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Physics Letters B 578 (2004) 156-164

www. elsevier. com/locate/physletb

Majorana phase in minimal S3 invariant extension of

the standard model

Jisuke Kubo1

Max-Planck-Institut für Physik, Werner-Heisenberg-Institut, D-80805 Munich, Germany Received 14 September 2003; received in revised form 14 October 2003; accepted 14 October 2003

Editor: P.V. Landshoff

Abstract

The leptonic sector in a recently proposed minimal extension of the standard model, in which the permutation symmetry S3 is assumed to be an exact flavor symmetry at the weak scale, is revisited. We find that S3 with an additional Zn symmetry allows one CP violating phase in the neutrino mixing. The leptonic sector contains six real parameters besides the phase to describe charged lepton and neutrino masses and the neutrino mixing. The model predicts: an inverted spectrum of neutrino mass, the absence of CP violating Dirac phase, tan623 = 1 + O(m2/m2ß) and sin613 = 0.0034 + O(mem^/m2). Neutrino mass as well as the effective Majorana mass {mee) in the neutrinoless double-^ decay can be expressed in a closed form as a function of ^v,Am2^1 ,Am^3 and tan612. The model also predicts (mee) ^ (0.036-0.066) eV. © 2003 Elsevier B.V. All rights reserved.

PACS: 11.30.Hv; 12.15.Ff; 14.60.Pq

The Yukawa sector of the standard model (SM), which is responsible for the generation of the mass of leptons and quarks, and their mixing, has too many redundant parameters. This not only weakens the predictivity of the SM, but also makes ambiguous how to go beyond the SM. An exact flavor symmetry could reduce this redundancy, thereby giving useful hints about how to unify the flavor structure of the SM.

Recently, a minimal S3 invariant extension of the SM was suggested in [1], while assuming that the Higgs, quark and lepton including the right-handed neutrino fields belong to the three-dimensional reducible representation of the permutation group S3.2 This smallest non-Abelian symmetry based on S3 is only spontaneously broken, because the electroweak gauge symmetry SU(2)L x U(1)Y is spontaneously broken. It was found in [1] that this flavor symmetry is consistent with experiments, and that in the leptonic sector an additional discrete symmetry Z2 can be introduced. It was argued there that due the additional discrete Z2 symmetry the neutrino mixing matrix VMns

E-mail address: kubo@mppmu.mpg.de (J. Kubo).

1 Permanent address: Institute for Theoretical Physics, Kanazawa University.

2 A partial list for permutation symmetries is [2-17]. See for instance [8] for a review. The basic idea of [1] is similar to that of [2,5,6].

0370-2693/$ - see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2003.10.048

cannot contain any CP violating phase.3 We now believe this is incorrect, and we would like to re-investigate the leptonic sector of the model in this Letter.

We will find that it is possible to introduce one independent Majorana CP violating phase [19] in the neutrino mixing even with an additional ZN symmetry in the leptonic sector. The permutation symmetry S3 with ZN allows three real mass parameters for the charged lepton mass matrix, and three real parameters and one phase for the neutrino mass matrix. The model predicts:4 an inverted spectrum of neutrino mass, the absence of CP violating Dirac phase 5, tan023 = 1 + 0(m1/m2 ) and sin0i3 = me/mfl*j2 + Neutrino mass as well as the

effective Majorana mass {mee) in the neutrinoless double-^ decay can be expressed in a closed form as a function of , Am^1, AmS|3 and tan012. We find that the minimum of mv2 as well as (mee) occurs at = 0, which is

approximately ^Am^/sin20i2.

Before we will come to our main purpose of the Letter, let us briefly summarize the basic ingredient of the S3 invariant SMof [1]. The quark, lepton and Higgs fields are denoted by QT = (uL,dL), uR, dR, LT = (vL,eL), eR, vR, H. Each of them forms a reducible representation 1 + 2 of S3. The doublets carry capital indices I, J which run from 1 to 2, and the singlets are denoted by Q3, u3R, u3R, L3, e3R, v3R, H3. The most general renormalizable Yukawa interactions are given by

