Are the small neutrino oscillation parameters all related?

Neutrino oscillations reveal several small parameters, namely, $\theta_{13}$, the solar mass splitting {\em vis-\`{a}-vis} the atmospheric one, and the deviation of $\theta_{23}$ from maximal mixing. Can these small quantities all be traced to a single source and, if so, how could that be tested? Here a see-saw model for neutrino masses is presented wherein a dominant term generates the atmospheric mass splitting with maximal mixing in this sector, keeping $\theta_{13} = 0$ and zero solar splitting. A Type-I see-saw perturbative contribution results in non-zero values of $\theta_{13}$, $\Delta m^2_{solar}$, $\theta_{12}$, as well as allows $\theta_{23}$ to deviate from $\pi/4$ in consistency with the data while interrelating them all. CP-violation is a natural consequence and is large ($\delta \sim \pi/2, 3\pi/2$) for inverted mass ordering. The model will be tested as precision on the neutrino parameters is sharpened.

Information on neutrino mass and mixing have been steadily emerging from oscillation experiments.Among them the angle 1 θ 13 is small (sin θ 13 ∼ 0.1) [1] while global fits to the solar, atmospheric, accelerator, and reactor neutrino oscillation data indicate that θ 23 is near maximal (∼ π/4) [2,3].On the other hand, the solar mass square difference is two orders smaller than the atmospheric one.These mixing parameters and the mass ordering are essential inputs for identifying viable models for neutrino masses.Here a proposal is put forward under which the atmospheric mass splitting and maximal mixing in this sector, θ 23 = π/4, arise from a zero-order mass matrix while the smaller solar mass splitting and realistic θ 13 and θ 23 are generated by a Type-I see-saw [4] which acts as a perturbation.θ 12 also arises out of the same perturbation and as a consequence of degeneracy is not constrained to be small.Attempts to generate some of the neurino parameters by perturbation theory are not new [5], but to our knowledge there is no work in the literature that indicates that all the small paramters could have the same perturbative origin.
The unperturbed neutrino mass matrix in the mass basis is 3 } with the mixing matrix of the form (1) Here ∆m 2 atm = (m are taken to be real and positive.The columns of U 0 are the unperturbed flavour eigenstates. 2As stated, ∆m 2 solar = 0 and θ 13 = 0. Since the first two states are degenerate in mass, at this stage it is not meaningful to define θ 12 .It is possible to generate this mass matrix from a Type-II see-saw.
In the flavour basis the mass matrix is ( The perturbation is obtained by a Type-I see-saw.In the flavour basis the Dirac mass term is taken to be proportional to the identity, i.e., M D = m D I. In this basis, out of the three right-handed neutrinos the first two form an effective Dirac state while the third is decoupled from the rest.Thus where x, y are dimensionless constants of O(1).No generality is lost by keeping the Dirac mass real.
As a warm-up consider first the real case, i.e., φ 1 = 0 or π, φ 2 = 0 or π.For notational convenience in the following the phase factors are not displayed; instead x (y) is taken as positive or negative depending on whether φ 1 (φ 2 ) is 0 or π.Negative x and y offer interesting variants which are stressed at the appropriate points.
The Type-I see-saw contribution in the mass basis is: The effect on the solar sector is governed by the submatrix of M ′mass in the subspace of the two degenerate states, To first order in the perturbation: For y/x = 1 one obtains the tribimaximal mixing value of θ 12 which, though allowed by the data3 at 3σ, is beyond the 1σ region.Since for the entire range of θ 12 one has tan 2θ 12 > 0, x and y must be chosen of the same sign.Therefore, either From the global fits to the experimental results one finds: 0.682 < y x < 1.075 at 3σ .
Further, from eq. ( 5), To first order in the perturbation the corrected wave function |ψ 3 is: where For positive x the sign of κ is fixed by that of m − .Since by convention all the mixing angles θ ij are in the first quadrant, from eq. ( 9) one must identify: where for x > 0 δ = 0 for normal mass ordering (NO) and δ = π for inverted mass ordering (IO).Needless to say, both these cases are CP conserving.If x is negative then NO (IO) would correspond to δ = π (0).
