Logical Dreams

We discuss the past and future of set theory, axiom systems and independence results. We deal in particular with cardinal arithmetic

Metrically homogeneous graphs of diameter 3

We classify countable metrically homogeneous graphs of diameter 3

Hypernatural Numbers as Ultrafilters

In this paper we present a use of nonstandard methods in the theory of ultrafilters and in related applications to combinatorics of numbers

Leptonic and charged kaon decay modes of the ϕ\phi meson measured in heavy-ion collisions at the CERN SPS

We report a measurement of $\phi$ meson production in central Pb+Au collisions at E$_{lab}$/A=158 GeV. For the first time in heavy-ion collisions, $\phi$ mesons were reconstructed in the same experiment both in the K$^+$K$^-$ and the dilepton decay channel. Near mid-rapidity, this yields rapidity densities, corrected for production at the same rapidity value, of 2.05 +- 0.14(stat) +- 0.25(syst) and 2.04 +- 0.49(stat)+-{0.32}(syst), respectively. The shape of the measured transverse momentum spectra is also in close agreement in both decay channels. The data rule out a possible enhancement of the $\phi$ yield in the leptonic over the hadronic channel by a factor larger than 1.6 at 95% CL.Comment: 11 pages, 4 figures,submitted to Phys. Rev. Let

Elliptic flow of charged pions, protons and strange particles emitted in Pb+Au collisions at top SPS energy

Differential elliptic flow spectra v2(pT) of \pi-, K0short, p, \Lambda have been measured at \sqrt(s NN)= 17.3 GeV around midrapidity by the CERN-CERES/NA45 experiment in mid-central Pb+Au collisions (10% of \sigma(geo)). The pT range extends from about 0.1 GeV/c (0.55 GeV/c for \Lambda) to more than 2 GeV/c. Protons below 0.4 GeV/c are directly identified by dE/dx. At higher pT, proton elliptic flow v2(pT) is derived as a constituent, besides \pi+ and K+, of the elliptic flow of positive pion candidates. The retrieval requires additional inputs: (i) of the particle composition, and (ii) of v2(pT) of positive pions. For (i), particle ratios obtained by NA49 were adapted to CERES conditions; for (ii), the measured v2(pT) of negative pions is substituted, assuming \pi+ and \pi- elliptic flow magnitudes to be sufficiently close. The v2(pT) spectra are compared to ideal-hydrodynamics calculations. In synopsis of the series \pi- - K0short - p - \Lambda, flow magnitudes are seen to fall with decreasing pT progressively even below hydro calculations with early kinetic freeze-out (Tf= 160 MeV) leaving not much time for hadronic evolution. The proton v2(pT) data show a downward swing towards low pT with excursions into negative v2 values. The pion-flow isospin asymmetry observed recently by STAR at RHIC, invalidating in principle our working assumption, is found in its impact on proton flow bracketed from above by the direct proton flow data, and not to alter any of our conclusions. Results are discussed in perspective of recent viscous dynamics studies which focus on late hadronic stages.Comment: 38 pages, 27 figures, 2 tables. Abstract and parts of introduction made more comprehensible; corrected typos; acknowledgement added. To appear in Nucl.Phys.

Semi-Hard Scattering Unraveled from Collective Dynamics by Two-Pion Azimuthal Correlations in 158 A GeV/c Pb + Au Collisions

Elliptic flow and two-particle azimuthal correlations of charged hadrons and high-$p_T$ pions ($p_T>$ 1 GeV/$c$) have been measured close to mid-rapidity in 158A GeV/$c$ Pb+Au collisions by the CERES experiment. Elliptic flow ($v_2$) rises linearly with $p_T$ to a value of about 10% at 2 GeV/$c$. Beyond $p_T\approx$ 1.5 GeV/$c$, the slope decreases considerably, possibly indicating a saturation of $v_2$ at high $p_T$. Two-pion azimuthal anisotropies for $p_T>$ 1.2 GeV/$c$ exceed the elliptic flow values by about 60% in mid-central collisions. These non-flow contributions are attributed to near-side and back-to-back jet-like correlations, the latter exhibiting centrality dependent broadening.Comment: Submitted to Phys. Rev. Letters, 4 pages, 5 figure

Azimuthal dependence of pion source radii in Pb+Au collisions at 158 A GeV

We present results of a two-pion correlation analysis performed with the Au+Pb collision data collected by the upgraded CERES experiment in the fall of 2000. The analysis was done in bins of the reaction centrality and the pion azimuthal emission angle with respect to the reaction plane. The pion source, deduced from the data, is slightly elongated in the direction perpendicular to the reaction plane, similarly as was observed at the AGS and at RHIC.Comment: 5 pages, 2 figure

e+e--pair production in Pb-Au collisions at 158 GeV per nucleon

We present the combined results on electron-pair production in 158 GeV/n {Pb-Au} ($\sqrt{s}$= 17.2 GeV) collisions taken at the CERN SPS in 1995 and 1996, and give a detailed account of the data analysis. The enhancement over the reference of neutral meson decays amounts to a factor of 2.31$\pm0.19 (stat.)\pm0.55 (syst.)\pm0.69 (decays)$ for semi-central collisions (28% $\sigma/\sigma_{geo}$) when yields are integrated over $m>$ 200 MeV/$c^2$ in invariant mass. The measured yield, its stronger-than-linear scaling with $N_{ch}$, and the dominance of low pair $p_t$ strongly suggest an interpretation as {\it thermal radiation} from pion annihilation in the hadronic fireball. The shape of the excess centring at $m\approx$ 500 MeV/$c^2$, however, cannot be described without strong medium modifications of the $\rho$ meson. The results are put into perspective by comparison to predictions from Brown-Rho scaling governed by chiral symmetry restoration, and from the spectral-function many-body treatment in which the approach to the phase boundary is less explicit.Comment: 39 pages, 40 figures, to appear in Eur.Phys.J.C. (2005

Recent results from Pb-Au collisions at 158 GeV/c per nucleon obtained with the CERES spectrometer

During the 1996 lead run time, CERES has accumulated 42 million events, corresponding to a factor of 5 more statistics than in 1995 and 2.5 million events of a special photon-run. We report on the results of the low-mass e$^+$e$^-$-pair analysis. Since the most critical item is the poor signal-to-background ratio we also discuss the understanding of this background, in absolute terms, with the help of a detailed Monte Carlo simulation. We show preliminary results of the photon analysis and summarize the results of the hadron analysis preliminarily reported on already at QM'97Comment: 10 pages, 9 figures, Proceedings of the XIV Int. Conf. on Nucleus-Nucleus Collisions,Quark Matter 99, Torino, Italy, May 10 - 15, 199