Cobimaximal lepton mixing from soft symmetry breaking

JID:PLB AID:33230 /SCO Doctopic: Phenomenology

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Physics Letters B ••• (••••) •••–•••

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Cobimaximal lepton mixing from soft symmetry breaking

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W. Grimus a , L. Lavoura b a b

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University of Vienna, Faculty of Physics, Boltzmanngasse 5, A-1090 Vienna, Austria Universidade de Lisboa, Instituto Superior Técnico, CFTP, 1049-001 Lisboa, Portugal

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Article history: Received 6 September 2017 Received in revised form 26 September 2017 Accepted 27 September 2017 Available online xxxx Editor: A. Ringwald

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45◦ and

±90◦ in the lepton mixing matrix V , arises as a

Cobimaximal lepton mixing, i.e. θ23 = δ= consequence of S V = V ∗ P , where S is the permutation matrix that interchanges the second and third rows of V and P is a diagonal matrix of phase factors. We prove that any such V may be written in the form V = U R P , where U is any predefined unitary matrix satisfying SU = U ∗ , R is an orthogonal, i.e. real, matrix, and P is a diagonal matrix satisfying P 2 = P . Using this theorem, we demonstrate the equivalence of two ways of constructing models for cobimaximal mixing—one way that uses a standard C P symmetry and a different way that uses a C P symmetry including μ–τ interchange. We also present two simple seesaw models to illustrate this equivalence; those models have, in addition to the C P symmetry, flavour symmetries broken softly by the Majorana mass terms of the right-handed neutrino singlets. Since each of the two models needs four scalar doublets, we investigate how to accommodate the Standard Model Higgs particle in them. © 2017 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

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1. Introduction

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Cobimaximal lepton mixing [1–7], i.e. θ23 = 45◦ and δ = ±90◦ , where θ23 is the atmospheric mixing angle and δ is the Dirac phase in the lepton mixing (PMNS) matrix, is still a viable option—for recent fits to the neutrino oscillation data see ref. [8]. Although the best-fit value of θ23 is slightly off 45◦ , θ23 = 45◦ is still possible at the 3σ level. Moreover, the most recent data prefer δ = −90◦ to δ = +90◦ . Cobimaximal mixing is equivalent to the condition

    V μ j  = V τ j 

∀ j = 1, 2, 3

(1)

in the lepton mixing matrix V [2]. The condition (1) does not restrict the mixing angles θ12 and θ13 . In model-building practice, condition (1) is more easily achieved through the condition

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S V = V ∗P ,

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where S is the permutation matrix given by

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E-mail addresses: [email protected] (W. Grimus), [email protected] (L. Lavoura).

(2)

1 0 S =⎝0 0 0 1

⎞

0 1⎠ 0

102 103

(3)

and P is a diagonal matrix of phase factors. Notice that S is real and unitary and that it satisfies S 2 = 1. (In this paper, 1 denotes the 3 × 3 unit matrix.) In the model-building literature one finds two approaches to cobimaximal mixing which can be distinguished by the C P transformation property of the Majorana mass Lagrangian of the three light neutrinos. The first approach [2–5] is based on the invariance of that mass Lagrangian under a non-standard C P transformation including the μ–τ interchange; the second approach [1, 6,7] uses the standard C P transformation. In section 2 of this paper we demonstrate that the two approaches are completely equivalent in their consequences for lepton mixing. In sections 3 and 4 we present two very simple models for cobimaximal mixing that illustrate the equivalence of both approaches. Those models include several Higgs doublets; it is thus non-trivial to accommodate in them a Standard Model (SM)-like Higgs boson with mass mh = 125 GeV [9]; a discussion of this issue is found in section 5. Section 6 presents our conclusions. In order to facilitate the reading of sections 3 and 4, we display the general formulas for weak-basis changes in Appendix A. Appendix B discusses, in a general multi-Higgs-doublet model, the conditions for the alignment

https://doi.org/10.1016/j.physletb.2017.09.082 0370-2693/© 2017 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

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W. Grimus, L. Lavoura / Physics Letters B ••• (••••) •••–•••

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of the vacuum expectation values (VEVs) needed to accommodate the SM Higgs boson.

