Renormalization of the 2PI Hartree-Fock approximation on de Sitter background in the broken phase

The infrared effects for light minimally coupled scalar fields with quartic self-interaction in de Sitter space is investigated using the 2PI effective action formalism. This formalism partially resums infinite series of loop diagrams, and enables us to circumvent the IR divergence problem for a massless minimally coupled scalar field in de Sitter space. It is anticipated that nonperturbative infrared effects generate a curvature-induced mass and self-regulate the IR divergence. However, due to its nonperturbative nature, the renormalization prescription is a nontrivial task. To calculate physical quantities, an appropriate renormalization prescription is required. In this paper, we will show that the MS-like scheme is possible at the Hartree-Fock truncation of the 2PI effective action, and infinite series of divergent terms are needed as counterterms. The phase structure and the quantum backreaction to Einstein's field equation are calculated.Comment: 25 pages, 2 figures, title changed, section 1 and 6 revised, typos corrected, results not change

On uniform lower bound of the Galois images associated to elliptic curves

Let p be a prime and K be a number field. Let rho_{E,p}:G_K \longrightarrow Aut(T_p E)\cong GL_2(Z_p) be the Galois representation given by the Galois action on the p-adic Tate module of an elliptic curve E over K. Serre showed that the image of rho_{E,p} is open if E has no complex multiplication. For an elliptic curve E over K whose j-invariant does not appear in an exceptional finite set, we give an explicit uniform lower bound of the size of the image of rho_{E,p}.Comment: 41 page

On Loops in the Hyperbolic Locus of the Complex H\'enon Map and Their Monodromies

We prove John Hubbard's conjecture on the topological complexity of the hyperbolic horseshoe locus of the complex H\'enon map. Indeed, we show that there exist several non-trivial loops in the locus which generate infinitely many mutually different monodromies. Our main tool is a rigorous computational algorithm for verifying the uniform hyperbolicity of chain recurrent sets. In addition, we show that the dynamics of the real H\'enon map is completely determined by the monodromy of a certain loop, providing the parameter of the map is contained in the hyperbolic horseshoe locus of the complex H\'enon map.Comment: 17 pages, 9 figures. For supplemental materials, see http://www.math.kyoto-u.ac.jp/~arai

Mathematical Analysis of a Generalized Chiral Quark Soliton Model

A generalized version of the so-called chiral quark soliton model (CQSM) in nuclear physics is introduced. The Hamiltonian of the generalized CQSM is given by a Dirac type operator with a mass term being an operator-valued function. Some mathematically rigorous results on the model are reported. The subjects included are: (i) supersymmetric structure; (ii) spectral properties; (iii) symmetry reduction; (iv) a unitarily equivalent model.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA