Introductory clifford analysis

In this chapter an introduction is given to Clifford analysis and the underlying Clifford algebras. The functions under consideration are defined on Euclidean space and take values in the universal real or complex Clifford algebra, the structure and properties of which are also recalled in detail. The function theory is centered around the notion of a monogenic function, which is a null solution of a generalized Cauchy–Riemann operator, which is rotation invariant and factorizes the Laplace operator. In this way, Clifford analysis may be considered as both a generalization to higher dimension of the theory of holomorphic functions in the complex plane and a refinement of classical harmonic analysis. A notion of monogenicity may also be associated with the vectorial part of the Cauchy–Riemann operator, which is called the Dirac operator; some attention is paid to the intimate relation between both notions. Since a product of monogenic functions is, in general, no longer monogenic, it is crucial to possess some tools for generating monogenic functions: such tools are provided by Fueter’s theorem on one hand and the Cauchy–Kovalevskaya extension theorem on the other hand. A corner stone in this function theory is the Cauchy integral formula for representation of a monogenic function in the interior of its domain of monogenicity. Starting from this representation formula and related integral formulae, it is possible to consider integral transforms such as Cauchy, Hilbert, and Radon transforms, which are important both within the theoretical framework and in view of possible applications

Study of Cabibbo Suppressed Decays of the Ds Charmed-Strange Meson involving a KS

We study the decay of Ds meson into final states involving a Ks and report the discovery of Cabibbo suppressed decay modes Ds -> Kspi-pi+pi+ (179 +/- 36 events) and Ds -> Kspi+ (113 +/-26 events). The branching ratios for the new modes are Gamma(Ds -> Kspi-pi+pi+)/Gamma(Ds -> KsK-pi+pi+) = 0.18 +/- 0.04 +/- 0.05 and Gamma(Ds -> Kspi+)/Gamma(Ds -> KsK+) = 0.104 +/- 0.024 +/- 0.013.Comment: 11 pages, 6 figure

The Target Silicon Detector for the FOCUS Spectrometer

We describe a silicon microstrip detector interleaved with segments of a beryllium oxide target which was used in the FOCUS photoproduction experiment at Fermilab. The detector was designed to improve the vertex resolution and to enhance the reconstruction efficiency of short-lived charm particles.Comment: 18 pages, 14 figure

Dalitz plot analysis of D_s+ and D+ decay to pi+pi-pi+ using the K-matrix formalism

FOCUS results from Dalitz plot analysis of D_s+ and D+ to pi+pi-pi+ are presented. The K-matrix formalism is applied to charm decays for the first time to fully exploit the already existing knowledge coming from the light-meson spectroscopy experiments. In particular all the measured dynamics of the S-wave pipi scattering, characterized by broad/overlapping resonances and large non-resonant background, can be properly included. This paper studies the extent to which the K-matrix approach is able to reproduce the observed Dalitz plot and thus help us to understand the underlying dynamics. The results are discussed, along with their possible implications on the controversial nature of the sigma meson.Comment: To be submitted to Phys.Lett.B A misprint corrected in formula

Study of the D^0 \to pi^-pi^+pi^-pi^+ decay

Using data from the FOCUS (E831) experiment at Fermilab, we present new measurements for the Cabibbo-suppressed decay mode $D^0 \to \pi^-\pi^+\pi^-\pi^+$. We measure the branching ratio $\Gamma(D^0 \to\pi^+\pi^- \pi^+\pi^-)/\Gamma(D^0 \to K^-\pi^+\pi^-\pi^+) = 0.0914 \pm 0.0018 \pm 0.0022$. An amplitude analysis has been performed, a first for this channel, in order to determine the resonant substructure of this decay mode. The dominant component is the decay $D^0 \to a_1(1260)^+ \pi^-$, accounting for 60% of the decay rate. The second most dominant contribution comes from the decay $D^0 \to \rho(770)^0\rho(770)^0$, with a fraction of 25%. We also study the $a_1(1260)$ line shape and resonant substructure. Using the helicity formalism for the angular distribution of the decay $D^0 \to \rho(770)^0\rho(770)^0$, we measure a longitudinal polarization of $P_L = (71 \pm 4\pm 2)$%.Comment: 38 pages, 8 figures. accepted for publication in Physical Review

