Complex Orthogonal Matrix Research Articles

Type-III see-saw: Phenomenological implications of the information lost in decoupling from high-energy to low-energy

The type-III see-saw seems to explain the very minuteness of neutrino masses readily and naturally. The high-energy see-saw theories usually involve larger number of effective parameters than the physical and measurable parameters appearing in the low-energy neutrino phenomenology. Casas-Ibarra parametrisation facilitates to encode the information lost in integrating the heavy fermions out in an arbitrary complex orthogonal matrix. The CMS collaboration has already searched for triplet fermions in the type-III see-saw model with only one generation of triplet fermion flavour democratically decaying into SM leptons. We reinterpret this CMS search in the context of a realistic type-III see-saw model with two or three generations of triplet fermions, and endeavour to comprehend the implications of the foregoing matrix on the 95% CL lower limit on the mass of the triplet fermions. We also discuss the phenomenological implication of the aforesaid matrix in view of charged lepton flavour violating observables and displaced decays of the triplet fermions at colliders.

A direct link between unflavored leptogenesis and low-energy CP violation via the one-loop quantum corrections

In the type-I seesaw mechanism the Casas-Ibarra (CI) parametrization provides a convenient description of the Dirac neutrino mass matrix in terms of the light and heavy Majorana neutrino masses, the lepton flavor mixing matrix U and an unknown complex orthogonal matrix O. If O is assumed to be real, it will be impossible to generate unflavored thermal leptogenesis via the lepton-number-violating and CP-violating decays of the lightest heavy Majorana neutrino. We find that this observation can be invalidated after small but unavoidable quantum corrections to the CI parametrization are taken into account with the help of the one-loop renormalization-group equations (RGEs) between the seesaw and electroweak scales. We illustrate a novel and viable unflavored leptogenesis scenario of this kind based on the RGEs in the seesaw-extended standard model, and show its direct link to the CP-violating phases of U at low energies.

Big-bang nucleosynthesis and leptogenesis in the CMSSM

We have investigated the constrained minimal supersymmetric standard model with three right-handed Majorana neutrinos whether there still is a parameter region which is consistent with all existing experimental data/limits such as Leptogenesis and the dark matter abundance and we also can solve the Lithium problem. Using Casas-Ibarra parameterization, we have found that a very narrow parameter space of the complex orthogonal matrix elements where the lightest slepton can have a long lifetime, that is necessary for solving the Lithium problem. Further, under this condition, there is a parameter region that can give an explanation for the experimental observations. We have studied three cases of the right-handed neutrino mass ratio \mbox{\em (i)} $M_{2}=2 \times M_{1}$, \mbox{\em (ii)} $M_{2}=4 \times M_{1}$, \mbox{\em (iii)} $M_{2}=10 \times M_{1}$ while $M_{3}=40 \times M_{1}$ is fixed. We have obtained the mass range of the lightest right-handed neutrino mass that lies between $10^9$ GeV and $10^{11}$ GeV. The important result is that its upper limit is derived by solving the Lithium problem and the lower limit comes from Leptogenesis. Calculated low-energy observables of these parameter sets such as BR($\mu \to e \gamma$) is not yet restricted by experiments and will be verified in the near future.

The subspaces spanned by Householder vectors associated with an orthogonal or a symplectic matrix

The Cartan–Dieudonné–Scherk Theorem guarantees that every complex orthogonal matrix can be written as a product of matrices of the form HS,u≡I−uuTS, where S=I and u∈Cn satisfies uTu=2; moreover, every complex symplectic matrix can be written as a product of matrices of the form HS,u≡I−uuTS where S=J=[0I−I0] and u≠0. Let a nonempty V⊆Cn be given. The S-orthogonal complement of V is VS={z∈Cn | wTSz=0 for all w∈V}. The image of an n-by-n complex matrix A is the set of all z∈Cn for which there is an x∈Cn such that z=Ax and is denoted by Im(A). Let S=I or S=J. Suppose that Q=HS,u1HS,u2⋯HS,ur. Set U=span{u1,u2,…,ur}. We study the relationship between Q, U, and Im(Q−I). Suppose that r is minimal. We show that if dim⁡(U)=r, then Im(Q−I)=U. We also show that S(Q−I) is not skew symmetric if and only if dim⁡(U)=r. Let W=Im(Q−I). We show that a relationship between W and WS determines the Jordan structure of Q, in particular, we show that (Q−I)2=0 if and only if W⊆WS.

