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Nuclear Physics B 824 (2010) 95-110

www.elsevier.com/locate/nuclphysb

Tri-bimaximal neutrino mixing from A4 and #13 ~ Oc

Yin Lin

Dipartimento di Fisica 'G. Galilei', Universita di Padova, INFN, Sezione di Padova, Via Marzolo 8, I-35131 Padua, Italy

Received 2 July 2009; accepted 20 August 2009

Available online 26 August 2009

Abstract

It is a common believe that, if the tri-bimaximal mixing (TBM) pattern is explained by vacuum alignment in an A4 model, only a very small reactor angle, say O13 ~ O(kCc) being kc = Oc the Cabibbo angle, can be accommodated. This statement is based on the assumption that all the flavon fields acquire VEVs at a very similar scale and the departures from exact TBM arise at the same perturbation level. From the experimental point of view, however, a relatively large value O13 ~ O(kc) is not yet excluded by present data. In this paper, we propose a seesaw A4 model in which the previous assumption can naturally be evaded. The aim is to describe a O13 ~ O(kc) without conflicting with the TBM prediction for O12 which is rather close to the observed value (at kC level). In our model the deviation of the atmospherical angle from maximal is subject to the sum-rule: sin2 O23 ^ 1/2 + V2/2cos S sin O13 which is a next-to-leading order prediction of our model.

© 2009 Elsevier B.V. All rights reserved.

1. Introduction

The present data [1], at 1a, on solar and atmospherical angles:

O12 = (34.5 ± 1.4)°, O23 = (42.3-3;4)0, (1)

are fully compatible with the TBM matrix:

/ /73 1//3 0 \

utb = -1/V6 1/V3 -1A/2 , (2)

V-1/V6 1//3 +1//2Z

E-mail address: yinlin@pd.infn.it.

0550-3213/$ - see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2009.08.018

which corresponds to sin2 612 = 1/3 (612 = 35.3°) and sin2 623 = 1/2. The TBM pattern also predicts 613 = 0, however, recent analysis based on global fits of the available data leads to hints for 613 > 0 [2,3]. Differently from solar or atmospherical mixing angles, the reactor one is less constrained and its value can still be relatively large, even at 1a level, say ~ XC:

sin2 613 = 0.016 ± 0.010 [2], sin2 613 = 0.010—0.016 [3].

For this reason the future experimental sensitivity on the reactor angle is fundamental for theoretical understanding of the TBM, which, due to its highly symmetric structure, strongly suggests an underlying non-Abelian flavour symmetry. A natural and economical class of models based on A4 flavour symmetry [4-8] has been proposed in describing TBM pattern. There are also more involved models based on other discrete groups [9,10] or continuous flavour symmetries [11]. However, if the TBM pattern results from a spontaneously broken flavour symmetry, higher order corrections should introduce deviation from exact TBM, generally all of the same order. Since the experimental departures of 6i2 from its tri-bimaximal value are at most of order ^C, a future observed value of 613 near its present upper bound should impose severe constraints on model buildings [12]. If this is the case, one apparently has to renounce the nice symmetric nature of 612 and simply imagines that its tri-bimaximal value may be completely accidental [13].

In this paper we will show that the TBM pattern can be explained by A4 symmetry without necessarily implying a very small value for 613. The fundamental new ingredient of our construction is to allow a moderate hierarchy, of order , between the VEVs of flavon fields of the charged lepton and the neutrino sectors. The paper is organized as follows. In the next section, we characterize some general conditions under which a hierarchy between VEVs of flavon fields of different sectors can be accommodated without fine tuning. In Section 3, we introduce the field content of our seesaw model based on A4 x Z3 x Z4 flavour group with great emphasis on vacuum alignment and its stability. In Section 4, we explain how the charged lepton hierarchy can be reproduced by a particular symmetry breaking pattern of A4. In Section 5, we describe the neutrino mass at leading order by seesaw mechanism and obtain an exact TBM at this level. Then in Section 6 we include all subleading corrections up to terms suppressed by 1 /A2 to our model and analyze possible deviation from TBM. In the end, in Section 7, we comment on other possible phenomenological consequences of our model and conclude.

