Narrow Results

The Cabibbo–Kobayashi–Maskawa (CKM) parameters are investigated in detail in recent predictive models which are based on low-energy non-Abelian discrete family symmetries. Some of the models can already be excluded at the present precision of the determination of the CKM parameters, while some of them seem to survive. We find that to make the uncertainties of the theoretical values comparable with the assumed uncertainties of ~1° and ~2° in ϕ₂(α) and ϕ₃(γ), respectively, at about 50 inverse atto barn achieved at a future B factory, it is necessary to reduce the uncertainties in the quark masses, especially that of the strange quark mass by more than 60%.

In our work, we study the texture zeros of $m_{\nu }$ in the minimal extended type-I seesaw (MES) with incorporating one extra gauge singlet field “$S$”. The MES models deal with the Dirac neutrino mass matrix $(M_D)$, the right-handed Majorana mass matrix $(M_R)$ and the sterile neutrino mass matrix $(M_S)$. We carry out the mapping of all possible zero textures of $M_D$, $M_R$ and $M_S$ for phenomenologically predictive cases having total eight zeros of $M_D$ and $M_R$ studied in the literature. With this motivation, we consider (a) $(4+4)$ scheme, (b) $(5+3)$ scheme and (c) $(6+2)$ scheme, where the digits of a pair represent the number of zeros of $M_D$ and $M_R$, respectively along with the one zero and two zero textures of $(M_S)$. There are a large number of possibilities of zeros of fermion mass matrices but the implementation of $S_3$ transformations reduces it to a very minimum number of basic structures. Interestingly out of four allowed one zero textures of $m_{\nu }$ without sterile neutrino, only three cases ($m_{e\tau }=0$, $m_{\mu \mu }=0$ and $m_{\tau \tau }=0$) are allowed in MES mechanism for the $(4+4)$ and $(5+3)$ schemes. We find some correlations for different combination of $M_D$, $M_R$ and $M_S$ on enforcement of zeros. We examined all the correlations under the recent neutrino oscillation data and find that only $m_{\tau \tau }=0$ survives while both $m_{e\tau }=0$ and $m_{\mu \mu }=0$ are ruled out. Interestingly the one zero textures inherently represent the inverted hierarchy of the mass ordering of light neutrinos. No two zero textures of $m_{\nu }$ survive in MES although there are a number of allowed structures phenomenologically. The allowed texture zeros are finally realized using a discrete Abelian flavor symmetry group $Z_7$ with the extension of standard model to include some scalar fields.

We study the texture zeros of $4\times 4$ neutrino mass matrices $M_{\nu }$ in the minimal extended type-I seesaw (MES) mechanism, incorporating one extra gauge singlet field “$S$”. The $4\times 4$  MES model deals with $3\times 3$  $(M_D)$, $3\times 3$  $(M_R)$ and $1\times 3$ mass matrix $(M_S)$ which couples the right-handed neutrinos and the singlet field “$S$”. We carry out the mapping of all possible zero textures of $M_D$, $M_R$ and $M_S$ with the restriction to phenomenologically predictive cases having total eight zeros of $M_D$ and $M_R$ studied in the literature. If $m_s$, the sterile neutrino mass, is subject to any limit, further block diagonalization of $M_{\nu }$ shall not be allowed to reduce it to a $3\times 3$ matrix. In $4\times 4$  $M_{\nu }$ scenario, the study of texture zero is totally different and interesting. With this motivation, we consider the $(4+4)$ scheme where the digits of the pair represent the number of zeros of $M_D$ and $M_R$, respectively, along with the one/two-zero textures of $M_S$. There are a large number of possibilities of zeros of fermion mass matrices, but the implementation of $S_3$ transformations reduces it to a very minimum number of basic structures. As the $4\times 4$ MES matrix is a matrix of rank 3, so we consider only those textures with two zeros which are of rank 3 whereby the number of feasible zero textures reduces to 12, out of 15. On realizing these 12 textures under MES mechanism with $(4+4)$ picture, we arrive at certain correlations for each texture. We examine the viability of each texture by scanning their respective correlations under recent neutrino oscillation data. Also, we discuss the interplay of Dirac and Majorana CP phases in determining the viability of a texture. The allowed two-zero textures are finally realized using a discrete Abelian flavor symmetry group $Z_9$ with the extension of Standard Model to include some scalar fields.

The Standard Model does not constrain the form of the Yukawa matrices and thus the origin of fermion mass hierarchies and mixing pattern remains puzzling. On the other hand, there are intriguing relations between the quark masses and their weak mixing angles, such as the well-known relationship $tan{\theta }_C=\sqrt{m_d/m_s}$ for the Cabibbo angle, which may point towards specific textures of Yukawa matrices hypothesized by Harald Fritzsch at the end of the 70s. Though the original ansatz of Fritzsch is excluded by the experimental data, one can consider its minimal modification which consists in introducing an asymmetry between the 23 and 32 entries in the down-quark Yukawa matrix. We show that this structure is perfectly compatible with the present precision data on quark masses and CKM mixing matrix, and theoretically it can be obtained in the context of $\mathrm{\text{SU}}(5)$ model with inter-family $\mathrm{\text{SU}}{(3)}_H$ symmetry. We also discuss some alternative approaches which could give a natural description of the fermion mass spectrum and weak mixing pattern.

The Standard Model does not constrain the form of the Yukawa matrices and thus the origin of fermion mass hierarchies and mixing pattern remains puzzling. On the other hand, there are intriguing relations between the quark masses and their weak mixing angles, such as the well-known relationship tan ${\theta }_C=\sqrt{m_d/m_s}$ for the Cabibbo angle, which may point towards specific textures of Yukawa matrices hypothesized by Harald Fritzsch at the end of the 70s. Though the original ansatz of Fritzsch is excluded by the experimental data, one can consider its minimal modification which consists in introducing an asymmetry between the 23 and 32 entries in the down-quark Yukawa matrix. We show that this structure is perfectly compatible with the present precision data on quark masses and CKM mixing matrix, and theoretically it can be obtained in the context of SU(5) model with inter-family SU(3)_H symmetry. We also discuss some alternative approaches which could give a natural description of the fermion mass spectrum and weak mixing pattern.