Search for a strongly decaying neutral charmed pentaquark

We present a search for a charmed pentaquark decaying strongly to $D^{(*)-}p$. Finding no evidence for such a state, we set limits on the cross section times branching ratio relative to $D^{*-}$ and $D^-$ under particular assumptions about the production mechanism.Comment: To be published in Physics Letters

Search for CP violation in D0 and D+ decays

A high statistics sample of photoproduced charm particles from the FOCUS (E831) experiment at Fermilab has been used to search for CP violation in the Cabibbo suppressed decay modes D+ to K-K+pi+, D0 to K-K+ and D0 to pi-pi+. We have measured the following CP asymmetry parameters: A_CP(K-K+pi+) = +0.006 +/- 0.011 +/- 0.005, A_CP(K-K+) = -0.001 +/- 0.022 +/- 0.015 and A_CP(pi-pi+) = +0.048 +/- 0.039 +/- 0.025 where the first error is statistical and the second error is systematic. These asymmetries are consistent with zero with smaller errors than previous measurements.Comment: 12 pages, 4 figure

Search for Λc+→pK+π−\Lambda_c^+ \to p K^+ \pi^- and Ds+→K+K+π−D_s^+ \to K^+ K^+ \pi^- Using Genetic Programming Event Selection

We apply a genetic programming technique to search for the double Cabibbo suppressed decays $\Lambda_c^+ \to p K^+ \pi^-$ and $D_s^+ \to K^+ K^+ \pi^-$. We normalize these decays to their Cabibbo favored partners and find $\text{BR}($\Lambda_c^+ \to p K^+ \pi^-$)/\text{BR}($\Lambda_c^+ \to p K^- \pi^+$) = (0.05 \pm 0.26 \pm 0.02)$ and $\text{BR}($D_s^+ \to K^+ K^+ \pi^-$)/\text{BR}($D_s^+ \to K^+ K^- \pi^+$) = (0.52\pm 0.17\pm 0.11)$ where the first errors are statistical and the second are systematic. Expressed as 90% confidence levels (CL), we find $< 0.46$ and $< 0.78$ respectively. This is the first successful use of genetic programming in a high energy physics data analysis.Comment: 10 page

Observation of a 1750 MeV/c^2 Enhancement in the Diffractive Photoproduction of K^+K^-

Using the FOCUS spectrometer with photon beam energies between 20 and 160 \gev, we confirm the existence of a diffractively photoproduced enhancement in $K^+K^-$ at 1750 \mevcc with nearly 100 times the statistics of previous experiments. Assuming this enhancement to be a single resonance with a Breit-Wigner mass shape, we determine its mass to be $1753.5\pm 1.5\pm 2.3$ \mevcc and its width to be $122.2\pm 6.2\pm 8.0$ \mevcc. We find no corresponding enhancement at 1750 \mevcc in $K^*K$, and again neglecting any possible interference effects we place limits on the ratio $\Gamma (X(1750) \to K^*K)/\Gamma (X(1750) \to K^+K^-)$. Our results are consistent with previous photoproduction experiments, but, because of the much greater statistics, challenge the common interpretation of this enhancement as the $\phi (1680)$ seen in $e^+e^-$ annihilation experiments.Comment: 10 pages, 5 figure

Study of the decay asymmetry parameter and CP violation parameter in the Lambdac+ --> Lambda pi+ decay

Using data from the FOCUS (E831) experiment at Fermilab, we present a new measurement of the weak decay-asymmetry parameter alpha(Lambdac) in Lambdac --> Lambda pi decay. Comparing particle with antiparticle decays, we obtain the first measurement of the CP violation parameter : A = [alpha(Lambdac)+alpha(antiLambda_c)]/[alpha(Lambdac)-alpha(antiLambda_c)]. We obtain alpha(Lambdac)=-0.78+-0.16+-0.13 and A = -0.07+-0.19+-0.12 where errors are statistical and systematic.Comment: 18 pages, to be submitted to Phys. Lett. B For a list of the FOCUS collaboration, see http://www-focus.fnal.gov/authors.htm