Reducing \theta_13 to 9 degrees

We propose to consider the possibility that the observed value of $\theta_{13}$ is not the result of a correction from an initially vanishing value, but rather the result of a correction from an initially larger value. As an explicit example of this approach, we consider analytically and numerically well-known CKM-like charged lepton corrections to a neutrino diagonalization matrix that corresponds to a certain mixing scheme. Usually this results in generating $\theta_{13} = 9^\circ$ from zero. We note here, however, that 9 is not only given by $0 + 9$, but also by $18 - 9$. Hence, the extreme case of an initial value of 18 degrees, reduced by charged lepton corrections to 9 degrees, is possible. For some cases under study new sum rules for the mixing parameters, and correlations with CP phases are found.


I. INTRODUCTION
Remarkable experimental activity in the past decades has established that the phenomenon of neutrino flavor transition is described by neutrino oscillations.Recent measurements of the smallest mixing angle θ 13 at reactor [1][2][3][4] and accelerator [5] neutrino experiments have finally led to an emerging picture where the order of magnitude of all elements of the PMNS matrix is known.Theorists now face the task to understand and/or explain that structure.Most flavor symmetry models [6][7][8] were constructed when only an upper limit on θ 13 was known, and therefore aimed at explaining θ 13 = 0. Corrections to generate a non-zero value are then applied.In the present paper we depart from the historically motivated approaches to generate non-zero θ 13 from an initially vanishing value, and consider the possibility that initially θ 13 is already large.Now the usual corrections to model predictions can reduce the initial value of θ 13 to its observed value.Of course, the phenomenology will then be different from the standard case.As an explicit example on the consequences that follow, we consider charged lepton corrections.
No matter if neutrinos are Majorana or Dirac particles, the lepton flavor mixing matrix stems from the mismatch between the diagonalization of the charged lepton mass matrix m ℓ and the neutrino mass matrix m ν , i.e.
where U ℓ and U ν are the unitary matrices diagonalizing m ℓ and m ν , respectively.Now one can apply the following strategy to generate non-zero θ 13 = arcsin |U e3 |.Assuming that (U ν ) 13 = 0, as well as (U ν ) 23 = (U ν ) 33 , and that U ℓ is related to the CKM matrix, i.e. essentially the unit matrix except for (U ℓ ) 12 = λ = sin θ C , it follows that |U e3 | = λ/ √ 2, or θ 13 = 9 • = 0 + 9 • .Numerically, this is basically the observed value of about θ 13 = 9 • , and the fact that this lepton mixing parameter is numerically connected to quark parameters seems to support this argument, but is of course not a proof 1 .Nevertheless, relating the charged lepton diagonalization to the CKM matrix can be arranged in grand unified models, especially based on SU (5), for which m ℓ = m T d is a typical outcome.Such a relation has to be viewed as an approximation due to the distinct mass spectra of leptons and quarks, and is modified by higher order corrections or Clebsch-Gordon coefficients.Nevertheless, models predicting U CKM ≃ U ℓ have been constructed, which in addition have (U ν ) 13 = 0 [9][10][11][12][13].Hence, the above strategy to generate |U e3 | = λ/ √ 2, where λ ≃ sin θ C ≃ 0.23, is based on actual model building foundations.We will use for the sake of simplicity and definiteness While the relation 9 • = 0+9 • has its virtues and attraction, one should not ignore the possibility that 9 • = 18 • −9 • .This means that initially U ν contains a too large value of its 13-element, which is reduced to its observed value by a sizable charged lepton correction, a CKM-like one in our case.Since the remaining lepton mixing angles are necessarily non-zero both in U and in U ℓ , the question arises whether θ 13 should initially be non-zero in the first place.This so far overlooked possibility is what we investigate here, by performing a general analysis of Eq. (1) when U ℓ is fixed to the CKM matrix.The case of initially vanishing (U ν ) 13 = 0 has been analyzed countless times, but the cases when have never been considered.As a result we find new interesting sum rules, and also note the already mentioned extreme case of reducing θ 13 from 18 degrees to 9 degrees, where the initial value could be obtained from flavor symmetries, as 18 • = π/10 is related to symmetries of geometrical objects.
The remainder of this paper is organized as follows.In Sec.II, we present the general formalism and derive the charged lepton corrections to an arbitrary U ν .Interesting sum rules between neutrino mixing parameters are summarized.In Sec.III, a detailed numerical analysis of the model parameters and predictions is performed.Finally, in Sec.IV, we state our conclusions.

