Basis Invariants Research Articles

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.

A fully basis invariant symmetry map of the 2HDM

We derive necessary and sufficient conditions for all global symmetries of the most general two Higgs doublet model (2HDM) scalar potential entirely in terms of reparametrization independent, i.e. basis invariant, objects. This culminates in what we call a “Symmetry Map” of the parameter space of the model and the fundamental insight that there are, in general, two algebraically distinct ways of how symmetries manifest themselves on basis invariant objects: either, basis invariant objects can be non-trivially related, or, basis covariant objects can vanish. These two options have different consequences on the resulting structure of the ring of basis invariants and on the number of remaining physical parameters. Alongside, we derive for the first time necessary and sufficient conditions for CP conservation in the 2HDM entirely in terms of CP-even quantities. This study lays the methodological foundation for analogous investigations of global symmetries in all other models that have unphysical freedom of reparametrization, most notably the Standard Model flavor sector.

CP odd weak basis invariants in minimal see-saw model and leptogenesis

Abstract In this paper, we derive the relationship between the weak basis invariants (WB) related to CP violation responsible for leptogenesis and CP violation relevant at low energy. We examine all the experimentally viable cases of Frampton-Glashow and Yanagida (FGY) model, in order to construct the WB invariants in terms of left handed Majorana neutrino mass matrix elements, and thus finding the necessary and sufficient condition for CP conservation. Further for all the viable FGY texture zeros, we have shown the explicit dependence of WB invariants on Dirac type and Majorana type CP violating phases. In the end, we discuss the implication of such interrelationships on leptogenesis.

Two-axis two-spin squeezed states

The states generated by the two-spin generalization of the two-axis countertwisting Hamiltonian are examined. We analyze the behavior at both short and long timescales, by calculating various quantities such as squeezing, spin expectation values, probability distributions, entanglement, Wigner functions, and Bell correlations. In the limit of large spin ensembles and short interaction times, the state can be described by a two-mode squeezed vacuum state; for qubits, Bell state entanglement is produced. We find that the Hamiltonian approximately produces two types of spin-EPR states, and the time evolution produces aperiodic oscillations between them. In a similar way to the basis invariance of Bell states and two-mode squeezed vacuum states, the Fock state correlations of spin-EPR states are basis invariant. We find that it is possible to violate a Bell inequality with such states, although the violation diminishes with increasing ensemble size. Effective methods to detect entanglement are also proposed, and formulas for the optimal times to enhance various properties are calculated.

Implications of CP invariants for the flavored hybrid neutrino mass matrix

Abstract We re-examine the weak basis invariants at low energies proposed by C. Jarlskog and Branco et al. in their earlier analyses, after confronting them with the assumptions of two zeros and an equality between arbitrary non-zero elements in the Majorana neutrino mass matrix in the flavored basis. This particular conjecture is found to be experimentally feasible, as shown by S. Dev and D. Raj in their recent work. The present analysis attempts to find the necessary and sufficient condition for CP invariance for each experimentally viable ansatz pertaining to the model, along with some important implications.

On the systematic construction of basis invariants

We describe a new, generally applicable strategy for the systematic construction of basis invariants (BIs). Our method allows one to count the number of mutually independent BIs and gives controlled access to the interrelations (syzygies) between mutually dependent BIs. Due to the novel use of orthogonal hermitian projection operators, we obtain the shortest possible invariants and their interrelations. The substructure of non-linear BIs is fully resolved in terms of linear, basis-covariant objects. The substructure distinguishes real (CP-even) and purely imaginary (CP-odd) BIs in a simple manner. As an illustrative example, we construct the full ring of BIs of the scalar potential of the general Two-Higgs-Doublet model.

Type-I seesaw with eV-scale neutrinos

We consider seesaw type-I models including at least one (mostly-)sterile neutrino with mass at the eV scale. Three distinct situations are found, where the presence of light extra neutrinos is naturally justified by an approximately conserved lepton number symmetry. To analyse these scenarios consistently, it is crucial to employ an exact parametrisation of the full mixing matrix. We provide additional exact results, including generalised versions of the seesaw relation and of the Casas-Ibarra parametrisation, valid for every scale of seesaw. We find that the existence of a light sterile neutrino imposes an upper bound on the lightest neutrino mass. We further assess the impact of light sterile states on short- and long-baseline neutrino oscillation experiments, emphasise future detection prospects, and address CP Violation in this framework via the analysis of CP asymmetries and construction of weak basis invariants. The proposed models can accommodate enough active-sterile mixing to play a role in the explanation of short-baseline anomalies.