2. In the second approach [1,6,7], M ν is real, i.e. the standard C P symmetry



3 4





νL x0 , x → i γ0 C ν¯ LT x0 , −x

2. Two equivalent approaches to cobimaximal mixing



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Lmass = −¯L M   R +

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1 2

νLT C −1 M ν νL + H.c.



†



ˆ , U  L M  U  R = diag me , mμ , mτ ≡ m

(5a)

ˆ ν. U νT M ν U ν = diag (m1 , m2 , m3 ) ≡ m

(5b)

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1 Uω = √ ⎝ 1 3 1 1

1

1

or U  L = U  , where [7]

⎛

1 U = ⎝ 0 0

0

 

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(10)

78 79 80

1

with  = √ . 2

(11)

Theorem 1. Any two 3 × 3 unitary matrices U 1 and U 2 that fulfil U k∗ = SU k (k = 1, 2) differ only by an orthogonal matrix R, i.e. U 2 = U 1 R. † Proof. U 1 U 2 = † Thus, U 1 U 2 ≡ R

 † U1 S† S U2

= ( SU 1 ) ( SU 2 ) = †

U 1T U 2∗

=

is a real matrix, hence it is orthogonal.

† U1U2

∗

.

2

1. In the first approach [2–5], the charged-lepton mass matrix is ˆ  U † R with didiagonal (or, more generally, of the form M  = m ˆ  and unitary U  R ) while the Majorana neutrino mass agonal m term enjoys the generalized C P symmetry2





νL x , x → i S γ0 C ν

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0

¯ LT





0

x , −x .

(6)

Because of the symmetry (6), the mass matrix M ν fulfils the condition

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S M ν S = M ν∗ .

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(7)

It has been shown in ref. [3] that equation (7) entails U ν = U P , where U has the form of Theorem 1, i.e. SU = U ∗ , and P is a diagonal matrix of phase factors. (Moreover, if m j = 0 then P j j can only be either ±1 or ±i. Thus, equation (7) leads to cobimaximal mixing plus a prediction for the Majorana phases.) Now, since the charged-lepton mass matrix is diagonal, the lepton mixing matrix V coincides—apart from multiplication from the left with a diagonal matrix of phase factors—with U ν and is thus given by

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V = U P.

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(8)

Equation (1) is then fulfilled and cobimaximal mixing obtained. Note that equation (8) together with SU = U ∗ lead to S V = V ∗ P 2 , which is a special case of equation (2).

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(a = ω, ),

(12)

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∗ S D = W D S D W D = W D S W D∗ = W D S ( S W D ) = W D W D = 1,

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†

†

†

†

(13) i.e. the C P transformation becomes the standard C P transforma∗ = SU ∗ tion of equation (9). Now, since U ω ω and U  = SU  , both

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Now, Theorem 1 tells us that both approaches lead to the same predictions for the mixing matrix, namely cobimaximal mixing and Majorana phases in neutrinoless ββ -decay which are either zero or 180◦ , while the mixing angles θ12 and θ13 are not restricted.5 Indeed, consider the first approach, leading to the mixing matrix of equation (8). Then, from SU = U ∗ , we deduce with Theorem 1 that there are orthogonal matrices R and R such that U = U ω R = U  R . Thus, equation (8) may be converted into the form of equation (12) and both approaches are, therefore, equivalent. If both approaches refer to the same model, then the mixing matrices in equations (8) and (12) are the same because they just correspond to the same C P symmetry in that model but written in different weak bases. For the general formulas of a weak-basis change, of a C P transformation, and of the transformation property under a weak-basis change of the C P -transformation matrix in flavour space, we refer the reader to equations (A.1), (A.3), and (A.4), respectively, in Appendix A. In particular, if S D is the matrix in flavour space that operates the C P transformation of the left-handed lepton doublets, we see that the C P transformation (6) corresponds to S D = S. According to equation (A.4), a change of weak basis performed by a unitary matrix W D sat∗ = S W changes S = S to isfying W D D D

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With Theorem 1 it is easy to understand that the following two approaches to cobimaximal mixing found in the literature are equivalent.

