We study the moduli stabilization by the radiative corrections due to the moduli dependent vector-like masses invariant under the finite modular symmetry. The radiative stabilization mechanism can stabilize the modulus $\textit{τ}$ of the finite modular symmetry Γ ($\textit{N}$ ∈ ℕ) at Im $\textit{τ}$ ≫ 1, where the shift symmetry $\textit{τ → τ}$ + 1 remains unbroken approximately. The shift symmetry can be considered as the residual ℤ symmetry which realizes the Froggatt-Nielsen mechanism with the hierarchy parameter $\textit{e}$ ≪ 1. In this work, we study the stabilization of multiple moduli fields, so that various hierarchical values of the modular forms coexist in a model. For example, one modulus stabilized at Im $\textit{τ}$ ∼ 3 is responsible for the hierarchical structure of the quarks and leptons in the Standard Model, and another modulus stabilized at Im $\textit{τ}$ ∼ 15 can account for the flatness of the Re $\textit{τ}$ direction which may be identified as the QCD axion.

Japan

Tetsutaro Higaki

South Korea

Junichiro Kawamura

Tatsuo Kobayashi

Kaito Nasu

Riku Sakuma

Large and small hierarchies from finite modular symmetries

We study the vacua of modulus potentials consisting of multiple contributions of modular forms in a finite modular symmetry. If the potential is given by a single modular form, the Minkowski vacuum is realized at the fixed point of the modular symmetry. We show that the de Sitter vacuum is realized with a multiple modular form case and obtain a nontrivial vacuum away from the fixed point, i.e. a large modulus vacuum expectation value, depending on the choice of the weight and representation of the modular forms. We study these vacua numerically and analytically. It is also found that the vacua obtained in this paper preserve CP symmetry.

United States

Yoshihiko Abe

Komei Goto

Modulus Stabilization in Finite Modular Symmetric Models

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.

Hajime Otsuka

Spontaneous CP violation and partially broken modular flavor symmetries

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 e 2πiτ  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∞.

Finite modular axion and radiative moduli stabilization

We propose models to explain the hierarchies of the quark masses and mixing by utilizing the  $$S_4^\prime $$  <math> <msubsup> <mi>S</mi> <mn>4</mn> <mo>′</mo> </msubsup> </math>   modular flavor symmetry. The hierarchy is realized by the modulus  $$\tau $$  <math> <mi>τ</mi> </math>   stabilized at  $$\textrm{Im}\,\tau \gg 1$$  <math> <mrow> <mtext>Im</mtext> <mspace width="0.166667em"></mspace> <mi>τ</mi> <mo>≫</mo> <mn>1</mn> </mrow> </math>  , where the residual  $$\mathbb {Z}_4^T$$  <math> <msubsup> <mi>Z</mi> <mn>4</mn> <mi>T</mi> </msubsup> </math>   symmetry is approximately unbroken and the Froggatt–Nielsen mechanism works. It is found that the quark hierarchies are realized only in a few cases of quark representations. We study two models with assigning the modular weights, so that the observed quark hierarchies are explained in the cases of both small and large ratios of the top to bottom Yukawa couplings. We also argue that  $$\mathcal {O}\left( {0.1}\right) $$  <math> <mrow> <mi>O</mi> <mfenced> <mrow> <mn>0.1</mn> </mrow> </mfenced> </mrow> </math>   hierarchies of the  $$\mathcal {O}\left( {1}\right) $$  <math> <mrow> <mi>O</mi> <mfenced> <mn>1</mn> </mfenced> </mrow> </math>   coefficients and the spontaneous CP violation can be realized by imposing another  $$S_3$$  <math> <msub> <mi>S</mi> <mn>3</mn> </msub> </math>   modular symmetry.

Quark masses and CKM hierarchies from  $$S_4'$$  <math> <msubsup> <mi>S</mi> <mn>4</mn> <mo>′</mo> </msubsup> </math>   modular flavor symmetry

We construct an SU(5) grand unified model in which the hierarchies of the quark and lepton masses and mixing are explained by the   <math> <msubsup> <mi>Γ</mi> <mn>6</mn> <mo>′</mo> </msubsup> </math>  $$ {\Gamma}_6^{\prime } $$  modular flavor symmetry. The hierarchies are realized by the Froggatt-Nielsen-like mechanism due to the residual   <math> <msubsup> <mi>Z</mi> <mn>6</mn> <mi>T</mi> </msubsup> </math>  $$ {Z}_6^T $$  symmetry, approximately unbroken at τ ~ i∞. We argue that the   <math> <msubsup> <mi>Γ</mi> <mn>6</mn> <mfenced> <msup> <mrow></mrow> <mo>′</mo> </msup> </mfenced> </msubsup> </math>  $$ {\Gamma}_6^{\left({}^{\prime}\right)} $$  symmetry is the minimal possibility to realize the up-type quark mass hierarchies, since the Yukawa matrix is symmetric. We find a combination of the representations and modular weights and then show numerical values of (1) coefficients for the realistic fermion hierarchies.

