[Paper Review] Modular Invariant Models of Leptons at Level 7

Gui-Jun Ding, Stephen F. King+2 more|arXiv (Cornell University)|Apr 27, 2020

Neutrino Physics Research53 references9 citations

TL;DR

This paper presents the first modular invariant flavor models based on the finite modular group Γ7 ≅ PSL(2, Z7), which has 168 elements and is isomorphic to Σ(168). It constructs flavor models at level 7 where lepton masses and mixing arise from modular forms transforming under irreducible representations of Γ7, with 26 linearly independent modular forms at weight 2 (triplet, septet, and two octets), and derives predictions for neutrino masses via the Weinberg operator or type-I seesaw mechanism, matching experimental data with minimal assumptions and no flavon fields.

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.

Motivation & Objective

• To construct the first modular invariant flavor models based on the finite modular group Γ7 at level 7, which is isomorphic to PSL(2, Z7) and has 168 elements.
• To derive a complete list of modular forms up to weight 8 for Γ7, identifying their transformation properties under irreducible representations.
• To build flavon-less models where lepton masses and mixing originate solely from modular invariance and modular forms, avoiding auxiliary scalar fields.
• To compare model predictions with experimental neutrino mixing angles and masses, using either the Weinberg operator or type-I seesaw mechanism for neutrino mass generation.

Proposed method

• The paper uses the finite modular group Γ7 ≅ PSL(2, Z7), a non-Abelian finite group of order 168, as the flavor symmetry group.
• It computes all linearly independent modular forms of level 7 and weights up to 8 using the SageMath algebra system, classifying them into irreducible representations: triplet, septet, and two octets at weight 2.
• The effective Lagrangian is constructed using modular forms as couplings, with no flavon fields required, ensuring modular invariance under Γ7.
• Neutrino masses are generated via either the dimension-5 Weinberg operator or the type-I seesaw mechanism, with the latter involving heavy right-handed neutrinos.
• The model's predictions are tested against experimental data on neutrino mixing angles and mass splittings.
• The modular forms are systematically derived using algebraic number theory and relations between theta functions and Dedekind eta functions, with constraints reduced via computational algebra.

Experimental results

Research questions

• RQ1 Can modular invariant flavor models be constructed at level 7 using the finite modular group Γ7 ≅ PSL(2, Z7), without introducing flavon fields?
• RQ2 What is the complete spectrum of modular forms of level 7 up to weight 8, and how are they decomposed into irreducible representations of Γ7?
• RQ3 Can such models reproduce the observed neutrino mixing patterns and mass hierarchies using only modular invariance and the Weinberg operator or type-I seesaw mechanism?
• RQ4 How do the predictions of these models compare quantitatively with current experimental data on neutrino mixing angles and mass splittings?

Key findings

• At weight 2, there are 26 linearly independent modular forms for Γ7, transforming as a triplet (3), a septet (7), and two octets (8a, 8b), forming a complete basis for the model construction.
• The full list of modular forms up to weight 8 is derived, including weight-10 forms such as Y(10)3a, Y(10)3b, Y(10)3c, Y(10)3̄a, and Y(10)3̄b, which transform under the 3 and 3̄ representations.
• The models are constructed without flavon fields, with lepton masses and mixing generated entirely by modular forms transforming under Γ7 representations.
• The Weinberg operator and type-I seesaw mechanisms both yield viable neutrino mass predictions that are consistent with experimental data on mixing angles and mass splittings.
• The model predicts tribimaximal-like mixing patterns in the neutrino sector, with deviations from exact tribimaximality arising from modular symmetry breaking.
• The framework successfully realizes a predictive flavor structure with only a few input parameters, demonstrating the viability of level 7 modular symmetry in flavor physics.