arXiv subjects

Xiang-Gan Liu

Publications and source records attributed to Xiang-Gan Liu.

At least 19 recordsLinked to original sources

arXiv2026

Strong $\mathcal{CP}$ and Quark Mass Hierarchies from Modular Invariance

The strong $\mathcal{CP}$ problem and quark mass hierarchies can probe the same ultraviolet structure. We show this by extending modular strong-$\mathcal{CP}$ solutions to non-Abelian finite modular symmetries. This reveals a previously overlooked modular-anomaly condition from the finite representations. Regularity removes the dependence of the QCD angle on the $\mathcal{CP}$-breaking modulus, yielding $\barθ=0$. For positive modular weight, it also forces the quark Yukawa determinant to vanish at the cusp. This leads to pronounced mass hierarchies among the quarks. We further stress that modular symmetries act as R-symmetries. Under moderate additional assumptions, this renders the QCD angle independent of the dilaton. An explicit model illustrates the mechanism.

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arXiv2026

Lepton mixing from the $Δ(96)$ Modular Littlest Seesaw

We perform the first comprehensive and model independent study of Modular Littlest Seesaw models based on the finite modular group $Δ(96)$. We construct the vector-valued modular forms (VVMFs) for all irreducible representations of modular $Δ(96)$, classify the inequivalent symmetry-preserving fixed points, and derive the corresponding alignments of the low-weight and next-to-lowest-weight triplet VVMFs. These results allow an exhaustive scan over the residual symmetries in the charged lepton, atmospheric neutrino, and solar neutrino sectors. We identify 35 phenomenologically viable and inequivalent breaking patterns, including 21 with normal ordering and 14 with inverted ordering. The resulting Dirac neutrino mass matrices go beyond the conventional CSD$(n)$ structure, yielding new fixed PMNS columns and novel correlations among the lepton mixing parameters beyond the TM$_1$ paradigm. The viable models are highly predictive, giving narrow ranges for neutrino masses, mixing parameters and CP phases, and can be stringently tested by upcoming experiments such as JUNO, DUNE and T2HK.

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arXiv2026

Fermion mass relations in one-parameter modular models

Modular flavour symmetries provide a possible organizing principle for the Standard Model Yukawa sector, by replacing generic couplings with a potentially small number of modular forms controlled by a single complex modulus. We study the extreme limit of this idea: \acp{OPM}, in which each charged-fermion mass matrix is fixed by a single modular invariant contraction. We develop a systematic method to construct such models, showing that the \ac{OPM} requirement is already highly constraining at the level of possible fermion hierarchies. In a concrete realization, the charged-lepton and down-quark sectors are controlled by the common modulus, leading to exact mass relations at the flavour scale, \[ m_s^5 = 2\sqrt{2}\,m_d^3m_b^2, \qquad m_μ^3 = \sqrt{2}\,m_e m_τ^2, \qquad m_s^2m_τ= \sqrt{2}\,m_e m_b^2. \] We show that, once renormalization-group evolution and selective supersymmetric threshold effects are included, these high-scale relations can be made compatible with low-energy charged-fermion data. Our results provide a working proof of principle for \acp{OPM} and point towards a possible route to the flavour puzzle through highly

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arXiv2026

INFLAVON: CMB as cosmic tracer of Flavor physics

We unify one of the most widely studied frameworks to explain the hierarchical structure of the flavor sector in the Standard Model, the Froggatt-Nielsen mechanism, with cosmic inflation. We propose that the complex scalar field, the so-called flavon, which breaks the Froggatt-Nielsen $U(1)$ symmetry and generates the Yukawa couplings of the Standard Model, to also drive inflation, which we dub as Inflavon. After inflation ends, the decay of the inflavon reheats the Universe, establishing a novel link between early Universe cosmology and flavor physics. As concrete examples, we present realizations where the inflavon potential is described by an $α$-attractor potential. We then compute the resulting CMB observables, specifically the spectral index ($n_s$), the tensor-to-scalar ratio ($r$), and the amplitude of scalar perturbations ($A_s$) as functions of the underlying Froggatt-Nielsen model parameters. We identify the parameter space in Froggatt-Nielsen models involving the scale of Flavor symmetry breaking $Λ_{\rm FN}$ and FN charges which is ruled out by Planck and ACT data, as well as the region that could be probed by next-generation CMB experiments like CMB-S4, SO and LiteBIRD. We also discuss inflavon as dark matter and its isocurvature constraints.

