Bi-large neutrino mixing from bilinear R-parity violation with non-universality

We investigate how the bi-large mixing required by the recent neutrino data can be accommodated in the supersymmetric standard model allowing bilinear R-parity violation and non-universal soft terms. In this scheme, the tree-level contribution and the so-called Grossman-Haber one-loop diagrams are two major sources of the neutrino mass matrix. The relative size of these two contributions falls into the right range to generate the atmospheric and solar neutrino mass hierarchy. On the other hand, the bi-large mixing is typically obtained by a mild tuning of input parameters to arrange a partial cancellation among various contributions.

Recently, impressive progress has been made in atmospheric and solar neutrino experiments [1,2]. They provided us convincing evidences for three active neutrino oscillations requiring two large and one small mixing angles [3]. The resulting neutrino mixing matrix [4] takes the form; where c ij = cos θ ij , s ij = sin θ ij , and s 13 ≈ θ 13 < ∼ 0.2. Here we put θ 23 = π/4 for the nearly maximal atmospheric neutrino mixing angle. The solar neutrino mixing angle θ 12 takes the value tan θ 12 ≈ 0.65 for the so-called LMA solution which is strongly favored by the recent SNO data [2]. The mass-squared differences explaining the atmospheric and solar neutrino data are ∆m 2 atm ≈ 2.5 × 10 −3 eV 2 and ∆m 2 sol ≈ 5 × 10 −5 eV 2 , respectively. Even though less favored, the so-called LOW solution with tan θ 12 ≈ 0.77 and ∆m 2 sol ∼ 10 −7 eV 2 is still viable.
One of attractive schemes to generate neutrino masses and mixing is to invoke R-parity and lepton-number violation allowed in the supersymmetric standard model [5]. The purpose of this paper is to address the question whether the bi-large mixing of three active neutrinos can arise naturally from the bilinear R-parity violation. The superpotential of the supersymmetric standard model may contain the following bilinear terms; generalizing the usual µ-term, µH 1 H 2 . Then, there are also six soft supersymmetry breaking terms in the scalar potential; where we used the same notations for the superfields and their scalar components. Let us note that B i in the first term is dimension-one and the corresponding term for the Higgs biliear is µBH 1 H 2 .
If the universal boundary condition is imposed on the soft-terms, the differences between the soft-terms of the Higgs boson H 1 and slepton L i such as with small U e3 . It is easy to understand it qualitatively as one can expect that the three parameters ǫ i control all the mixing angles. A small θ 13 and a large θ 23 requires ǫ 1 ≪ ǫ 2 ≈ ǫ 3 leading to θ 12 ≈ θ 13 [6,7]. Thus, in order to accommodate the bi-large neutrino mixing, one has to go beyond this minimal scheme. One way is to allow trilinear couplings while keeping the universality. In this case, the five couplings related to the third generation fermions may play a major role to generate the desired neutrino mass matrix [6,7]. Another way is to allow non-universal soft-terms [8,9,10,11]. Introduction of general flavor-mixing soft-masses is, of course, tightly constrained by the flavor changing neutral current processes, such as µ → eγ or τ → µγ [12]. However, the non-universality in the flavor-diagonal soft-parameters is not severely constrained. Generically, one could expect ∆m 2 i /m 2 H 1 and ∆B i /B to be of order one. One can also have m 2 In this paper, we investigate how the desired neutrino mass and mixing pattern can arise under such a generic non-universality condition. We will see that the right values of the mixing angles and the mass hierarchy can be obtained in reasonable ranges of parameter space without severe fine-tunning. In the below, we will first quantify all the tree-level and one-loop contributions to the neutrino mass matrix and identify the dominant contributions.
Obtaining a rather simple form of the leading neutrino mass matrix, we will make qualitative discussions to understand how the desired masses and mixing arise. This will be completed by presenting our numerical analysis.
