Contents lists available at SciVerse ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Minimal see-saw model predicting best fit lepton mixing angles

Stephen F. King

School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, United Kingdom

CrossMark

ARTICLE INFO

Article history: Received 23 May 2013 Accepted 5 June 2013 Available online 7 June 2013 Editor: M. CvetiC

ABSTRACT

We discuss a minimal predictive see-saw model in which the right-handed neutrino mainly responsible for the atmospheric neutrino mass has couplings to (ve,vp,vT) proportional to (0,1,1) and the right-handed neutrino mainly responsible for the solar neutrino mass has couplings to (ve, vp,vT) proportional to (1,4, 2), with a relative phase n = —2n/5. We show how these patterns of couplings could arise from an A4 family symmetry model of leptons, together with Z3 and Z5 symmetries which fix n = —2n/5 up to a discrete phase choice. The PMNS matrix is then completely determined by one remaining parameter which is used to fix the neutrino mass ratio m2/m3. The model predicts the lepton mixing angles 012 & 34",023 & 41",013 & 9.5°, which exactly coincide with the current best fit values for a normal neutrino mass hierarchy, together with the distinctive prediction for the CP violating oscillation phase S & 106".

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Daya Bay [1] and RENO [2] have measured a non-zero reactor angle 013 ^ 0.15 which excludes Tri-Bimaximal (TB) mixing [3]. Recent global fits also hint at deviations of the atmospheric and solar angles from their TB values (for a recent review see e.g. [4]). Such deviations may be expressed in terms of the deviation parameters (s, a and r) from TB mixing [5] (for a related parametrisation see [6]):

sin 012 = —= (1 + s),

sin 023 = -—= (1 + a),

sin 013 =

With zero solar and atmospheric deviations from TB mixing, s = a = 0, and Cabibbo-like reactor mixing described by r = X, with X = 0.225 being the Wolfenstein parameter, one is led to Tri-Bimaximal Cabibbo (TBC) mixing [7]. However, as mentioned above, current global fits prefer non-zero solar and atmospheric TB deviation parameters,

s = -X2/2, a =-X/3, r = X, corresponding to the angles,

012 = 34.2°

023 = 40.8°

013 = 9.15°

These angles are close to the best fit values for all three global fits in the case of a normal neutrino mass ordering [4]. Assuming a normal neutrino mass hierarchy with m1 = 0, one is led to [13],

m2 3 x

E-mail address: king@soton.ac.uk.

corresponding to m2/m3 & 0.17, close to the best fit value [4]. The deviation parameters in Eq. (2) have the feature that the atmospheric mixing angle is in the first octant and the solar mixing angle is somewhat less than its tri-maximal value, in agreement with the latest global fits for the case of a normal neutrino mass ordering. In particular it reproduces the best fit values of angles of all three global fits [4] to within one standard deviation.

There have been many attempts to describe the lepton mixing angles based on the type I see-saw model [8] combined with sequential dominance (SD) [9] in which the right-handed neutrinos contribute with sequential strength. Constrained sequential dominance (CSD) [10] involves the right-handed neutrino mainly responsible for the atmospheric neutrino mass having couplings to (ve, v^, vT) proportional to (0,1,1) and the right-handed neutrino mainly responsible for the solar neutrino mass having couplings to (ve, v^, vT) proportional to (1,1, —1) and it led to TB mixing. CSD2 [11] was proposed to give a non-zero reactor angle and is based on the same atmospheric alignment but with right-handed neutrino mainly responsible for the solar neutrino mass having couplings to (ve, Vp, vT) proportional to (1, 0, —2) or (1, 2, 0) yielding a reactor angle 013 & 6" which unfortunately is too small, although the situation can be rescued by invoking charged lepton corrections [12]. The CSD3 model in [13] involves the right-handed neutrino mainly responsible for the solar neutrino mass having

0370-2693/$ - see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016Zj.physletb.2013.06.013

Table 1

Lepton, Higgs and flavon superfields and how they transform under the symmetries relevant for the Yukawa sector of the model. The only non-trivial charged lepton charges are in the upper left of the table and the only non-trivial neutrino charges in the lower right of the table. Note that the only the lepton doublets L and A4 symmetry, are common to both charged lepton and neutrino sectors and are given near the central column and row. The Standard Model gauge symmetries and U (1)R symmetry, under which all the leptons have a charge of unity while the Higgs and flavons have zero charge, are not shown in the table.

