c CbLh* h* ff1H.c. 5 f ab8 L3 $naL c @nbLh0* h0* 1 l bLh0* h1 11~ l c!bLh0* h2 2# 1~ l aL!c@nbLh1 1h0* 1 l bLh1 1h1 11~ l c!bLh1 1h2 2# 1 l aR@nbLh0* h2 21 l bLh2 2h1 11~ l c!bLh2 2h2 2#%ff 1H.c. ~8! Notice that this operator conserves the total lepton num since L(h2 1)522 and L(h1 2)5L(h0)50 @we recall that L(x2)5L(x22)52 and hence the interactions in Eq.~4! are also L conserving#. The dimension-five operato CaL c CbLh* h* which violates the lepton number explicitl can be forbidden by introducing discrete symmetries as will show below. For the moment we will avoid it. If we want this quantum number to be broken spontaneously should begin withL-conserving interactions and let a no zero VEV, in this casêf&, break this symmetry spontane ously. A dangerous Majoron-like Goldstone can be avoid by breaking softly or explicitly the total lepton number in th scalar potential@see Eq.~11! below# or by assuring that the Majoron is almost singlet@19#. Hence, after the scalars in volved in Eq.~8! develop their respective VEVs the neutrin masses are given by Mab n 5 f ab8 L3 ^h&2^f&2. ~9! Inserting the values of the VEV given in Eq.~6! the expres- sion above reads 3-2 t r le a e w of on pr th w m t is le , pt e ef- tri- f as ain ym- n. tor - e ctor, sec- al ki- t LEPTON MASSES FROM A TeV SCALE IN A 3-3-1 MODEL PHYSICAL REVIEW D66, 113003 ~2002! Mab n .3.9f ab8 S ^f& 1 GeVD 2 31029 GeV. ~10! According to this the VEV involved above,^f&, has to be around 531022 GeV if f ab8 'O(1), in order to generate the expected order of magnitude of the neutrino masses, tha of the 1022 eV order. That value of the VEV for the scala singlet could imply a fine-tuning since we expect the sing scalar boson to be very heavy, i.e.,mf'L. However, in order to get such a small VEV in a more natural way we c implement a type II seesaw mechanism with the scalar fi f @20#. In fact, it is possible to implement this mechanism as will show in the following. Let us consider, for the sake simplicity, the discrete symmetryh→2h, f→2f ~other fields are even under this symmetry!. We also allow terms in the scalar potential that violate explicitly the total lept number. In this case the most complete scalar potential senting these terms is V~h,r,x,f!5mh 2h†h1mr 2r†r1mx 2x†x1mff†f 1l1~h†h!21l2~r†r!21l3~x†x!2 1l4~f†f!21~h†h!@l5~r†r!1l6~x†x!# 1l7~r†r!~x†x!1l8~r†h!~h†r!1l9~x†h! 3~h†x!1l10~r†x!~x†r!1~f†f! 3@l11~h†h!1l12~r†r!1l13~x†x!# 1@l14ehrxf1l15x †hr†h1H.c.#, ~11! where the last two terms are those that explicitly violate lepton number. From this scalar potential we find the follo ing constraint equation over^f&: ^f&@mf 2 1l11̂ h&21l12̂ r&21l13̂ x&21l4^f&2# 1l14̂ h&^r&^x&50. ~12! Supposing thatmf 2 ,0 is the dominant parameter in the ter within square brackets, we have, ^f&.2l14 ^h&^r&^x& mf 2 . ~13! Using the values in Eq.~6! in Eq. ~13!, and assumingumfu 'L we obtain^f&.531022 GeV, if l1450.25. We recall that it was already shown in the literature that it is possible have a heavy scalar with a corresponding small VEV@10,19# as in the present case. From Eq. ~10! we find that the neutrino mass matrix given by the following expression: Mab n .1022f ab8 eV. ~14! We see from the discussion above that both charged tons and neutrinos gain mass through effective operators can be seen from Eqs.~5! and~10! or ~7! and~14!. However, the scale of neutrino masses relative to the charged le 11300 is, t n ld e e- e - o p- as on masses arises as a consequence of the dimension of th fective operator and of the VEV of the scalar involved. III. REALISTIC SCENARIOS If we want to obtain a definite texture on the mass ma ces in Eqs.