Quark mixing in the standard model and space rotations

The rotation matrix and the Cabibbo–Kobayashi–Maskawa (CKM) matrix are discussed. The CKM matrix is viewed as the rotation matrix in Euler angles with pitch–roll–yaw convention for the angles and as the angle–axis representation of the rotation matrix. A comparison of the exponential parameterisation of the CKM matrix with the matrix exponent generator of the space rotations is made. How to account for the CP violating phase in CKM and the O(3) rotation matrix in the angle–axis form is discussed in the context of such a view of the mixing matrix. The generation of the new parameterisations of the CKM matrix in an exponential form with distinguished CP violating part is demonstrated.     

Spontaneous CP violating phase as the phase in PMNS matrix

We study the possibility of identifying the CP violating phases in the PMNS mixing matrix in the lepton sector and also that in the CKM mixing matrix in the quark sector with the phase responsible for the spontaneous CP violation in the Higgs potential, and some implications. Since the phase in the CKM mixing matrix is determined by experimental data, the phase in the lepton sector is also fixed. The mass matrix for neutrinos is constrained, leading to constraints on the Jarlskog CP violating parameter J, and the effective mass 〈m _ββ 〉 for neutrinoless double beta decay. The Yukawa couplings are also constrained. Different ways of identifying the phases have different predictions for \(\mu\to e e\bar{e}\) and \(\tau\to l_{1} l_{2} \bar{l}_{3}\). Future experimental data can be used to distinguish different models.     

Higgs and neutrino sector,EDM and ε<Subscript>K</Subscript> in a spontaneously CP and R-parity breaking supersymmetric model

We construct an extension of the supersymmetric standard model where both CP symmetry and R-parity are spontaneously broken. We study the electroweak symmetry breaking sector of the model and find minima consistent with the experimental bounds on Higgs boson masses. Neutrino masses and mixing angles are generated through both seesaw and bilinear R-parity violation. We show that the hierarchical mass pattern is obtained, and mixings are consistent with measured values. Due to the spontaneous CP and R-parity violation, the neutrino sector is CP violating, and we calculate the corresponding phase. We further restrict the parameter space to agree with the limits on the electric dipole moment of the neutron. Finally, we study the CP violation parameter ε_K in the kaon system and show that we obtain results consistent with the experimental value.     

MSSM Higgs sector CP violation at photon colliders: revisited

The CP violating phases in soft SUSY breaking terms may induce indirect CP violation in the neutral Higgs boson sector through SUSY particle loops as well as direct CP violation in the γγ→H_i (i=1,2,3) through CP violating vertices. We present a complete analysis of the MSSM Higgs sector CP violation at photon colliders including the chargino loop contributions in the regime , where the CP violating mixing in the neutral Higgs sector is very strongly influenced by the scalar top loop, more so than the chargino and neutralino ones. However, the CP violating phases of the higgsino and wino mass parameters may still generate direct CP violation in γγ→H_i. In this process, the CP violating phenomena become very rich, and thus we study the production rates and polarization asymmetries in the Higgs production in detail. Photon colliders with high luminosity and well controlled polarized initial photon beams are indispensable in determining the CP properties of neutral Higgs bosons. PACS 14.80.Cp; 12.60.Jv; 11.30.Er     

The β angle as the CP violating phase in the CKM matrix

The CKM matrix describing quark mixing with three generations can be parameterized by three Euler mixing angles and one CP violating phase. In most of the parameterizations, the CP violating phase chosen is not a directly measurable quantity and is parameterization dependent. In this work, we propose to use the most accurately measured CP violating angle β in the unitarity triangle as the phase in the CKM matrix, and construct an explicit β parameterization. We also derive an approximate Wolfenstein-like expression for this parameterization.     

Gravitational Waves,baryon asymmetry of the universe and electric dipole moment in the CP-violating NMSSM

In this work, we make the first study of electroweak baryogenesis(EWBG) based on the LHC data in the CP-violating next-to-minimal supersymmetric model(NMSSM) where a strongly first order electroweak phase transition(EWPT) is obtained in the general complex Higgs potential. With representative benchmark points which pass the current LEP and LHC constraints, we demonstrate the structure of EWPT for those points and how a strongly first order EWPT is obtained in the complex NMSSM where the resulting gravitational wave production properties are found to be within the reaches of future space-based interferometers like BBO and Ultimate-DECIGO. We further calculate the generated baryon asymmetries where the CP violating sources are(1): higgsino-singlino dominated,(2): higgsino-gaugino dominated or(3): from both sources. It is shown that all three representing scenarios could evade the strong constraints set by various electric dipole moments(EDM) searches where cancellations among the EDM contributions occur at the tree level(higgsino-singlino dominated) or loop level(higgsino-gaugino dominated).The 125 GeV SM like Higgs can be either the second lightest neutral Higgs H_2 or the third lightest neutral Higgs H_3. Finally, we comment on the future direct and indirect probe of CPV in the Higgs sector from the collider and EDM experiments.     