2. General consideration

The difficulty in the standard formulation of A4 models [4,5] in generating a relatively large value of 613 is related to the vacuum alignment problem which plays a fundamental rule in order to naturally describe the TBM pattern from spontaneously broken flavour symmetries. The group A4 (see Appendix A) has two important subgroups: GS, which is a reflection subgroup generated by S and GT, which is the group generated by T, isomorphic to Z3. A4 can be spontaneously broken by VEVs of two sets of flavon fields, 0 for the neutrino sector and for the charged lepton sector. The direction of (0) should leave the subgroup GS unbroken leading to the TBM. However one generally has two options for the alignment of 0'. (0') can be such that GT is preserved leading to diagonal charged lepton masses but their hierarchy is usually generated by an independent Froggatt-Nielsen (FN) mechanism [14]. The second option is to consider a vacuum alignment of 0' which entirely breaks A4 and in this case the mass hierarchy is directly related to (0')/A, being A the cut-off scale, without an extra FN component [6-8,15]. A natural mechanism for the vacuum alignment of 0 and 0' in different directions requires the existence of an Abelian factor GA in addition to A4. The aim of GA is to guarantee the following decomposition

of the scalar potential as:

V(0, 0') = Vv(0) + Ve(0') + VNLO(0, 0') + ..., (3)

where we see that the interaction term between 0 and 0' appears from next-to-leading order (NLO). We will refer this situation as a "partial" separation in the scalar potential which is tightly related on the fact that only one of the sets 0 and 0' is charged under GA, a standard choice in the literature [4,5,8]. At leading order, the two scalar sectors are actually separated, however, the vacuum alignments are affected by NLO corrections encoded in VNLO(0, 0'). The order of magnitude of the corrections to the VEVs (0) and (0') depends on (0)/A and (0')/A and they are subject to some conditions. First of all, the corrections to the tri-bimaximal value of 612 are at most of order X2C. Furthermore, the corrections to (0') are required to be smaller than mx/mT ~ O(l2C) or more restrictively smaller than me/mx ~ O(X3C); otherwise, the generated charged lepton hierarchy should not be stable. These conditions shall translate to upper bounds on the scale of flavour symmetry breaking with respect of the cut-off scale:

(0)/A, (0')/A < X2C (4)

(0')/A < X2C. In conclusion, a value of 613 near its present experimental bound cannot be described if the scalar potential is "partially" separated as quoted in (3).

In this paper we will exploit the possibility of a "fully" separated scalar potential which corresponds to (3) with VNLO(0, 0') = VNLO(0) or VNLO(0, 0') = VNLO(0'). The "fully" separated scalar potential can be obtained if GA is a direct product of two Abelian factors GvA and GeA which separately acts on 0 and 0'. In this case, since Vv(0) and Ve(0') can be minimized in a completely independent way, even including NLO corrections, we are not necessarily subject to the strict condition (4). In fact, it is possible to construct a completely natural model for TBM based on the A4 symmetry in which (0')/A ~ O(k2C) and (0)/A ~ O(XC) can be compatible with all experimental constraints. The model belongs the constrained A4 models considered in [6,7] in which the leading order neutrino TBM and the charged lepton mass hierarchy are simultaneously reproduced by the vacuum alignment. Our choice for GA in order to guarantee a "fully" separated scalar potential is given by Z3 x Z4. We are particularly interested in analyzing the possibility to have a relatively large value of 613 without fine-tuning. We will show indeed that 613 can be of order XC while 612 is corrected by subleading effects arising at order ^C. Furthermore, deviations from TBM can be more intriguing since they obey a definite sum-rule which can be in principle tested.

3. Field content and vacuum alignment

In this section we introduce the field content of the model and analyze the most general scalar potential which is invariant under the flavour symmetry A4 x Z3 x Z4. The lepton SU(2) doublets li (i = e, /, t) are assigned to the triplet A4 representation, while the lepton singlets ec, /xc and tc are all invariant under A4. The neutrino sector is described by seesaw mechanism with 3 heavy right-handed neutrinos vC which also form an A4 triplet. The symmetry breaking sector consists of the scalar fields neutral under the SM gauge group, divided in two sets as advanced before: 0 = {(pS,^,Z) and 0' = {<pT,%'}. As anticipated before, in addition to A4, we also have an Abelian symmetry GA = Z3 x Z4 which is a distinguishing feature of our construction. All the fields of the model, together with their transformation properties under the flavour group, are listed in Table 1. We observe that 0 is charged under Z3 while 0' is charged under Z4.