~5! and ~9! we have to enlarge the number o scalar multiplets, for instance one triplet of the typer andx and a singletf for each generation. We denote them r1,2,3,x1,2,3,f1,2,3. In this case we have L5 f ab i j L CaL c CbLx i* r j* 1 f ab8 i j L3 CaL c CbLh* h* f if j1H.c., ~15! and we will impose discrete symmetries in order to obt appropriate mass matrices. For instance, consider the s metryZ9^ Z2 with fields transforming under theZ9 factor as Ce→v1Ce , Cm→v3Cm , Ct→v2Ct , r1→v3r1 , x1→v1 21r1 , r2→v3 21r2 , x2→v0x2 , r3→v4r3 , x3→v4 21x3 , h→v0h, f1→v1 21f1 , f2→v3 21f2 , f3→v2 21f3 , ~16! with vk5e2p ik/9, k50, . . . ,4 andunder theZ2 factor the fieldsCm ,r1 ,x1 ,f2,3 are odd, while the other ones are eve ~This implies appropriate transformation in the quark sec if the scalar multiplets also couple to quarks.! Notice that the Z9 symmetry forbids the interactionseCaL c CbLh and CaL c CbLh* h* . From the interactions in Eq.~15! and the discrete symme tries in Eq. ~16! we obtain mass matrices which mix th second and third generations in the charged lepton se and the first and the second generations in the neutrino tor, i.e., they are diagonalized by the following orthogon transformations: UL l >S 1 0 0 0 c0 s0 0 2s0 c0 D , UL n>S c08 s08 0 2s08 c08 0 0 0 1 D ~17! for the charged lepton and neutrinos, respectively. The Ma Nakagawa-Sakata~MNS! mixing matrix is of the form@21# U5UL l†UL n5S c08 s08 0 2c0s08 c0c08 s0 s0s08 2s0c08 c0 D , ~18! and we have omitted Dirac or MajoranaCP violating phases. Notice thatUe3 is zero at the tree level, and it mus 3-3 n ti ch in s rix ive lu g ix ar e to th ra ve al ne eral tor ince the - hat t the her , nal he . J. C. MONTERO, C. A. de S. PIRES, AND V. PLEITEZ PHYSICAL REVIEW D66, 113003 ~2002! arise from radiative corrections. After taking into accou these corrections we expect thatU→V: V'S c( s( Ve3 2catms( catmc( 2satm 2satms( satmc( catm D .S c( s( Ve3 2 1 A2 s( 1 A2 c( 2 1 A2 2 1 A2 s( 1 A2 c( 1 A2 D . ~19! In this context, the small value ofuVe3u (,0.16) @22# is natural in the present model because it arises from radia corrections involving a term in the scalar potential whi softly breaks those discrete symmetries. For instance, generalization of the expression in Eq.~11! a term likex1 †x2 softly breaks theZ9 symmetry. Hence, radiative correction should induce a small value for this entry in the MNS mat and~small! corrections to the angles in Eqs.~17! and~18!. In this case (c,s)0→(c,s)atm,(c8,s8)0→(c,s,)( , Ue3(50) →Ve3(<0.16). Since in general we expect that radiat correction does not amplify the mixing angles in Eqs.~17! and ~18! then (c,s)0.(c,s)atm and (c8,s8)0.(c,s)( . No- tice that the interactions that contribute to a nonzero va for Ve3 may induce a complex value for this entry inducin in this way aCP violating phase@23#. Another possible scenario, where there is bimaximal m ing among the neutrinos but not in the MNS matrix, appe as a result of a new symmetryL8 ~in the context of the standard modelL8 can be identified with Le-Lm-Lt @12,24,25# but here it must be an independent global symm try!. In this case we need only two sorts ofr and x scalar triplets and two singlets. We assignL8(Ce)52L8(Cm)5 2L8(Ct)51, L8(r1)5L8(x1)52L8(r2)52L8(x2)51, andL8(h)5L8(f)50. With this symmetry we obtain from Eq. ~15! a general mass matrix in the charged lepton sec and the following neutrino mass matrix at the tree level: M n5S 0 f em8 12 f et8 12 f em8 12 0 0 f et8 12 0 0 D ^h&2^f1&^f2& L3 ~20! with f em8 12; f et8 12. In this case the neutrino mass matrix has inverse hierarchy (m,2m,0) where m}@( f em8 12)2 1( f et8 12)2#1/2. The mass splitting has to be generated by diative corrections, as we will show below, and we ha um3u!um1u.um2u. Assuming thatf em8 125 f et8 12 there is a bi- maximal neutrino mixing pattern@24#: 11300 t ve a e - s - r, e - UL n5S 1 A2 2 1 A2 0 1 2 1 2 2 1 A2 1 2 1 2 1 A2 D . ~21! Since the bimaximal mixing is not favored by the actu neutrino solar data, i.e., tanu(,1 @5#, we have to explain the deviation from this bimaximal scenario. This can be do since from Eq.