When the neutrino mass eigenstate $ν_2$ is trimaximally mixed, the mixing matrix is called trimaximal. The middle column of trimaximal mixing matrix is identical to tri-bimaximal mixing and the other two columns are subject to unitarity constraints. This corresponds to a mixing matrix with four independent parameters in the most general case. Apart from the two Majorana phases, the mixing matrix has only one free parameter in the CP conserving limit. Trimaximality results into interesting interplay between mixing angles and CP violation. A notion of maximal CP violation naturally emerges here: CP violation is maximal for maximal 2-3 mixing. Similarly, there is a natural constraint on the deviation from maximal 2-3 mixing which takes its maximal value in the CP conserving limit.
Trimaximal mixing is the mixing hypothesis with maximal symmetry. In trimaximal mixing there remains a still worrying conflict between the large values of $Δm^2$ preferred by the atmospheric fits (possibly supported by the early K2K data) and reactor limits on $ν_e$-mixing. However, the latest solar results do seem to point to energy-independent (ie. `no-scale') solar solutions, like the trimaximal solution.
We consider trimaximal lepton mixing, defined by |U_{alpha 2}|^2 = 1/3 for all alpha = e, mu, tau. This corresponds to a two-parameter lepton mixing matrix U. We present a model for the lepton sector in which trimaximal mixing is enforced by softly broken discrete symmetries; one version of the model is based on the group Delta(27). A salient feature of our model is that no vacuum alignment is required.
We propose a flavour theory of leptons implementing an $A_4$ family symmetry. Our scheme provides a simple way to derive trimaximal neutrino mixing from first principles, leading to simple and testable predictions for neutrino mixing and CP violation. Dark matter mediates neutrino mass generation, as in the simplest scotogenic model.
We analyze in detail the predictions of "trimaximal" neutrino mixing, which is defined by a mixing matrix with identical second column elements. This column is therefore identical to the second column in the case of tri-bimaximal mixing. We also generalize trimaximal mixing by assuming that the other rows and columns of the mixing matrix individually have the same forms as for tri-bimaximal mixing. The phenomenology of these new mixing scenarios is studied. We emphasize how trimaximal mixings can be distinguished experimentally from broken tri-bimaximal mixing.
We present a model which employs the seesaw mechanism with five right-handed neutrinos, leading to trimaximal and CP-conserving lepton mixing. Tri-bimaximal mixing is a natural limiting case of our model which occurs when one particular vacuum expectation value is real and preserves the mu-tau interchange symmetry of the Lagrangian. Our model allows for leptogenesis even in the case of exact tri-bimaximal mixing.
We construct a supersymmetric $S_4$ flavor symmetry model with one of the trimaximal neutrino mixing patterns, the so-called TM$_1$, by using the novel way to stabilize flavons, which we proposed recently. The flavons are assumed to have tachyonic supersymmetry breaking mass terms and stabilized by higher-dimensional terms in the potential. We can obtain the desired alignment structure of the flavon vacuum expectation values to realize neutrino masses and mixings consistent with the current observations. This mechanism naturally avoids the appearance of dangerous cosmological domain walls.