Table 1

The transformation properties of leptons, electroweak Higgs doublets and flavons under A4 x Z3 x Z4.

Field l ec Hc Tc vc hu hd PT $' PS Z

A4 3 1 1 1 3 1 1 3 1' 3 1 1

Z3 1 1 1 1 m 1 1 1 1 m m m2

Z4 1 -1 -i 1 1 1 -i i i 1 1 1

The vacuum alignment problem of the model can be solved by the supersymmetric driving field method introduced in [5]. This approach exploits the continuous ^(1)R symmetry in the superpotential w under which matter fields have R = +1, while Higgses and flavons have R = 0. The spontaneous breaking of A4 can be employed by adding to fields already present in Table 1 a new set of multiplets, called driving fields, with R = 2. We introduce a driving field $0, fully invariant under A4, and two driving fields pT and pS, triplet of A4. The driving fields $0 and pS, which are responsible for the alignment of ps, have a charge m under Z3 and are invariant under Z4. pT has a charged -1 under Z4, invariant under Z3, and drives a non-trivial VEV of pT. The most general driving superpotential wd invariant under A4 x GA with R = 2 is a sum of two independent parts wd = wvd($0,pS ,&) + wed(pT ,&') where

wvd = gtp^pl + g2$(p0Ps) + g3$0(PsPs) + g4M2 + g5$0$f + g6$0$2 + glMzH0Z, (5) wd = h\$ '{(IPT) " + h2(pTPTVr)- (6)

The "fully" separated superpotential is guaranteed by GA = Z3 x Z4. Eqs. (5) and (6) gives two decoupled sets of F-terms for driving fields which characterize the supersymmetric minimum. In other words, wvd and wed independently determine the vacuum alignment of @ and @', respectively. From (5) we have:

dw , 2 \

= g2$PS1 + 2g\{ps 1 - PS2PS3) = 0,

dP0S1 dw

„ s = g2$PS3 + 2g\(ps2 - PS1PS3) = 0, dp0

s = g2$PS2 + 2g\(pps2 - PS1PS2) = 0, 03

^ = g4$ 2 + g5^f + g6$2 + glMzZ + g3 (ps 1 + 2ps2 PS3) = 0. (7)

In a finite portion of the parameter space, we find the following stable solution

<$)=0, <§ ) = u, <Z ) = v,

2_ g4U2 + glMz

(ps) = (vs,vs,vs), vs =----, (8)

with u and v undetermined. Since (f) = 0,1 we have ignored the existence of f in the rest of the paper. Setting to zero the F-terms from Eq. (6), we obtain:

T = h1f 'pt3 + 2h2(pT2 - PT2PT3) = 0, '01

= h1f 'pt2 + 2h2p2 - PT 1PT3) = 0,

dw , f 2 \

—T = h1f PT 1 + 2h2(PT3 - PT 1PT2) = 0, dP03

and the stable solution to these four equations is:

(f ') = u' = 0, (PT) = (0,VT, 0), VT = -h1^, (9)

with u' undetermined. The flat directions can be removed by the interplay of radiative corrections to the scalar potential and soft SUSY breaking terms. It is worth to observe that, thanks to GA, the VEV alignments (8) and (9) are independent even at NLO.

Since the VEVs of the scalar fields in 0 (0') are related each other by adimensional constants of order one, we should expect that they have a common scale indicated by (0) ((0')). However, (0)/A and (0')/A can be in principle different and they are subject to phenomenological constraints. As we will see in the next section, (0') is responsible for charged lepton hierarchy so we have to require

me r, (0') 0 mu

— — 4 < — < x2C —

m/ A mT

The superpotential wed is affected by non-renormalizable terms (see Appendix B for the detail) from the neutrino sector 0 suppressed by 1/A2. Requiring that the sub-leading corrections to (0') are smaller than m^/mt — O(k"C), we obtain the condition

0 < C A

The vacuum alignment with a "fully" separated scalar potential allows a hierarchy between the VEVs of the scalars in different sectors (0') ^ (0).

