Golden Ratio Prediction for Solar Neutrino Mixing

It has recently been speculated that the solar neutrino mixing angle is connected to the golden ratio phi. Two such proposals have been made, cot theta_{12} = phi and cos theta_{12} = phi/2. We compare these Ansatze and discuss a model leading to cos theta_{12} = phi/2 based on the dihedral group D_{10}. This symmetry is a natural candidate because the angle in the expression cos theta_{12} = phi/2 is simply pi/5, or 36 degrees. This is the exterior angle of a decagon and D_{10} is its rotational symmetry group. We also estimate radiative corrections to the golden ratio predictions.


Introduction
The question what kind of flavor model underlies the peculiar features of lepton mixing is one of the dominating ones in contemporary theoretical neutrino physics. One hopes that precision measurements of the flavor parameters will provide hints towards the symmetry principle behind the apparent regularities. We will in this paper discuss one intriguing example of this line of thought. All these issues are linked to the structure of the neutrino (and the charged lepton) mass matrix. Typically, the smallness of |U e3 | and the close-to-maximality of θ 23 are -in the charged lepton basis -attributed to the presence of an approximate µ-τ symmetry: The eigenvector to the eigenvalue D − E indeed is (0, −1, 1) T , but solar neutrino mixing is unconstrained by the matrix given above. If in addition to µ-τ symmetry the condition A + B = D + E holds, then the value sin 2 θ 12 = 1 3 is obtained: the infamous tri-bimaximal mixing [1], which dominates the current theoretical literature on lepton flavor model building. However, comparing the tri-bimaximal mixing parameters (sin 2 θ 12 = 1 3 , sin 2 θ 13 = 0 and sin 2 θ 23 = 1 2 ) with the current best-fit, 1, 2 and 3σ ranges [2] (very similar results are found in [3]) sin 2 θ 12 sin 2 θ 13 sin 2 θ 23 0.304 0.01 0.50 0.288 ÷ 0.326 ≤ 0.026 0.44 ÷ 0.57 0.27 ÷ 0.35 ≤ 0.040 0.39 ÷ 0.63 0.25 ÷ 0.37 ≤ 0.056 0.36 ÷ 0.67 (2) one notes that there is ample room for mixing scenarios other than tri-bimaximal mixing. In this short note we consider alternatives to tri-bimaximal mixing and focus on the fascinating possibility to link solar neutrino mixing with the golden ratio ϕ = ϕ 2 − 1 = 1 2 (1 + √ 5). Two such proposals have recently been made. The first one is [4][5][6] (A): cot θ 12 = ϕ ⇒ sin 2 θ 12 = 1 1 + ϕ 2 = The second possibility is [7] (B): cos θ 12 = ϕ 2 ⇒ sin 2 θ 12 = 1 4 (3 − ϕ) = 5 − It can be seen that the both predictions lie within the current 2σ range 1 . The possibility (A) has first been noted in Ref. [4], and discussed in more detail in [5], where it was also mentioned that A 5 might be a candidate for the underlying flavor symmetry group. In this spirit a model based on A 5 , which can lead to cot θ 12 = ϕ, has been outlined in Ref. [6]. The reason why A 5 is the candidate symmetry is because this group is isomorphic to the rotational group of the icosahedron and its geometrical features can be linked to the golden ratio. For instance, the 12 vertices of an icosahedron with edge-length 2 have Cartesian coordinates (0, ±1, ±ϕ), (±1, ±ϕ, 0) and (±ϕ, 0, ±1). A peculiar feature of cot θ 12 = ϕ is that the angle gives also tan 2θ 12 = 2, and this can be obtained from a simple matrix proportional to [5] m ν ∝ 0 1 1 1 .
