Next-to-leading order short distance QCD corrections to the effective ΔS=2\Delta S = 2 Hamiltonian, implications for the KLK_L-KSK_S mass difference

We report on the results of a calculation of next-to leading order short distance QCD corrections to the coefficient $\eta_1$ of the effective $\Delta S = 2$ Lagrangian in the standard model and discuss the uncertainties inherent in such a calculation. As a phenomenological application we comment on the contributions of short distance physics to the ${\rm K}_{\rm L}$--${\rm K}_{\rm S}$ mass difference. This report is based on research work done in collaboration with Ulrich Nierste.Comment: 4 pages and 2 figures, uses LaTeX espcrc2 documentstyle option (all necessary files included in an uuencoded compressed tar file). Invited talk at the conference 'QCD94', Montpellier, France, 7-13 July 1994, to appear in the proceedings. TUM-T31-72/9

On some peculiar aspects of the constructive theory of point-free spaces

This paper presents several independence results concerning the topos-valid and the intuitionistic (generalized) predicative theories of locales. In particular, certain consequences of the consistency of a general form of Troelstra's uniformity principle with constructive set theory and type theory are examined

Expressive Power in First Order Topology

A first order representation (fo.r.) in topology is an assignment of finitary relational structures of the same type to topological spaces in such a way that homeomorphic spaces get sent to isomorphic structures. We first define the notions one f.o.r. is at least as expressive as another relative to a class of spaces and one class of spaces is definable in another relative to an f.o.r. , and prove some general statements. Following this we compare some well-known classes of spaces and first order representations. A principal result is that if X and Y are two Tichonov spaces whose posets of zero-sets are elementarily equivalent then their respective rings of bounded continuous real-valued functions satisfy the same positiveuniversal sentences. The proof of this uses the technique of constructing ultraproducts as direct limits of products in a category theoretic setting

The Complete |Delta S|=2 Hamiltonian in the Next-To-Leading Order

We present the complete next-to-leading order short-distance QCD corrections to the effective \dstwo -hamiltonian in the Standard Model. The calculation of the coefficient $\eta_3$ is described in great detail. It involves the two-loop mixing of bilocal structures composed of two \dsone\ operators into \dstwo\ operators. The next-to-leading order corrections enhance $\eta_3$ by 27\% to \eta_3=0.47 \errorpm{+0.03}{-0.04} thereby affecting the phenomenology of $\epsilon_K$ sizeably. $\eta_3$ depends on the physical input parameters $m_t$, $m_c$ and \laMSb only weakly. The quoted error stems from renormalization scale dependences, which have reduced compared to the old leading log result. The known calculation of $\eta_1$ and $\eta_2$ is repeated in order to compare the structure of the three QCD coefficients. We further discuss some field theoretical aspects of the calculation such as the renormalization group equation for Green's functions with two operator insertions and the renormalization scheme dependence caused by the presence of evanescent operators.Comment: 68 pages, requires LaTeX2e and the standard LaTeX-packages epsf.sty, rotate.sty, a4.sty, subeqn.sty, cite.sty, array.sty, dcolumn.sty; Figures are submitted as a seperate tar.gz-file. A complete PostScript version may be obtained from ftp://feynman.t30.physik.tu-muenchen.de/pub/preprints/tum-86-96.ps.gz or ftp://feynman.t30.physik.tu-muenchen.de/pub/preprints/tum-86-96.ps2.gz (scaled down and rotated version to print two pages on one sheet of paper). Source available at ftp://feynman.t30.physik.tu-muenchen.de/pub/preprints/tum-86-96.tar.g

Estimates of Long-distance Contributions to the Bs→γγB_s \to \gamma \gamma Decay

We present first calculations of new long-distance contributions to $B_s \to \gamma \gamma$ decay due to intermediate $D_s$ and $D_s^*$ meson states. The relevant $\gamma$ vertices are estimated using charge couplings and transition moment couplings. Within our uncertainties, we find that these long-distance contributions could be comparable to the known short-distance contributions. Since they have different Cabibbo-Kobayashi-Maskawa matrix-element factors, there may be an interesting possibility of observing CP violation in this decay.Comment: 9 pages, 2 figures (embedded), all style files include

Choice principles in elementary topology and analysis

summary:Many fundamental mathematical results fail in {\bf{ZF}}, i.e., in Zermelo-Fraenkel set theory without the Axiom of Choice. This article surveys results --- old and new --- that specify how much ``choice'' is needed {\it precisely} to validate each of certain basic analytical and topological results