Fermion mass hierarchies, large lepton mixing and residual modular symmetries

Abstract

In modular-invariant models of flavour, hierarchical fermion mass matrices may arise solely due to the proximity of the modulus τ to a point of residual symmetry. This mechanism does not require flavon fields, and modular weights are not analogous to Froggatt-Nielsen charges. Instead, we show that hierarchies depend on the decomposition of field representations under the residual symmetry group. We systematically go through the possible fermion field representation choices which may yield hierarchical structures in the vicinity of symmetric points, for the four smallest finite modular groups, isomorphic to S3, A4, S4, and A5, as well as for their double covers. We find a restricted set of pairs of representations for which the discussed mechanism may produce viable fermion (charged-lepton and quark) mass hierarchies. We present two lepton flavour models in which the charged-lepton mass hierarchies are naturally obtained, while lepton mixing is somewhat fine-tuned. After formulating the conditions for obtaining a viable lepton mixing matrix in the symmetric limit, we construct a model in which both the charged-lepton and neutrino sectors are free from fine-tuning.

Modular invariant dynamics and fermion mass hierarchies around τ = i

“…Few weeks after completion of this work, ref. [43] appeared on the arXiv, in which the authors performed a systematic study of all fixed points, τ S = i, τ ST = e 2πi/3 , and τ T = i∞, assuming a minimal form of the Kähler potential. The points τ ST and τ T enjoy residual Z 3 × Z 2 and Z N × Z 2 symmetries, respectively (where N is the level of modular forms used in the construction, N = 3 in our work).…”

Modular origin of mass hierarchy: Froggatt-Nielsen like mechanism

“…1 Indeed, various modular invariant models which successfully reproduce the SM have been constructed in these years, e.g. for Γ 2 [14][15][16][17], Γ 3 [1,2,14,[17][18][19][20][21][22][23][24], Γ 4 [17,20,[25][26][27][28], Γ 5 [17,28,29], Γ 7 [30], and the double covering groups of the modular groups [17,[31][32][33][34][35][36][37]. A combined symmetry of the modular symmetry with the conventional flavor symmetries or CP-symmetry is also considered in [38][39][40][41][42][43][44][45].…”

“…For reviews, see[9][10][11][12] and other approaches including continuous flavor symmetry and the GUT are shortly reviewed in[13].2 Another approach to the mass hierarchy by the residual modular symmetry can be found in[17,50].…”

Modular symmetry at level 6 and a new route towards finite modular groups

“…The inhomogeneous finite modular group Γ N is the quotient group of the modular group PSL(2, Z) ∼ = Γ over the principal congruence subgroup Γ(N ). The phenomenologically viable modular invariant models have been widely discussed by using the inhomogeneous finite modular group Γ N in the literature, such as models based on the finite modular groups Γ 2 ∼ = S 3 [11][12][13][14][15], Γ 3 ∼ = A 4 [10][11][12], Γ 4 ∼ = S 4 [29,[44][45][46][47][48][49][50][51][52][53][54][55], Γ 5 ∼ = A 5 [49,56,57] and Γ 7 ∼ = PSL(2, Z 7 ) [58] have been studied. The modular forms of integral weights will be decomposed into irreducible representations of the homogeneous finite modular JHEP10(2021)238 group Γ N which is the double covering of Γ N [59].…”