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with R a orthogonal and P a having the same properties as P in the previous paragraph. (The matrix U ν = R a P a diagonalizes the real matrix M ν .4 ) Since R a is real and the second and third rows in both U ω and U  are the complex conjugates of each other, equation (1) holds and the matrix V of equation (12) displays cobimaximal mixing.



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The PMNS matrix is V = U  L U ν . We now prove1

V = Ua Ra P a

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3

⎞

0 −i  ⎠ i

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ω ω2 ⎠ with ω = exp (2i π /3), ω2 ω

†

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⎞

Then, in both cases the PMNS matrix has the form

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⎛

†

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†

such that either U  L = U ω , where

(4)

for the chiral charged-lepton and neutrino fields  and ν , respectively. In equation (4), C is the charge-conjugation matrix in Dirac space. The mass matrices in flavour space M  and M ν are both 3 × 3 in order to accommodate the three families; M ν is a symmetric but in general complex matrix since it corresponds to Majorana mass terms. The diagonalization of the mass matrices proceeds via

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applies to the neutrino Majorana mass terms. Furthermore, there is some symmetry in the charged-lepton mass terms

In order to fix the notation we write down the mass terms

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(9)

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A lemma demonstrated in ref. [5] states that if a unitary matrix U 1 fulfils U 1∗ = SU 1 , then the unitary matrix U 2 = U 1 R, where R is an arbitrary real orthogonal matrix, fulfils U 2∗ = SU 2 . Our theorem is the converse of that lemma. 2 For the sake of clarity, we spell out the transformation (6) at length: 1

















T νeL x0 , x → i γ0 C ν¯ eL x0 , −x , νμ L x0 , x → i γ0 C ν¯ τT L x0 , −x ,

ντ L



   T x0 , x → i γ0 C ν¯ μ x . L x0 , −

3

In ref. [7] the notation U 2 is used for our U  . 4 The matrices P a are needed in order to obtain positive neutrino masses m1,2,3 . Indeed, if M ν is real then it is diagonalized by the orthogonal matrix R a , but the diagonal matrix resulting from that diagonalization may contain some negative diagonal entries; when m j < 0 one needs ( P a ) j j = ±i to correct for that. 5 We stress that this statement only refers to the conditions laid out in this section. In specific models the mixing matrix could be more predictive.

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U ω and U  are well suited to perform changes of weak basis that lead from S D = S to S D = 1. The resulting PMNS matrix V = U P of equation (8) cannot change when the weak basis is changed, hence, since SU = U ∗ , one must have U = U ω R ω = U  R  as in equation (12); this is precisely what Theorem 1 guarantees us.

1 M ν = − M TD M − R MD = −

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LY = −

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12



(14)

α =e,μ,τ

In equation (14), the D α L = (να L , α L ) T are the leptonic gaugeSU (2) doublets, the α R are the right-handed charged-lepton fields, and the να R are three right-handed neutrinos that we have introduced in order to enable a seesaw mechanism. The Lagrangian (14) has only two Yukawa coupling constants, y 1 and y 2 . The Lagrangian (14) may be enforced, for instance, through an S 3 permutation symmetry among the α indices together with the six Z2 symmetries [10] (α ,1)

Z2

(α ,2)

Z2

: DαL → −DαL , : φα → −φα , (α ,1)

φα → −φα ,

να R → −να R ,

α R → −α R .

