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Register a new userDiagonal reflection symmetries, maximal CP violation, and three-zero texture
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Masaki J. S. Yang
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In this paper, we consider the diagonal reflection symmetries and three-zero texture in the SM. The three-zero texture has two less assumptions ($(M_{u})_{11} , (M_{ u})_{11} eq 0$) than the universal four-zero texture for mass matrices $(M_{f})_{11} = (M_{f})_{13,31} = 0$ for $f = u,d, u, e$. The texture and symmetries reproduce the CKM and MNS matrices with accuracies of $O(10^{-4})$ and $O(10^{-3})$. By assuming a $d$-$e$ unified relation ($M_{d} sim M_{e}$), this system predicts the normal hierarchy, the Dirac phase $delta_{CP} simeq 202^{circ},$ the Majorana phases $alpha_{12} = 11.3^{circ}, alpha_{13} = 6.90^{circ}$ up to $pi$, and the lightest neutrino mass $m_{1} simeq 2.97,-,4.72,$[meV]. The effective mass of the double beta decay $|m_{ee}|$ is found to be $1.24 sim 1.77 ,$[meV].
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In this paper, we consider a set of new symmetries in the SM, {it diagonal reflection} symmetries $R , m_{u, u}^{*} , R = m_{u, u}, ~ m_{d,e}^{*} = m_{d,e}$ with $R =$ diag $(-1,1,1)$. These generalized $CP$ symmetries predict the Majorana phases to be $alpha_{2,3} /2 sim 0$ or $pi /2$. A realization of reflection symmetries suggests a broken chiral $U(1)_{rm PQ}$ symmetry and a flavored axion. The axion scale is suggested to be $langle theta_{u,d} rangle sim Lambda_{rm GUT} , sqrt{m_{u,d} , m_{c,s}} / v sim 10^{12} , $[GeV]. By combining the symmetries with the four-zero texture, the mass eigenvalues and mixing matrices of quarks and leptons are reproduced well. This scheme predicts the normal hierarchy, the Dirac phase $delta_{CP} simeq 203^{circ},$ and $|m_{1}| simeq 2.5$ or $6.2 , $[meV]. In this scheme, the type-I seesaw mechanism and a given neutrino Yukawa matrix $Y_{ u}$ completely determine the structure of right-handed neutrino mass $M_{R}$. An $u- u$ unification predicts mass eigenvalues to be $ (M_{R1} , , M_{R2} , , M_{R3}) = (O (10^{5}) , , O (10^{9}) , , O (10^{14})) , $[GeV].
In this paper, we impose a magic symmetry on the neutrino mass matrix $M_{ u}$ with universal four-zero texture and diagonal reflection symmetries. Due to the magic symmetry, the MNS matrix has trimaximal mixing inevitably. Since the lepton sector has only six free parameters, physical observables of leptons are all determined from the charged leptons masses $m_{ei}$, the neutrino mass differences $Delta m_{i1}$, and the mixing angle $theta_{23}$. As new predictions, we obtain $sin theta_{12} = 0.584$ and $sin theta_{13} = 0.149$. The latter one is almost equal to the latest best fit.
Using the seesaw mechanism, we construct a model for the light-neutrino Majorana mass matrix which yields trimaximal lepton mixing together with maximal CP violation and maximal atmospheric-neutrino mixing. We demonstrate that, in our model, the light-neutrino mass matrix retains its form under the one-loop renormalization-group evolution. With our neutrino mass matrix, the absolute neutrino mass scale is a function of |U_e3| and of the atmospheric mass-squared difference. We study the effective mass in neutrinoless double beta-decay as a function of |U_e3|, showing that it contains a fourfold ambiguity.
In this letter, we consider exact $mu-tau$ reflection symmetries for quarks and leptons. Fermion mass matrices are assumed to be four-zero textures for charged fermions $f = u,d,e$ and a symmetric matrix for neutrinos $ u_{L}$. By a bi-maximal transformation, all the mass matrices lead to $mu-tau$ reflection symmetric forms, which seperately satisfy $T_{u} , m_{u, u}^{*} , T_{u} = m_{u, u}$ and $T_{d} , m_{d,e}^{*} , T_{d} = m_{d,e}$. Reconciliation between the $mu-tau$ reflection symmetries and observed $sin theta_{13}$ predicts $delta_{CP} simeq 203^{circ}$. Moreover, imposition of universal texture $(m_{f})_{11} = 0$ for $f=u,d, u,e$ predicts the normal hierarchy with the lightest neutrino mass $|m_{1}| = 6.26$ or $2.54$ meV.
CP-odd invariants, independent of basis and valid for any choice of CP transformation are a powerful tool in the study of CP. They are particularly convenient to study the CP properties of models with family symmetries. After interpreting the consequences of adding specific CP symmetries to a Lagrangian invariant under $Delta(27)$, I use the invariant approach to systematically study Yukawa-like Lagrangians with an increasing field content in terms of $Delta(27)$ representations. Included in the Lagrangians studied are models featuring explicit CP violation with calculable phases (referred to as explicit geometrical CP violation) and models that automatically conserve CP, despite having all the $Delta(27)$ representations.
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