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We propose a model for the quark masses and mixings based on an A_4 family symmetry. Three scalar SU(2) doublets form a triplet of A_4. The three left-handed-quark SU(2) doublets are also united in a triplet of A_4. The right-handed quarks are singlets of A_4. The A_4-symmetric scalar potential leads to a vacuum in which two of the three scalar SU(2) doublets have expectation values with equal moduli. Our model makes an excellent fit of the observed |V_ub/V_cb|. The symmetry CP is respected in the charged gauge interactions of the quarks.
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The hierarchical quark masses and small mixing angles are shown to lead to a simple triangular form for the U- and D-type quark mass matrices. In the basis where one of the matrices is diagonal, each matrix element of the other is, to a good approximation, the product of a quark mass and a CKM matrix element. The physical content of a general mass matrix can be easily deciphered in its triangular form. This parameterization could serve as a useful starting point for model building. Examples of mass textures are analyzed using this method.
We propose a model that all quark and lepton mass matrices have the same zero texture. Namely their (1,1), (1,3) and (3,1) components are zeros. The mass matrices are classified into two types I and II. Type I is consistent with the experimental data in quark sector. For lepton sector, if seesaw mechanism is not used, Type II allows a large $ u_mu - u_tau$ mixing angle. However, severe compatibility with all neutrino oscillation experiments forces us to use the seesaw mechanism. If we adopt the seesaw mechanism, it turns out that Type I instead of II can be consistent with experimental data in the lepton sector too.
Modular symmetries have been impeccable in neutrino and quark sectors. This motivated us, therefore, to propose a variant of scotogenic model based on modular $A_4$ symmetry to realize the neutrino mass generation at one-loop level through radiative mechanism. Alongside, the lepton flavour violating process $mu to e gamma$ and the muon $g-2$ anomaly are also addressed. The lightest Majorana fermions turn out to be potential dark matter candidates, made stable by suitable assignment of modular weights. The relic density of the same has been computed with annihilations mediated by inert scalars and new $U(1)$ gauge boson.
We look for all weak bases that lead to texture zeroes in the quark mass matrices and contain a minimal number of parameters in the framework of the standard model. Since there are ten physical observables, namely, six nonvanishing quark masses, three mixing angles and one CP phase, the maximum number of texture zeroes in both quark sectors is altogether nine. The nine zero entries can only be distributed between the up- and down-quark sectors in matrix pairs with six and three texture zeroes or five and four texture zeroes. In the weak basis where a quark mass matrix is nonsingular and has six zeroes in one sector, we find that there are 54 matrices with three zeroes in the other sector, obtainable through right-handed weak basis transformations. It is also found that all pairs composed of a nonsingular matrix with five zeroes and a nonsingular and nondecoupled matrix with four zeroes simply correspond to a weak basis choice. Without any further assumptions, none of these pairs of up- and down-quark mass matrices has physical content. It is shown that all non-weak-basis pairs of quark mass matrices that contain nine zeroes are not compatible with current experimental data. The particular case of the so-called nearest-neighbour-interaction pattern is also discussed.
A new idea for neutrino mass was proposed recently, where its smallness is not due to the seesaw mechanism, i.e. not inversely proportional to some large mass scale. It comes from a one-loop mechanism with dark matter in the loop consisting of singlet Majorana fermions $N_i$ with masses of order 10 keV and neutrino masses are scaled down from them by factors of about $10^{-5}$. We discuss how this model may be implemented with the non-Abelian discrete symmetry $A_4$ for neutrino mixing, and consider the phenomenology of $N_i$ as well as the extra scalar doublet $(eta^+,eta^0)$.
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