(15a) (15b)

(α ,2)

Both Z2 and Z2 act trivially on fields with flavour β = α . If one imposes the C P symmetry

















⎞

D eL eR D L = ⎝ D μL ⎠ ,  R = ⎝ μ R ⎠ , Dτ L τR

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0 0

⎛

⎞

⎛

⎞

νe R φe ν R = ⎝ νμ R ⎠ , φ = ⎝ φμ ⎠ . ντ R φτ

Clearly, the transformation (16a) includes, in particular, the transformation (6). Let the VEVs be

vl

0 |φl | 0 = √

2

0 1

(18)

for l = e , μ, τ , ν . The charged-lepton mass matrix is diagonal:

⎛

ve y1 M = √ ⎝ 0 2 0

0 vμ 0

⎞

0 0 ⎠. vτ

(19)

The breaking of C P is spontaneous, via v μ = v τ , corresponding to mμ = mτ . In order to complete the model, one introduces the bare Majorana mass terms

LMaj =

1 2

ν RT C −1 M ∗R ν R + H.c.

(22)

0 y1 0

⎛



⎞

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0 0 ⎠, 0

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⎞

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(23)

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  ¯ α L φl (l )α β β R + φ˜l (l )α β νβ R D

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+ H.c.

(24)

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The other Yukawa-coupling matrices are

e = μ = τ = 0,

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0 0 0 0 0 ⎠ , ν = ⎝ 0 0 0 ⎠ , y1 0 0 0



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c r ⎠ , ∗ c

⎛

⎞

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⎞

α ,β=e,μ,τ l=e,μ,τ ,ν

(16d)

(17)

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0

(16c)

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0

τ = ⎝ 0 0

LY = −

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⎛

(16b)

⎛

r

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0 0 0 0 0 ⎠ , μ = ⎝ 0 0 0 0

 R x0 , x → i S γ0 C ¯TR x0 , −x     ν R x0 , x → i S γ0 C ν¯ RT x0 , −x ,     φ x0 , x → S φ ∗ x0 , −x ,     φν x0 , x → φν∗ x0 , −x , ⎞

c

c

⎞

in the notation

⎛

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where r and r are real while c and c are complex. Equation (7) 1 then holds, because M ν ∝ M − R . One thus has a model corresponding to the first approach of section 2. Note that the neutrino masses are not predicted in this model. We now seek to transform this model into an equivalent one that accords with the second approach of the previous section. In equation (14), the Yukawa-coupling matrices l responsible for the charged-lepton masses are

(16a)

then one obtains real y 1 and y 2 . In transformation (16),

r

c∗

M R = S M R S = ⎝ c∗

y1 e = ⎝ 0 0

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⎛

⎛

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Due to the C P symmetry one has

¯ TL x0 , −x , D L x0 , x → i S γ0 C D

(16e)

(21)

are explicitly but softly broken through a nondiagonal M R , • the C P symmetry (16c) is conserved.

∗

¯ α L y 1 φα α R + y 2 φ˜ ν να R + H.c. D

1 M− R .

• the Z2

3. A simple model for cobimaximal mixing We construct an extension of the SM with gauge group SU (2) × U (1) and four scalar SU (2) doublets φα (α = e , μ, τ ) and φν . The Yukawa Lagrangian is

2

(α ,1)

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y ∗2 2 v 2ν

One further requires that in LMaj

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3

102

ν = y 2 1.

(25)

103

Now we transform the fields into a basis where the C P symmetry has the standard form. Comparing equations (16) and (A.3), one sees that

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S D = S = Sν = S,

Sφ =

S 01×3

03×1 1

W D = W  = W ν = Uω,

Wφ =

Uω 01×3

y1

e = √ 1,  μ = √ E T , 3 3 ⎞ ⎛ 0 1 0 where E ≡ ⎝ 0 0 1 ⎠ . 1 0 0

y1

03×1 1

 τ = √ E , 3

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,

(27)

with U ω of equation (10). One may now write the Yukawacoupling matrices in the new basis by using equations (A.2). One obtains

y1

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(26)

.

We want to use equation (A.4) to obtain S D = S  = S ν = 1 and S φ = diag (1, 1, 1, 1). Equation (13) suggests to choose6



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 ν = y 2 1,

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(28)

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(20)

and one assumes a (type 1) seesaw mechanism √ [11]. Clearly, the neutrino Dirac mass matrix is M D = y ∗2 v ν 2 1 and

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In equation (27) we might have utilized U  instead of U ω . Indeed, we are free to utilize any matrix U that satisfies SU = U ∗ instead of U ω in equation (27). If we had utilized U  , we would have obtained the matrices  μ,τ in equation (33) below, with y 3 → y 1 .