~15! and theL8 symmetry the mass matrix in the charged lepton sector, as we said before, is now a gen one. Hence the left-handed mixing matrixUL l can be written in terms of three angles (c,s)12,(c,s)23,(c,s)13. In this case assuming thats12@s23@s13 the mixing matrixUL l is a hier- archical one similar to the mixing matrix in the quark sec and the charged lepton mass matrix is almost diagonal. S the MNS matrix is defined asV5UL l†UL n the analysis of Ref. @26# follows. In this almost bimaximal scenario we should address mass splitting betweenn1 andn2. To get such a mass split ting we have to consider terms in the scalar potential t breakL explicitly, as is the case of the lastl15 term in the scalar potential in Eq.~11!. However, with the symmetryZ9 introduced above, this term involvingr i , x i , andh is for- bidden. Hence, it should be necessary thath transforms non- trivially underZ9 ~or a higher discrete symmetry! or, only as an illustration, we can add a fourth pair of tripletsr4 andx4 which transform likeh ~and for this reason they do no couple directly with leptons! and withL8(x4)5L8(r4)50. Thus we have the terml15(x4 †h)(h†r4). The scalar poten- tial produces a general mixing among all the scalar of same charge and we have interactions likex1r2^h&2, where x1,r2 denote symmetry eigenstates. That term, toget with the interaction in Eq.~15!, will generate corrections through the one-loop diagrams, for example to the diago entries in the mass matrix in Eq.~20!, providing the mass splitting betweenn1 and n2. The loop diagram which will generate the diagonal entries in Eq.~20! is depicted in Fig. 1. FIG. 1. One-loop contribution to the diagonal entries for t neutrino mass matrix in the second almost bimaximal scenario 3-4 s- in eu a h ve u- e a ica in er to e ro le to ic x- a pr th a nc ive ain ed - po- - on ch lep- ton tive uire h a hen, veral ent For in olar ess od- the LEPTON MASSES FROM A TeV SCALE IN A 3-3-1 MODEL PHYSICAL REVIEW D66, 113003 ~2002! It gives, up to logarithmic corrections, the following expre sion for such entries: Maa n . l15f aa 3 ^h&2^r&2^x&2 mx 2L3 . ~22! All the parameters above exceptmx and l15 were already previously fixed in this work. Assuming now thatmx5^x&, and inserting Eq.~6! in Eq. ~22!, we have Maa n .1.131028l 15f aa 3 GeV. ~23! The large mixing angle Mikheyev-Smirnov-Wolfenste ~MSW! solution to the solar neutrino problem requires a n trino mass scale of the order of 2.831023 eV @27#. The only free parameter in Eq.~23! is l15, while the diagonalf pa- rameters are already fixed by the charged lepton masses their values are given in Sec. II A~although in this case eac entry of the matrixf ab has several contributions that we ha not written explicitly!. We obtain this mass scale for the ne trinos by consideringl15;2.531024. Hence to generate th n1-n2 mass splitting we must fine-tunel15. In the next section we analyze what main ingredients underlying theory should have to realize, in an econom way, the effective operators and generate the two mix scenarios considered above. IV. A POSSIBLE UNDERLYING THEORY The minimal scenario we can imagine is the one wh the effective dimension-five operator in Eq.