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

Japan

Komei Goto

Tetsutaro Higaki

Tatsuo Kobayashi

Kaito Nasu

Modulus Stabilization in Finite Modular Symmetric Models

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.

South Korea

Junichiro Kawamura

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

We study black hole extremality in nonlinear electrodynamics motivated by the Weak Gravity Conjecture (WGC) and the Festina Lente (FL) bound. For illustration, we consider the Euler-Heisenberg model and the Dirac-Born-Infeld model in asymptotically flat spacetime, de Sitter spacetime, and anti-de Sitter spacetime. We find that in all cases the extremal condition enjoys a certain monotonicity expected by the WGC. This provides evidence for the conjecture beyond the leading order corrections to the Einstein-Maxwell theory. We also study how light charged particles modify the mass-charge relation of Nariai black holes in de Sitter spacetime and discuss possible implications for the FL bound. Besides, we point out an interesting similarity between our black hole analysis and gravitational positivity bounds on scattering amplitudes.

Toshifumi Noumi

Kaho Yoshimura

Black hole extremality in nonlinear electrodynamics: a lesson for weak gravity and Festina Lente bounds

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

Hajime Otsuka

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

In this work, the current stability is discussed for cosmic strings with the bosonic superconductivity. A non-vanishing curvature of string generally induce the quantum instability of the current-carrying particle. Its decay rates are explored for various types of model parameters, curved string shapes, and decay processes. As a cosmological application, the stability is examined for superconducting strings in the string network and also for cosmic vortons by evaluating their cosmological evolution. The zero mode and hence the vorton cannot be stable in various cases, e.g., with a hierarchy between the current-carrying particle mass off the string and the string tension or with sizable couplings of the current-carrying particle to light species such as the Standard Model particles.

Yu Hamada

Kota Saji

Koichi Yoshioka

Quantum current dissipation in superconducting strings and vortons

We consider a situation where right-handed neutrinos couple to a light scalar which is possibly a Nambu-Goldstone boson resulting from high-energy symmetry breaking. Its coupling is typically complex-valued and flavor-dependent. In this work, we investigate the possibility of the leptonic asymmetry generation in the Universe from the right-handed neutrino decay to flavorful light scalar. Furthermore a new source of asymmetry generation from a single decay process is pointed out, which is characteristic of the present setting.

Toshimasa Ito

Leptonic CP asymmetry and Light flavored scalar

Abstract The gradient flow exact renormalization group (GFERG) is a framework for defining the Wilson action via a gradient flow equation. We study the fixed point structure of the GFERG equation associated with a general gradient flow equation for scalar field theories, and show that it is the same as that of the conventional Wilson–Polchinski (WP) equation in general. Furthermore, we see that the GFERG equation has a similar RG flow structure around a fixed point to the WP equation. We illustrate these results with the O(N) non-linear sigma model in 4 − ϵ dimensions and the Wilson–Fisher fixed point.

Junichi Haruna

Fixed point structure of the gradient flow exact renormalization group for scalar field theories Critical phenomena for field theorists

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.