Large Solar Neutrino Mixing and Radiative Neutrino Mechanism

We find that the presence of a global $L_e-L_\mu-L_\tau$ ($\equiv L^\prime$) symmetry and an $S_2$ permutation symmetry for the $\mu$- and $\tau$-families supplemented by a discrete $Z_4$ symmetry naturally leads to almost maximal atmospheric neutrino mixing and large solar neutrino mixing, which arise, respectively, from type II seesaw mechanism initiated by an $S_2$-symmetric triplet Higgs scalar $s$ with $L^\prime=2$ and from radiative mechanism of the Zee type initiated by two singly charged scalars, an $S_2$-symmetric $h^+$ with $L^\prime=0$ and an $S_2$-antisymmetric $h^{\prime +}$ with $L^\prime=2$. The almost maximal mixing for atmospheric neutrinos is explained by the appearance of the democratic coupling of $s$ to neutrinos ensured by $S_2$ and $Z_4$ while the large mixing for solar neutrinos is explained by the similarity of $h^+$- and $h^{\prime +}$-couplings described by $f^h_+\sim f^h_-$ and $\mu_+\sim\mu_-$, where $f^h_+$ ($f^h_-$) and $\mu_+$ ($\mu_-$) stand for $h^+$ ($h^{\prime +}$)-couplings, respectively, to leptons and to Higgs scalars.

Neutrino oscillations have been long recognized to occur if neutrinos have masses [1]. The experimental confirmation of such neutrino oscillations has been given by the Super-Kamiokande collaboration [2] for atmospheric neutrinos and the clear evidence of the solar neutrino oscillations has been released by the SNO collaboration [3]. These observed oscillation phenomena can be explained by the mixings between ν e and ν µ with ∆m 2 ⊙ < ∼ 10 −4 eV 2 for solar neutrinos and between ν µ and ν τ with ∆m 2 atm ∼ 3 × 10 −3 eV 2 for atmospheric neutrinos [4]. Their masses are implied to be as small as O(10 −2 ) eV and the smallness of neutrino masses can be explained by either the seesaw mechanism [5] or the radiative mechanism [6,7]. The mixing specific to atmospheric neutrinos is found to prefer maximal mixing [8]. It has also been suggested for solar neutrinos that solutions with large mixing angles are favored while solutions with small mixing angles are disfavored [9]. Therefore, both neutrino oscillations are characterized by large neutrino mixings.
One of the promising theoretical assumptions to account for the observed mixing pattern is to use the bimaximal mixing scheme [10,11]. The radiative mechanism of the Zee-type [6] provides the natural explanation on bimaximal neutrino mixing [12] when combined with a global L e − L µ − L τ (≡ L ′ ) symmetry [13,14] since the Zee model only supplies flavor-off-diagonal mass terms. However, the recent extensive analyses on solar neutrino oscillation data imply that the maximal solar neutrino mixing is not well compatible with the data, which prefer sin 2 2θ 12 ∼ 0.8 for the large mixing angle (LMA) MSW solution [15]. If this observed tendency of solar neutrino oscillations with large mixing but not with maximal mixing is really confirmed, the bimaximal structure in the Zee model should be modified [16].
In this report, we discuss a possible modification of the Zee model with the L ′ symmetry to accommodate the LMA solution without the maximal solar neutrino mixing [17]. The original Zee model requires the presence of a Higgs scalar of φ ′ , the duplicate of the standard Higgs scalar φ, which initiates radiative neutrino mechanism together with a singly charged scalar of h + , and assumes φ ′ to couple to no leptons. One of the modifications is to relax this constraint such that φ ′ couples to leptons. By allowing φ ′ to generate lepton masses, the authors of Ref. [18] have found that solar neutrinos can exhibit sin 2 2θ 12 ∼ 0.8 but their realization of the large solar neutrino mixing entails various fine-tunings, which seem unnatural. We, instead, rely upon a certain underlying symmetry to constrain the interactions of φ ′ with leptons and utilize an S 2 permutation symmetry for the µ-and τ -families, which is responsible for the appearance of the almost maximal atmospheric neutrino mixing [19]. Under S 2 , φ transforms as a symmetric state and φ ′ transforms as an antisymmetric state.