Let us start our main discussion by describing the structure of neutrino mass matrix coming from R-parity violation. Adopting the notations of Ref. [6], the most general oneloop renormalized neutrino mass matrix can be written as where Here, the first term is the neutrino mass matrix arising at tree-level, the second terms containing δ i come from the one-loop correction to the neutrino-neutralino mixing masses projected on to the neutrino direction, and the last term Π ν ij is the one-loop correction to the ν i -ν j Majorana mass matrix. The non-zero values of ξ i ≡ L 0 i / H 0 1 − ǫ i arise due to non-universal soft terms in the slepton-Higgs sector as follows; where the sneutrino mass-squared is m 2 As is well-known, the tree-level mass matrix makes massive only one neutrino in the direction of ξ, which is typically the heaviest one, ν 3 . In fact, the quantity ξ i controls the neutrino-neutralino mixing and thus could be probed by lepton flavor violating decays of the lightest neutralino in the future colliders [7,13,14]. Here, let us introduce another quantity, which governs the mixing between the sleptons and Higgs bosons. As we will see, the flavor structure of the neutrino mass matrix depends on these two R-parity violating parameters, ξ i and η i , as well as non-universal slepton masses.
A simplication of the full neutrino mass matirx comes from the obseravtion that the second term on the right-hand side of Eq. (4) can be ignored in our case [15]. This can be seen immediately by going to the basis where the tree-level mass matrix is diagonalized by the eigenvectorξ and any two orthogonal unit vectors. In this basis, one finds that the second mass matrix has vanishing components in the 1-2 plane orthogonal toξ. Thus, leaving the heaviest ν 3 untouched, approximate see-saw diagonalization can be applied to get the contribution to the 1-2 plane of the order of M Z δ 2 . This is like a two-loop contribution much smaller than the (non-vanishing) 1-2 components of the last term Π ν . Thus, there is no need to compute the second mass term in most cases even though we included it in our analysis.
The main contribution to the last term Π ν of Eq. (4) comes from the one-loop diagrams exchanging sneutrinos/Higgs bosons and gauginos [15,16] in the case of generic non-universality under consideration. Here we present the explicit formula of this one-loop mass matrix which is calculated by the use of approximate see-saw rotation [6]; where N ab is the 4x4 neutralino diagonalization matrix,χ 0 denotes the neutralino mass eigenstates, φ represents the neutral Higgs bosons (φ = h, H and A), and the loop-function The effect of the bilinear R-parity violating terms are encoded in the coefficients θ iφ and Z ij which are given by where η i is defined in Eq. (6), α is the usual diagonalization angle of two CP even Higgs . A few remarks are in order: (i) The coefficients θ iφ are the linear combinations of θ S ij 's defined in Eq. (9) of Ref. [7]. They are related by the Higgs mass diagonalization. In Eq. (8), the quantity ξ i appears to include the effect of neutrino-neutralino mixing by ǫ i . This ξ i dependence can be easily understood if one goes to the basis where ǫ i vanishes [6]. (ii) The same diagrams have been considered in Ref. [15] using the mass-insertion method which must yield the equivalent results to ours. These diagrams involve two mass-insertions which can be seen here as products of two induced R-parity odd ν − φ − χ 0 vertices, θ iφ θ jφ , and as individual sneutrino vertices, Z ij , which is R-parity even. (iii) Among various contributions in θ iφ θ jφ , the term proportional to ξ i ξ j can be absorbed into the tree-level mass term giving a negligible effect. The term proportional to ξ i η j is suppressed due to the similar reason discussed before, but cannot be neglected completely. (iv) The term Z ii is nothing but the contribution due to the sneutrino-anti-sneutrino mass splitting induced by R-parity violation, a lá Grossman-Haber [16], and Z ij with i = j comes from the effective sneutrino The terms with Z ij are proportional to M 2 Z c 2 β /m 2 ν i , and thus give smaller contributions than the terms with η i η j from θ iφ θ jφ in a reasonable range of parameters. However, they can give a sizable effect in general. Now, let us consider the other one-loop contributions and show that (7) dominates over them in the case of the general non-universality. Among various contributions, we take the well-known diagram with squark-quark exchange to be compared with (7). Considering the trilinear couplings induced from bottom quark Yukawa couplings h b such as Taking the ratio of the above two contributions, one typically gets (9)/(7) ≈ 5×10 −6 t 3 β (ǫ/η) 2 with mχ0 = 100 GeV, µ = mb = 250 GeV. Therefore, (9) can be neglected as far as tan β is not too large and ǫ i ∼ η i . In the similar way, one can find that the other diagrams are also sub-leading to (7). In Ref. [10], a slight deviation of non-universality has been assumed to yield ǫ/η ∼ 10 3 and thus (9) was considered as the main one-loop correction. In fact, this is a typical situation in the case of universality. The importance of the contribution (7) in the case of large deviation from universality has been notified in Ref. [15] and its impact on viable neutrino mass matrices has been considered in Refs. [9,11].
