O(3) Flavor Symmetry and an Empirical Neutrino Mass Matrix

Based on a new approach to quark and lepton masses, where the mass spectra originate in vacuum expectation values of O(3)-flavor 1+5 (gauge singlet) scalars, a neutrino mass matrix of a new type is speculated. The mass matrix is described in terms of the up-quark and charged lepton masses, and, by assuming a special flavor basis, it can be accommodated to a nearly tribimaximal mixing without explicitly assuming a discrete symmetry. Quark mass relations are also discussed based on the new approach.


Introduction
One of the most challenging problems in contemporary particle physics is to clarify the origin of flavors. For this purpose, searching for a unified description of the observed quark and lepton mass spectra will provide a promising clue to us. In conventional mass matrix model, the quark and lepton mass matrices M f are given by the forms (M f ) ij = (Y f ) ij v H , where (Y f ) ij are coupling constants of the Yukawa interactionsf Li f Rj H 0 and v H is a vacuum expectation value (VEV) of the neutral component of the Higgs scalar H, v H = H 0 . Against this conventional approach, there is another idea: the origin of the mass spectra is due to VEV structures of Higgs scalars H ij [1,2], i.e. (M f ) ij = y f (H 0 ) ij . In the present paper, we will investigate an extended model by separating the role of H ij into two roles: one of the roles is to cause SU(2) L symmetry breaking at the energy scale µ ∼ 10 2 GeV, and the conventional SU(2) L doublet Higgs scalars H u and H d still play the role in this scenario; another one is to give an origin of the mass spectra, and we consider gauge-singlet scalars (Y f ) ij whose VEVs give effective Yukawa coupling constants (Y f ) ij /Λ (Λ is an energy scale of the effective theory). As a typical model with such gauge-singlet scalars (Y f ) ij , there has been a model [3] with U(3)-nonet scalars, where quarks and leptons are assigned to 3 and3 of U(3) F , so that Y f (f = u, d, ν, e) are assigned to nonet of U(3) F . However, this U(3) scenario with this assignment cannot apply to a grand unification theory (GUT) scenario, because, for example, in SU(5)-GUT, the SU(2) L doublet and singlet quark fields Q and U should be assigned to the same multiplet 3, so that Y u must be6 (not nonet). Then, the model considerably become complicated, because we need fields 6 in addition to fields6 in order to make singlets of U(3).
In this paper, considering applicability of the scenario to a GUT scenario, we assume an O(3) flavor symmetry instead of U(3). We consider the following superpotential terms: where Y f (f = u, d, ν, e) and Y R are O(3)-flavor 1+5 (gauge singlet) fields , and Q and L are quark and lepton SU(2) L doublet fields of O(3) F triplets, and U , D, N , and E are SU(2) L singlet matter fields of O(3) F triplets. Therefore, the fields Y f and Y R are symmetric. Our basic assumption is as follows: the fields Y f and Y R always behave as a combination of 1+5, so that, for example, 5 alone never appears in the interaction terms. Hereafter, for convenience, we will denote 1+5 as 6. u, d, ν, e), and Q X (U ) = −q u to U , Q X (E) = −q e to E, and so on. The field Y R has the charge Q X (Y R ) = 2q ν . In this paper, we will write down our superpotential W under the O(3) F and U(1) X symmetries.
In the present approach, we will investigate relations among Y f and Y R by using supersymmetric (SUSY) vacuum conditions for the superpotential W u, d, ν, e) and W R determine the VEV structures of Y f and Y R , respectively. (Since we can easily show Q = L = U = D = N = E = 0, hereafter, we will drop the term W Y from W when we investigate the VEV structures of Y f . ) Such an approach to quark and lepton mass matrices has first been adopted by Ma [4] and has been developed by the author within a context of U(3)-flavor nonet model [3]. In the conventional mass matrix approach, the investigation has now been on a level with theoretically reliable ground via a long period of phenomenological investigations. However, the present approach is still in its beginning stage, so that we need more phenomenological investigations. Therefore, we adopt the following strategy in this approach: (i) First, we search for a possible form of the superpotential W which can successfully provide relations among the observed masses and mixings from the phenomenological point of view; (ii) Next, we investigate what symmetries or quantum number assignments can explain such a specific form of W . In this paper, we will investigate a possible form of W by putting weight on the step (i).
