CP Transformation Research Articles

CP transformed mixed μτ antisymmetry for neutrinos and its consequences

We propose a complex extension of mixed $\mu\tau$ antisymmetry in the neutrino Majorana mass matrix $M_\nu$. This can be implemented in the Lagrangian by a generalized CP transformation (labeled by a mixing parameter $\theta$) on the left-chiral flavor neutrino fields. We investigate its implications for leptonic CP violation and neutrino phenomenology in general. Interestingly, the $\mu\tau$ mixing parameter $\theta$ gets correlated with the Dirac CP phase $\delta$ and the atmospheric mixing angle $\theta_{23}$ through an analytical relation. In general, for arbitrary $\theta$, both $\theta_{23}$ and $\delta$ are nonmaximal. We discuss the corresponding results for the CP asymmetry parameter $A_{\mu e}$ in neutrino oscillation experiments. For a nonmaximal $\delta$, one of the two Majorana phases is different from $0$ or $\pi$, thereby leading to nonvanishing Majorana CP violation with observable consequences for the neutrinoless double beta ($\beta\beta0\nu$) decay process. We numerically work out in detail the predictions for that process in relation to various ongoing and forthcoming experiments. We also work out the predictions of our scheme on flavor flux ratios at neutrino telescopes. While exact CP transformed $\mu\tau$ interchange antisymmetry ($\theta=\pi/2$) leads to an exact equality among those ratios, taking a value $0.5$, a tiny deviation can cause a drastic change in them. Careful measurement of these flux ratios in future will further constrain the parameter $\theta$.

Quark and lepton mixing patterns from a common discrete flavor symmetry with a generalized CP symmetry

We have studied two approaches to predict quark and lepton mixing patterns from the same flavor symmetry group in combination with CP symmetry. The first approach is based on the residual symmetry $Z_2$ in the charged lepton sector and $Z_2\times CP$ in the neutrino sector. All lepton mixing angles and CP violation phases depend on three real parameters $\theta_{l}$, $\delta_{l}$ and $\theta_{\nu}$. This approach is extended to the quark sector, the up and down quark mass matrices are assumed to be invariant under a $Z_2$ subgroup and $Z_2\times CP$. The necessary and sufficient conditions for the equivalence of two mixing patterns are derived. The second approach has an abelian subgroup and a single CP transformation as residual symmetries of the charged lepton and neutrino sectors respectively. The lepton mixing would be determined up to a real orthogonal matrix multiplied from the right hand side. Analogously we assume that a single CP transformation is preserved by the down (or up) quark mass matrix, and the residual symmetry of the up (or down) quark sector is be an abelian subgroup. As an example, we analyze the possible mixing patterns which can be obtained from the breaking of $\Delta(6n^2)$ and CP symmetries. We find $\Delta(294)$ combined with CP is the smallest flavor group which can give a good fit to the experimental data of quark and lepton mixing in both approaches.

Neutrino predictions from generalized CP symmetries of charged leptons

We study the implications of generalized CP transformations acting on the mass matrices of charged leptons in a model-independent way. Generalized e − μ, e − τ and μ−τ symmetries are considered in detail. In all cases the physical parameters of the lepton mixing matrix, three mixing angles and three CP phases can be expressed in terms of a restricted set of independent “theory parameters” that characterize a given choice of CP transformation. This leads to implications for neutrino oscillations as well as neutrinoless double beta decay experiments.