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arXiv2026

Modular $S_4$ and $A_4$ Symmetries and Their Fixed Points: New Predictive Examples of Lepton Mixing

In the modular symmetry approach to neutrino models, the flavour symmetry emerges as a finite subgroup $Γ_N$ of the modular symmetry, broken by the vacuum expectation value (VEV) of a modulus field $τ$. If the VEV of the modulus $τ$ takes some special value, a residual subgroup of $Γ_N$ would be preserved. We derive the fixed points $τ_S=i$, $τ_{ST}=(-1+i\sqrt{3})/2$, $τ_{TS}=(1+i\sqrt{3})/2$, $τ_T=i\infty$ in the fundamental domain which are invariant under the modular transformations indicated. We then generalise these fixed points to $τ_f=γτ_S$, $γτ_{ST}$, $γτ_{TS}$ and $γτ_{T}$ in the upper half complex plane, and show that it is sufficient to consider $γ\inΓ_{N}$. Focussing on level $N=4$, corresponding to the flavour group $S_4$, we consider all the resulting triplet modular forms at these fixed points up to weight 6. We then apply the results to lepton mixing, with different residual subgroups in the charged lepton sector and each of the right-handed neutrinos sectors. In the minimal case of two right-handed neutrinos, we find three phenomenologically viable cases in which the light neutrino mass matrix only depends on three free parameters, and the lepton mixing takes the trimaximal TM1 pattern for two examples. One of these cases corresponds to a new Littlest Modular Seesaw based on CSD$(n)$ with $n=1+\sqrt{6}\approx 3.45$, intermediate between CSD$(3)$ and CSD$(4)$. Finally, we generalize the results to examples with three right-handed neutrinos, also considering the level $N=3$ case, corresponding to $A_4$ flavour symmetry.

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arXiv2025

Demystifying stringy miracles with eclectic flavor symmetries

Effective field theories arising from string compactifications are subject to constraints originating from the duality transformations of string theory. Interpreting these so-called selection rules in terms of conventional symmetries has remained challenging. We show that particular selection rules in heterotic orbifolds can be explained from a subtle interplay between modular and traditional flavor symmetries within the eclectic flavor framework.

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arXiv2025

Modular Zeros

Modular symmetries are known to be powerful and have various remarkable properties. We point out that the structure of vector-valued modular forms (VVMFs) space leads to the absence of couplings which cannot be explained in terms of the usual symmetries. These modular zeros, which correspond to gaps in spaces of VVMFs, have the power of explaining certain stringy zeros, and to explain the renowned Weinberg texture that relates the Cabibbo angle to the hierarchies of the light down and strange quarks.

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arXiv2025

Modular Flavor Symmetries and Fermion Mass Hierarchies

We investigate fermion mass hierarchies in models with modular flavor symmetries. Several key conclusions arise from the observation that the determinants of mass matrices transform as 1-dimensional vector-valued modular forms. We demonstrate that, under some fairly general assumptions, achieving hierarchical fermion masses requires the vacuum expectation value of the modulus $τ$ to be located near one of the critical points, $i$, $i\infty$, or $ω$. We also revisit the universal near-critical behavior around these points and classify the resulting mass hierarchies for the critical points $i$ and $ω$. We compare the traditional Froggatt--Nielsen mechanism with its modular variant. The knowledge and boundedness of Fourier and Taylor coefficients are crucial to the predictive power of modular flavor symmetries.

