Phenomenological implications of the Friedberg-Lee transformation in a neutrino mass model with μτ-flavored CP symmetry

We propose a neutrino mass model with μτ-flavored CP symmetry, where the effective light neutrino Lagrangian enjoys an additional invariance under a Friedberg-Lee (FL) transformation on the left-handed flavor neutrino fields that leads to a highly predictive and testable scenario. While both types of the light neutrino mass ordering, i.e., Normal Ordering (NO) as well as the Inverted Ordering (IO) are allowed, the absolute scale of neutrino masses is fixed by the vanishing determinant of light Majorana neutrino mass matrix Mν. We show that for both types of mass ordering, whilst the atmospheric mixing angle θ23 is in general nonmaximal (θ23 ≠ π/4), the Dirac CP phase δ is exactly maximal (δ = π/2, 3π/2) for IO and nearly maximal for NO owing to cos δ ∝ sin θ13. For the NO, very tiny nonvanishing Majorana CP violation might appear through one of the Majorana phases β; otherwise the model predicts vanishing Majorana CP violation. Thus, despite the fact, that from the measurement of θ23, it is difficult to rule out the model, any large deviation of δ from its maximality, will surely falsify the scenario. For a comprehensive numerical analysis, beside fitting the neutrino oscillation global fit data, we also present a study on the νμ → νe oscillation which is expected to show up Dirac CP violation in different long baseline experiments. Finally, assuming purely astrophysical sources, we calculate the Ultra High Energy (UHE) neutrino flavor flux ratios at neutrino telescopes, such as IceCube, from which statements on the octant of θ23 could be made in our model.


Introduction
In spite of the spectacular developments in last couple of decades, the theoretical origin of neutrino masses, flavor mixing and CP violation [1] in the leptonic sector remain unresolved. In addition, models with definitive statements about the mass ordering and the absolute scale of three light neutrino masses are yet to be tested. Experiments so far with solar, atmospheric, reactor and accelerator neutrinos have determined the three mixing angles and the two independent mass-squared differences to a reasonably decent accuracy, while the current cosmological upper bound on the sum of the three light neutrino masses is fairly robust: i m i < 0.17 eV [2]. The octant of the atmospheric mixing angle θ 23 remains unknown though the best-fit values are reported as 47.2 • for NO and 48.1 • for IO [3,4]. Therefore, a precise prediction of θ 23 can be used to exclude and discriminate models in the light of forthcoming precision measurements. On the other hand, the current best-fit values of the Dirac CP phase δ, are close to 234 • for NO and 278 • for IO. While the possibility of CP conservation (sin δ = 0) is allowed at slightly above 1σ, one of the CP violating value δ = π/2 is disfavored at 99% CL. Thus, the remaining CP violating value δ = 3π/2 and deviations around it still remain potentially viable and tantalizing possibilities. Beside all these, it still remains a baffling conundrum for neutrino experts whether the light neutrinos are Dirac or Majorana in nature. Till date, despite relentless searches, no experimental signature of the neutrinoless double β−decay signal have been observed. However, the rapid development in the long baseline experiments such as T2K [5], NOνA [6][7][8] and also 0νββ experiments such as KamLandZen [9], GERDA [10,11] is expected to shed light JHEP03(2019)081 on the above issues shortly. Thus, from a theoretical perspective, this is a moment of paramount importance in neutrino mass model building, since many of the existing models that have predictions of θ 23 , δ and the neutrino mass ordering are likely to be challenged through precise measurements of these quantities in ongoing and forthcoming experiments.
A particular generalization [38,48] of (1.1) is CP µτ θ which is implemented in the neutrino Majorana mass term with the field transformation In the neutrino flavor space G µτ θ has the generic form with 'θ' being an arbitrary mixing angle that mixes the ν Lµ and ν Lτ flavor fields. The negative signs in (1.5) are to comply with the PDG convention. It is worth noticing that θ = π/2 reduces the mixing symmetry G µτ θ lm to the interchange symmetry G µτ lm and any nonzero value of θ−π/2 has the potential to account for the deviation from CP µτ . Eq. (1.5)

