CP Transformation Research Articles

CP-like symmetry with discrete and continuous groups and CP violation/restoration

We study physical implications of general CP symmetry including CP-like symmetry. Various scattering amplitudes of CP asymmetry are calculated in CP-like symmetric models. We explicitly show that the CP-like transformation leads to a specific relation between different CP asymmetries. The resultant relation is similar to the one obtained in GUT baryogenesis and sphaleron processes, where we also obtain a required condition for generating particle number asymmetry in CP-like symmetric models. In addition, we propose a generalization of a CP-like transformation for continuous symmetry groups. Since the CP transformation is an outer automorphism, which depends on the internal symmetry group, it turns out that the physical CP and CP-like symmetries can be mutually converted through the spontaneous symmetry breaking (SSB) of the internal symmetry. We investigate properties of physical CP asymmetry in both CP and CP-like symmetric phases, and find that the spontaneous CP violation and restoration can be observed even in models with continuous groups. We demonstrate that CP-like symmetric models with continuous Lie groups can be naturally realized in physical CP symmetric models through the SSB.

The basis invariant flavor puzzle

The flavor puzzle of the Standard Model quark sector is formulated in a non-perturbative way, using basis invariants that are independent of the choice of quark field basis. To achieve this, we first derive the algebraic ring of 10 CP even (primary) and 1 CP odd (secondary) basis invariants, using the Hilbert series and plethystic logarithm. An orthogonal basis in the ring of basis invariants is explicitly constructed, using hermitian projection operators derived via birdtrack diagrams. The thereby constructed invariants have well defined CP transformation behavior and give the most direct access to the flavor symmetric alignments of basis covariants. We firstly “measure” the orthogonal basis invariants from experimental data and characterize their location in the available parameter space. The experimentally observed orthogonal basis invariants take very close to maximal values and are highly correlated. Explaining the location of the invariants at close to maximal points, including the associated miniscule and highly correlated deviations, corresponds to solving the flavor puzzle in the invariant language. Once properly normalized, the orthogonal basis invariants are close to scale (RGE) invariant, hence, provide exquisite targets for fits of both, low- and high-scale (bottom-up and top-down) flavor models. Our result provides an entirely new angle on the flavor puzzle, and opens up ample opportunities for its ultimate exploration.

Neutrino Flavor Model Building and the Origins of Flavor and CP Violation

The neutrino sector offers one of the most sensitive probes of new physics beyond the Standard Model of Particle Physics (SM). The mechanism of neutrino mass generation is still unknown. The observed suppression of neutrino masses hints at a large scale, conceivably of the order of the scale of a rand unified theory (GUT), which is a unique feature of neutrinos that is not shared by the charged fermions. The origin of neutrino masses and mixing is part of the outstanding puzzle of fermion masses and mixings, which is not explained ab initio in the SM. Flavor model building for both quark and lepton sectors is important in order to gain a better understanding of the origin of the structure of mass hierarchy and flavor mixing, which constitute the dominant fraction of the SM parameters. Recent activities in neutrino flavor model building based on non-Abelian discrete flavor symmetries and modular flavor symmetries have been shown to be a promising direction to explore. The emerging models provide a framework that has a significantly reduced number of undetermined parameters in the flavor sector. In addition, such a framework affords a novel origin of CP violation from group theory due to the intimate connection between physical CP transformation and group theoretical properties of non-Abelian discrete groups. Model building based on non-Abelian discrete flavor symmetries and their modular variants enables the particle physics community to interpret the current and anticipated upcoming data from neutrino experiments. Non-Abelian discrete flavor symmetries and their modular variants can result from compactification of a higher-dimensional theory. Pursuit of flavor model building based on such frameworks thus also provides the connection to possible UV completions: in particular, to string theory. We emphasize the importance of constructing models in which the uncertainties of theoretical predictions are smaller than, or at most compatible with, the error bars of measurements in neutrino experiments. While there exist proof-of-principle versions of bottom-up models in which the theoretical uncertainties are under control, it is remarkable that the key ingredients of such constructions were discovered first in top-down model building. We outline how a successful unification of bottom-up and top-down ideas and techniques may guide us towards a new era of precision flavor model building in which future experimental results can give us crucial insights into the UV completion of the SM.

