Hybrid textures in minimal seesaw mass matrices

Physics Letters B 693 (2010) 249–254

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Physics Letters B www.elsevier.com/locate/physletb

Hybrid textures in minimal seesaw mass matrices Srubabati Goswami a,∗ , Subrata Khan a , Atsushi Watanabe b a b

Physical Research Laboratory, Navrangpura, Ahmedabad 380009, India Department of Physics, Niigata University, Niigata 950-2181, Japan

a r t i c l e

i n f o

Article history: Received 18 June 2010 Received in revised form 16 August 2010 Accepted 16 August 2010 Available online 19 August 2010 Editor: A. Ringwald Keywords: Neutrino Neutrino mass and mixing

a b s t r a c t In the context of the minimal seesaw framework, we discuss the implications of Dirac and Majorana mass matrices in which two properties coexist, namely, equalities among matrix elements and texture zeros. Among the large number of general possibilities, only 12 patterns are found to be consistent with the global neutrino oscillation data at the level of the most minimal number of free parameters. The predictions of the allowed textures for mass hierarchy, θ13 and effective mass governing neutrino-less double beta decay are discussed. We also explore the possibility of having non-zero CP violation for each allowed solution. We find that only one allowed solution can accommodate both low and high energy CP violation. We discuss the prediction of this solution for leptogenesis and explore the correlation, between leptogenesis and low energy CP violation. © 2010 Elsevier B.V. All rights reserved.

1. Introduction It is well known that in the standard electroweak theory, masses and mixing angles of fermions originate from the Yukawa interactions with the Higgs boson, responsible for the electroweak symmetry breaking. While the gauge interactions are controlled by the gauge invariance, Yukawa couplings are not governed by any principle. They bring a multitude of free parameters into the theory, and even have ambiguities in reconstructing their values from experiments. A viable approach, therefore, is to search for mass matrices which take suggestive forms in the light of model building ingredients, such as symmetry among generations. A direct scheme in this spirit is the texture zero in the fermion mass matrices, which was first attempted in the quark sector [1]. In this approach, it is assumed that the mass matrices have elements which are anomalously small compared to the others. The texture zero in the lepton sector [2,3] have been discussed also, and their results have stimulated the model building activities for the lepton masses and mixings. Another way to reduce the number of parameters is to impose equalities among the elements of the mass matrix [4]. The coexistence of both zeros and equality relations has been explored in [5]. Although in the Standard Model neutrinos were assumed to be massless the data from neutrino oscillation experiments have undoubtedly proved that neutrinos have very small mass. The 3σ ranges of the two independent mass squared differences as ob-

*

Corresponding author. E-mail addresses: [email protected] (S. Goswami), [email protected] (S. Khan), [email protected] (A. Watanabe). 0370-2693/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2010.08.033

tained from global oscillation analysis are presented in Table 1 [6]. One of the most attractive mechanism for generating such small neutrino masses is the seesaw mechanism [7]. In the so-called type-I seesaw mechanism small neutrino masses arise naturally by the suppression effect of large mass of a heavy right-handed neutrino. By integrating out the heavy field the Majorana mass matrix of the light neutrinos is obtained as 1 M = −mTD M − R mD ,

(1.1)

where m D denotes the Dirac mass matrix after the electroweak symmetry breaking, and M R is the Majorana mass matrix for the heavy neutrinos. The mixing matrix in the lepton sector, the socalled Pontecorvo–Maki–Nakagawa–Sakata (PMNS) matrix is the unitary matrix which diagonalizes the Majorana mass matrix M, in the basis where the charged-lepton mass matrix is diagonal. This is characterized by the three mixing angles whose best-fit values and 3σ allowed ranges are presented in Table 1. In addition there are three phases which can induce CP violation in the lepton sector. At present there are no constraints on these phases. In this work, we consider the equalities and zeros in the Dirac (m D ) and the right-handed Majorana mass matrices (M R ) in Type-I seesaw mechanism. We perform a thorough classification of such hybrid textures, and identify the matrix forms which are compatible with the neutrino oscillation data. We also discuss the possibility of having leptogenesis in these textures and explore the connection between high and low energy CP violation, if any. The layout of the Letter goes as follows. In Section 2, we discuss the possible hybrid textures and present the allowed solutions. In Section 3, we discuss the possible connections between leptogen-

