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Physics Letters B 579 (2004) 113-122

www. elsevier. com/locate/physletb

A non-standard CP transformation leading to maximal atmospheric neutrino mixing

Walter Grimusa, Luís Lavourab

a Institut für Theoretische Physik, Universität Wien, Boltzmanngasse 5, A-1090 Wien, Austria b Universidade Técnica de Lisboa, Centro de Física das Interacgöes Fundamentais, Instituto Superior Técnico, P-1049-001 Lisboa, Portugal

Received 8 September 2003; received in revised form 10 October 2003; accepted 23 October 2003

Editor: G.F. Giudice

Abstract

We discuss a neutrino mass matrix Mv originally found by Babu, Ma, and Valle (BMV) and show that this mass matrix can be characterized by a simple algebraic relation. From this relation it follows that atmospheric neutrino mixing is exactly maximal while at the same time an arbitrary mixing angle 613 of the lepton mixing matrix U is allowed and—in the usual phase convention—CP violation in mixing is maximal; moreover, neither the neutrino mass spectrum nor the solar mixing angle are restricted. We put forward a seesaw extension of the Standard Model, with three right-handed neutrinos and three Higgs doublets, where the family lepton numbers are softly broken by the Majorana mass terms of the right-handed neutrino singlets and the BMV mass matrix results from a non-standard CP symmetry. © 2003 Elsevier B.V. All rights reserved.

1. Introduction

The atmospheric neutrino problem, with mixing angle 023, requires sin2 2023 > 0.92 at 90% CL, with a best fit value sin2 2023 = 1, i.e., maximal mixing [1]. There are many models and textures in the literature which attempt to explain large—not necessarily maximal—atmospheric neutrino mixing—for reviews see Ref. [2]. But, the closer the experimental lower bound on sin2 2023 comes to 1, the more urgent it becomes to find a rationale for maximal atmospheric mixing. Unfortunately this is not an easy task. Maximal mixing means | U^31 = | UT31, where U is the lepton mixing matrix, and this in general requires a \x-x interchange symmetry, which on the other hand must be broken since m^ = mT. For a recent discussion of this point see Ref. [3].

Two models for maximal atmospheric mixing have been suggested by us, one of them [4] based on lepton-number symmetries softly broken at the seesaw scale, the other one [5] based on a discrete symmetry spontaneously broken at the same scale. Both models yield an effective mass matrix for the light left-handed neutrinos at the

E-mail addresses: walter.grimus@univie.ac.at (W. Grimus), balio@cfif.ist.utl.pt (L. Lavoura).

0370-2693/$ - see front matter © 2003 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2003.10.075

seesaw scale

Mv =( y z w], (1)

where x, y, z, and w are in general complex. The matrix

H = M*MV (2)

then has an eigenvector (0,1, -1)r, and therefore the models predict Ue3 = 0 besides |Ux3| = |Ur3|; they will have to be discarded if | Ue31 is experimentally found to be non-zero.

A different approach has been suggested by Babu, Ma, and Valle (BMV) [6]. Starting from a degenerate neutrino mass matrix at the seesaw scale, and using the renormalization group in the context of softly broken supersymmetry with a general slepton mass matrix, they have obtained at the weak scale

(a r r * \

r s b), (3)

r* b s */

where r and s are in general complex while a and b remain real. BMV have found that, under some approximations, this Mv yields maximal atmospheric mixing and, furthermore, an imaginary Ue3 ("maximal CP violation") in the standard phase convention (to be specified shortly).

It is important to note that the mass matrix in Eq. (3) is not a generalization of the one in Eq. (1). If r and s in Eq. (3) are real, then that mass matrix coincides with the one in Eq. (1) with x, y, z, and w real. On the other hand, the mass matrix of Eq. (3) in general yields a non-zero Ue3 as soon as r and s are complex, while (1) always yields Ue3 = 0, even when x, y, z, and w are complex.