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Note that  ν = e =  μ =  τ = 0. The Yukawa-coupling matrices are real before and after the basis transformation because of the C P symmetry. Obviously, the Yukawa-coupling matrices look simpler in the first approach—equation (23)—than in the second approach— equation (28). On the other hand, M R = U ω M R U ω is simply a real matrix in the second approach, and therefore the neutrino Majorana mass terms look simpler in the second approach than in the first one.

5. Accommodation of the SM Higgs

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We have two models, both of them with four scalar doublets, namely the φl with l = e , μ, τ , ν . We must use those doublets to give mass to the quarks. It suggests itself to use φν for this purpose. Then | v ν | should be large, because the top-quark Yukawa coupling cannot be much larger than 1. Consequently, it is natural to assume (for definiteness, in the model of section 3)

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LY = − D¯ eL y 1 φe e R + y 2 φ˜ ν νe R

 ¯ α L y 3 φα α R + y 4 φ˜ ν να R + H.c. − D

50

This Lagrangian may be enforced, for instance, through a μ–τ permutation symmetry together with the Z2 symmetries (15). We once again impose the C P symmetry (16), making y 1,2,3,4 real. The charged-lepton mass matrix is

y1 v e M = √ ⎝ 0 2 0

0 y3 v μ 0

1

⎞

0 0 ⎠. y3 v τ

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(30)

The breaking of C P occurs via v μ = v τ . We use once again the seesaw mechanism with the matrix M R of equation (22). We thus obtain another model for cobimaximal mixing, very similar to the one of the previous section but with a Yukawa Lagrangian with four coupling constants instead of just two. We next transform this model into a form that accords with the second approach of section 2. The initial Yukawa-coupling matrices are

⎛

y1 e = ⎝ 0 0

⎛

0 0 0

0 0

τ = ⎝ 0 0 0 0

⎞

⎛

0 0 0 ⎠ , μ = ⎝ 0 0 0

⎞

⎛

⎞

0 y3 0

0 y2 0 ⎠ , ν = ⎝ 0 y3 0

0 y4 0

W D = W  = W ν = U,

Wφ =

U 01×3

0 0 ⎠, y4

0 y3  μ = √ ⎝ 0 2 0

0 1 0

⎞

0 0 ⎠, 1

⎛

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(34)

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,

(32)

⎞

0 0 0 y3  τ = √ ⎝ 0 0 −1 ⎠ . 2 0 1 0

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charged-lepton masses. We know that a scalar field h has been discovered at LHC. We also know that the couplings of that scalar to the heavy fermions and to the gauge bosons are close to the couplings of the SM Higgs. The couplings of h to the light fermions may be at variance with the ones in the SM. We use the formalism in refs. [12–14] for the neutral scalars. The neutral components of the doublets φl are written

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1

φl0 = √



2

vl +

8 

 (35)

b =1

Re V Im V

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98

(36)

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is 8 × 8 orthogonal. The first column of V , which corresponds to the Goldstone boson, is given by Vl1 = i v l / v, where



+ μ− L = · · · + g mW W μ W +

(37)

mZ Zμ Z μ 2c w

 8

b1

b =2

102 103 104 105 106 107



109 110 111

(38)

⎞

ve 1⎜ vμ ⎟ ⎜ ⎟ v= ⎝ vτ ⎠ v vν

101

108



S b0 Im V † V .