Neutrino propagation in an electron background with an inhomogeneous magnetic field

We study the electromagnetic coupling of a neutrino that propagates in a two-stream electron background medium. Specifically, we calculate the electromagnetic vertex function for a medium that consists of a normal electron background plus another electron stream background that is moving with a velocity four-vector v^mu relative to the normal background. The results can be used as the basis for studying the neutrino electromagnetic properties and various processes in such a medium. As an application, we calculate the neutrino dispersion relation in the presence of an external magnetic field (mathbf {B}), focused in the case in which B is inhomogeneous, keeping only the terms of the lowest order in 1/m^2_W and linear in the B and its gradient. We show that the dispersion relation contains additional anisotropic terms involving the derivatives of mathbf {B}, such as the gradient of {hat{k}}cdot (mathbf {v}times mathbf {B}), which involve the stream background velocity, and a term of the form {hat{k}}cdot (nabla times mathbf {B}) that can be present in the absence of the stream background, in addition to a term of the form {hat{k}}cdot mathbf {v} and the well known term {hat{k}}cdot mathbf {B} that arises in the constant mathbf {B} case. The derivative-dependent terms are even under a CP transformation. As a result, in contrast to the latter two just mentioned, they depend on the sum of the particle and antiparticle densities and therefore can be non-zero in a CP-symmetric medium in which the particle and antiparticle densities are equal.

CP symmetry and thermal effects on Dirac bi-spinor spin–parity local correlations

Intrinsic quantum correlations supported by the SU(2)⊗SU(2) structure of the Dirac equation used to describe particle/antiparticle states, optical ion traps and bilayer graphene are investigated and connected to the description of local properties of Dirac bi-spinors. For quantum states driven by Dirac-like Hamiltonians, quantum entanglement and geometric discord between spin and parity degrees of freedom – sometimes mapped into equivalent low energy internal degrees of freedom – are obtained. Such spin–parity quantum correlations and the corresponding nonlocal intrinsic structures of bi-spinor fermionic states can be classified in order to relate quantum observables to the (non)local behavior of these correlations. It is shown that free particle mixed states do not violate the Clauser–Horne–Shimony–Holt inequality: the correlations in such mixed bi-spinors, although quantum, can be reproduced by a suitable local hidden variable model. Additionally, the effects due to a non-minimal coupling to a homogeneous magnetic field, and to the inclusion of thermal effects are evaluated, and quantum correlations of associated quantum mixtures and of the thermal states are all quantified. The above-mentioned correlation quantifiers are then used to measure the influence of CP transformations on spin–parity quantum correlations, and our results show that quantum entanglement is invariant under CP transformations, although the geometric discord is highly sensitive to the CP symmetry.

The Complete Chloroplast Genome of Catha edulis: A Comparative Analysis of Genome Features with Related Species.

Qat (Catha edulis, Celastraceae) is a woody evergreen species with great economic and cultural importance. It is cultivated for its stimulant alkaloids cathine and cathinone in East Africa and southwest Arabia. However, genome information, especially DNA sequence resources, for C. edulis are limited, hindering studies regarding interspecific and intraspecific relationships. Herein, the complete chloroplast (cp) genome of Catha edulis is reported. This genome is 157,960 bp in length with 37% GC content and is structurally arranged into two 26,577 bp inverted repeats and two single-copy areas. The size of the small single-copy and the large single-copy regions were 18,491 bp and 86,315 bp, respectively. The C. edulis cp genome consists of 129 coding genes including 37 transfer RNA (tRNA) genes, 8 ribosomal RNA (rRNA) genes, and 84 protein coding genes. For those genes, 112 are single copy genes and 17 genes are duplicated in two inverted regions with seven tRNAs, four rRNAs, and six protein coding genes. The phylogenetic relationships resolved from the cp genome of qat and 32 other species confirms the monophyly of Celastraceae. The cp genomes of C. edulis, Euonymus japonicus and seven Celastraceae species lack the rps16 intron, which indicates an intron loss took place among an ancestor of this family. The cp genome of C. edulis provides a highly valuable genetic resource for further phylogenomic research, barcoding and cp transformation in Celastraceae.

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.