Differently from wed, wvd receives NLO corrections which are suppressed only by 1/A but don't depend on the charged lepton sector 0':

w = A [(P0SPS )z2 + f0fz2].

One may wonder if a large VEV of 0 with (0)/A — XC could introduce a too large correction to the leading order vacuum alignment (8) destroying the stability of the TBM prediction. Fortunately, this is not the case. Since there is no fundamental distinction between Z2 and f the NLO correction Swvd should induce terms which have the same form of those already present in wvd. In fact, including 8wd in the minimization, one easily find that the (pS) receives only a small shift

1 Since there is no fundamental distinction between the singlets f and f we have defined f as the combination that

couples to (pSPs) in the superpotential wd. The introduction of an additional singlet is essential to recover a non-trivial

solution.

in the same direction of the leading order alignment. For this reason we will no longer consider VEV shifts of ps in the following.

4. Charged lepton hierarchy

In the present section, we illustrate how a fully broken A4 symmetry can generate the charged lepton hierarchy. The key ingredient is the alignment <pT) ~ (0, 1,0). Such a VEV breaks the permutation symmetry of the second and third generation of neutrinos in a maximal way in the sense that

<PT )%-3<PT) = 0,

/1 0 0^ S2-3 = 0 0 1

\0 1 0,

The A4 group is fully broken2 in the charged lepton sector by 0' with the vacuum structure quoted in (9) and only the tau mass is generated at leading order. The muon and electro masses are generated respectively by <pT)2 a (0, 0, 1) and <pT)3 a (1, 0, 0). Then the correct hierarchy between the charged lepton masses me ^ mM ^ mT is reproduced if we assume X2C < <0')/A <

Since 0' carries a charge i under Z4 we have to assign different Z4 charges for lepton singlets. Considering only insertions of 0', the charged lepton masses are described by we, given by, up to 1/A3:

we = a\Tc(lpT)hd/A

+ '(lpT)"hd/A2 + P2^c(lpTPT)hd/A2

+ Y1 ec ($ ")2(lpT)'hd/A 3 + Y2ec$ '(lprPT)"hd/A3 + Y3ec (lpTPTPT)hd/A3.

After electroweak symmetry breaking, <hu,d) = vu,d, given the specific orientation of <pT) a (0, 1, 0), we give rise to diagonal and hierarchical mass terms for charged leptons. Defining the expansion parameter vT/A = X2 ^ 1 (it is not restrictive to consider vT to be positive) and the Yukawa couplings y (l = e, /x,t) as

= |a 11,

= Ipiu'/vt + 2^2 |X2, ye = \ y1(u'/vt)2 - y2u'/vt - 2Y3 i, the charged lepton masses are given by

ml = yll2vd(l = e,^,t). (10)

As already pointed out in the previous section and analyzed in detail in Appendix B, the vacuum alignment for pT receives correction of order <0 )2/A2 ~ different for each component:

PT = (st 1, vt + 8t2,8t3).

2 Similarly as explained in [6,8], a residual symmetry A4 x Z3 from A4 x Z4 survives in the charged lepton sector guaranteeing the stability of the vacuum alignment.

Including correction to the vacuum alignment for pT, the diagonal form of the charged lepton mass should slightly change and small off-diagonal entries appear:

/ me meO(XC)

O(X2C)\

m^O(X2C) mM m^O(X2C)

\mx O(X2C) mTO(X2C)

The transformation needed to diagonalize me is VjmeUe = diag(me,m^,mt) and the unitary matrix Ue is given by

O(XC) \O(X2)

O(XC) 1

O(X2C)\

O(X2c) 1

Another source of off-diagonal correction to charged leptons comes from the interaction with the neutrino sector. In fact, the products fZ and pSZ are invariant combination under GA and we can include them on top of each term in we. However, we find that the introduction of these additional terms changes the charged lepton mass me exactly in the same way as the corrections induced by VEV shifts of pT, i.e. (11). Then (12) is the most general structure of the charged lepton contribution to TB mixing.