Generalized matter parities from finite modular symmetries

AbstractWe classify a supersymmetric extension of the Standard Model by discrete symmetries originating from finite modular symmetries ΓN. Since all the couplings in supersymmetric theories of finite modular symmetries ΓN are described by holomorphic modular forms with even modular weights, renormalizable and non-renormalizable operators such as baryon- and/or lepton-number violating operators are severely constrained. From the modular transformation of matter multiplets with modular weight 1/M, we find $\mathbb {Z}_{2M}$ symmetries, including the generalized baryon and lepton parities, R-parity, $\mathbb {Z}_3$ baryon triality and $\mathbb {Z}_6$ proton hexality. Such $\mathbb {Z}_{2M}$ symmetries are enlarged to $\mathbb {Z}_{2M} \rtimes \mathbb {Z}_2^{\text{CP}}$ symmetries together with the CP transformation.

Shift-Invariant Orders of an Axionlike Particle.

It is generally believed that global symmetries, in particular, axion shift symmetries, can only be approximate. This motivates one to quantify the breaking of the shift invariance that characterizes the flavorful effective couplings of an axionlike particle to standard model fermions. We identify a minimal set of Jarlskog-like flavor invariants that vanish if and only if the axion is shift symmetric. Therefore, they represent order parameters for the breaking of the axion shift symmetry. We illustrate properties of the invariants by matching to a UV model, studying the CP transformation of the invariants, calculating their renormalization group evolution, and investigating similar conditions in the low-energy effective field theory below the electroweak scale. Finally, we discuss the order parameter associated to the nonperturbative shift-breaking induced by the axion-gluons coupling, which is also flavorful.

23, 381, 6242, 103268, 1743183, … : Hilbert series for CP-violating operators in SMEFT

We introduce a systematic method to classify the Standard Model Effective Field Theory (SMEFT) operators based on their CP properties with the Hilbert series techniques. Our method makes it possible to enumerate operators violating CP symmetry in a few seconds. We present the complete classification of dimension eight operators under CP transformation, and the number of CP-odd or CP-violating operators are listed up to dimension 14. We also provide a companion code in Form that allows anybody to reproduce our results.

Time reversal and CP invariance in Calabi-Yau compactifications

We revisit the question of time reversal and CP invariance in Calabi-Yau compactifications. We show that time reversal invariance is respected by quantum corrections to the prepotential. In particular, field independent θ angles whose presence is dictated by requiring integrality of relevant monodromy transformations can take precisely the quantized values compatible with time reversal invariance. Furthermore, monodromy symmetry enlarges the region on moduli space on which time reversal is not spontaneously broken. We define the action of the CP transformation for multi-parameter models and argue that on the slice of moduli space where it is defined, CP is trivially a symmetry of the theory. For supersymmetric vacua that lie in this slice, we derive a condition on the third cohomology of the compactification manifold which determines whether CP preserving fluxes exist that stabilize the moduli to such points. In the case of one-parameter models, the condition is always satisfied.

The chloroplast genome of Salix floderusii and characterization of chloroplast regulatory elements.

Salix floderusii is a rare alpine tree species in the Salix genus. Unfortunately, no extensive germplasm identification, molecular phylogeny, and chloroplast genomics of this plant have been conducted. We sequenced the chloroplast (cp) genome of S. floderusii for the first time using second-generation sequencing technology. The cp genome was 155,540 bp long, including a large single-copy region (LSC, 84,401 bp), a small single-copy region (SSC, 16,221 bp), and inverted repeat regions (IR, 54,918 bp). A total of 131 genes were identified, including 86 protein genes, 37 tRNA genes, and 8 rRNA genes. The S. floderusii cp genome contains 1 complement repeat, 24 forward repeats, 17 palindromic repeats, and 7 reverse repeats. Analysis of the IR borders showed that the IRa and IRb regions of S. floderusii and Salix caprea were shorter than those of Salix cinerea, which may affect plastome evolution. Furthermore, four highly variable regions were found, including the rpl22 coding region, psbM/trnD-GUC non-coding region, petA/psbJ non-coding region, and ycf1 coding region. These high variable regions can be used as candidate molecular markers and as a reference for identifying future Salix species. In addition, phylogenetic analysis indicated that the cp genome of S. floderusii is sister to Salix cupularis and belongs to the Subgenus Vetrix. Genes (Sf-trnI, Sf-PpsbA, aadA, Sf-TpsbA, Sf-trnA) obtained via cloning were inserted into the pBluescript II SK (+) to yield the cp expression vectors, which harbored the selectable marker gene aadA. The results of a spectinomycin resistance test indicated that the cp expression vector had been successfully constructed. Moreover, the aadA gene was efficiently expressed under the regulation of predicted regulatory elements. The present study provides a solid foundation for establishing subsequent S. floderusii cp transformation systems and developing strategies for the genetic improvement of S. floderusii.

