Lepton Flavor Mixing Research Articles

Neutrino mass ordering and μ-τ reflection symmetry breaking**Supported by National Natural Science Foundation of China (11135009, 11375207)

If the neutrino mass spectrum turns out to be m3<m1<m2, it may be relabeled as such that all the masses of fundamental fermions with the same electrical charges are in order. In this case the columns of the 3×3 lepton flavor mixing matrix U should be reordered accordingly, and the resulting pattern U′ may involve one or two large mixing angles in the standard parametrization or its variations. Since the Majorana neutrino mass matrix remains unchanged in such a mass relabeling, a possible μ-τ reflection symmetry is respected in this connection and its breaking effects are model-independently constrained at the 3σ level by using current experimental data.

The \u03bc-\u03c4 reflection symmetry of Dirac neutrinos and its breaking effect via quantum corrections

Given the Dirac neutrino mass term, we explore the constraint conditions which allow the corresponding mass matrix to be invariant under the μ-τ reflection transformation, leading us to the phenomenologically favored predictions θ23 = π/4 and δ = 3π/2 in the standard parametrization of the 3 × 3 lepton flavor mixing matrix. If such a flavor symmetry is realized at a superhigh energy scale Λμτ , we investigate how it is spontaneously broken via the one-loop renormalization-group equations (RGEs) running from Λμτ down to the Fermi scale ΛF. Such quantum corrections to the neutrino masses and flavor mixing parameters are derived, and an analytical link is established between the Jarlskog invariants of CP violation at Λμτ and ΛF. Some numerical examples are also presented in both the minimal supersymmetric standard model and the type-II two-Higgs-doublet model, to illustrate how the octant of θ23, the quadrant of δ and the neutrino mass ordering are correlated with one another as a result of the RGE-induced μ-τ reflection symmetry breaking effects.

Broken S3L×S3R flavor symmetry and leptonic CP violation**Supported by NNSFC (11325525), National Recruitment Program for Young Professionals and CAS Center for Excellence in Particle Physics (CCEPP)

In the framework of the canonical seesaw model, we present a simple but viable scenario to explicitly break an S3L×S3R flavor symmetry in the leptonic sector. It turns out that the leptonic flavor mixing matrix is completely determined by the mass ratios of the charged leptons (i.e., me/mμ and mμ/mτ) and those of light neutrinos (i.e., m1/m2 and m2/m3). The latest global-fit results of the three neutrino mixing angles {θ12, θ13, θ23} and two neutrino mass-squared differences at the 3σ level are used to constrain the parameter space of {m1/m2,m2/m3}. The predictions for the mass spectrum and flavor mixing are highlighted: (1) the neutrino mass spectrum shows a hierarchical pattern and a normal ordering, e.g., m1≈2.2 meV, m2≈8.8 meV and m3≈52.7 meV; (2) only the first octant of θ23 is allowed, namely, 41.8°≲θ23≲43.3°; (3) the Dirac CP-violating phase δ≈−22° deviates significantly from the maximal value −90°. All these predictions are ready to be tested in ongoing and forthcoming neutrino oscillation experiments. Moreover, we demonstrate that the cosmological matter-antimatter asymmetry can be explained via resonant leptogenesis, including the individual lepton-flavor effects. In our scenario, leptonic CP violation at low- and high-energy scales is closely connected.

Implications of residual CP symmetry for leptogenesis in a model with two right-handed neutrinos

We analyze the interplay between leptogenesis and residual symmetry in the framework of two right-handed neutrino model. Working in the flavor basis, we show that all the leptogenesis CP asymmetries are vanishing for the case of two residual CP transformations or a cyclic residual flavor symmetry in the neutrino sector. If a single remnant CP transformation is preserved in the neutrino sector, the lepton mixing matrix is determined up to a real orthogonal matrix multiplied from the right side. The $R$-matrix is found to depend on only one real parameter, it can take three viable forms, and each entry is either real or purely imaginary. The baryon asymmetry is generated entirely by the CP violating phases in the mixing matrix in this scenario. We perform a comprehensive study for the $\Delta(6n^2)$ flavor group and CP symmetry which are broken to a single remnant CP transformation in the neutrino sector and an abelian subgroup in the charged lepton sector. The results for lepton flavor mixing and leptogenesis are presented.