Z 6 Z3

Z / Z3

vsol Z5

xc t c ve v/ Vt H1 L H 2 ^atm <Psol Natm Nsol £atm isol

w2 1 1 1 1 1 1 1 1 1 1 1 1 1

1 1 w 1 1 1 1 1 1 1 1 1 1 1

W2 1 1 w 1 1 1 1 1 1 1 1 1 1

1 w2 1 1 w 1 1 1 1 1 1 1 1 1

1 1 3 3 3 1 3 1 3 3 1 1 1 1

1 1 1 1 1 1 1 1 p3 1 P2 1 p 1

1 1 1 1 1 1 1 1 1 P3 1 P2 1 p

vatm Z5

couplings to (ve, vvr) proportional to (1, 3,1) or (1,1, 3) with a relative phase ^n/3 yielding a reactor angle 613 ^ 8.5° close to the observed value. However CSD3 predicts approximate TBC mixing with an almost maximal atmospheric mixing angle disfavoured by the latest global fits, and so it may soon be challenged.

In this Letter we shall propose a model based on a new possibility called CSD4 which predicts the above best fit angles in Eq. (3) of the PMNS lepton mixing matrix and also makes predictions for the physical CP violating phases. Similar to all SD models, the CSD4 model involves effectively two right-handed neutrinos and a normal neutrino mass hierarchy, leading to m1 = 0. As in CSD2 and CSD3, the CSD4 model only requires one input parameter, namely the ratio of neutrino masses which is selected to be m2/m3 ^ 3x/4, which is a natural value that one would expect in such models. Also as in CSD2 and CSD3, once this value is chosen, the entire PMNS mixing matrix is then fixed by the theory (up to a discrete choice of phases) with no remaining free parameters. In the CSD4 model, the right-handed neutrino mainly responsible for the atmospheric neutrino mass has couplings to (ve,V/,VT) proportional to (0, 1,1) and the right-handed neutrino mainly responsible for the solar neutrino mass has couplings to (ve, V/,VT) proportional to (1, 4, 2), with a relative phase n = —2n/5. These couplings and phase relation were first discovered in [13] and shown to lead to lepton mixing angles in good agreement with the latest global fits, but no model has been proposed based on CSD4. The goal of this Letter is to show how CSD4 can arise from an A4 family symmetry, together with additional discrete Z5 and Z3 symmetries, and to present the first model of leptons along these lines. This is necessary since it is far from clear whether alignments such as (1, 4, 2) are possible to achieve within a realistic model. The CSD4 model presented here predicts the best fit PMNS angles in Eq. (3) with the distinctive prediction for the oscillation phase S ^ 106°.

2. A minimal predictive A4 model of leptons

In this section we outline a supersymmetric (SUSY) A4 model of leptons with CSD4 along the lines of the A4 models of leptons discussed in [11,14]. The basic idea is that the three families of lepton doublets L form a triplet of A4 while the right-handed charged leptons ec, xc, tc, right-handed neutrinos Natm, Nso and the two Higgs doublets H1, H2 required by SUSY are all singlets of A4. In addition the model employs an additional Ze3 family symmetry in order to account for the charged lepton mass hierarchy.

The vacuum alignment that is required for the model is discussed in Appendix A. In Table 1 we have displayed the symmetries and superfields relevant for the Yukawa sector only. In Appendix A the transformation properties of the remaining super-fields under Z3 x Zresponsible for vacuum alignment is discussed and are consistent with the charges shown in Table 1,

where we have written 0atm — Vv3 and 0soi — Vv4 and hence

1 — Zv3 and Z1°' — Zv4

5 _ 5 aiiu ^5 = ^5 .

The charged lepton sector of the model employs the A4 triplet flavons ye, V/, Vt whose alignment is discussed in Appendix A. With the lepton symmetries in the upper left of Table 1 we may enforce the following charged lepton Yukawa superpotential at leading order

'Yuk ~ —H 1(vr ■ L)tc + — 6Hi&x ■ L)/c

+ —3 62Ht(Ve ■ L)ec,

which give the charged lepton Yukawa couplings after the flavons develop their vevs. A is a generic messenger mass scale, but in a renormalisable model the messengers scales may differ. The charged lepton symmetries include three lepton flavour symme-

tries Z3'x' under which ye, V/, yT and ec, /c, tc transform respectively as w and w2, together with a lepton family symmetry Z3 under which ec, /c, tc transform as w, w2, 1 respectively (where w = ean/3) with the family symmetry breaking flavon 6 transforming as w and otherwise being a singlet under all other symmetries. Hi and L and all other fields are singlets under Z^^ and Z®. With these charge assignments the higher order corrections are very suppressed.