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S. Goswami et al. / Physics Letters B 693 (2010) 249–254

Table 1 The present best-fit values and the 3σ ranges of oscillation parameters [6] m2i j =

Table 2 1 The allowed textures of m D and M − R .

m2i − m2j .

mD

m221 [10−5 eV2 ]

best fit

3σ range

7.59

7.03–8.27

|m231 | [10−3 eV2 ]

2.40

2.07–2.75

sin2 θ12 sin2 θ23 sin2 θ13

0.318 0.50 0.013

0.27–0.38 0.36–0.67 0.053



a



b

a

a



a

b

b

b

a

a

a

b

0

a

b

a

0 a

b

a

a

b

0 b

a

0 a

b

a

0

I

II

III

 IV



esis and CP violation in neutrino oscillation for the viable textures and finally conclude in Section 4.

V

2. Hybrid textures and allowed mass matrices

VI



We consider the minimal seesaw model with two right-handed neutrinos. The advantage of this choice is that there are less number of free parameters. Thus the mass matrices get simple forms and rich predictions compared to the standard three heavy neutrino models. In this scenario the Dirac mass matrix m D and the Majorana mass matrix M R will be 2 × 3 and 2 × 2 matrices respectively. In this framework, the rank of the induced matrix M becomes at most 2, so that at least one neutrino is predicted to be massless at the outset. The total number of elements in the Dirac mass matrix is 6. The textures with equalities are defined by imposing equality relations among the matrix elements. For example, the relation (m D )11 = (m D )12 means

 mD =

a

a

b

c

d

e

 (2.1)

.

Since there are 6 matrix elements, the maximum number of equality relations that can be imposed is 5. However, in the case of 5 equalities, all the elements of m D are equal and the neutrino spectrum contains two massless states irrespective of M R . This is not consistent with evidences of neutrino oscillations which require at least two states to be massive. Thus m D ’s with 5 equalities are excluded. The patterns with less than 5 equalities in m D cannot be excluded a priori. By exhausting all possibilities, total 201 patterns were found as general possibilities for m D with equalities. The Majorana mass matrix M R contains 3 independent matrix elements and can accommodate at most 2 equalities. However, 2 equalities in M R imply a vanishing determinant. With such an M R , there appears a state which does not receive seesaw suppression in mass. In this work, we do not consider such spectrum and in what follows we will simply exclude the cases where M R has two 1 equalities. The three alternatives for M R (or M − R ) with 1 equality 1 are : 1 M− R =



A

B

B

A





,

A

A

A

B





,

A

B

B

B



.

(2.2)

We will consider these three textures as general possibilities for M R in the following discussions. Note that all the three textures in (2.2) contain 1 phase that cannot be removed by redefinition of fields. The hybrid textures are the matrices in which vanishing elements and equalities among elements coexist. The procedure adopted to impose the equalities and texture zeros on the mass

1

Note that in 2 × 2 case, the equalities in M R are directly connected to the equal-

1 ities in M − R .

a a

0 a a

b

1 M− R

























A

A

A

B

A

A

A

0

0

A

A

A

A

0

0

A

0

A

A

0

A

A

A

0

     

matrices is as follows. We impose texture zeros on the mass matrices after introducing the equalities among the matrix elements. For instance if we consider the following m D and put b = 0 we get,



a

a

b

b

b

a





→

a

a

0

0

0

a



.

(2.3)

The resultant texture belongs to 4 equalities and 1 zero. Note that we do not impose texture zeros on each entry, but rather force the parameter b to be zero. On the other hand, if we consider the following m D and put b = 0 and c = 0 we get,



a

a

c

b

b

a





→

a

a

0

0 0

a



.