Jordan Semi-Triple Multiplicative Maps on the Symmetric Matrices

In this paper, we show that if an injective map on symmetric matrices Sn(C) satisfies then for all , where f is an injective homomorphism on C, S is a complex orthogonal matrix and Af is the image of A under f applied entrywise.

CP violation and lightest neutrino mass effects in thermal leptogenesis

Effects of the lightest neutrino mass in ‘flavoured’ leptogenesis when the CP-violation necessary for the generation of the baryon asymmetry of the Universe is due exclusively to the Dirac and/or Majorana phases in the neutrino mixing matrix U are discussed. The type I see-saw scenario with three heavy right-handed Majorana neutrinos having hierarchical spectrum is considered. The ‘orthogonal’ parametrisation of the matrix of neutrino Yukawa couplings, which involves a complex orthogonal matrix R, is employed. Results for light neutrino mass spectrum with normal and inverted ordering (hierarchy) are reviewed.

The interplay between “low” and “high” energy CP-violation in leptogenesis

We analyse, within the “flavoured” leptogenesis scenario of baryon asymmetry generation, the interplay of “low energy” CP-violation, originating from the PMNS neutrino mixing matrix U, and “high energy” CP-violation, which can be present in the matrix of neutrino Yukawa couplings, λ, and can manifest itself only in “high” energy scale processes. The type I see-saw model with three heavy right-handed Majorana neutrinos having a hierarchical spectrum is considered. The “orthogonal” parameterisation of the matrix of neutrino Yukawa couplings, which involves a complex orthogonal matrix R, is employed. In this approach the matrix R is the source of “high energy” CP-violation. Results for normal hierarchical (NH) and inverted hierarchical (IH) light neutrino mass spectrum are derived in the case of decoupling of the heaviest right-handed Majorana neutrino. It is shown that taking into account the contribution to Y B due to the CP-violating phases in the neutrino mixing matrix U can change drastically the predictions for Y B , obtained assuming that only “high energy” CP-violation from the R-matrix is operative in leptogenesis. In the case of the IH spectrum, in particular, there exist significant regions in the corresponding parameter space where the purely “high energy” contribution in Y B plays a subdominant role in the production of baryon asymmetry compatible with the observations.

Effects of lightest neutrino mass in leptogenesis

The effects of the lightest neutrino mass in “flavoured” leptogenesis are investigated in the case when the CP-violation necessary for the generation of the baryon asymmetry of the Universe is due exclusively to the Dirac and/or Majorana phases in the neutrino mixing matrix U. The type I see-saw scenario with three heavy right-handed Majorana neutrinos having hierarchical spectrum is considered. The “orthogonal” parametrisation of the matrix of neutrino Yukawa couplings, which involves a complex orthogonal matrix R, is employed. Results for light neutrino mass spectrum with normal and inverted ordering (hierarchy) are obtained. It is shown, in particular, that if the matrix R is real and CP-conserving and the lightest neutrino mass m 3 in the case of inverted hierarchical spectrum lies the interval 5 × 10 −4 eV ≲ m 3 ≲ 7 × 10 −3 eV , the predicted baryon asymmetry can be larger by a factor of ∼100 than the asymmetry corresponding to negligible m 3 ≅ 0 . As consequence, we can have successful thermal leptogenesis for 5 × 10 −6 eV ≲ m 3 ≲ 5 × 10 −2 eV even if R is real and the only source of CP-violation in leptogenesis is the Majorana and/or Dirac phase(s) in U.

