NEUTRINO MASS AND MIXING IN THE 3-3-1 MODEL AND S3 FLAVOR SYMMETRY WITH MINIMAL HIGGS CONTENT
V. V. Viena* H. N. Longh**
Department of Physics, Toy Nguyen University Buon Ma Thuot, DakLak, Vietnam
b Institute of Physics, VAST Ba Dinh, Hanoi, Vietnam
Received December 20 2013
A new S3 flavor model based on the SU(3)c®SU(3)l l"( I) \ gauge symmetry responsible for fermion masses and mixings different from our previous work [14.17] is constructed. The new feature is a two-dimensional representation of a Higgs anti-sextet under S3, which is responsible for neutrino masses and mixings. The neutrinos acquire small masses from only an anti-sextet of SU(3), which is in a doublet under S3. If the difference of components of the anti-sextet is regarded as a small perturbation, S3 is equivalently broken into identity, the corresponding neutrino mass mixing matrix acquires the most general form, and the model can fit the latest data on neutrino oscillations. This way of symmetry breaking helps us reduce a content in the Higgs sector, to only one anti-sextet instead of two as in our previous work [14]. Our results show that the neutrino masses are naturally small and a small deviation from the tri-bimaximal neutrino mixing form can be realized. The Higgs potential of the model as well as the minimization conditions and gauge boson masses and mixings are also considered.
DOI: 10.7868/S0044451014060044
1. INTRODUCTION
The experiments 011 neutrino oscillations have indicated that the neutrinos have small masses and mixings fl 4], and therefore the standard model of fundamental particles and interactions must be extended. Among this direction, there have been various models proposed, such as [5, C] and others. An alternative is to extend the electroweak symmetry SU(2)i OU(l)y to SU(3)l O U(1).y, in which to complete the fundamental representations of SU(3)l with the standard-model doublets so as to obtain the neutral fermions. This proposal, which has nice features and has been extensively-studied over the last two decades, is called 3 3 1 models [7 9], with the number of fermion families having been proved to be three [7,10].
The parameters of neutrino oscillations such as the
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**E-mail: hnlong'fliop.vast.ac.vn
squared IIlclSS differences and mixing angles are now very constrained. The data in Ref. [4] imply that
sin2(20i2) = 0.857± 0.024 (tV2 « 0.6717), sin2(20i3) = 0.098 ± 0.013 (.s13 « 0.1585), sill2 (2023 )> 0.95, (1)
AjttJi = (7.50 ± 0.20) • 10-5 eV2, Am2, = (2.32ig;J2) • 10-3 eV2.
These large neutrino mixing angles are completely different from the quark mixing ones defined by the Cabibbo Kobayashi Maskawa (CKM) matrix. Therefore, it is very important to find a natural model that leads to these mixing patterns of quarks and leptons with good accuracy. Small non-Abclian discrete symmetries are considered to be the most attractive choice for the flavor sector fll 13]. The simplest explanation for these conclusions is probably due to an S3 flavor symmetry, which is the smallest 11011-Abelian discrete group [14,15]. I11 fact, there is an approximately maximal mixing of two flavors //. and r as given above, which can be connected by the 2 irreducible representation of S3. Besides the 2, the S3 group can provide two in-equivalent singlet representations 1 and 1', which play
a crucial role in reproducing consistent fermion masses and mixings [14]. The S3 models have been studied extensively over the last decade [13]. In [14], we proposed two 3 3 1 models, with either neutral fermions or right-handed neutrinos, based on the S3 flavor symmetry, in which a large number of Higgs triplets was required. In this paper, we propose a new S3 flavor symmetry in the 3 3 1 model with neutral fermions, in which the number of Higgs triplets required is less and the Higgs potential of the model is therefore much simpler than the previous ones.
The motivation for extending the above application to the 3 3 1 models with the neutral fermions Nr is mentioned in [14,16,17]. In this paper, we investigate simpler choices for Higgs multiplets of S3 in which the unique anti-sextet responsible for the neutrino mass and mixing lying in 2 under S3 and the difference between two VEV components of the anti-sextet play the role of perturbation. It is also noted that the numbers of fermion families in the 3 3 1 models originate from the anomaly-free gauge symmetry and naturally meet our criteria 011 the dimensions of flavor group representations such as S3, unlike the others in the literature, mostly imposed by hand [11 13].
The rest of this work is as follows. In Sec. 2, we present the necessary elements of the 3 3 1 model with neutral fermions Nr under the S3 symmetry and introduce the necessary Higgs fields responsible for the charged-lepton and quark masses. Section 3 is devoted to the neutrino mass and mixing. In Sec. 4, we consider the Higgs potential and minimization conditions. We summarize our results and make conclusions in Sec. 6.
