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In this paper, we study the connection between the tribimaximal and bitrimaximal mixing patterns. In doing so, we are forced to work in a non-diagonal charged lepton basis. This leads to several relations that must hold between the lepton mixing angles. After a short discussion, we analyze the underlying flavor symmetry responsible for this prediction. Finally, we add CP violation to bitrimaximal mixing and study its effect on the flavor symmetry group., Comment: 18 pages, 1 figure. v2: References added, text clarified, figure improved, content matches version published in PLB

By assuming there exist three massive non-degenerate Majorana neutrinos, it is possible to describe neutrino mixing with a residual, unbroken discrete Klein subgroup of a larger spontaneously broken flavor symmetry group. Motivated by forthcoming measurements of leptonic CP violation, we revisit this framework by applying group presentation rules to it. We develop a method that is able to reproduce all previous results in the literature and may also hint at a possible group theoretical origin of CP violation in the Klein symmetry elements. This is due to the explicit appearance of a phase in them. However, for the cases considered in this analysis, it turns out that this phase can be removed. Still, this new method warrants further study., Comment: 14 pages. v3: References added, text further clarified, content matches version published in IJMPA

We extend the recently proposed $SU(5) \times \mathcal{T}_{13}$ model for the asymmetric texture to the up-type quark and seesaw sectors. The hierarchical up-type quark masses are generated from higher-dimensional operators involving family-singlet Higgses, gauge-singlet familons, and vectorlike messengers. The complex-tribimaximal (TBM) seesaw mixing arises from the vacuum structure of a minimal number of familons, resulting in an alignment between the Yukawa and Majorana matrices of the seesaw formula. Introducing four right-handed neutrinos, normal ordering of the light neutrino masses is obtained, with $m_{\nu_1} = 27.6\ \mathrm{meV}$, $m_{\nu_2} = 28.9\ \mathrm{meV}$ and $m_{\nu_3} = 57.8\ \mathrm{meV}$. Their sum almost saturates Planck's cosmological upper bound ($120$ $\text{meV}$). The right-handed neutrino masses are expressed in terms of two parameters for a particular choice of familon vacuum alignment. We predict the $\require{cancel}\cancel{CP}$ Jarlskog-Greenberg invariant to be $|\mathcal{J}| = 0.028$, consistent with the current PDG estimate, and Majorana invariants $|\mathcal{I}_1| = 0.106$ and $|\mathcal{I}_2| = 0.011$. A sign ambiguity in the model parameters leads to two possibilities for the invariant mass parameter $|m_{\beta \beta}|$: $13.02$ or $25.21$ $\text{meV}$, both within an order of magnitude of the most rigorous experimental upper limit ($61$--$165$ $\text{meV}$)., Comment: 36 pages, 2 figures, minor changes to match with the published version

We explore the theoretical constraints on the observable parameters of neutrino mixing on predictions for the leptonic Dirac CP-violating phase within a well-studied class of simple theoretical models that includes a single source of CP violation due to charged lepton corrections. The approach guarantees that a physically meaningful prediction for the most likely values for the leptonic Dirac CP-violating phase is obtained., Comment: 19 pages, 6 figures, 4 tables. v3: Removed section on independent model parameters. Edited for clarity. Version submitted to IJMPA

We propose $\mathcal{T}_{13} = \mathcal{Z}_{13} \rtimes \mathcal{Z}_3$ as the underlying non-Abelian discrete family symmetry of the asymmetric texture presented in arXiv:1805.10684 [hep-ph]. Its mod 13 arithmetic distinguishes each Yukawa matrix element of the texture. We construct a model of effective interactions that singles out the asymmetry and equates, without fine-tuning, the products of down-quark and charged-lepton masses at a GUT-like scale., Comment: Major change from previous version: the symmetry $\mathcal{Z}_4$ has been changed to $\mathcal{Z}_5$ to prohibit a new unwanted vertex, 26 pages, version accepted for publication in Physical Review D

We explore the implications of recent results relating the Dirac CP-violating phase to predicted and measured leptonic mixing angles within a standard set of theoretical scenarios in which charged lepton corrections are responsible for generating a non-zero value of the reactor mixing angle. We employ a full set of leptonic sum rules as required by the unitarity of the lepton mixing matrix, which can be reduced to predictions for the observable mixing angles and the Dirac CP-violating phase in terms of model parameters. These sum rules are investigated within a given set of theoretical scenarios for the neutrino sector diagonalization matrix for several known classes of charged lepton corrections. The results provide explicit maps of the allowed model parameter space within each given scenario and assumed form of charged lepton perturbations., Comment: 37 pages, 58 figures. V2: Typos corrected, references added, content matches version published in PRD

We study the effects of considering nontrivial unphysical lepton sector phases on the group theoretical properties of the flavor and generalized CP symmetry elements in the case where there are three light, distinct Majorana neutrino species. We highlight the similarities and differences between the charged lepton and neutrino sectors and further elucidate the group properties of the flavor and generalized CP symmetry elements. We show how the inclusion of these leptonic phases affects the bottom-up constructions of these symmetry elements and discuss the implications for top-down model building based on discrete symmetry groups., Comment: 21 pages. v2: Typos corrected, discussion clarified throughout, Section 4 added with examples, main results unchanged, version to appear in PRD

