Modular Symmetry Research Articles

Finite modular symmetries and the strong CP problem

Recently, it was shown that modular symmetry may solve the strong CP problem without axions, by producing a vanishing QCD angle while generating a large quark CP violation phase. We extend this framework to finite modular groups, systematically identifying the allowed mass textures. We find quark fields must furnish 1D representations and scan the minimal model landscape.

Modular invariant slow roll inflation

We propose new classes of inflation models based on the modular symmetry, where the modulus field τ serves as the inflaton. We establish a connection between modular inflation and modular stabilization, wherein the modulus field rolls towards a fixed point along the boundary of the fundamental domain. We find the modular symmetry strongly constrain the possible shape of the potential and identify some parameter space where the inflation predictions agree with cosmic microwave background observations. The tensor-to-scalar ratio is predicted to be smaller than 10-6 in our models, while the running of spectral index is of the order of 10-4.

The eclectic flavor symmetries of T2/ℤK\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbbm{T}}^2/{\\mathbb{Z}}_K $$\\end{document} orbifolds

Only four T2/ℤK\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbbm{T}}^2/{\\mathbb{Z}}_K $$\\end{document} orbifold building blocks are admissible in heterotic string compactifications. We investigate the flavor properties of all of these building blocks. In each case, we identify the traditional and modular flavor symmetries, and determine the corresponding representations and (fractional) modular weights of the available massless matter states. The resulting finite flavor symmetries include Abelian and non-Abelian traditional symmetries, discrete R symmetries, as well as the double-covered finite modular groups (S3 × S3) ⋊ ℤ4, T′, 2D3 and S3 × T′. Our findings provide restrictions for bottom-up model building with consistent ultraviolet embeddings.

Modular properties of massive scalar partition functions

We compute the exact thermal partition functions of a massive scalar field on flat spacetime backgrounds of the form ℝd−q × \U0001d54bq+1 and show that they possess an SL(q + 1, ℤ) symmetry. Non-trivial relations between equivalent expressions for the result are obtained by doing the computation using functional, canonical and worldline methods. For q = 1, the results exhibit modular symmetry and may be expressed in terms of massive Maass-Jacobi forms. In the complex case with chemical potential for U(1) charge turned on, the usual discussion of relativistic Bose-Einstein condensation is modified by the presence of the small dimensions.

Spontaneous CP violation and partially broken modular flavor symmetries

We study the realization of spontaneous CP violation through moduli stabilization. In modular flavor models, the source of CP violation is the vacuum expectation values of the complex structure moduli of toroidal compact space. We demonstrate that the combined effects of Type IIB flux compactifications with modular invariant couplings between the moduli and matter fields can induce spontaneous CP violation without or with supersymmetry breaking. Furthermore, some general properties of CP and modular invariant scalar potentials are presented. It is found that certain modifications or partial breakings of modular symmetry are useful in generating spontaneous CP violation.

Minimal type-I Dirac seesaw and leptogenesis under A4 modular invariance

We present a Dirac mass model based on A4 modular symmetry within Type-I seesaw framework. This extension of Standard Model requires three right-handed neutrinos and three heavy Dirac fermions superfields, all singlet under SU(2)L symmetry. The scalar sector is extended by the inclusion of a SU(2)L singlet superfield, χ. Here, the modular symmetry plays a crucial role as the Yukawa couplings acquire modular forms, which are expressed in terms of Dedekind eta function η(τ). Therefore, the Yukawa couplings follow transformations akin to other matter fields, thereby obviating the necessity of additional flavon fields. The acquisition of vev by complex modulus τ leads to the breaking of A4 modular symmetry. We have obtained predictions on neutrino oscillation parameters, for example, the normal hierarchy for the neutrino mass spectrum. Furthermore, we find that heavy Dirac fermions, in our model, can decay to produce observed baryon asymmetry of the Universe through Dirac leptogenesis.

Non-holomorphic modular flavor symmetry

The formalism of non-holomorphic modular flavor symmetry is developed, and the Yukawa couplings are level N polyharmonic Maaß forms satisfying the Laplacian condition. We find that the integer (even) weight polyharmonic Maaß forms of level N can be decomposed into multiplets of the finite modular group ΓN′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\Gamma}_N^{\\prime } $$\\end{document} (ΓN). The original modular invariance approach is extended by the presence of negative weight polyharmonic Maaß forms. The non-holomorphic modular flavor symmetry can be consistently combined with the generalized CP symmetry. We present three example models for lepton sector based on the Γ3 ≅ A4 modular symmetry, the charged lepton masses and the neutrino oscillation data can be accommodated very well, and the predictions for the leptonic CP violation phases and the effective Majorana neutrino mass are studied.