From the previous discussions, we can write down the leading contributions to the full mass matrix (4) as follows: where f a ij derivable from Eqs. (7) and (8) is the function of the masses of neutralinos, sneutrinos and Higgs bosons and its flavor dependence comes from the non-universal slepton masses.
We are ready to discuss how the desirable neutrino masses and mixing can be realized by the bilinear R-parity violation with generic non-universal soft masses. For this, we will take the following representative set of R-parity conserving parameters; throughout this paper. This choice gives the light and heavy neutral Higgs boson masses, m h = 84 GeV and m H = 302 GeV, respectively. Other choices will not change the main features of our results. Concerning the R-parity violating parameters, we allow the general flavor dependence for the supersymmetric ǫ i and soft B i parameters. To make our discussion simpler, we will take m 2 L i H 1 = 0 in this paper. This would be a plausible choice for the minimal lepton flavor violation as it may arise due to some mechanism of generating the µ and ǫ i µ terms. Now, let us start with the simplest case: (A) the "minimal" deviation from the universality, that is, sleptons have a universal soft-mass: This was the scheme employed in the analysis of Refs. [9,15]. In this case, the lepton flavor dependence in f ij disappears and thus the neutrino mass matrix (10) takes the following simple form: where Here,x andŷ are nothing but the unit vectors in the direction of ξ and η, respectively. As analyzed in Ref. [17], the mass matrix (12) has two non-vanishing eigenvalues, m 3 ≈ m x and m 2 ≈ m y s 2 ϕ , whose eigenvectors are in the directions ofx andx × (x ×ŷ), respectively. Here the angle ϕ is defined by c ϕ =x ·ŷ.
From these, one finds that the desired neutrino mixing matrix (1) is obtained for with an arbitrary number k. The ratio of two mass eigenvalues is given by Note that one can easily obtain its right value to accommodate the atmospheric and solar neutrino (LMA) mass scales; namely, m 2 /m 3 ≈ ∆m 2 sol /∆m 2 atm ∼ 0.16 putting mχ0 = F N = 200 GeV, t β = 5, |η|/|ξ| = 1 and s 2 ϕ = 1. Furthermore, the relation (13) can also be arranged by an appropriate choice of two independent set of parameters ξ i and η j . In the similar way, the LOW solution can also be easily accommodated. However, it remains to be seen how such an arrangement for ξ i and η i can be made in terms of the input parameters, In order to answer this question, let us choose the following set of values; which give rise to the desired bi-large mixing of the atmospheric and solar neutrino oscillations. Note that the above choice corresponds to c ϕ = 0. The normalization of η will be chosen to reproduce a right value of ∆m 2 sol /∆m 2 atm ∼ 2 × 10 −2 . Since we will calculate the ratios of neutrino mass eigenvalues and mixing angles, we put, e.g., ξ 2 = ξ 3 = 1. In order to obtain the mass scale of m 3 = 0.05 eV, one can take an overall rescaling of R-parity violating variables, ξ, η and ǫ, by factor of 5 × 10 −6 . We now give three examples realizing the above choice of ξ i and η i as follows.