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is a special case of eq. 8 of ref. [49] with α = π, β = −π and γ = 0. Though, in general, CP symmetries are highly predictive in terms of mixing angles and CP-violating phases, for most of the cases, it lacks information regarding light neutrino masses and mass ordering unless one invokes additional flavor symmetries to reduce the number of parameters [14], e.g, by the means of 'texture zeros' in the light neutrino mass matrix [34,45].
In this work, to have testable predictions in each sector (masses as well as mixing) instead of any additional flavor symmetry, in combination with (1.4), we consider a Friedberg-Lee (FL) transformation [50][51][52][53][54][55] This leads to where η l (l = e, µ, τ ) are three arbitrary complex numbers, η = (η e η µ η τ ) T and ξ is a fermionic Grassmann field [50]. Note that, (1.6) is a simple CP generalization of the ordinary (general) FL transformation (also known as twisted FL symmetry [56,57]) We would like to stress that in this work we mainly focus on the effective field transformation (1.6) and its low energy phenomenological consequences without an explicit top down model realization like in the cases of CP combined with flavor symmetries [32,33,36]. Nevertheless, the generalized µτ and FL could arise from a discrete flavor symmetries such D 4 [58] and singlet scalar extension to the Standard Model [53] respectively. Since the residual symmetries in the charged lepton sector and the neutrino sector decide the low energy predictions for the neutrino parameters, from the phenomenological point of view it is a challenging task to identify proper residual symmetries which are predictive while being consistent with the extant neutrino data. Individually, flavor symmetries, CP symmetries or FL symmetries would not suffice to lead to residual symmetries which are predictive in mass as well as mixing sectors. That is why certain combinations of these symmetries are always attractive at least at the phenomenological level. For example, various models discussed in [14] deal with a combined theory of CP and flavor at high energy as well as at low energy (after spontaneous symmetry breaking, the low energy effective symmetries are still a combined theory of CP and flavor). Refs. [34,45] combines a U(1) global symmetry and its discrete subgroups such as Z 8 with µτ reflection to have texture zeros in light neutrino mass matrices so that the model could predict neutrino parameters in both the sectors, masses as well as mixing. Due to the blindness in the mixing sector, a combination of µτ symmetry with FL symmetry has been proposed in [56]. Similar to these models, in our work, FL symmetry could be thought of as a complementary symmetry to the generalized µτ reflection and vice versa, rather than treating any of them (FL or general µτ ) as an expedient partner of each other.

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Amongst many of the interesting results (which we shall discuss in the next section) that emerge as a consequence of the transformation in (1.6), it is worthwhile to stress two important departures from CP µτ .
• First of all, as mentioned earlier, G µτ θ lm in (1.5) is a µτ mixing symmetry. It reduces to 'µτ -interchange' in the limit θ → π/2 which we address in rest of this paper as 'µτinterchange limit (MTIL)'. It is now trivial to anticipate that the mixing parameter θ( = π/2) conspires for the departure from maximal δ and θ 23 . However, we show in this paper that despite the generalization from CP µτ to CP µτ θ , the additionally imposed FL symmetry only allows a tiny deviation from the maximality of δ in this model. For a consistent phenomenological analysis, apart from fitting the neutrino oscillation global-fit data, we study here the impact of CP µτ θ symmetry on ν µ → ν e oscillation in the long baseline experiments such as NOνA, T2K and DUNE. In addition, in the context of recent discovery of high energy neutrino events at IceCube [59][60][61][62][63], assuming high energy neutrinos originate purely from distant astrophysical sources, 1 we also calculate the fluxratios which will be measured with enhanced statistics at advanced neutrino telescopes (e.g. IceCube and ANTARES [64]) in near future. These calculations show that any potential deviation from the democratic 1:1:1 distribution of flux ratios [65][66][67][68] can lead to predictions on the octant of θ 23 in our model.
The rest of the paper is organized as follows. Section 2 contains the most general parametrization of M ν that is invariant under (1.6), thereby satisfying the conditions of (1.7). Section 3 deals with the evaluation of Majorana phases α, β and the leptonic Dirac CP phase δ for both types of mass ordering analysed in two different subsections. The numerical analysis in section 4 comprises of four subsections. Subsection 4.1 entails the extraction of the allowed parameter space and the prediction of light neutrino masses, whereas subsection 4.2 deals with the prediction on neutrinoless double beta decay process. Subsection 4.3 discusses of the range of variation of the oscillation probability P µe and the CP asymmetry parameter A µe in experiments such as T2K, NOνA and DUNE for both NO and IO. Subsection 4.4 comments on the possibility of determining the octant of θ 23 from futuristic measurements of flavor flux ratios in neutrino telescopes such as IceCube.