Leptogenesis and low energy CP-violation in neutrino physics

Taking into account the recent progress in the understanding of the lepton flavor effects in leptogenesis, we investigate in detail the possibility that the CP-violation necessary for the generation of the baryon asymmetry of the Universe is due exclusively to the Dirac and/or Majorana CP-violating phases in the PMNS neutrino mixing matrix U, and thus is directly related to the low energy CP-violation in the lepton sector (e.g., in neutrino oscillations, etc.). We first derive the conditions of CP-invariance of the neutrino Yukawa couplings λ in the see-saw Lagrangian, and of the complex orthogonal matrix R in the “orthogonal” parametrization of λ. We show, e.g. that under certain conditions (i) real R and specific CP-conserving values of the Majorana and Dirac phases can imply CP-violation, and (ii) purely imaginary R does not necessarily imply breaking of CP-symmetry. We study in detail the case of hierarchical heavy Majorana neutrino mass spectrum, presenting results for three possible types of light neutrino mass spectrum: (i) normal hierarchical, (ii) inverted hierarchical, and (iii) quasi-degenerate. Results in the alternative case of quasi-degenerate in mass heavy Majorana neutrinos, are also derived. The minimal supersymmetric extension of the standard theory with right-handed Majorana neutrinos and see-saw mechanism of neutrino mass generation is discussed as well. We illustrate the possible correlations between the baryon asymmetry of the Universe and (i) the rephasing invariant J CP controlling the magnitude of CP-violation in neutrino oscillations, or (ii) the effective Majorana mass in neutrinoless double beta decay, in the cases when the only source of CP-violation is respectively the Dirac or the Majorana phases in the neutrino mixing matrix.

The riemann boundary problem with a complex orthogonal matrix

The Riemann boundary problem is studied under the assumption that the coefficient of the problem is a complex orthogonal matrix. In this case a property of the partial indices of the problem is established together with certain properties of the canonical matrices, which are then used to construct the canonical matrix of a complex orthogonal matrix of second order.

On a Generalized Distribution of the Poles of the Unitary Collision Matrix

An expression for the joint distribution of the complex poles of the unitary collision matrix is derived for the single-channel case, which is valid for all values of the ratio of the width to the spacing. The derivation uses the statistical distribution of the parameters of the real R-matrix theory. We find that unitarity gives rise to the statistical correlations between the width and the spacing of the collision matrix. It is shown that the distribution of the poles of the unitary collision matrix using Feshbach's unified theory of nuclear reactions is the same as the one obtained using R-matrix theory, provided we make a particular choice of the arbitrary boundary condition in the latter theory. A remark is made about the use of the random complex orthogonal matrix in the study of the parameters of the statistical collision matrix.

Statistical Description of the Complex-Boundary-Value Problem. II. Distribution of the Parameters of the Collision Matrix

A detailed statistical study is made of the parameters of the statistical collision matrix using the $N$-dimensional random complex orthogonal matrix. It is shown that, even without a complete knowledge of weight function which has to be introduced for the convergence of the normalization integral, certain relations between the average values of the parameters of the statistical collision matrix can be obtained and a statement can be made that the channel correlations of the parameters are always positive. A suitable form of the weight function is guessed, and the distributions of the parameters are also given. It is shown that under certain conditions the distribution of the parameters is close to the Porter-Thomas distribution except for small values. The resonance correlations of the parameters are also studied. Excellent agreement has been obtained between the values predicted by the present theoretical formulation and those obtained by a numerical calculation using the parameters of the real-boundary-value problem and a certain transformation matrix.

Simple Theorem on Hermitian Matrices and an Application to the Polarization of Vector Particles

It is proven that a necessary and sufficient condition for an $n$-dimensional Hermitian matrix $\ensuremath{\rho}$ to be positive definite is that it be expressible in the form $\ensuremath{\rho}=OEO\ifmmode\dagger\else\textdagger\fi{}$, where $O$ is a complex orthogonal matrix and $E$ is a diagonal matrix with positive elements. This accomplishes a parametrization since $O$ has ${n}^{2}\ensuremath{-}n$ real parameters and $E$ has $n$ of them. The proof is constructive, giving $O$ and $E$. It is further shown that the limit forms of this expression yield all the non-negative definite matrices. The parametrization for the polarization matrix of a spin-one particle is given explicitly.