Leptonic flavour mixing influenced by flavon cross couplings

We develop an economical approach to connect flavon cross couplings with flavour mixing. A non-zero reactor mixing angle and almost maximal CP violation arise from cross couplings between flavons in the charged lepton and neutrino sectors. Within this approach, we analyze how cross couplings contribute to charged lepton-flavour-violating decays. We find branching ratio sum rules of these processes and observe that the electroweak-scale flavour symmetry is consistent with current experimental data.

Aspects of Leptonic Flavour Mixing

Since the discovery of neutrino oscillations many ideas have been put forward to explain the special features of the leptonic mixing and the differences with respect to the quark sector. In this talk I review some of these proposals, emphasizing especially their predictability. In the light of the new data, I first revisit fixed-point relations among mixing angles and phases. Then I briefly comment on radiative neutrino masses. Finally I discuss the role of flavour symmetries. Given the very many existing models I focus on two classes of models. On the one hand I illustrate the ability of models based on a generalization of the anarchy idea in reproducing the main features of both the quark and the lepton spectrum, also in a GUT framework. On the other hand I discuss less ambitious but more predictive models based on discrete flavour symmetries, centered on the properties of the leptonix mixing matrix.

Overview on Neutrino Theory and Phenomenology

In this talk, I give an overview on recent theoretical and phenomenological studies of massive neutrinos. First of all, the present status of neutrino mixing parameters is summarized. The phenomenology of neutrino oscillations is then discussed, and current understanding of lepton flavor mixing is presented. Finally, I consider the seesaw models of neutrino masses and briefly mention the direct searches at the CERN Large Hadron Collider (LHC).

Symmetric formulation of neutrino oscillations in matter and its intrinsic connection to renormalization-group equations

In this article, we point out that the effective Hamiltonian for neutrino oscillations in matter is invariant under the transformation of the mixing angle and the exchange of first two neutrino masses , if the standard parametrization of lepton flavor mixing matrix is adopted. To maintain this symmetry in perturbative calculations, we present a symmetric formulation of the effective Hamiltonian by introducing an η-gauge neutrino mass-squared difference for , where for , and show that only , or is allowed. Furthermore, we prove that is the best choice to derive more accurate and compact neutrino oscillation probabilities, by implementing the approach of renromalization-group equations. The validity of this approach becomes transparent when an analogy is made between the parameter η herein and the renormalization scale μ in relativistic quantum field theories.

Flavon-induced connections between lepton flavour mixing and charged lepton flavour violation processes

In leptonic flavour models with discrete flavour symmetries, couplings between flavons and leptons can result in special flavour structures after they gain vacuum expectation values. At the same time, they can also contribute to the other lepton-flavour-violating processes. We study the flavon-induced LFV 3-body charged lepton decays and radiative decays and we take as example the $A_4$ discrete symmetry. In $A_4$ models, a $Z_3$ residual symmetry roughly holds in the charged lepton sector for the realisation of tri-bimaximal mixing at leading order. The only processes allowed by this symmetry are $\tau^-\to \mu^+ e^- e^-, e^+ \mu^- \mu^-$, and the other 3-body and all radiative decays are suppressed by small $Z_3$-breaking effects. These processes also depend on the representation the flavon is in, whether pseudo-real (case i) or complex (case ii). We calculate the decay rates for all processes for each case and derive their strong connection with lepton flavour mixing. In case i, sum rules for the branching ratios of these processes are obtained, with typical examples $\text{Br}(\tau^-\to \mu^+ e^- e^-) \approx \text{Br}(\tau^-\to e^+ \mu^- \mu^-)$ and $\text{Br}(\tau^-\to e^- \gamma) \approx \text{Br}(\tau^-\to \mu^- \gamma)$. In case ii, we observe that the mixing between two $Z_3$-covariant flavons plays an important role. All processes are suppressed by charged lepton masses and current experimental constraints allow the electroweak scale and the flavon masses to be around hundreds of GeV. Our discussion can be generalised in other flavour models with different flavour symmetries.