(2.4)

Thus, after putting the zeros, the resultant matrix is the same in both cases though (2.4) is obtained by setting two different parameters to be zero. In that sense (2.4) belongs to 3 equalities and 2 zeros denoting the fact that the zeros have originated from different parameters. Such a classification is justified because strictly speaking, when we impose texture zeros then it does not imply exact zero element but some matrix element which is anomalously small compared to the other elements.2 Therefore in the most general scenario the two matrices can belong to different categories although the total number of reductions remain the same. However, it is to be noted that in our present work we have treated a zero as an exact zero and from this viewpoint both (2.3) and (2.4) will give identical results for the predictions of masses and mixing angles. Therefore once we consider the case of 4 equalities and 1 zero we need not redo the calculations for 3 equalities and 2 zeros. At the level of the most minimal number of free parameters, there are 6 independent textures which are compatible with the 3σ ranges of the current data. These are summarized in Table 2. Note that all these textures consist of total 6 reductions together in m D and M R . These can be grouped as (i) 5 equalities + 1 zero or (ii) 4 equalities + 2 zeros. The total number of textures that we analyzed in the above two categories were 456. In Table 3 we present numerical values of the five oscillation parameters, the averaged neutrino mass mee  governing neutrinoless double beta (0ν 2β ) decay, the neutrino mass mβ probed in tritium beta decay and the mass ordering for the six allowed textures. We also indicate whether the basis-independent measures of low and high energy CP violation are zero or non-zero. In obtaining the values of the mixing angles we take the mass squared

2 From the viewpoint of model building it is difficult to obtain exact zeros for instance due to quantum corrections.

S. Goswami et al. / Physics Letters B 693 (2010) 249–254

Table 3 Representative values of the five oscillation parameters for the allowed solutions. We also present the mass mβ probed in β -decay and the effective mass mee  governing neutrino less double beta decay. In the items for the mass ordering, the symbol “” means each texture accommodates the corresponding mass patterns, and “×” means it does not. The last two items show finiteness of the invariant measures for the CP violation [9] at low and high energy respectively. I

II

III

IV

V

VI

m221 [10−5 eV2 ] |m231 | [10−3 eV2 ] sin2 θ12 sin2 θ23 sin2 θ13 mee  [eV]

7.65

7.30

7.65

7.20

7.35

7.10

2.40

2.50

2.40

2.60

2.60

2.70

0.33

0.27

0.31

0.26

0.27

0.27

0.50

0.50

0.52

0.49

0.46

0.54

0

0

0.039

0.055

0.045

0.044

0.016

0.024

0.016

0.005

0

0

mβ [eV]

0.048

0.05

0.048

0.013

0.011

0.012

Normal hierarchy Inverted hierarchy

× 

× 

Il

0

0

×  = 0

 × = 0

 × = 0

Ih

= 0

= 0

0

0

0

 × = 0 = 0

differences from their allowed 3σ ranges of Table 1 and check if the values of all the 3 mixing angles lie within the allowed range or not. The mass squared differences presented in Table 3 are some representative values which can give the permissible mixing angles within each texture. In the items for the mass ordering, the symbol “” means each texture is consistent with the corresponding mass orderings, and “×” means it is not. The textures I, II and III are consistent with the inverted hierarchy, while IV, V and VI accommodate the normal hierarchy. The resultant mass matrices after seesaw diagonalization for both solution I and II obey the so-called scaling property between the second and third rows and second and third columns [8]. This matrix has μ–τ symmetry and a zero eigenvalue such that the mass ordering is inverted. Moreover, the reactor and the atmospheric angles are given as θ13 = 0 and θ23 = 45◦ . The two nonzero eigenvalues of M for the solution I are

λ± =

1 2

7 A + 2B ±





57 A 2 + 20 A B + 4B 2 ,

(2.5)

where we introduce the parameter A ≡ a2 A and B ≡ a2 B to simplify should be identified as  the notation. These eigenvalues 

λ+ =

|m231 | + m221 and λ− = − |m231 | in order to fit the ob-

servations. The parameters A and B are therefore fixed in terms of the two mass differences as

  1



A  − m231 

3

+



1 18

α + O α2





,

    7 4   B  m231  + α + O α2 , 6

(2.6)

9

where α ≡ m221 /|m231 |. The solar angle gets fixed by the two mass differences as,

sin θ12 =  where

x≡

1+x 8x2

+ (1 + x)2

1 √ 1+α−1− 18



√



2 + 34 1 + α + α .

By expanding (2.7) in powers of

sin θ12  0.58 − 0.096α .