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where c θ ≡ cos θ, s θ ≡ sin θ and t θ/2 = tan θ 2 . For simplicity, we restrict to a reasonable choice that η l are a priori arbitrary complex numbers with same phases, so that the ratios η 1 η 1 , η 2 η 3 and η 3 η 1 are all real. In (2.1), there are five real free parameters: a 1 , a 2 , c 1 , η 1 η 2 and θ which can be well constrained by existing neutrino oscillation global-fit data. It is to be noted that (2.1) does not contain the parameter η 3 owing to a consistency relation of the form η 2 The mass matrix M ν in (2.1) can be diagonalized by a similarity transformation with a unitary matrix U : where m i (i = 1, 2, 3) are real and we assume that m i ≥ 0. Without any loss of generality, we work in the diagonal basis of the charged lepton so that U can be related to the P M N S mixing matrix U P M N S as is an unphysical diagonal phase matrix and c ij ≡ cos θ ij , s ij ≡ sin θ ij with the mixing angles θ ij ∈ [0, π/2]. We work within the PDG convention [69] but denote our Majorana phases by α and β. CP-violation enters through nontrivial values of the Dirac phase δ and of the Majorana phases α, β where δ, α, β ∈ [0, 2π].

Impact of mass ordering on mixing angles and CP properties
Eqs. (1.7) and (2.2) jointly imply [26] G θ U * = Ud, is either +1 or −1, and therefore (3.1) can be written in the following explicit form: Eq. (3.2) is equivalent to nine equations for the three rows: It is useful to construct the following two rephasing invariant quantities, that are independent of the unphysical phases, for calculating the Majorana phases:

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From the first row of (3.3), we get, Again, using the above different expressions for I 1,2 , in (3.4) and (3.5), we find the following relations, c 12 s 12 c 2 13 e −iα/2 =d 1d2 c 12 s 12 c 2 13 e iα/2 (3.6) and From (3.6) and (3.7), we find, i.e., either α = 0 or α = π, and either β = 2δ or β = 2δ − π. Therefore, there are four possible distinct pairs of values for the Majorana phases. From the third row of (3.3), taking the absolute square, we obtain, Similarly, the absolute square of the second relation in the third row in (3.3) is devoid of the unphysical phase difference (φ 2 − φ 3 ), and we get, Note that, both the relations, i.e., (3.10) and (3.11) reduce to the co-bimaximal prediction of CP µτ in the MTIL, as expected. We also stress that the relations (3.8), (3.10) and (3.11) hold irrespective of the neutrino mass ordering. Now, due to FL invariance, M ν has a vanishing eigenvalue with corresponding normalized eigenvector given by where γ is an arbitrary phase signifying that the normalized eigenvector is unique up to an overall phase. If the zero eigenvalue is associated with m 1 = 0 (m 3 = 0), we discover additional consequences for the normal (inverted) ordering.

Normal ordering
Here, v is associated with the first column of PMNS. Equating v with the first column of U in (2.3), we get, P DF (δ) dδ = 0.795. Thus upon a large number of random trial (we choose that number to be 10 6 ), there is 80 % probability that δ will be in the range 270 ± 0.2.

