Radiative Neutrino Masses in a SUSY GUT Model

Radiatively-induced neutrino mass matrix is investigated within the framework of an SU(5) SUSY GUT model. The model has matter fields of three families \bar{5}_{L(+)i}+5_{L(+)i} in addition to the ordinary matter fields \bar{5}_{L(-)i}+10_{L(+)i} and Higgs fields H_{(+)}+\bar{H}_{(0)}, where (+,0,-) denote the transformation properties (\omega^{+1},\omega^0,\omega^{-1}) (\omega^3=+1) under a discrete symmetry Z_3. R-parity violating terms are given by \bar{5}_{L(+)} \bar{5}_{L(+)} 10_{L(+)}, while the Yukawa interactions are given by \bar{H}_{(0)} \bar{5}_{L(-)} 10_{L(+)}, i.e. the \bar{5}-fields in both are different from each other. The Z_3 symmetry is only broken by the terms \bar{5}_{L(+)i}5_{L(+)i} softly, so that the \bar{5}_{L(+)i}\leftrightarrow \bar{5}_{L(-)i} mixings appear at \mu<M_X. Of the R-parity violating terms \bar{5}_{L(+)} \bar{5}_{L(+)} 10_{L(+)}, only the terms (e_L\nu_-\nu_Le_)e_R^c sizably appear at \mu<M_X.


Introduction
The idea of the radiative neutrino mass [1] is an antithesis to the idea of the neutrino seesaw mechanism [2]: in the latter model, the neutrinos acquire Dirac masses of the same order as quark and charged lepton masses and the smallness of the observed neutrino masses is explained by the seesaw mechanism due to large Majorana masses of the right-handed neutrinos ν R , while, in the former model, there are no right-handed neutrinos, so that there are no Dirac mass terms, and small Majorana neutrino masses are generated radiatively. Currently, the latter idea is influential, because it is hard to embed the former model into a grand unification theory (GUT). A supersymmetric (SUSY) model with R-parity violation can provide neutrino masses [3], but the model cannot be embedded into GUT, because the R-parity violating terms induce proton decay inevitably [4].
Recently, Sato and the author [5] have proposed a model with R-parity violation within the framework of an SU(5) SUSY GUT. In the model, there are no R-parity violating terms 5 L 5 L 10 L (5 L and 10 L denote 5 L -plet and 10-plet matter fields in SU(5) SUSY GUT), which are forbidden by a discrete symmetry Z 2 . At µ < M X (M X is a unification scale of the SU(5) GUT), the Z 2 symmetry is softly broken, and H d ↔ 5 L mixing is induced, so that the R-parity violation terms 5 ′ L 5 ′ L 10 L are effectively induced from the Yukawa interactions H d 5 L 10 L . Although the model is very interesting as an R-parity violation mechanism, it is too restricted for neutrino mass matrix phenomenology, because the coefficients λ of 5 In contrast to the above scenario, in the present paper, we propose another 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 R-parity 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 order to make such a scenario, i.e. in order to allow the interactions 5 ′ L 5 ′ L 10 L , but to forbid 5 L 5 L 10 L and 5 L 5 ′ L 10 L , we introduce a discrete symmetry Z 3 . (We cannot build such a model by using Z 2 symmetry.) We denote fields with the transformation properties Ψ → ω +1 Ψ, Ψ → ω 0 Ψ and Ψ → ω −1 Ψ (ω 3 = +1) as Ψ (+) , Ψ (0) and Ψ (−) , respectively. We consider matter fields 5 L(−)i + 10 L(+)i (i = 1, 2, 3: family indices) which contribute the Yukawa interactions as 1) and additional matter fields 5 L(+)i + 5 L(+)i which contribute the R-parity interactions as The R-parity violating interactions 5 L(−) 5 L(−) 10 L((+) and 5 L(−) 5 L(+) 10 L((+) are forbidden by the Z 3 symmetry. In order to give 5 L(−) ↔ 5 L(+) mixings, where s i = sin θ i and c i = cos θ i , we consider a superpotential where Φ (0) is a 24-plet Higgs field with the vacuum expectation value (VEV) Φ (0) = v 24 diag(2, 2, 2, −3, −3), which gives doublet-triplet splitting in the mass terms 5 L(−)i 5 L(+)i at µ < M X , i.e.
The discrete symmetry Z 3 is softly broken by the M SB i -terms in (1.4). Then, we obtain where the index (a) denotes that the field Ψ (a) with a = 2 (a = 3) is a doublet (triplet) component of SU ( (1.7) The field 5