In this paper we study a parametrized description of neutrino mixing from a phenomenological point of view. We concentrate on the parametrization in terms of higher order corrections to the leading order mixing matrix. A method to describe subleading contributions and its applications to tri-bimaximal mixing are discussed. We show that mixing matrices similar to tri-bimaximal-Cabibbo mixing can be obtained by straightforward choices of parameters. To achieve better agreement with the experimental data without increasing the number of free parameters, we impose a simple phenomenological relatio…
We discuss a minimal flavour model with twin modular symmetries, leading to trimaximal TM$_1$ lepton mixing in which the first column of the tri-bimaximal lepton mixing matrix is preserved. The model involves two modular $S_4$ groups, one acting in the neutrino sector, associated with a modulus field value $τ_{SU}$ with residual $Z^{SU}_2$ symmetry, and one acting in the charged lepton sector, associated with a modulus field value $τ_{T}$ with residual $Z^{T}_3$ symmetry. Apart from the predictions of TM$_1$ mixing, the model leads to a new neutrino mass sum rule which implies lower bounds on …
Recent T2K results indicate a sizeable reactor angle theta_13 which would rule out exact tri-bimaximal lepton mixing. We study the vacuum alignment of the Altarelli-Feruglio A4 family symmetry model including additional flavons in the 1' and 1" representations and show that it leads to trimaximal mixing in which the second column of the lepton mixing matrix consists of the column vector (1,1,1)^T/sqrt{3}, with a potentially large reactor angle. In order to limit the reactor angle and control the higher order corrections, we propose a renormalisable S4 model in which the 1' and 1" flavons of A4…
Recent T2K, MINOS and Double CHOOZ results, together with global fits of mixing parameters, indicate a sizeable reactor angle theta13 ~ 8 deg which, if confirmed, would rule out tri-bimaximal (TB) lepton mixing. Recently two of us studied the vacuum alignment of the Altarelli-Feruglio A4 family symmetry model including additional flavons in the 1' and 1" representations, leading to so-called "trimaximal" neutrino mixing and allowing a potentially large reactor angle. Here we show how such a model may arise from a Supersymmetric (SUSY) Grand Unified Theory (GUT) based on SU(5), leading to sum r…
In this paper, we make an attempt to combine the neutrino trimaximal (TM1 and TM2) mixings and $μ$-$τ$ reflection symmetry in the type-I seesaw model. Such a scenario is highly restrictive and predictive: in addition to three right-handed neutrino masses, there are only five real parameters; all the lepton flavor mixing parameters except for $θ^{}_{13}$ will be predicted. The relations between the model parameters and the measurable neutrino parameters will be derived. The implications of this scenario for leptogenesis will be explored in detail. A further reduction of this scenario to the mor…
The measurement of the reactor angle by the Daya Bay and RENO experiments in 2012 has ruled out the tri-bimaximal paradigm. Adopting an S4 family symmetry, we propose direct models of the trimaximal type TM1 in which the tri-bimaximal Klein symmetry of the neutrino sector is broken to a residual Z2 symmetry. In such a scenario, the solar mixing angle is decreased compared to its tri-bimaximal value by about one degree, thus bringing it in excellent agreement with experimental observation. The atmospheric mixing angle, on the other hand, depends on the CP violating Dirac phase delta. Imposing C…
In this paper, we consider the imbedding of the popular and well-motivated trimaximal mixing and $μ$-$τ$ reflection symmetry (which can help us shape the forms of the neutrino mass matrix) in the minimal seesaw model (which contains much fewer parameters than the general seesaw model) with two TeV-scale right-handed neutrinos (for realizing a low-scale seesaw) of nearly degenerate masses (for realizing a resonant leptogenesis). However, either for the trimaximal mixing scenario (which is realized through the Form Dominance approach here) or for the $μ$-$τ$ reflection symmetry scenario, leptoge…
In the seesaw framework, in the basis of $M^{}_{\rm R}$ being diagonal, we explore the simplified textures of $M^{}_{\rm D}$ that can naturally yield the trimaximal neutrino mixings and their consequences for the neutrino parameters and leptogenesis. We first formulate the generic textures of $M^{}_{\rm D}$ that can naturally yield the trimaximal mixings and then examine if their parameters can be further reduced. Our analysis is restricted to the simple but instructive scenario that there is only one phase parameter $ϕ$ responsible for both the CP violating effects at low energies and leptoge…
We analyze neutrino mass matrices having one texture zero, assuming that the neutrino mixing matrix has either its first (TM$_1$) or second (TM$_2$) column identical to that of the tribimaximal mixing matrix. We found that all the six possible one texture zero neutrino mass matrices are compatible with the present neutrino oscillation data when combined with TM$_1$ or TM$_2$ mixing. These textures have interesting predictions for the presently unknown parameters such as the effective Majorana neutrino mass, the Dirac CP violating phase and the neutrino mass scale. We also present a way to theo…