CP-symmetry of order 4 and its consequences

Extended Higgs sectors offer rich opportunities for various forms of CP-violation. Here, we describe a new form of CP-conservation and discuss its consequences. We give a concrete example of a three-Higgs-doublet model dubbed CP4-3HDM with a CP-symmetry of order 4 and no other other accidental symmetries. If the vacuum conserves this symmetry, the model is CP-conserving with pairwise mass-degenerate extra neutral Higgs bosons. These fields cannot be classified as CP-even or CP-odd but they can be combined into complex physical fields which are CP-half-odd, that is, they pick up the i factor upon CP transformation. These CP-half-odd scalars can be Yukawa-coupled to the fermion bilinears in a CP-conserving way. We discuss fundamental and phenomenological features of the model, and stress a peculiar clash between the CP-symmetry and any convention for the particle-antiparticle assignment.

CP as a Symmetry of Symmetries

It is explained that the Standard Model combined charge conjugation and parity transformation (CP) is a simultaneous complex conjugation outer automorphism transformation of gauge and space-time symmetries. Simple examples are given for the general concept of outer automorphisms (“symmetries of symmetries”), as well as for their possible actions on physical theories. It is highlighted that complex conjugation outer automorphisms do not, in general, exist for all symmetries. Examples are given for cases in which the physical CP transformation is violated as a consequence of requiring another symmetry. A toy model is illustrated in which CP is spontaneously violated in the broken phase of a continuous gauge symmetry, while an unbroken outer automorphism protects the topological vacuum angle at θ = 0.

Lepton mixing patterns from the group Σ(36×3) with a generalized CP transformation

The group $\Sigma(36\times3)$ with the generalized CP transformation is introduced to predict the mixing pattern of leptons. Various combinations of abelian residual flavor symmetries with CP transformations are surveyed. Six mixing patterns could accommodate the fit data of neutrinos oscillation at $3\sigma$ level. Among them, two patterns predict the nontrivial Dirac CP phase, around $\pm 57^{\circ}$ or $\pm 123^{\circ}$, which is in accordance with the result of the literature and the recent fit data. Furthermore, one pattern could satisfy the experimental constraints at $1\sigma$ level.

Flavor and CP symmetries for leptogenesis and 0νββ decay

We perform a comprehensive analysis of the phenomenology of leptonic low and high energy CP phases in a scenario with three heavy right-handed neutrinos in which a flavor and a CP symmetry are non-trivially broken. All CP phases as well as lepton mixing angles are determined by the properties of the flavor and CP symmetry and one free real parameter. We focus on the generation of the baryon asymmetry YB of the Universe via unflavored leptogenesis and the predictions of mee, the quantity measurable in neutrinoless double beta decay. We show that the sign of YB can be fixed and the allowed parameter range of mee can be strongly constrained. We argue on general grounds that the CP asymmetries ϵi are dominated by the contribution associated with one Majorana phase and that in cases in which only the Dirac phase is non-trivial the sign of YB depends on further parameters. In addition, we comment on the case of flavored leptogenesis where in general the knowledge of the CP phases and light neutrino mass spectrum is also not sufficient in order to fix the sign of the CP asymmetries. As examples we discuss the series of flavor groups Δ(3n2) and Δ(6n2), n≥2 integer, and several classes of CP transformations.

The remnant CP transformation and its implications

In the context of remnant CP transformations, I briefly discuss a generalized μ‒τ reflection symmetry, where the “Majorana” phases have CP conserving values, which are directly related with the CP parities of neutrino states. Also, one finds that the “Dirac-like” CP violation phase is correlated with the atmospheric mixing angle, giving important phenomenological implications for current and future long baseline oscillation neutrino experiments.

GeneralizedCPandΔ(3n2)family symmetry for semidirect predictions of the PMNS matrix

The generalized CP transformations can only be consistently defined in the context of $\Delta(3n^2)$ lepton symmetry if a certain subset of irreducible representations are present in a model. We perform a comprehensive analysis of the possible automorphisms and the corresponding CP transformations of the $\Delta(3n^2)$ group. It is sufficient to only consider three automorphisms if $n$ is not divisible by 3 while additional eight types of CP transformations could be imposed for the case of $n$ divisible by 3. We study the lepton mixing patterns which can be derived from the $\Delta(3n^2)$ family symmetry and generalized CP in the semi-direct approach. The PMNS matrix is determined to be the trimaximal pattern for all the possible CP transformations, and it can only take two distinct forms.

