54 research outputs found

    The critical Ising model via Kac-Ward matrices

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    The Kac-Ward formula allows to compute the Ising partition function on any finite graph G from the determinant of 2^{2g} matrices, where g is the genus of a surface in which G embeds. We show that in the case of isoradially embedded graphs with critical weights, these determinants have quite remarkable properties. First of all, they satisfy some generalized Kramers-Wannier duality: there is an explicit equality relating the determinants associated to a graph and to its dual graph. Also, they are proportional to the determinants of the discrete critical Laplacians on the graph G, exactly when the genus g is zero or one. Finally, they share several formal properties with the Ray-Singer \bar\partial-torsions of the Riemann surface in which G embeds.Comment: 30 pages, 10 figures; added section 4.4 in version

    Measurement of event-shape observables in Z→ℓ+ℓ− events in pp collisions at √ s=7 TeV with the ATLAS detector at the LHC

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    Event-shape observables measured using charged particles in inclusive ZZ-boson events are presented, using the electron and muon decay modes of the ZZ bosons. The measurements are based on an integrated luminosity of 1.1fb11.1 {\rm fb}^{-1} of proton--proton collisions recorded by the ATLAS detector at the LHC at a centre-of-mass energy s=7\sqrt{s}=7 TeV. Charged-particle distributions, excluding the lepton--antilepton pair from the ZZ-boson decay, are measured in different ranges of transverse momentum of the ZZ boson. Distributions include multiplicity, scalar sum of transverse momenta, beam thrust, transverse thrust, spherocity, and F\mathcal{F}-parameter, which are in particular sensitive to properties of the underlying event at small values of the ZZ-boson transverse momentum. The Sherpa event generator shows larger deviations from the measured observables than Pythia8 and Herwig7. Typically, all three Monte Carlo generators provide predictions that are in better agreement with the data at high ZZ-boson transverse momenta than at low ZZ-boson transverse momenta and for the observables that are less sensitive to the number of charged particles in the event.Comment: 36 pages plus author list + cover page (54 pages total), 14 figures, 4 tables, submitted to EPJC, All figures including auxiliary figures are available at http://atlas.web.cern.ch/Atlas/GROUPS/PHYSICS/PAPERS/STDM-2014-0
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