Neutrino Masses Induced by R-Parity Violation in a SUSY SU(5) Model with Additional \bar{5}'_L+5'_L

Within the framework of an SU(5) SUSY GUT model, a possible general form of the neutrino mass matrix induced by R-parity violation is investigated. The model has matter fields \bar{5}'_{L}+5'_{L} in addition to the ordinary matter fields \bar{5}_{L}+10_{L} and Higgs fields H_u+\bar{H}_d. The R-parity violating terms are given by \bar{5}_{L} \bar{5}_{L} 10_{L}, while the Yukawa interactions are given by \bar{H}_d \bar{5}'_{L} 10_{L}. Since the matter fields \bar{5}'_L and \bar{5}_L are different from each other at the unification scale, the R-parity violation effects at a low energy scale appear only through the \bar{5}'_L \leftrightarrow \bar{5}_{L} mixings. In order to make this R-parity violation effect harmless for proton decay, a discrete symmetry Z_3 and a triplet-doublet splitting mechanism analogous to that in the 5-plet Higgs fields are assumed.


Introduction
As an origin of the neutrino masses, the idea of the radiative neutrino mass [1] is very interesting as well as the idea of the neutrino seesaw mechanism [2]. However, currently, the latter idea is influential, because it is hard to embed the former model into a grand unification theory (GUT). For example, a supersymmetric (SUSY) model with R-parity violation can provide radiative neutrino masses [3], but the model cannot be embedded into GUT, because the R-parity violating terms induce proton decay inevitably [4].
Recently, the author [5] has proposed a model with R-parity violation within the framework of an SU(5) SUSY GUT: we have quark and lepton fields 5 L +10 L , which contribute to the Yukawa interactions as H u 10 L 10 L and H d 5 L 10 L ; we also have additional matter fields 5 ′ L + 5 ′ L which contribute to the Rparity violating terms 5 ′ L 5 ′ L 10 L . Since the two 5 L and 5 ′ L are different from each other, the R-parity violating interactions are usually invisible. The R-parity violating effects become visible only through 5 L ↔ 5 ′ L mixings in low energy phenomena. In the previous model [5], a discrete symmetry Z 3 has been assumed, and their quantum numbers have been assigned as 5 L(−) + 10 L(+) + 5 ′ L(+) + 5 ′ L(+) and H d(0) + H u(+) , where we have denoted fields with the transformation properties Ψ → ω +1 Ψ, Ψ → ω 0 Ψ and Ψ → ω −1 Ψ (ω = e i2π/3 ) as Ψ (+) , Ψ (0) and Ψ (−) , respectively. Therefore, in the set 5 L + 10 L , the fields 5 L(−) and 10 L(+) have different transformation properties each other. In contrast to the previous model, in the present paper, we will propose a model with alternative assignments Although the mechanism of the harmless R-parity violation is the same as the previous model, since the Z 3 quantum number assignment is different from the previous one, the structure of the model is completely different from the previous one.
In the present paper, we will investigate not only the radiatively-induced neutrino masses, but also the contributions from the vacuum expectation values (VEV) of the sneutrinos, ν , although in the previous paper the estimate of ν was merely based on an optimistic speculation.

