No abstract

We employ a variational basis with a number of " qq and lattice interpolating fields with quantum numbers of the resonance to extract the discrete energy spectrum in a finite volume. In the elastic region, this spectrum is related to the phase shift of the continuum scattering amplitude by Lüscher's formula, and the relation allows the extraction of resonance parameters from the spectrum calculation. The simulations are performed at three different total momenta of the coupled "qq À system, which allows us to extract the p-wave scattering phase at five values of pion relative momenta near the resonance region. The effective range formula describes the phase-shift dependence nicely, and we extract the resonance mass m ¼ 792ð7Þð8Þ MeV and the coupling g ¼ 5:13ð20Þ at our m ' 266 MeV. The coupling g is directly related to the width of the meson, and our value is close to the value derived from the experimental width. The simulations are performed using dynamical gauge configurations with two massdegenerate flavors of tree-level improved clover-Wilson fermions. Correlation functions are calculated using the recently proposed distillation method with Laplacian-Heaviside smearing of quarks, which enables flexible calculations, in many cases with unprecedented accuracy.

In this paper we develop the theory of the W and Z scale functions for right-continuous (upwards skip-free) discrete-time, discrete-space random walks, along the lines of the analogous theory for spectrally negative Lévy processes. Notably, we introduce for the first time in this context the one- and two-parameter scale functions Z, which appear for example in the joint deficit at ruin and time of ruin problems of actuarial science. Comparisons are made between the various theories of scale functions as one makes time and/or space continuous.

We extract the P-wave ππ phase shift for five values of pion relative momenta, which gives information on the ρ resonance. The Breit-Wigner formula describes the ππ phase shift dependence nicely and we extract m ρ = 792(7)(8) MeV and the coupling g ρππ = 5.13(20) at our m π = 266 MeV. We extract the P-wave scattering length a ππ l=1 = 0.082(10)(3) fm 3 from the state with the lowest pion relative momenta. We also determine the S-wave ρπ phase shift for two values of relative momenta, which provides parameters of the lowest axial resonance a 1 (1260). Using the Breit-Wigner fit we extract m a1 = 1.44(4) GeV and the coupling g a 1 ρπ = 1.1(3) GeV. From the lowest state we also extract the ρπ scattering length a ρπ l=0 = 0.23( 12) fm for our m π . The simulation is performed using one N f = 2 ensemble of gauge configurations with cloverimproved Wilson quarks. The phase shifts are determined from the lowest two energy-levels, which are obtained by the variational analysis with a number of quark-antiquark and mesonmeson interpolators. The correlation functions are calculated using the distillation method with the Laplacian Heaviside (LapH) smearing of quarks.

We extract the P-wave ππ phase shift for five values of pion relative momenta, which gives information on the ρ resonance. The Breit-Wigner formula describes the ππ phase shift dependence nicely and we extract m ρ = 792(7)(8) MeV and the coupling g ρππ = 5.13(20) at our m π = 266 MeV. We extract the P-wave scattering length a ππ l=1 = 0.082(10)(3) fm 3 from the state with the lowest pion relative momenta. We also determine the S-wave ρπ phase shift for two values of relative momenta, which provides parameters of the lowest axial resonance a 1 (1260). Using the Breit-Wigner fit we extract m a1 = 1.44(4) GeV and the coupling g a 1 ρπ = 1.1(3) GeV. From the lowest state we also extract the ρπ scattering length a ρπ l=0 = 0.23(12) fm for our m π. The simulation is performed using one N f = 2 ensemble of gauge configurations with cloverimproved Wilson quarks. The phase shifts are determined from the lowest two energy-levels, which are obtained by the variational analysis with a number of quark-antiquark and mesonmeson interpolators. The correlation functions are calculated using the distillation method with the Laplacian Heaviside (LapH) smearing of quarks.