This matrix is invariant under a Z 2 symmetry generated by [5] where invariance is fulfilled when S T m ν S = m ν . Now consider the second golden ratio prediction cos θ 12 = ϕ/2, which corresponds simply to θ 12 = π/5. A mixing scenario based on this value was proposed with a purely phenomenological purpose in Ref. [7]. A unified parametrization of both the CKM and the PMNS matrix was constructed by choosing in addition to the lepton mixing angle θ 12 = π/5 a similar expression for the quark sector, namely θ q 12 = π/12. The resulting value for the sine of the Cabibbo angle, sin θ q 12 = ( √ 3 − 1)/ √ 8, is also a simple algebraic and irrational number. The point made in Ref. [7] was that at zeroth order the CKM matrix is a 12-rotation with angle π/12, while the PMNS matrix is a 12-rotation with angle π/5 multiplied with an additional maximal (atmospheric) 23-rotation. To correct the 12-angles of the quark and lepton sectors to their respective best-fit values, one needs to multiply both zeroth order mixing matrices with a small 12-rotation. It turns out that one can achieve this with a universal (i.e., the same for quarks and leptons) angle ǫ 12 ≃ −0.03 [7]. In the present letter we concentrate on the possible theoretical origin of the golden ratio prediction (B). We stress that flavor models based on the symmetry group D 10 are natural candidates to generate θ 12 = π/5. The dihedral group D 10 is the rotational symmetry group of a decagon and the exterior angle in a decagon is nothing but π/5, or 36 degrees. Indeed, we will present a model based on D 10 in the next Section 2. We remark that also D 5 , the rotational symmetry group of a regular pentagon, could be possible. In a pentagon the length of a diagonal is ϕ times the length of a side. The triangle formed by the diagonal and two sides has one angle of 108 • (the internal angle) and two angles with 36 • each. However, here we focus on D 10 because it turns out that the vacuum alignment we need in our model is simplified due to the larger number of representations in D 10 . Note that just as considering A 5 for the golden ratio prediction (A) was motivated by geometrical considerations, the use of the (mathematically simpler) pentagon or decagon symmetry group is here motivated by prediction (B). These are examples for the hope mentioned in the beginning, namely that precision measurements may give us hints towards the underlying symmetry behind flavor physics 2 . The present paper is build up as follows: after discussing general symmetry properties of mass matrices with cos θ 12 = ϕ/2 and an explicit D 10 model in Sec. 2 we will in Section 3 deal with renormalization group corrections to both golden ratio predictions (A) and (B), before we conclude in Section 4.
2 Golden Ratio Prediction θ 12 = π/5 and Dihedral Groups We have seen in the Introduction that there is a simple Z 2 under which a mass matrix generating cot θ 12 = ϕ is invariant, see Eqs. (5) and (6). The second golden ratio proposal (B) in Eq. (4) corresponds to tan 2θ 12 = 1 + ϕ 2 /(ϕ − 1), and therefore it diagonalizes a less straightforward matrix. Nevertheless, in this case one can make use of Z 2 invariance as well, however the charged lepton sector has also to be taken into account. We will first discuss this for the simplified 2-flavor case with symmetric mass matrices, before making the transition to dihedral groups and then to the explicit model based on D 10 that we will construct.
The generators of the Z 2 under which the neutrino mass matrix m ν and the charged lepton mass matrix m ℓ have to be invariant are respectively. The matrices m ν and m ℓ are invariant when they have the following structure: The inner matrix can be written as The total diagonalization matrices of m ν and m ℓ are U ν,ℓ = P ν,ℓŨν,ℓ and the physical mixing matrix is their product 2 Tri-bimaximal mixing is usually obtained with models based on A 4 , the symmetry group of a tetrahedron [8]. Here the angle between two faces (the dihedral angle) is 2θ TBM , where sin 2 θ TBM = 1 3 .
The fact that a non-trivial phase matrix lies in between the two maximal rotationsŨ † ℓ andŨ ν is crucial. Obviously, at this stage any mixing angle can be generated. However, the observation made in Refs. [9] was that the phase factors in Eq. (7) can be linked to group theoretical flavor model building with dihedral groups D n . To make the connection from Eq. (10) to dihedral groups, we note that the flavor symmetry D n has 2-dimensional representations 2 j , with j = 1, . . . , n 2 − 1 (j = 1, . . . , n−1 2 ) for integer (odd) n, generated by Z 2 subgroups are generated by with integer k. This is just the required form of a Z 2 generator in Eq. (7). It is now possible to construct models in which the two fermions transform under the representation 2 j of D n , and D n is broken such that m ν is left invariant under B A kν and m ℓ is left invariant under B A k ℓ [9]. Consequently, the relation in Eq. (10) is obtained and we can identify Hence, a natural candidate to implement the requested value of π/5 is e.g., D 10 . This is no surprise given the observation that we made in the Introduction, namely that π/5 is the exterior angle of a decagon and that D 10 is its rotational symmetry group.