In Fig. 1 the global-fit 3σ range of sin θ 13 and tan 2θ 12 is shown as the blue dot-dashed box with the best-fit value indicated by a violet dot.Once the atmospheric and solar mass splittings are fixed, for  12) with m 0 = 2.5 meV when the best-fit values of the two mass-splittings are used.The portion below the green solid (dashed) straight line is excluded by θ 23 at 3σ -eq.( 16) -for the first (second) octant for real M ′ .For inverted ordering no solution of eq. ( 12) is allowed for real M ′ .
any point within this region eq.( 14) determines a z, or equivalently an m 0 , which leads to the correct solar splitting.
From the data [2] ω min = 0 for both octants and ω max = 6.6 • (−8.3 • ) for the first (second) octant at 3σ.As | cos δ| = 1 for the real M ′ case, in this model one has from eq. ( 16) for both octants |ω| ≥ 5.14 • at 3σ.Thus the range of θ 23 that can be obtained is rather limited 4 .The green solid (dashed) straight line is from eq. ( 16) for ω max for the first (second) octant.The region below this line is excluded in this model.Note that the best-fit point is permitted only for θ 23 in the second octant.
In summary, for real M ′ the free parameters are m 0 , m 2 D /xm R and y with which the solar mass splitting, θ 12 , θ 13 , θ 23 are reproduced for normal mass ordering.Inverted ordering cannot be accommodated.
Reverting now to the complex M R in eq. ( 3) one has in the mass basis in place of eq. ( 4): x and y are now positive.M ′ is no longer hermitian.This is addressed, as usual, by defining the hermitian combination (M 0 + M ′ ) † (M 0 + M ′ ) and treating M 0 † M 0 as the unperturbed term and (M 0 † M ′ + M ′ † M 0 ) as the perturbation to lowest order.The zero order eigenvalues are now (m and the complex yet hermitian perturbation matrix is where The subsequent analysis is similar to the one for real M ′ . The perturbation which splits the degenerate solar sector is the 2 × 2 block of eq. ( 18).The solar mixing angle now is The limits of eq. ( 7) apply on this combination.Also, (cos φ 1 / cos φ 2 ) must be positive.
Including first order corrections the wave function |ψ 3 is One immediately has sin θ 13 cos δ = κ cos φ 1 , The sign of cos δ is the same as (opposite of) sgn(cos φ 1 ) for normal (inverted) mass ordering.Further, sin φ 1 determines the combination sin θ 13 sin δ that appears in the Jarlskog parameter, J, a measure of CP-violation.Note, φ 2 plays no role in fixing the CP-phase δ.
It is seen that for normal ordering the quadrant of δ is the same as that of φ 1 .For inverted ordering δ is in the first (third) quadrant if φ 1 is in the second (fourth) quadrant and vice-versa.
A straight-forward calculation after expressing m D and m R in terms of sin θ 13 cos δ, yields which bears a strong similarity with eq. ( 12) for real M ′ .Eqs. ( 13) and ( 14) continue to hold.To ensure the positivity of ∆m 2 solar , noting the factors determining the sign of cos δ, one concludes that sgn(cos φ 1 cos φ 2 ) must be positive for both mass orderings.Thus, satisfying the solar mass splitting leaves room for either octant of θ 23 for both mass orderings.The allowed range of δ can be easily read off if we rexpress eq. ( 14) as: In the following m 0 , θ 13 , and θ 12 are taken as inputs and δ and θ 23 are obtained using eqs.( 26) and (24).From these the CP-violation measure, J, and the combination |m νeνe | which determines the rate of neutrinoless double beta decay are calculated.
In the left panel of Fig. 2 is shown (thick curves) the dependence of θ 23 on the lightest neutrino mass m 0 when the neutrino mass square splittings and the angles θ 13 and θ 12 are varied over their allowed range at 3σ.The thin curve corresponds to taking the best-fit values.In all plots in this figure the green (pink) curves are for normal (inverted) mass ordering while solid (dashed) curves are for solutions in the first (second) octant.For inverted ordering the thick and thin curves are very close and cannot be distinguished in this panel.Notice that the 3σ predictions from this model are not consistent with θ 23 = π/4.As expected from eq. ( 24), θ 23 values are symmetrically distributed around π/4.Its range for inverted ordering falls outside the 1σ global fits but are consistent at 3σ.An improvement in the determination of θ 23 will be the easiest way to exclude one of the orderings unless one is in the quasidegenerate regime.
