CP symmetry and symplectic modular invariance

We present a detailed analysis of the eclectic flavor structure of the two-dimensional ℤ2 orbifold with its two unconstrained moduli T and U as well as SL(2, ℤ)T× SL(2, ℤ)U modular symmetry. This provides a thorough understanding of mirror symmetry as well as the R-symmetries that appear as a consequence of the automorphy factors of modular transformations. It leads to a complete picture of local flavor unification in the (T, U) modulus landscape. In view of applications towards the flavor structure of particle physics models, we are led to top-down constructions with high predictive power. The first reason is the very limited availability of flavor representations of twisted matter fields as well as their (fixed) modular weights. This is followed by severe restrictions from traditional and (finite) modular flavor symmetries, mirror symmetry, $$ \mathcal{CP} $$ CP and R-symmetries on the superpotential and Kähler potential of the theory.

Section: Discussionmentioning

confidence: 75%

Completing the eclectic flavor scheme of the ℤ2 orbifold

Baur

Kade

Nilles³

et al. 2021

“…The Double Covering groups T [34,35] and S 4 [36,37] were also realized in the modular symmetry. Furthermore, phenomenological studies have been developed in many works while theoretical investigations have been also proceeded [89][90][91][92][93][94][95][96][97][98][99][100][101][102][103][104][105][106].…”

Section: Introductionmentioning

confidence: 99%

Modulus τ linking leptonic CP violation to baryon asymmetry in A4 modular invariant flavor model

Okada

Shimizu

Tanimoto

et al. 2021

We propose an A4 modular invariant flavor model of leptons, in which both CP and modular symmetries are broken spontaneously by the vacuum expectation value of the modulus τ. The value of the modulus τ is restricted by the observed lepton mixing angles and lepton masses for the normal hierarchy of neutrino masses. The predictive Dirac CP phase δCP is in the ranges [0°, 50°], [170°, 175°] and [280°, 360°] for Re [τ] < 0, and [0°, 80°], [185°, 190°] and [310°, 360°] for Re [τ] > 0. The sum of three neutrino masses is predicted in [60, 84] meV, and the effective mass for the 0νββ decay is in [0.003, 3] meV. The modulus τ links the Dirac CP phase to the cosmological baryon asymmetry (BAU) via the leptogenesis. Due to the strong wash-out effect, the predictive baryon asymmetry YB can be at most the same order of the observed value. Then, the lightest right-handed neutrino mass is restricted in the range of M1 = [1.5, 6.5] × 1013 GeV. We find the correlation between the predictive YB and the Dirac CP phase δCP. Only two predictive δCP ranges, [5°, 40°] (Re [τ] > 0) and [320°, 355°] (Re [τ] < 0) are consistent with the BAU.

“…Then a unified description of leptons and quarks can be achieved from a common finite discrete modular group [22,36,40,60,62,64]. In order to further improve the predictive power of the modular invariance approach, the setup of combining modular symmetry with the generalized CP (gCP) symmetry has been discussed [67,68]. The consistency of modular and gCP symmetries requires that the complex modulus τ transforms as τ CP −→ −τ * up to modular transformations [67][68][69][70][71][72].…”

Section: Higher Weight Modular Forms Of Level N = 6 Under T 1 Introductionmentioning

confidence: 99%

“…In order to further improve the predictive power of the modular invariance approach, the setup of combining modular symmetry with the generalized CP (gCP) symmetry has been discussed [67,68]. The consistency of modular and gCP symmetries requires that the complex modulus τ transforms as τ CP −→ −τ * up to modular transformations [67][68][69][70][71][72]. In this context, the VEV of τ is the unique source of both modular and gCP symmetries.…”

Section: Higher Weight Modular Forms Of Level N = 6 Under T 1 Introductionmentioning

confidence: 99%

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

Liu

Ding

2021

Self Cite

We propose to construct the finite modular groups from the quotient of two principal congruence subgroups as Γ(N′)/Γ(N″), and the modular group SL(2, ℤ) is ex- tended to a principal congruence subgroup Γ(N′). The original modular invariant theory is reproduced when N′ = 1. We perform a comprehensive study of $$ {\Gamma}_6^{\prime } $$ Γ 6 ′ modular symmetry corresponding to N′ = 1 and N″ = 6, five types of models for lepton masses and mixing with $$ {\Gamma}_6^{\prime } $$ Γ 6 ′ modular symmetry are discussed and some example models are studied numerically. The case of N′ = 2 and N″ = 6 is considered, the finite modular group is Γ(2)/Γ(6) ≅ T′, and a benchmark model is constructed.