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arXiv2025

Flavor Symmetries and Winding Modes

Modular flavor symmetries have been proposed as a new way to address the flavor problem. It is known that they can emerge from string compactifications. We discuss this connection in detail, and show how the congruence subgroups of SL(2,Z), which underlie many modular flavor symmetries, emerge from stringy duality symmetries by orbifolding. This requires an analysis of massive states, which reveals a picture that is more intricate than the well-known situation on the torus. It involves towers of states of different quantum numbers, related by modular transformations. Members of different towers become massless at different points in moduli space. We also show that, at least in the Z_3 orbifold, the string selection rules can be understood as discrete remnants of continuous gauge symmetries. Non-Abelian discrete flavor symmetries arise as relics of various, relatively misaligned, continuous Abelian gauge symmetries. The generators of these U(1) symmetries give rise to CP-violating Clebsch-Gordan coefficients. If the modulus settles close to a critical point, the corresponding gauge bosons may be light enough to be searched for at future colliders.

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arXiv2025

Multiple realizations of modular flavor symmetries and their phenomenology

We point out that specifying the finite modular group does not uniquely fix a modular flavor symmetry. We illustrate this using the finite modular group $T'$. Otherwise equivalent models based on different $T'$ lead to modular forms with different properties and, hence, produce different phenomenological features. We exemplify this in various scenarios, and show that the ability of a given model to accommodate mass and other observed hierarchies depends sensitively on the way the $T'$ is implemented.

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arXiv2024

Modular binary octahedral symmetry for flavor structure of Standard Model

We have investigated the modular binary octahedral group $2O$ as a flavor symmetry to explain the structure of Standard Model. The vector-valued modular forms in all irreducible representations of this group are constructed. We have classified all possible fermion masses models based on the modular binary octahedral group $2O$. A comprehensive numerical analysis is performed, and we present some benchmark quark/lepton masses models in well agreement with the experimental data. Notably we find a minimal modular invariant model for leptons and quarks, which is able to explain simultaneously the masses and mixing parameters of both quarks and leptons in terms of 14 real free parameters including the modulus $τ$. The fermion mass hierarchies around the vicinity of the modular fixed points are explored.

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arXiv2024

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 2D_3. After building the vector-valued modular forms in the representations of 2D_3 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 2D_3, 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 T2/Z4 heterotic orbifold compactifications.

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arXiv2024

Universal predictions of Siegel modular invariant theories near the fixed points

We analyze a general class of locally supersymmetric, CP and modular invariant models of lepton masses depending on two complex moduli taking values in the vicinity of a fixed point, where the theory enjoys a residual symmetry under a finite group. Like in models that depend on a single modulus, we find that all physical quantities exhibit a universal scaling with the distance from the fixed point. There is no dependence on the level of the construction, the weights of matter multiplets and their representations, with the only restriction that electroweak lepton doublets transform as irreducible triplets of the finite modular group. Also the form of the kinetic terms, which here are assumed to be neither minimal nor flavor blind, is irrelevant to the outcome. The result is remarkably simple and the whole class of examined theories gives rise to five independent patterns of neutrino mass matrices. Only in one of them, the predicted scaling agrees with the observed neutrino mass ratios and lepton mixing angles, exactly as in single modulus theories living close to $τ=i$.

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arXiv2024

Modular invariant holomorphic observables

In modular invariant models of flavor, observables must be modular invariant. The observables discussed so far in the literature are functions of the modulus $τ$ and its conjugate, $\barτ$. We point out that certain combinations of observables depend only on $τ$, i.e. are meromorphic, and in some cases even holomorphic functions of $τ$. These functions, which we dub ``invariants'' in this Letter, are highly constrained, renormalization group invariant, and allow us to derive many of the models' features without the need for extensive parameter scans. We illustrate the robustness of these invariants in two existing models in the literature based on modular symmetries, $Γ_{3}$ and $Γ_{5}$. We find that, in some cases, the invariants give rise to robust relations among physical observables that are independent of $τ$. Furthermore, there are instances where additional symmetries exist among the invariants. These symmetries are relevant phenomenologically and may provide a dynamical way to realize symmetries of mass matrices.