Ly = LyD + Lyv + LyE + Lyv ,

Lyd = -Qj2 Y%HidR + h.c., Lyv = -Q[i02)Y_] Y\Hur + h.c., i = 1 i=1

Lye = -Lj2 Y%HieR + h.c., Lyv =-L(ia2^ \vHHivR + h.c.,

and the Yukawa coupling matrices are given by [1]

jk lH3

0 k Y 2 k Y 5

k Y 2 0 0

k Y 4 0 0

k Y1 0 0

0 k Y1 0

0 0 yk Y3

fk lH2

k y2 0 0

0 Yk - y 2 Yk y5

0 k y4 0

Further, the Majorana mass terms for the right-handed neutrinos is given by

Lm = -M^IRCVR - M3v^RCv3R,

where C is the charge conjugation matrix.5

Pakvasa and Sugawara [2] analyzed the Higgs potential. The potential they analyzed has not only an Abelian discrete symmetry (which we will use for selection rules of the Yukawa couplings), but also a permutation symmetry S2: H1 ^ H2, which is not a subgroup of the flavor group S3 of the model. We assume throughout this Letter that the vacuum can respect this accidental symmetry of the Higgs potential, and

(H1) = (H2)

3 See for instance [18] for recent reviews on CP violation in the leptonic sector.

4 Similar but different predictions are obtained from different types of discrete symmetry [10-15]. See also [16,17].

5 Supersymmetrization ofthe present model has been proposed in [9].

is satisfied. ((Hi) = — (H2) would yield the same physics.) Then the Yukawa interactions (1) yield the mass matrices of the general form

«2 m5 m1 — «2 «5

«4 «3,

The Majorana mass for vL can be obtained from the see-saw mechanism [20], and the corresponding mass matrix is given by mv = mvb m—^m^)7, where m = diag(M1,M1,M3). The mass matrices are diagonalized by the unitary matrices U' s

d(u,e)LMd(u,e)Ud(u,e)R,

u! mvuv.

The diagonal masses can be complex, and so the physical masses are their absolute values, which we denote by rnV1, rnV2, «V3, «e, «T, etc.

It would be certainly desirable to classify, in a similar way as in [21,22], all possible mass matrices that are consistent with an additional discrete Abelian symmetry and with experimental data. We, however, leave this program to feature work. Here we simply adopt the result of [1] that

Yf = Y^ = Yf = Y5v = 0,

and consequently

«1 = «3 = «1 = «5 = o

follows from a Z2 symmetry. We emphasize that there are a number of different charge assignments of ZN that can yield (7):6 provided that the charge of H3, Q(H3), is different from Q(H1,2), only the conditions

Q(L3) = Q(Lh2) = Q(e3R) + Q(H1,T) = Q(e1,2R) + Q(Hh2) = Q( V>1,2r) — Q(Hh2) = Q( V3R) — Q(H3)

modulo N should be satisfied to forbid Y1e,Y3e,Y1v and Y5V. Unfortunately, none of the abelian discrete symmetries above is a symmetry in the quark sector. Note that if ZN is chiral, it is broken by QCD anyway (S3 is not broken by QCD, because it is not a chiral symmetry). The symmetry violating effect of the quark sector appears first at the two-loop level in the leptonic sector, so that the violation of ZN in the leptonic sector may be assumed to be negligibly small. Therefore, we throughout neglect that violating effect.7 To proceed with our discussion, we calculate the unitary matrix UeL from

u]LMe MeUeL = diag( m

Me me =

2«e)2+(«5) «5)2 2rn2rn4

«5)2 2«2)2 + «5)2 0

2rn2rn4

2«4)2

6 We do not consider U(1) to avoid the appearance of a (nearly) massless particle.

7 That is, we assume that the relation (7) is satisfied at the weak scale. If one assumes that it is satisfied at some higher scale, one should take into account the renormalization group running of the parameters [23]. See also [24] for further references. We however expect that the corrections will be small in the present model, in contrast to models, in which a large neutrino mixing is not related to a symmetry ofthe theory.

and all the mass parameters appearing in (11) are real. We find that UeL can be approximately written as [9]

2(1J-il _ +-zi^ -J-

2V1 xl) 4+2x2/ ./?

where x = me5/m2 ~ and y = meA/me2 ~ \flmelmtl.