The charged lepton Yukawa matrix is diagonal at leading order due to the alignment of the charged lepton-type flavons in Eq. (25) (where the driving fields responsible for the alignment in Eq. (24) absorb the charges under the newly introduced symmetries Z^^

and Z3) and has the form,

Ye = diag(ye, yyx) ~ diag(e1 yx

where we choose e ~ <6)/A ~ X2 in order to generate the correct order of magnitude charged lepton mass hierarchy, with precise charged lepton masses also dependent on order-one coefficients which we have suppressed here.

With the neutrino symmetries in the lower right part of Table 1 we may enforce the following leading order neutrino Yukawa superpotential

— H2(0atm ■ L)Natm + — ^((fel ■ L)NSo|.

Again the higher order corrections are completely negligible. The neutrino sector of the model exploits the A4 triplet flavons 0atm — Vv3, and — yv4 whose alignment is discussed in Appendix A.

As is typical in models of this kind [11,14], the RH neutrinos have no mass terms at the renormalisable level, but they become massive after some A4 singlet flavons £atm and develop their vevs due to the renormalisable superpotential,

Wr — latm N2tm + |sol N

When the right-handed neutrino flavons develop their vevs (latm ) ~ Ma together with (| sol) — Mb, then the RH neutrino mass matrix is diagonal as required,

MA 0 0 MB

To ensure that the mixed terms are absent at renormalisable order we have imposed a right-handed neutrino flavour symmetry Zatm under which Natm and |atm transform as p2 and p (where p = el2n/5) while 0atm transforms as p3 with all other fields being singlets. We have also imposed a similar symmetry Z f1 under which the "solar" fields transform in an analogous way. We remark that these charge assignments are consistent with the flavon superpotential in Eq. (29), where we identify 0atm = (V3, and 0soi = (V4, with suitable charges assigned to the driving fields.

With all masses, couplings and messenger scales set approximately equal, the driving flavon superpotentials would predict l(£atm)l & l(£soi)l and hence approximately equal right-handed neutrino masses MA & MB. Similarly, from Appendix A with 0atm = (v3 and 0soi = (v4, we see that (02tm)

^soi> would lead to

(0atm> = | Vatm,

(0sol> =

V2^v2'v s°i, (10)

where vatm & Vsoi.

The above charge assignments allow higher order non-renor-malisable mixed terms such as

&Wr — 1 (0atm • 0sol)NatmNsol, (11)

which contribute off-diagonal terms to the right-handed neutrino mass matrix of a magnitude which depends on the absolute scale of the flavon vevs (0atm) and (0soi) compared to (|atm) and (|soi). If all flavon vevs and messenger scales are set equal then these terms are suppressed by e — X2 according to the estimate below Eq. (6), however they may be even more suppressed. We shall ignore the contribution of such off-diagonal mass terms in the following.

Implementing the see-saw mechanism, the effective neutrino mass matrix has the form,

V v2 ((0atm)(0atm)T , (^solX^sol)7

A2 V (latm)

(Isol)

where v2 = (H2). Hence it can be parameterised, up to an overall irrelevant phase, as,

'0 0 0 0 1 1 ^0 1 1

4 2 16 8 8 1

where ma and eV are real mass parameters which determine the physical neutrino masses m3 and m2 and we written the relative phase difference between the two terms as 2n. Using Eq. (10) the see-saw mechanism naturally leads to the neutrino mass matrix in Eq. (13) with eV & 2/21 & 0.1. Hence the desired value of eV & 0.06 assumed below is not unreasonable, and may be achieved for example by taking MB & 2MA. As discussed in [13], we shall also require a special phase relation n = —2n/5 in order to achieve our goal of predicting the best fit values of the lepton mixing angles.

The phase difference n = —2n/5 between flavon vevs can be obtained in the context of spontaneous CP violation from discrete symmetries as discussed in [12], and we shall follow the strategy outlined there. The basic idea is to impose CP conservation on the theory so that all couplings and masses are real. Note that the A4

assignments in Table 1 do not involve the complex singlets 1', 1" or any complex Clebsch-Gordan coefficients so that the definition of CP is straightforward in this model and hence CP may be defined in different ways which are equivalent for our purposes (see [12] for a discussion of this point). The CP symmetry is broken in a discrete way by the form of the superpotential terms. We shall follow [12] and suppose that the flavon vevs (0atm) and (0soi) to be real with the phase n in Eq. (13) originating from the solar right-handed neutrino mass due to the flavon vev (|soi) — MBe4in/5 having a complex phase of 4n/5, while the flavon vev (|atm) — MA is real and positive. This can be arranged if the right-handed neutrino flavon vevs arise from the superpotential,

WflaroM = gpf ^ _ M:

+ gP'( -tOT _ M

„' 2

where, as in [12], the driving singlet fields P, P' denote linear combinations of identical singlets and all couplings and masses are real due to CP conservation. The F-term conditions from Eq. (14) are,

(latm)5 _ M2

(lsol)5 _ m'2

These are satisfied by (|atm) = |(a3 M2)1/5| and (|sol) = |(a'3M'2)1/5|e4n/5 where we arbitrarily select the phases to be zero and 4n/5 from amongst a discrete set of possible choices in each case. More generally we require a phase difference of 4n/5 since the overall phase is not physically relevant, which would happen one in five times by chance. In the basis where the right-handed neutrino masses are real and positive this is equivalent to having a phase difference n = _2n/5 between flavon vevs in Eq. (10) according to the see-saw result in Eq. (12).

Similarly the flavons appearing in Eqs. (36) each have a discrete choice of phases. The charged lepton flavons ( may take any phases since such phases are unphysical. In fact the only physically significant flavon phases from the previous subsection are those of 0atm = (V3, and 0soi = (V4 whose phases are selected to be equal. As before, this would occur one in five times by chance.

We emphasise that, with the alignments including the phase n fixed, the neutrino mass matrix is completely determined by only two parameters, namely an overall mass scale ma, which may be taken to fix the atmospheric neutrino mass m3 = 0.048-0.051 eV, the ratio of input masses eV, which may be taken to fix the solar to atmospheric neutrino mass ratio m2/m3 = 0.17-0.18. In particular the entire PMNS mixing matrix and all the parameters therein are then predicted as a function of m2/m3 controlled by the only remaining parameter eV. In Table 2 we show the predictions for CSD4 as a function of eV and hence m2/m3.

We remark that an accuracy of one degree in the angles is all that can be expected due to purely theoretical corrections in a realistic model due to renormalisation group running [16] and canonical normalisation corrections [17]. In addition, there may be small contributions from a heavy third right-handed neutrino [18] which can affect the results.

As in the case of CSD2, the neutrino mass matrix implies the TM1 mixing form [20] where the first column of the PMNS matrix is proportional to (2, —1, 1)T. The reason is simply that (V1) a (2, —1, 1)T is an eigenvector of mV in Eq. (13) with a zero eigenvalue corresponding to the first neutrino mass m1 being zero. The reason for this is that mV in Eq. (13) is a sum of two terms, the first being proportional to AAT a ((V3)((V3)T and the second being proportional to BBT a ((V4)((V4)T. Since (V1) a (2, —1,1)T is orthogonal to both ((V3) and ((V4) it is then clearly annihilated by the neutrino mass matrix, i.e. it is an eigenvector with zero eigenvalue. Therefore we immediately expect mV in Eq. (13) to be

Table 2

The predictions for PMNS parameters and m2/m3 arising from CSD4 as a function of ev. Note that these predictions assume n = —2n/5. The predictions are obtained numerically using the Mixing Parameter Tools package based on [15]. The leading order analytic results are not reliable as discussed in Appendix B.

cy m2/m3 »12 »13 »23 S P

0.057 0.166 34.2° 9.0° 40.8° 107° —84°

0.058 0.170 34.2° 9.2° 40.9° 107° —83°

0.059 0.174 34.1° 9.4° 41.0° 106° —82°

0.060 0.177 34.1° 9.6° 41.1° 105° —80°

0.061 0.181 34.1° 9.7° 41.3° 104° — 79°

diagonalised by the TM1 mixing matrix [20] where the first column is proportional to (yVl) a (2, —1, 1)T. Therefore we already know that CSD4 must lead to TM1 mixing exactly to all orders according to this general argument.

Exact TM1 mixing angle and phase relations are obtained by equating moduli of PMNS elements to those of the first column of the TB mixing matrix (see also [20]):

C12C13 =

|c23 S12 + S13 S23Cl2e I = ^,

1 1 V6

IS23S12 - S13C23C12eiSI =

From Eq. (16) we see that TM1 mixing approximately preserves the successful TB mixing for the solar mixing angle »12 « 35° as the correction due to a non-zero but relatively small reactor angle is of second order. While general TM1 mixing involves an undetermined reactor angle »13, we emphasise that CSD4 fixes this reactor angle. For n = —2n/5 the reactor angle is in the correct range as shown in Table 2.

In an approximate linear form, the relations in Eqs. (16)-(18) imply the atmospheric sum rule relation a = r cos S [11], hence,

e23 « 45° + V2e13 cos s.

