DeltaGamma_s Measurement at the Upsilon(5S) from Belle

Using the full Belle Upsilon(5S) data sample of 121 fb^-1we have measured exclusive branching fractions for the decays B_s^0 -> D_s^(*)+D_s^(*)-. Assuming these decay modes saturate decays to CP-even final states, the branching fraction determines the relative width difference between the CP-odd and -even eigenstates of the B_s.Comment: Proceedings of CKM 2012, the 7th International Workshop on the CKM Unitarity Triangle, University of Cincinnati, USA, 28 September - 2 October 201

Observation of Bs0→Ds∗−π+B_s^{0} \to D_s^{*-} \pi^{+}, Bs0→Ds(∗)−ρ+B_s^{0} \to D_s^{(*)-} \rho^{+} and Bs0→Ds(∗)+Ds(∗)−B_s^{0} \to D_s^{(*)+} D_s^{(*)-} and Estimate of ΔΓCP\Delta \Gamma_{CP} at Belle

The large data sample being recorded with the Belle detector at the $\Upsilon$(5S) energy provides a unique opportunity to study the less-well-known $B_s^{0}$ meson decays. Following our recent measurement of $B_s^{0}\to D_s^{-}\pi^{+}$ in a sample of 23.6 fb$^{-1}$, we extend the analysis to include decays with photons in the final state. Using the same sample, we report the first observation of three other dominant exclusive $B_s^{0}$ decays, in the modes $B_s^{0}\to D_s^{*-}\pi^+$, $B_s^{0}\to D_s^{-}\rho^+$ and $B_s^{0}\to D_s^{*-}\rho^+$. We measure their respective branching fractions and, using helicity-angle distributions, the longitudinal polarization fraction of the $B_s^{0}\to D_s^{*-}\rho^+$ decay. We also present a measurement of the branching fractions for the decays $B_s^{0} \to D_s^{(*)+}D_s^{(*)-}$. In the heavy quark limit, this branching fraction is directly related to the width difference between the $B_s$ CP-even and CP-odd eigenstates.Comment: 4 pages, 2 figures, To appear in the proceedings of the "35th International Conference On High Energy Physics: ICHEP 2010", 21-28 July 2010, Paris, Franc

Updated Measurement of B(Bs→Ds(∗)+Ds(∗)−){\cal B}(B_s \to D_s^{(*)+}D_s^{(*)-}) and Determination of ΔΓs\Delta \Gamma_{s}

Using fully reconstructed $B_{s}$ mesons, we measure exclusive branching fractions for the decays $B_s \to D_s^{(*)+}D_s^{(*)-}$. The results are {\cal B}(B^0_s\ra D^+_s D^-_s)=(0.58\,^{+0.11}_{-0.09}\,\pm 0.13)%, {\cal B}(B^0_s\ra D^{*\pm}_s D^{\mp}_s)=(1.8\, \pm 0.2\,\pm 0.4)%, and {\cal B}(B^0_s\ra D^{*+}_s D^{*-}_s)=(2.0\,\pm 0.3\,\pm 0.5)%; the sum is {\cal B}(B^0_s\ra D^{(*)+}_s D^{(*)-}_s)=(4.3\,\pm 0.4\,\pm 1.0)%. Assuming these decay modes saturate decays to CP-even final states, the branching fraction determines the relative width difference between the $B_s$ CP-odd and CP-even eigenstates. Taking \cp\ violation to be negligibly small, we obtain \dgs/\gs = 0.090\,\pm 0.009\,{\rm(stat.)}\,\pm 0.022 \,{\rm (syst.)}, where \gs is the mean decay width. The results are based on a data sample collected with the Belle detector at the KEKB $e^+ e^-$ collider running at the $\Upsilon(5S)$ resonance with an integrated luminosity of 121.4 fb$^{-1}$.Comment: 5 pages, 2 figures, 3 tables, for the proceedings of the DPF-2011 conference, Providence, RI, August 9-13, 201