The Majorana masses of the right-handed neutrinos, M1 and M3 in (4)which may be complex, can be absorbed by a redefinition of mv2, mv4 and m^, and we may therefore assume that M1 and M3 are real. After rescaling of mv2, mv4 and mv3 as

(m2) ^ p2 = k>m

we obtain

mv = Mvd m-1 (Mvd)t =

(m4) ^ p4 = (m4)/Mj/2, (m£) ^ p3V = (m£)/M.

'2(p2v)2 0

,2p2vp4v

2(p2v)2 0

2p2vp4v 0

2(p4v)2 ■

All the phases in (14), except for one, can be absorbed.Without loss of generality, we may assume that pV is complex. We find that mv can be diagonalized as

ut MvUv =

' mV1 ei01-i0v 0 0

C12ei*v

J02 +i0v 0

mV3 sin0V = mV2 sin02 = mV1 sin01, and c12 = cos 012 and s12 = sin 012. The mixing angle is given by

tan2 012 =

(m2 - m2 sin2 0v)1/2 -'

cos 0V

(ml - m23 sin2 0V)1/2 + m

from which we find

(1 + 2i?2 + £ -

cos 0V

Am23 4if2(1 + if2)(1 + if2 - rif2) cos2 0V

- tan2 0V

sin22012cos2 0V

- tan2 0v for |r |<1,

where i12 = tan012,r = Am21/Am23. As in [1], we find that only an inverted mass spectrum

mv3 < mv1 ,mv2

is consistent with the experimental constraint | Am211 < | Am231 in the present model. To see this, we first derive

mV1 cos01 - mV3 cos0V = -2pVp\A1, mv2 cos02 - mv3 cos0v = 2pVp\A2,

(22) (23)

Fig. 1. «V2 versus sin012 for = 6 9 x 10 5 eV2, A«13 = 2.5 x 10 3 eV2 and sin0V = 0 (solid), 0.6 (dotted) and 0.96 (dot-dashed).

Ai = sin2^i2 + cos2 6>12/tan2^12, A2 = sin26>i2 - sin2 0U/tan26>i2.

Then we use the fact that if A1 is positive (negative), then A2 is always positive (negative). Suppose that 2pVp\ A2 is positive, which implies that «V2 cos 02 > «V3 cos 0 v and «V1 cos 01 < «V3 cos 0 v. In this case, Eq. (17) can be satisfied, only if «V2 > «V3 or «V1 > «V3. Similarly, if — 2pV p\ A1 is positive, then «V2 > «V3 or «V1 > «V3 has to be satisfied. Therefore, «V3 cannot be the largest among «Vi's.8

In Fig. 1 we plot «V2 versus sin012 for A«11 = 6.9 x 10—5 eV2, A«13 = 2.5 x 10—3 eV2 (best-fit values reported in [25-27]) and sin 0V = 0 (solid), 0.6 (dotted) and 0.96 (dashed). The sin0V dependence of «V2 is shown in Fig. 2 for tan012 = 0.68 and the same values of A«11 and A«13 as in Fig. 1. As we see from (20) and also from Fig. 2, «V2 assumes at sin 0V = 0 its minimal value

V2 ,min ■

~J Am23/sin2612 = (0.036-0.066) eV,

where we have used Am223 = (1.3-3.0) x 10-3 eV2 and sin2612 = 0.83-1.0 [25-28].

Now the product u]lU V defines a neutrino mixing matrix, which we bring by an appropriate phase transformation to a popular form

Vmns =

C12 C13 £12 C13 £13e \ 1 0 0

-£12C23 - C12£23£13e*5 C12C23 - £12^23£13e*5 £23 C13 1 ( 0 eia 0

£12£23 - C12C23£13e*5 -C12£23 - £12C23£13e* C23C13 / 0 0 e*ß

8 Of course, mV1 > mv3 > mV2 or mV2 > mv3 > mV1 is mathematically allowed, but is excluded by experiments.

Fig. 2. mV2 as a function of sin 0V for tan O12 = 0.68,Am22l = 6.9 x 10 5 eV2 and Am^3 = 2.5 x 10 3 eV2.

We find:

Sl3 = -L=^L + 0 (memß/m2T) ~ 0.0034, t23 = — = 1 - ^ ( — ) K/^?)