Pati-Salam models with A4 modular symmetry

The flavor structure of quarks and leptons and quark-lepton unification are studied in the framework of Pati-Salam models with A4 modular symmetry. The three generations of the left-handed and right-handed fermions are assigned to be triplet or singlets of A4. The light neutrino masses are generated through the type-I seesaw mechanism. We perform a systematic classification of Pati-Salam models according to the transformations of matter fields under the A4 modular symmetry, and the general form of the fermion mass matrix is given. We present four phenomenologically viable benchmark models which provide excellent descriptions of masses and flavor mixing of quarks and leptons, including neutrinos. In such models we find that the normal ordered neutrino mass spectrum is preferred over the inverted case, with neutrinoless double beta decay predicted to be too small to be observed by the next generation of experiments.

Modular forms and hierarchical Yukawa couplings in heterotic Calabi-Yau compactifications

We study the modular symmetry in heterotic string theory on Calabi-Yau threefolds. In particular, we examine whether moduli-dependent holomorphic Yukawa couplings are described by modular forms in the context of heterotic string theory with standard embedding. We find that SL(2, ℤ) modular symmetry emerges in asymptotic regions of the Calabi-Yau moduli space. The instanton-corrected holomorphic Yukawa couplings are then given by modular forms under SL(2, ℤ) or its congruence subgroups such as Γ0(3) and Γ0(4). In addition to the modular symmetry, it turns out that another coupling selection rule controls the structure of holomorphic Yukawa couplings. Furthermore, the coexistence of both the positive and negative modular weights for matter fields leads to a hierarchical structure of matter field Kähler metric. Thus, these holomorphic modular forms and the matter field Kähler metric play an important role in realizing a hierarchical structure of physical Yukawa couplings.

Neutrino phenomenology in the modular S3 seesaw model

We have studied neutrino phenomenology in the supersymmetric type-I seesaw model endowed with the Γ2≃S3 modular symmetry. We have identified different realizations of the S3 modular symmetry, referred to as models A, B, C, and D. The four models are compatible with neutrino mass being inverted ordering (IO). Moreover, models A, B, and D can also accommodate normal ordering (NO) neutrino masses. We identify parameter space for each model compatible with neutrino oscillation at the 2−σ level. We then proceed to study the neutrino phenomenology of each model. We find that the lightest neutrino mass can be as light as 0.64 meV in the case of NO in model A and 50 meV in the case of IO in model D. The smallest effective electron neutrino mass attainable in our analysis is 8.8 meV in the case of NO (model A) and 50 meV for IO (model D). Finally, we note that the effective Majorana mass can be as small as 0.33 meV in the case of NO (model A) and 22 meV for IO (model D). Published by the American Physical Society 2024

Finite modular majoron

We point out that the accidental U(1)B−L symmetry can arise from a finite modular symmetry ΓN in the type-I seesaw. The finite modular symmetry is spontaneously broken in such a way that the residual ℤNT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbb{Z}}_N^T $$\\end{document} discrete symmetry, associated with the T-transformation which shifts the modulus τ → τ + 1, remains unbroken. This discrete ℤNT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\mathbb{Z}}_N^T $$\\end{document} symmetry mimics U(1)B−L, and hence the majoron appears as a pseudo Nambu-Goldstone boson of U(1)B−L. Without introducing additional interactions, the modulus τ can be stabilized by the Coleman-Weinberg (CW) potential given by the Majorana mass terms of the right-handed neutrinos. We study cosmological implications of the majoron, with particular interests in the dark matter and dark radiation, where the latter may alleviate the Hubble tension. We also find that the CW potential can have a wide range of nearly exponential shape which prevents τ from overshooting, and makes the amount of dark radiation not too large.

Modular average and Weyl anomaly in two-dimensional Schwarzian theory

The gauge formulation of Einstein gravity in AdS3 background leads to a boundary theory that breaks modular symmetry and loses the covariant form. We examine the Weyl anomaly for the cylinder and torus manifolds. The divergent term is the same as the Liouville theory when transforming from the cylinder to the sphere. The general Weyl transformation on the torus also reproduces the Liouville theory. The Weyl transformation introduces an additional boundary term for reproducing the Liouville theory, which allows the use of CFT techniques to analyze the theory. The torus partition function in this boundary theory is one-loop exact, and an analytical solution to disjoint two-interval Rényi-2 mutual information can be obtained. We also discuss a first-order phase transition for the separation length of two intervals, which occurs at the classical level but is smoothed out by non-perturbative effects captured by averaging over a modular group in the boundary theory.

Texture zeros realization in a three-loop radiative neutrino mass model from modular A4 symmetry

We derive verifiable two-zero textures in the framework of a three-loop induced neutrino mass model applying a modular A4 symmetry for the lepton sector. Interestingly, the neutrino mass structure is determined by assignments of the right-handed charged-lepton particles only. We show all the possible texture patterns and demonstrate their analytical and numerical analyses. Finally, we mention a possibility of detection for doubly-charged boson via colliders that decays into specific modes due to the flavor symmetry.