We study neutrino mass matrices with one texture equality and the neutrino mixing matrix having either its first (TM$_1$) or second (TM$_2$) column identical to that of the tribimaximal mixing matrix. We found that out of total fifteen possible neutrino mass matrices with one texture equality, only seven textures are compatible with $TM_1$ mixing and six textures are compatible with $TM_2$ mixing in the light of the current neutrino oscillation data. These textures have interesting implications for the presently unknown parameters such as the neutrino mass scale, effective Majorana neutrino ma…
We construct phenomenologically viable models of lepton masses and mixing based on modular $A_4$ invariance broken to residual symmetries $\mathbb{Z}^{T}_3$ or $\mathbb{Z}^{ST}_3$ and $\mathbb{Z}^S_2$ respectively in the charged lepton and neutrino sectors. In these models the neutrino mixing matrix is of trimaximal mixing form. In addition to successfully describing the charged lepton masses, neutrino mass-squared differences and the atmospheric and reactor neutrino mixing angles $θ_{23}$ and $θ_{13}$, these models predict the values of the lightest neutrino mass (i.e., the absolute neutrino …
We construct two flavor models based on $S_4$ family symmetry and generalised CP symmetry. In both models, the $S_4$ family symmetry is broken down to the $Z^{SU}_2$ subgroup in the neutrino sector, as a consequence, the trimaximal $\text{TM}_1$ lepton mixing is produced. Depending on the free parameters in the flavon potential, the Dirac CP is predicted to be either conserved or maximally broken, and the Majorana CP phases are trivial. The two models differ in the neutrino sector. The flavon fields are involved in the Dirac mass terms at leading order in the first model, and the neutrino mass…
In this paper, we impose a magic symmetry on the neutrino mass matrix $m_ν$ with universal four-zero texture and diagonal reflection symmetries. Due to the magic symmetry, the MNS matrix has trimaximal mixing inevitably. Since the lepton sector has only six free parameters, physical observables of leptons are all determined from the charged leptons masses $m_{ei}$, the neutrino mass differences $Δm_{i1}^{2}$, and the mixing angle $θ_{23}$. This scheme predicts $\sin θ_{13} = 0.149$, that is almost equal to the latest best fit, as a function of the lepton masses $m_{e, μ}$ and the mass differen…
Following the recent T2K indication of a sizeable reactor angle, we present a class of models which fix θ_{13} while preserving trimaximal solar mixing. The models are based on a new type of constrained sequential dominance involving new vacuum alignments, along the (1,2,0)^T or (1,0,2)^T directions in flavour space. We show that such alignments are easily achieved using orthogonality, and may replace the role of the subdominant flavon alignment (1,1,1)^T in constrained sequential dominance. In such models, with a normal hierarchical spectrum, the reactor angle is related to a ratio of neutrin…
The $μ$-$τ$ reflection symmetry $(ν_{e},\thinspaceν_μ,\thinspaceν_τ)\rightarrow(\overlineν_{e},\thinspace\overlineν_τ,\thinspace\overlineν_μ)$ and the TM1 mixing (a PMNS matrix with the first column fixed to the TBM form) are both well compatible with experiments. If both approaches are simultaneously assumed, all lepton mixing parameters except for $θ_{13}$ are predicted. In particular, one expects maximal CP violation ($|δ|=90^{\circ}$), maximal atmospheric mixing ($θ_{23}=45^{\circ}$), a slightly less-than-TBM solar mixing angle ($θ_{12}\approx34^{\circ}$), as well as values of $0$ or $π$ f…
Encouraged by the latest SNO results, we consider the lepton mixing matrix in the approximation that the nu_2 mass eigenstate is trimaximally (democratically) mixed. This suggests a new parameterization of the remaining mixing degrees of freedom, which eschews mixing angles, dealing instead, directly with the complex parameter U_e3 of the mixing matrix. Unitarity triangles then take a particularly simple form, which we hope will faciltate comparison with experiment.
Recently T2K gives hint in favor of large reactor angle. Most of the models, with tri-bimaximal mixing at the leading order, can not reproduce such a large mixing angle since they predict typically corrections for the reactor angle of the order of the Cabibbo angle. In this letter, we discuss the possibility to have large reactor angle within the T2K region with maximal atmospheric mixing angle and trimaximal solar mixing angle, through the deviation from tri-bimaximal. We derive the structure of neutrino mass matrix that leads to the large reactor angle leaving maximal atmospheric angle and t…
We analyze the quantum consistency of anomalous abelian models and of their effective field theories, rendered anomaly-free by a Wess-Zumino term, in the case of multiple abelian symmetries. These models involve the combined Higgs-Stuckelberg mechanism and predict a pseudoscalar axion-like field that mixes with the goldstones of the ordinary Higgs sector. We focus our study on the issue of unitarity of these models both before and after spontaneous symmetry breaking and detail the set of Ward identities and the organization of the loop expansion in the effective theory. The analysis is perform…
We present a classification of the unitarity quadrangles in the four-neutrino mixing scheme. We find that there are totally thirty-six distinct topologies among twelve different unitarity quadrangles. Concise relations are established between the areas of those unitarity quadrangles and the rephasing invariants of CP and $T$ violation.