~4! is realized at the tree level, while the effective dimension-seven opera in Eq. ~8! is realized through the one-loop level. For this w only need to add to the minimal 3-3-1 model~three quark generations and three scalar tripletsh, r, andx) at least two sextetsSp, p51,2, which do not necessarily gain a nonze VEV, two heavy lepton singlets,E1,2(L,R);(1,1,21) @15,16#, and three scalar singletsf r , r 51,2,3, like thef introduced in Sec. II B. We will assume that these sing leptons have the following assignments of the total lep number: L(E1)51 and L(E2)50. We recall that it was shown in Ref.@28# that a tree level realization of a symmetr bilinear CL cCL is implemented by introducing a scalar se tet. With the representation content discussed above we h the leptonic interactions L5Gab p ~Cc!aiLCb jLSi j p 1G1aCaLE1Rr1G2axTN1LCaL c 1gr8E1LE2Rf r1grE2LE1Rf r* 1M1E1LE1R 1M2E2LE2R1H.c., ~24! with M1 ,M2.L and we have omittedSU(3) indices and summation over the repeated indices. By imposing an ap priate discrete symmetry we can get the result that one of scalar sextets couples only to the first leptonic generation the other one to the second and third generations. He 11300 - nd n l g e r t n ve o- e nd e, with the interactions above we can realize the effect dimension-five operator in Eq.~15! as is shown in Fig. 2~a!, where the trilinear vertex arises from a term likegxTS†r whereg is a constant with dimension of mass. So, we obt a mass matrix for the charged lepton which is diagonaliz by an orthogonal matrix, likeUL l in Eq. ~17!, and from Eq. ~4! we see that 1/L5g/MS 2 . For the realization of the effec tive dimension-seven operator the last term of the scalar tential in Eq.~11! is also important. This realization is de picted in Fig. 2~b!. V. CONCLUSIONS In this work we developed a simple mechanism based effective operators in the context of a 3-3-1 model whi generates masses for the neutrinos and for the charged tons as well. By using the same bilinear the charged lep masses are generated in this mechanism by an effec dimension-five operator, while the neutrino masses req an effective dimension-seven operator in conjunction wit type II seesaw mechanism applied on a scalar singlet. T we use the effective operator mechanism to generate se mixing matrices in the lepton sector which are consist with the solar, atmospheric, and reactor neutrino data. this we have considered the discrete symmetryZ9^ Z2 or a global one,L8, in the effective operators given in Eq.~15!. In this case, without resort to large fine-tuning we obta neutrino masses compatible with some solutions to the s and atmospheric neutrino anomalies. We would like to str that in this 3-3-1 model the bilinearCcC gives mass to both lepton sectors. This issue is characteristic of the 3-3-1 m els: in the standard model the bilinearLcL can only contrib- ute to the neutrino masses. FIG. 2. Tree level and one-loop diagrams contributing to effective operators defined in Eqs.~4! and ~8!, respectively. 3-5 w in n th l, J. C. MONTERO, C. A. de S. PIRES, AND V. PLEITEZ PHYSICAL REVIEW D66, 113003 ~2002! Although we have not considered the quark masses would like to call attention to an interesting mechanism, the context of a 3-3-1 model, for generating the top a bottom masses at the tree level, while the masses of the o quarks and charged leptons arise at the one-loop leve proposed in Ref.@29#. 8 lio is . M , L y, . Re a, v 11300 e d er as ACKNOWLEDGMENTS This work was supported by Fundac¸ão de Amparo a` Pes- quisa do Estado de Sa˜o Paulo~FAPESP!, Conselho Nacional de Ciência e Tecnologia~CNPq!, and Programa de Apoio a Núcleos de Exceleˆncia ~PRONEX!. . D . D z, . sa, , . nd ys. . D @1# Y. Fukudaet al., Phys. Rev. Lett.81, 1562 ~1998!; 81, 1158 ~1998!; Phys. Lett. B436, 33 ~1998!; K. S. Hirataet al., ibid. 280, 146 ~1992!; R. Becker-Szendyet al., Phys. Rev. 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