In Ref. [7], Harrison and Scott (HS) suggested that the lepton mixing matrix may satisfy

|Uw | = \ UXj | for j = 1, 2, 3, (4)

which is still a particular case of maximal atmospheric mixing, but has the advantage of being more general than the extra condition Ue3 = 0. HS showed that, if U satisfies Eq. (4) and the neutrinos are Dirac particles, then U may be parametrized as

/ u1 U2 U3 \

U =1 W1 W2 W3 I , (5)

\w* w* w*7

where the uj (j = 1, 2, 3) are real and non-negative, while the wj are complex and satisfy the orthogonality conditions

2Re(w;w*) = Sjk - ujuk. (6)

HS also introduced the concept of "¡-t reflection," which they defined as "the combined operation of ¡-t flavor exchange [■ ■ ] and CP transformation on the leptonic sector" and which is embodied in the mixing matrix of Eq. (5).

It is the purpose of this Letter to, firstly, prove that the mass matrix of BMV always yields Eq. (4) and, as a consequence, exact maximal atmospheric neutrino mixing and maximal CP violation. We shall also show that the BMV mass matrix leads to a mixing matrix of the form in Eq. (5), even while the neutrinos are Majorana particles. Secondly, we shall put forward a model, based on softly broken lepton numbers and on the non-standard CP symmetry called "¡-t reflection" by HS, which obtains the BMV mass matrix at the seesaw scale—without the need for the renormalization group, for supersymmetry, or for an extended fermion spectrum like in the original BMV model. In this way we conclude that maximal atmospheric mixing is compatible with a non-zero Ue3 and can be obtained in an extension of the Standard Model.

2. The mass matrix

Let Mv be a symmetric complex 3 x 3 matrix, the Majorana mass matrix of the light neutrinos, defined by

Anass = \vTLC~lMvVL + H.C. (7)

(C is the Dirac-Pauli charge conjugation matrix), in the basis where the charged-lepton mass matrix is diagonal. The lepton mixing matrix U is the diagonalizing matrix of Mv, defined by

UT MvU = rn = diag(m1,m2,m3), (8)

where the masses mj are real and non-negative.

Lemma 1. Suppose U and U' satisfy Eq. (8) and the masses are non-degenerate. Then there is a diagonal unitary matrix X such that U' = UX. Furthermore, Xjj is an arbitrary phase factor if mj = 0, while Xjj =±1 for mj = 0.

Proof. Since both U and U' fulfill Eq. (8), Mv = U*mUt = U'*mU't, or

W*m = mW with W = U'tU. (9)

This equation, together with the non-degeneracy of the masses, forces W to be diagonal, i.e., W* = X. It is moreover clear that Wjj is real when mj is non-zero. □

The matrix Mv of Eq. (3) is characterized by

(1 0 0 \

SMvS = mv with S = 0 0 1 . (10)

\0 1 0/

Let us write U = (c1,c2,c3) with column vectors cj. Eq. (8) means that

MvCj = m jc*. (11)

Starting from this equation and using Eq. (10) we see that

Mv (Sc*) = mj (Sc* )*. (12)

We thus have a second diagonalizing matrix U' = SU*. Using the lemma above we find that, if the masses are non-degenerate,

SU* = UX. (13)

Consequently, Eq. (4) holds.

An alternative proof of Eq. (4) starts from the observation that the matrix H corresponding to the Mv of Eq. (3) has

Hxx = Htt and Hex = H**t. (14)

As a consequence,

(h")a = (H")tt (15)

for any positive integer n. Using H = Urn U t , it follows fromEq. (15) for two distinct values of n that either there are degenerate neutrinos or Eq. (4) holds.