(A1) ∆m 2 i /m 2 H 1 = 0.7: This corresponds to the sneutrino mass, mν i = 67 GeV, and gives the neutrino mass matrix, Therefore For the cases (A1) and (A2), our general parameter scan showed that the realistic neutrino masses and mixing can be obtained within the range of input parameters: 1 < ∼ |ǫ i | < ∼ 10 and 0.1 < ∼ |∆B i /B| < ∼ 1 leading to |ξ|, |η| ∼ 1. From the above samples, one can see that there need certain arrangements in the flavor structure of the input parameters realizing the required mixing angles. This would be the case in many class of models. In our case, the smallness of |ξ 1 | is arranged not by the smallness of |ǫ 1 | but by a partial cancellation between two terms: ∆m 2 1 ≈ −∆B 1 µt β leading to ∆B 1 /B ≈ 0.22 and −0.31 for (A1) and (A2), respectively. This pattern arises also in more general cases as we will see shortly. Since |ǫ 1 | is not necessarily smaller than |ǫ 2,3 |, it is favored to have ∆B 1 /B ∼ ∆B 2,3 /B. Thus, a vanishingly small |U e3 | cannot be naturally realized in our scheme. In the case (A3), the universality is maintained to a certain degree, As we can see, this requires |ǫ i | ≫ |η i |, |ξ i | and a strong correlation for the fine-tunned values of |∆B i /B| ≪ 1. In fact, this is a characteristic property of the universality case where the small deviation of ∆m 2 i /m 2 H 1 and ∆B i /B arises due to RGE of soft parameters. We excluded such cases in our analysis.
Let us now relax the universality condition of the slepton and Higgs boson masses, which leads to the following form of the neutrino mass matrix; Again, one needs a relation ∆m 2 1 ≈ −∆B 1 µt β . We find that this case (B) is not particularly fine-tunned compared to the previous case (A) and can be a viable option. for which the cancellation in ξ 1 happens as discussed before. Anther solution set is allowed around x 1 = 3.4 or 0.4 for which the sneutrino mass is close to the heavy or light Higgs mass, respectively. In this region, the mixing elements (8) and thus the coefficients c ij in Eq. (18) become large to enhance the one-loop contribution. As a consequence, U e3 can be arranged to be small without making ξ 1 small. In FIG. 2, one sees that the points (x 2 , p 2 ) close to These two figures show that there appears the pattern, |ξ 1 | ≪ |ξ 2 | ≈ |ξ 3 |, which gives rise to θ 13 ≪ 1 s 23 ≈ c 23 ≈ 1/ √ 2 as shown in Eq. (13) for the tree-dominance case.
To conclude, we showed how naturally the realistic neutrino mass matrix can arise from bilinear R-parity violation assuming non-universal soft-terms. When generic non-universality is allowed and tan β is not too large, the neutrino mass matrix is dominated by two contributions; the tree-level mass and the one-loop mass from the so-called Grossman-Haber diagrams arising due to the sneutrino-Higgs mixing. This was checked by our numerical calculation taking the full one-loop renormalized neutrino mass matrix. In this scheme, the loop-to-tree mass ratio falls naturally into the right range to generate the desired values for ∆m 2 sol /∆m 2 atm . Considering nine input parameters, ǫ i , ∆B i and ∆m 2 i , we analyzed the parameter space accommodating two large (θ 12 and θ 23 ) and one small (θ 13 ) mixing angles.
Typically, the smallness of θ 13 is realized by a cancellation between the terms contributing to ξ 1 . This was shown by some examples and also by the scatter plot of FIG. 1. Such an arrangement would not be a severe fine-tunning of input parameters. However, our scheme cannot provide a natural reason for vanishingly small θ 13 if it turns out so. We presented the results accommodating only the LMA solution, but the similar conclusion can be drawn also in the case of the LOW solution as can be inferred from our discussions.