An implementation of estimating the two-to-two K-matrix from finite-volume energies based on the Lüscher formalism and involving a Hermitian matrix known as the "box matrix" is described. The method includes higher partial waves and multiple decay channels. Two fitting procedures for estimating the K-matrix parameters, which properly incorporate all statistical covariances, are discussed. Formulas and software for handling total spins up to S = 2 and orbital angular momenta up to L = 6 are obtained for total momenta in several directions. First tests involving ρ-meson decay to two pions include the L = 3 and L = 5 partial waves, and the contributions from these higher waves are found to be negligible in the elastic energy range.
The elastic I = 1/2, s-and p-wave kaon-pion scattering amplitudes are calculated using a single ensemble of anisotropic lattice QCD gauge field configurations with N f = 2 + 1 flavors of dynamical Wilson-clover fermions at m π = 230MeV. A large spatial extent of L = 3.7fm enables a good energy resolution while partial wave mixing due to the reduced symmetries of the finite volume is treated explicitly. The p-wave amplitude is well described by a Breit-Wigner shape with parameters m K * /m π = 3.808(18) and g BW K * Kπ = 5. 33(20) which are insensitive to the inclusion of d-wave mixing and variation of the s-wave parametrization. An effective range description of the near-threshold s-wave amplitude yields m π a 0 = −0.353(25).
The Roper state is extracted with valence overlap fermions on a 2 + 1-flavor domain-wall fermion lattice (spacing a = 0.114 fm and mπ = 330 MeV) using both the Sequential Empirical Bayes (SEB) method and the variational method. The results are consistent, provided that a large smearing-size interpolation operator is included in the variational calculation to have better overlap with the lowest radial excitation. The SEB and variational calculation with large smearing size are also carried out for an anisotropic clover lattice with similar parameters (spatial lattice spacing as = 0.12 fm and pion mass mπ = 396 MeV) and obtain consistent results. However, these calculations with clover fermions give a Roper mass of mR = 1.92(6) GeV, while the same approach with overlap fermions finds the Roper ≈ 280 MeV lower, at mR = 1.64(9) GeV, for identical valence pion mass. The fact that the prediction of the Roper state by overlap fermions is consistently lower than those of clover fermions, chirally improved fermions, and twisted-mass fermions over a wide range of pion masses has been dubbed a "Roper puzzle."To understand the origin of this difference, we study the hairpin Z-diagram in the isovector scalar meson (a0) correlator in the quenched approximation. The lack of quark loops in the quenched approximation turns the a0 correlator negative; giving rise to a ghost "would-be" ηπ state. Comparing the a0 correlators for valence clover and overlap fermions, at a valence pion mass of 290 MeV, on three quenched Wilson-gauge lattices, we find that the spectral weight of the ghost state with clover fermions is smaller than that of the overlap at a = 0.12 fm and 0.09 fm −− the ratios of the Wilson ghost-state magnitudes (correlator minima) are about half of those of overlap −− whereas, the whole a0 correlators of clover and overlap at a = 0.06 fm coincide within errors. This suggests that chiral symmetry is restored for clover at a ≤ 0.06 fm and that the Roper mass should agree between clover and overlap fermions toward the continuum limit.We conclude that the present work supports a resolution of the "Roper puzzle" due to Z-graph type chiral dynamics. This entails coupling to higher components in the Fock space (e.g. N π, N ππ states) to induce the effective flavor-spin interaction between quarks as prescribed in the chiral quark model, resulting in the parity-reversal pattern as observed in the experimental excited states of N, ∆ and Λ.
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