(2.7)

,

(2.8)

α , we find

251

trip-bi maximal mixing scenario. Similarly for the solution II also one can follow the same procedure. The results for this case are more complicated and lengthy and so we just give the final answer for the solar mixing angle obtained,

sin θ12 = 0.54 − 0.041α ,

(2.10)

which agrees with the current 3σ range although it is close to its lower bound. The final mass matrix after seesaw diagonalization for solution III has Mτ τ = 0 [3]. The two nonzero eigenvalues of the resultant matrix M obtained after seesaw diagonalization are

λ± =

1 2 a + 4a b + b 2 2





9a 4 + 8a 3 b + 14a 2 b 2 + 8a b 3 + 5b 4 , (2.11) √ √ where a ≡ a B and b ≡ b   B. Identifying these as λ− = − |m231 | + m221 and λ+ = |m231 | the parameters a and b

±

are fixed in terms of the two mass differences. These can be expressed powers of α as,

1/4     a  − m231  0.848 + 0.116α + O α 2 , 1/4     b  m231  0.227 + 0.201α + O α 2 .

(2.12)

The reactor and the atmospheric angles no longer satisfy θ13 = 0 and θ23 = 45◦ , and all nonzero values of the mixing angles can be described as functions of the mass differences as,

sin θ12  0.56 + 0.0078α ,

(2.13)

sin θ23  0.72 + 0.017α ,

(2.14)

sin θ13  0.19 + 0.15α .

(2.15)

This texture gives a relatively large θ13 which can be measured in forthcoming experiments like Double-Chooz. It is to be noted that for the textures I, II and III, the possible ranges of the mixing angles are narrow; the values sin θi j ’s vary at most a few percent. The solutions I–III are consistent with IH and m3 = 0. The effective mass governing neutrino-less double beta decay for this case can be approximated as [10]

mee  =

  m231 1 − sin2 2θ12 sin2 α2 /2

(2.16)

where α2 denotes the Majorana Phase. The values of mee  presented in Table 3 are obtained by using the above expression with α2 = π . There also exist three viable textures IV, V and VI with normal hierarchy.3 For the solution V and VI the element Mee in the resultant Majorana mass matrix is vanishing. This predicts the effective mass governing neutrino-less double beta decay to be zero. In general for NH in the limit m1 → 0 the expression for mee  is [10],

      mee  = sin2 θ12 m221 + sin2 θ13 m231 e i α32 

(2.17)

where α32 = α3 − α2 . Assuming the phase factor to be 1 the √ solar and reactor angles correlate with α as sin2 θ13 / sin2 θ12 = α . For the current allowed values of θ12 and α , this relation implies large (just below the current 3σ bound) θ13 .

(2.9)

It is interesting to observe that, in √ the 0-th order, the solar mixing angle turns out to be 0.58 = 1/ 3, which is the same as in the

3

We note that with the updated values of the oscillation parameters the values

of sin2 θ12 and sin2 θ13 for texture IV presented in Table 3 are marginally disallowed.

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S. Goswami et al. / Physics Letters B 693 (2010) 249–254

We note that the neutrino mass mβ probed by direct searches in tritium β -decay experiments for IH (in the limit m3 → 0) is given as [10]

mβ =



m231 .

(2.18)

For NH in the limit m1 → 0, mβ can be expressed as [10]

mβ =



sin2 θ12 m221 + sin2 θ13 m231 .

(2.19)