Recently, as a byproduct in such approach, an interesting neutrino mass matrix form [5] has been reported: the form is given by Neutrino mass matrix models which leads to the so-called tribimaximal neutrino mixing [6] have usually been proposed based on discrete symmetries, while, if we assume a specific relation between a diagonal basis of the charged lepton mass matrix (we call it "e-basis") and a diagonal basis of the down-quark mass matrix (we call it "d-basis"), the mass matrix (1.2) can be accommodate to a nearly tribimaximal neutrino mixing without explicitly assuming a discrete symmetry. On the other hand, in general, if a neutrino mass matrix M ν can give reasonable masses and mixing, a neutrino mass matrixM ν with an inverse form of M ν ,M ν = m 2 0 M −1 ν , can also give reasonable predictions, because, by taking the inverse of Y −1 e , we will investigate a neutrino mass matrix with a seesawtype where Y R and Y ν are given by 4) and Y ν ∝ Y e , respectively. In the model (1.2), the matrix M ν was for Dirac neutrinos, while the present M ν is for Majorana neutrinos. The mass matrix (1.2) could not provide a reasonable mass spectrum without adjusting the parameter ξ 0 , while, in this paper, we will give a small value of ∆m 2 21 /|∆m 2 32 | without the ξ 0 -term. Note that, in the present scenario, since the Dirac neutrino mass matrix Y ν is identical with the charged lepton mass matrix Y e , the nearly tribimaximal mixing originates in the structure of Y R .
In the next section, we will derive the neutrino mass matrix (1.3) with the form (1.4) of Y R by using SUSY vacuum conditions for an O(3) F and U(1) X invariant superpotential, and we will evaluate the mass matrix M ν by using the observed values of up-quark and charged lepton masses. However, in order to evaluate the neutrino mixing matrix, we must know the form of (1.3) on the e-basis, especially the form of Y 1/2 u on the e-basis. Therefore, in the present paper, we will put a phenomenological assumption on the relation between e-and d-bases. Then, we will find that, by using the observed Cabibbo-Kobayashi-Maskawa (CKM) matrix parameters, the mass matrix (1.3) with the form (1.4) can be accommodated to the observed nearly tribimaximal mixing. However, since the result is dependent on a phenomenological assumption on the form of Y u on the basis "e-basis", the mass matrix is still an empirical one. Nevertheless, we consider that the result is very suggestive.
In Sec.3, we will discuss the structure of Y d lightly. Finally, Sec.4 will be devoted to concluding remarks.

Neutrino mass matrix without a discrete symmetry
In order to give the operator Y 1/2 u in the expression (1.4), we introduce additional O(3) F 6 fields Φ u and X u with the U(1) X charges 1 2 q u and −q u , respectively. Then, we can write down the superpotential for the u-sector From SUSY vacuum conditions (for the moment, we regard W u as W ), we obtain From the condition (2.2), we obtain a bilinear relation does not always mean where D denotes that the matrix is on its diagonal basis. As we see later, we need the relation (2.7). Therefore, we assume the field Φ u (and also Y f ) is real, so that the matrix U u is orthogonal matrix. From the condition (2.3), we obtain Therefore, from the condition (2.4), we obtain ∂W Φu /∂Φ u = 0. We assume that three eigenvalues of Φ u can completely be determined by this condition ∂W Φu /∂Φ u = 0. However, for this purpose, the superpotential term W Φu will include U(1) X symmetry breaking terms. In this paper, we do not discuss the explicit form of W Φu . We assume that the VEV values are suitably given by Eq.(2.7) with the observed up-quark masses m ui . For convenience, for the e-sector, we also assume superpotential term W e similar to the u-sector: W e = λ e Tr[Φ e Φ e X e ] + m e Tr[Y e X e ] + W Φe (Φ e ), (2.9) where Φ e , X e and Y e have U(1) X charges 1 2 q e , −q e and q e , respectively, so that we obtain relations where we have again assumed that the field Φ e is real.