2. THE MODEL
The fermion content of the model is similar to that in [14]: the fermions in the model transform under the respective [SU(3)l,U(1)_y,U(1)£,53] symmetries as
L = {P1LJIL,N[R)T ^ [3,-1/3,2/3,1], -[1,^1,1,1], Kl = (VaLJaL,NrnR)T ~ [3,-1/3,2/3,2], iaR~ [1,-1,1,21,
QlL = {u1L,d1L,UL)T ~ [3,1/3,^1/3,1], u1R ~ [1,2/3,0,1], thn ~ [1,-1/3,0,1], UR~ [1,2/3,-1,1],
QaL = (daL, -UaL, DaLf ~ [3* , 0, 1/3, 2], uaR ~ [1,2/3,0,2], daR ~ [1, —1/3, 0,2], DaIi~ [1,-1/3,1,2],
where a = 2,3 is a family index of the last two lepton and quark families, which are defined as the components of the 2 representations. We note that the 2 for quarks satisfies the requirement of anomaly cancellation, where the last two left-quark families are in 3* and the first one as well as the leptons are in 3. All the £ charges of the model multiplets are listed in square brackets. In what follows, we consider possibilities for generating the fermion masses. The scalar multiplets needed for this purpose are to be introduced accordingly
To generate masses for the charged leptons, we introduce two SU(3)l scalar triplets <f> and <j>' respectively-lying in 1 and 1' under S3, with the VEVs {<j>) = (0 v 0)T and {(jJ) = (0 (/0)T [14]. From the invariant Yukawa couplings for the charged leptons, we obtain in, = hi v, in¡, = hv — h'v', in7 = he + h'v', and the mixing matrices of the left- and right-handed charged leptons are diagonal, Uu = Um = 1. The charged leptons /1;2,3 are therefore by themselves the physical mass eigenstates and the lepton mixing matrix depends on only that of the neutrinos, which is studied in the next section.
In similarity to the charged lepton sector, to generate the quark masses, we additionally introduce the three scalar Higgs triplets \.and if respectively lying in 1,1, and 1' under S3. Quark masses can be derived from the invariant Yukawa interactions for quarks, assuming that the VEVs of and \ are u = (j/J), u' = {ifi), and w = (\!j) and the other VEVs {;/§), {//'3), and vanish due to the lepton parity conservation. The exotic quarks therefore acquire the masses idu = /1«' and m£>1-2 = /«>• The masses of ordinary up-quarks and down-quarks are
id „ = I)" 11. ID, = I) "c + //"(•'. 11)1 = I) "c — //"(•'.
id,1 = hfv, mh = I)'111 + I)"111'. nib = I)'111 — I)"' u'.
The unitary matrices that couple the left-handed quarks uL and (1l to those in the IIlclSS bases are unit ones. The CKM quark mixing matrix at the tree level is then Uckm = I In/ 1,1. = 1- The lepton parity breaking due to the odd VEVs {;/§), {//3), ( x j), or a violation of £ and/or S3 symmetry in terms of Yukawa interactions would disturb the tree-level matrix, resulting in a mixing between the SM and exotic quarks and/or possibly providing the desirable quark mixing pattern /. \ f 1 /(. QlX*<1r, QilXUii. with a mixing between SM and exotic quarks. To obtain a realistic pattern of the SM quark mixing, we should add radiative corrections or use the effective six-dimensional operators (see Rcf. [18] for the details). However, we leave this problem for the future work. A detailed study of charged
lopton and quark masses can be found in Rcf. [14]. In this paper, we consider a new representation for the anti-sextet responsible for neutrino masses and mixings that are different from those in Ref. [141.
3. NEUTRINO MASSES AND MIXING
The neutrino masses arise from the couplings of tf^LiJ>aL, 4'Il4'il and 4'lL4'aL to scalars, where '4'caL'4'aL transforms as 3* 9 6 under SU(3)l and as 1 1' 2 under S3; 4'il:4'il transforms as 3* 9 6 under SU(3)l and as 1 under S3, and 4'iL'4'aL transforms as 3* 9 6 under SU(3)l and as 2 under S3. For the known scalar triplets (<;>. <;/. \. //. //'). the available interactions are only and (i^^aL )<£', but are explicitly-
suppressed because of the £-symmetry. We therefore propose a new SU(3)l anti-sextet coupling to '4<cl:4'l i'o-sponsible for the neutrino masses lying in either 1, 1', or 2 under S3. To obtain a realistic neutrino spectrum with the minimal Higgs content, we introduce the Higgs anti-sextet
/ s° •sll S+ "12 s° \
Hi = "12 "22 S+
\ s.o •s13 S+ "23 s.0 •\33
[6*, 2/3, -4/3, 2],
i = 1.2.
where numerical subscripts 011 the component scalars are the SU(3)l indices, whereas i = 1,2 is that of S3. The VEV of .1 is set as ({.s'i), (,s2)) under S3, with
( A i 0
\ Vi
Vi \ 0
A« /
■i = 1,2.
(3)
Following the potential minimization conditions, we have several VEV alignments. The first one is that {•*>' 1) = {«2); then S3 is broken into Z-2 consisting of the identity- element and one transposition (out of the three) of S3. The second one is that {,s'i) ^ 0 = (,s2) or {.s'i) = 0 ^ (,s2); then S3 is broken to Z3 as in the case of the charged lopton sector. The third one is that (.s'i) ^ (,s2); then S3 is broken to the identity. In our previous work [14], we have argued that both breakings S3 —¥ Z-2 and S3 —¥ Z3 must take place, and hence, to obtain a realistic neutrino
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