Radiative Generation of Non-zero $\theta_{13}$ in MSSM with broken $A_4$ Flavor Symmetry

We study the renormalization group effects on neutrino masses and mixing in Minimal Supersymmetric Standard Model (MSSM) by considering a $\mu-\tau$ symmetric mass matrix at high energy scale giving rise to Tri-Bi-Maximal (TBM) type mixing. We outline a flavor symmetry model based on $A_4$ symmetry giving rise to the desired neutrino mass matrix at high energy scale. We take the three neutrino mass eigenvalues at high energy scale as input parameters and compute the neutrino parameters at low energy by taking into account of renormalization group effects. We observe that the correct output values of neutrino parameters at low energy are obtained only when the input mass eigenvalues are large $|m_{1,2,3}| = 0.08-0.12$ eV with a very mild hierarchy of either inverted or normal type. A large inverted or normal hierarchical pattern of neutrino masses is disfavored within our framework. We also find a preference towards higher values of $\tan{\beta}$, the ratio of vacuum expectation values (vev) of two Higgs doublets in MSSM in order to arrive at the correct low energy output. Such a model predicting large neutrino mass eigenvalues with very mild hierarchy and large $\tan{\beta}$ could have tantalizing signatures at oscillation, neutrino-less double beta decay as well as collider experiments.


I. INTRODUCTION
Exploration of the origin of neutrino masses and mixing has been one of the major goals of particle physics community for the last few decades. The results of recent neutrino oscillation experiments have provided a clear evidence favoring the existence of tiny but non-zero neutrino masses [1]. Recent neutrino oscillation experiments like T2K [2], Double ChooZ [3], Daya-Bay [4] and RENO [5] have not only confirmed the earlier predictions for neutrino parameters, but also provided strong evidence for a non-zero value of the reactor mixing angle θ 13 . The latest global fit values for 3σ range of neutrino oscillation parameters [6] are as follows: where NH and IH refers to normal and inverted hierarchy respectively. Another global fit study [7] reports the 3σ values as The observation of non-zero θ 13 which is evident from the above global fit data can have non-trivial impact on neutrino mass hierarchy as studied in recent papers [8]. Non-zero θ 13 can also shed light on the Dirac CP violating phase in the leptonic sector which would have remained unknown if θ 13 were exactly zero. The detailed analysis of this non-zero θ 13 have been demonstrated both from theoretical [9], as well as phenomenological [10] point of view, prior to and after the confirmation of this important result announced in 2012. It should be noted that prior to the discovery of non-zero θ 13 , the neutrino oscillation data were compatible with the so called TBM form of the neutrino mixing matrix [11] given by which predicts sin 2 θ 12 = 1 3 , sin 2 θ 23 = 1 2 and sin 2 θ 13 = 0. However, since the latest data have ruled out sin 2 θ 13 = 0, there arises the need to go beyond the TBM framework. In view of the importance of the non-zero reactor mixing and hence, CP violation in neutrino sector, the present work demonstrates how a specific µ-τ symmetric mass matrix (giving rise to TBM type mixing) at high energy scale can produce non-zero θ 13 along with the desired values of other neutrino parameters ∆m 2 21 , ∆m 2 23 , θ 23 , θ 12 at low energy scale through renormalization group evolution (RGE). We also outline how the µ − τ symmetric neutrino mass matrix with TBM type mixing can be realized at high energy scale within the framework of MSSM with an additional A 4 flavor symmetry at high energy scale. After taking the RGE effects into account, we observe that the output at TeV scale is very much sensitive to the choice of neutrino mass ordering at high scale as well as the value of tanβ = vu v d , the ratio of vev's of two MSSM Higgs doublets H u,d . We point out that this model allows only a very mild hierarchy of both inverted and normal type at high energy scale. We scan the neutrino mass eigenvalues at high energy and constrain them to be large |m 1,2,3 | = 0.08 − 0.12 eV in order to produce correct neutrino parameters at low energy. We consider two such input values for mass eigenvalues, one with inverted hierarchy and the other with normal hierarchy and show the predictions for neutrino parameters at low energy scale. We also show the evolution of effective neutrino mass m ee = | i U 2 ei m i | (where U is the neutrino mixing matrix) that could be interesting from neutrino-less double beta decay point of view. Finally we consider the cosmological upper bound on the sum of absolute neutrino masses ( i |m i | < 0.23 eV) reported by the Planck collaboration [12] to check if the output at low energy satisfy this or not.