(Hereafter, for simplicity, we will sometimes express VEV matrices A as simply A.) In order to obtain the relation Y ν ∝ Y e , we assume the following structure of W ν : where φ ν and φ e are gauge-and flavor-singlet fields, and we assign U(1) X charges as Q X (X ν ) = x ν , Q X (φ ν ) = −(q ν + x ν ) and Q X (φ e ) = −(q e + x ν ). From ∂W/∂φ ν = 0 and ∂W/∂φ e = 0, we obtain X ν = 0. From ∂W/∂X ν = 0, we obtain Next, let us investigate a possible form of W R . In order to obtain the relation (1.4) from the phenomenological point of view, we assume the following form of W R : where we have assumed U(1) X charges Q X (Y R ) = −Q X (X R ) = 2q ν and 1 2 q u +q e −2q ν = 0, From SUSY vacuum conditions ∂W /∂Y R = 0, we obtain X R = 0. Then, the requirement ∂W /∂Y e = 0 leads to the condition ∂W e /∂Y e = 0, so that we obtain the relation (2.10). From ∂W /∂X R = 0, we obtain (2.14) Thus, we can obtain the desirable form (1.4) of Y R (without the ξ 0 -term). For convenience, let us define a name of a flavor basis as follows: when a VEV matrix Y f takes a diagonal form on a basis, we call the basis "f -basis", and we denote a form of a matrix A on the f -basis as A f . In order to obtain the neutrino mixing matrix form on the e-basis, we must know a matrix form of Φ u on the e-basis, i.e. Φ u e , although the form Φ u u on the u-basis is given by Eq.(2.7). Let us defined a transformation of a VEV matrix Y f from a b-basis to an a-basis as where U ab are unitary matrices, and they satisfy U † ab = U ba and U ab U bc = U ac . (These operators U ab are not always members of O (3) is the standard expression of CKM matrix. The simplest assumption is to consider that the d-basis is identical with the e-basis, so that we can regard U ue as U ue = V because U ud = V . Then, we can evaluate the neutrino mass matrix (1.3) with Y R e ∝ Φ u e Y e e + Y e e Φ u e by using the form

The relation (2.5) is valid only on a basis which is transformed from the u-basis by an orthogonal transformation.)
In the numerical calculation of M ν , we adopt the standard phase convention V (δ) [8] of the CKM matrix, and use the following input values: the up-quark masses [7] at the energy scale µ = M Z , m u1 = 0.00127 GeV, m u2 = 0.619 GeV, m u3 = 171.7 GeV, and the CKM parameters [8], |V us | = 0.2257, |V cb | = 0.0416, |V ub | = 0.00431. (Here, we have used the quark mass values at µ = M Z because we have used the CKM parameter values at µ = M Z . For the energy scale dependency of the mass ratios and CKM parameters, for example, see Ref. [9].) As seen in Table  1, the results are dependent on the CP violating phase parameter δ. The present experimental data [8] on the CKM matrix favor δ ≃ π/3. However, as seen in Table 1, the predicted value of sin 2 2θ 23 at δ ≃ π/3 is in poor agreement with the observed value sin 2 2θ 23 = 1.00 −0.13 [10], although the predicted value of tan 2 θ 12 is roughly in agreement with the observed value  [11]. Therefore, we cannot regard that the e-basis is identical with the d-basis.
However, as seen in Table 1, note that the case with δ ≥ 2π/3 can give a nearly tribimaximal mixing. Especially, we notice that the case δ = π highly realizes the tribimaximal mixing. Since we have assumed that Φ u and Φ e (therefore, Y u and Y e ) are real matrices, so that the u-and e-bases are connected each other by an orthogonal transformation. This guarantees to use the relations (2.5), (2.12) and (2.14), which are obtained from the O(3)-invariant superpotential, on the u-and e-bases. If we still suppose U ue ≃ U ud = V (δ), the possible candidates of the orthogonal matrix U ue will be U ue = V (0) and/or U ue = V (π). (Indeed, we can show [5] U ed = U † ue U ud = V † (δ ue )V (δ) = 1+O(|V ub |), so that we can still consider U ue ≃ U ud ). Therefore, the case U ue = V (π) is likely. However, in this paper, we a priori assume the form U ue = V (δ ue ) with δ ue ≥ 2π/3 as a phenomenological requirement suggested in Table 1. Again, we summarize our phenomenological neutrino mass matrix which can lead to a nearly tribimaximal mixing for |δ ue | ≥ 2π/3 as follows: (For the phenomenological reason why the mass matrix (2.17) can give a nearly tribimaximal mixing, see Ref. [5].) As seen in Table 1, the predicted value of R = ∆m 2 21 /∆m 2 32 is considerably small compared to the observed value |R| = 0.028 ± 0.004, where we have used the observed values ∆m 2 21 = (7.59 ± 0.21) × 10 −5 eV 2 [11] and |∆m 2 32 | = (2.74 +0.44 −0.26 ) × 10 −3 eV 2 [10]. The value R can be adjusted by taking the ξ 0 -term in Eq.(1.4) into consideration. (It is easy to bring the ξ 0 -term into the present model.) However, the smallness of ∆m 2 21 can also become mild by considering the renormalization group equation (RGE) effects. Since, so far, we have not fixed the energy scale Λ, the values without RGE effects have been listed in Table 1. The RGE effects will be able to give a reasonable value of R without the ξ 0 term. By the way, the present neutrino masses are normal hierarchical, so that, if regard m ν3 as m ν3 = ∆m 2 32 = 0.0523 eV, we can obtain neutrino masses m ν1 = 0.78 meV, m ν2 = 1.52 meV and m ν3 = 52.3 meV for the case δ ue = π.

