Toponogov comparison theorem for open triangles

Dedicated to Professor Gromoll: The aim of our article is to generalize the Toponogov comparison theorem to a complete Riemannian manifold with smooth convex boundary. A geodesic triangle will be replaced by an open (geodesic) triangle standing on the boundary of the manifold, and a model surface will be replaced by the universal covering surface of a cylinder of revolution with totally geodesic boundary. Applications of our theorem are found in our article "Applications of Toponogov's comparison theorems for open triangles" (arXiv:1102.4156).Comment: This version 4 (36 pages, no figures) is a version to appear in Tohoku Mathematical Journa

Probing new intra-atomic force with isotope shifts

In the development of atomic clocks, some atomic transition frequencies are measured with remarkable precision. These measured spectra may include effects of a new force mediated by a weakly interacting boson. Such effects might be distilled out from possible violation of a linear relation in isotope shifts between two transitions, as known as King's linearity, with relatively suppressed theoretical uncertainties. We discuss the experimental sensitivity to a new force in the test of the linearity as well as the linearity violation owing to higher order effects within the Standard Model. The sensitivity to new physics is limited by such effects. We have found that for Yb$^+$, the higher order effect is in the reach of future experiments. The sensitivity to a heavy mediator is also discussed. It is analytically clarified that the sensitivity becomes weaker than that in the literature. Our numerical results of the sensitivity are compared with other weak force search experiments.Comment: 16 pages, 3 figures; published versio

Collider signatures of the SO(5) x U(1) gauge-Higgs unification

Collider signatures of the $SO(5) \times U(1)$ gauge-Higgs unification model in the Randall-Sundrum warped space are explored. Gauge couplings of quarks and leptons receive small corrections from the fifth dimension whose effects are tested by the precision data. It is found that the forward-backward asymmetries in $e^+e^-$ collisions on the $Z$ pole are well explained in a wide range of the warp factor $z_L$, but the model is consistent with the branching fractions of $Z$ decay only for large z_L \simg 10^{15}. Kaluza-Klein (KK) spectra of gauge bosons, quarks, and leptons as well as gauge and Higgs couplings of low-lying KK excited states are determined. Right-handed quarks and leptons have larger couplings to the KK gauge bosons than left-handed ones. Production rates of Higgs bosons and KK states at Tevatron, LHC and ILC are evaluated. The first KK $Z$ has a mass 1130 GeV with a width 422 GeV for $z_L = 10^{15}$. The current limit on the $Z'$ production at Tevatron and LHC indicates $z_L >10^{15}$. A large effect of parity violation appears in the difference between the rapidity distributions of $e^+$ and $e^-$ in the decay of the first KK $Z$. The first KK gauge bosons decay into light and heavy quarks evenly.Comment: 45 pages, 6 figures. Contributions from the first KK photon are taken into accoun