For n = —2n/5 the predictions shown in Table 2 for the small deviations of the atmospheric angle from maximality are well described by the sum rule in Eq. (19). In the present model this sum rule is satisfied by particular predicted values of angles and CP phase which only depend on the neutrino mass ratio m2/m3. Over the successful range of m2/m3 we predict CP violation with S & 106° and 023 & 41° which satisfy the sum rule. Note that according to this sum rule, non-maximal atmospheric mixing is linked to non-maximal CP violation.

3. Conclusions

There is long history of attempts to explain the neutrino mixing angles starting from the type I see-saw mechanism and using SD, first using CSD to account for TB mixing, then using CSD2 to obtain a small reactor angle before going to CSD3 where the correct reactor angle can be reproduced along with maximal atmospheric mixing. We have discussed a minimal predictive see-saw model based on CSD4 in which the right-handed neutrino mainly responsible for the atmospheric neutrino mass has couplings to (ve ,v^,vx) proportional to (0, 1,1) and the right-handed neutrino mainly responsible for the solar neutrino mass has couplings to (ve, v^,vx) proportional to (1, 4, 2), with a relative phase n = —2n/5. We have shown how these patterns of couplings and phase could arise from an A4 family symmetry model of leptons.

We remark that the type of model presented here is referred to as "indirect" according to the classification scheme of models

in [4], meaning that the family symmetry is completely broken by flavons and its only purpose is to generate the desired vacuum alignments. By contrast, the "direct" models where the symmetries of the neutrino and charged lepton mass matrices is identified as a subgroup of the family symmetry, requires rather large family symmetry groups in order to account for the reactor angle [21]. It is possible to have "semi-direct" models, either at leading order or emerging due to higher order corrections [4], but these are inherently less predictive. In the light of the observed reactor angle, "indirect models" therefore offer the prospect of full predic-tivity at the leading order from a small family symmetry group. Spontaneous CP violation seems to be an important ingredient in the "indirect" approach since a particular phase relation between flavons a crucial requirement.

The particular indirect model presented here, in which CSD4 emerges from an A4 family symmetry, offers a highly predictive framework involving only one free parameter which is used to fix the neutrino mass ratio m2/m3, together with an overall neutrino mass scale which is used to fix the atmospheric neutrino mass m3. Remarkably, the model then predicts the PMNS angles 012 & 34°, 023 & 41°, 013 & 9.5°, which exactly coincide with the current best fit values for a normal neutrino mass hierarchy, together with the distinctive prediction for the CP violating oscillation phase S & 106° . These predictions will surely be tested by current and planned high precision neutrino oscillation experiments.

Acknowledgements

S.F.K. would like to thank A. Merle for help with MPT, Christoph Luhn and Stefan Antusch for discussions and A. Kusenko and T. Yanagida and the IPMU for hospitality and support. S.F.K. also acknowledges partial support from the STFC Consolidated ST/J000396/1 and EU ITN grants UNILHC 237920 and INVISIBLES 289442.

Appendix A. Vacuum alignment

In this appendix we shall discuss how to achieve the following vacuum alignment,

(0atm)

(0sol)

which we refer to as CSD4.

The vacuum alignments associated with TB mixing have been very well studied. Here we shall focus on the family symmetry A4 as it is the smallest non-Abelian finite group with an irreducible triplet representation. The generators of the A4 group, can be written as S and T with S2 = T3 = (ST)3 = I. A4 has four irreducible representations, three singlets 1, 1' and 1" and one triplet. The products of singlets are:

1 ® 1 = 1, 1'® 1" = 1, 1'® 1' = 1'', 1''® 1'' = 1'. We work in the basis [19], 1

0 0 0 1 0

— 1 0 , T = 0 0 1

0 —1 1 0 0

In this basis one has the following Clebsch rules for the multiplication of two triplets,

(ab)1 = a\b\ + <3262 + a3b3,

(ab) 1' = a\b\ + wa2b2 + M2a3b3, (ab)= aibi + a)2a2b2 + wa3b3, (ab)31 = (a2b3, a3bi, ab),

(ab)32 = (a3b2, aib3, a2b1), (23)

where w3 = 1, a = (a1, a2, a3) and b = (b1, b2, b3).

Following the methods of [14] it is straightforward to obtain the vacuum alignments for charged lepton flavon alignments suitable for a diagonal charged lepton mass matrix. The charged lepton flavon alignments used to generate a diagonal charged lepton mass matrix are obtained from the renormalisable superpotential [14],

{Ve} = Vel 0 \0

{Vt} = vr[ 0 ).

{Vß} = vß ( 1 ) ,

Of more interest to us in this Letter are the new neutrino flavon alignments. The starting point for the discussion is the usual standard TB neutrino flavon alignments proportional to the respective columns of the TB mixing matrix,

{Vvi} = v v

{vV3 } = vV3 ( 1 ) .