A popular representation of U [8] is given by

U = diag(eia1 ,eia2 ,eia3 )U23UnU12diag(1,eift ,eip2), (16)

\1 0 0 x

U23 =1 0 C23 523 I , (17)

\ 0 -523 C23I

\ C13 0 513e-i 5 \

U13 = 0 1 0 , (18)

\-513e 5 0 C13 / \ C12 512 0 \

U12 = -512 C12 0 . (19)

\ 0 01/

The phases aj (j = 1,2, 3) are unphysical (unobservable); 5 is the Dirac phase and p1,2 are the Majorana phases. Computing the product of matrices one gets

\ C12C13 512C13 513e-i5 \

U23U13U12 = I -512C23 - C12523513ei5 C12C23 - 512523513ei5 523C13 |. (20)

N 512523 - C12C23513ei5 -C12523 - 512C23513ei5 C23C13 /

Eq. (4) applies to this product. From that equation with j = 3 one obtains 1

C2i= S2i = —p- (21)

Now we inspect Eq. (4) with j = 1,2. We know experimentally that c12512 = 0, since solar neutrinos oscillate [9]. It follows that

513 cos 5 = 0, (22)

i.e., either Ue3 = 0 or CP violation is maximal. Conversely, if we require maximal CP violation (e's = ±i) and maximal atmospheric neutrino mixing (|Um3| = |UT3|) with the parameterization of Eq. (20), it is easy to see that Eq. (4) follows [7].

We stress that the mass matrix of Eq. (3) restricts neither the neutrino mass spectrum nor the solar neutrino mixing angle 012. In the general case cos 5 = 0 also 613 remains free. Note that, with Eq. (21), the parameter measured in atmospheric neutrino oscillations is sin2 20atm = 41 Um3 |2 (1 -| Um3 |2) = 1 - 543.

Now we want to discuss the relation between the BMV mass matrix and the parameterization of the mixing matrix in Eq. (5). We stick to the—experimentally justified—assumption that the neutrinos are non-degenerate, and we employ again Eq. (13). If mj = 0, then we know that the Xjj are either +1 or -1. If Xjj =+1 then we see from Eq. (13) that

with real uj .If Xjj = -1 then

It remains to consider the possibility mj — 0. In that case we have Sc* — cj Xjj with an arbitrary phase factor Xjj

Since the massless case allows rephasing of the Majorana neutrino field, one can absorb a factor (Xjj) 1/2 into

j = wj

that field. It is easy to see that, then, cj assumes the form in Eq. (23). In the case of Eq. (24), we may multiply the physical neutrino field by a factor i, thereby passing from Eq. (24) to Eq. (23), but also changing the sign in front of mj in Eq. (11). We may also, if needed, multiply the neutrino fields by factors — 1 so that all three uj become non-negative. We thus obtain the following result: if the mass matrix is of the BMV type (and since neutrinos are non-degenerate) then the lepton mixing matrix U is of the form in Eq. (5), but the Majorana phase factors nj may be either 1 or i; or, in other words, with a U of the form in Eq. (5), the BMV mass matrix is diagonalized as UT MvU = diag(n2m1,n2>m2,n^m3).

Let us finally consider the conditions under which Ue3 will be zero with a mass matrix of the BMV type. Suppose r2s* inEq. (3) is real. Then,

( a |r | |r | \

Mv = Y |r | ±|s| b Y with Y = diag(1,e! argr,e—! argr). (25)

|r| b ±|s|

This means that Mv is essentially identical to the matrix in Eq. (1), and we conclude that if r2s is real then Ue3 = 0. Conversely, let us now suppose that the neutrino masses are non-degenerate. Then we know, from Eq. (13), that Sc| = ±c3. (This relation, with the plus sign, also holds for m3 = 0, as we have argued in the previous paragraph.) If Ue3 = 0 this means c3 = (0,w3, ±w|)T. Now, fromEq. (11), Mvc3 = m3c|. This gives

rw3 ± (rw3)* = 0, (26)

SW3 ± bw3 = m3w3. (27)

Eq. (27) implies that sw32 is real. Eq. (26) implies that r2w32 is real. As w3 = 0, we conclude that provided the neutrinos are non-degenerate, Ue3 = 0 implies a real r2s . We have thus shown that, with the mass matrix of BMV, Ue3 being zero is equivalent to r 2s * being real.

