Status and prospects of `bi-large' leptonic mixing

Bi-large patterns for the leptonic mixing matrix are confronted with current neutrino oscillation data. We analyse the status of these patterns and determine, through realistic simulations, the potential of upcoming long-baseline experiment DUNE in testing bi-large \emph{ansatze} and discriminating amongst them.


INTRODUCTION
The lepton mixing matrix is the leptonic analogue of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. It is given as U = U † l U ν where U l and U ν represent the diagonalization matrices corresponding to the charged-leptons and neutrino sectors, respectively. The peculiar form taken by its matrix elements, which follows from current experimental data, has puzzled theorists ever since oscillations were established. Even more so after reactor experiments established that θ 13 is nonzero [1,2], and T2K has provided a hint for a nearly maximal value of the neutrino oscillation CP phase δ [3]. Many recent attempts to account for non-zero θ 13 have been proposed [4][5][6][7][8], including deviations from the conventional tribimaximal neutrino mixing pattern [9,10], that can provide realistic descriptions of neutrino oscillation data [11,12].
It is well-known that the magnitudes of the elements of the quark mixing matrix can be nicely described in terms of the Cabibbo angle [13]. Motivated by the fact that the smallest lepton mixing angle is similar in magnitude to the largest of the quark mixing angles, it has been suggested that the Cabibbo angle may act as the universal seed for quark and lepton mixings. This idea prompted alternative approaches for describing the structure of the lepton mixing matrix involving bi-large neutrino mixing [14][15][16]. Specially constraining variants of this approach have been recently suggested [17]. Bi-large neutrino mixing implies that the lepton and quark sectors may be related with each other, possibly serving as a new starting point in the quest for quark-lepton symmetry and unification.
Here we examine the status of bi-large proposals for the leptonic mixing matrix by confronting them with current neutrino oscillation data. They require, in addition to the bilarge assumption on the matrix U ν representing the neutrino diagonalization matrix, also the charged-leptons correction factor U l which we take to be "CKM like", motivated from Grand Unified Theory (GUT). By carefully simulating upcoming long baseline oscillation experiments we also determine their potential in testing the bi-large ansatze. For definiteness we focus on the Deep Underground Neutrino Experiment (DUNE) [18,19]. In order to determine its potential to test various bi-large predictions we simulate the experimental features according to their design specifications. What we describe in this letter follows the same general strategy as other symmetry-based studies in the context of DUNE such as those in Refs. [20][21][22][23][24][25], but focusing on the study of bi-large lepton mixing. We determine the sensitivity regions of oscillation parameters within different bi-large ansatze for the lepton mixing matrix. We also discuss the possibility of discriminating amongst different bi-large options.

BEFORE AND AFTER DUNE: THE GENERAL CASE
The proposal and technical details of the next generation superbeam neutrino oscillation experiment DUNE are described in [18,19]. The collaboration plans on using the Long-Baseline Neutrino Facility (LBNF) beam, which will be constructed over the next few years at Fermilab as a neutrino source. The first detector will record particle interactions near the beam source, at Fermilab, while the second, much larger, underground detector at the Sanford Underground Research Facility (SURF) in South Dakota, at about 1300 km distance, will use four 10 kton volume of liquid argon time-projection chambers (LArTPC). The expected design flux corresponds to 1.07 MW beam power which gives 1.47 × 10 21 protons on target per year for an 80 GeV proton beam energy. For our numerical DUNE simulation we use the GLoBES package [26,27] along with the auxiliary files in Ref. [19]. We assume 3.5 years running time in both neutrino and antineutrino modes with a 40 kton detector volume.
We also take into account both the appearance and disappearance channels of neutrinos and antineutrinos in the numerical simulation. Both the signal and background normalization uncertainties for the appearance as well as disappearance channels have been adopted in our analysis as mentioned in the DUNE CDR [19]. Furthermore, as the latest global analysis of neutrino oscillation data tends to favor normal mass hierarchy (i.e., ∆m 2 31 > 0) over inverted mass hierarchy (i.e., ∆m 2 31 < 0) at more than 3σ [28,29], we focus on the first scenario throughout this work. The remaining numerical details that have been adopted here are same as [25] unless otherwise mentioned. Before considering the bi-large schemes we first recall the potential of the DUNE setup in probing the neutrino oscillation parameters within a generic oscillation scenario. The results of our simulation are summarized in Fig. 1.