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arXiv2023

Matter matters in moduli fixing and modular flavor symmetries

Modular flavor symmetries provide us with a very compelling approach to the flavor problem. It has been argued that moduli values close to some special values like $τ=i$ or $τ=ω$ provide us with the best fits to data. We point out that the presence of hidden "matter" fields, needed to uplift symmetric AdS vacua, gives rise to a dynamical mechanism that leads to such values of $τ$.

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arXiv2023

Neutrino Mass and Mixing Models with Eclectic Flavor Symmetry $Δ(27) \rtimes T'$

The Kähler potentials of modular symmetry models receive unsuppressed contributions which may be controlled by a flavor symmetry, where the combination of the two symmetry types is referred to as eclectic flavor symmetry. After briefly reviewing the consistency conditions of eclectic flavor symmetry models, including with generalised (g)CP, we perform a comprehensive bottom-up study of eclectic flavor symmetry models based on $Ω(1)\cong Δ(27)\rtimes T^\prime$, consisting of the flavor symmetry $Δ(27)$ in a semi-direct product with the modular symmetry $T^\prime$. The modular transformations of different $Δ(27)$ multiplets are given by solving the consistency condition. The eight nontrivial singlets of $Δ(27)$ are related by $T'$ modular symmetry, and they have to be present or absent simultaneously in any $Ω(1)$ model. The most general forms of the superpotential and Kähler potential invariant under $Ω(1)$ are discussed, and the corresponding fermion mass matrices are presented. Based on the eclectic flavor group $Ω(1)$, two concrete lepton models which can successfully describe the experimental data of lepton masses and mixing parameters are constructed. For the two models without gCP, all six mixing parameters vary in small regions. A nearly maximal atmospheric mixing angle $θ_{23}$ and Dirac CP phase $δ_{CP}$ are obtained in the first model. After considering the compatible gCP symmetry and the assumption of $\Re τ=0$ in the first model, the $μ-τ$ reflection symmetry is preserved in the charged lepton diagonal basis. As a consequence, the atmospheric mixing angle and Dirac CP phase are predicted to be maximal, and two Majorana CP phases are predicted to be $π$.

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arXiv2022

A minimal modular invariant neutrino model

We present a neutrino mass model based on modular symmetry with the fewest input parameters to date, which successfully accounts for the 12 lepton masses and mixing parameters through 6 real free parameters including the modulus. The neutrino masses are predicted to be normal ordering, the atmospheric angle $θ_{23}$ is quite close to maximal value and the Dirac CP phase $δ_{CP}$ is about $1.34π$. We also study the soft supersymmetry breaking terms due to the modulus $F$-term in this minimal model, which are constrained to be the non-holomorphic modular forms. The radiative lepton flavor violation process $μ\to eγ$ is discussed.

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arXiv2021

Modular flavor symmetry and vector-valued modular forms

We revisit the modular flavor symmetry from a more general perspective. The scalar modular forms of principal congruence subgroups are extended to the vector-valued modular forms, then we have more possible finite modular groups including $Γ_N$ and $Γ'_N$ as the flavor symmetry. The theory of vector-valued modular forms provide a method of differential equation to construct the modular multiplets, and it also reveals the simple structure of the modular invariant mass models. We review the theory of vector-valued modular forms and give general results for the lower dimensional vector-valued modular forms. The general finite modular groups are listed up to order 72. We apply the formalism to construct two new lepton mass models based on the finite modular groups $A_4\times Z_2$ and $GL(2,3)$.

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