3. A model

We now want to produce a model that leads to the mass matrix of Eq. (3). In doing this we find inspiration in our model of maximal atmospheric neutrino mixing of Ref. [4]. Thus, we supplement the Standard electroweak Model with three right-handed neutrinos and two extra Higgs doublets. We denote the three lepton families e, ¡, and t by the general index a; thus, we have three left-handed lepton doublets

Da = ^ ^ ^ a = e,x, t, (28)

together with three right-handed charged-lepton singlets aR and three right-handed neutrino singlets vaR. In the scalar sector we employ three Higgs doublets

= (, J = 1, 2, 3. (29)

These Higgs doublets acquire vacuum expectation values (VEVs) (0|<p°|0) = Vj / \fl. and v =

Vl^il2 + \v2\2 + \v3\2 — 246 GeV represents the Fermi scale.

We introduce the three U(1) lepton-number symmetries La. These symmetries are meant to be broken only softly at the high seesaw scale. We also introduce a Z2 symmetry under which ¡r, tr, 02, and 03 change sign. This symmetry Z2 is broken only spontaneously by the VEVs v2 and v3. Because of the lepton-number symmetries and of the Z2 symmetry, the Yukawa Lagrangian of the leptons is (see also Refs. [4,5])

\fl , q ^—\ _ -sf^TYlg , _ O^A — ^—^ ^—^ / _ O^A —

=--VPl'-Vl) ¿^ faVaRDa--—\ipx , ipx )eRDe - ^ ^ {<Pj , Vj )g.jauRDa, (30)

V1 a=e,x,T V1 j=2 a=x,T

where the three fa and the four gja are complex numbers (me is real without loss of generality and represents the electron mass). Notice that, through the first line of Eq. (30), the smallness of the neutrino masses may be correlated with the smallness of the electron mass. The Z2 above is analogous to the auxiliary Z2 of Refs. [4,5]. The right-handed neutrinos have Majorana mass terms given by

Cm =\(vTRC-1M*rvr-vrMRCvTR), (31)

where MR is a 3 x 3 symmetric matrix in flavor space. Now, MR is not diagonal since the terms in Eq. (31) have dimension three and we allow the lepton-number symmetries La to be broken softly [4]. Indeed, MR is the sole source of lepton mixing in this framework. According to the seesaw formula [10], when the eigenvalues of x/M*rMr are all of order mR~^>v one has

Mv = -mtdm-1md, (32)

where MD = diag(fe,fl,fT).It has been shown in Ref. [11] that this framework, in which the tree-level Yukawa couplings are diagonal but MR is not, leads at the one-loop level to a renormalized theory with flavor-changing neutral Yukawa interactions, in which flavor-changing processes like |± ^ e±y or Z° ^ e±|T are suppressed by inverse powers of mR while processes like |± ^ e±e+e- are unsuppressed by any inverse powers of mR —they are suppressed only by small Yukawa couplings—since they may be mediated by neutral scalar particles.

We now want to enforce maximal atmospheric neutrino mixing through an Mv like in Eq. (3). We do this by imposing the following generalized CP transformation [7,12]:

VLa ^ iSapY°cv[p, aL ^ iSapY0cp[, VRa ^ iSapY°CvR,p, aR ^ iSapY°CpT,

01,2 ^ 01* 2, 03 . (33)

This CP symmetry makes fe real and f^ = f *, while g2| = g|T and g3| = — g*r. Without loss of generality we assume that v1 is real and positive. Then we find

MD = SMDS and MR = SMRS, (34)

where the second relation follows from the CP invariance of £M. Therefore Mv, too, fulfills Eq. (10), just as we wanted.

