Modular invariant models of leptons at level 7

Abstract

We consider for the first time level 7 modular invariant flavour models where the lepton mixing originates from the breaking of modular symmetry and couplings responsible for lepton masses are modular forms. The latter are decomposed into irreducible multiplets of the finite modular group Γ 7 , which is isomorphic to PSL(2, Z 7), the projective special linear group of two dimensional matrices over the finite Galois field of seven elements, containing 168 elements, sometimes written as PSL 2 (7) or Σ(168). At weight 2, there are 26 linearly independent modular forms, organised into a triplet, a septet and two octets of Γ 7. A full list of modular forms up to weight 8 are provided. Assuming the absence of flavons, the simplest modular-invariant models based on Γ 7 are constructed, in which neutrinos gain masses via either the Weinberg operator or the type-I seesaw mechanism, and their predictions compared to experiment.

Spontaneous CP violation by modulus τ in A4 model of lepton flavors

“…Furthermore, modular forms for ∆ (96) and ∆(384) were constructed [38], and the extension of the traditional flavor group was discussed with modular symmetries [39]. The level 7 finite modular group Γ 7 PSL(2, Z 7 ) was also presented for the lepton mixing [40]. Based on those works, phenomenological studies have been developed in many works while theoretical investigations have been also proceeded [81][82][83][84][85][86].…”

SU(5) GUTs with A4 modular symmetry

“…Additional relations are needed to render the group finite make for N > 5 [48]. In the present we are interested in the minimal finite modular group Γ 3 = Γ/Γ(3) ∼ = A 4 which…”

“…The modular form of level N and integral weight k can be arranged into some modular multiplets of the homogeneous finite modular group Γ N ≡ Γ/Γ(N ) [7], and they can be organized into modular multiplets of the inhomogeneous finite modular group Γ N ≡ Γ/Γ(N ) if k is an even number [6]. The inhomogeneous finite modular group Γ N of lower levels N = 2 [8][9][10][11], N = 3 [6,8,9,, N = 4 [25,[38][39][40][41][42][43][44][45], N = 5 [43,46,47] and N = 7 [48] have been considered and a large number of models have been constructed. All the modular forms of integral weights can be generated from the tensor products of weight one modular forms and the odd weight modular forms are in the representations with ρ r (S 2 ) = −1.…”

Symmetries and stabilisers in modular invariant flavour models

“…Interesting models which are trying to explain neutrino masses and lepton mixing have been constructed in different finite modular symmetries, specifically in Γ 2 S 3 [11,12], Γ 3 A 4 [8,9,[12][13][14][15][16][17][18], Γ 4 S 4 [10,[19][20][21][22][23], and Γ 5 A 5 [24,25]. Discussion was extended to the modular symmetry with a higher level N = 7, Γ 7 P SL 2 (Z 7 ) Σ(168) which includes complex triplet representations [26]. The modular invariance approach was also generalised to include odd-weight modular forms [27][28][29] and half-integer modular forms [30], which can be arranged into irreducible representations of the homogeneous finite modular groups Γ N and finite metaplectic group Γ 4N , respectively.…”

“…For N > 5, additional conditions have to imposed to make the group finite. e.g., (Sτ T 3 τ )4 = e for N = 7[26].…”