A modular SU (5) littlest seesaw

We extend the littlest modular seesaw to a Grand Unified scenario based on SU (5) endowed with three modular S4 symmetries. We leverage symmetry protected zeroes in the leptonic and down quark sectors to suppress deviations to the littlest modular seesaw predictions, but not contributions to the quark mixing. The model is supplemented by two weighton fields, such that the hierarchical nature of the charged-lepton masses, as well as the quark masses and mixing, stem from the content and symmetries of the model, rather than a hierarchical nature of the Yukawa coefficients.

Quark and lepton modular models from the binary dihedral flavor symmetry

Inspired by the structure of top-down derived models endowed with modular flavor symmetries, we investigate the yet phenomenologically unexplored binary dihedral group 2D3. After building the vector-valued modular forms in the representations of 2D3 with small modular weights, we systematically classify all (Dirac and Majorana) mass textures of fermions with fractional modular weights and all possible 2 + 1-family structures. This allows us to explore the parameter space of fermion models based on 2D3, aiming at a description of both quarks and leptons with a minimal number of parameters and best compatibility with observed data. We consider the separate possibilities of neutrino masses generated by either a type-I seesaw mechanism or the Weinberg operator. We identify a model that, besides fitting all known flavor observables, delivers predictions for six not-yet measured parameters and favors normal-ordered neutrino masses generated by the Weinberg operator. It would be interesting to figure out whether it is possible to embed our model within a top-down scheme, such as \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathbb{T}}^{2}/{\\mathbb{Z}}_{4}$$\\end{document} heterotic orbifold compactifications.

Modular flavor symmetric models

We review the modular flavor symmetric models of quarks and leptons focusing on our works. We present some flavor models of quarks and leptons by using finite modular groups and discuss the phenomenological implications. The modular flavor symmetry gives interesting phenomena at the fixed point of modulus. As a representative, we show the successful texture structure at the fixed point [Formula: see text]. We also study CP violation, which occurs through the modulus stabilization. Finally, we study SMEFT with modular flavor symmetry by including higher dimensional operators.

Finite modular axion and radiative moduli stabilization

We propose a simple setup which can stabilize a modulus field of the finite modular symmetry by the Coleman-Weinberg potential. Our scenario leads to a large hierarchy suppressing instanton-like corrections e2πiτ and to a light axion identified as Reτ, where τ is the modulus field. This stabilization mechanism provides the axion solution to the strong CP problem. The potential has a minimum at a large Imτ which suppresses explicit U(1)PQ violation terms proportional to e−2πImτ, and hence the quality of the axion is ensured by the residual symmetry associated with the T-transformation, τ → τ + 1, around the fixed point τ ∼ i∞.

The flavor puzzle: Textures and symmetries

We discuss aspects of a promising top-down origin of flavor symmetries in particle physics. Modular transformations originating from string theory dualities are shown to play a crucial role. We introduce the notion of an “eclectic” flavor scheme that unifies traditional flavor symmetries, modular symmetries and [Formula: see text]-transformations. It exhibits the phenomenon of “Local Flavor Unification” with enhanced flavor symmetries at fixed points or lines in moduli space. Successful fits of masses and mixing angles of quarks and leptons are found in the vicinity of these points and lines.

Flavor's Delight.

Discrete flavor symmetries provide a promising approach to understand the flavor sector of the standard model of particle physics. Top-down (TD) explanations from string theory reveal two different types of such flavor symmetries: traditional and modular flavor symmetries that combine to the eclectic flavor group. There have been many bottom-up (BU) constructions to fit experimental data within this scheme. We compare TD and BU constructions to identify the most promising groups and try to give a unified description. Although there is some progress in joining BU and TD approaches, we point out some gaps that have to be closed with future model building.

Predictions from scoto-seesaw with A4 modular symmetry

This paper's novelty lies in introducing a hybrid scoto-seesaw model rooted in A4 discrete modular symmetry leading to several interesting phenomenological implications. The scoto-seesaw framework leads to generation of one mass square difference (Δmatm2) using the type-I seesaw mechanism at the tree level. Additionally, the scotogenic contribution is vital in obtaining the other mass square difference (Δmsol2) at the loop level, thus providing a clear interpretation of the two different mass square differences. The non-trivial transformation of Yukawa couplings under the A4 modular symmetry helps to explore neutrino phenomenology with a specific flavor structure of the mass matrix. In addition to predictions for neutrino mass ordering, mixing angles and CP phases, this setup leads to precise predictions for ∑mi as well as |mee|. In particular, the model predicts ∑mi∈(0.073,0.097) eV and |mee|∈(3.15,6.66)×10−3 eV range; within reach of upcoming experiments. Furthermore, our model is also promising for addressing lepton flavor violations, i.e., ℓα→ℓβγ, ℓα→3ℓβ and μ−e conversion rates while staying within the realm of current experimental limits.