2 -1 1

{VV2 }=v V2 ( 1

We will also employ the alternative TB alignments which are related by phase redefinitions,

<vvi> = v'Vl ( 1 ), {vv2} = vv2 I -1 ), {vv3 } = vv^ -1 ).

In the remainder of this subsection we shall show how to obtain the neutrino flavon alignments including the new alignment,

{vV4} = V V4 ( 4 ) ,

which corresponds to the CSD4 solar flavon alignment in Eq. (20). We shall identify 0atm = pV3, and 0soi = pV4. The renormalisable superpotential involving the driving fields necessary for aligning the neutrino-type flavons is given as

Wflvon'v = av2 (gl^V2 VV2 + g2 VV2 V ) + av2 (g'l Vv>2 Vv'2 + g2 pv2

+ Oev3 g3pePv3 + O v2v3 g4pv2 pv3 + O v1v2 g5pv1 pv2 + O V1V3 g6Pvi PV3 + Oev3 g3PePv'^ + O v<2v3 g4pv2 pv3

+ 0 v1 v2 g5 (Pv\ Pv!2 + 0 v' v3 g6(Pv\ (v3

+ 0 MV5 g7PixPv5 + 0 V' V5 g8pv' PV5 + 0IXV6g9PixPv6

+ 0 V5V6 gl0pv5 Pv6 + 0 V6V4 gllpv6 Pv4

+ 0 V'V4 gl2pvi PV4 , (29)

where AV2 is a triplet driving field and 0jj are singlet driving fields whose F-terms lead to orthogonality relations between the accompanying flavon fields. Here gi are dimensionless coupling constants. The first line of Eq. (29) produces the vacuum alignment

(pV2) a (1, 1, —1)T of Eq. (26) and (pV,) a (1, —1, 1)T of Eq. (27) as can be seen from the F -term conditions1

W AtoM — Ae pe pe + A^pp + Axpxpx + 0 e^Pe Pi

+ 0ex(e PT + 0 ITPIPT. (24) 2g1

The triplet driving fields Ae,i,T give rise to flavon alignments (pe,i,T) with two zero components, and the singlet driving fields 0 ij require orthogonality among the three flavon vevs so that we arrive at the vacuum structure [14],

{Vv2 }2{Vv2 }A /{Vv2 }A \ 0 \

{VV2 }3{VV2 }1 I + g2 {^V2 } I {VV2 }2 I = I 0 I {VV2} 1 {Vv2 h/ \{VV2 }3/

plus similar conditions involving the primed flavons. The first two terms in the second line of Eq. (29) give rise to orthogonality conditions which uniquely fix the alignment (pV3) a (0,1,1)T of Eq. (26), 3

{Ve}' '{Vv3} = {Vv2}' • {Vv3} = 0

{Vv3 }a| 1 ). (31) 1

The last two terms in the second line of Eq. (29) give rise to orthogonality conditions which uniquely fix the alignment (PV1) a (2, -1, 1)T of Eq. (26),

(pvi)T -(pV2) = (pvi)T • (pv3) = 0 ^ (pVi. (32)

Similarly the terms in the third line of Eq. (29) give rise to orthogonality conditions which fix the alternative TB alignments in Eq. (27) corresponding to a different choice of phases.

The terms in the fourth line of Eq. (29) give rise to orthogonality conditions which fix the alignments of the auxiliary flavon fields pV5 and pV6,

(P,l)T • (pV5 ) = (pv^)T• (pV5 ) = 0 ^ (pV5 >a( o) , (33)

{Vß}T • {vv6 } = {vv5 }'• {vv6 } = 0 ^ {vv6}«( 0 I. (34)

2 0 -1

The neutrino-type flavon of interest labelled as pV4 gets aligned by the remaining terms in the fifth line of Eq. (29), leading to the desired alignment in Eq. (28),

{vV1 }T • {vV4 } = {vV6 }T • {vV4} = 0

{vV4} « ( 4 2

So far we have only shown how to align the flavon vevs and have not enforced them to be non-zero. In order to do this we shall introduce the additional non-renormalisable superpotential terms which include,

1 We remark that the general alignment derived from these F -term conditions is (pV2) a (±1, ±1, ±1)T. One can, however, show that all of them are equivalent up to phase redefinitions. Note that (1,1, —1) is related to permutations of the minus sign as well as to (—1, —1, —1) by A4 transformations. The other four choices can be obtained from these by simply multiplying an overall phase (which would also change the sign of the £V2 vev).