Lepton and quark mixing patterns with generalized CP transformations

In this study, we modify a scenario, originally proposed by Grimus and Lavoura, in order to obtain maximal values for the atmospheric mixing angle and , violating the Dirac phase of the lepton sector. To achieve this, we employ and some discrete symmetries in a type II seesaw model. To make predictions about the neutrino mass ordering and smallness of the reactor angle, we establish some conditions on the elements of the neutrino mass matrix of our model. Finally, we study the quark masses and mixing pattern within the framework of our model.

Dual Circularly Polarized 3-D Printed Broadband Dielectric Reflectarray With a Linearly Polarized Feed

A broadband dual circularly polarized (dual-CP) reflectarray based on 3-D printed dielectric materials is proposed in this article. A novel 3-D dielectric array element that enables the broadband linearly polarization (LP) to CP transformation is proposed. The unit cell consists of two orthogonal dielectric cuboids that adjust the phases of the two orthogonal LP waves independently and then combine them into a CP wave. The innovative unit cell design provides an extra degree of freedom in varying the geometries of the array elements in all three dimensions, which enables us to independently control the phases of the two LP waves. This maintains an equal amplitude and 90° phase difference condition across the entire reflectarray surface, realizing a broadband and high gain LP–CP reflectarray. The placement of the feed is also optimized to achieve the highest aperture efficiency. Finally, an off-set reflectarray was designed and fabricated using lost-cost 3-D printing. The reflectarray is able to provide both left-hand circular polarization (LHCP) and right-hand circular polarization (RHCP), with just an LP feed. The measurements agree well with simulated results where the maximum realized gain and directivity at 34 GHz are measured as 27.9 and 28.1 dBi, respectively. The measured 3 dB gain bandwidth and aperture efficiency are 30% and up to 38%, respectively. More importantly, a broad 3 dB axial ratio (AR) bandwidth greater than 40% has been achieved for both LHCP and RHCP, covering almost the entire frequency band of interest, ranging from 26 to 40 GHz.

Identifying a minimal flavor symmetry of the seesaw mechanism behind neutrino oscillations

In the canonical seesaw framework flavor mixing and CP violation in weak charged-current interactions of light and heavy Majorana neutrinos are correlated with each other and described respectively by the 3 × 3 matrices U and R. We show that the very possibility of |Uμi| = |Uτi| (for i = 1, 2, 3), which is strongly indicated by current neutrino oscillation data, automatically leads to a novel prediction |Rμi| = |Rτi| (for i = 1, 2, 3). We prove that behind these two sets of equalities and the experimental evidence for leptonic CP violation lies a minimal flavor symmetry — the overall neutrino mass term keeps invariant when the left-handed neutrino fields transform as νeL→ (νeL)c, νμL→ (ντL)c, ντL→ (νμL)c and the right-handed neutrino fields undergo an arbitrary unitary CP transformation. Such a generalized μ-τ reflection symmetry may help constrain the flavor textures of active and sterile neutrinos to some extent in the seesaw mechanism.

Double cover of modular S4 for flavour model building

We develop the formalism of the finite modular group Γ4′≡S4′, a double cover of the modular permutation group Γ4≃S4, for theories of flavour. The integer weight k>0 of the level 4 modular forms indispensable for the formalism can be even or odd. We explicitly construct the lowest-weight (k=1) modular forms in terms of two Jacobi theta constants, denoted as ε(τ) and θ(τ), τ being the modulus. We show that these forms furnish a 3D representation of S4′ not present for S4. Having derived the S4′ multiplication rules and Clebsch-Gordan coefficients, we construct multiplets of modular forms of weights up to k=10. These are expressed as polynomials in ε and θ, bypassing the need to search for non-linear constraints. We further show that within S4′ there are two options to define the (generalised) CP transformation and we discuss the possible residual symmetries in theories based on modular and CP invariance. Finally, we provide two examples of application of our results, constructing phenomenologically viable lepton flavour models.