Fermion hierarchies in SU(5) grand unification from   <math> <msubsup> <mi>Γ</mi> <mn>6</mn> <mo>′</mo> </msubsup> </math>  $$ {\Gamma}_6^{\prime } $$  modular flavor symmetry

We study slow-roll inflation model controlled by the modular flavor symmetry. In the model, the modulus field plays a role of inflaton and the introduction of the stabilizer field coupled to a modular form in the superpotential produces the inflaton potential. In order to generate the flat direction for the slow-roll inflation, we consider the Kähler potential corrected by the modular form. It is noted that the modulus field perpendicular to the inflaton direction is stabilized during the inflation. The model turns out to be consistent with the current observations and behaves similarly to the α-attractor models in some parameter spaces. The inflaton rolls down to the CP-symmetric vacuum at the end of inflation.

Fumiya Kaneko

Moduli inflation from modular flavor symmetries

We propose models in which the hierarchical structures of the masses and mixing in both quark and lepton sectors are explained by the <math><msubsup><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math> modular flavor symmetry near the fixed point <math><mi>τ</mi><mo>∼</mo><mi>i</mi><mo>∞</mo></math>. The model provides the first explicit example which explains hierarchies of both quarks and leptons from single modular flavor symmetry. The hierarchies are realized by powers of <math><mi>ϵ</mi><mo>=</mo><msup><mrow><mi>e</mi></mrow><mrow><mn>2</mn><mi>π</mi><mi>i</mi><mi>τ</mi><mo>/</mo><mn>4</mn></mrow></msup><mo>=</mo><mi>O</mi><mo>(</mo><mn>0.01</mn><mo>)</mo></math> and <math><mn>2</mn><mspace width="0.2em"></mspace><mrow><mi>Im</mi></mrow><mspace width="0.2em"></mspace><mi>τ</mi><mo>∼</mo><mn>5</mn></math>, where τ being the modulus. The small parameter ϵ plays a role of flavon in the Froggatt-Nielsen mechanism under the residual <math><msubsup><mrow><mi>Z</mi></mrow><mrow><mn>4</mn></mrow><mrow><mi>T</mi></mrow></msubsup></math> symmetry, and powers of <math><mn>2</mn><mspace width="0.2em"></mspace><mrow><mi>Im</mi></mrow><mspace width="0.2em"></mspace><mi>τ</mi></math> in the Yukawa couplings are controlled by modular weights via the canonical normalization. The doublet quarks are identified to a <math><msubsup><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math> triplet to explain the hierarchical structure of the quark mixing angles, while the doublet leptons are composed of three singlets for the large mixing angles in the lepton sector. We show that the <math><msubsup><mrow><mi>S</mi></mrow><mrow><mn>4</mn></mrow><mrow><mo>′</mo></mrow></msubsup></math> modular symmetry alone can explain the hierarchies in both quark and lepton sectors by <math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> coefficients.

Quark and lepton hierarchies from Modular Flavor Symmetry

The chiral soliton lattice (CSL) is a lattice structure composed of domain walls aligned in parallel at equal intervals, which is energetically stable in the presence of a background magnetic field and a finite (baryon) chemical potential due to the topological term originated from the chiral anomaly. We study its formation from the vacuum state, with describing the CSL as a layer of domain-wall disks surrounded by the vortex or string loop, based on the Nambu-Goto-type effective theory. We show that the domain wall nucleates via quantum tunneling when the magnetic field is strong enough. We evaluate its nucleation rate and determine the critical magnetic field strength with which the nucleation rate is no longer exponentially suppressed. We apply this analysis to the neutral pion in the two-flavor QCD as well as the axionlike particles (ALPs) with a finite (baryon) chemical potential under an external magnetic field. In the former case, even though the CSL state is more energetically stable than the vacuum state and the nucleation rate becomes larger for a sufficiently strong magnetic field, it cannot be large enough so that the nucleation of the domain walls is not exponentially suppressed and promoted, without suffering from the tachyonic instability of the charged pion fluctuations. In the latter case, we confirm that the effective interaction of the ALPs generically includes the topological term required for the CSL state to be energetically favored. We show that the ALP CSL formation is promoted if the magnetic field strength and the chemical potential of the system is slightly larger than the scale of the axion decay constant.

Kohei Kamada

Kentaro Nishimura

Formation of a chiral soliton lattice

We study a systematic derivation of four-dimensional <math><mi>N</mi><mo>=</mo><mn>1</mn></math> supersymmetric effective theory from a ten-dimensional non-Abelian Dirac-Born-Infeld action compactified on a six-dimensional torus with magnetic fluxes on the D-branes. We find a new type of matter Kähler metric, while gauge kinetic function and superpotential are consistent with previous studies. For the ten-dimensional action, we use a symmetrized trace prescription and focus on the bosonic part up to <math><mi>O</mi><mo>(</mo><msup><mi>F</mi><mn>4</mn></msup><mo>)</mo></math>. In the presence of the supersymmetry, four-dimensional chiral fermions can be obtained via index theorem. The new matter Kähler metric is independent of flavor but depends on the fluxes, 4D dilaton, Kähler moduli, and complex structure moduli and will be always positive definite if an induced Ramond-Ramond charge of the D-branes on which matters are living are positive. We read the superpotential from an F-term scalar quartic interaction derived from the ten-dimensional action and the contribution of the new matter Kähler metric to the scalar potential which we derive turns out to be consistent with the supergravity formulation.