Equation (38) indicates that we should have Im V † V b1 = 1 for a neutral scalar with index b > 1 that has couplings to pairs of gauge bosons identical to the ones of the SM Higgs.7 Since Vl1 =   i v l / v, Im V † V b1 = 1 is equivalent to Vlb = v l / v. Thus, in order to exactly reproduce the SM Higgs, the vector

⎛

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matrix V has the property that

V˜ =

82

90

Vlb S b0 ,

where the eight fields S b0 (b = 1, 2, . . . , 8) are neutral eigenstates of mass. By definition, S 10 is the neutral “would-be” Goldstone boson and the S b0 for b = 2, . . . , 8 are physical scalars. The 4 × 8 complex



80

89



112 113 114 115 116 117 118 119 120

(39)

121 122 123 124

(33)

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70

 2 | v e |2   v μ   | v τ |2 hold due to the strong hierarchy among the

(31)

with U  of equation (11). Note that in equation (32) we use the matrix U  instead of U ω as in equation (27). This is just an arbitrary choice—we may use any unitary matrix U such that SU = U ∗ , but using U  in this case yields simpler results. We write down the Yukawa-coupling matrices in the new basis: e , ν = 0, and the four l remain invariant while

⎛

 | v ν | ≈ (246 GeV) .

The interaction of the neutral scalars with a pair of SM gauge bosons is given by the Lagrangian [14]

⎞

03×1 1

2

  2 v = | v e |2 +  v μ  + | v τ |2 + | v ν |2 .

0 0 ⎠, 0

and ν = e = μ = τ = 0. We transform the fields into a basis where the C P symmetry has the standard form by using

51 52

(29)

α =μ,τ

⎛

y 21

2

(In this section, as elsewhere in this paper, the indices α , β , and γ vary in the range {e , μ, τ }.) Obviously, the inequalities



48 49

|v α | =

In our second model the Yukawa Lagrangian is

24 25

69

74

2m2α

2

4. Another simple model for cobimaximal mixing

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The Yukawa-coupling matrices are still real, but now this follows from the standard C P symmetry. The matrix M R = U  M R U  is real too.

125 7

In the C P -conserving version of the two-Higgs-doublet model there are two physical neutral scalars named h and H . Their couplings to pairs of gauge  † bosons  8 0 are given by a Lagrangian similar to equation (38), with b=2 S b Im V V b1 → h sin (β − α ) + H cos (β − α ), where β − α is a suitably defined angle [15]. The condition for the neutral scalar h to have couplings to pairs of gauge bosons identical to the ones of the SM Higgs is thus sin (β − α ) = 1.

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must be one of the columns of the matrix V , namely the one corresponding to a neutral-scalar mass eigenstate with mass mh = 125 GeV. We discuss here the possibility that v is an exact mass eigenstate. One may eventually consider an approximate mass eigenstate, taking into account the inequality (34). The conditions that the scalar potential must satisfy in order for the vector v to correspond to a neutral mass eigenstate with mass mh are given by equations (B.10). It is easy to fulfil equation (B.10a) by fine tuning. One simply has to assume for the matrix μ2 in the quadratic part of the scalar potential that

11 12 13 14 15 16 17 18 19 20

2

μ =−

mh2 2

vv + · · · ,

where the dots indicate a part of the matrix μ2 operating in the space orthogonal to v. We make the following simplifying assumptions concerning the scalar potential V = V 2 + V 4 :

• In the quartic part V 4 we assume invariance under any per-

μ, and τ ; in addition, we assume that all the

doublets occur in pairs φl , φl . Thus,

V4 = λ



2

†

φα φα



†

+ λν φ ν φ ν

2

2

28 29

+

30

α =β





λ ν φν φν †









 † † † φα φα + λ

ν φν φα φα φν . (41)

32

36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53

This may be achieved by observing that the Yukawa Lagrangian in equation (14) is invariant under

⎛

⎞

⎛

⎞

λαααα = λ,

λαα ββ

λα ββ α = λβ αα β = λαανν = λνναα =

64 65

⎛

eR

⎞

⎝ μ R ⎠ → ⎝ ωμ R ⎠ , 2

τR ⎛

ω τR ⎞

⎛

⎞

2

λ

ν 2

λνααν = λαννα =

2

λ

(α = β),

2

λ ν 2

λ

ν 2

72

(α = β),

(44c)

vν vα.