II. METHODOLOGY
In the picture of three-flavor neutrino oscillations, the lepton flavor mixing is described by a 3 × 3 unitary matrix U , which is conventionally parametrized by three mixing angles (θ 12 , θ 23 and θ 13 ), and three CP violating phases out of which one is the Dirac phase (δ) and the other two are the Majorana phases (ρ and σ).In the standard parametrization, the lepton mixing matrix is given by where s ij ≡ sin θ ij and c ij ≡ cos θ ij (for ij = 12, 23, 13).In case of Dirac neutrinos the phases ρ and σ will be irrelevant.The results of this paper are independent on the nature of the neutrino.The latest global analysis of current neutrino oscillation data yields [14] sin2 θ 12 = 0.313 +0.013 −0.012 , sin 2 θ 23 = 0.444 +0.036 −0.031 , sin 2 θ 13 = 0.0244 +0.0020 −0.0019 , where short baseline reactor data with baseline shorter than 100 m are not included.Another recent fit result is obtained in [15], with similar results.There are also non-trivial results on the CP phase δ, with best-fit results around 3π/2, or cos δ ≃ 0. However, the 1σ ranges are very large, including essentially also the case cos δ ≃ −1.We note that for some of the cases that we will discuss it is actually crucial whether cos δ is 0 or −1, and therefore we use only the obtained ranges of the mixing angles in our fits.
The concrete form of U ℓ cannot be fixed unless a specific mode is considered.Motivated by the connection between the CKM matrix and U ℓ in many grand unified models we assume here for definiteness U ℓ = U CKM .As for the unitary matrix U ν diagonalizing the neutrino mass matrix, one can parametrize it in analogy to U by using three rotation angles θ12 , θ23 , and θ13 together with a phase φ.Note that we have ignored the Majorana-like phases in this parametrization, since they are located on the right-hand-side of U ν and hence do not affect our discussions on the mixing angles and Dirac CP phase.Now the lepton flavor mixing matrix is given by 2 Here P = diag(e ix , e iy , 1) is a phase matrix stemming from the mismatch between U e and U ν [17].
We proceed to expand the mixing matrix U in order to obtain the charged lepton corrections.Different from the lepton sector, the CKM matrix takes a nearly diagonal form, and is typically parametrized by using four parameters (λ, A, ρ and η) in the Wolfenstein parametrization.Since we are mainly interested in the lepton flavor mixing which has not been measured as precisely as U CKM , we will keep the Wolfenstein parametrization only up to λ 2 , i.e.
Now, by inserting Eq. ( 5) into ( 4) we obtain the matrix elements of U to order λ as3 where ϕ = x − y has been defined, and the notation sij ≡ sin θij , cij ≡ cos θij is adopted.Since the charged lepton mixing matrix takes the CKM form, only the 12-rotation plays a role.Consequently, one can rotate away one of the phases, leaving only the difference between two CP phases x and y in the above results.
Comparing with the standard parametrization given in Eq. ( 2), we find where the O(λ 2 ) terms are only kept for sin 2 θ 13 , since θ 13 is relatively smaller compared to the other mixing angles.
As for the Dirac phase δ, to leading order we have where s φ = sin φ, s ϕ = sin ϕ and so on.It might also be useful to express the Jarlskog invariant [19,20] in terms of the model parameters, i.e.
where, as usual, JCP is defined as Of course, even if θ13 = 0 is assumed, CP violation can still be induced by the λ correction, when sin ϕ = sin(x−y) = 0.Both θ 13 and θ 23 are independent of θ12 at leading order.The leading corrections to θ 13 and θ 23 are proportional to λ s13 , whereas the leading correction to θ 12 is proportional to λ.This indicates that a larger deviation of θ12 from θ 12 than for the other mixing angles is allowed.However, there are terms including cosines of phases in the expressions, which can suppress the corrections.Note that the same combination of phases appears in the expressions for sin 2 θ 23 and sin 2 θ 13 , which implies a correlation between both observables, if the second order term in sin 2 θ 13 can be ignored.It reads In case φ = 0, there is a correlation between the 23-and 12-sectors: However, the general case is complicated and depends on many parameters.The obvious extreme cases are θ13 = 0, θ13 > θ 13 and θ13 ≃ θ 13 .We will in the following discuss these cases analytically, before performing a general numerical analysis.
In the tri-bimaximal based case, δ has to be close to π/2 (or 3π/2) in order to suppress the θ 13 correction to sin 2 θ 12 = 1/3.The situation is however different in the bimaximal case, in which a sizable and negative θ 13 -correction is required in order to reduce the maximal mixing value sin 2 θ12 = 1/2.Hence, δ ≃ π or 2π has to be fulfilled.This interplay of the mixing scheme (bimaximal/tri-bimaximal) in U ν and the Dirac phase in neutrino oscillations has first been noticed in [29].Recall that the fit results from Ref. [14,15] include at 1σ essentially both cases, δ ≃ 2π and δ ≃ 3π/2, where the latter value is close to the best-fit one.