We continue with an explicit model: we work in the framework of the MSSM without explicitly introducing right-handed neutrinos. Majorana masses for the light neutrinos are thus generated by an effective operator coupling to two Higgs vacuum expectation values (VEVs). We augment the MSSM by a flavor symmetry D 10 × Z 5 . The symmetry D 10 is used for our prediction of the solar mixing angle, while the auxiliary Abelian symmetry Z 5 separates the charged lepton and neutrino sectors. Due to the flavor symmetry, no renormalizable Yukawa couplings are allowed for the charged leptons and the dimension 5 operator giving mass to the neutrinos also vanishes. Mass for the leptons is generated by coupling them to gauge singlet flavons, which acquire VEVs and thereby break the flavor group. The charged lepton masses are thus generated by dimension 5 operators, the neutrino masses by dimension 6 operators 3 . The transformation properties of the MSSM leptons and Higgs fields, as well as the representations under which the flavons transform, are given in Table 1. The multiplication  table and 1 2 2 1 1 1 1 1 1 2 2 2 3 2 4 1 1 2 1 2 2 2  is the fifth root of unity.
the fermions and the flavons that couple to them are all in unfaithful representations of D 10 (i.e., in 2 2 and 2 4 ), so that here a D 5 structure would have sufficed. However, the full D 10 structure is needed to achieve the desired vacuum alignment. We can continue by constructing the Yukawa superpotential, giving the leading order terms for both charged lepton and neutrino masses: As we will show below in Appendix B, introducing appropriate "driving fields" and minimizing the flavon superpotential leads to the following VEVs for the flavons: where k is an odd integer between 1 and 9, and The VEVs of the singlet flavons σ e = x e and σ ν = x ν are assumed to be also nonvanishing. The VEV structure leads to the following mass matrices: To see that indeed the golden ratio prediction is obtained from the above two matrices, note that for the choice k = 3 the relevant matrix m ℓ m † ℓ takes the form The  13), where we have to set j = 4 because the first and second left-handed lepton doublets transform as 2 4 , we expect |U e1 | 2 = | cos 6 5 π| 2 , which is indeed equivalent to an angle of π/5. We will explicitly check this in the following.
The diagonal phase matrix on the left is crucial. The rotation angle in the 23-axis is given by and the charged lepton masses are given by The neutrino mass matrix is diagonalized via The eigenvalues have in general non-trivial phases which are taken into account in the diagonal matrix P , and their absolute values are We note that the model makes no predictions about the neutrino masses or their ordering. Nevertheless, one can easily convince oneself that the number of free parameters in the model is enough to fit the neutrino and charged lepton masses, as well as the large atmospheric neutrino mixing angle θ 23 . The model does in general not predict θ 23 to be maximal, which is not an issue given the fact that it is the lepton mixing parameter with the largest allowed range. However, maximal mixing is compatible with the model. We have θ 23 = π/4 when G = A + B, in which case m 2 µ,τ = A + B ∓ √ 2 D and m 2 e as in Eq. (21). The fact that there is not more predictivity can be traced to the fact that there is a comparably large number of flavon fields required in order to make the model work. This is the price one unfortunately has to pay if one insists in the rather peculiar value of θ 12 . Given the fact that current data allows for this very interesting possibility, one should nevertheless pursue the task of constructing models leading to it. The final PMNS matrix is One finds that U e3 is vanishing and that atmospheric neutrino mixing is governed by tan 2θ 23 given by Eq. (20). As mentioned above, the PMNS matrix has a non-trivial phase matrix including Φ in between the two maximal 12-rotations, one of which stems from U ℓ , the other from U ν . As discussed above, this is the origin of the required result. Indeed, the 12-element of U is and due to U e3 = 0 this is just sin 2 θ 12 . We have thus achieved our goal of predicting θ 12 = π/5. As discussed in Appendix B, higher order corrections to the scenario, as well as flavor changing neutral currents, can be estimated to give only very small contributions.