5. A seesaw realization of the constrained A4 model

The masses of light neutrinos of our model is described by seesaw superpotential with 3 heavy right-handed neutrinos vf, triplet of A4. Terms in the superpotential which contain vc invariant under the flavour group are given by:

= y(vcl)zhu/A + xa^(vcvc) + xb((psvcvc) + h.c. + ••

In the heavy neutrino sector A4 x Z3 is broken by (pS) = (vS,vS,vS) and (f) = u down to GS (with Z4 unbroken) with an accidental extra G2-3 symmetry. Then the residual symmetry of the right-handed neutrino masses is Gtb — Gs x G2-3. Gtb can be transfered to the light neutrino sector if the Dirac neutrino mass commute its generators. This is in fact the case. After electroweak and A4 symmetry breaking from (13) we obtain the following leading contribution to the Dirac and Majorana masses:

'a + 2b

M= I -b

b = xb

-b 2b a — b

-b a — b 2b

We immediately see that [mD,S] = 0. The leading order lepton mixing matrix is entirely encoded in the right-handed neutrino mass matrix M which is diagonalized by the transformation:

UMUo = diag(la + 3b|, \a\, \a — 3b\)u,

with U0 = Utb^, where Q = diag(e!^1/2,e!^2/2,fe!^3/2| and are respectively phases

of a + 3b, a, a — 3b. Naturally <1 and <3 depend on <2 and A, the relative phase between a and b.

The light neutrino masses are given by the type I seesaw mechanism: mv = (mff) which is invariant under GTB and then also diagonalized by U0.3 Denoting the physical masses of vC as M1 = |a + 3b|, M2 = |a| and M3 = |a - 3b|, we obtain

UÏm° Uo =

■ 1 1 1 ■ dia^ M 'M2 = diag{m1'm2'm3}-

m2 > m1 implies t = \3b\/\a\ > —2 cos A and in principle both normal and inverted hierarchies in the neutrino spectrum can be reproduced. The normal hierarchy is realized for t/2 < cos A < 1 whereas an inverted spectrum requires —t/2 < cos A < 0. The ratio r = Am2^/Am:;tm (where Am2un = m2 — m2 and Am:;tm = \m| — m1\) is given in our model by:

(t + 2cos A)(1 +12 - 2t cos A) 4 cos A '

One can show that for the normal hierarchy, a small value of r & 1 /30 can be reproduced only for cos A & t & 1. In particular, a normal ordered spectrum can never be degenerate. Then we can expand t = 1 + St with St ^ t obtaining the following approximate spectrum:

mi « yAm2um/3, m2 « 2mx, ^Am\Xm - Am2um/3.

The inverted hierarchy can be realized only for t &-2 cos A and in this case we can expand cos A = -t/2 + St' with St' ^ t. Expressing St in function of r we obtain

m1 = Am 2tm

m2 = Am 2tm

m3 = Am 2tm

— (- 1

1 + 2t2 + 112 - T+2Î2

1 + 2t2 + I1 + t2 - T+2Î2

t2 1 + 2t2

In principle, the previous expansion is valid also for a degenerate spectrum realized by t ^ 1 which is, however, parametrically fine-tuned4 in our model.

Before going beyond the leading order result obtained in this section, we can estimate the natural mass scale of the lightest right-handed neutrino v| considering, for simplicity, a normal hierarchy for light neutrinos. In this case, the right-handed neutrinos are also hierarchical according to M3 « Vr/3M1 and M2 « (1/2)M1. By taking neutrino mass scale as ~ 0.05 eV and the scale of mD as vuXC with vu = 174 GeV one obtains M3 ~ 3 x 1013 GeV. FromEq. (16) we see that the right-handed neutrinos have a same mass scale as >. Then the hierarchy among

3 The overall phase appearing in the Dirac neutrino mass mD can be absorbed by the redefinition of 02 and there are only two independent Majorana phases.