Harmless R-parity violation mechanism
Under the Z 3 quantum number assignment (1.1), the Z 3 invariant tri-linear terms in the superpotential are only the following three terms: ( 2.1) Similarly, the Z 3 invariant bi-linear terms are only two: H d(−) H u(+) and 5 L(0) H L(0) . In order to give doublet-triplet splitting, we assume the following "effective" bi-linear terms where Φ (0) is a 24-plet Higgs field with the VEV Φ (0) = v 24 diag (2, 2, 2, −3, −3), so that, for example, L denote doublet and triplet components of the fields 5 L , respectively) are given by (2. 3) The last term in Eq. (2.2) has been added in order to break the Z 3 symmetry softly. We define the 5 L ↔ 5 ′ L mixing as follows: Then, the effective R-parity violating terms at µ < M GUT are given by (2.7) In order to suppress the unwelcome term d c R d c R u c R in the effective R-parity violating terms (2.7), we assume a fine tuning Note that in the present model the observed down-quarks d c On the other hand, radiative neutrino masses are generated by the R-parity violating term (e L ν L − ν L e L )e c R with a factor s (2) s (2) ≃ 1. The up-quark masses are generated by the Yukawa interactions (2.1), so that we obtain the up-quark mass matrix M u as (M u We also obtain the down-quark mass matrix M d and charged lepton mass matrix M e as where where based on a mixing between two 5 L has been discussed, for example, by Bando and Kugo [6] in the context of an E 6 model.) 3 General form of the neutrino mass matrix First, we investigate a possible form of the radiatively-induced neutrino mass matrix M rad . In the present model, since we do not have a term which induces e + R ↔ H + d(−) mixing, there is no Zee-type diagram [1], which is proportional to the Yukawa vertex (Y d ) ij and R-parity violating vertex λ ijk .
Only the radiative neutrino masses in the present scenario come from a charged-lepton loop diagram: the radiative diagram with (ν L ) j → (e R ) l + ( e c L ) n and (e L ) k + ( e c L ) m → (ν c L ) i . The contributions (M rad ) ij from the charged lepton loop are given, except for the common factors, as follows: where s i = s (2) i , and M e and M 2 eLR are charged-lepton and charged-slepton-LR mass matrices, respectively. (In the present paper, we define the charged lepton mass matrix M e and the neutrino mass matrix M ν as e L M e e R and ν L M ν ν c L , respectively, so that the complex conjugate quantities λ * ijk and so on have appeared in the expression (3.1) Since the coefficient λ ijk is antisymmetric in the permutation i ↔ j, it is useful to define and where S = diag(s 1 , s 2 , s 3 ). Then, the radiative neutrino mass matrix is given by The coefficient m −1 0 is calculated from one-loop diagram (Fig.1) as where Next, let us investigate the contributions from the VEVs of sneutrinos ν i . In general, the sneutrinos ν i can have VEVs v i ≡ ν i = 0 [7], if there are one or more of the following terms: µ i 5 Li H u in superpotential W , and B i 5 Li H u + m 2 HLi 5 LiH † d in the bilinear soft SUSY breaking terms V sof t . In the present model, there is no such a term at tree level, because these terms are forbidden by the Z 3 symmetry.  (Fig. 2). The contribution m 2 HLi is proportional to On the other hand, the contribution M V EV from ν i = 0 to the neutrino mass matrix is proportional to and v i ≡ ν i are proportional to the values (m 2 HLi ) * , so that the mass matrix M V EV is given by where ξ is a relative ratio of M V EV to M rad . In conclusion, the neutrino mass matrix M ν in the present model is given by the form i.e.

