Abstract We compute the one-loop vacuum energy in 10-dimensional (10D) super Yang–Mills theory compactified on  in the presence of the Abelian magnetic fluxes. Regularization of the infinite Kaluza–Klein (KK) sum is achieved by the use of Barnes -functions, which can be applied to the case in which the KK mass spectrum has the Landau level structure. We apply the technique to two different models of the 10D super Yang–Mills theory: One is to introduce the magnetic flux background to the 10D super Yang–Mills action. The other is to first embed the 10D super Yang–Mills action into 4D  superspace and to introduce magnetic fluxes. The two models built result in different KK mass spectra as well as tree level potentials, and we compute the one-loop vacuum energy of each case. Nevertheless, both of the KK mass spectra have the Landau level structure and we can apply the regularization method in the same way. It turns out that the former model shows UV finite one-loop vacuum energy whereas the latter shows UV divergent one-loop potential that vanish when 4D  supersymmetry is not broken by magnetic fluxes. We discuss the origin of the differences from a higher-dimensional-supersymmetry viewpoint.

Japan

Hiroyuki Abe

Akinari Koichi

Yusuke Yamada

One-Loop Vacuum Energy in 10D Super Yang–Mills Theory on Magnetized Tori with/without 4D = 1 Supersymmetric Completion

We discuss the effects of electric fields along compact directions within (supersymmetric) gauge theory in   <math> <msup> <mi>ℝ</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>3</mn> </mrow> </msup> <mo>×</mo> <msup> <mi>T</mi> <mn>2</mn> </msup> <mfenced> <mrow> <mo>×</mo> <msup> <mi>T</mi> <mn>2</mn> </msup> <mo>×</mo> <msup> <mi>T</mi> <mn>2</mn> </msup> </mrow> </mfenced> </math>  $$ {\mathbb{R}}^{1,3}\times {\mathbbm{T}}^2\left(\times {\mathbbm{T}}^2\times {\mathbbm{T}}^2\right) $$ . The electric field along compact directions is equivalent to time-dependent homogeneous configuration of Wilson line moduli, which would be relevant to physics in the early universe. In particular, we consider models with and without background magnetic fluxes, which lead to completely different effects of the electric field due to the difference in the Kaluza-Klein (KK) level structure. We show that, in the case without magnetic fluxes, the deceleration of KK momenta may cause non-perturbative KK particle production dubbed as the KK Schwinger effect, whereas in the case with magnetic flux such KK particle production does not take place but flavor structure of low energy effective theory may be affected.

Roles of electric field/time-dependent Wilson line in toroidal compactification with or without magnetic fluxes

We pursue the time evolution of the domain walls in 5D gravitational theory with a compact extra dimension by numerical calculation. In order to avoid a kink-antikink pair that decays into the vacuum, we introduce a topological winding in the field space. In contrast to the case of non-gravitational theories, there is no static domain-wall solution in the setup. In the case that the minimal value of the potential is non-negative, we find that both the 3D space and the extra dimension will expand at late times if the initial value of the Hubble parameter is chosen as positive. The wall width almost remains constant during the evolution. In other cases, the extra dimension diverges and the 3D space shrinks to zero at a finite time.

Shuntaro Aoki

Yu Asai

Yutaka Sakamura

Fate of domain walls in 5D gravitational theory with compact extra dimension

We study Fayet-Iliopoulos (FI) terms of six-dimensional supersymmetric Abelian gauge theory compactified on a <math><msup><mi>T</mi><mn>2</mn></msup><mo>/</mo><msub><mi>Z</mi><mn>2</mn></msub></math> orbifold. Such orbifold compactifications can lead to localized FI-terms and instability of bulk zero modes. We study 1-loop correction to FI-terms in more general geometry than the previous works. We find induced FI-terms depend on the complex structure of the compact space. We also find the complex structure of the torus can be stabilized at a specific value corresponding to a self-consistent supersymmetric minimum of the potential by such 1-loop corrections, which is applicable to the modulus stabilization.

Tatsuo Kobayashi

Shohei Uemura

Junji Yamamoto

Loop Fayet-Iliopoulos terms in <math><msup><mi>T</mi><mn>2</mn></msup><mo>/</mo><msub><mi>Z</mi><mn>2</mn></msub></math> models: Instability and moduli stabilization

We study the supersymmetry breaking patterns in four-dimensional   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 gauged supergravity. The model contains multiple (Abelian) vector multiplets and a single hypermultiplet which parametrizes SO(4, 1)/SO(4) coset. We derive the expressions of two gravitino masses under general gaugings and prepotential based on the embedding tensor formalism, and discuss their behaviors in some concrete models. Then we confirm that in a single vector multiplet case, the partial breaking always occurs when the third derivative of the prepotential exists at the vacuum, which is consistent with the result of ref. [1], but we can have several breaking patterns otherwise. The discussion is also generalized to the case of multiple vector multiplets, and we found that the full (  <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 0) breaking occurs even if the third derivative of the prepotential is nontrivial.