Down-quark sector
So far, we did not discuss a structure of W d . Although the purpose of the present paper is not to give a structure of W d , here, let us discuss a possible structure of W d lightly. As we have assumed that the fields Φ u and Φ e are real and since we know that the CP is broken in the quark sector, we must consider that Y d is complex. By way of trial, let us assume the following superpotential W d : Since we consider Φ u and Φ X are real, the factor e iα in Eq. (3.1) (3.1) is merely a phenomenological assumption, and the form (3.1) is not a general form. Also note that the cubic term detΦ X breaks the U(1) X symmetry. From the condition ∂W /∂Φ X = 0, we obtain Since we have already taken ∂W u /∂Φ u = 0 in Eq. (2.4), we obtain X d = 0 for Φ X = 0 from the condition ∂W /∂Φ u = λ du (Tr[Φ X ]X d + e iα Tr[Φ X X d ]1) + ∂W u /∂Φ u = 0. Then, from the condition ∂W /∂Φ X = 0, we obtain where we have assumed e iα ≃ 1 and (x 2 /x 3 ) 2 ≪ u u2 /m u3 . The observed values are m s /m b ≃ 0.019 and m c /m t ≃ 0.060 at µ = M Z [7], so that the relation (3.5) is in roughly agreement with the observed value. (We can adjust the predicted value to the observed value by taking a suitable choice of x i and α.) Also we can obtain m d /m s ≃ m u /m c , but the result is sensitive to the values of x i /x j and α, so that we do not discuss no more details of m di /m dj in this paper.
In conclusion, we have proposed a new approach to the masses and mixings of quarks and leptons. In the new approach, we write a superpotential W for O(3)-flavor 1 + 5 fields Y f whose VEVs give effective Yukawa coupling constants (Y f ) ij /Λ and we obtain relations among masses and mixings from the SUSY vacuum conditions. In this approach, we cannot predict the absolute values of the masses and mixings, but we can obtain relations among the VEV matrices Y f and Y R . Under this approach, we have found an empirical neutrino mass matrix (2.17). The form (2.17) was found as a byproduct when we assumed that Y e and Y d can simultaneously be diagonalized. Regrettably, the idea U ue = U ud = V (δ) with δ ≃ π/3 was failed to explain the observed fact sin 2 2θ 23 ≃ 1, but we have found that U ue = V (δ ue ) with δ ue ≥ 2π/3 can successfully give the observed values sin 2 2θ 23 ≃ 1 and tan 2 θ 12 ≃ 0.5. At present, there is no theoretical reason for the form U ue = V (δ ue ). (Since we have assumed that Φ u and Φ e are real, U ue must be real.) Nevertheless, it is worthwhile noticing because the form is of a new type which is related to the up-quark masses and which successfully leads to the nearly tribimaximal mixing without assuming any discrete symmetry. (However, we do not consider that this denies applicability of a discrete symmetry to the neutrino sector. Rather, we consider that this suggests that the discrete symmetry is applicable not only to the lepton sector, but also to the quark sector.) Since the present approach is still in its beginning stage, we have many tasks to investigate: for example, (i) investigation of the explicit structures of W Φu (Φ u ) and W Φe (Φ e ), which completely determine the eigenvalues of Φ u and Φ e , i.e. ( √ m u , √ m c , √ m t ) and ( √ m e , √ m µ , √ m τ ), respectively; (ii) investigation of the explicit structure of W d in order to give more definite quark mass relations and CKM matrix parameters (Y d in this paper has included free parameters x i , so that we cannot derive definite conclusions because we can adjust the parameters x i to the observed values freely); (iii) investigation of symmetries and quantum number assignments which can uniquely derive the present specific (phenomenological) form of W . In the present scenario, most of the fields Φ u , Φ e , Y f (f = u, d, e), Y R , and so on, take VEV of the order of Λ, and their masses are also of the order Λ. However, some components of those fields are massless in the SUSY limit, and, under the SUSY breaking at µ ∼ 1 TeV, they have masses of the order µ ∼ 1 TeV. Since those particles are gauge singlets, in principle, they are harmless in the low energy phenomenology. However, in TeV region physics, we may expect fruitful phenomenology about flavor-mediated (but gauge-singlet) processes. This approach will shed a new light on the quark and lepton masses and mixings and on a TeV scale flavor physics.