The quark mixing matrix is parameterised such that its "Cabibbo substructure" is emphasised. One can choose one of the parameters to be an arbitrarily chosen angle of the unitarity triangle, for example the angle $β$ (also called $Φ_1$).
The non-unitarity of the leptonic mixing matrix is a generic signal of new physics aiming at the generation of the observed neutrino masses. We discuss the Minimal Unitarity Violation (MUV) scheme, an effective field theory framework which represents the class of extensions of the Standard Model (SM) by heavy neutral leptons, and discuss the present bounds on the non-unitarity parameters as well as estimates for the sensitivity of the Circular Electron Positron Collider (CEPC), based on the performance parameters from the preCDR.
The solar and baseline neutrino oscillation data suggest bimaximal neutrino mixing among the first two generations, and trimaximal mixing between all three neutrino flavors. It has been conjectured that this indicates the existence of an underlying symmetry for the leptonic fermion mass textures. The experimentally measured quantities however, are associated to the latter indirectly and in a rather complicated way through the mixing matrices of the charged leptons and neutrinos. Motivated by these facts, we derive exact analytical expressions which directly link the charged lepton and neutrino…
In the light of the recent LHC data, we study precision tests sensitive to the violation of lepton universality, in particular the violation of unitarity in neutrino mixing. Keeping all data we find no satisfatory fit, even allowing for violations of unitarity in neutrino mixing. Leaving out sin$^2 θ_{\scriptsize \mbox{eff}}$ from the hadronic forward-backward asymmetry at LEP, we find a good fit to the data with some evidence of lepton universality violation at the $\mathcal{O}(10^{-3})$ level.
We investigate the possibility of unitarity violation in the sequential neutrino mixing matrix in a scenario with extra compact spacelike dimensions. Gauge singlet neutrinos are assumed to propagate in one extra dimension, giving rise to an infinite tower of states in the effective four-dimensional theory. It is shown that this leads to small lepton-number violating entries in the neutrino mass matrix, which can violate unitarity on the order of one per cent.
Although the measurement of the reactor angle has killed tribimaximal lepton mixing, its descendants survive with charged lepton corrections, or in less constrained forms such as trimaximal mixing and/or mu-tau symmetry, each with characteristic predictions. Such patterns may be enforced by a remnant of some non-Abelian discrete family symmetry, possibly together with a generalised CP symmetry, which could originate from continuous gauge symmetry and/or superstring theory in compactified extra dimensions, as a finite subgroup of the modular symmetry.
In the Standard Model of elementary particles, quark-mixing is expressed in terms of a 3 x 3 unitary matrix V, the so called Cabibbo-Kobayashi-Maskawa (CKM) matrix. Significant unitarity checks are so far possible for the first row of this matrix. This article reviews the experimental and theoretical information on these matrix elements. On the experimental side, we find a 2.2 sigma to 2.7 sigma deviation from unitarity, which conflicts with the Standard Model.
The non-unitarity effects in the leptonic flavor mixing are regarded as one of the generic features of the type-I seesaw model. Therefore, we explore these effects in the TeV-scale type-I seesaw model, and show that there exist non-trivial correlations among the non-unitarity parameters, stemming from the typical flavor structure of the low-scale seesaw model. In general, it follows from analytical discussions and numerical results that all the six non-unitarity parameters are related to three model parameters, while the widely studied parameters η_{eτ} and η_{μτ} cannot be phenomenologically …
We present a new method for the determination of the Cabibbo-Kobayashi-Maskawa quark mixing matrix element $|V_{tb}|$ from electroweak loop corrections, in particular those affecting the process $Z\to b\bar{b}$. From a combined analysis of results from the LEP, SLC, Tevatron, and neutrino scattering experiments we determine $|V_{tb}| = 0.77^{+0.18}_{-0.24}$. This is the first determination of $|V_{tb}|$ which is independent of unitarity assumptions.