In order to realize the large mixing, the natural resolution is to include flavor-diagonal mass terms because the main source of sin 2 2θ 12 ≈ 1 in the Zee model comes from the constraint on neutrino masses of m 1,2,3 given by m 1 + m 2 + m 3 =0 specific to flavor-off-diagonal mass terms. It is known that flavor-diagonal mass terms can be supplied by an SU (2) L -triplet Higgs scalar [20] denoted by whose vacuum expectation value (VEV), 0|s 0 |0 , generates neutrino masses via interactions of ψ c L sψ L , where the subscript c denotes the charge conjugation including the G-parity of SU (2) L . The smallness of the neutrino masses can be ascribed to that of 0|s 0 |0 , which is given by ∼ µ( 0|φ|0 /m s ) 2 produced by the combined effects of µφ † sφ c and m 2 s Tr(s † s), where µ and m s are mass parameters. The type II seesaw mechanism [21] can ensure tiny neutrino masses by the dynamical requirement of | 0|φ|0 | ≪ m s with µ ∼ m s .
To see which masses of flavor neutrinos give contributions to yield sin 2 2θ 12 = 1, we examine a possible neutrino mass texture that can be diagonalized by U MN S with two mixing angles, θ 12 and θ 23 , which, respectively, connect (ν 1 , ν 2 ) with (ν e , ν µ ) and (ν 2 , ν 3 ) with (ν µ , ν τ ), where (ν 1 , ν 2 , ν 3 ) T (= |ν mass ) with (m 1 , m 2 , m 3 ) and (ν e , ν µ , ν τ ) T (=|ν weak ) are related by |ν weak = U MN S |ν mass . The resulting mass matrix denoted by M ν takes the form of where the atmospheric neutrino mixing is specified by t 23 = sin θ 23 / cos θ 23 [18,19,22]. The masses and the solar neutrino mixing angle of θ 12 are calculated to be: where c = −t 23 b and |m 1 | < |m 2 | is always maintained by adjusting the sign of η (= ±1). The result shows that the significant deviation of sin 2 2θ 12 from unity is only possible if (a − d + t 23 e) 2 = O(b 2 + c 2 ). In our subsequent discussions, we take the "ideal" solution [23] with t 23 = ±1 (≡ σ) given by which provides m 1 = m 2 = 0 and m 3 = 2d. The deviation from this solution that yields b = 0 and c = 0 is caused by radiative effects, which also add additional contributions to d and e, and (d − t 23 e) 2 = O(b 2 + c 2 ) is ensured by S 2 . Then, the splitting between m 1 and m 2 is induced to yield the LMA solution with sin 2 2θ 12 ∼ 0.8. To realize the "ideal" solution, we introduce a permutation symmetry, S 2 , for the µ-and τ -families [24] as have been announced, which is compatible with the requirement of L ′ . For s with L ′ = 2, s only couple to the µ-and τ -families, which provide an S 2 -symmetric democratic mass texture [25] for ν µ,τ with an additional Z 4 discrete symmetry to ensure the "ideal" structure of Eq.(5), leading to one massless neutrino (ν 2 ) and one massive neutrino (ν 3 ). These two neutrinos radiatively mixed with ν e finally give observed neutrino mixings.
All interactions are taken to conserve L ′ and to be invariant under the transformation of S 2 as well as Z 4 . The scalars of φ, s and h + are assigned to S 2 -symmetric states. The other scalar of φ ′ is assigned to an S 2 -antisymmetric state and we introduce an additional copy of φ ′ and h + as S 2 -antisymmetric states denoted by φ ′′ and h ′+ . The inclusion of φ ′′ and h ′+ , respectively, allows us to meet the mass hierarchy of m µ ≪ m τ and the large solar neutrino mixing satisfying (d − t 23 e) 2 = O(b 2 + c 2 ). To distinguish these copies from the original fields, it is sufficient to use a discrete symmetry of Z 4 . The quantum numbers of the participating fields are tabulated in TABLE I, where ψ ±L = (ψ τ L ± ψ µ L )/ √ 2 and ℓ ±R = (τ R ± µ R )/ √ 2. The assignment of the Z 4 -charges of ψ ±L and s forbids the coexistence of ψ +L sψ +L and ψ −L sψ −L , which disturbs the democratic structure of the "ideal" solution. The present assignment corresponds to the σ = 1 solution of Eq. (5). Since charged leptons simultaneously couple to the Higgs scalars of φ, φ ′ and φ ′′ , flavor-changing interactions are induced by the exchanges of these Higgs scalars, which will be shown to give well-suppressed contributions at the phenomenologically consistent level. It is obvious that quarks that are S 2 -symmetric can have couplings to φ but not to φ ′ and φ ′′ ; therefore, quarks do not have this type of dangerous flavor-changing interactions.