Reconstructing the chargino system at e^+e^- linear colliders

In most supersymmetric theories charginos, , belong to the class of the lightest supersymmetric particles. The chargino system can be reconstructed completely in collider experiments: . By measuring the total cross sections and the asymmetries with polarized beams, the chargino masses and the gaugino–higgsino mixing angles of these states can be determined accurately. If only the lightest charginos are kinematically accessible in a first phase of the machine, transverse beam polarization or the measurement of chargino polarization in the final state is needed to determine the mixing angles. From these observables the fundamental SUSY parameters can be derived: the SU(2) gaugino mass , the modulus and the cosine of the CP–violating phase of the higgsino mass parameter , and , the ratio of the vacuum expectation values of the two neutral Higgs doublet fields. The remaining two–fold ambiguity of the phase can be resolved by measuring the normal polarization of the charginos. Sum rules of the cross sections can be exploited to investigate the closure of the two–chargino system. Received: 2 February 2000 / Published online: 17 March 2000     

Complex CKM from spontaneous CP violation without flavor changing neutral current

We analyse the general constraints on unified gauge models with spontaneous CP breaking that satisfy the conditions that (i) CP violation in the quark sector is described by a realistic complex CKM matrix, and (ii) there is no significant flavor changing neutral current effects in the quark sector. We show that the crucial requirement in order to conform to the above conditions is that spontaneous CP breaking occurs at a very high scale by complex vevs of standard model singlet Higgs fields. Two classes of models are found, one consisting of pure Higgs extensions and the other one involving fermionic extensions of the standard model. We give examples of each class and discuss their possible embeddings into higher unified theories. One of the models has the interesting property that spontaneous CP violation is triggered by spontaneous P violation, thereby linking the scale of CP violation to the seesaw scale for neutrino masses.     

Neutral Higgs sector of the MSSM with explicit CP violation and non-holomorphic soft breaking

Using the effective potential method, we computed one-loop corrections to the mass matrix of neutral Higgs bosons of the Non-Holomorphic Supersymmetric Standard Model (NHSSM) with explicit CP violation, where the radiative corrections due to the quarks and squarks of the third generation were taken into account. We observed that the non-holomorphic trilinear couplings can compete with the holomorphic ones in CP violating issues for the mass and mixing of the neutral Higgs bosons.     

Phenomenological features in a model with non-universal gaugino CP phases

We study phenomenological features in an extended gauge mediation SUSY breaking model that has non-universal gaugino masses and CP phases. We show that large CP phases in soft SUSY breaking parameters can be consistent with the constraints coming from the electric dipole moment (EDM) of an electron, a neutron, and also a mercury atom. Masses of the superpartners are not necessarily required to be larger than 1 TeV but allowed to be O(100) GeV. We also investigate the mass spectrum of Higgs scalars and their couplings to gauge bosons in that case. Compatibility of this model with the present experimental data on the Higgs sector is discussed.     

Supersymmetric contributions to B→DK and the determination of angle γ

We analyze supersymmetric contributions to the branching ratios and CP asymmetries of B^-→D⁰K^- and B^-→D̄⁰K^- processes. We investigate the possibility that supersymmetric CP violating phases can affect our determination for the angle γ in the unitary triangle of Cabibbo–Kobayashi–Maskawa mixing matrix. We calculate the gluino and chargino contributions to b→u(c̄s) and b→c(ūs) transitions in a model independent way by using the mass insertion approximation method. We also revise the D⁰–D̄⁰ mixing constraints on the mass insertions between the first and second generations of the up sector. We emphasize that in case of negligible D⁰–D̄⁰ mixing, one should consider simultaneous contributions from more than one mass insertion in order to be able to obtain the CP asymmetries of these processes within their 1σ experimental range. However, with a large D⁰–D̄⁰ mixing, one finds a significant deviation between the two asymmetries and it becomes natural to have them of the order of the central values of their experimental measurements.     