Muon $$g-2$$ g - 2 through a flavor structure on soft SUSY terms

In this work we analyze the possibility to explain the muon anomalous magnetic moment discrepancy within theory and experiment through lepton flavor violation processes. We propose a flavor extended MSSM by considering a hierarchical family structure for the trilinear scalar Soft-Supersymmetric terms of the Lagranagian, present at the SUSY breaking scale. We obtain analytical results for the rotation mass matrix, with the consequence of having non-universal slepton masses and the possibility of leptonic flavour mixing. The one-loop supersymmetric contributions to the leptonic flavour violating process $\tau \to \mu\gamma$ are calculated in the physical basis, with slepton flavour mixed states, instead of using the well known Mass Insertion Method. We present the regions in parameter space where the muon g-2 problem is either entirely solved or partially reduced through the contribution of these flavor violating processes.

Viability of exact tri-bimaximal, golden-ratio and bimaximal mixing patterns and renormalization-group running effects

In light of the latest neutrino oscillation data, we examine whether the leptonic flavor mixing matrix can take on an exact form of tri-bimaximal (TBM), golden-ratio (GR) or bimaximal (BM) mixing pattern at a superhigh-energy scale, where such a mixing pattern could be realized by a flavor symmetry, and become compatible with experimental data at the low-energy scale. Within the framework of the Minimal Supersymmetric Standard Model (MSSM), the only hope for realizing such a possibility is to count on the corrections from the renomalization-group (RG) running. In this work we focus on these radiative corrections, and fully explore the allowed parameter space for each of these mixing patterns. We find that when the upper bound on the sum of neutrino masses $\Sigma^{}_\nu \equiv m^{}_1 + m^{}_2 + m^{}_3 < 0.23~\text{eV}$ at the $95\%$ confidence level from Planck 2015 is taken into account, none of these mixing patterns can be identified as the leptonic mixing matrix below the seesaw threshold. If this cosmological upper bound on the sum of neutrino masses were relaxed, the TBM and GR mixing patterns would still be compatible with the latest neutrino oscillation data at the $3\sigma$ level, but not at the $1\sigma$ level. Even in this case, no such a possibility exists for the BM mixing.

Analysis of the Lepton Mixing Matrix in the Two Higgs Doublet Model

In the theoretical framework of Two Higgs Doublet Model (2HDM) plus three right-handed neutrinos we consider a universal treatment for the mass matrices, aside from that the active neutrinos acquire their small mass through the type-I seesaw mechanism. Then, as long as a matrix with four-zero texture is used to represent the right-handed neutrinos and Yukawa matrices, we obtain a unified treatment where all fermion mass matrices have four-zero texture. We obtain analytical and explicit expressions for the lepton flavour mixing matrix PMNS in terms of fermion masses and parameters associated with the 2HDM-III. Further, we compare these expressions of the PMNS matrix with the most up to date values of masses and mixing in the lepton sector, via a likelihood test $\chi^{2}$. We find that the analytical expressions that we derived reproduce remarkably well the most recent experimental data of neutrino oscillations.

Radiative corrections to the solar lepton mixing sum rule

The simple correlation among three lepton flavor mixing angles $(\theta^{}_{12}, \theta^{}_{13}, \theta^{}_{23})$ and the leptonic Dirac CP-violating phase $\delta$ is conventionally called a sum rule of lepton flavor mixing, which may be derived from a class of neutrino mass models with flavor symmetries. In this paper, we consider the solar lepton mixing sum rule $\theta^{}_{12} \approx \theta^{\nu}_{12} + \theta^{}_{13} \cos \delta$, where $\theta^\nu_{12}$ stems from a constant mixing pattern in the neutrino sector and takes the value of $\theta^\nu_{12} = 45^\circ$ for the bi-maximal mixing (BM), $\theta^\nu_{12} = \tan^{-1}(1/\sqrt{2}) \approx 35.3^\circ$ for the tri-bimaximal mixing (TBM) or $\theta^\nu_{12} = \tan^{-1}\left[2/(\sqrt{5} + 1)\right] \approx 31.7^\circ$ for the golden-ratio mixing (GR), and investigate the renormalization-group (RG) running effects on lepton flavor mixing parameters when this sum rule is assumed at a superhigh-energy scale. For illustration, we work within the framework of the minimal supersymmetric standard model (MSSM), and implement the Bayesian approach to explore the posterior distribution of $\delta$ at the low-energy scale, which becomes quite broad when the RG running effects are significant. Moreover, we also discuss the compatibility of the above three mixing scenarios with current neutrino oscillation data, and observe that radiative corrections can increase such a compatibility for the BM scenario, resulting in a weaker preference for the TBM and GR ones.