This article is organized as follows. In section II, we discuss briefly the A 4 model at high energy scale. In section III we outline the RGE's of mass eigenvalues and mixing parameters.
In section IV we discuss our numerical results, and finally conclude in section V.

II. A 4 MODEL FOR NEUTRINO MASS
Type I seesaw framework is the simplest mechanism for generating tiny neutrino masses and mixing. In this seesaw mechanism neutrino mass matrix can be written as Within this framework of seesaw mechanism neutrino mass has been extensively studied by discrete flavor groups by many authors [13] available in the literature. Among the different discrete groups the model by the finite group of even permutation, A 4 also can explain the µ − τ symmetric mass matrix obtained from this type I seesaw mechanism. This group has 12 elements having 4 irreducible representations, with dimensions n i , such that i n 2 i = 12. The characters of 4 representations are shown in table I. The complex number ω is the cube root of unity. In the present work we outline a neutrino mass model with A 4 symmetry given in the ref. [14]. This flavor symmetry is also accompanied by an additional Z 3 symmetry in order to achieve the desired leptonic mixing. In this model, the three families of left-handed lepton doublets l = (l e , l µ , l τ ) transform as triplets, while the electroweak singlets e c , µ c , τ c and the electroweak Higgs doublets H u,d transform as singlets under the A 4 symmetry. In order to break the flavor symmetry spontaneously, two A 4 triplet scalars φ l = (φ l1 , φ l2 , φ l3 ), φ ν = (φ ν1 , φ ν2 , φ nu3 ) and three scalars ζ 1 , ζ 2 , ζ 3 transforming as 1, 1 ′ , 1 ′′ under A 4 are introduced. The Z 3 charges for l, H u,d , φ l , φ ν , ζ 1,2,3 are ω, 1, 1, ω, ω respectively.
Under the electroweak gauge symmetry as well as the flavor symmetry mentioned above, the superpotential for the neutrino sector can be written as where Λ is the cutoff scale and y ′ s are dimensionless couplings. Decomposing the first term Similarly, the decomposition of the last three terms into A 4 singlet gives llζ 1 = (l e l e + l µ l τ + l τ l µ )ζ 1 llζ 2 = (l µ l µ + l e l τ + l τ l e )ζ 2 llζ 3 = (l τ l τ + l e l µ + l µ l e )ζ 3 Assuming the vacuum alignments of the scalars as φ ν = α ν Λ(1, 1, 1), ζ 1 = α ζ Λ, ζ 2,3 = 0, the neutrino mass matrix can be written as where d = y νφ α ν , a = y νζ1 α ζ and v u is the vev of H u . The above mass matrix has eigenvalues . Without adopting any un-natural fine tuning condition to relate the mass eigenvalues further, we wish to keep all the three neutrino mass eigenvalues as free parameters in the A 4 symmetric theory at high energy and determine the most general parameter space at high energy scale which can reproduce the correct neutrino oscillation data at low energy through renormalization group evolution (RGE).
Such a parameterization of the neutrino mass matrix however, does not disturb the generic features of the model for example, the µ − τ symmetric nature of m LL , TBM type mixing as well the diagonal nature of the charged lepton mass matrix, which at leading order (LO) is given by [14,15]  Here v d is the vev of H d ; y e , y µ , y τ and α l are dimensionless couplings. These matrices in the leptonic sector given by (6) and (7) are used in the next section for numerical analysis.

