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 neutrino masses close to current limits from neutrinoless double beta decay experiments and cosmology.
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 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…
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…
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.
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…
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:…
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 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 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 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…
Following recent results from the SNO solar neutrino experiment and the K2K long-baseline neutrino experiment, the combined existing data on neutrino oscillations now point strongly to a specific form for the lepton mixing matrix, with effective bimaximal mixing of $ν_μ$ and $ν_τ$ at the atmospheric scale and effective trimaximal mixing for $ν_e$ with $ν_μ$ and $ν_τ$ at the solar scale (hence `tri-bimaximal' mixing). We give simple mass-matrices leading to tri-bimaximal mixing, and discuss its relation to the Fritzsch-Xing democratic ansatz.
We develop a general formalism for multiple moduli and their associated modular symmetries. We apply this formalism to an example based on three moduli with finite modular symmetries $S_4^A$, $S_4^B$ and $S_4^C$, associated with two right-handed neutrinos and the charged lepton sector, respectively. The symmetry is broken by two bi-triplet scalars to the diagonal $S_4$ subgroup. The low energy effective theory involves the three independent moduli fields $τ_A$, $τ_B$ and $τ_C$, which preserve the residual modular subgroups $Z_3^A$, $Z_2^B$ and $Z_3^C$, in their respective sectors, leading to t…
We perform a systematical and analytical study of lepton mixing which can be derived from the subgroups of $SU(3)$ under the assumption that neutrinos are Dirac particles. We find that type D groups can predict lepton mixing patterns compatible with the experimental data at $3σ$ level. The lepton mixing matrix turns out to be of the trimaximal form, and the Dirac CP violating phase is trivial. Moreover, we extend the flavor symmetry to the quark sector. The Cabibbo mixing between the first two generations of quarks can be generated by type D groups. Since all the finite subgroups of $U(3)$ whi…
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.
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…
Recent stringent experiment data of neutrino oscillations induces partial symmetries such as $Z_{2}$, $Z_{2}\times CP$ to derive lepton mixing patterns. New partial symmetries expressed with elements of group algebras are studied. A specific lepton mixing pattern could correspond to a set of equivalent elements of a group algebra. The transformation which interchanges the elements could express a residual $CP$ symmetry. Lepton mixing matrices from $S_{3}$ group algebras are of the trimaximal form with the $μ-τ$ reflection symmetry. Accordingly, elements of $S_{3}$ group algebras are equivalent…
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.
We propose a next-to-minimal supersymmetric Standard Model (NMSSM) extended by an $\mathbb{A}_{4}\times \boldsymbol{Z}_{3}$ flavor symmetry and three right-handed neutrinos providing a detailed study of the neutrino sector and a solution to the domain wall problem. In this proposal, neutrino masses are generated through Type I seesaw mechanism while the mixing angles are described by the trimaximal mixing realized using the NMSSM singlet S and only two flavon fields. The phenomenology of neutrino parameters is studied for normal and inverted mass hierarchies. In particular, we numerically eval…
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…
We obtain predictions of lepton mixing parameters for direct models based on Delta(6n^2) family symmetry groups for arbitrarily large n in which the full Klein symmetry is identified as a subgroup of the family symmetry. After reviewing and developing the group theory associated with Delta(6n^2), we find many new candidates for large n able to yield reactor angle predictions within 3 sigma of recent global fits. We show that such Delta(6n^2) models with Majorana neutrinos predict trimaximal mixing with reactor angle theta_{13} fixed up to a discrete choice, an oscillation phase of either zero …
We propose a new mixing pattern for neutrinos with a nonzero mixing angle $θ_{13}$. Under a simple form, it agrees well with current neutrino oscillation data and displays a number of intriguing features including the $μ$-$τ$ interchange symmetry $|U_{μi}|=|U_{τi}|$, $(i=1,2,3)$, the trimaximal mixing $|U_{\e 2}|=|U_{μ2}|=|U_{τ2}|=1/\sqrt{3}$, the self-complementarity relation $θ_1+θ_3=45°$, together with the maximal Dirac CP violation as a prediction.