The Yukawa interactions for leptons are, now, given by where f 's stand for coupling constants. Higgs interactions are described by usual Hermitian terms composed of ϕϕ † (ϕ = φ, φ ′ , φ ′′ , h + , h ′+ , s) and by non-Hermitian terms in where λ 1,2 are Higgs couplings, which conserves L and L ′ . The soft breaking terms of L and L ′ can be chosen to be: where µ's represent mass scales and V 1,2 and V 3 are, respectively, used to activate the radiative mechanism and the type II seesaw mechanism. Although L and L ′ are spontaneously broken by 0|s|0 , L+L ′ (∝ L e ) is still conserved.
In terms of the L e -conservation, V 2 and V 3 are classified as L e -conserving interactions and its explicit breaking is provided by V 1 . L e -breaking interactions such as those causing µ, τ → eee and → eγ necessarily involve V 1 . All other interactions are forbidden by the conservation of L e and Z 4 . Especially, (h + h + ) † det s could give a divergent term of ν e ν e at the two loop level as depicted in FIG.1 (a), which then would require a tree level mass term of the ν e ν e -term as a counter term. The appearance of this counter term is not consistent with the absence of the tree-level ν e ν e -term in Eq. (5). Since L ′ is explicitly broken, ν e ν e is induced by interactions shown in FIG.1 Fortunately, this diagram leads to the finite convergent term. Charged lepton masses are generated via the Higgs couplings to leptons, which are specified by the following matrix of M ℓ 0 (φ, φ ′ , φ ′′ ): The masses for leptons denoted by M ℓ 0 are, thus, described by with where v = 0|φ 0 |0 , v ′ = 0|φ ′0 |0 and v ′′ = 0|φ ′′0 |0 . To be consistent with the pattern of the observed hierarchy of m e ≪ m µ ≪ m τ , we simply adopt the parameterization based on the "hierarchical" one [26]. It is straight forward to reach U ℓ (V ℓ ) that links the original states of |ℓ 0 L(R) to the states with the diagonal masses of |ℓ L(R) : |ℓ 0 where c α = cos α etc. defined by The µ and τ masses are calculated to be m 2 µ = λ − and m 2 τ = λ + with λ ± given by where The hierarchical mass pattern of m µ ≪ m τ can be realized by the hierarchical conditions of |s α |, |s β | ≪ 1. It is convenient for our later discussions to relate m ℓ ij with m µ,τ , which are described by where By combining Eqs. (11) and (17), we find that should be satisfied for |s α |, |s β | ≪ 1.
Even after the rotation that gives the diagonal mass matrix of ) still contain flavor-off-diagonal couplings. In fact, Eq. (10) is transformed into M ℓ (φ, φ ′ , φ ′′ ), whose elements denoted by M ij are calculated to be: One can readily find that the identification of α φ ′ and α φ ′′ with α φ corresponding to the case of the standard model gives diagonal interactions, leading to the diagonal lepton masses. The flavor-changing interactions involving τ and µ such as τ → µγ and τ → µµµ are roughly controlled by the coupling of m i /m H (=ξ i ), where i = e, τ and m H is a mediating Higgs boson mass. We find constraints on ξ e,τ to suppress these interactions to the phenomenologically consistent level, which are given by examining the following typical processes: 1. for τ − → µ − e − e + mediated by φ, |s α,β ξ τ ξ e /m 2 where the data are taken form Ref. [27]. Since m H > ∼ v weak is anticipated, where v weak = (2 √ 2G F ) −1/2 =174 GeV for the weak boson masses, ξ τ ∼ m τ /v weak and ξ e ∼ m e /v weak with |s α,β | ≪ 1 readily satisfy these constraints. As stated previously, there are no such Higgs interactions for quarks that only couple to φ. The similar flavor-changing interactions caused by h + and h ′+ [28] are sufficiently suppressed because of the smallness of their-couplings to leptons to be estimated in Eq.(36).