(44d)

λ

⎞   2 v ν ν v β  ⎠ ,

2

80

(45a)

v

β

2

2

λ

= λν | v ν |2 + ν 2

= λ |v α | +

2

β=α

v

λ

2

V˜ 4 = a ⎝



φl φl ⎠ + b

93

(46b)

2



†





l=e ,μ,τ ,ν l =e ,μ,τ ,ν

†

φl φl



†



φl φl .

(43b) (43c) (43d)

,

(43e)

,

(43f)

λ = λν = 2a,



λ = λν = 2b.

2

97 98 99 100 101 102 103 104 105 106 108 109 110

(48)

Equations (46a) and (46b) then merge into

mh2

96

107

Comparison of V˜ 4 with V 4 of equation (41) yields



94 95

(47)

λ = λν = a + b ,

91 92

β=α



87

90

(46a)

  2 λ + λ

ν v β  + ν | v ν |2 .

⎞2



85

89

β

l=e ,μ,τ ,ν

84

88

  2 ν v β  ,

+ λ

83

86

+ λ

2

81 82

⎞

Equation (46b) separately holds for α = e , μ, τ . The right-hand side of equation (46b) has, in general, a depenof α . Consisdence on α , while its left-hand side  is independent  tency is achieved by assuming λ = λ + λ

2. Equations (46) may be satisfied by assuming a custodial-type symmetry [16] in V 4 . Since we have four Higgs doublets, we may choose U (4). The U (4)-symmetric quartic potential is then

⎛

77 79

+ λ

λ + λ

  2 λ ν + λ

ν vα | v ν |2 ⎠ vβ + . (v)α = ⎝λ | v α |2 +

mh2

76 78

⎛

2

74 75

∗

(v)ν = ⎝λν | v ν |2 + ν

mh2

71 73

v ∗β v α

⎛

(42)

(43a)

λνννν = λν , λ

= λββ αα = (α = β),

59

63

⎞

In order to use the notation of equation (B.1) we identify, in the V 4 of equation (41),

57

62

eR

and then by requiring V 4 to be invariant under this transformation too. • We allow for a general quadratic part V 2 . This breaks softly both symmetries (15) and (42). The soft symmetry breaking also occurs in the matrix M R anyway, therefore we must assume its presence in V 2 too.

56

61

⎞

⎛

⎛

νe R φe νe R φe ⎝ νμ R ⎠ → ⎝ ω2 νμ R ⎠ , ⎝ φμ ⎠ → ⎝ ωφμ ⎠ , ντ R φτ ωντ R ω 2 φτ

55

60

⎞

⎛

D eL D eL ⎝ D μ L ⎠ → ⎝ ω2 D μ L ⎠ , Dτ L ωDτ L

54

58

λ

α

31

35

α β =

(44b)

γ

Equation (B.10b) then reads





 1   † †

† † + λ φα φα φβ φβ + λ

φα φβ φβ φα

27

70

(45b)

α

26

34

2

68 69

  2 v γ  , νν = λν | v ν |2 + ν

†

23

(44a)

2

γ =α

Therefore,

22

33

2

67

λ

(40)

mutation of e,

25

66

  2 λ

 v γ  + ν | v ν |2 ,

λ

αα = λ | v α |2 +

να = αν =

21

24

whence we find

∗

†

5

111 112 113 114

2

= (a + b) v .

(49)

115 116 117

6. Conclusions

118 119

Neutrino oscillation data indicate that cobimaximal mixing may be a viable scenario, at least as a first approximation, for lepton mixing. In the literature there are two approaches to cobimaximal mixing. They may be characterized by the underlying C P symmetry of the mass Lagrangian of the light neutrinos. The first approach features a generalized C P symmetry that includes a μ–τ interchange, in the basis where the charged-lepton mass matrix is diagonal. In the second approach, the C P symmetry is the standard one, but the charged-lepton mass matrix is non-diagonal and † † provides a factor U  L in the mixing matrix V = U  L U ν such that †

SU  L = U TL , with S given by equation (3). We have demonstrated

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W. Grimus, L. Lavoura / Physics Letters B ••• (••••) •••–•••

