Asymptotic scaling in the two-dimensional O(3) Nonlinear sigma model: a Monte Carlo study on parallel computers

We investigate the 2d O(3) model with the standard action by Monte Carlo simulation at couplings β up to 2.05. We measure the energy density, mass gap and susceptibility of the model, and gather high statistics on lattices of size L ≤ 1024 using the Floating Point Systems T-series vector hypercube and the Thinking Machines Corp.'s Connection Machine 2. Asymptotic scaling does not appear to set in for this action, even at β = 2.10, where the correlation length is 420. We observe a 20% difference between our estimate m/Λ^─_(Ms) = 3.52(6) at this β and the recent exact analytical result . We use the overrelaxation algorithm interleaved with Metropolis updates and show that decorrelation time scales with the correlation length and the number of overrelaxation steps per sweep. We determine its effective dynamical critical exponent to be z' = 1.079(10); thus critical slowing down is reduced significantly for this local algorithm that is vectorizable and parallelizable.

We also use the cluster Monte Carlo algorithms, which are non-local Monte Carlo update schemes which can greatly increase the efficiency of computer simulations of spin models. The major computational task in these algorithms is connected component labeling, to identify clusters of connected sites on a lattice. We have devised some new SIMD component labeling algorithms, and implemented them on the Connection Machine. We investigate their performance when applied to the cluster update of the two dimensional Ising spin model.

Finally we use a Monte Carlo Renormalization Group method to directly measure the couplings of block Hamiltonians at different blocking levels. For the usual averaging block transformation we confirm the renormalized trajectory (RT) observed by Okawa. For another improved probabilistic block transformation we find the RT, showing that it is much closer to the Standard Action. We then use this block transformation to obtain the discrete β-function of the model which we compare to the perturbative result. We do not see convergence, except when using a rescaled coupling β_E to effectively resum the series. For the latter case we see agreement for m/ Λ^─_(Ms) at , β = 2.14, 2.26, 2.38 and 2.50. To three loops m/Λ^─_(Ms) = 3.047(35) at β = 2.50, which is very close to the exact value m/ Λ^─_(Ms) = 2.943. Our last point at β = 2.62 disagrees with this estimate however.

Experimental study study of antiproton-proton annihilation into a pair of charged Pi-Mesons or K-Mesons for incident antiproton mementa in the range from 0.72 GeV/c to 2.62 GeV/c

The cross sections for the two antiproton-proton annihilation-in-flight modes,

ˉp + p → π⁺ + π^-

ˉp + p → k⁺ + k^-

were measured for fifteen laboratory antiproton beam momenta ranging from 0.72 to 2.62 GeV/c. No magnets were used to determine the charges in the final state. As a result, the angular distributions were obtained in the form [dσ/dΩ (Θ_C.M.) + dσ/dΩ (π – Θ_C.M.)] for 45 ≲ Θ_C.M. ≲ 135°.

A hodoscope-counter system was used to discriminate against events with final states having more than two particles and antiproton-proton elastic scattering events. One spark chamber was used to record the track of each of the two charged final particles. A total of about 40,000 pictures were taken. The events were analyzed by measuring the laboratory angle of the track in each chamber. The value of the square of the mass of the final particles was calculated for each event assuming the reaction

ˉp + p → a pair of particles with equal masses.

About 20,000 events were found to be either annihilation into π ^±-pair or k ^±-pair events. The two different charged meson pair modes were also distinctly separated.

The average differential cross section of ˉp + p → π⁺ + π^- varied from ~ 25 µb/sr at antiproton beam momentum 0.72 GeV/c (total energy in center-of-mass system, √s = 2.0 GeV) to ~ 2 µb/sr at beam momentum 2.62 GeV/c (√s = 2.64 GeV). The most striking feature in the angular distribution was a peak at Θ_C.M. = 90° (cos Θ_C.M. = 0) which increased with √s and reached a maximum at √s ~ 2.1 GeV (beam momentum ~ 1.1 GeV/c). Then it diminished and seemed to disappear completely at √s ~ 2.5 GeV (beam momentum ~ 2.13 GeV/c). A valley in the angular distribution occurred at cos Θ_C.M. ≈ 0.4. The differential cross section then increased as cos Θ_C.M. approached 1.

The average differential cross section for ˉp + p → k⁺ + k^- was about one third of that of the π^±-pair mode throughout the energy range of this experiment. At the lower energies, the angular distribution, unlike that of the π^±-pair mode, was quite isotropic. However, a peak at Θ_C.M. = 90° seemed to develop at √s ~ 2.37 GeV (antiproton beam momentum ~ 1.82 GeV/c). No observable change was seen at that energy in the π^±-pair cross section.

The possible connection of these features with the observed meson resonances at 2.2 GeV and 2.38 GeV, and its implications, were discussed.