B. The case of θ13 > θ13
If θ13 is larger than the observed value of θ 13 , the term proportional to λ 2 term in Eq. ( 10) can be neglected, leaving us with a set of novel sum rules.Appealing values of the initial value are e.g.θ13 = π/10 or θ13 = π/12.Assuming θ13 = π/10 (or θ13 = 18 • ) and for simplicity also θ23 = π/4, the following sum rules can be deduced: Thus, using the measured value θ 13 ≃ 9 • and Eq. ( 20), one predicts θ 23 ≃ 47.3 • .Another interesting example is θ13 = π/12 (or θ13 = 15 • ), which leads to the following rum rules, By inserting θ 13 = 9 • into Eq.( 23) we obtain the prediction θ 23 ≃ 46.3 • .As in the previous example, we find θ 23 in the second octant.It is obvious from Eq. ( 13) or from (6−9) that in case ( Ũν ) e3 > U e3 at leading order δ ≃ φ holds.In addition, from (22) it is clear that the first and second terms should cancel to a large extent in order to reduce to the observed value of |U e3 | 2 .To this end, the cosine in (22) should be close to 1, which gives Similar to the discussion in the previous subsection, if sin 2 θ12 = 1/3 holds, δ ≃ π/2 (or 3π/2) is required to suppress its corrections to θ 12 .In contrast, for sin 2 θ12 = 1/2, δ ≃ π is expected in order to avoid a too large solar mixing angle.Amusingly, the correlation between sin 2 θ12 and CP violation is identical to the one for vanishing θ13 .Both cases can in principle be distinguished by their prediction for θ 23 , see the blue and red points in the lower left plot in Fig. 6.

C. The case of θ13 ≃ θ13, or sin θ13 ≃ sin θ23λ
This is obviously the most complicated case, and does not allow much analytical results.The Dirac CP phase is determined by δ = −Arg(s 13 e −iφ − s23 c13 e −iϕ λ) .(26) or by Eq. ( 13).In principle, any value for δ is possible.As an interesting example, we look at the scenario with θ13 = 9 • (or θ13 = π/20).In this special case, the sum of the first and third term of Eq. ( 10) is about 0.05, the same size as the second term if the cosine would not be there.Since the measured θ 13 is also very close to 9 • , one would naturally expect that the phase difference between φ and ϕ is around ±π/3.Concretely, we have the following relation Note also that corrections to θ 12 are not sensitive to θ13 as shown in the general formula (11), which implies that the CP phase δ is restricted to be close to ±π/6 and ±π/3 for s2 12 = 1/3 and s2 12 = 1/2, respectively.

III. NUMERICS
In this section we fit the five parameters ( θ12 , θ23 , θ13 , φ and ϕ) to the experimental data using the exact form of Eq. (4).To figure out the allowed parameter spaces of the model parameters, we compare the latest global-fit data with a χ 2 -function defined as where θ 0 ij represents the experimental data given in Eq. ( 3), σ ij denote the corresponding 1σ absolute errors, and θ ij are the predictions of the model and can be expressed in terms of the model parameters.For the color contours, we have fixed θ23 = 45 • .In the upper panel, we allow all phases to freely vary between 0 and 2π.In the middle panel, we switch off φ but not ϕ, whereas in the lower panel, all CP phases are set to zero.

A. θ12-θ13 plane
We start from projecting the parameter space to the θ12 -θ13 plane.The parameter ranges for θ12 and θ13 are shown in Fig. 1 using contour lines for the most general case.We also consider the case of maximal θ23 using colored contours, and make assumptions about the CP phases.
From Fig. 1 we see that θ13 can be as large as 19.2 • , which inspires us with mixing patterns such as sin 2 (π/10) = (3 − √ 5)/8 and sin 2 (π/12) = (2 − √ 3)/4.Such values of π divided by n can be obtained in flavor symmetry models such as in Refs.[35,36].The range of θ12 is wide and a maximal θ12 can be accommodated.If θ23 is fixed to π/4, the parameter space shrinks only slightly, which is a consequence of the suppressed (by both λ and θ13 ) correction terms to θ23 , see Eq. ( 12).In the limit φ = 0, for which the 12-and 13 sectors are correlated, see Eq. ( 17), a sizable θ13 demands a relatively large value of cos ϕ in order to suppress its contribution to θ 13 .This in turn requires θ12 to be FIG.2: The parameter ranges of θ12 and θ23 at 1, 2 and 3σ.For the color contours θ13 = 0 (upper row) or θ13 = π/10 (lower row) is fixed, but the choices of the phases are different.In the left column, we allow all phases to freely vary between 0 and 2π, whereas in the right column, all phases are set to zero.close to maximal.In contrast, if θ13 is tiny, the constraint on θ12 becomes less stringent, which can be seen clearly from our analytical results Eq. (11).Explicitly, for a vanishing θ13 , one has the approximate relation sin θ 13 ≃ λ sin θ 23 .In such a case, the leading order correction to θ12 is flexible since it is proportional to cos ϕ.If all phases are zero, a significant and negative correction to θ12 is expected, and consequently only the nearly maximal value θ12 ≃ π/4 can be accommodated.