The radiative neutrino masses, δm rad ij , are generated by interactions corresponding to FIG.2. Let us denote by M vertex 0 the amplitude involving contributions from the vertices connected by the mediating Higgs scalar, either one of φ, φ ′ and φ ′′ , and kinematical factors due to one-loop contributions denoted by P , P ′ and P ′′ : with whereĥ andĥ ′ project out the contributions of h + and h ′+ withĥĥ =ĥ ′ĥ′ = 1 andĥĥ ′ = 0. The U -term arises from the interaction of µ − φ c † φ ′′ h ′+ † giving µ −ĥ ′ and 0|φ ′′0 |0 (=v ′′ ) and of f φ ψ e L φe R giving f φ with the mediating φ + and h ′+ involved in P and similarly for other terms. The one-loop factors of P 's are defined by where m's are masses of the relevant scalars and m h = m h + (m h ′+ ) if P 's accompanyĥ (ĥ ′ ) in Eq. (21) and similarly for P ′ with m 2 φ → m 2 φ ′ and P ′′ with m 2 φ → m 2 φ ′′ , By considering the rotation effects due to U ℓ that transforms the original states of |ν 0 into |ν weak : |ν weak = U † ℓ |ν 0 , we find that δm rad ij can be parameterized by The radiative neutrino masses given by δm ν ii = 2δm rad ii and δm ν ij = δm rad ij + δm rad ji (i = j) are calculated to be: where r = v ′ /v ′′ and r ′′ = v ′′ /v and we have neglected the non-leading contributions of O(s α,β ) and O(m µ,e /m τ ). The tree level masses, m ν ij (i, j = µ, τ ), are given by the type II seesaw mechanism to be:

2s α ) and
A µτ = A τ µ = c 2 α − s 2 α (∼ 1). Our mass matrix of Eq.(2) has the following mass parameters: where we have used the exact expressions for d, e and f as far as the tree-level contributions are concerned. The possible contribution to the mass parameter of a from the two-loop convergent diagram of FIG.1 (b) is well suppressed by m s arising from the propagator of s and does not jeopardize a=0.
We are now in a position to estimating various neutrino oscillation parameters. In the course of calculations, we assume for the simplicity that m 2 h + = m 2 h ′+ ≫ m 2 φ = m 2 φ ′ = m 2 φ ′′ , leading to P = P ′ = P ′′ . The mixing angle t 23 for the atmospheric neutrino oscillations is computed to be: from t 23 = −c/b(= −δm ν eτ /δm ν eµ ). In the limit of δm ν µτ = 0, t 23 is also given by 23 − t 23 )e, which ensures the almost maximal atmospheric neutrino mixing characterized by t 23 ≈ 1 because |s α | ≪ 1 for the hierarchical µ and τ mass texture. To be consistent, we require that r 2 = −t α , thereby, v ′′ ≫ v ′ for r 2 ≪ 1 leading to g 2 − /f 2 − ≪ 1 from Eq. (19). By including δm ν µτ , we find that r 2 = −t α is modified into up to O(s α ), where we have replaced f h − P µ − m 2 τ by δm ν µτ defined in Eq. (25). The mixing angle sin 2 2θ 12 for the solar neutrino oscillations is given by sin 2 2θ 12 = 8/(8 + x 2 ) of Eq.(4) with x calculated to be: up to r 2 and |s α |, where we have used the relation of supplied by Eq. (25). The tree-level contributions to x involving s α vanish in Eq.(30) because of the use of Eq. (29). This cancellation is realized by the "ideal" structure of the tree-level mass terms thanks to the presence of S 2 and Z 4 and can be traced back to the fact that the tree-level contributions alone give x=0 since the relations of t 23 = (1 − t α )/(1 + t α ) and x ∝ −(c α − s α ) 2 + t 23 (c 2 α − s 2 α ) yields x = 0. The masses of neutrinos that of course depend on x satisfy the "normal" mass hierarchy of |m 1 | < |m 2 | ≪ m 3 determined by Eq.