Neutrino masses,mixings, and FCNC’s in an <Emphasis Type="Italic">S</Emphasis><Subscript>3</Subscript> flavor symmetric extension of the standard model

By introducing threeHiggs fields that are SU(2) doublets and a flavor permutational symmetry, S ₃, in the theory, we extend the concepts of flavor and generations to the Higgs sector and formulate a Minimal S ₃-Invariant Extension of the Standard Model. The mass matrices of the neutrinos and charged leptons are re-parameterized in terms of their eigenvalues, then the neutrino mixing matrix, V _PMNS, is computed and exact, explicit analytical expressions for the neutrino mixing angles as functions of the masses of neutrinos and charged leptons are obtained in excellent agreement with the latest experimental data. We also compute the branching ratios of some selected flavor-changing neutral current (FCNC) processes, as well as the contribution of the exchange of neutral flavor-changing scalars to the anomaly of the magnetic moment of the muon, as functions of the masses of charged leptons and the neutral Higgs bosons. We find that the S ₃ × Z ₂ flavor symmetry and the strong mass hierarchy of the charged leptons strongly suppress the FCNC processes in the leptonic sector, well below the present experimental bounds by many orders of magnitude. The contribution of FCNC’s to the anomaly of the muon’s magnetic moment is small, but not negligible.     

<Emphasis Type="Italic">CP</Emphasis> violation due to multi-Froggatt–Nielsen fields

We study how to incorporate CP violation in the Froggatt–Nielsen (FN) mechanism. To this end, we introduce non-renormalizable interactions with a flavor democratic structure to the fermion mass generation sector. It is found that at least two iso-singlet scalar fields with a discrete symmetry imposed are necessary to generate CP violation due to the appearance of the relative phase between their vacuum expectation values. In the simplest model, the ratios of quark masses and the Cabibbo–Kobayashi–Maskawa (CKM) matrix including the CP violating phase are determined by the CKM element |V_us| and the ratio of two vacuum expectation values of FN fields, R=|R|e^iα (a magnitude and a phase). It is demonstrated how the angles φ_i (i=1,...,3) of the unitarity triangle and the CKM off-diagonal elements |V_ub| and |V_cb| are predicted as a function of |V_us|, |R| and α. Although the predicted value of the CP violating phase does not agree with the experimental data within the simplest model, the basic idea of our scenario would be promising if one wants to construct a more realistic model of flavor and CP violation. PACS 11.30.Er; 12.60.-i     

Complex CKM matrix, spontaneous CP violation and generalized symmetry

The multi-Higgs models having spontaneous CP violation (SPCPV) and natural flavor conservation (NFC) lead to a real CKM matrix V contradicting current evidence in favour of a complex V. This contradiction can be removed by using a generalized μ–τ (called 23) symmetry in place of the discrete symmetry conventionally used to obtain NFC. If the 23 symmetry is exact then the Higgs induced flavour changing neutral currents (FCNC) vanish as in the case of NFC. 23 breaking introduces SPCPV, a phase in V and suppressed FCNC among quarks. The FCNC couplings between i and j generations show a hierarchy with the result that the FCNC can have observable consequences in B mixing without conflicting with the mixing. Detailed fits to the quark masses and the CKM matrix are used to obtain the (complex) couplings and . Combined constraints from flavour and CP violations in the K, B_d, B_s, D mesons are analyzed within the model. They allow (i) relatively light Higgs, 100–150 GeV (ii) measurable extra contributions to the magnitudes and phases of the mixing amplitudes and (iii) the mixing at the current sensitivity level.     

Exceptional Point of the Neutral Heavy Higgs System H2–H3

We study the singularity of the surface that represents the masses of the isolated doublet of heavy, neutral Higgs bosons, H ₂–H ₃, in a toy model based on the MSSM with CP violation, in parameter space. These two heavy, neutral Higgs bosons are coherent and, for large values of the masses, nearly degenerate. In this scenario, mixing between the mass eigenstates of the H ₂–H ₃ system could be very large and exact degeneracy is possible. As function of the Lagrangian parameters, the physical mass of the doublet has an algebraic branch point of rank one at the exceptional point where the two masses are equal. The real and imaginary parts of the masses in the doublet have branch cuts that start at the same branch point but extend in opposite directions in parameter space. Associated with this branch point, the propagator of the mixing doublet of neutral heavy Higgs bosons has a double pole in the complex s-plane of the energy squared. We computed the mass surface of the isolated doublet of H ₂–H ₃ bosons as function of the Lagrangian parameters in the neighbourhood of the exceptional point in a toy model of the system H ₂–H ₃. We also computed the trajectories of the poles of the transition matrix for values of the Lagrangian parameters close to the exceptional point and explained the characteristic change of identity seen in these trajectories in the s-plane as a manifestation of the topology of the two-sheeted mass surfaces in the space of Lagrangian parameters.     