In Table 3 we present the values of mβ for the six allowed patterns. For all the scenarios mβ is below the sensitivity of 0.2 eV of KATRIN experiment [11]. This implies that a positive signal in KATRIN experiment can rule out all the scenarios discussed in this Letter. Besides these 6 combinations, the patterns obtained by permuting 2–3 column of m D for each combination in the above list will also be compatible with the data. This is because the 2–3 column exchange keeps θ23 close to π /4 without disturbing the good agreement of θ12 with the data. The predicted θ13 also remains the same. Although such 6 counterparts are independent textures, we present only 6 textures in the table because physical predictions of the 6 counterparts are almost the same as the original ones. At this level of minimality, patterns other than these 12 textures are not consistent with the 3σ ranges of parameters presented in Table 1. The texture forms should be assumed at some high-energy scale where the right-handed neutrinos are not decoupled from the theory. Since the seesaw scale is expected to be around 1014 GeV for O (1) Yukawa couplings, extrapolation to the low energy scale by Renormalization Group (RG) equations is necessary. However, within the SM, the RG effects on the neutrino parameters are appreciable only if the light neutrino spectrum is quasidegenerate [12]. In the seesaw scenario with two heavy neutrinos, we necessarily have one massless neutrino and hence a hierarchical spectrum. Thus the RG effects are not strong enough to disfavor an allowed texture or allow a disfavored texture within the present setup. 3. CP violation at high and low-energy scales Besides the predictions for the oscillation parameters and the 0ν 2β decay, distinctive features of each texture also appear in CP violating phenomena. In this section, we discuss leptogenesis [13] and CP violation at the neutrino oscillation for the allowed hybrid textures. We would like to mention that in this Letter we deal with non-flavoured leptogenesis [14]. In the leptogenesis scenario, the right-handed neutrinos decay into the Higgs fields and the lepton doublets so that the lepton– antilepton asymmetry is produced. The CP asymmetry generated by the i-th generation right-handed neutrino is described by †

i =

˜ D )2i j ] ˜ Dm Im[(m †

˜ D )ii v 2 ˜ Dm (m √

  f

M 2j M i2



 +g

M 2j M i2

 (3.1)

, √

where f (x) = x(1 − (1 + x) ln[(1 + x)/x]) and g (x) = x/(1 − x). ˜ D are the Dirac mass matrix in the baThe mass parameters m sis where the right-handed Majorana mass matrix M R is diagonal; M R = diag( M 1 , M 2 ). The resultant lepton asymmetry η L is evaluated as

ηL 

1 g∗

κi i

†

˜ Dm ˜ i = (m ˜ D )ii / M i . If the right-handed neuon the effective mass m ˜ i  103 eV, an trinos are strongly coupled to the thermal bath m approximate analytic expression for the factor κi is available [15]



where κi are the factor which represent wash-out effect calculated by solving the Boltzmann equations, and g ∗ is the number of relativistic degrees of freedom. The wash-out effect strongly depends

 ln

˜i m

 − 0 .6

˜i m 10−3 eV

(3.3)

.

Finally a part of B − L asymmetry is converted to the baryon asymmetry by the sphaleron processes in thermal equilibrium and leads to the relation η B = −28/51η L in the SM. In the neutrino oscillations, CP violation may be observed in the difference between the appearance probabilities P (να → νβ ) − P (ν¯ α → ν¯ β ) with α = β . The difference is proportional to the leptonic version of the Jarlskog invariant [16]





J CP = Im V α 1 V α∗ 2 V β∗1 V β 2 ,

(3.4)

where V stands for the PMNS matrix. A promising channel for the discovery of CP violation is P (νμ → νe ) − P (ν¯ μ → ν¯ e ) in longbaseline experiments such as T2K and NOν A [17]. In order to ascertain if there is CP violation in the oscillation experiments or leading to successful leptogenesis it is convenient to evaluate the weak basis invariant quantities [9]



† 3

I l = Tr MM† , ml ml



I h = Im Tr

(3.5)

,

 † † m D mTD M R M R M ∗R m D m D M R , ∗

(3.6)

where ml is the charged-lepton mass matrix. The necessary conditions for successful leptogenesis and the observation of CP violation in oscillation experiments are I h = 0 and I l = 0 respectively. It is to be noted that in general there are three weak basis invariants that can be related to CP violation responsible for leptogenesis [9]. However for models with two right handed neutrinos these three are not independent and are proportional to each other. The † other weak basis invariants obtained by substituting m∗D ml ml mTD

in place of m∗D mTD in I h are not related to leptogenesis as shown in [9]. Therefore it is sufficient to consider only I h to check if the CP violation at high energy is related to leptogenesis or not. The low-energy invariant I l is related to J CP as [9]