Inverted ordering
In this case, v is associated with the third column of PMNS. Equating v with the third column of U in (2.3), we get, 20) Note that, (3.21) and (3.22) together imply which is consistent with the relation (3.10). Note that, since the unphysical phase difference (φ 2 − φ 3 ) = π, it follows from (3.11) that the Dirac CP violation is maximal irrespective of the value of θ 23 i.e., cos δ = 0. (3.24) Clearly, since the Dirac CP phase deviates slightly from its maximal value only for the NO, and both types of mass ordering in this model predict arbitrary nonmaximality in θ 23 , it is difficult to make comments on the mass ordering, only from the measurement of these two parameters. Though any large nonmaximality in δ will rule out CP µτ as well as this model (CP µτ θ + FL), however, if the experiments favour nonmaximal θ 23 along with a maximal value of δ the latter model will survive while the former will be in tension.
One might wonder whether the minimal seesaw, which also leads to a vanishing eigenvalue, will lead to the same predictions as above when combined with general µτ symmetry. Though eq. (3.11) holds for both the cases (combination of the generalized µτ reflection symmetry with minimal seesaw or FL symmetry), a closer inspection of eq. (3.18) reveals in general predictions for cos δ need not be the same. This is because in each case the model parameters are different and will be constrained differently by the neutrino oscillation data.

Parameter estimation
We present a comprehensive numerical analysis to demonstrate the phenomenological viability of our proposal, and explore its implications on neutrino phenomenology in general. It is organized as follows. We utilize the (3σ) ranges of the globally fitted neutrino oscillation data [4] together with the upper bound of 0.17 eV [2] on the sum of the light neutrino masses from PLANCK and other cosmological observations in table 1. The allowed range of parameters of M ν are tabulated in table 2. Subsequently, we discuss the predictions in our model on neutrinoless double beta decay, CP asymmetry in ν µ → ν e oscillations and flavor flux ratios at neutrino telescopes in three separate subsections. Parameters  Table 2. Output values of the parameters of M ν .

Neutrinoless double beta (0νββ) decay process
For certain nuclei such as Ge-76, it is energetically favorable to undergo a double beta decay (2νββ) instead of a singular β−decay emitting two electrons and two neutrinos. Moreover, if the neutrino is a Majorana particle those two neutrinos can annihilate each other to give rise to a neutrinoless double beta decay (0νββ): which clearly violates the lepton number by 2 units. Observation of such decay will firmly establish the Majorana nature of the neutrinos. The half-life corresponding to the above decay is given by 1 T 0ν where G 0ν denote the two-body phase space factor, M is the nuclear matrix element (NME), m e is the mass of the electron and M ee is the (1,1) element of the effective light neutrino mass matrix M ν . Using the PDG parametrization convention for U P M N S , the M ee can be written as For the inverted ordering, δ = π/2 or 3π/2, and m 3 = 0. Here, due to the latter condition, the expression (4.3) becomes independent of β and reduces to two different possibilities:

Effect of CP asymmetry in neutrino oscillations
In this section, we work out the effect of the presence of leptonic Dirac CP violation δ in neutrino oscillation experiments. The phase δ will appear in the asymmetry parameter A lm , defined as where l, m = (e, µ, τ ) are flavor indices and the P 's are transition probabilities. First, let us consider oscillation in vacuum. The ν µ → ν e transition probability is given by P µe ≡ P (ν µ → ν e ) = P atm + P sol + 2 P atm P sol cos(∆ 32 + δ), (4.5) where ∆ ij = ∆m 2 ij L/4E is the kinematic phase factor (L being the baseline length and E being the beam energy) and P atm , P sol are respectively defined as Here a = G F N e / √ 2 with G F as the Fermi constant and N e is the number density of electrons in the medium of propagation, so that a take into account the matter effects in neutrino propagation through the earth. An approximate value of a for the earth is (3500km) −1 [49,76]. In the limit a → 0, (4.5) leads to the oscillation probability in vacuum. With this, the CP asymmetry parameter is given by where δ is given by (3.18) and (3.24) for NO and IO respectively. In figure 3 represents the variation of P µe and A µe against the baseline length L for IO, i.e., for δ = 3π/2, while in figure 5 we give same plots for δ given by (3.18) i.e., for NO. The baseline lengths T2K, NOνA and DUNE are indicated in these figures by vertical lines. In figures 4 and 6 the CP asymmetry A µe is plotted against the beam energy E for the same three experiments for IO and NO respectively.