SUSY CP problem in gauge mediation model

SUSY CP problem in the gauge mediation supersymmetry breaking model is reconsidered. We pay particular attention to two sources of CP violating phases whose effects were not seriously studied before; one is the effect of the breaking of the GUT relation among the gaugino masses due to the field responsible for the GUT symmetry breaking, and the other is the supergravity effect on the supersymmetry breaking parameters, in particular, on the bi-linear supersymmetry breaking Higgs mass term. We show that both of them can induce too large electric dipole moments of electron, neutron, and so on, to be consistent with the experimental bounds.     

Extended minimal flavour violating MSSM and implications for B physics

The recently reported measurements of the CP asymmetry by the BABAR and BELLE collaborations, obtained from the rate differences in the decays etc., and their charge conjugates, are in good agreement with the standard model (SM) prediction of the same, resulting from the unitarity of the CKM matrix. The so-called minimal flavour violating (MFV) supersymmetric extensions of the standard model, in which the CKM matrix remains the only flavour changing structure, predict similar to the one in the SM. With the anticipated precision in and other CP asymmetries at the B factories and hadron colliders, one hopes to pin down any possible deviation from the SM. We discuss an extension of the MFV-supersymmetric models which comfortably accommodates the current measurements of the CP asymmetry , but differs from the SM and the MFV-supersymmetric models due to an additional flavour changing structure beyond the CKM matrix. We suggest specific tests in forthcoming experiments in B physics. In addition to the CP-asymmetries in B-meson decays, such as and , and the mass difference in the system, we emphasize measurements of the radiative transition as sensitive probes of the postulated flavour changing structure. This is quantified in terms of the ratio , the isospin violating ratio , and the CP-asymmetry in the decay rates for and its charge conjugate. Interestingly, the CKM–unitarity analysis in the Extended–MFV model also allows solutions for the Wolfenstein parameter, as opposed to the SM and the MFV-supersymmetric models for which only solutions are now admissible, implying , where . Such large values of are hinted by the current measurements of the branching ratios for the decays and . Received: 20 May 2001 / Revised version: 5 August 2001 / Published online: 31 August 2001     

Quark flavour mixing and the exponential form of the Kobayashi–Maskawa matrix

The form of the mixing matrix of quarks is discussed. The exponential parameterisation of the Cabibbo–Kobayashi–Maskawa (CKM) matrix is identified as the most general form of the mixing matrix. Its properties are reviewed in the context of the unitarity requirement. A recurrence series representation of the exponential form of the mixing matrix that is easy to handle is obtained, allowing for a direct and simple method of calculation of the CKM matrix. The generation of the new parameterisations of the CKM matrix by the exponential form is demonstrated.     

Determining SUSY parameters in chargino pair-production in e^+e^- collisions

In most supersymmetric theories, charginos , mixtures of charged color-neutral gauginos and higgsinos, belong to the class of the lightest supersymmetric particles. They are easy to observe at colliders. By measuring the total cross sections and the left–right asymmetries with polarized electron beams in , the chargino masses and the gaugino–higgsino mixing angles can be determined. From these observables the fundamental SUSY parameters can be derived: the SU(2) gaugino mass , the modulus and of the higgsino mass parameter, and , the ratio of the vacuum expectation values of the two neutral Higgs doublet fields. The solutions are unique; the CP-violating phase can be determined uniquely by analyzing effects due to the normal polarization of the charginos. Received: 3 December 1998 / Published online: 27 April 1999     

Bi-large Neutrino Mixing See-Saw Mass Matrix with Texture Zeros and Leptogenesis

We study constraints on neutrino properties for a class of bi-large mixing See-Saw mass matrices with texture zeros and with the related Dirac neutrino mass matrix to be proportional to a diagonal matrix of the form diag(ε,1,1). Texture zeros may occur in the light (class a) or in the heavy (class b) neutrino mass matrices. Each of these two classes has 5 different forms which can produce non-trivial three generation mixing with at least one texture zero. We find that two types of texture zero mass matrices in both class a and class b can be consistent with present data on neutrino masses and mixing. None of the neutrinos can have zero masses and the lightest of the light neutrinos has a mass larger than about 0.046 eV for class a and 0.0027 eV for class b. In these models although the CKM CP violating phase vanishes, the non-zero Majorana phases can exist and can play an important role in producing the observed baryon asymmetry in our universe through leptogenesis mechanism. The requirement of producing the observed baryon asymmetry can further distinguish different models and also restrict the See-Saw scale to be in the range of 10¹²～10¹⁵ GeV. We also discuss RG effects on V₁₃.