4 The fine-tuning required in order to reproduce a small r becomes more severe if we include in wv also the five-dimensional operator lhulhu/A' which leads to amass matrix structure similar to the term %vcvc. Indeed, if the Weinberg operator has a cutoff scale A' ~ A, its contribution becomes larger than the seesaw one. This situation is equivalent to go to the limit a ^ b and then it is disfavored. In order to avoid this problem we will assume that the lepton number is violated only by Majorana mass term up to A. In other words, we require A' ^ A and a direct five-dimensional operator can be neglected.

the scales is

{Ф') — {Ф}Xc — ЛХ2с

with {Ф) — M3-M1. Correspondingly the cut-off scale Л will range between about 1014 GeV and 1015 GeV. Beyond this energy scale, new physics like grand unified theories should come into play.

6. Deviation from TBM and 613 ~ Xc

In this section we show how a relatively large reactor angle, say в13 — вс, can naturally arise in our model, without conflicting with the precise value of в12 predicted by TBM. The neutrino mass described in the previous section predicts an exact TBM. Including sub-leading contributions dictated by higher-dimensional operators, the leading order lepton mixing matrix should be modified. As we shall see in a moment, not all deviations from TBM arise at the same perturbation level, this is one of the most important feature of the model. We find that the NLO corrections generate a non-vanishing reactor angle which is correlated with deviation of atmospherical angle from maximal. While the corrections to solar angle appear only at next-to-next to leading order (NNLO).

First of all we focus on higher order corrections to the right-handed Majorana neutrino mass up to terms suppressed by 1 /Л2. At NLO, there is only one additional contribution to heavy Majorana mass: Z2vcvc/Л. Since Z2 has exactly the same property of f, this term can be absorbed by a redefinition of a. The NNLO contributions arise from adding the products fZ and psZ, invariant combination under GA, on top of the leading order terms. In this case, not all the corrections have the same structure of the terms already present in wv and consequently cannot be regarded as small shifts of a and b, for example (vcvc)'(psps)" and (vcvc)"(psps)'. However, these terms can be absorbed by parameters y1 and y2 in the NLO correction to the Dirac mass SmD as will be clear in a moment.

Now we move to consider the correction to Dirac neutrino mass: SmD beginning with terms suppressed by 1/Л2. There are many independent terms of the type (vclpp)hu, with p e {ps,f}, invariant of A4 which contribute to SmD at this order:

Swv = hu^2 {vci)\psps)n + hu Л {vcl)"(psps)' + hu-y32 vc(lps)Af

+ huy2 {vcl)1(psps)1 + huy2 (vcl)f2 + huy2 vc(l<ps)sf. (19)

Observe that the operators with coefficients y',y",y'2 give contribution to Dirac mass matrix in a form invariant under GTB exactly as right-handed neutrino mass. Then these corrections can be adsorbed into a redefinition of the leading-order coefficients. The relevant correction to the Dirac mass comes from the first three terms in Eq. (19) and has the following form:

(0 y1 + y3 y2 — y3 \ 2

y1 — y3 y2 y3 \vuA2, (20)

y2 + y3 — y3 y1 '

where y1,y2,y3 = y3u/vs are generally complex number of order 1. Before discussing the important consequence when we include the NLO correction to the Dirac neutrino mass, we comment possible NNLO effects on mD. Here the NNLO contributions are suppressed by 1 /A3

and they are of the type (vclz2y)hu. All these terms can be absorbed by a redefinition of y3 and y2, then we can forget them in the following analysis.

In order to find the correction to the leading neutrino mixing matrix U0 = UTB^, it is convenient to define

mD = ulmDu0,

where mD = mD + SmD. The light neutrino mass is then formally given by mv = Uoni vU^

where mv = (tnD)TM—gmD with M"^ = diag{1/M1,1/M2,1M3}. If mv can be diagonalized

by the unitary matrix SU ~ I as T

SUm vSU = diag{m\,m 2,m 3}, where mi ^ mi, the full PMNS mixing matrix will be given by

Upmns = UjUoSU. (21)

In our case, the matrix mD has a very simple expression:

1 0 ei*31 c+es

mD 0 1 0 ]yvuV, (22)

J*31 c_e 0 -1

where *31 = (*3 — *1)/2, c+(_) = i\/3/2(y2 - yi + (—)2y3) and e = vj/(vA) ~ . Then we get

m1 0 e1*31 (c+m1 + c_m3)e\

0 m2 0 |+ O (e2). (23)

e'*31 (c+m1 + c_m3)e 0 m3 /

This result means that a correction (SU) 13 ~ \C can be present and we can expect that a deviation of 012 from it tri-bimaximal value arises only at order X2C. However, observe that if m1 « m3 i.e. the spectrum becomes degenerate, a fine-tuning will be required in order to reproduce a small (S U) 13. From this viewpoint, a degenerate spectrum is disfavored if we require that the deviation from TBM is naturally small.