BI-LARGE MIXING PATTERNS
As already mentioned, in contrast to quark mixing, the observed pattern of lepton mixing is described by two large mixing angles characterizing solar and atmospheric oscillations. A specially intriguing possibility is provided by bi-large mixing ansatze [14]. Apart from fitting observation, another reason for these ansatze is that they fit well within the framework of unified theories [15]. In addition, they open the door to the possibility of having the Cabibbo angle as a universal seed for quark as well as lepton mixings [16]. Here we give an overview of the predictions of bi-large mixing patterns for the lepton mixing matrix. For definiteness we focus on pattern T1, T2 from the recent paper in [17], while T3 and T4 are taken from a previous paper [16]. Following Ref. [16] we assume that the bi-large patterns arise from the simplest Grand Unified Theories, where the charged-lepton and the down-type quarks have roughly the same mass.

Type-1 (T1)
The first bi-large case T1 assumes that the neutrino mixing angles are related with the Cabibbo angle as follows [17] sin θ 23 = 1 − λ, sin θ 12 = 2λ, sin θ 13 = λ . (1) Truncated to O(λ 2 ), the neutrino part of mixing matrix U BL1 is given by In order to construct U BL1 we assume that the phase δ = π. Notice that the observed leptonic mixing angles are obtained once the charged-lepton corrections have been incorporated (see discussion below). Clearly U BL1 on its own cannot be the full leptonic mixing matrix. In order to get the leptonic mixing matrix consistent with current oscillation data we need to take into account appropriate charged-lepton corrections to U BL1 . The SO(10) GUTmotivated, CKM-type charged-lepton corrections for this case are given by Here, sin θ CKM 12 = λ and sin θ CKM 23 = Aλ 2 , where λ = 0.22453 ± 0.00044, A = 0.836 ± 0.015 are the Wolfenstein parameters [13], and Φ = diag{e −iφ/2 , e iφ/2 , 1} with φ being a free parameter. The lepton mixing matrix for T1 is simply U = U † l 1 U BL1 , with the mixing angles and the Jarlskog invariant J CP , given by Note the presence of the next-to-leading order term −2λ 3 sin φ in J CP .
The neutrino diagonalization matrix is approximately of the form As in the previous U BL1 case, in order to construct U BL2 we assumed δ = π. Here, motivated by the SU (5) unification, we take the contribution from the charged-lepton diagonalization matrix to be of the form where θ CKM
where ψ is a free parameter whose value can be fitted from neutrino oscillation data. The cases T3 and T4 differ from each other in the type of charged-lepton corrections used. The exact form of the neutrino diagonalization matrix U BL3 is In order to determine the lepton mixing matrix for this case here we choose δ = 0. For the case T3 the charged-lepton diagonalization matrix is taken to be same as the U l 1 matrix of Eq. (3). The resulting expressions of the mixing parameters are sin 2 θ 13 λ 2 − 2ψλ 3 cos φ + −1 + 2Ac cos φ + ψ 2 λ 4 , Type-4 (T4) The T4 case also assumes the neutrino part of the bi-large mixing matrix to be the same as U BL3 , as given by Eq. (10). The T4 case differs from the T3 case in the type of correction assumed for the charged-lepton diagonalization matrix. In this case, the latter has the form U l 2 given in Eq. (7). The corresponding expressions for the mixing parameters are sin 2 θ 13 λ 2 + 2ψλ 3 cos φ − 1 − ψ 2 λ 4 , sin 2 θ 12 c 4 + 2c 2 ψ cos φ + ψ 2 λ 2 + c 4 − ψ 2 λ 4 , sin 2 θ 23 ψ 2 λ 2 + 2 (Ac − cos φ) ψλ 3 + 1 − ψ 2 − 2Ac cos φ + A 2 c 2 λ 4 , After introducing the various types of bi-large mixing patterns, we now proceed to study their current status in the next section.

