Neutrino Mixing by modifying the Yukawa coupling structure of constrained sequential dominance

In the constrained sequential dominance (CSD), tri-bimaximal mixing (TBM) pattern in the neutrino sector has been explained, by proposing a certain Yukawa coupling structure for the right-handed neutrinos of the model. Since the current nomenological model where we consider Yukawa couplings which are modified that of CSD. Essentially, we add small complex parameters to the Yukawa couplings of CSD. using these modified Yukawa couplings, we demonstrate that neutrino mixing angles can deviate from their TBM values. We also construct a model, based on flavour symmetries, in order to justify the modified form of Yukawa couplings of our work.


Introduction
From various experimental observations it is known that neutrinos have very small mass [1]. In a Type I seesaw mechanism, through the mediation of heavy right-handed neutrinos, smallness of neutrino masses can be understood [2,3]. To test this mechanism at the LHC the mass of the right-handed neutrinos should be around 1 TeV. However, with 1 TeV masses for right-handed neutrinos some tuning in the Yukawa couplings may be required in order to fit the tiny masses of neutrinos. Moreover, due to large number of seesaw parameters this mechansim may not be predicted from the experimental data. To alleviate the above mentioned problems, models based on sequential dominance [4] with two right-handed neutrinos and one texture zero in the neutrino Yukawa matrix have been proposed [5,6]. These models are named as CSD(n), which we describe them briefly below.
It is known that the neutrinos mix among them [1] and the current oscillation data [7] suggest that the neutrino mixing angles are close to the TBM pattern [8]. To explain these mixing angles in the models of CSD(n), the two right-handed neutrinos are proposed to have certain particular Yukawa couplings with the three lepton doublets. To be specific, the two right-handed neutrinos, up to proportionality factors, are proposed to have the following Yukawa couplings: (0, 1, 1) and (1, n, n − 2). Here, n is a positive integer but can be taken to be real as well. For the case of n = 1, the model predicts that the three mixing angles will take the following TBM values: sin θ 12 = 1 √ 3 , sin θ 23 = 1 √ 2 , sin θ 13 = 0. This case of n = 1 is originally named as constrained sequential dominance (CSD), which was viable six years ago. But now the current oscillation data suggests that θ 13 = 0 and hence this case is ruled out. Among the other integer values for n, only the models with n = 3, 4 are compatible with the current neutrino oscillation data [6].
In this work, we study on a possibility where we consider modifications to model parameters of CSD and demonstrate that the neutrino observables from the oscillation data can be explained. As explained above that CSD is nothing but CSD(n = 1) and hence the Yukawa couplings in this model are proportional to (0, 1, 1) and (1, 1, −1). In the next section we will describe that with this particular form for Yukawa couplings, the mixing angles for neutrinos can be predicted to have the TBM values. Now, in order to get deviations in neutrino mixing angles away from the TBM values, we consider the Yukawa couplings of the two right-handed neutrinos to be proportional to (ǫ 1 , 1 + ǫ 2 , 1 + ǫ 3 ) and (1 + ǫ 4 , 1 + ǫ 5 , −1 + ǫ 6 ). Here, ǫ i , i = 1, · · · , 6, are complex numbers. By proposing above mentioned Yukawa couplings for neutrinos, we are considering here a phenomenological model. Now, in this phenomenological model, in the limit where all ǫ i → 0, our model should give the results of CSD. As a result of this, we can expect that for small parametric values of ǫ i we should get deviations in neutrino mixing angles away from the TBM values.
The reason for considering all ǫ i to be small is due to the fact that the observed mixing angles are close to the TBM values. After assuming that ǫ i to be small, we study if we can consistently fit the neutrino masses and mixing angles, whose values are obtained from oscillation data. Like in the model of CSD, in our model also only two right-handed neutrinos are proposed. As a result of this, in our model, one neutrino would be massless and the other two can have non-zero masses. Hence, in this model, we will show that only normal hierarchy is possible for neutrino masses. We can fit the non-zero masses of our model to square root of solar ( ∆m 2 sol ) and atmospheric ( ∆m 2 atm ) mass squared differences. From the global fits to neutrino oscillation data we can see that there is a hierarchy between ∆m 2 sol and ∆m 2 atm [7]. In fact, from the results of ref. [7], one can notice that Because of this, we take ∆m 2 sol ∆m 2 atm and sin θ 13 to be small, whose values can be around 0.15.