Let us now consider the masses of the i and t leptons. Those masses are given by

= mt = -^=\g2tv*+g3tvl\ = -^=\g*^v*-gl^vl\. (35)

With the notation v2,3 = |v2,3|e!^2-3, we obtain

ml_l = ^=\g2l_l\v2\+g3l_l\v3\ei^-^\, mT = -j=\g2vi\v2\-g3»\v3\eiV3-fi2)\. (36)

Therefore, mT = mM requires

e'(»2—»3) = —e'(»3—»2)_ (37)

Now let us check the case of CP conservation. There we have [13]

v| = V2, V* = —V3, (38)

and therefore

CP conservation ^ ei(§2—§3) =±i. (39)

Thus, if CP is conserved we have mf = mT; CP violation is necessary for mf = mT (for an earlier model of this type see Ref. [14]).

We have thus shown that, provided CP is spontaneously broken, we are able to obtain mf = mT while Mv satisfies Eq. (10). We thus have a model with maximal atmospheric mixing but a free | Ue31.

4. Obtaining m^ « mT

Let us again consider Eq. (35). Since mf and mT are both given by essentially the same VEVs and Yukawa couplings, it seems natural to expect that they will be of the same order of magnitude, even if spontaneous CP breaking ensures that they are different. However, in reality one has mf < mT. This strong inequality requires in our model the almost complete cancellation of two different products of a VEV and a Yukawa coupling.

In general, if the muon mass is generated by the Yukawa coupling , where $ is some Higgs doublet,

then there is a natural explanation for the smallnessof mf: one just introduces the symmetry fR — —fR,$ — —$. This symmetry in general restricts the Higgs potential in such a way that it allows a vacuum with (0|^°|0) = 0. The muon mass then turns out to be zero. If one now lets the symmetry $ — — $ be softly broken by some terms of dimension two in the Higgs potential ($^$', where $' is some other Higgs doublet), then we obtain a technically natural explanation for the smallness of (0|^0|0) and thus of mf. Of course, this only works if $ does not have any Yukawa couplings besides the one to the f, which is not the case in the Standard Model since that model only contains one Higgs doublet.

This idea may be implemented in the context of our model in the previous section. Let us introduce the extra symmetry [5]

K: fR -—MR, $2 ^ $3 (40)

into that model, then one obtains g2f = — g3f and

| g2f Li |g2f |

mfl = —j=-\v2-v3\, mT = —j=-\v2 + v3\. (41)

On the other hand, the symmetry K will also restrict the scalar potential, and this in such a way that there will in general be a range of parameters of the potential for which the vacuum leaves that symmetry unbroken, i.e., v2 = v3. This immediately leads to mf = 0. (Notice that v2 = v3 constitutes a maximal spontaneous breaking of our non-standard CP symmetry, cf. Eq. (39).) In order to obtain a non-zero but small mf it is now enough to introduce into the potential terms which break K softly. In our model, considering its other symmetries Z2 and CP, there are two such terms:

f s1 ($2$2 — $3$3) + if s2($2$3 — $3$2). (42)

The constants fs1 and fs2 are real. A detailed analysis of the potential is outside the scope of this Letter [15], but it is intuitive to expect that, provided f S1 and f S2 are small, | V2 — V31 will also be kept small and thus mf mT will prevail. The crucial point is that this inequality is now protected by a softly-broken symmetry, and it is therefore technically natural.

5. Leptogenesis

Seesaw models offer an attractive possibility for explaining the observed baryon asymmetry of the universe by leptogenesis [16,17]. In order to analyse leptogenesis one works in the weak basis where the Majorana mass matrix MR of the right-handed neutrinos is diagonal, with matrix elements M1, M2, and M3. The Dirac mass matrix MD is not diagonal in that weak basis. Consider the Hermitian matrix R = MDM^D in that weak basis. Then, the

CP -violating asymmetry relevant for leptogenesis is

3Mi ^lm[(Ru)2l

UTtViRu 4-i Mi

8nviRii

where for simplicity we have assumed M1 ^ M2,M3.