III. RGE FOR NEUTRINO MASSES AND MIXING
The left-handed Majorana neutrino mass matrix m LL which is generally obtained from seesaw mechanism at high scale M R , is usually expressed in terms of K(t), the coefficient of the dimension five neutrino mass operator [16,17] in a scale-dependent manner [18], where t = ln(µ/1GeV ) and the vev is v u = v 0 sin β with v 0 = 174 GeV in MSSM. The neutrino mass eigenvalues m i and the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix U P M N S [19] are then extracted through the diagonalization of m LL (t) at every point in the energy scale t using the equations (8), is usually parameterized in terms of the product of three rotations R(θ 23 ), R(θ 13 ) and R(θ 12 ), (neglecting CP violating phases) by where U l is unity in the basis where charge lepton mass matrix is diagonal, s ij = sin θ ij and c ij = cos θ ij respectively. The RGE's for v u and the eigenvalues of coefficient K(t) in equation (8), defined in the basis where the charged lepton mass matrix is diagonal, can be expressed as [20,22] Neglecting h 2 µ and h 2 e compared to h 2 τ , and taking scale-independent vev as in equation (8), we have the complete RGE's for three neutrino mass eigenvalues, The above equations together with the evolution equations for mixing angles (23)(24), are used for the numerical analysis in our work. The approximate analytical solution of equation (14) can be obtained by taking static mixing angle U 2 τ i in the integration range as [21] m The integrals in the above expression are usually defined as [18,21] and where i = 1, 2, 3 and f = t, b, τ respectively. For a two-fold degenerate neutrino masses that (15) is further simplified to the following expressions While deriving the above expressions, the following approximations are used The sign of the quantity δ τ in MSSM depends on the neutrino mixing matrix parameters and the approximation on δ τ taken here is valid only if t 0 is associated with the top quark mass. From equations (18) and (19), the low energy solar neutrino mass scale is then obtained as  Table II A. Evolution equations for mixing angles The corresponding evolution equations for the PMNS matrix elements U f i are given by [20] dU where f = e, µ, τ and i, k = 1, 2, 3 respectively. Here H e is the Yukawa coupling matrices of the charged leptons in the diagonal basis and  Table II Neglecting h 2 µ and h 2 e as before and denoting A ki = m k +m i m k −m i , equation (22) simplifies to [20] These equations are valid for a generic MSSM with the minimal field content and are independent of the flavor symmetry structure at high energy scale.  Table II

IV. NUMERICAL ANALYSIS AND RESULTS
For the analysis of the RGE's, equations (14), (23)- (25) for neutrino masses and mixing angles, here we follow two consecutive steps (i) bottom-up running [21] in the first place, and then (ii) top-down running [18] in the next. In the first step (i), the running of the RGE's for the third family Yukawa couplings (h t , h b , h τ ) and three gauge couplings  Table II existence of a one-light Higgs doublet and five quark flavors below m t scale [21,22]. Using CERN-LEP data, M Z = 91.187GeV ,α s (M Z ) = 0.118 ± 0.004, α −1 1 (M Z ) = 127.9 ± 0.1, sin 2 θ W (M Z ) = 0.2316 ± 0.0003, and SM relations,  Table II h where ητ . The one-loop RGE's for top quark, bottom quark and τ -lepton Yukawa couplings in the MSSM in the range of mass scales m t ≤ µ ≤ M R are given by  Table II   d where The two-loop RGE's for the gauge couplings are similarly expressed in the range of mass  Table III where Values of h t , h b , h τ , g 1 , g 2 , g 3 evaluated for tanβ = 55 at high scale M R = 1.6 × 10 16 from equation (28)- (30) and (32) Table III high scale t R (= lnM R ) to low scale t o . In this case, we use the input values of Yukawa and gauge couplings evaluated earlier at scale t R from the first stage running of RGE's in case (i). In principle, one can evaluate neutrino masses and mixing angles at every point of the energy scale. It can be noted that in the present problem, the running of other SUSY parameters such as M 0 , M 1/2 , µ, are not required and hence, it is not necessary to supply their input values.
We are now interested in studying radiative generation θ 13 for the case when m 1,2,3 = 0 and s 13 = 0 at high energy scale. Such studies can give the possible origin of the reactor angle in a broken A 4 model. During the running of mass eigenvalues and mixing angles from high to low scale, the non-zero input value of mass eigenvalues m 1,2,3 will induce radiatively a non-zero values of s 13 . Similar approach was followed in [23] considering m 3 = 0. The authors in [23] used inverted hierarchy neutrino mass pattern (m, −m, 0) at high scale.  Table III Such a specific structure of mass eigenvalues however, require fine tuning conditions in the flavor symmetry model at high energy. Instead of assuming a specific relation between mass eigenvalues at high energy scale, here we attempt to find out the most general mass eigenvalues at high energy which can give rise to the correct neutrino data at low energy scale. The only assumption in our work is the opposite CP phases i.e. (m 1 , −m 2 , m 3 ). In another work [24], authors have shown the radiative generation of △m 2 21 considering the non-zero θ 13 at high scale and tanβ values lower than 50. They have also shown that ∆m 2 21 can run from zero at high energy to the observed value at the low energy scale, only if θ 13 is relatively large and the Dirac CP-violating phase is close to π. The running effects can be observed only when θ 13 is non-zero at high-energy scale as per their analysis. In the present work, θ 13 is assumed to be zero at high scale consistent with a TBM type mixing within A 4 symmetric model. We also examine the running behavior of neutrino parameters in a neutrino mass model obeying special kind of µ-τ symmetry at high scale, which was not  Table III studied in the earlier work mentioned above.  