Assuming that the observed pattern of 3-neutrino mixing is related to the existence of a (lepton) flavour symmetry, corresponding to a non-Abelian discrete symmetry group $G_f$, and that $G_f$ is broken to specific residual symmetries $G_e$ and $G_\nu$ of the charged lepton and neutrino mass terms, we derive sum rules for the cosine of the Dirac phase $\delta$ of the neutrino mixing matrix $U$. The residual symmetries considered are: i) $G_e = Z_2$ and $G_{\nu} = Z_n$, $n > 2$ or $Z_n \times Z_m$, $n,m \geq 2$; ii) $G_e = Z_n$, $n > 2$ or $Z_n \times Z_m$, $n,m \geq 2$ and $G_{\nu} = Z_2$; iii) $G_e = Z_2$ and $G_{\nu} = Z_2$; iv) $G_e$ is fully broken and $G_{\nu} = Z_n$, $n > 2$ or $Z_n \times Z_m$, $n,m \geq 2$; and v) $G_e = Z_n$, $n > 2$ or $Z_n \times Z_m$, $n,m \geq 2$ and $G_{\nu}$ is fully broken. For given $G_e$ and $G_\nu$, the sum rules for $\cos\delta$ thus derived are exact, within the approach employed, and are valid, in particular, for any $G_f$ containing $G_e$ and $G_\nu$ as subgroups. We identify the cases when the value of $\cos\delta$ cannot be determined, or cannot be uniquely determined, without making additional assumptions on unconstrained parameters. In a large class of cases considered the value of $\cos\delta$ can be unambiguously predicted once the flavour symmetry $G_f$ is fixed. We present predictions for $\cos\delta$ in these cases for the flavour symmetry groups $G_f = S_4$, $A_4$, $T^\prime$ and $A_5$, requiring that the measured values of the 3-neutrino mixing parameters $\sin^2\theta_{12}$, $\sin^2\theta_{13}$ and $\sin^2\theta_{23}$, taking into account their respective $3\sigma$ uncertainties, are successfully reproduced., Comment: 62 pages, includes 6 figures and 14 tables; results unchanged; sections 7 and 8 revised; typos corrected; matches published version

We perform a model-independent analysis of the possible residual Klein and generalized CP symmetries associated with arbitrary lepton mixing angles in the case that there are three light Majorana neutrino species. This approach emphasizes the unique role of the Majorana phases and provides a useful framework in which to discuss the origin of the Dirac CP phase in scenarios with spontaneously broken flavor and generalized CP symmetries. The method is shown to reproduce known examples in the literature based on tribimaximal and bitrimaximal mixing patterns, and is used to investigate these issues for the case of a particular (GR1) golden ratio mixing pattern., Comment: 18 pages, references added, minor typos corrected, matches version published in JHEP

We perform a comprehensive study of family symmetry models based on $A_4$ combined with the generalised CP symmetry $H_{\rm{CP}}$. We investigate the lepton mixing parameters which can be obtained from the original symmetry $A_4\rtimes H_{\rm{CP}}$ breaking to different remnant symmetries in the neutrino and charged lepton sectors. We find that only one case is phenomenologically viable, namely $G^{\nu}_{\rm{CP}}\cong Z^{S}_2\times H^{\nu}_{\rm{CP}}$ in the neutrino sector and $G^{l}_{\rm{CP}}\cong Z^{T}_3\rtimes H^{l}_{\rm{CP}}$ in the charged lepton sector, leading to the prediction of no CP violation, namely $\delta_{CP}$ and the Majorana phases $\alpha_{21}$ and $\alpha_{31}$ are all equal to either zero or $\pi$. We then propose an effective supersymmetric model based on the symmetry $A_4\rtimes H_{\rm{CP}}$ in which trimaximal lepton mixing is predicted together with either zero CP violation or $\delta_{CP}\simeq\pm \pi/2$ with non-trivial Majorana phases. An ultraviolet completion of the effective model yields a neutrino mass matrix which depends on only three real parameters. As a result of this, all three CP phases and the absolute neutrino mass scale are determined, the atmospheric mixing angle is maximal, and the Dirac CP can either be preserved with $\delta_{CP}=0,\pi$ or maximally broken with $\delta_{CP}=\pm \pi /2$ and sharp predictions for the Majorana phases and neutrinoless double beta decay., Comment: 38 pages, 3 figures