A Novel Scaling Process for the Computation of Multiconductor Transmission-Line Modal Properties Using the Basis Invariance of Total Current and Power

Some modal (or decoupled) transmission-line properties such as per-unit-length impedance, admittance, or characteristic impedance have long been held to be, in general, non-unique. This ambiguity arises from the nature of the similarity transformations used to relate the terminal and modal domains, for which the voltage transformation matrix has been shown to be only loosely related to the corresponding current transformation matrix. Modern methods have attempted to relate the two, but these relations typically rely on arbitrary normalizations, leading to strictly incorrect and/or non-unique results. This work introduces relations between the two transformations, derived from the physical equivalence of total power and currents between the two domains, by which the transformation matrices can be unambiguously related to each other, and the modal properties uniquely solved. This technique allows for the correct extraction of the modal transmission-line properties for any arbitrary system of conductors. Multiple examples are studied to validate the proposed solution process.

Data-driven scalar-flux model development with application to jet in cross flow

The classical gradient-diffusion hypothesis has known deficiencies when applied to cooling applications. In this paper, the gene-expression programming (GEP) method, a machine learning approach, has been applied to develop scalar-flux models via symbolic regression. The scalar-flux, the unclosed term of the mean passive-scalar transport equation, is treated by considering the polynomial basis and scalar invariants available from computable Reynolds-averaged quantities. This method has been applied to develop and then assess a model for the test case of jet in crossflow. A high-fidelity database was first probed for insight into which of the candidate bases are the most suitable as modelling terms. The high dimensionality of the function space, spanned by the basis, was then reduced by basic statistical techniques. The resulting data-driven model is presented and tested for a range of different jet in crossflow cases. Compared with eddy-diffusivity models, the new model is shown to produce reliably more accurate results. This demonstrates that the current framework can be used for scalar-flux modelling in complex three-dimensional flows and has potential to provide generalized form closures with improved predictive accuracy for the same classes of flows they were trained on.

Central elements of the Jennings basis and certain Morita invariants

From Morita theoretic viewpoint, computing Morita invariants is important. We prove that the intersection of the center and the [Formula: see text]th (right) socle [Formula: see text] of a finite-dimensional algebra [Formula: see text] is a Morita invariant; this is a generalization of important Morita invariants — the center [Formula: see text] and the Reynolds ideal [Formula: see text]. As an example, we also studied [Formula: see text] for the group algebra FG of a finite [Formula: see text]-group [Formula: see text] over a field [Formula: see text] of positive characteristic [Formula: see text]. Such an algebra has a basis along the socle filtration, known as the Jennings basis. We prove certain elements of the Jennings basis are central and hence form a linearly independent subset of [Formula: see text]. In fact, such elements form a basis of [Formula: see text] for every integer [Formula: see text] if [Formula: see text] is powerful. As a corollary we have [Formula: see text] if [Formula: see text] is powerful.

Systematic construction of basis invariants in the 2HDM

A new systematic method for the explicit construction of (basis-)invariants is introduced and employed to construct the full ring of basis invariants of the Two-Higgs-Doublet-Model (2HDM) scalar sector. Co- and invariant quantities are obtained by the use of hermitian projection operators. These projection operators are constructed from Young tableaux via birdtrack diagrams and they are used in two steps. First, to extract basis-covariant quantities, and second, to combine the covariants in order to obtain the actual basis invariants. The Hilbert series and Plethystic logarithm are used to find the number and structure of the complete set of generating invariants as well as their interrelations (syzygies). Having full control over the complete ring of (CP-even and CP-odd) basis invariants, we give a new and simple proof of the necessary and sufficient conditions for explicit CP conservation in the 2HDM, confirming earlier results by Gunion and Haber. The method generalizes to other models, with the only foreseeable limitation being computing power.

Beyond basis invariants

Physical observables cannot depend on the basis one chooses to describe fields. Therefore, all physically relevant properties of a model are, in principle, expressible in terms of basis-invariant combinations of the parameters. However, in many cases it becomes prohibitively difficult to establish key physical features exclusively in terms of basis invariants. Here, we advocate an alternative route in such cases: the formulation of basis-invariant statements in terms of basis-covariant objects. We review several examples where the basis-covariant path is superior to the traditional approach in terms of basis invariants. In particular, this includes the formulation of necessary and sufficient basis-invariant conditions for various physically distinct forms of textit{CP} conservation in two- and three-Higgs-doublet models.

ОБЩАЯ ТЕОРИЯ РЕШЕНИЯ ЗАДАЧ ПРИМЕНИТЕЛЬНО К НАЧЕРТАТЕЛЬНОЙ ГЕОМЕТРИИ

One of the drawbacks of the domestic educational system is the inadequate ability of the graduates of educational institutions to apply the obtained knowledge in the solution of practical problems. The general theory of problems solution can help them in overcoming it, and at the same time it is the basis of a new science - the science of ratiology. This theory is concerned with the precise formulation of problems and the schematic methods of their solution invariantly to subject domains. Due to it, the apparatus of this theory (R-methodology of problems solution) can be applied in any of them. An example of using R-methodology for the solution of rather specific problems of descriptive geometry is considered in the article. The good scientific basis, modern nature and subject invariance of R-methodology allow using it as a unified tool for teaching to solve a wide variety of problems in educational institutions of different levels and directions. At the same time, a number of interesting possibilities of the modernization of the entire educational process appear: the formation and wide use of local and integral databases, an emphasis on students’ independent work with them, decrease in the part of insufficiently effective forms of teaching and so on.