As mentioned above, in our work, we are modifing the neutrino Yukawa couplings of CSD model by introducing small complex ǫ i parameters. To be consistent with the oscillation data, we assume that the real and imaginary parts of ǫ i to be less than or of the order of ∆m 2 sol ∆m 2 atm ∼ sin θ 13 . After assuming this, we diagonalise the seesaw formula for active neutrinos in our model, by following an approximation procedure, where we expand the seesaw formula in power series of ǫ i . A related work in this direction can be seen in ref. [9]. Following our diagonalisation procedure, we derive expressions for neutrino masses and mixing angles in terms of ǫ i . We show that by keeping terms up to first order in ǫ i of our analysis, we get sin θ 13 and and sin θ 23 − 1 √ 2 to be non-zero but sin θ 12 − 1 √ 3 is found to be undetermined. In order to know if sin θ 12 − 1 √ 3 can be determined, we compute expressions in our analysis up to second order in ǫ i . Thereafter we demonstrate that sin θ 12 − 1 √ 3 can also be determined by ǫ i parameters. We study the above described work in a phenomenological model, where the neutrino Yukawa couplings of this model are modified from that of CSD model. One would like to know how such modified form for Yukawa couplings could be possible in our model. In order to address this point, towards the end of this paper, we construct a model, based on flavour symmetries, where we explain the smallness of ǫ i parameters. In fact, through this model we justify the structure of Yukawa couplings of our phenomenological model. The paper is organised as follows. In the next section we describe sequential dominance and the CSD model. In section 3, we describe our phenomenological model and also explain the approximation procedure for diagonalising the seesaw formula for neutrinos of this model. Using this approximation procedure we demonstrate that the neutrino mixing angles in our model deviate away from the TBM pattern. In the same section, we compute expressions for neutrino masses and mixing angles up to first order in our approximation scheme. Second order corrections to the above mentioned neutrino observables have been computed in section 4. In section 5, we construct a model in order to justify the structure of Yukawa couplings of our phenomenological model. We conclude in the last section.

Sequential dominance and CSD
The idea for CSD is motivated from sequential dominance, which is briefly described below. Consider a minimal extension to the standard model, where the additional fields are three singlet right-handed neutrinos. After electroweak symmetry breaking, charged leptons and neutrinos acquire mixing mass matrices. We can consider a basis in which both charged leptons and right-handed neutrinos have been diagonalised. In this basis, the mass matrix for right-handed neutrinos and the mixing mass matrix between left-and right-handed neutrinos can be written, respectively, as In the equation for m D , elements such as a, b, c, etc can be viewed as neutrino Yukawa coupling multiplied by vacuum expectation value (vev) of the Higgs field. Assuming that the masses for right-handed neutrinos are much larger than the elements of Dirac mass matrix, the seesaw formula for active neutrinos would be From the seesaw formula we get three masses for active neutrinos, which may be denoted by m 1 , m 2 and m 3 . The objective of sequential dominance is to achieve m 1 ≪ m 2 ≪ m 3 , and thereby the model can predict normal mass hierarchy for neutrinos. In order to achieve this objective of sequential dominance, following assumptions on the masses of right-handed neutrions and the elements of the Dirac mass matrix have been made [4] M atm ≫ M sol ≫ M dec , Here, x, y ∈ a, b, c and x ′ , y ′ ∈ a ′ , b ′ , c ′ .
With the above mentioned assumptions of sequential dominance, leading order expressions for neutrino masses and mixing angles have been computed in ref. [10]. Using these expressions, following set of conditions on the model parameters have been proposed, in order to obtain the TBM pattern for neutrino mixing angles [5].
Here, φ ′ b and φ ′ c denote sum of a combination of phases of the elements in the Driac mass matrix [5]. From the above mentioned conditions we can notice that the elements in the third column of m D and M R play no part in determining the TBM pattern for neutrino mixing angles. In fact, from the leading order expressions for neutrino masses and mixing angles given in ref. [10], we can see that the third column elements of m D and M R determine only the lightest neutrino mass m 1 . Since the current experimental data can be satisfied with m 1 = 0, in order to reduce the number of degrees of freedom in this model, we can decouple away the third column elements of m D and M R . Essentially this decoupling can be done by reducing the number of right-handed neutrinos from three to two in the above described model.
After performing the above mentioned decoupling, in the resultant model, to satisy the conditions of Eq. (4), the Dirac and right-handed neutrino mass matrices can be taken, respectively, as [5] By plugging the above mentioned m D and M R in the seesaw formula of Eq. (2), we can check that the m ν can be diagonalised as From the unitary matrix U TBM , one can extract the three neutrino mixing angles and we see that they will have the TBM values.