In our model, in the weak basis where MD is diagonal, which is the one that we have used before, one has MD = MD = diag(fe,f^,f*) with fe real, while MR = SMRS. Now, we know from Section 2 that a matrix MR satisfying MR = SMRS is diagonalized by a unitary matrix V of the Harrison-Scott type, i.e.,

the pj being real. We now move to the weak basis where MR is diagonal (MR = MR) and MD is not. In that weak basis Md = VtMd. Thus,

It is clear that the matrix R is not only Hermitian but, as a matter of fact, real. Therefore, in our model leptogenesis is not possible. The root of this fact can be traced directly to the existence of a CP symmetry in our model, which is spontaneously broken only at the weak scale, i.e., much below the scale at which a net lepton number is supposed to be generated. Since CP is unbroken at that super-high scale, e is necessarily zero and leptogenesis cannot proceed.

6. Conclusions

In this Letter we have discussed the mass matrix of Eq. (3), originally found by Babu, Ma, and Valle [6] in the context of a model based on the group A4 and on softly broken supersymmetry with additional heavy charged-lepton singlets, an enlarged scalar sector, and the seesaw mechanism. In that model, the relations a = b and r = s = 0 hold at the seesaw scale and the full mass matrix of Eq. (3) arises at the weak scale after the renormalization-group evolution of Mv. (Note, however, that subsequently a much simpler non-supersymmetric A4 model was proposed by Ma [6], where the radiative corrections to a = b and r = s = 0 are generated by an A4 triplet of charged scalars.)

Firstly, we have shown that the mass matrix (3) can be characterized by the algebraic relation in Eq. (10). If we consider all the parameters of the matrix (3) as independent, it follows readily from this characterization that atmospheric neutrino mixing is maximal and that either 013 = 0 (and then the CP phase in lepton mixing is physically meaningless), or 013 is arbitrary and—in the phase convention of Eq. (20)—the CP phase is given by n/2—see Eq. (20); this is the more general case. Moreover, the neutrino mass matrix of Eq. (3) fixes neither the neutrino masses nor the solar mixing angle.1

Secondly, we have derived this mass matrix in the context of a model based on the lepton sector of the Standard Model with three right-handed neutrinos, the seesaw mechanism, and three—instead of one—Higgs doublets. We have constructed our model in two steps:

(1) Inspired by Refs. [4,5], we have imposed the three U(1)La symmetries associated with the family lepton numbers, which are softly broken by the LM of Eq. (31).

1 As hinted at above, in the A4 models of Ref. [6] the parameters of Mv are not completely independent and, therefore, statements about the neutrino mass spectrum and $13 can be made.

(2) As suggested by the relation (10) and by Ref. [7], we have imposed the non-standard CP symmetry of Eq. (33) in order to get the neutrino mass matrix of Eq. (3).

In this way, we have obtained a renormalizable model where lepton mixing arises solely from the Majorana mass matrix MR of the heavy neutrino singlets and where mM = mT is a consequence of the spontaneous breaking of the non-standard CP symmetry.

Our model contains a Z2 symmetry which is spontaneously broken at the Fermi scale (see the discussion of the Yukawa Lagrangian (30)), and this may lead to a cosmological problem through the formation of domain walls at that scale. An additional symmetry (see Eq. (40)) solves the problem of mM ^ mT in a technically natural way.

We stress that in our model the neutrino masses are completely free, contrary to what happens in the A4 models which predict neutrinos to be approximately degenerate. We also stress that in our model the Mv of Eq. (3) holds at the seesaw scale and its form will be slightly changed by the renormalization-group (RG) evolution down to the Fermi scale, while in the A4 models Eq. (3) holds precisely after the RG evolution. Our realization of the mass matrix (3) is an interesting illustration of the fact that exact maximal atmospheric neutrino mixing and a non-zero mixing angle 013 can coexist, enforced by a symmetry. This was not the case in the models of Refs. [4,5], where the mass matrix (1) was obtained.

Acknowledgement

The work of L.L. was supported by the Portuguese Fundagao para a Ciencia e a Tecnologia under the contract CFIF-Plurianual.

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