Sosuke Imai

Behaviors of two supersymmetry breaking scales in   <math> <mi>N</mi> </math>  $$ \mathcal{N} $$  = 2 supergravity

We study Fayet-Iliopoulos (FI) terms of 5-dimensional supersymmetric <math><mi>U</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> gauge theory compactified on <math><msup><mi>S</mi><mn>1</mn></msup><mo>/</mo><msub><mi>Z</mi><mn>2</mn></msub></math>. In this model, loop diagrams including matter hypermultiplets and brane chiral multiplets induce FI-terms localized at the fixed points. Localized FI-terms lead instabilities of bulk modes. The form of the induced FI-terms strictly depends on wave function profiles of matter multiplets. It is a nontrivial question whether the vacuum of 1-loop corrected potential is stable under radiative corrections. We investigate this issue and it is found that the stable configuration is obtained when the bulk zero modes shield the brane charge completely.

Revisiting instabilities of <math><mrow><msup><mrow><mi>S</mi></mrow><mrow><mn>1</mn></mrow></msup><mo>/</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></math> models with loop-induced Fayet-Iliopoulos terms

We consider magnetized orbifolds, where the supersymmetric mass term for a pair of up- and down-type Higgs (super)fields, called μ-term, is localized at the orbifold fixed points, and study the effects on the zero-mode spectra. The zero-mode degeneracy to be identified as the generation in four-dimensional (4D) effective theories is determined by the magnetic fluxes. It is known that multiple Higgs zero-modes appear in general in magnetized orbifold models. We derive the analytic form of the μ-term matrix in the 4D effective theory generated by the localized sources on <math><msup><mrow><mi>T</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>/</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub></math> orbifold fixed points, and find that this matrix can lead to a distinctive pattern of the eigenvalues that yields hierarchical μ-terms for the multiple Higgs fields. The lightest ones can be exponentially suppressed due to the localized wavefunctions of zero-modes determined by the fluxes, while the others are of the order of the compactification scale, which can provide a dynamical origin of the electroweak scale as well as a simultaneous decoupling of extra Higgs fields. We also show that a certain linear combination of the lightest Higgs fields could generate the observed mass ratios of down-type quarks through their Yukawa couplings determined by the wavefunctions.

Makoto Ishida

Germany

Yoshiyuki Tatsuta

Effects of localized -terms at the Fixed Points in Magnetized Orbifold Models

We discuss an <math><mi>N</mi><mo>=</mo><mn>2</mn></math> supergravity model that interpolates the full and the partial supersymmetry breakings. In particular, we find the conditions for an <math><mi>N</mi><mo>=</mo><mn>0</mn></math> Minkowski vacuum, which is continuously connected to the partial-breaking (<math><mi>N</mi><mo>=</mo><mn>1</mn></math> preserving) one. The model contains multiple (Abelian) vector multiplets and a single hypermultiplet, and is constructed by employing the embedding tensor technique. We compute the mass spectrum on the Minkowski vacuum, and find some non-trivial mass relations among the massive fields. Our model allows us to choose the two supersymmetry-breaking scales independently, and to discuss the cascade supersymmetry breaking for the applications to particle phenomenology and cosmology.

Interpolation of partial and full supersymmetry breakings in Supergravity

We study magnetized orbifold models. We assume the localized Fayet-Iliopoulos terms and the corresponding gauge background. Such terms lead to the strong localization of zero-mode wave functions. In this setup, we compute quark mass matrices.

Shintaro Takada

Shio Tamba

Takuya H. Tatsuishi

Quark mass matrices in magnetized orbifold models with localized Fayet-Iliopoulos terms

We propose a four-dimensional N = 1 supergravity-based Starobinsky-type inflationary model in terms of a single massive vector multiplet, whose action includes the Dirac-Born-Infeld-type kinetic terms and a generalized (new) Fayet-Iliopoulos-type term, without gauging the R-symmetry. The bosonic action and the scalar potential are computed. The inflaton is the superpartner of the Goldstino in our model, and supersymmetry is spontaneously broken after inflation by the D-type mechanism, whose scale is related to the value of the cosmological constant.