Spontaneous CP violation in multi-Higgs potentials with triplets of \u0394(3n2) and \u0394(6n2)

Motivated by discrete flavour symmetry models, we analyse Spontaneous CP Violation (SCPV) for potentials involving three or six Higgs fields (both electroweak doublets and singlets) which fall into irreducible triplet representations of discrete symmetries belonging to the Δ(3n2) and Δ(6n2) series, including A4, S4, Δ(27) and Δ(54). For each case, we give the potential and find various global minima for different regions of the parameter space of the potential. Using CP-odd basis Invariants that indicate the presence of Spontaneous CP Violation we separate the VEVs into those that do or do not violate CP. In cases where CP is preserved we reveal a CP symmetry of the potential that is preserved by those VEVs, otherwise we display a non-zero CP-odd Invariant. Finally we identify interesting cases where there is Spontaneous Geometrical CP Violation in which the VEVs have geometrical phases.

CP-odd invariants for multi-Higgs models and applications with discrete symmetry

CP-odd invariants are useful for studying the CP properties of Lagrangians in any basis. We explain how to build basis invariants for the scalar sector, and how to distinguish CP-odd invariants from CP-even invariants. Up to a certain order, we use these methods to systematically build all the CP-odd invariants. The CP-odd invariants signal either explicit or spontaneous violation of CP. Making use of the CP-odd invariants, we determine the CP properties of potentials with 3 and with 6 Higgs fields arranged as triplets of specific discrete symmetries in the Δ(3n2) or Δ(6n2) series (inlcuding A4, S4, Δ(27) and Δ(54) as well as the cases for n > 3).

CP-odd invariants for multi-Higgs models: Applications with discrete symmetry

CP-odd invariants provide a basis independent way of studying the CP properties of Lagrangians. We propose powerful methods for constructing basis invariants and determining whether they are CP-odd or CP-even, then systematically construct all of the simplest CP-odd invariants up to a given order, finding many new ones. The CP-odd invariants are valid for general potentials when expressed in a standard form. We then apply our results to scalar potentials involving three (or six) Higgs fields which form irreducible triplets under a discrete symmetry, including invariants for both explicit as well as spontaneous CP violation. The considered cases include one triplet of Standard Model (SM) gauge singlet scalars, one triplet of SM Higgs doublets, two triplets of SM singlets, and two triplets of SM Higgs doublets. For each case we study the potential symmetric under one of the simplest discrete symmetries with irreducible triplet representations, namely $A_4$, $S_4$, $\Delta(27)$ or $\Delta(54)$, as well as the infinite classes of discrete symmetries $\Delta(3n^2)$ or $\Delta(6n^2)$.

Basis invariant measure of CP-violation and renormalization

We analyze, in the context of a simple toy model, for which renormalization schemes the CP-properties of bare Lagrangian and its finite part coincide. We show that this is the case for the minimal subtraction and on-shell schemes. The CP-properties of the theory can then be characterized by CP-odd basis invariants expressed in terms of renormalized masses and couplings. For the minimal subtraction scheme we furthermore show that in CP-conserving theories the CP-odd basis invariants are zero at any scale but are not renormalization group invariant in CP-violating ones.

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.

Invariant approach to [formula omitted] in unbroken Δ(27)

The invariant approach is a powerful method for studying CP violation for specific Lagrangians. The method is particularly useful for dealing with discrete family symmetries. We focus on the CP properties of unbroken Δ(27) invariant Lagrangians with Yukawa-like terms, which proves to be a rich framework, with distinct aspects of CP, making it an ideal group to investigate with the invariant approach. We classify Lagrangians depending on the number of fields transforming as irreducible triplet representations of Δ(27). For each case, we construct CP-odd weak basis invariants and use them to discuss the respective CP properties. We find that CP violation is sensitive to the number and type of Δ(27) representations.

Quark sectorCPviolation of the universal seesaw model

We study the CP violation of universal seesaw model, especially its quark sector. The model is based on SU(2)_L \times SU(2)_R \times U(1)_{Y^\prime}. In order to count the number of parameters in quark sector, we use the degree of freedom of weak basis transformation. For N(3)-generation model, the number of CP violating phase in quark sector is identified as 3N^2-3N+1 (19). We also construct nineteen CP violating weak basis invariants of Yukawa coupling matrices and SU(2) singlet quark mass matrices in the three-generation universal seesaw model. The quark interaction terms induced by neutral currents are given as an exact formula. Both of the charged current and the neutral current are expressed in terms of the mass basis by finding the transformations from weak basis to mass basis. Finally, we calculate the mixing matrix element approximately assuming that the SU(2)_R breaking scale v_R is much larger than the electro-weak breaking scale v_L.