Modular TM$_1$ mixing in light of precision measurement in JUNO
School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS（杭州高等研究院基础物理与数学学院，UCAS）；Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）；University of Chinese Academy of Sciences（中国科学院大学）
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…
Experimentally viable lepton mixing parameters can be predicted in so-called direct flavour models with Majorana neutrinos using $Δ(6n^2)$ groups as a flavour group. In direct models, in which the flavour group is broken to a $Z_2\times Z_2$ subgroup in the neutrino sector, mixing angles and Dirac CP phase are purely predicted from symmetry. General predictions of direct models with $Δ(6n^2)$ flavour groups are that all mixing angles are fixed up to a discrete choice and that the Dirac CP phase is $0$ or $π$; Furthermore, the middle column of the mixing matrix is trimaximal which yields the su…
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…
Generalised CP transformations are the only known framework which allows to predict Majorana phases in a flavour model purely from symmetry. For the first time generalised CP transformations are investigated for an infinite series of finite groups, $Δ(6n^2)=(Z_n\times Z_n)\rtimes S_3$. In direct models the mixing angles and Dirac CP phase are solely predicted from symmetry. $Δ(6n^2)$ flavour symmetry provides many examples of viable predictions for mixing angles. For all groups the mixing matrix has a trimaximal middle column and the Dirac CP phase is 0 or $π$. The Majorana phases are predicte…
We motivate the simplest ansatz for the neutrino mass matrix consistent with the data from neutrino oscillation experiments, and admitting CP violation. It has only two free parameters: an arbitrary mass-scale and a small dimensionless ratio. This mass matrix exhibits two symmetries, Democracy and Mutativity, which respectively ensure trimaximal mixing of the |nu_2> mass eigenstate, and mixing parameter values |theta_{23}|=45 degrees and |delta|=90 degrees, consistent with bimaximal mixing of the |nu_3> mass eigenstate. A third constraint relates the smallness of |U_{e3}|^2 to that of the mass…
We discuss the correlation between the CP violating Dirac phase of the lepton mixing matrix and the cosmological baryon asymmetry based on the leptogenesis in the minimal seesaw model with two right-handed Majorana neutrinos and the trimaximal mixing for neutrino flavors. The sign of the CP violating Dirac phase at low energy is fixed by the observed cosmological baryon asymmetry since there is only one phase parameter in the model. According to the recent T2K and NO$ν$A data of the CP violation, the Dirac neutrino mass matrix of our model is fixed only for the normal hierarchy of neutrino mas…
A simple three-flavor neutrino-oscillation model is discussed which has both nonzero mass differences and timelike Fermi-point splittings, together with a combined bi-maximal and trimaximal mixing pattern. One possible consequence would be new effects in ν_μ \to ν_{e} oscillations, characterized by an energy-dependent effective mixing angle Θ_{13}. Future experiments such as T2K and NOvA, and perhaps even the current MINOS experiment, could look for these effects.
We explore the feasibility of the discrete flavor symmetry $S_3$ to explain the pattern of neutrino masses and mixings. In the flavor basis, the neutrino mass matrix is taken to be invariant under $S_3$ symmetry at the zeroth order. The effects of breaking $S_3$ symmetry by matrices invariant under different $S_2$ subgroups of $S_3$ are studied. The resulting retrocirculant perturbation matrix leads to the perturbed $S_3$ neutrino mass matrix having a trimaximal eigenvector as suggested by the solar neutrino data. It is found that after the third order perturbation the neutrino mixings only de…
Tri/bi-maximal mixing (TBM) is a specific lepton mixing ansatz, which describes the trend of the current neutrino oscillation data, in particular the recent SNO and KAMLAND results. The significant feature of TBM in this respect is |U_e2|^2=|U_m2|^2=|U_t2|^2=1/3, and we say that the nu_2 is tri-maximally mixed. We have generalised the TBM ansatz to a generic mixing matrix with the nu_2 trimaximally mixed, whereby the neutrino mass matrix in the lepton flavour basis takes the form of a general S3 group matrix (3 x 3 `magic-square'). In exact TBM the charged-lepton mass matrix in the neutrino ma…
Recent results from the Daya Bay and RENO reactor experiments have measured the smallest lepton mixing angle and found it to have a value of theta_13 approximately 9 degrees. This result presents a new challenge for the existing paradigms of discrete flavour symmetries which attempt to describe all quark and lepton masses and mixing angles. Here we propose a Supersymmetric Grand Unified Theory of Flavour based on Delta(96) x SU(5), together with a U(1) x Z3 symmetry, including a full discussion of Delta(96) in a convenient basis. The Grand Delta(96) x SU(5) Flavour Model relates the quark mixi…
We present a model of leptonic mixing based on two $A_5$ modular symmetries using the Weinberg operator. The two modular symmetries are broken to the respective diagonal $A_5$ subgroup. At the effective level, the model behaves as a model with a single $A_5$ modular flavour symmetry, but with two moduli. Using both as stabilisers, different residual symmetries are preserved, leading to golden ratio mixing that is perturbed by a rotation.
We extend the even weight modular forms of modular invariant approach to general integral weight modular forms. We find that the modular forms of integral weights and level $N$ can be arranged into irreducible representations of the homogeneous finite modular group $Γ'_N$ which is the double covering of $Γ_N$. The lowest weight 1 modular forms of level 3 are constructed in terms of Dedekind eta-function, and they transform as a doublet of $Γ'_3 \cong T'$. The modular forms of weights 2, 3, 4, 5 and 6 are presented. We build a model of lepton masses and mixing based on $T'$ modular symmetry.
This is a review article about neutrino mass and mixing and flavour model building strategies based on discrete family symmetry. After a pedagogical introduction and overview of the whole of neutrino physics, we focus on the PMNS mixing matrix and the latest global fits following the Daya Bay and RENO experiments which measure the reactor angle. We then describe the simple bimaximal, tri-bimaximal and golden ratio patterns of lepton mixing and the deviations required for a non-zero reactor angle, with solar or atmospheric mixing sum rules resulting from charged lepton corrections or residual t…
We construct lepton flavour models based on two $A_4$ modular symmetries. The two $A_4$ are broken by a bi-triplet field to the diagonal $A_4$ subgroup, resulting in an effective modular $A_4$ flavour symmetry with two moduli. We employ these moduli as stabilisers, that preserve distinct residual symmetries, enabling us to obtain Tri-Maximal 2 (TM2) mixing with a minimal field content (without flavons).
We consider a flavor model based on A4 modular group to account for both lepton and quark parameters (masses and mixing). The inverse seesaw mechanism is considered to produce the light neutrino masses. Lepton masses and mixing are obtained in terms of Yukawa coupling ratios and values of the modulus τ nearby some fixed points for inverted neutrino mass hierarchy. The quark masses and mixing are arisen at the same τ values used in inverted neutrino mass hierarchy and are in agreements with the recent data.
