Fermion hierarchies in SU(5) grand unification from $$ {\Gamma}_6^{\prime } $$ modular flavor symmetry

Abstract

We construct an SU(5) grand unified model in which the hierarchies of the quark and lepton masses and mixing are explained by the $$ {\Gamma}_6^{\prime } $$ Γ 6 ′ modular flavor symmetry. The hierarchies are realized by the Froggatt-Nielsen-like mechanism due to the residual $$ {Z}_6^T $$ Z 6 T symmetry, approximately unbroken at τ ~ i∞. We argue that the $$ {\Gamma}_6^{\left({}^{\prime}\right)} $$ Γ 6 ′ symmetry is the minimal possibility to realize the up-type quark mass hierarchies, since the Yukawa matrix is symmetric. We find a combination of the representations and modular weights and then show numerical values of (1) coefficients for the realistic fermion hierarchies.

CP symmetry and symplectic modular invariance

“…Cosmological evolution can offer a mechanism for moduli trapping near the points enjoying an enhanced symmetry [52][53][54]. In general, (τ − τ 0 ) provides a useful expansion parameter to understand the hierarchy among charged fermion masses and mixing angles [55][56][57][58][59][60][61][62][63][64][65][66].…”

Quarks at the modular S4 cusp

“…• models of flavour based on the "double cover" groups Γ ′ 3 ≃ T ′ [93][94][95][96][97], Γ ′ 4 ≃ S ′ 4 [14,98,99] and Γ ′ 5 ≃ A ′ 5 [100][101][102][103] and on Γ ′ 6 ≃ S 3 ×T ′ [22,104] (see also [105], considering a generalisation to other finite modular groups),…”

“…• models of quark-lepton unification [20,22,79,98,101,108,[111][112][113][114][115][116][117][118][119][120][121][122][123][124][125][126][127][128],…”

“…This mechanism was recently employed in refs. [16][17][18][19][20][21][22] and will also be explored in the present work. It relies on the VEV of the modulus taking a value close to one of the symmetric points.…”

Modular binary octahedral symmetry for flavor structure of Standard Model

“…The generalized CP symmetry can be consistently combined with (symplectic) modular symmetries [24][25][26][27], the coupling constants would be enforced to be real so that the predictive power would be improved further. The modular symmetry framework can also be incorporated into various Grand Unified Theories (GUT) [28][29][30][31][32][33][34][35][36][37]. Modular symmetry can not only explain the fermion masses and flavor mixing but also address the strong CP problem [38].…”