Neutrino mass sum rules relate the three neutrino masses within generic classes of flavour models, leading to restrictions on the effective mass parameter measured in experiments on neutrinoless double beta decay as a function of the lightest neutrino mass. We perform a comprehensive study of the implications of such neutrino mass sum rules, which provide a link between model building, phenomenology, and experiments. After a careful explanation of how to derive predictions from sum rules, we discuss a large number of examples both numerically, using all three global fits available for the neutrino oscillation data, and analytically wherever possible. In some cases, our results disagree with some of those in the literature for reasons that we explain. Finally we discuss the experimental prospects for many current and near-future experiments, with a particular focus on the uncertainties induced by the unknown nuclear physics involved. We find that, in many cases, the power of the neutrino mass sum rules is so strong as to allow certain classes of models to be tested by the next generation of neutrinoless double beta decay experiments. Our study can serve as both a guideline and a theoretical motivation for future experimental studies., Comment: v2: typos corrected, slight modifications, one sum rule added, one more figure

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 or pi and the atmospheric angle sum rules theta_{23}=45 degrees -/+ theta_{13}/sqrt{2}, respectively, which are consistent with recent global fits and will be tested in the near future., Comment: 12 pages, 1 figure, minor changes made, version published in Physics Letters B

We construct models of leptons based on $S_4$ family symmetry combined with a generalised CP symmetry $H_{CP}$. We show how the flavon potential can spontaneously break the symmetry $S_4 \rtimes H_{CP}$ down to $Z_2 \times H^{\nu}_{CP}$ in the neutrino sector, where the choice of preserved CP symmetry $H^{\nu}_{CP}$ is controlled by free (real) parameters in the flavon potential. We propose two realistic models of this kind, one at the effective level and one at the renormalisable level. Both models predict trimaximal lepton mixing with CP being either fully preserved or maximally broken, with the intermediate possibility forbidden by the structure of the models., Comment: 34 pages, 2 figures

We propose a new A_5 model of leptons which corrects the LO predictions of Golden Ratio mixing via a minimal NLO Majorana mass correction which completely breaks the original Klein symmetry of the neutrino mass matrix. The minimal nature of the NLO correction leads to a restricted and correlated range of the mixing angles allowing agreement within the one sigma range of recent global fits following the reactor angle measurement by Daya Bay and RENO. The minimal NLO correction also preserves the LO inverse neutrino mass sum rule leading to a neutrino mass spectrum that extends into the quasi-degenerate region allowing the model to be accessible to the current and future neutrinoless double beta decay experiments., Comment: 39 pages, 6 figures, version published in Nuclear Physics B

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 mixing angles and masses in the form of the Gatto-Sartori-Tonin relation and realises the Georgi-Jarlskog mass relations between the charged leptons and down-type quarks. We predict a Bi-trimaximal (not Tri-bimaximal) form of neutrino mixing matrix, which, after including charged lepton corrections with zero phase, leads to the following GUT scale predictions for the atmospheric, solar, and reactor mixing angles: theta_23=36.9 degrees, theta_12=32.7 degrees and theta_13=9.6 degrees, in good agreement with recent global fits, and a zero Dirac CP phase delta~0., Comment: 35 pages, version to appear in NPB

We provide a systematic and thorough exploration of lepton flavor models in which the solar mixing angle is related to the golden ratio. For scenarios in which the solar mixing angle is given by the inverse cotangent of the golden ratio, we demonstrate that $A_5$ is the smallest non-Abelian finite group that contains all of the symmetries necessary to enforce this specific lepton mixing pattern. Within this context, we propose two lepton flavor models that yield this mixing pattern through the breaking of $A_5$ at leading order to the Klein four subgroup in the neutrino sector. Both models have triplet embeddings of the lepton doublets as well as the charged lepton singlets. In the charged lepton sector, the residual symmetry is $Z_5$ in the first model, while in the second model, $A_5$ is broken completely at leading order. For the second model, the reactor mixing angle vanishes at leading order and is of the order of the square of the Cabibbo angle at next-to-leading order, which is allowed by the global analysis of current lepton data., Comment: 37 pages, 1 figure

We investigate the idea that the double cover of the rotational icosahedral symmetry group is the family symmetry group in the quark sector. The icosahedral (A5) group was previously proposed as a viable family symmetry group for the leptons. To incorporate the quarks, it is highly advantageous to extend the group to its double cover, as in the case of tetrahedral (A4) symmetry. We provide the basic group theoretical tools for flavor model-building based on the binary icosahedral group I' and construct a model of the quark masses and mixings that yields many of the successful predictions of the well-known U(2) quark texture models., Comment: 10 pages, references added, typos in up quark mass matrix corrected

We investigate the possibility of using icosahedral symmetry as a family symmetry group in the lepton sector. The rotational icosahedral group, which is isomorphic to A5, the alternating group of five elements, provides a natural context in which to explore (among other possibilities) the intriguing hypothesis that the solar neutrino mixing angle is governed by the golden ratio. We present a basic toolbox for model-building using icosahedral symmetry, including explicit representation matrices and tensor product rules. As a simple application, we construct a minimal model at tree level in which the solar angle is related to the golden ratio, the atmospheric angle is maximal, and the reactor angle vanishes to leading order. The approach provides a rich setting in which to investigate the flavor puzzle of the Standard Model., Comment: 22 pages, version to be published in Phys. Rev. D