The role of flavon cross couplings in leptonic flavour mixing

In models with discrete flavour symmetries, flavons are critical to realise specific flavour structures. Leptonic flavour mixing originates from the misalignment of flavon vacuum expectation values which respect different residual symmetries in the charged lepton and neutrino sectors. Flavon cross couplings are usually forbidden, in order to protect these symmetries. Contrary to this approach, we show that cross couplings can play a key role and give raise to necessary corrections to flavour-mixing patterns, including a non-zero value for the reactor angle and CP violation. For definiteness, we present two models based on $A_4$. In the first model, all flavons are assumed to be real or pseudo-real, with 7 real degrees of freedom in the flavon sector in total. A sizable reactor angle associated with nearly maximal CP violation is achieved, and, as both originate from the same cross coupling, a sum rule results with a precise prediction for the value of the Dirac CP-violating phase. In the second model, the flavons are taken to be complex scalars, which can be connected with supersymmetric models and multi-Higgs models. The complexity properties of flavons provide new sources for generating the reactor angle. Models in this new approach introduce very few degrees of freedom beyond the Standard Model and can be more economical than those in the framework of extra dimension or supersymmetry.

Leptonic unitarity triangles and effective mass triangles of the Majorana neutrinos

Given the best-fit results of six neutrino oscillation parameters, we plot the Dirac and Majorana unitarity triangles (UTs) of the 3×3 lepton flavor mixing matrix to show their real shapes in the complex plane. The connections of the three Majorana UTs with neutrino–antineutrino oscillations and neutrino decays are explored, and the possibilities of right or isosceles UTs are discussed. In the neutrino mass limit of m1→0 or m3→0, which is definitely allowed by current data, we show how the six triangles formed by the effective Majorana neutrino masses 〈m〉αβ (for α,β=e,μ,τ) and their corresponding component vectors look like in the complex plane. The relations of such triangles to the Majorana phases and to the lepton-number-violating decays H++→α+β+ in the type-II seesaw mechanism are also illustrated.

Generalized μ–τ reflection symmetry and leptonic CP violation

We propose a generalized μ–τ reflection symmetry to constrain the lepton flavor mixing parameters. We obtain a new correlation between the atmospheric mixing angle θ23 and the “Dirac” CP violation phase δCP. Only in a specific limit our proposed CP transformation reduces to standard μ–τ reflection, for which θ23 and δCP are both maximal. The “Majorana” phases are predicted to lie at their CP-conserving values with important implications for the neutrinoless double beta decay rates. We also study the phenomenological implications of our scheme for present and future neutrino oscillation experiments including T2K, NOνA and DUNE.

Testing the minimal S 4 model of neutrinos with the Dirac and Majorana phases

We propose two new simple lepton flavor models in the framework of the $S_4$ flavor symmetry. The neutrino mass matrices, which are given by two complex parameters, lead to the inverted mass hierarchy. The charged lepton mass matrix has the 1-2 lepton flavor mixing, which gives the non-vanishing reactor angle $\theta_{13}$. These models predict the Dirac phase and the Majorana phases, which are testable in the future experiments. The predicted magnitudes of the effective neutrino mass for the neutrino-less double beta decay are in the regions as $32~\text{meV}\lesssim |m_{ee}|\lesssim 49~\text{meV}$ and $34~\text{meV}\lesssim |m_{ee}|\lesssim 59~\text{meV}$, respectively. These values are close to the expected reaches of the coming experiments. The total sum of the neutrino masses are predicted in both models as $0.0952~\text{eV}\lesssim \sum m_i\lesssim 0.101~\text{eV}$ and $0.150~\text{eV}\lesssim \sum m_i\lesssim 0.160~\text{eV}$, respectively.