Forgetting for a moment Ue which arises only at NNLO, from Eq. (21), one find that

Ue3 =/fei*13 (SU)13, UM3 = _+ y6e!*13 ^13, (24)

and Ul2, l = e, ix, t, remain unchanged. As a result, the solar angle 012 remains rather close to its tri-bimaximal value. However, (SU) 13 simultaneously induces a departure of 013 and of 023 — n/4 from zero. Defining S' as the phase of (SU)13, the CP-violating Dirac phase is given by — S = S' + *13. Since sin 013 = V 2/3|(SU)131, the deviation of the atmospherical angle from maximal is subject to the following sum-rule:

sin2023 = 1 —U[|2 ^ 2 + ^^cosS sin013 + O(023), (25)

this is a prediction of our model. This is a special feature of the present seesaw A4 model. The presence of the Abelian factor GA in our model, not only allows a relatively large value of 013,

at OC level, also strongly suppresses possible higher order contributions giving rise correlation between them.

Independently from the seesaw sector, TBM and in particular the solar angle receives corrections from charged lepton sector. Adopting the standard parametrization of ^PMNS, from (21) and (12) one finds that all the mixing angles receive a correction of order X2C. Then we in particular obtain

sin2 O12 = 1 + O (4 )■

As claimed in the beginning, O13 can be of order Xc since it arises from corrections at NLO in the neutrino sector while O12 receives corrections only of order X2C which are subleading effects at NNLO.

7. Conclusion and discussion

In this paper we have addressed one of the most important issues in the A4 realization of TBM, i.e. if a 013 ~ 0C can be allowed without fine tuning. We have discussed a framework, referred as constrained A4 model, in which the vacuum alignment is realized by a fully separated scalar potential. The model is based on the A4 x Z3 x Z4 flavour symmetry and (Type I) seesaw mechanism. In the charged lepton sector, the A4 group is entirely broken by the set of scalar field @' = WtiH'}. The symmetry breaking parameter ')/A ~ X2C directly controls the charged lepton mass hierarchy without requiring a U(1)FN symmetry. In the neutrino sector, the set of scalar fields @ = breaks the A4 group to its subgroup Gs guaranteeing the TBM at leading

order. The symmetry breaking parameter )/A, however, can be chosen at order of the Cabibbo angle XC without altering the required vacuum alignment for @'. Moreover, a non-vanishing 613 and a deviation of 023 from n/4 are simultaneously generated at order O(XC) leaving 012 unchanged. Subsequently, a deviation of the solar angle from its TBM value is generated at order O(k2C) which just corresponds to its 1a experimental sensitivity.

The model is called constrained A4 model because, differently from its standard formulation widely studied in literature, the NLO corrections are also dictated by A4 symmetry itself. This is another interesting feature of our model. There is, indeed, a correlation between the deviation of 023 from maximal and the value of generated 913: sin2 023 « 1/2 + \[2/2 cos S sin 013 + O(023) which can be in principle tested by future experiments. Concerning the neutrino spectrum, it can be either of normal hierarchy or inverted one. However, a degenerated spectrum is parametrically fine tuned and is disfavored requiring that the deviation from TBM is naturally small. For this reason, we should also expect that the effect of running on mixing angles is negligible. Since the solar angle has been measured more precisely than the others, its running can be potentially important if the neutrino spectrum were degenerate.