(a)
Li are massless. We regard 5 Li + 10 L(+)i as the observed quarks and leptons at low energy scale (µ < M X ). Then, the effective R-parity violating terms at µ < M X are given by (1.8) In order to suppress the unwelcome term d c R d c R u c R in the effective R-parity violating terms (1.8), we assume a fine tuning 9) where m SUSY denotes the SUSY breaking scale (m SUSY ∼ 1 TeV) and M SB i are difined by (1.4) (i.e. the Z 3 symmetry breaking terms are given by On the other hand, radiative neutrino masses are generated by the R-parity violating term (1.11) Since m(5

′(3)
Li can basically contribute to the renormalization group equation (RGE) effects at µ < M X . However, since we consider M SB i ∼ M X × 10 −1 , the numerical effect does almost not spoil the gauge-coupling-constant unification at µ = M X ∼ 10 16 GeV.
The up-quark masses are generated by the Yukawa interactions (1.1), so that we obtain the up-quark 12) where v u = H 0 (+) . From the Yukawa interaction (1.1), we also obtain the down-quark mass matrix M d and charged lepton mass matrix M e as i.e. where can be different from each other. (The idea M T d = M e 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.) In order to give doublet-triplet splitting for the Higgs fields H (+) and H (0) , we assume the "missing partner mechanism" [7]: for example, we consider  (0) are 50-plet and 75-plet Higgs fields, respectively.
In this section, we investigate a possible form of the radiatively-induced neutrino mass matrix M rad . Contribution from non-zero VEVs of sneutrinos ν = 0 to the neutrino mass matrix will be discussed in the next section. In the present model, since we do not have a term which induces e + R ↔ H + (0) 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 . (The e + R ↔ H + (0) mixing can come from interactions of a type H H 10 L(+) . However, in the present model, 5-plet Higgs fields are only on type H (0) . Therefore, the combination H (0) H (0) 10 L(+) is forbidden because of the antisymmetric property of SU (5) 10-plet fields 10 L(+) . Even after the SU (5) is broken, H (0) cannot couple to the SU(2) singlet e + R because SU (2) singlet composed of 2 × 2 must be antisymmetric. Therefore, we cannot bring the H (0) H (0) e + R term even as a soft supersymmetry breaking term.) 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 where Since the coefficient λ ijk is antisymmetric in the permutation i ↔ j, it is useful to define and Then, we can rewrite (2.4) as The expression (2.8) is explicitly given as follows: where M ij ≡ (M rad ) ij and we have dropped a common factor m rad 0 /m 2 3 . As discussed in (1.10), in a phenomenological investigation in the next section, we will take s 1 = s 2 = s 3 = 1 for simplicity.