For normal ordering the smallest value of m 0 is determined by the 3σ limits of θ 23 in the two octants.Eq. ( 25) permits arbitrarily small m 0 for inverted mass ordering (see below).In the inset of this panel |m νeνe | has been plotted.The sensitivity of direct neutrino mass measurements is expected to reach around 200 meV [6] in the near future.Planned neutrinoless double beta decay experiments will also probe the quasidegenerate range of m 0 [7].As can be seen from Fig. 2 to distinguish the two mass orderings at least a further one order improvement in sensitivity will be needed.Long baseline experiments or large atmospheric neutrino detectors such as INO will settle the mass ordering more readily.
In the right panel is displayed the variation of δ with m 0 for both mass orderings while J is shown in the inset.The conventions are the same as in the left panel.The sign of J is positive if δ is in the first or second quadrant and is negative for the other cases.As noted, the quadrant of δ (and the associated sign of J) can be altered by the choice of the quadrant of φ 1 .However, from eq. ( 26) for these alternatives, namely, ±δ and (π ± δ), the dependence of | cos δ| on m 0 is the same for a particular mass ordering.With this proviso in mind, Fig. 2 (right panel) has been plotted keeping δ in the first quadrant and J has been taken as positive.
J, which is proportional to sin 2θ 23 , has no dependence on the octant of θ 23 as the latter is symmetrical around π/4.In both panels, for normal ordering a slightly larger range of m 0 is allowed when θ 23 is in the second octant.For the region where both octants are allowed the curves in the right panel completely overlap.For inverted mass ordering both δ and J remain nearly independent of m 0 .
For m 0 smaller than 10 meV, the CP-phase δ is significantly larger for inverted ordering.This could provide a clear test of this model when the mass ordering is known and CP-violation in the neutrino sector is measured.The real limit is seen to be admissible, as expected from Fig. 1, only for normal ordering and that too not for the entire 3σ range, with the second octant allowing a larger region.
Since 0 ≤ z ≤ 1/2 for NO and 1/2 ≤ z ≤ 1 for IO, the allowed values of δ in the two orderings as seen from eq. ( 26) are complementary tending towards a common value as z → 1/2, the quasidegenerate limit, which begins to set in from around m 0 = 100 meV.The main novelty from the real M ′ case is that in eq. ( 14) by choosing cos δ sufficiently small one can make z ≡ m − m (0) 1 /∆m 2 atm ∼ 1 so that solutions exist for m 0 for inverted mass ordering corresponding to even vanishing m 0 unlike the case of normal ordering where the lower limit of m 0 is set by cos δ = 1, i.e., real M ′ .
We have checked that the size of the perturbation is at most around 20% of the unperturbed contribution for all cases.
In conclusion, a model for neutrino masses has been proposed in which the atmospheric mass splitting together with θ 23 = π/4 has an origin different from that of the solar mass splitting, θ 12 , θ 13 , and ω = π/4 − θ 23 , all of which arise from a single perturbation resulting from a Type-I see-saw.The global fits to the mass splittings, θ 12 and θ 13 completely pin-down the model and the CP-phase δ and the octant of θ 23 are predicted in terms of the lightest neutrino mass m 0 .Both mass orderings are allowed, the inverted ordering being associated with near-maximal CP-violation.Both octants of θ 23 can be accommodated.Further improvements in the determination of θ 23 , a measurement of the CP-phase δ, along with a knowledge of the neutrino mass ordering will put this model to tests from several directions.
Acknowledgements: SP acknowledges a Junior Research Fellowship from CSIR, India.AR is partially funded by the Department of Science and Technology Grant No. SR/S2/JCB-14/2009.

Figure 1 :
Figure1: The blue dot-dashed box is the global-fit 3σ allowed range of sin θ 13 and tan 2θ 12 .The best-fit point is shown as a violet dot.The red dotted curve is from eq. (12) with m 0 = 2.5 meV when the best-fit values of the two mass-splittings are used.The portion below the green solid (dashed) straight line is excluded by θ 23 at 3σ -eq.(16) -for the first (second) octant for real M ′ .For inverted ordering no solution of eq.(12) is allowed for real M ′ .

Figure 2 :
Figure 2: Different observables as a function of the lightest neutrino mass m 0 (in eV).The green (pink) curves arefor the normal (inverted) mass ordering.For every plot the region allowed at 3σ is between the thick curves while the thin curves are for the best-fit values of the inputs.The solid (dashed) curves correspond to the first (second) octant of θ 23 .The variation of θ 23 is displayed in the left panel.In the inset is shown the effective mass |m νeνe | (in eV) contributing to neutrinoless double beta decay.The CP-phase δ is presented in the right panel and the inset shows the leptonic CP-violation measure J.