1 / me

2 V m.

8 = 0,

mv, sin 0V , / ~ ~ / ö ö \

sin 2a = sin(0i 02) =-U/mvo sin 0V + J m% m%. sur 0V )

m V1 m V2 vv 2 3 V 1 3 '

~ 2 sin0v(mV3/mV2)^ 1 (mV3/mV2)2 sin2 0V,

sin 0y , / ~ ~ « x

sin2)0 = sm(02 0v) =--1 rn; m^ smz0v mV3 cos0V),

m 2 2 3

where 01,02 and 0 v are defined in (17). As announced, the Dirac phase £ is absent in the present model. Therefore, no CP violating process should be observed in neutrino oscillation experiments. Since sin22013 ~ 4.6 x 10-5 in addition to 8 = 0, future oscillation experiments such as J-Park experiment [28] can easily exclude the model. In Fig. 3 we plot sin 2a (solid) and — sin 2^ (dotted) as a function of sin 0v for tan 012 = 0.68, Am21 = 6.9 x 10-5 eV2 and Am23 = 2.5 x 10—3 eV2. As we can see, sin2a reaches its maximal value 1 at sin0V ~ 0.936. Similarly, the maximal value of — sin2^, which is about 0.46, occurs at sin0V ~ 0.85. We then consider the effective Majorana mass

(mee ) =

mviyei

I m V1 c22 + m V2 s^exp i 2a |

cos 0 v

[sin-2 2^12 - sin2 0v ]1/2

sin2 2012

(cos 2a — 1)

Q l_l_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_I_l_

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

sin d)

Fig. 3. sin2a (solid) and — sin2p (dotted) versus sinfor tan612 = 0.68, Am2i = 6.9 x 10-5 eV2 and Am23 = 2.5 x 10-3 eV2.

which can be measured in neutrinoless double p decay experiments. (a is given in (29).) In Fig. 4 we plot {mee) as a function of sin0v for tan612 = 0.68,Am21 = 6.9 x 10—5 eV2 and Am23 = 2.5 x 10—3 eV2. As we can see from Fig. 4, the effective Majorana mass stays at about its minimal value {mee)min for a wide range of sin0v. Since {mee)min is approximately equal to mv2,min (which is given in (25)), it is consistent with recent experiments [29,30] and is within an accessible range of future experiments [31]. An experimental verification of (20), (21) and (27)-(33) would strongly indicate the existence of the smallest non-Abelian symmetry based on the permutation group S3 along with an Abelian discrete symmetry ZN at the electroweak scale, where ZN is only an approximate symmetry of the whole theory, but the effect of its violation is of two-loop order in the leptonic sector.

S3 is obviously a possible answer to the question why there exist three generations of leptons and quarks. S3, of course, cannot explain the hierarchy of the fermion mass spectrum, but S3 with ZN in the leptonic sector can relate the mass spectrum and mixing in this sector, making testable predictions, which have been re-investigated in the present Letter. Therefore, S3 solves partially the flavor problem of the SM. Since there are three SU(2)L doublet Higgs fields in the model, there exit FCNC processes at the tree level. In [1] the magnitude of various tree level FCNC amplitudes have been estimated, and it has been found that they are sufficiently suppressed. The suppression follows from the smallness of the corresponding Yukawa couplings, where S3 plays an important role for that smallness. However, we find that AmK, the difference of the mass of KL and KS, exceeds the experimental value, unless the mixing of the Higgs fields is fine tuned. This problem is currently under investigation, and we will report the result elsewhere.

It is straightforward to keep the discrete flavor symmetries, S3 in the hadronic sector and S3 x ZN in the leptonic sector, in a supersymmetric extension of the standard model [9]. The supersymmetric flavor problem has been investigated there, and it has been explicitly found that thanks to the flavor symmetries the dangerous FCNC and CP violating processes, that originate from soft supersymmetry breaking terms, are sufficiently suppressed, in a similar manner as it was found in [32].

A 0.08

Q Q5_i_i_I_i_i_I_i_i_I_i_i_I_i_i_I_i_i_I_i_i_I_i_i_I_i_i_I_i_i_

' 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

sin <b

Fig. 4. {mee) as a function of sin for tan 612 = 0.68, Am21 = 6.9 x 10—5 eV2 and Am23 = 2.5 x 10—3 eV2. Acknowledgements

This work is supported by the Grants-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (No. 13135210).

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