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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

that the two approaches yield the same consequences for lepton mixing, on the one hand by using a simple mathematical theorem which proves the equivalence of the mixing matrices in both approaches, and on the other hand by explicitly stating the weakbasis transformation that transforms the generalized C P symmetry into the standard one. Moreover, we have displayed two renormalizable models for cobimaximal mixing which illustrate the relationship between the two approaches. In these seesaw models, the mixing angles θ12 and θ13 as well as the neutrino masses are undetermined. They use not only the above-mentioned C P symmetry but also flavour symmetries which are softly broken in the Majorana mass terms of the right-handed neutrino singlets; in this way, cobimaximal mixing is obtained straightforwardly. Finally, since each of our models has four Higgs doublets, the accommodation of the SM Higgs boson is non-trivial. Using the general discussion of this issue for any number of Higgs doublets presented in Appendix B, we have formulated some necessary conditions for the existence of a neutral scalar with SM-like couplings in the two models.

20 21

24 25

Let the Yukawa Lagrangian be as in equation (24). We perform the basis transformation





27

The transformed Yukawa-coupling matrices are



l =

l

30 31 32

l =

34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

W φ∗

† W D l W ν . l l

(A.2)

58

 R → i S  γ0 C ¯TR ,

¯ RT ,

ν R → i S ν γ0 C ν

∗

φ → Sφ φ .

(A.3)

The C P -transformation matrices in the new basis are

S D =

† ∗ WD SDWD , † ∗

Sν = W ν Sν W ν ,

S  = W  S  W ∗ , †

S φ

=

W φ S φ W φ∗ . †

(A.4)

Appendix B. VEV alignment for the SM Higgs We use in this appendix the notation and results of appendix A of ref. [13]. Suppose there are n H Higgs doublets φi (i = 1, 2, . . . , n H ). The scalar potential is

V =

nH 





nH 

μ2i j φi† φ j +

i , j =1



†

λi jkl φi φ j



†



φk φl ≡ V 2 + V 4 ,

i , j ,k,l=1

(B.1) where V 2 is the quadratic part of the potential and V 4 is the quartic part. From now on we employ the summation convention. We define the matrices , K , and K through ∗

59

i j ≡ λi jkl v k v l ,

60

We also define the vector

61 62 63 64 65

2

2 ∗ ij vi v j

μ

+

1 4

70

∗

∗

λi jkl v i v j v k v l .

(B.5)

∂V0 1 2 = μi j v j + λi jkl v j v k∗ v l ∗ ∂ vi 2

73 74

(B.6)

77 78



2

μ +  v = 0.

(B.7)

As explained after equation (39), we want v to be a column of the matrix V corresponding to a neutral scalar with mass mh , therefore it has to fulfil (see equation (A18) of ref. [13])





μ2 +  + K v + K v∗ = mh2 v.



v≡

n H  i =1

K ik ≡ λi jkl v j v l ,

−1/2 |v i |

2

⎛ ⎜ ⎜ ⎜ ⎝

75 76

(note that λi jkl = λkli j by definition). We thus have



71 72

Since the vacuum state is a stationary point of V 0 ,

0=

68

79 80 81 82 83 84

(B.8)

85 86 87



88

μ2 + 3 v = mh2 v.

(B.9)

89 90

We then employ equation (B.7) to obtain

v =

mh2 2

mh2 2

91 92

v,

(B.10a)

93 94 95

v.

(B.10b)

96 97 98 99

54

57

V0 =

1

67 69

References

53

56

(B.4)

The expectation value of the potential in the vacuum state is

μ2 v = −



¯ TL , D L → i S D γ0 C D

52

55

v = K v = K v.

Let the C P symmetry be

50 51

(A.1)



l

33

νR = W ν νR , φ = W φ φ .

† W φ l l W D l W  ,





R = W R ,



DL = W D DL,

29

66



Because of equation (B.4), equation (B.8) may be rewritten

26 28

∗

Appendix A. Basis transformation

22 23

It is clear from these definitions that

∗

K il ≡ λi jkl v j v k .

(B.2)

⎞

v1 v2 ⎟ ⎟

.. ⎟ . . ⎠

vnH



(B.3)

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