Current Status of bi-large Mixing Schemes
We first discuss the current status of the four patterns. One sees that, T1 and T2 are one-parameter patterns for the lepton mixing matrix, depending only on the parameter φ, to be adjusted to reproduce all angles and the Dirac CP phase δ. Thus, all the mixing angles and δ are strongly correlated. The allowed range of φ can be obtained by requiring all the mixing angles and Dirac phase δ to lie within their current 3σ experimental ranges [28]. Thus, we find φ/π ∈ [0.749, 0.779] and [0.218, 0.251] for T1 and T2, respectively 1 .   On the other hand T3 and T4 depend on two parameters, ψ and φ. The allowed parameter ranges for ψ and φ can be found by fitting the current oscillation data with respect to them as shown in Fig. 2. This Figure provides updated variants of the Figure given in [16]. As seen from Fig. 2, current oscillation data restrict the range of both ψ and φ to narrow ranges. In particular the value of ψ is restricted to a small band around ψ = 3. In order to easily compare all the four bi-large cases on the same footing, henceforth we will fix ψ = 3 for both T3 and T4 ansatze. This allows us to put all the results for the four bi-large ansatze in a common figure. It should be noted however that ψ = 3 is merely a convenient benchmark choice. Other values of ψ within the narrow allowed band of Fig. 2 are equally allowed and will lead to different predicted values for the mixing parameters. In this work we will only discuss the implications of picking the benchmark value ψ = 3.
In Fig. 3, we show the parameter regions allowed by the current global-fit of neutrino oscillation data in the sin 2 θ 23 −δ plane at 1σ (yellow), 2σ (dark-yellow) and 3σ (darker-yellow) confidence level. These are indicated by the filled contours, respectively. The predictions for sin 2 θ 23 and δ are shown by the magenta, cyan, pink and brown curves for the T1, T2, T3 and T4, respectively. However, we emphasize here that the numerical analysis performed for the different bi-large schemes are based on exact formulae and not on the leading order approximations given at the beginning of this section.
Note that each bi-large ansatz predicts two values of δ, symmetrically placed across the δ = π line. Since the current oscillation data disfavors δ < π at more than 3σ, in what follows we only take the upper predicted value of the δ for the various bi-large scenarios. From the figures one can see that the predicted values (the one in upper half of CP Plane) of the scenarios T1, T2 and T4 all have good agreement with the latest global-fit of neutrino oscillation data. However, the predicted value of T3 case is more than 3σ away from the current value, clearly disfavoured by the latest global-fit [28]. Thus, henceforth we will disregard T3 as a true value.
Notice that the dark-blue marks in the curves are obtained by requiring sin 2 θ 13 to lie within its current 3σ range [28]. For example, one sees that the solutions of sin 2 θ 23 , δ with (sin 2 θ 23 , δ) ∼ (0.57, 1.32π), (0.51, 1.28π) and ∼ (0.51, 1.78π) are allowed by the current oscillation data for T1, T2 and T4, respectively. Having discussed the current status of the different bi-large mixing schemes, in the next section we examine their testability at the upcoming DUNE experiment.

TESTING BI-LARGE SCENARIOS AT DUNE
Adopting these values of (sin 2 θ 23 , δ) as the seed points (see section 3 for details), we examine sensitivity regions of DUNE. We show the allowed parameter space considering 1σ (cyan), 2σ (light-cyan) and 3σ (green) filled-color contours. From Fig. 4, one can see that if T1 predicted value is true value, then DUNE after 3.5 + 3.5 years running time can rule out all the other bi-large schemes at more than 3σ.
From Fig. 5, one can see that if the T2 predicted value is the true value, then DUNE after 3.5 + 3.5 years run can rule out all of the other bi-large schemes at more than 3σ. The T1 and T3 cases would be ruled out as their predicted value for sin 2 θ 23 are very different, while for the case of T4 the CP phase is very different. As can be seen from Fig. 5 DUNE has enough sensitivity to the CP phase to rule out T4 as well.
By adopting the T4 predicted value of sin 2 θ 23 , δ as our true value, we present DUNE's sensitivity region in Fig. 6. We find that, again, in this case DUNE has sensitivity to rule out all the other schemes. T3 would be ruled out by its predicted value for sin 2 θ 23 while for the cases of T1 and T2 the CP phase also plays an important role. For example, in the case of T2 the phase gives the main sensitivity.
Finally, by assuming the current best fit value of the global neutrino oscillation fit as the true value, we show the sensitivity in the (sin 2 θ 23 , δ) plane after 3.5 + 3.5 years run DUNE in Fig. 7. One sees that DUNE can rule out the T3, T4 bi-large scheme at more than 3σ, while the T1 predicted seed point can be disfavored at more than 2σ. The T2 scheme can only be disfavoured at the 1σ level.  true value, as shown by the 'star-mark', against which the other proposals are tested. The filled-color contours represent the DUNE sensitivity contours after 3.5 + 3.5 years of run.

FIG. 7:
Capability of DUNE to test bi-large proposals is shown by the filled-color contours. Here current global best-fit value [28] is taken as true value, as shown by the 'star-mark', against which the other proposals are tested.

SUMMARY
We described four bi-large patterns for the lepton mixing matrix. While T1 and T2 are one-parameter patterns, with a single parameter φ to account for all leptonic mixing angles and Dirac CP phase δ, T3 and T4 depend on two parameters ψ and φ. We have described the current status of the four bi-large patterns obtained by confronting them with current oscillation data. One of the noticeable points is that when we combine the latest constraint of sin 2 θ 13 with the bi-large scheme predictions we obtain very stringent restrictions. For definiteness in the T3 and T4 cases we fixed ψ = 3, close to the actual allowed value preferred by the analysis of the existing oscillation data. This allows us to put all the results for the four bi-large ansatze on the same footing for comparison. Finally, we described the potential of the DUNE experiment in testing these patterns for the lepton mixing matrix.