We have demonstrated above that in a model with two right-handed neutrinos, which is motivated by sequential dominance, TBM pattern for neutrino mixing is possible. This has been named as CSD [5]. One can notice that in this process of obtaining TBM pattern, the columns of Dirac mass matrix need to be aligned in some particular directions. This problem of alignment has been addressed in a supersymmetric model which has some flavour symmetries and flavon fields [5].

Our model and deviations from TBM pattern
In the previous section we have described on how CSD can predict TBM pattern for neutrino mixing angles. Since this pattern is currently ruled out, we need to modify the model of CSD. To achieve this, we initaily consider a phenomenological model where the field content is same as that of CSD. But the difference between our model and the CSD is that we propose a modified structure for Dirac mass matrix, which is given below.
Here, ǫ i , i = 1, · · · , 6, are complex parameters. At this stage we are suggesting the above form for Dirac mass matrix, purely from phenomenoligical point of view. We justify this form of matrix by constructing a model for this in section 5. Regarding the Dirac mass matrix, we have explained in the previous section that the elements of this matrix should be viewed as a producut of neutrino Yukawa couplings and vev of the Higgs field. As a result of this, the above Dirac mass matrix corresponds to the fact that the Yukawa couplings of the two right-handed neutrinos are proportional to (ǫ 1 , 1 + ǫ 2 , 1 + ǫ 3 ) and (1 + ǫ 4 , 1 + ǫ 5 , −1 + ǫ 6 ). As we have argued in section 1, with this form for Yukawa couplings we should expect to get deviations for neutrino mixing angles away from the TBM values.
As explained above that in our model, the form for Dirac mass matrix is given by m ′ D and hence the seesaw formula for active neutrinos is Since we are in a basis where charged leptons are diagonalised, this seesaw formula should be diagonalised by Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. The PMNS matrix can be parametrised by the neutrino mixing angles and the CP violating Dirac phase δ CP . We follow the PDG convention for this parametrisation [11], which is given below.
Here, c ij = cos θ ij and s ij = sin θ ij . As explained above that in our model, with the form for m ′ D of Eq. (7), we should get deviations in the neutrino mixing angles away from the TBM values. As a result of this, we should expect s 13 , s 12 − 1/ √ 3 and s 23 − 1/ √ 2 to become non-zero. In order to simplify our calculations, we parametrise s 12 and s 23 as [12] We have known the 3σ ranges for the square of the sine of the neutrino mixing angles, which are obtained from the global fits to oscillation data [7]. From these 3σ ranges, we can find the corresponding ranges for r and s, which are found, respectively, as: (−8.8 × 10 −2 , 2.5 × 10 −2 ) and (−8.2 × 10 −2 , 0.13). The corresponding allowed range for s 13 is found to be narrow, whose values are around 0.15. From the above mentioned ranges, we can notice that the values for r and s are less than or of the order of s 13 . As explained before that r, s and s 13 will be become non-zero in our model, if we allow non-zero values for ǫ i parameters in m ′ D . As a result of this, to be consistent with our analysis, we can assume that real and imaginary parts of ǫ i to be less than or of the order of s 13 .
As described previously, seesaw formula for active neutrinos in our model is given by Eq. (8) and this matrix should be diagonalised by U PMNS . The relation for this diagonalisation can be written as Here, the matrices m s ν and U PMNS depend on variables ǫ i , r, s and s 13 , which are small. As a result of this, we can expand m s ν and U PMNS as power series in terms of these small variables. First we expand m s ν and U PMNS up to first order in ǫ i , r, s and s 13 . After doing that one can see that m d ν need not be in diagonal form. But, since we expect this to be of diagonal form, we demand that the off-diagonal elements of m d ν to be zero. Thereby we get three relations among ǫ i , r, s and s 13 . Solving these relations, we can determine ǫ i in terms of r, s and s 13 . Now, from the diagonal elements of m d ν we get expressions for the three neutrino masses in terms of model parameters. We follow the above described methodology for diagonalising the seesaw formula of our model. However, while doing so, one needs to take care of the small numbers that may arise due to hierarchy in neutrino masses. Discussion related to this is explained below.
In the limit where ǫ i , r, s and s 13 tend to zero, from Eq. (11) we get the leading order expressions for neutrino masses, which are given below.