Phenomenology
In general, the sneutrinos ν i can have VEVs v i ≡ ν i = 0 [8]. Since the mass matrix for (ν 1 , ν 2 , ν 3 , W 0 ) (except for the radiative masses) is given by where, for simplicity, we have dropped the elements for B 0 , the contribution Mν from ν i = 0 to the neutrino masses is expressed by under the seesaw approximation. Note that the matrix M ν is a rank-1 matrix. Therefore, in the present model, the neutrino mass matrix M ν is given by We have estimated the absolute magnitudes of the radiative masses in (2.3)-(2.5). On the other hand, it is hard to estimate the absolute values of ν i , because, in the present model, there is neither a term corresponding to the so-called "µ-term" µH d H u nor 5 L(−)i ↔ H (0) mixing terms, so that the sneutrinos ν i cannot have the VEVs ν i at the tree level. The non-zero VEVs appears only through the renormalization group equation (RGE) effect [9]. The contribution highly depends on an explicit model of the SUSY breaking. Therefore, in the present paper, we will deal with the relative ratio of the contributions Mν to M rad as a free parameter.
The recent neutrino data [10,11,12] have indicated that sin 2 2θ atm ≃ 1 and tan 2 θ solar ≃ 0.5. In response to these observations, He and Zee have found a phenomenological neutrino mass matrix [13] (3.4) which leads to a nearly bimaximal mixing i.e. sin 2 2θ atm = 1 , (3.6) (Although He and Zee gave the mass matrix (3.4) with x = 0 in Ref. [13], since a term which is proportional to a unit matrix does not affect the mixing matrix form, the most general form of the He-Zee mass matrix is given by (3.4).) The mass matrix (3.4) gives the following mass eigenvalues: . (3.11) (Therefore, the parameter y has to be y = −1 and y = 1 + x.) In the present model, there are many adjustable parameters for the neutrino mass matrix phenomenology. Let us seek for an example with simple and plausible forms of M rad and Mν with a clue of the successful He-Zee mass matrix form. First, we think that it is likely that the VEVs ν i are democratic on the basis on which the charged lepton mass matrix is diagonal, i.e.
so that we can regard the second term in the He-Zee matrix (3.4) as Mν which originates in the sneutrino VEVs. Then, it is interesting whether our radiative mass matrix (2.8) can give the first term in the He-Zee mass matrix (3.4) or not.
Corresponding to the assumption (3.12), we may also suppose that the coefficients h ij are invariant under the permutation among ℓ Li = (ν Li , e Li ) which belong to 5 Li (not among e c Ri which belong to 10 Li ). The most simple case will be (3.13) Then, we obtain the radiative neutrino mass matrix which corresponds the first term in the He-Zee mass matrix (3.4) with x = −2 and y = −2, so that we get From the best fit values of ∆m 2 ij [10,11,12], R obs = 6.9 × 10 −5 eV 2 2.5 × 10 −3 eV 2 = 2.76 × 10 −2 , we obtain and m ν1 = 0 , m ν2 = 0.0083 eV , m ν3 = 0.050 eV , (m 0 = 0.025 eV , (3.20) where we have used the best fit values [10,11,12] ∆m 2 atm = 2.5 × 10 −3 eV 2 and ∆m 2 solar = 6.9 × 10 −5 eV 2 . In the present model (3.14), the absolute magnitude of m ν3 , which is radiatively generated, is given by whereÂ andF are numerical values of the parameters A and F in unit of TeV, which are defined by M 2 eLR = AM e and the equation (2.5), respectively. If we, for example, take A ≃ 1 TeV andm 2 eL ≃ m 2 eR ≃ 0.5 TeV, we obtain m ν3 ≃ 0.075λ 2 eV. Thus, roughly speaking, the choice m SUSY ∼ 1 TeV and λ ∼ 1 can give a reasonable magnitude of m ν3 .
In conclusion, we have proposed a model with R-parity violation within the framework of an SU(5) SUSY GUT. In the model, we have matter fields 10 L(+) + 5 L(−) + 5 L(+) + 5 L(+) and Higgs fields H (+) and H (0) , where (+, 0, −) denote their transformation properties (ω +1 , ω 0 , ω −1 ) under a discrete symmetry Z 3 , respectively. Although 5 L(−) 5 L(+) acquires a heavy mass M 5 at µ = M X , the effective masses of the triplet and doublet components 5  In the present model, the radiative neutrino masses are generated only through the charged lepton loop. The general radiative mass matrix form M rad is given by the expression (2.8) [(2.9)-(2.14)]. If there are contributions M ν from VEVs of the sneutrinos ν = 0 to the neutrino mass matrix M ν with suitable magnitudes relative to M rad , especially, with a democratic form (3.12), we can obtain the He-Zee neutrino mass matrix form (3.4), which leads to a nearly bimaximal mixing with sin 2 2θ atm = 1 and tan 2 θ solar = 1/2. Of course, this is only an example of the explicit mass matrix form and the He-Zee matrix with forms (3.12) and (3.14) are not a logical consequence of the present model. We have to assume something of an anzatz for a flavor symmetry. Maybe, a more plausible ansatz for the flavor symmetry will give a more elegant mass matrix form which gives beautiful explanations for the observed neutrino and lepton-flavor-violation phenomena. Search for such a flavor symmetry is one of our future tasks.
In the present paper, we did not discuss the quark and charged lepton mass matrices. In the present model, the down-quark mass matrix M d is related to the charged lepton mass matrix M e as M T d = CM e with C = 1. Investigation of a possible structure of C is also a future task in the model.
In conclusion, the present model will bring fruitful results not only in phenomenology, but also in a theoretical extension.