Table III in the range 0.01 − 0.12 eV and generate the neutrino parameters at low energy. We restrict the neutrino parameters θ 13 , θ 12 , θ 23 and ∆m 2 21 at low energy to be within the allowed 3σ range and show the variation of ∆m 2 23 (IH), ∆m 2 31 (NH) at low energy with respect to the input mass eigenvalues at high energy. We show the results in figure 1, 2, 3, 4, 5 and 6 for a specific value of tanβ = 55. It can be seen from these figures that the correct value of neutrino parameters at low energy can be obtained only for large values of mass eigenvalues at high energy scale |m 1,2,3 | = 0.08 − 0.12 eV. We then choose two specific sets of mass eigenvalues at high energy scale corresponding to inverted hierarchy and normal hierarchy respectively and show the evolution of several neutrino observables including oscillation parameters, effective neutrino mass m ee = | i U 2 ei m i |, sum of absolute neutrino masses i |m i | in figure 7 , 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20. It can be seen from figure 7 and 14 that the correct value of θ 13 can be obtained at low energy only for very  Table III high values of tan β = 55. The other neutrino parameters also show a preference for higher tan β values. The output values of neutrino parameters at low energy are given in table II and III for both sets of input parameters. The large deviation of θ 13 at low energy from its value at high energy (θ 13 = 0 for TBM at high energy) whereas smaller deviation of other two mixing angles can be understood from the RGE equations for mixing angles (23), (24), (25). Using the input values given in table II and III, the slope of sinθ 13 can be calculated to be h 2 τ 16π 2 (−5.88) and h 2 τ 16π 2 (5.23) for inverted and normal hierarchies respectively. On the other hand, the slope of sinθ 23 at high energy scale is found to be h 2 τ 16π 2 (2.95) and h 2 τ 16π 2 (−2.63) for inverted and normal hierarchies respectively. Thus, the lower value of slope for sinθ 23 results in smaller deviation from TBM values compared to that of sinθ 13 . We also note from figure 12, 19 that the sum of the absolute neutrino masses at low energy is 0.315 eV and 0.3555 eV for inverted and normal hierarchy respectively. This lies outside the limit set by  Table III the Planck experiment Σ|m i | < 0.23 eV [12]. However, there still remains a little room for the sum of absolute mass to lie beyond this limit depending on the cosmological model, as suggested by several recent studies [25]. Ongoing as well as future cosmology experiments should be able to rule out or confirm such a scenario.
It is interesting to note that, our analysis shows a preference for very mild hierarchy of either inverted or normal type at high energy scale which also produces a very mild hierarchy at low energy. This can have interesting consequences in the ongoing neutrino oscillation as well as neutrino-less double beta decay experiments. Also, the large tan β region of MSSM (which gives better results in our model) will undergo serious scrutiny at the collider experiments making our model falsifiable both from neutrino as well as collider experiments.
We note that the present analysis will be more accurate if the two loop contributions [26] RGE's are taken into account.  • Very mild hierarchy (with all neutrino mass eigenvalues having same order of magnitude values and |m 1,2,3 | = 0.08 − 0.12 eV) give correct results at low energy provided the tan β values are kept high, close to 55. Such a preference towards large mass eigenvalues with all eigenvalues having same order of magnitude values can have tantalizing signatures at oscillation as well as neutrino-less double beta decay experiments.
• No significant changes in running of sin 2 θ 23 , sin 2 θ 12 with tan β are observed.
• Sum of absolute neutrino masses at low energy lie above the Planck upper bound Σ|m i | < 0.23 eV [12] hinting towards non-standard cosmology to accommodate a larger Σ|m i | or more relativistic degrees of freedom [25].
• The preference for high tan β regions of MSSM could go through serious tests at collider experiments pushing the model towards verification or falsification.
Although we have arrived at some allowed parameter space in our model giving rise to correct phenomenology at low energy with the additional possibility that many or all of these parameter space might get ruled out in near future, we also note that it would have been more interesting if the running of the Dirac and Majorana CP violating phases [27] were taken into account. We also have not included the seesaw threshold effects and considered all the right handed neutrinos to decouple at the same high energy scale. Such threshold effects could be important for large values of tan β as discussed in [28]. We leave such a detailed study for future investigations.