The above result agree with that of CSD which is given in section 2. Here, up to the leading order, the lightest neutrino mass m 1 is zero. However, we will show later that even at subleading orders, m 1 is still zero. This result is due to the consequence of the fact that in our model we have proposed only two right-handed neutrinos. As a result of this, neutrino masses in our model can only have normal mass hierarchy. Due to this, we can fit the expressions for m 2 and m 3 to square root of solar ( ∆m 2 sol ) and atmospheric ( ∆m 2 atm ) mass squared differences, respectively. Although the expressions in Eq. (12) are valid at leading order, at subleading orders, expressions for m 2 and m 3 get corrections which are proportional to ǫ i , r, s and s 13 . Since ǫ i , r, s and s 13 are small values, we can expect the following, when we fit the expressions for m 2 and m 3 to ∆m 2 sol and ∆m 2 atm .
We use the above mentioned order of estimations in the diagonalisation process of the seesaw formula of our model. Regarding this, a point to be noticed here is that from the global fits to neutrino oscillation data [7], a hierarchy is found between ∆m 2 sol and ∆m 2 atm . In fact, from the results of ref. [7], one can notice that . We explain below about this series expansion and also the results obtained from such expansion.
Up to first order in ǫ i , m s ν can be expanded as Similarly, up to first order in r, s and s 13 , the expansion for U PMNS is Here, the form of U TBM can be seen in Eq. (6) (14), we can get the expressions for the three neutrino masses, which are given below From the above equations we can see that only m 3 get correction at the first order level.
Among the off-diagonal elements of 1 √ ∆m 2 atm m d ν , we have found that 12-element is zero up to first order level. However, 13-and 23-elements are not found to be zero at this order.
After demanding that they need to be zero, they would lead to the below expressions From the above two equations we can see that, in our model, sin θ 13 will be non-zero if we take ǫ 1 = 0. Similarly, sin θ 23 will deviate from its TBM value if we take either ǫ 2 or ǫ 3 to be non-zero. However, the deviation of sin θ 12 from its TBM value, which is quantified in terms of r, is undetermined at the first order level corrections to the diagoanalisation of our seesaw formula. As a result of this, the parameters ǫ 4 , ǫ 5 and ǫ 6 are undetermined at this level. We will show in the next section that these parameters can be determined in terms of neutrino mixing angles by considering second order level corrections to the diagonalisation of our seesaw formula.
In the model of PCSD, the structure of neutrino Yukawa couplings is similar to that in our model. The Yukawa couplings in PCSD can be obtained from that of our model by taking

Second order corrections
In the previous section, after considering first order corrections to the diagonalisation of the seesaw formula for neutrinos, it is found that the deviation of sin θ 12 from its TBM value is found to be undetermined. To know if this deviation can be determined in terms of model parameters, we study here the second order corrections to the diagonalisation of the seesaw formula for neutrino masses. In order to do this we need to expand terms Expansion for m s ν and U PMNS , up to second order in ǫ i , r, s and s 13 are given below Here, the expressions for m s ν(0) , m s ν (1) and ∆U can be found in Eqs.
After demanding that the off-diagonal elements of 1 √ ∆m 2 atm m d ν should be zero, we get the following three relations.
Here, φ is the Majorana phase difference in the neutrino masses m 2 and m 3 . While obtaining the results up to second order level, which are given above, we have used relations in Eq. (20).
From the expressions for neutrinos masses which are given in Eq. (24), we can see that the lightest neutrino mass is m 1 = 0. Hence in our model only normal mass hierarchy is possible for neutrino masses. The expressions for m 2 and m 3 can be fitted to ∆m 2 sol and ∆m 2 atm respectively. While doing this fitting, we can notice that terms involving ǫ i , s 13 and s give small corrections. Hence, we can see that a 2 M sol and e 2 Matm can be of the order of ∆m 2 sol and ∆m 2 atm respectively. This result is consistent with the assumption we have made in Eq. (13). Another point to be noticed here is that both the expressions for m 2 and m 3 depend on the complex ǫ i parameters. As a result of this, both m 2 and m 3 can be complex. But since neutrino masses should be real, the complex phases in m 2 and m 3 can be absorbed in to Majorana phases. Or else, another possibility is that we can choose the parameters a and e to be complex so that m 2 and m 3 can be real, and in this case the Majorana phases will become zero.
Regarding the neutrino mixing angles, we have explained in the previous section that the deviation in sin θ 12 from its TBM value is undetermined at the first order level corrections to diagonalisation of the seesaw formula for neutrinos. But now after considering second order corrections, from Eq. (25) we can see that this deviation can be determined in terms of ǫ 4 , ǫ 5 and ǫ 6 . In fact, out of these three ǫ-parameters, only two can be deter-

A model for our Dirac mass matrix
In the last two sections we have explained that deviations from TBM pattern is possible in our model, where we have considered a specific structure for Dirac mass matrix which is given in Eq. (7). In the Dirac mass matrix we have introduced small ǫ i parameters in order to get right amount of deviations from TBM pattern. In this section, we construct a model to explain small values of ǫ i , and in the same model, we justify the sturcuture of our proposed Dirac mass matrix.