Octant of θ 23 from flavor flux measurement at neutrino telescope
Recent discovery [59][60][61][62][63] of Ultra High Energy (UHE) neutrino events at IceCube has opened a new era in the neutrino astronomy. Including track+shower, IceCube has reported 82 high-energy starting events (HESE) which constitute more than 7σ excess over the atmospheric background and thus points towards an extraterrestrial origin of the UHE neutrinos(for a recent update see ref. [77]). In addition, no significant spatial clustering has been found and the recent data seems to be consistent with isotropic neutrino flux from uniformly distributed point sources [78] and hints towards extra galactic nature of the observed events. Although the HESE events are not consistent with the standard astrophysical one component unbroken isotropic power-law spectrum Φ(E ν ) ∝ E −2 ν and also suffer constraints from multi-messenger gamma-ray observation [79], two component explanation of the observed neutrino flux from purely astrophysical sources is still a plausible scenario [80].  Figure 5. Plots of the transition probability (P µe ) and CP asymmetry parameter (A µe ) with baseline length L for NO (E = 1GeV). The bands are due to 3σ ranges of the mixing angles and also the ranges for the parameters 79.6 • < θ < 101.6 • and 1.79 < |η 1 /η 2 | < 2.11. In this case, δ is not fixed, but varies over a range predicted from (3.18) with the same ranges of the mixing angles, and model parameters θ and η 1 /η 2 . The three vertical dashed lines and the horizontal dotted line specify the same as figure 3. on which could be made from enhanced statistics at neutrino telescopes (e.g., IceCube) and fits like [80], we first lay out a short summary of the subject as a necessary prerequisite.
The dominant source of UHE cosmic neutrinos are pp (hadro-nuclear) collisions in cosmic ray reservoirs such as galaxy clusters and pγ (photo-hadronic) collisions in cosmic ray accelerators [81,82] such as gamma-ray bursts, active galactic nuclei and blazars. In pp collisions, protons of TeV-PeV range produce neutrinos via the decays π + → µ + ν µ , π − → µ −ν µ , µ + → e + ν eνµ and µ − → e −ν e ν µ . Therefore, the normalized flux distributions over flavor are [67] where the superscript S denotes 'source'. On the other hand, the pγ collisions involve relatively less energetic γ−rays (GeV-10 2 GeV range). Therefore, the center-of-mass energy JHEP03(2019)081 of γp system is such that it can only produce γp → ∆ + → π + n, which in turn give rise to decays π + → µ + ν µ and µ + → e + ν eνµ . The corresponding normalized flux distributions over flavor (4. 10) In either case, if we take φ S l = φ S ν l + φ S ν l with l = e, µ, τ , (4.11) As neutrino oscillations will change flavor distributions from source (S) to telescope (T) [83] the flux reaching the telescope will be given by (4.12) Since the source-to-telescope distance is much greater than the oscillation length, the flavor oscillation probability averaged over many oscillations is given by (4.13) Thus the flux reaching the telescope is given by where φ 0 is the overall flux normalization. The unitarity of the PMNS matrix implies where ∆ i = |U µi | 2 − |U τ i | 2 . Existence of exact µτ (anti)symmetry, therefore dictates that ∆ i = 0, and φ T e = φ T µ = φ T τ . With the above background, one can define certain flavor flux ratios R l (l = e, µ, τ ) at the neutrino telescope as where l, m = e, µ, τ and U is given in (2.3). Each R l depends on all three mixing angles and cos δ. For NO, θ 23 and cos δ are given by (3.19) and (3.18) while for IO the corresponding quantities are given by (3.23) and (3.24) respectively. For both types of ordering, we display in figure 7 the variation of R e,µ,τ w.r.t. θ in its phenomenologically allowed ranges (table 2) using the exact expressions in (4. 16). For NO, θ 23 can be eliminated in favor of θ and η 1 /η 2 . Keeping the latter fixed at a value 1.5, we show in figure 7 (left panel) the contour corresponding to the best-fit values of θ 12 and θ 13 , while the bands arise when θ 12 and θ 13 are allowed to vary in their current 3σ JHEP03(2019)081 ranges. It should be emphasized that the contours corresponding to cos δ > 0 and cos δ < 0 are practically indistinguishable, and therefore, we show the contours and bands only for the case cos δ > 0.
Next, in case of IO, θ 23 can be eliminated in favor of θ only. The resulting variation of R e,µ,τ with θ are shown in the right panel of figure 7. In generating these plots, the mixing angles θ 12 and θ 13 are again allowed to vary in their experimental 3σ ranges. The contours within the bands represent the case when θ 12 and θ 13 are kept fixed at their best-fit values. Unlike NO, the expressions for R l in case of IO are relatively simple and can be used to explain the nature of the plots. The expressions for R e,µ,τ for IO are: where we have used (3.24), (3.23) and neglected terms of O(s 2 13 ). It is evident from the approximate expressions (4.17) that in the exact µτ interchange limit θ = π 2 , all the flavor flux ratios converge to the value 1 2 . It is clear from the figure as well as from the approximate expression of R e that for R e < 1 2 (R e > 1 2 ), we have θ < π 2 (θ > π 2 ). Since (3.23) implies 2θ 23 = π − θ, observed value of R e will give a definite value of θ 23 . In particular, θ > π 2 implies θ 23 < π 4 and vice versa. Similar conclusion can be made from the observed value of R µ . Although, the expression for R µ in (4.17) is quadratic in cos θ, only one of the roots of this equation belongs to the numerically allowed range of θ (table 2). However, a definite observational value of R τ cannot unambiguously predict the value of θ. This is because of the quadratic dependence of R τ on c θ which is clearly visible from figure 7, specifically for θ < π/2. For consistency, the unique value of θ determined from the future precision measurement of R e (or R µ ) lead to a theoretical prediction of the ranges of R µ (or R e ) and R τ which should in turn match the observed values of R µ (or R e ) and R τ . Conversely, if θ 23 is measured with significant precision in a complementary experiment (e.g. long baseline experiments), the range of each R l can be uniquely predicted for all l, which can again be compared with the observations in IceCube.