General features of the neutrino mass matrix
In the present model, if the charged lepton mass matrix M e and the structure of λ ijk (i.e. L ij ) are given, then we can obtain K = (SM e L T ) * , so that we can predict neutrino masses and mixings. However, at present, we have many unknown parameters, so that in order to give explicit predictions of the neutrino masses and mixings, we must put a further assumption on the parameters K ij . In the present section, we investigate general features of the neutrino mass matrix (3.11) [or (3.12)] without making any explicit assumptions about flavor symmetries. So far, the expression of M ν , (3.12), has been given in the initial under a rotation of the flavor basis the matrix K transforms as We have a great interest in the form of M ′ ν in the flavor basis with M ′ e = D e ≡ diag(m e , m µ , m τ ). Hereafter, we denote the quantities M ′ ν , K ′ , and so on in the M ′ e = D e basis as M ν , K and so on, respectively. The matrix K is expressed as 5) where U 5 = U e L and U 10 = U e R , and we have put S ≃ 1 because of S ≃ 1 as we have assumed in Eq. (2.9). Here, let us summarize general features of the present neutrino mass matrix (3.12). (i) If the matrix K defined by Eq. (3.4) satisfies K T = K in the initial basis, the matrix K ′ in the arbitrary basis also satisfies K ′T = K ′ , so that the present model gives ν ′ i = 0 in the arbitrary basis. For such a case, the neutrino mass matrix is simply given by (ii) When K is symmetric under the flavor 2 ↔ 3 permutation, the neutrino mass matrix M ν is also symmetric under the 2 ↔ 3 permutation. It is well-known [8] that when the neutrino mass matrix M ν is symmetric under the 2 ↔ 3 permutation, the mass matrix M ν gives a nearly bimaximal mixing, i.e. sin 2 2θ 23 = 1 and |U 13 | 2 = 0, which are favorable to the observed atmospheric [9], K2K [10] and CHOOZ [11] data. In the present model, the 2 ↔ 3 symmetry of M ν means that the parameters are symmetric under the 2 ↔ 3 permutation. In other words, the 2 ↔ 3 symmetry of M ν is due to special structures of U e L and K. For example, when K and U e L are given by the textures the matrix K is 2 ↔ 3 symmetric: (4.10) so that the neutrino mass matrix M ν is also 2 ↔ 3 symmetric: and K in the initial basis is given by Finally, let us show a simple example which is suggested by above comments (i) and (ii). We assume that M e M † e on the initial basis is 2 ↔ 3 symmetric: (4.13) so that U e L has a form of a nearly bimaximal mixing. For simplicity, we assume that U e L is given by the full bimaximal mixing form 14) which demands the constraint F = B+G on the matrix (4.13). Then, the eigenvalues D 2 e = diag(m 2 e , m 2 µ , m 2 τ ) are given by On the other hand, we assume that K in the initial basis is given by the form (4.8) with K 23 = K 32 , so that we obtain a = a ′ and (4.16) Note that the mass matrix (4.16) does not include the contributions (ξ-terms) from nonvanishing sneutrino VEVs because of K T = K. The mass matrix (4.16) gives the following eigenvalues and mixings: so that we obtain together with sin 2 2θ atm = 1 and |U 13 | 2 = 0. For a further simple case with f = 0, which demands we obtain m ν1 = m ν2 /2 = 2(g 2 − b 2 )m −1 0 , so that where we have considered The result (4.22) is favorable to the recent solar [12] and KamLAND data [13]. By using the best fit values ∆m 2 solar = 7.2 × 10 −5 eV 2 [12,13] and ∆m 2 atm = 2.4 × 10 −3 eV 2 [9,10], we obtain where R = ∆m 2 solar /∆m 2 atm , and m ν1 = 0.0049 eV, m ν2 = 0.0098 eV, m ν3 = 0.050 eV, (4.26) where we have used the relation m ν1 /m ν2 = 1/2 and ∆m 2 atm = 3m 2 ν2 /4. Of course, this is only an example, and the result (4.22) is not a prediction which is inevitably driven from the general form of M ν .

Summary
In conclusion, within the framework of a SUSY GUT model, we have proposed an R-parity violation mechanism which is harmless for proton decay and investigated a general form of the neutrino mass matrix M ν . As we have given in Eq. (3.12), the form of M ν is described in terms of the matrix K defined in Eq. (3.4). (i) If K T = K, the VEVs of sneutrinos are exactly zero, ν i = 0, in the arbitrary basis, so that M ν is given only by the radiative contributions. (ii) If K is 2 ↔ 3 symmetric, then M ν is also 2 ↔ 3 symmetric, so that M ν can predict sin 2 2θ atm = 1 and |U 13 | 2 = 0.
In order to demonstrate that the general form indeed has a phenomenologically favorable parameter range, we have given a simple example of K and M e M † e in the last part of the section 4. Although such a simple form of K, (4.8), with the constraint (4.23) is likely, the investigation of the origin of the possible form K will be our next task. The purpose of the present paper is not to give a special model for neutrino phenomenology, and it is to demonstrate that it is indeed possible to build a neutrino mass matrix model with R-parity violation, i.e. without a seesaw mechanism, even if the model is within a framework of GUT.
The present model has assigned Z 3 quantum numbers to the superfields differently from those in the previous model [5] with 5 L ↔ 5 ′ L mixing: we have been able to assign the same Z 3 quantum number to the matter fields 5 L and 10 L (and also to 5 ′ L and 5 ′ L ). This re-assignment will give fruitful potentiality for a further extension of the present model.