We postulate that the neutrino mass matrix in the lepton flavour basis is an S3 group matrix in the natural representation of S3. This immediately requires one neutrino to be trimaximally mixed, as suggested by the solar neutrino data. We go on to postulate that the charged-lepton mass matrix in the neutrino mass-basis is an S3 class matrix in the natural representation of the S3 class-algebra, leading to exact tri-bimaximal mixing, which is compatible with data overall. The tri-bimaximal mixing matrix is seen to be closely related to the S3 character table, and is properly the S2 C S3 table o…
The study of charmless hadronic two-body decays of B mesons is one of the most fascinating topics in B physics. A construction of the unitarity triangle based on such decays is presented, which is independent of B-B(bar) and K-K(bar) mixing. It provides stringent tests of the Standard Model with small theoretical uncertainties.
We present results of a combined analysis in neutrino oscillations without unitarity assumption in the $3ν$ mixing picture. Constraints on neutrino mixing matrix elements are based on recent data from the reactor, solar and long-baseline accelerator neutrino oscillation experiments. The current data are consistent with the standard $3ν$ scheme. The precision on different matrix elements can be as good as a few percent at $3σ$ CL, and is mainly limited by the experimental statistical uncertainty. The $ν_e$ related elements are the most precisely measured among all sectors with the uncertainties…
The non-unitarity of the effective leptonic mixing matrix at low energies is a generic signal of extensions of the Standard Model (SM) with extra fermionic singlet particles, i.e. sterile or right-handed neutrinos, to account for the observed neutrino masses. The low energy effects of such extensions can be described in a model-independent way by the Minimal Unitarity Violation (MUV) scheme, an effective field theory extension of the SM. We perform a global fit of the MUV scheme parameters to the present experimental data, which yields the up-to-date constraints on leptonic non-unitarity. Furt…
A low-energy non-unitary leptonic mixing matrix is a generic effect of some theories of new physics accounting for neutrino masses. We show how the extra CP-phases of a general non-unitary matrix allow for sizeable CP-asymmetries in the $ν_μ\to ν_τ$ channel. This CP-asymmetries turns out to be an excellent probe of such new physics.
In this paper we explore the effect of non-unitary neutrino mixing on neutrino oscillation probabilities both in vacuum and matter. In particular, we consider the $ν_μ\to ν_τ$ channel and using a Neutrino Factory as the source for $ν_μ$'s discuss the constraints that can be obtained on the moduli and phases of the parameters characterizing the violation of unitarity. We point out how the new CP violation phases present in the case where the non-unitary mixings give rise to spurious ``degenerate'' solutions in the parameter space and discuss how the true solutions can be extricated by combining…
In analogy with the recently proposed lepton mixing sum rules, we derive quark mixing sum rules for the case of hierarchical quark mass matrices with 1-3 texture zeros, in which the separate up and down type 1-3 mixing angles are approximately zero, and $V_{ub}$ is generated from $V_{cb}$ as a result of 1-2 up type quark mixing. Using the sum rules, we discuss the phenomenological viability of such textures, including up to four texture zeros, and show how the right-angled unitarity triangle, i.e., $α\approx 90^\circ$, can be accounted for by a remarkably simple scheme involving real mass matr…
Using the seesaw mechanism, we construct a model for the light-neutrino Majorana mass matrix which yields trimaximal lepton mixing together with maximal CP violation and maximal atmospheric-neutrino mixing. We demonstrate that, in our model, the light-neutrino mass matrix retains its form under the one-loop renormalization-group evolution. With our neutrino mass matrix, the absolute neutrino mass scale is a function of |U_e3| and of the atmospheric mass-squared difference. We study the effective mass in neutrinoless double beta-decay as a function of |U_e3|, showing that it contains a fourfold…
We explore the implications of imposing the constraint that two neutrino flavors (which for definiteness we take to be muon and tau neutrinos) are similarly coupled to the mass basis in addition to the unitarity constraints. Implications of this scheme for specific experimental situations are discussed.
Geometric representation of CP phases $δ_{\rm PDG}, δ_{\rm KM}$ in flavor mixing matrix and its sum rule by alternative unitarity triangle and quadrangle
We derive a new set of sum rules between the neutrino mass and mixing parameters in vacuum and their effective counterparts in matter. The moduli of nine genuine lepton mixing matrix elements can then be calculated in terms of the matter-corrected ones, and the latter can directly be determined from a variety of long-baseline neutrino oscillations. We show that it is possible to reconstruct the leptonic unitarity triangles and CP violation in a similar parametrization-independent way.
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 genera…