Summary and conclusion
In this paper, we propose an invariance of the low energy neutrino Majorana mass term under a mixed µτ -flavored CP symmetry CP µτ θ compounded with a generic Friedberg-Lee (FL) transformation on the left-handed flavor neutrino fields. Both types of mass ordering are allowed with a nondegenerate neutrino mass spectrum and vanishing value for the smallest neutrino mass as a direct consequence of FL invariance. While the atmospheric mixing angle θ 23 is in general nonmaximal (θ 23 = π/4), the Dirac CP phase δ is exactly maximal (δ = π/2, 3π/2) for IO and nearly maximal for NO owing to cos δ ∝ sin θ 13 though JHEP03(2019)081 the deviation from maximality does not exceed 0.4 • on either side of the maximal value δ = 3π/2. It also turns out that one of the Majorana phases, α, is restricted to lie at its CP conserving values while the other, β, admits a simple linear relation with δ leading to a tiny Majorana CP violation. For the IO, θ 23 is, in general, nonmaximal but δ is maximal irrespective of the value of θ 23 . For the NO, the Majorana CP violation sneaking through the Majorana phase β is numerically insignificant so that the model essentially predicts vanishing Majorana CP violation. Evidently, any large departure of δ from 3π/2, will exclude our model. After fitting the neutrino oscillation global fit data, we also consider a numerical study of ν µ → ν e oscillation which is expected to show up Dirac CP violation in different long baseline experiments. Finally, assuming purely astrophysical sources, we calculate the Ultra High Energy (UHE) neutrino flavor flux ratios at neutrino telescopes such as IceCube. From this we comment on the predictability of the octant of θ 23 in our model.