Neutrino mixing fromCPsymmetry

The neutrino mass matrix has remnant CP symmetry expressed in terms of the lepton mixing matrix, and vice versa the remnant CP transformations allow us to reconstruct the mixing matrix. We study the scenario that all the four remnant CP transformations are preserved by the neutrino mass matrix. The most general parameterization of remnant CP transformations is presented. The lepton mixing matrix is completely fixed by the remnant CP, and its explicit form is derived. The necessary and sufficient condition for conserved Dirac CP violating phase is found. If the Klein four flavor symmetry generated by the postulated remnant CP transformations arises from a finite flavor symmetry group, the phenomenologically viable lepton flavor mixing would be the trimaximal pattern, both Dirac CP phase $\delta_{CP}$ and Majorana phase $\alpha_{31}$ are either $0$ or $\pi$ while another Majorana phase $\alpha_{21}$ is a rational multiple of $\pi$. These general results are confirmed to be true in the case that the finite flavor symmetry group is $\Delta(6n^2)$.

Deviation from bimaximal mixing and leptonic CP phases in S4 family symmetry and generalized CP

The lepton flavor mixing matrix having one row or one column in common with the bimaximal mixing up to permutations is still compatible with the present neutrino oscillation data. We provide a thorough exploration of generating such a mixing matrix from $S_4$ family symmetry and generalized CP symmetry $H_{CP}$. Supposing that $S_4\rtimes H_{CP}$ is broken down to $Z^{ST^2SU}_2\times H^{\nu}_{CP}$ in the neutrino sector and $Z^{TST^{2}U}_4\rtimes H^{l}_{CP}$ in the charged lepton sector, one column of the PMNS matrix would be of the form $\left(1/2, 1/\sqrt{2}, 1/2\right)^{T}$ up to permutations, both Dirac CP phase and Majorana CP phases are trivial in order to accommodate the observed lepton mixing angles. The phenomenological implications of the remnant symmetry $K^{(TST^2, T^2U)}_4\rtimes H^{\nu}_{CP}$ in the neutrino sector and $Z^{SU}_{2}\times H^{l}_{CP}$ in the charged lepton sector are studied. One row of PMNS matrix is determined to be $\left(1/2, 1/2, -i/\sqrt{2}\right)$, and all the three leptonic CP phases can only be trivial to fit the measured values of the mixing angles. Two models based on $S_4$ family symmetry and generalized CP are constructed to implement these model independent predictions enforced by remnant symmetry. The correct mass hierarchy among the charged leptons is achieved. The vacuum alignment and higher order corrections are discussed.

Predicting CP violation in deviation from tri-bimaximal mixing of neutrinos

We study the CP violation in the deviation from the tri-bimaximal (TBM) mixing of neutrinos. We examine nontrivial relations among the mixing angles and the CP violating Dirac phase in the typical four cases of the deviation from the TBM. The first two cases are derived by the additional rotation of the 2-3 or 1-3 generations of neutrinos in the TBM basis. The other two cases are given by the additional rotation of the 1-3 or 1-2 generations of charged leptons with the TBM neutrinos. These four cases predict different relations among three mixing angles and the CP violating Dirac phase. The rotation of the 2-3 generations of neutrinos in the TBM basis predicts sin2θ12&lt; 1/3, and the CP violating Dirac phase to be ±(0.09π–0.76π) for NH (±(0.15π–0.73π) for IH) depending on sin2θ23. The rotation of the 1-3 generations of neutrinos in the TBM basis gives sin2θ12&gt; 1/3. The CP violating Dirac phase is not constrained by the input of the present experimental data. For the case of the 1-3 and 1-2 rotations of charged leptons in the TBM basis, the CP violating Dirac phase is predicted in ±(0.35π–0.60π) depending on sin2θ12for both normal hierarchy (NH) and inverted hierarchy (IH) cases. We also discuss the specific case that θ13is related with the Cabibbo angle λ such as [Formula: see text], in which the maximal CP violation is preferred. The CP violating Dirac phase can distinguish the lepton flavor mixing patterns at T2K and NOνA experiments in the future.