I l = −6i Δμe Δτ μ Δτ e m221 m231 m232 J CP

(3.7)

where Δγ β = (m2γ − m2β ), with γ , β = e , μ, τ and m2i j are the light neutrino mass squared differences defined earlier. In Table 3, we demonstrate whether I l and I h are zero or nonzero for each of the solutions. Since for textures I and II θ13 comes out as zero I l is also zero. On the other hand, the solutions III–VI allow I l to be non-vanishing in general indicating the possibility of observing CP violation at low energy. As for the high-energy CP violation, the textures III, IV, and V give I h = 0 while solutions I, II, VI give I h = 0. Thus, only the texture VI accommodates non-vanishing values for both I l and I h so that there can be a correlation between low and high energy CP violation. In what follows we consider in detail the prediction of this interesting solution for leptogenesis and explore the correlation between CP violation at high and lowenergy scales. The relation between leptogenesis and CP violation at low energy stems from the fact that there is only one physical phase in the neutrino mass matrices, and the one phase controls both the baryon to photon ratio η B and the invariant measure J CP . Without loss of generality, one may take



(3.2)

i

10−3 eV

κi  0.3

mD = x

0

|a|e i ϕ

|a|e i ϕ |b| |b|

0



,

1 M− R =



| A| | A| | A|

0



.

(3.8)

The phase of the element A and b have been absorbed into the right-handed neutrino fields. With this phase convention, η B and

S. Goswami et al. / Physics Letters B 693 (2010) 249–254

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demonstrate in terms of the weak basis invariants if the solutions can give CP violation at low and high energy. We find that only one solution can have correlation between low and high energy CP violation. We discuss the implications of this solution for leptogenesis and explore the correlation between high and low energy CP violation. Acknowledgement The authors thank A. Mohanty for a careful reading of the draft. References

Fig. 1. The baryon-to-photon ratio η B as a function of the Jarlskog invariant J CP . The horizontal dashed line shows η B = 6 × 10−10 suggested by the WMAP observation [18]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)

J CP are given as the function of the phase ϕ with the condition that η B = J CP = 0 at ϕ = 0. Fig. 1 shows the parametric plot of η B and J CP = Im × ∗ V ∗ V ] as a function of the phase ϕ . The dashed (red), [ V μ1 V μ 2 e1 e2 dotted (green) and solid (blue) curves are the cases of the heavy mass scales A −1 = 1014.0 , 1013.75 and 1013.0 GeV respectively. The other parameters are determined so that the data in Table 1 are satisfied. From Fig. 1, it is seen that the absolute value of the baryon asymmetry |η B | is increased as the right-handed neutrino scale is increased with a fixed value of J CP . The observed baryon asymmetry is achievable only if the scale A −1 is larger than  1013.75 GeV = 5.6 × 1013 GeV which corresponds to the right-handed neutrino masses M 1 = 3.5 × 1013 GeV and M 2 = 9.1 × 1013 GeV. For a fixed successful number of the baryon asymmetry η B , there is a correlation between the invariant J CP and the right-handed neutrino scale A −1 . For A −1 > 1013.75 GeV, the relation is represented by

 J CP ∼ 0.01

1013.75 GeV A −1

2 .

(3.9)

It has been discussed that for J CP  0.0066, the CP violation can be established with a probability of more than 99% at the 99% CL in T2HK [19]. The texture VI thus predicts measurable CP violation at future oscillation experiment if the right-handed neutrino scale satisfies 5.6 × 1013  A −1  7 × 1013 GeV. 4. Conclusion In this Letter, we consider simultaneous presence of equality relations and texture zeros for Dirac and Majorana mass matrices in the minimal Type I seesaw mechanism with 2 heavy right-handed neutrinos. We study a large number of independent options and find that at the level of minimal number of free parameters only the 6 textures presented in Table 2 and 6 others obtained by permuting the 2–3 column of m D presented in this table stand out to be consistent with global neutrino oscillation data. The latter 6 have the same predictions as the original ones and hence we have not presented these separately. Three of the solutions in Table 2 are found to be consistent with normal mass hierarchy while three others with inverted mass hierarchy. We discuss the prediction of each solution for the oscillation parameters, 0ν 2β decay and the mass mβ probed by tritium β decay. It is noteworthy that the allowed values of mβ are below the sensitivity reach of KATRIN experiment. Hence a positive signal in this experiment can rule out all the allowed patterns discussed in this Letter. We also

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