(3) to be: with δm ν rad = δm ν2 eµ + δm ν2 eτ , where η is chosen such that ηx = −|x|. Then, ∆m 2 atm,⊙ are calculated to be: To get numerical estimations, let us fix |x| = √ 2 corresponding to sin 2 2θ 12 = 0.8 and also fix r ′′ = 1 (v = v ′′ ) and weak . In the end, we derive sin 2 2θ 12 = 0.78 from reasonable assumptions on the couplings. The conditions of Eq. (19) in turn require to be consistent with r 2 =1/9 and give the estimation of the Yukawa couplings to be: The tree level mass of m ν is estimated to be ∼ 0.03 eV for ∆m 2 atm = 3 × 10 −3 eV 2 and δm ν rad = 3.2×10 −3 eV is obtained for ∆m 2 ⊙ = 4.5 × 10 −5 eV 2 . Since r 2 in Eq. (29) is almost saturated by −s α for these values of m ν and δm ν rad , we observe that s α ∼ −1/9. The type II seesaw mechanism for m ν yields an estimate of the mass of s: m s (= µ) = 1.7×10 14 × (|f s + |/e) GeV, where e is the electromagnetic coupling. From the expression of δm ν eµ in Eq. (25), we find that the estimation of δm ν rad yields where µ + = m φ = v weak and m 2 h + = 10v 2 weak are used to compute the loop-factor of P . From Eqs. (30) and (31), we also find that x = 26δm ν µτ /27δm ν rad , which yields |x| = √ 2 for |δm ν µτ |/δm ν rad = 1.47, leading to from Eq.(31) with |δm ν eµ | ≈ δm ν rad . Finally, the masses of m 1,2,3 are predicted to be: It is remarkable to note that the result of these numerical estimates is consistent with the reasonable expectation of to yield the large solar neutrino mixing. This result should be contrasted with the requirement of "inverse" hierarchy for the original Zee model [29]. If the relation of f h + ∼ f h − and µ + ∼ µ − is assumed, one finds that |x| ∼ 1.53 for r 2 =1/9 leading to sin 2 2θ 12 ∼ 0.78 in good agreement with the observed data.
Summarizing our discussions, in the radiative mechanism based on the conservation of L ′ and the invariance under the S 2 -transformation as well as under the discrete Z 4 -transformation, we have demonstrated that the almost maximal atmospheric neutrino mixing is guaranteed by the S 2 -symmetric coupling of s to neutrinos and the large solar neutrino mixing is derived by the radiative effects only, where the tree-level contributions from s vanish owing to the presence of S 2 . Our model spontaneously breaks L and L ′ but preserves L + L ′ , namely L e . This remaining L e -conservation is used to select the Higgs interactions that include the key interactions for type II seesaw mechanism and radiative mechanism. The massless Nambu-Goldstone boson associated with the spontaneous breakdown of L − L ′ , namely L µ + L τ , can be removed by introducing a soft breaking such as φ ′c † φ ′′ h +′ . The model seems to suffer from the emergence of the dangerous flavor changing interactions that disturbs the well-established low-energy phenomenology of leptons because there are three Higgs scalars of φ, φ ′ and φ ′′ . However, the explicit calculations show that the lepton sector has couplings to those Higgs scalars at most of order m τ /v weak , which are shown to be sufficiently small to suppress these interactions to the phenomenologically consistent level.
In our scenario, properties of neutrino masses are summarized as follows: 1. The smallness of neutrino masses is ensured by type II seesaw mechanism for atmospheric neutrinos and by radiative mechanism for solar neutrino neutrinos.
2. The mixing angle of θ 12 for solar neutrinos is determined to be sin 2 2θ 12 = 8/(8 + x 2 ) by radiative mass terms, where x = δm ν µτ /|δm ν eτ |, which is subject to the cancellation of the tree-level contributions in x ensured by S 2 . The ratio |x| of O(1) needed for the explanation of the large solar neutrino mixing can be realized by the requirement of f h + ∼ f h − and µ + ∼ µ − , where f h + (f h − ) and µ + (µ − ) stand for h + (h ′+ )-couplings, respectively, to leptons and to Higgs scalars.
It should be noted that the hierarchical mass texture for µ and τ characterized by the finite mixing angle with sin 2 α (∼ sin 2 β) ≪ 1 is inevitable to be consistent with the almost atmospheric neutrino mixing.