Renormalization Corrections to the Golden Ratio Predictions
It is worth discussing renormalization group (RG) effects to the golden ratio predictions, because any symmetry leading to the predictions discussed in this paper could presumably be operating at a high energy scale Λ, and the observables have to be evolved down to the low energy scale λ. Note that RG corrections to |U e3 | and θ 23 are typically suppressed with respect to the running of θ 12 by a factor of ∆m 2 ⊙ /∆m 2 A . As the initial values of both |U e3 | and θ 23 need not to be specified here (other than being small or close to maximal, respectively) we do not comment on their RG-shift. We will stay here model-independent and estimate the corrections as a function of the unknown neutrino mass values and ordering. An expression forθ 12 , where the dot denotes the derivative with respect to t = ln µ/µ 0 with µ the renormalization scale, is given e.g., in [10]. One can therefrom estimate the shift for the solar neutrino mixing angle: where θ 0 12 is the initial value of θ 12 (here given by Eq. (3) or (4)) and with v u = 246 GeV, c = − 3 2 in the SM and (1 + tan 2 β) in the MSSM. Neutrino physics is included in Consequently, from Eq. (26) one finds 4 sin 2 θ 12 ≃ sin 2 θ 0 12 + k 12 ǫ RG sin 2θ 0 12 .
Note that the Majorana phase α can suppress the running. As well known, θ 12 decreases in the SM and increases in the MSSM, independent on the sign of ∆m 2 A . The following numerical estimates are done with sin 2 θ 0 23 = 1 2 , Λ/λ = 10 10 and with ∆m 2 ⊙ , ∆m 2 A fixed for simplicity at their current best-fit values [2]. In the normal hierarchy (NH, m 3 ≃ ∆m 2 A , m 2 ≃ ∆m 2 ⊙ ≫ m 1 ) the running in the SM is completely negligible. In case of the MSSM, even for tan β = 40 the shift in sin 2 θ 12 is not more than 1.5 %. This changes in the inverted hierarchy (IH, m 2 ≃ m 1 ≃ ∆m 2 A ≫ m 3 ), where in the MSSM and tan β = 10 the value of sin 2 θ 12 can increase by around 10 %. In the SM, again, the shift is with less than half a percent not measurable. For quasi-degenerate neutrinos with a common mass scale of 0.2 eV the SM allows shifts of around 3 %, whereas in the MSSM the shift can be as large as the value of sin 2 θ 12 , even for small values of tan β = 5. We illustrate this in Fig. 1, where we used Eq. (29) to show the RG-induced shifts of sin 2 θ 12 for a normal mass hierarchy (SM and MSSM with tan β = 40), an inverted hierarchy (SM and MSSM with tan β = 10), as well as quasi-degenerate neutrinos (QD, smallest mass 0.2 eV for the SM and MSSM with tan β = 5). In case of a normal and inverted hierarchy we have chosen (at high scale) 0.001 eV for the smallest neutrino mass. To a good approximation and unless in the MSSM tan β is very large, the running of the neutrino masses can be described by a rescaling, with basically no dependence on the other neutrino parameters [10]. Because k 12 from Eq. (28) has the masses appearing in the denominator and numerator, their running cancels in our approximation as long as |k 12 ǫ RG | ≪ 1. The range of the corrections in Fig. 1 is due to the unknown Majorana phases. For illustration, we also include the shifts for tri-bimaximal mixing. To bring θ 12 very close to the best-fit value, the prediction (A) requires the MSSM and IH or QD, while prediction (B) (and tri-bimaximal mixing) requires the SM with rather large neutrino masses. If future data leads to more precise determinations of sin 2 θ 12 and other neutrino parameters, one will be able to rule out some of the existing possibilities. 4 Inserting sin 2 θ 0 12 = 1 3 and sin 2 θ 0 23 = 1 2 in the following and the last expression reproduces the results from Ref. [11].  Figure 1: RG-induced shifts on sin 2 θ 12 , estimated from Eq. (29), for the two golden ratio proposals and for tri-bimaximal mixing (TBM). The current best-fit value, as well as the 1 and 2σ ranges are also indicated. In case of the MSSM we have taken tan β = 40 for a normal hierarchy (NH), tan β = 10 for the inverted hierarchy (IH) and tan β = 5 for quasi-degenerate neutrinos (QD) with a mass scale of 0.2 eV. The line for the SM and a normal hierarchy cannot be seen because the effect is too small.

