Modular invariant dynamics and fermion mass hierarchies around τ = i

Abstract

We discuss fermion mass hierarchies within modular invariant flavour models. We analyse the neighbourhood of the self-dual point τ = i, where modular invariant theories possess a residual Z4 invariance. In this region the breaking of Z4 can be fully described by the spurion ϵ ≈ τ − i, that flips its sign under Z4. Degeneracies or vanishing eigenvalues of fermion mass matrices, forced by the Z4 symmetry at τ = i, are removed by slightly deviating from the self-dual point. Relevant mass ratios are controlled by powers of |ϵ|. We present examples where this mechanism is a key ingredient to successfully implement an hierarchical spectrum in the lepton sector, even in the presence of a non-minimal Kähler potential.

Completing the eclectic flavor scheme of the ℤ2 orbifold

“…ref. [11]), the geometry of the T 2 /Z 2 orbifold adopts the form of a square raviolo, where the corners correspond to the singularities of the orbifold and the edges are perpendicular and have the same length. As just mentioned, in this case the traditional flavor group is enhanced byĈ S to the unified flavor group [128, 523], considering the Z 8 phases (−i) n U = e −πi nU /2 associated with the automorphy factors of the matter fields.…”

“…In the top-down approach (which we adopt here), this extension of the symmetry reflects the symmetries of the underlying string theory, which restrict the modular weights to welldefined specific values. 2 In the bottom-up approach to modular flavor symmetries, the choice of the modular weights of matter fields is part of model building and can be used to obtain so-called "shaping symmetries" that appear as additional accidental symmetries for specific choices of the modular weights [10,11].…”

Explicit perturbations to the stabilizer τ = i of modular $$ {A}_5^{\prime } $$ symmetry and leptonic CP violation

“…If τ is exactly located at one of the stabilizers, it seems unlikely to realize viable lepton flavor models by using only one modular symmetry [48,49] or a common value of τ in both the charged-lepton and neutrino sectors [46,50]. One can alternatively construct modularinvariant models with the modulus close to the stabilizers, with which the strong hierarchy of charged-lepton masses can be successfully realized [51,52].…”

Quark and lepton flavor model with leptoquarks in a modular $$A_4$$ symmetry

“…We then do not need the scalar fields to obtain a predictive mass matrix. Along the line of this idea, a vast reference has recently appeared in the literature, e.g., A 4 [24,, S 3 [59][60][61][62][63][64], S 4 [65][66][67][68][69][70][71][72][73][74][75][76], A 5 [70,77,78], double covering of A 5 [79][80][81], larger groups [82], multiple modular symmetries [83], and double covering of A 4 [84,85], S 4 [86,87], and the other types of groups [88][89][90][91][92][93] in which masses, mixing, and CP phases for the quark and/or lepton have been predicted. 2 Moreover, a systematic approach to understand the origin of CP transformations has been discussed in Ref.…”