We introduce a flavour symmetry SO(3) × SO(3) ′ and also the following scalar fields: These scalar fields are singlets under the standard model gauge group, but otherwise, charged under the above flavour symmetry. The lepton doublets L, where we have suppressed generation index, are charged under this flavour symmetry. The Higgs doublet H and the two right-handed neutrinos ν atm R , ν sol R are singlets under this symmetry. To get the masses for right-handed neutrinos, we introduce the following additional scalar fields, which are standard model gauge singlets: χ atm , χ sol . To forbid unwanted interactions in our model we introduce a discreet symmetry Z 3 . In table 1, the charges assignments of the fields, which are relevant to neutrino sector, are given. With these charge assignments, the invariant Lagrangian in the neutrino sector can be written Here, M P is the Planck scale, which is the cut-off scale of the model. We have taken M P as the cut-off scale but grand unified scale can also be taken as the cut-off of the model.
From the interactions terms in Eq. (28), we can see that neutrinos acquire Dirac mass terms, once the following scalar fields acquire vevs: φ atm , φ sol , φ ′ atm , φ ′ sol . The vevs of φ atm , φ sol spontaneously break the flavour symmetry SO (3), whereas, SO(3) ′ is spontaneously broken by φ ′ atm , φ ′ sol . Let us suppose that the vevs of φ atm , φ sol have the following pattern Here, y a , y s are dimensionless quantities. One can notice that we have assumed a particular alignment for vevs of φ atm and φ sol . Here, we are not proposing a solution to this alignment problem. But it is to be noted that this problem has already been addressed in ref. [5].
On the other hand, we need not assume any alignment for the vevs of φ ′ atm , φ ′ sol . Hence, after breaking the SO(3) ′ spontaneously, φ ′ atm and φ ′ sol may take the following form Here, y ′ a , y ′ s are dimensionless quantities. Substituing the Eqs. (29) & (30) in the first four terms of Eq. (28), we get the structure of Dirac mass matrix of Eq. (7), provided if the following conditions are satisfied: y a = y ′ a and y s = y ′ s . These conditions may be satisfied by suitabily choosing the parameters in the scalar potential among the fields φ atm , φ sol , φ ′ atm , φ ′ sol . Finally, the last two terms of Eq. (28) can generate diagonal masses for the two right-handed neutrinos, after giving vevs to χ atm , χ sol . Having explained the mass structures of both Dirac and right-handed neutrinos, below we explain how ǫ i parameters can be small in this model.
Let us assume that the symmetries SO(3) and SO(3) ′ are broken around the scales Λ and Λ ′ respectively. Hence, the vevs which break the above symmetries can have the following scales.
We propose that there is a little hierarchy between Λ and Λ ′ , where Λ ′ ∼ 0.1 × Λ. Using this in Eqs. (29) & (30), we can see that ǫ i ∼ 0.1. This is the required amount of smallness we want for ǫ i in order to get the right amount of deviations from the TBM pattern for neutrino mixing. Hence, to explain the neutrino mixing angles in this model, we need to assume that the breaking scale for the symmetry SO(3) ′ is about 0.1 times that of the SO (3) breaking. The little hierarchy between these two breaking scales may be explained by proposing a mechanism which depends on the physics at or beyond the Plack scale.
Proposing such mechanism is beyond the scope of this paper, which we postpone it for a later work.

Conclusions
In this work, we have attempted to explain the neutrino mixing in order to be consistent with the current neutrino oscillation data. From the current data, it is known that θ 13 = 0, and hence, the neutrino mixing angles deviate away from the TBM pattern. Earlier, to explain the TBM pattern in neutrino sector, CSD model has been proposed. Here, we have considered a phenomenological model, where we have modified the neutrino Yukawa couplings of CSD model, by introducing small ǫ i parameters which are complex. We have assumed real and imaginary parts of ǫ i to be less than or of the order of sin θ 13 ∼ ∆m 2 sol ∆m 2 atm . Thereafter, we have followed an approximation procedure in order to diagonalise the seesaw formula for light neutrinos in our model. We have computed expressions, up to second order level, to neutrino masses and mixing angles in terms of small ǫ i parameters.
Using these expressions we have demonstrated that neutrino mixing angles can deviate away from their TBM values by appropriately choosing the ǫ i values. Finally, we have constructed a model in order to justify the neutrino Yukawa coupling structure of our phenomenological model.