CP-violating phase on magnetized toroidal orbifolds

Abstract

We study the CP-violating phase of the quark sector on T-2/Z(N) (N = 2, 3, 4, 6) with non-vanishing magnetic fluxes, where properties of possible origins of the CP violation are investigated minutely. In this system, a non-vanishing value is mandatory in the real part of the complex modulus parameter tau of the two-dimensional torus in order to explain the CP violation in the quark sector. On T-2 without orbifolding, underlying discrete flavor symmetries severely restrict the form of Yukawa couplings and it is very difficult to reproduce the observed pattern in the quark sector including the CP-violating phase delta(CP). When multiple Higgs doublets emerge on T-2/Z(2), the mass matrices of the zero -mode fermions can be written in the Gaussian textures by choosing appropriate configurations of vacuum expectation values of the Higgs fields. When such Gaussian textures of mass matrices are realized, we show that all of the quark profiles, which are mass hierarchies among the quarks, quark mixing angles, and delta(CP) can be simultaneously realized

Supersymmetric models on magnetized orbifolds with flux-induced Fayet-Iliopoulos terms

“…Although the rough estimation in Section 3.3 shows some deviations from the observed values, we expect that they would be improved by the thorough analyses in future works, where CP-violating phases should also be studied (CPviolating phases of the quark sector in the magnetized orbifold models were recently studied in Ref. [32]). …”

Zero-modes on orbifolds: Magnetized orbifold models by modular transformation

“…(22) and (23). Then, we obtain phenomenological interesting results e.g., the realization of quark and lepton mass hierarchies and their mixing angles [20][21][22][23].…”

“…Indeed, Z 2 orbifold models have been studied on several aspects, e.g. model building [15][16][17][18][19], realization of quark and lepton masses and their mixing angles and CP phase [20][21][22][23]. In addition, it is possible to introduce some degree of freedom on orbifold fixed points, e.g.…”

Wavefunctions on S2 with flux and branes

“…where w ∈ Z is the Chern number of L → X and χ(X) = 0 is the Euler characteristic of X. 14) Since the zero mode wavefunction is expressed as a holomorphic section u of L → X (or its complex conjugate), its norm u|u hol is locally expressed as…”