Sum-Product Type Estimates over Finite Fields

Let $\mathbb{F}_q$ denote the finite field with $q$ elements where $q=p^l$ is a prime power. Using Fourier analytic tools with a third moment method, we obtain sum-product type estimates for subsets of $\mathbb{F}_q$. In particular, we prove that if $A\subset \mathbb{F}_q$, then $$|AA+A|,|A(A+A)|\gg\min\left\{q, \frac{|A|^2}{q^{\frac{1}{2}}} \right\},$$ so that if $A\ge q^{\frac{3}{4}}$, then $|AA+A|,|A(A+A)|\gg q$.Comment: Lemma 1.2 is written in a general form. I would like to thank Oliver Roche-Newton for pointing out that the main result of the paper can be improved by using point-line incidence theorem of Le Anh Vinh over finite field

Erd\H{o}s Type Problems in Modules over Cyclic Rings

In the present paper, we study various Erd\H{o}s type geometric problems in the setting of the integers modulo $q$, where $q=p^l$ is an odd prime power. More precisely, we prove certain results about the distribution of triangles and triangle areas among the points of $E\subset \mathbb{Z}_q^2$. We also prove a dot product result for $d$-fold product subsets $E=A\times \ldots \times A$ of $\mathbb{Z}_q^d$, where $A\subset \mathbb{Z}_q$

Jets, Lifts and Dynamics

We show that complete cotangent lifts of vector fields, their decomposition into vertical representative and holonomic part provide a geometrical framework underlying Eulerian equations of continuum mechanics. We discuss Euler equations for ideal incompressible fluid and Vlasov equations of plasma dynamics in connection with the lifts of divergence-free and Hamiltonian vector fields, respectively. As a further application, we obtain kinetic equations of particles moving with the flow of contact vector fields both from Lie-Poisson reductions and with the techniques of present framework

On Geometry of Schmidt Legendre Transformation

A geometrization of Schmidt-Legendre transformation of the second order Lagrangians is proposed by building a proper Tulczyjew's triplet. The symplectic relation between Ostrogradsky-Legendre and Schmidt-Legendre transformations is obtained. Several examples are presented

Lie Algebra of Hamiltonian Vector Fields and the Poisson-Vlasov Equations

We introduce natural differential geometric structures underlying the Poisson-Vlasov equations in momentum variables. We decompose the space of all vector fields over particle phase space into a semi-direct product algebra of Hamiltonian vector fields and its complement. The latter is related to dual space of Lie algebra. Lie algebra of Hamiltonian vector fields is isomorphic to the space of all Lagrangian submanifolds with respect to Tulczyjew symplectic structure. This is obtained as tangent space at the identity of the group of canonical diffeomorphisms represented as space of sections of a trivial bundle. We obtain the momentum-Vlasov equations as vertical equivalence of complete cotangent lift of Hamiltonian vector field generating particle motion. Vertical representatives can be described by holonomic lift from a Whitney product to a Tulczyjew symplectic space. A generalization of complete cotangent lift is obtained by a Lie algebra homomorphism from the algebra of symmetric contravariant tensor fields with Schouten concomitant to the Lie algebra of Hamiltonian vector fields. Momentum maps for particular subalgebras result in plasma-to-fluid map in momentum variables. We exhibit dynamical relations between Lie algebras of Hamiltonian vector fields and of contact vector fields, in particular; infinitesimal quantomorphisms. Gauge symmetries of particle motion are extended to tensorial objects including complete lift of particle motion. Poisson equation is then obtained as zero value of momentum map for the Hamiltonian action of gauge symmetries for kinematical description

Lagrangian Dynamics on Matched Pairs

Given a matched pair of Lie groups, we show that the tangent bundle of the matched pair group is isomorphic to the matched pair of the tangent groups. We thus obtain the Euler-Lagrange equations on the trivialized matched pair of tangent groups, as well as the Euler-Poincar\'e equations on the matched pair of Lie algebras. We show explicitly how these equations cover those of the semi-direct product theory. In particular, we study the trivialized, and the reduced Lagrangian dynamics on the group $SL(2,\mathbb{C})$

De Donder Form for Second Order Gravity

We show that the De Donder form for second order gravity, defined in terms of Ostrogradski's version of the Legendre transformation applied to all independent variables, is globally defined by its local coordinate descriptions. It is a natural differential operator applied to the diffeomorphism invariant Lagrangian of the theory