Transformation of short-chain chlorinated paraffins by the bacterial haloalkane dehalogenase LinB – Formation of mono- and di-hydroxylated metabolites

Short-chain chlorinated paraffins (SCCPs) are listed as persistent organic pollutants (POPs) under the Stockholm Convention. Such substances are toxic, bioaccumulating, transported over long distances and degrade slowly in the environment. Certain bacterial strains of the Sphingomonadacea family are able to degrade POPs, such as hexachlorocyclohexanes (HCHs) and hexabromocyclododecanes (HBCDs). The haloalkane dehalogenase LinB, expressed in certain Sphingomonadacea, is able to catalyze the transformation of haloalkanes to hydroxylated compounds. Therefore, LinB is a promising candidate for conversion of SCCPs. Hence, a mixture of chlorinated tridecanes was exposed in vitro to LinB, which was obtained through heterologous expression in Escherichia coli. Liquid chromatography mass spectrometry (LC-MS) was used to analyze chlorinated tridecanes and their transformation products. A chloride-enhanced soft ionization method, which favors the formation of chloride adducts [M+Cl]- without fragmentation, was applied. Mathematical deconvolution was used to distinguish interfering mass spectra of paraffinic, mono-olefinic and di-olefinic compounds. Several mono- and di-hydroxylated products including paraffinic, mono-olefinic and di-olefinic compounds were found after LinB exposure. Mono- (rt = 5.9–6.9 min) and di-hydroxylated (rt = 3.2–4.5 min) compounds were separated from starting material (rt = 7.7–8.5 min) by reversed phase LC. Chlorination degrees of chlorinated tridecanes increased during LinB-exposure from nCl = 8.80 to 9.07, indicating a preferential transformation of lower chlorinated (Cl<9) tridecanes. Thus, LinB indeed catalyzed a dehalohydroxylation of chlorinated tridecanes, tridecenes and tridecadienes. The observed hydroxylated compounds are relevant CP transformation products whose environmental and toxicological effects should be further investigated.

Fermion number 1/2 of sphalerons and spectral mirror symmetry

We present a rederivation of the baryon and lepton numbers frac{1}{2} of the SU(2)_L S sphaleron of the standard electroweak theory based on spectral mirror symmetry. We explore the properties of a fermionic Hamiltonian under discrete transformations along a noncontractible loop of field configurations that passes through the sphaleron and whose endpoints are the vacuum. As is well known, CP transformation is not a symmetry of the system anywhere on the loop, except at the endpoints. By augmenting CP with a chirality transformation, we observe that the Dirac Hamiltonian is odd under the new transformation precisely at the sphaleron, and this ensures the mirror symmetry of the spectrum, including the continua. As a consistency check, we show that the fermionic zero mode presented by Ringwald in the sphaleron background is invariant under the new transformation. The spectral mirror symmetry which we establish here, together with the presence of the zero mode, are the two necessary conditions whence the fermion number frac{1}{2} of the sphaleron can be inferred using the reasoning presented by Jackiw and Rebbi or, equivalently, using the spectral deficiency frac{1}{2} of the Dirac sea. The relevance of this analysis to other solutions is also discussed.

Discriminating between CP and family transformations in the bilinear space of the N -Higgs-doublet model

The scalar potential of the N-Higgs-doublet model (NHDM) is best analyzed not in the space of N complex doublets $\phi_a$ but in the $N^2$-dimensional space of real-valued bilinears constructed of $\phi_a^\dagger \phi_b$. In particular, many insights have been gained into CP violation in the 2HDM and 3HDM by studying how generalized CP transformations (GCPs) act in this bilinear space. These insights relied on the fact that GCPs, which involved an odd number of mirror reflection, could be clearly distinguished from Higgs family transformations by the sign of the determinant of the transformation matrix. It was recently pointed out that this criterion fails starting from 4HDM, where the reflection/rotation dichotomy does not exist anymore. In this paper, we restore intuition by finding a different quantity which faithfully discriminates between GCPs and Higgs family transformations in the bilinear space for any number of Higgs doublets. We also establish the necessary and sufficient conditions for an orthogonal transformation in the bilinear space to represent a viable transformation back in the space of N doublets, which is helpful if one prefers to build an NHDM directly in the bilinear space.