AWfl4 , -

£ A (im • Vi)pi — m3) + p(A — M2), (36)

aw^avon,v „ £ A (^ . ^ )pVi — M3) + p( A — MA, (37)

¡=1 ^ '

where, as in [11], the driving singlet fields P denote linear combinations of identical singlets and we have introduced explicit masses M to drive the non-zero vevs, as well as the messenger scales denoted as a. We have also introduced A4 singlets pi and pVi whose vevs are driven by the F-terms of the singlets P in the second terms in Eqs. (36) and (37). These singlet vevs enter the first terms in Eqs. (36) and (37) which drive the vevs of the triplet flavons.

The flavons and driving fields introduced in this appendix transform under Zl3 x ZVi symmetries whose purpose is to allow only the terms in Eqs. (24), (29) and (36) and forbid all other terms. The superfields yVi transform under ZVi as p3 (where p = ei2n/5) and are singlets under all other discrete symmetries. The super-fields pVi transform under ZVi as p4 and are singlets under all other discrete symmetries. Any superfield with a single subscript l transforms under Z3 as w (where w = ei2n/3) and is a singlet under all other discrete symmetries. The orthogonality driving su-perfields Oij with two subscripts transform under Z3 x ZVi in such a way as to allow the terms in Eqs. (24), (29). For example the OiVi driving fields transform under Z3 x Z^ as (w2,p2). In addition driving superfields are assigned a charge of two while flavon superfields have zero charge under a U(1)^ symmetry.

Appendix B. Leading order analytic results

For the case of atmospheric alignments of the form (0, z1,1) and solar alignments of the form (1, z2, z3), the leading order analytic results in [9,13] give,

tan023 &|Z1|, (38)

cot012m&c23|z2|cos(n2—2)—s23|z3|co<n3—2) (39) 013 & — s22C23||z3| + |Z2| tan023ei(n2—n3)|, (40)

where n2 = arg(z2/z1) and n3 = arg(z3), while /) is a Majorana phase. With z1 = 1 and arbitrarily assuming /) = 0 and real phases ±1 associated with n2 and n3 one finds the relations

tan »23 « 1, 1

cot»12 « — Z3|,

»13 :

m2 1 m3 3V2

|Z3 + Z2|.

We should say immediately that the assumption P = 0 is not justified so these results can be at best suggestive. With this caveat, we note that approximately tri-maximal solar mixing cot »12 « V2 results from the general condition |z2 — z31 = 2 which is satisfied by all the proposed forms of CSD.2 Moreover, CSD with z2 = 1, z3 = —1 leads to »13 « 0, CSD2 with z2 = 2, z3 = 0 leads to »13 « mi'CSD3 with z2 = 3, z3 = 1 leads to »13 « mf ^ CSD4 with z2 = 4, z3 = 2 leads to »13 « ^ V2.

2 I would like to thank Stefan Antusch (private communications) for emphasising the condition |z2 — z3| = 2.

Although the above leading order results provide a qualitative understanding of the results obtained for CSD, CSD2, CSD3 and CSD4, they have large corrections of order m2/m3, much larger than the errors in the global fits and so do not give reliable predictions. In addition there is a strong dependence on the phase difference between the solar and atmospheric alignments which these results ignore. Moreover, the phase n does not appear in the leading order formula for 013, but in practise the reactor angle depends strongly on n, as discussed in [13]. On the other hand, while the phases do appear in the solar angle formula, we have arbitrarily and incorrectly assumed /) = 0.

In summary, the leading order results, while providing a qualitative understanding, are quantitatively unreliable and cannot be used to estimate the mixing angles to the required accuracy. The general analysis as performed in [13] did not rely on the leading order results in any way and was not inspired by them. Starting from an exact master formula, the analysis [13] determined from first principles not only the moduli |zi| but also the phases ni which are required for a proper definition of any new type of CSD. For example CSD4 with solar alignment (1, 4, 2) is only properly defined once the phases n2 = n3 = —2n/5 are specified. In retrospect, it is rather fortuitous that the condition |z2 — z3| = 2 is satisfied for all the proposed forms of CSD. Indeed other more complicated but equally successful examples were found that violated the condition |z2 — z3|=2 and these were also tabulated in [13].

References

[1] F.P. An, et al., DAYA-BAY Collaboration, Phys. Rev. Lett. 108 (2012) 171803, arXiv:1203.1669;

Y. Wang, Talk at What is v? INVISIBLES'12, Galileo Galilei Institute for Theoretical Physics, Florence, Italy, 2012; available at http://indico.cern.ch/ conferenceTimeTable.py?confId=195985.

[2] J.K. Ahn, et al., RENO Collaboration, Phys. Rev. Lett. 108 (2012) 191802, arXiv:1204.0626.