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.

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 $\Delta(6n^2)$.

Invariant approach toCPin family symmetry models

We propose the use of basis invariants, valid for any choice of CP transformation, as a powerful approach to studying specific models of CP violation in the presence of discrete family symmetries. We illustrate the virtues of this approach for examples based on $A_4$ and $\Delta(27)$ family symmetries. For $A_4$, we show how to elegantly obtain several known results in the literature. In $\Delta(27)$ we use the invariant approach to identify how explicit (rather than spontaneous) CP violation arises, which is geometrical in nature, i.e. persisting for arbitrary couplings in the Lagrangian.

Adding CP to flavour symmetries

I propose the use of CP-odd invariants, which are independent of basis and valid for any choice of CP transformation, as a powerful approach to study CP in the presence of flavour symmetries. As examples of the approach I focus on Lagrangians invariant under Δ(27). I comment on the consequences of adding a specific CP symmetry to a Lagrangian and distinguish cases where several Δ(27) singlets are present depending on how they couple to the triplets. One of the examples included is a very simple toy model with explicit CP violation with calculable phases, which is referred to as explicit geometrical CP violation by comparison with previously known cases of (spontaneous) geometrical CP violation.

Neutrino mixing and CP phase correlations

A special form of the 3×3 Majorana neutrino mass matrix derivable from μ–τ interchange symmetry accompanied by a generalized CP transformation was obtained many years ago. It predicts θ23=π/4 as well as δCP=±π/2, with θ13≠0. Whereas this is consistent with present data, we explore a deviation of this result which occurs naturally in a recent proposed model of radiative inverse seesaw neutrino mass.

Lepton flavor mixing andCPsymmetry

The strategy of constraining the lepton flavor mixing from remnant CP symmetry is investigated in a rather general way. The neutrino mass matrix generally admits four remnant CP transformations which can be derived from the measured lepton mixing matrix in the charged lepton diagonal basis. Conversely, the lepton mixing matrix can be reconstructed from the postulated remnant CP transformations. All mixing angles and CP violating phases can be completely determined by the full set of remnant CP transformations or three of them. When one or two remnant CP transformations are preserved, the resulting lepton mixing matrix would depend on three real parameters or one real parameter respectively in addition to the parameters characterizing the remnant CP, and the concrete form of the mixing matrix is presented. The phenomenological predictions for the mixing parameters are discussed. The conditions leading to vanishing or maximal Dirac CP violation are studied.

Lepton mixing from [formula omitted] and [formula omitted] and CP

We perform a detailed study of lepton mixing patterns arising from a scenario with three Majorana neutrinos in which a discrete flavor group Gf=Δ(3n2) or Gf=Δ(6n2) and a CP symmetry are broken to residual symmetries Ge=Z3 and Gν=Z2×CP in the charged lepton and neutrino sectors, respectively. While we consider all possible Z3 and Z2 generating elements, we focus on a certain set of CP transformations. The resulting lepton mixing depends on group theoretical indices and one continuous parameter. In order to study the mixing patterns comprehensively for all admitted Ge and Gν, it is sufficient to discuss only three types of combinations. One of them requires as flavor group Δ(6n2). Two types of combinations lead to mixing patterns with a trimaximal column, while the third one allows for a much richer structure. For the first type of combinations the Dirac phase as well as one of the Majorana phases are trivial, whereas the other two types of combinations predict in general all CP phases to be non-trivial and also non-maximal. Already for small values of the index n of the group, n≤11, experimental data on lepton mixing can be accommodated well for particular choices of the parameters of the theory. We also comment on the relation of the used CP transformations to the automorphisms of Δ(3n2) and Δ(6n2).