The corrections beyond the leading order are important not only in describing deviations from TBM, but also give rise other interesting phenomenology. For example, the same breaking pattern for charged lepton sector can be easily extended to the quark sector. In this case, the VCKM arises when the correction to the vacuum alignment pT is taken into account. Then the resulting VCkm should have the same form of the unitary matrix diagonalizing charged leptons Ue given in (12). The inclusion of the sub leading corrections can also play an important role in explaining the baryon asymmetry of the universe (BAU) through leptogenesis [16]. As pointed out in [7], the generated BAU can be indeed directly trigged by low energy phases appearing Ue3. Moreover, the structure of A4 symmetry breaking pattern can be revealed by other physical

effects [17], not directly related to neutrino properties, such as lepton flavour violating process as well as the anomalous magnetic moments and the electric dipole moments of charged leptons. Such a possibility becomes realistic if there is new physics at a much lower energy scale around 1-10 TeV. All these issues merit a further and more detailed study.

Acknowledgements

We thank Ferruccio Feruglio for useful suggestions and for his encouragement in our work. We thank also Guido Altarelli, Davide Meloni and Luca Merlo for useful discussions. We recognize that this work has been partly supported by the European Commission under contracts MRTN-CT-2004-503369 and MRTN-CT-2006-035505.

Appendix A. The group A4

The group A4 has 12 elements and four non-equivalent irreducible representations: one triplet and three independent singlets 1,1' and 1". Elements of A4 are generated by the two generators S and T obeying the relations:

S2 = (ST)3 = T3 = 1.

S = 1, T = 1,

S = 1, T = ei4n/3 = co2, S = 1, T = ei2n/3 = o,

We will consider the following unitary representations of T and S:

for 1: for 1': for 1'':

and for the triplet representation (1 02

T = 0 0

\0 0 (OJ

The tensor product of two triplets is given by 3 x 3 = 1 + 1' + 1" + 3S + 3A. From (27) and (28), one can easily construct all multiplication rules of A4. In particular, for two triplets f = (f 1,f2,f3) and p = {(P\,(P2,(P3) one has:

flVl + f2P3 + f3P2 ~ 1,

f3P3 + f 1P2 + f2P1 ~ 1',

f2P2 + f3P1 + f 1P3 ~ 1'',

i2f 1P1 - f 2P3 - f 3P2 s 2f3P3 - f 1P2 - f2P1

\2f2P2 - f\P3 - t3P1 Appendix B. Correction to alignment of ^t and çs

f2P3 - f3P2" t\P2 - t2P1 f3P1 - f\P3 -

■3A.

In this appendix we will study correction to the leading order alignment of ps and pt when we include higher dimensionality operators up to the order 1 /A2.

In our model, the correction to the driving superpotential for <s, depends only on 0 at NNLO, then the obtained vacuum alignment (<ps) a (1, 1,1) is always stable since it preserves the subgroup Gs of A4. However a relative large (0)/A ~ kC may have some effects on the leading order alignment for <T a (0,1, 0). The products fZ and <sZ are invariant combination under Ga, then we can include them on top of each term in wed. With the introduction of these higher dimensionality operators, wd should be modified into wed + -wed where5

Swd = A [hZWivlvr)" + t2Zf{voVTVT)

+13 Zf' {voVTVs)" + UZivlvs) '(<t<t)"\ .

The alignment for <T should be shifted (the shift in f' is needless) and we can look for a solution that perturbs (<T) to second order in the 1 /A expansion:

(f ') = u, (<pt ) = (-t 1 ,vt + -T2,-T3). The minimum conditions from wd + -wed become equations in the shifts -vTi:

-4h2vTSvT3 + ( t4 -t3-— ) —p^vT = 0,

4hi\vvi .2

' hi ) A2 "

(4h2\vvS 2 / 2h2\ vu 2

t4 + t3 VA2VVT + - ^J A VT = 0

-4h2VT5VT 1 + ( t4 +13— I — vT = 0.

4h2 \ vvs ^ 2 'hi) A2 "

These equations are linear in SvTi and can be easily solved by:

SVt3 _ (__vvi

vT \4h2 h\) A2,

-vt2_ f t4 2tA vvs it1 t2\ vu — = V 2h2 + In) A2 + \ h1 - ~h2) A2, svt 1 ( t4 + t£\ vvs

vT 4h2 h1 A2

Observe that the shifts in three components are different but all of the same order of magnitude, as claimed in the text:

SvTi 2)

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