[3] P.F. Harrison, D.H. Perkins, W.G. Scott, Phys. Lett. B 530 ( 2002) 167, arXiv: hep-ph/0202074.

[4] S.F. King, C. Luhn, Rep. Prog. Phys. 76 (2013) 056201, arXiv:1301.1340 [hep-ph].

[5] S.F. King, Phys. Lett. B 659 (2008) 244, arXiv:0710.0530.

[6] S. Pakvasa, W. Rodejohann, T.J. Weiler, Phys. Rev. Lett. 100 (2008) 111801, arXiv:0711.0052.

[7] S.F. King, Phys. Lett. B 718 (2012) 136, arXiv:1205.0506 [hep-ph].

[8] P. Minkowski, Phys. Lett. B 67 (1977) 421;

T. Yanagida, in: O. Sawada, A. Sugamoto (Eds.), Proceedings of the Workshop on Unified Theory and Baryon Number of the Universe, KEK, 1979, p. 95; P. Ramond, Invited talk given at Conference: C79-02-25, CALT-68-709, February 1979, pp. 265-280, arXiv:hep-ph/9809459;

M. Gell-Mann, P. Ramond, R. Slansky, in: P. van Niewwenhuizen, D. Freedman (Eds.), Supergravity, North-Holland, Amsterdam, 1979, Conf. Proc. C790927, p. 315, PRINT-80-0576.

[9] S.F. King, Phys. Lett. B 439 (1998) 350, arXiv:hep-ph/9806440; S.F. King, Nucl. Phys. B 562 (1999) 57, arXiv:hep-ph/9904210; S.F. King, Nucl. Phys. B 576 (2000) 85, arXiv:hep-ph/9912492; S.F. King, JHEP 0209 (2002) 011, arXiv:hep-ph/0204360;

T. Blazek, S.F. King, Nucl. Phys. B 662 (2003) 359, arXiv:hep-ph/0211368; S. Antusch, S. Boudjemaa, S.F. King, JHEP 1009 (2010) 096, arXiv:1003.5498.

[10] S.F. King, JHEP 0508 (2005) 105, arXiv:hep-ph/0506297.

[11] S. Antusch, S.F. King, C. Luhn, M. Spinrath, Nucl. Phys. B 856 (2012) 328, arXiv: 1108.4278 [hep-ph].

[12] S. Antusch, S.F. King, M. Spinrath, arXiv:1301.6764 [hep-ph].

[13] S.F. King, arXiv:1304.6264 [hep-ph].

[14] S.F. King, C. Luhn, JHEP 1203 (2012) 036, arXiv:1112.1959 [hep-ph].

[15] S. Antusch, J. Kersten, M. Lindner, M. Ratz, M.A. Schmidt, JHEP 0503 (2005) 024, arXiv:hep-ph/0501272.

[16] S. Boudjemaa, S.F. King, Phys. Rev. D 79 (2009) 033001, arXiv:0808.2782 [hep-ph].

[17] S. Antusch, S.F. King, M. Malinsky, Phys. Lett. B 671 (2009) 263, arXiv:0711.4727 [hep-ph];

S. Antusch, S.F. King, M. Malinsky, JHEP 0805 (2008) 066, arXiv:0712.3759 [hep-ph];

S. Antusch, S.F. King, M. Malinsky, Nucl. Phys. B 820 (2009) 32, arXiv:0810.3863 [hep-ph].

[18] S. Antusch, S. Boudjemaa, S.F. King, JHEP 1009 (2010) 096, arXiv:1003.5498 [hep-ph].

[19] E. Ma, G. Rajasekaran, Phys. Rev. D 64 (2001) 113012, arXiv:hep-ph/0106291; G. Altarelli, F. Feruglio, Nucl. Phys. B 720 (2005) 64, arXiv:hep-ph/0504165;

C. Luhn, S. Nasri, P. Ramond, J. Math. Phys. 48 (2007 ) 073501, arXiv:hep-th/ 0701188.

[20] C.S. Lam, Phys. Rev. D 74 (2006) 113004, arXiv:hep-ph/0611017;

C.H. Albright, W. Rodejohann, Eur. Phys. J. C 62 (2009) 599, arXiv:0812. 0436;

C.H. Albright, A. Dueck, W. Rodejohann, Eur. Phys. J. C 70 (2010) 1099, arXiv: 1004.2798;

W. Rodejohann, H. Zhang, Phys. Rev. D 86 (2012) 093008, arXiv:1207.1225 [hep-ph].

[21] S.F. King, T. Neder, A.J. Stuart, arXiv:1305.3200 [hep-ph].