B. θ12-θ23 plane
The allowed parameter space in the θ12 -θ23 plane is shown in Fig. 2. As special cases, we choose θ13 = 0 and θ13 = π/10, both for the general case and for all phases being set to zero.
As expected from the suppressed corrections to θ23 , the parameter range of θ23 is similar to that of θ 23 .If we neglect the CP phases, θ13 = 0 leads to a large negative correction to θ12 , and a relatively larger θ12 is favored.In case of large θ13 , θ23 is driven towards smaller values, see Eq. ( 16).

C. θ13-θ23 plane
The allowed parameter space in the θ13 -θ23 plane is shown in Fig. 3.As special cases we choose sin 2 θ12 = 1/3 and As the figure shows, θ13 and θ23 are not sensitive to the choice of θ12 , which has already been shown in the analytical part above, cf.Eqs.(10,12).They are however very sensitive to the CP phases, i.e. φ = 0 restricts the range of θ13 down to −10 • < ∼ θ13 < ∼ 10 • in the case of sin 2 θ12 = 1/3, and in two distinct regions around 0 and 18 • in the case of sin 2 θ12 = 1/2, with θ13 ∼ 9 • being excluded.It is worth noting that, when all the phases are set to zero, there is no parameter space for sin 2 θ12 = 1/3, since the derived θ 12 is too small.FIG.3: The parameter range of θ13 and θ23 at 1, 2 and 3σ.For the color contours, we fix sin 2 θ12 = 1/3 in the upper row and sin 2 θ12 = 1/2 in the lower row.In the left column, we allow all phases to freely vary between 0 and 2π, whereas in the right column φ = 0 is fixed.

D. ϕ-θ12 plane
As pointed out in the analytical section, the phase difference ϕ = x − y is very crucial for certain mixing patterns, in particular for θ12 .Thus, we illustrate the relation between ϕ and θ12 in Fig. 4. The correlation between small phases for sin 2 θ12 = 1/2 and phases around π for sin 2 θ12 = 1/3 is reproduced.Note that this feature is present for all values of θ13 .

IV. CONCLUSIONS
Since for a long time only an upper limit on θ 13 existed, most neutrino models were constructed to generate zero θ 13 .The recent finding of a sizable value, θ 13 = 9 • , have led to many studies on generating that value from an initially zero value.We have noted here that this approach may be misleading, and that in fact θ 13 could have initially been larger.The routinely applied corrections in models will then reduce θ 13 to the observed value, a possibility usually not taken into account.We illustrated the consequences of this approach in an explicit example based on charged lepton corrections 4 .
An extreme case is that initially θ 13 corresponds to 18 • , or π/10.It is then corrected by sin θ C / √ 2 to the observed value of 9 • .Hence, here we do not have 0 + 9 = 9, but rather of 18 − 9 = 9.An analytical and numerical study of the general case was performed, revealing new correlations and sum rules, different from the usually considered charged lepton corrections, that are based on initially vanishing θ 13 .We find that the correlation of maximal CP violation (δ = π/2) for initial tri-bimaximal mixing and CP conservation (δ = π) for initial bimaximal mixing is present for both extreme cases, initial θ 13 = 18 • and θ 13 = 0.
We conclude that the possibility of a more complex mixing pattern than usually considered should not be ignored.The simple framework studied here is one example where a departure from the usual approaches results in interesting and novel phenomenology.

20 FIG. 1 :
FIG.1: Parameter ranges of θ12 and θ13 at 1, 2 and 3σ.For the color contours, we have fixed θ23 = 45 • .In the upper panel, we allow all phases to freely vary between 0 and 2π.In the middle panel, we switch off φ but not ϕ, whereas in the lower panel, all CP phases are set to zero.

FIG. 4 :
FIG.4:The parameter range of ϕ and θ12 at 1, 2 and 3σ.All other model parameters are marginalized.