Generalized P and CP transformations in the three-Higgs-doublet model

We study generalised P and CP transformations in the three-Higgs-doublet model (3HDM) with Higgs and gauge fields only. We find that there are two equivalence classes, with respect to flavour transformations, of generalised P transformations and there is only one class of CP transformations. We discuss the conditions the potential has to satisfy in order to be invariant under these transformations. We apply the method of bilinears which we briefly review. We discuss the relation to the conventional basis, where the potential is written in terms of scalar products of the doublet fields. In particular we reproduce the known result that a potential is invariant under CP transformations if and only if there is a conventional basis where all parameters are real. Eventually we study standard P and CP transformations in the $n$-Higgs-doublet model (nHDM). We show that for the bilinears of the nHDM the standard CP transformation corresponds to a diagonal linear transformation with only $\pm 1$ as diagonal elements. We give this matrix explicitly for arbitrary $n$.

Importance of generalized \u03bc\u03c4 symmetry and its CP extension on neutrino mixing and leptogenesis

Within the framework of residual symmetry, two ℤ2 type associate μτ inter- change symmetries robustly constrain the Dirac CP phase δ in a model independent way. Both of them predict simultaneous maximality of δ and the atmospheric mixing angle θ23. We show how these well known correlations will be changed if we generalize the μτ in- terchange symmetry to a μτ mixing symmetry. In particular, we show that the stringent condition of simultaneous maximality could be relaxed even with a very small departure from the exact μτ interchange. In addition, the present neutrino data on δ and θ23 can be explained better by the mixing symmetry. After discussing the impact of the μτ mix- ing in some realistic neutrino mass models, we show how the proposed mixing could be realized with two simultaneous CP transformations which also lead to novel and testable correlations between δ and the mixing angles θij . Next we discuss in particular, the ‘three flavour regime’ of leptogenesis within the CP extended framework and show, unlike the ordinary CP extended μτ interchange symmetry, a resonant leptogenesis is possible due the generalization of μτ interchange to the μτ mixing and the resulting baryon asymmetry always requires a nonmaximal θ23 owing to the fact that the baryon to photon ratio ηB vanishes in the exact limit of θ23 = π/4. This is one of the robust predictions of this frame- work. The CP extended μτ mixing is also a novel example of a low energy effective model that provides an important insight to the off-diagonal terms of the flavour coupling matrix which have usually been neglected in literature to compute the final baryon asymmetry, in particular in the models with flavour symmetries.

Seesaw mechanism and pseudo C-symmetry

It is shown that the specific “charge conjugation” transformation used to define the Majorana fermions in the conventional seesaw mechanism, namely (nu _{R})^{C}=Coverline{nu _{R}}^{T} for a chiral fermion nu _{R} (and similarly for nu _{L}), is a hidden symmetry associated with CP symmetry, and thus it formally holds independently of the P- and C-violating terms in the CP invariant Lagrangian and it is in principle applicable to charged leptons and quarks as well. This hidden symmetry, however, is not supported by a consistent unitary operator and thus it leads to mathematical (operatorial) ambiguities. When carefully examined, it also fails as a classical transformation law in a Lorentz invariant field theory. To distinguish it from the standard charge conjugation symmetry, we suggest for it the name of pseudo C-symmetry. The pseudo C-symmetry is effective to identify Majorana neutrinos analogously to the classical Majorana condition. The analysis of CP breaking in weak interactions is performed using the conventional CP transformation, which is defined independently of the pseudo C-transformation, in the seesaw model after mass diagonalization. A way to ensure an operatorially consistent formulation of C-conjugation is to formulate the seesaw scheme by invoking a relativistic analogue of the Bogoliubov transformation.

Testing generalized CP symmetries with precision studies at DUNE

We examine the capabilities of the DUNE experiment in probing leptonic CP violation within the framework of theories with generalized CP symmetries characterized by the texture zeros of the corresponding CP transformation matrices. We investigate DUNE's potential to probe the two least known oscillation parameters, the atmospheric mixing angle $\theta_{23}$ and the Dirac CP-phase $\delta_{\rm CP}$. We fix theory-motivated benchmarks for ($ \sin^2\theta_{23}, \delta_{\rm CP} $) and take them as true values in our simulations. Assuming 3.5 years of neutrino running plus 3.5 years in the antineutrino mode, we show that in all cases DUNE can significantly constrain and in certain cases rule out the generalized CP texture zero patterns.

Group–Theoretical Origin of CP Violation

This is a short review of the proposal that CP violation may be due to the fact that certain finite groups do not admit a physical CP transformation. This origin of CP violation is realized in explicit string compactifications exhibiting the Standard Model spectrum.