A Multiplication Rules and Clebsch-Gordan Coefficients of D 10
We present here the Clebsch-Gordan coefficients for D 10 . The multiplication rules for the Kronecker products are given in Table A1. For s i ∼ 1 i and (a 1 , a 2 ) T ∼ 2 j we find s 1 a 1 s 1 a 2 ∼ 2 j , s 2 a 1 −s 2 a 2 ∼ 2 j , s 3 a 2 s 3 a 1 ∼ 2 5-j and s 4 a 2 −s 4 a 1 ∼ 2 5-j .
The Clebsch-Gordan coefficients for the product of (a 1 , a 2 ) T with (b 1 , b 2 ) T , both in ∼ 2 i , read depending on whether i = 1, 2 or i = 3, 4. For the two doublets (a 1 , a 2 ) T ∼ 2 i and (b 1 , b 2 ) T ∼ 2 j we find for i + j = 5  If i + j = 5 holds the covariants read Again, the first case is relevant for k = i − j, while the second one is valid for k = j − i.

B VEV Alignment of the D 10 × Z 5 Model
To obtain the necessary vacuum alignment in the flavon potential, we need to introduce a U(1) R and driving fields [12]. Regular R-parity is a subgroup of the U(1) R . To ensure a supersymmetric Lagrangian, the superpotential must have a U(1) R charge of 2. The superfields containing the SM fermions have an R-charge of 1, while the Higgs fields have an R-charge of zero. Hence, for the Yukawa superpotential given in Eq. (14) to be viable, the flavons also need to have a vanishing R-charge. Consequently, for the flavon superpotential one needs to introduce additional flavor-charged fields, having an R-charge of 2.
As advocated above, these two sets of equations are uniquely solved by the VEV configurations given in Eqs. (15) and (16), where we have set a possible relative phase in the doublet of VEVs of the flavons in the charged lepton sector to zero. This can be done without loss of generality, as only the phase difference between the two sectors is phenomenologically relevant. We have also assumed that none of the parameters in the superpotential vanish. For the charged lepton sector, the flavon VEVs w e and x e are free parameters (which we take to be non-zero), while The driving fields themselves are only allowed vanishing VEVs, as can be inferred from considering the F-terms of the flavons. Note, that since we can not make the cutoff scale Λ arbitrarily large, we need to take into account NLO corrections to both the Yukawa and flavon superpotentials. We also should be careful in what regards potentially dangerous flavor changing neutral currents induced by the flavons. All this could be taken into account by carefully studying the mass spectrum of the scalars. Given the sizable number of fields this is a formidable task, but fortunately it suffices to make some general estimates, which agree well quantitatively with a lengthy explicit calculation in a similar model [13]: the τ lepton mass, see Eq. (21), is of order f v/Λ, where f is a flavon vev, v the Higgs vev (≃ 10 2 GeV) and Λ the cutoff scale. The neutrino mass, see Eq. (23), is of order f v 2 /Λ 2 .
With the charged lepton τ mass ≃ GeV and the neutrino mass ≃ 0.1 eV it follows Λ ≃ 10 12 GeV and f ≃ 10 10 GeV. Now we can estimate that the flavon mass is also of order of f . NLO corrections to the potential, and therefore to the neutrino and charged lepton mass matrices, are of order f /Λ ≃ 10 −2 and therefore under control. Any potentially dangerous flavor changing neutral currents are also suppressed by the heavy mass scale f .