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Register a new userIsoscalar $pipi, Koverline{K}, etaeta$ scattering and the $sigma, f_0, f_2$ mesons from QCD
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We present the first lattice QCD study of coupled isoscalar $pipi,Koverline{K},etaeta$ $S$- and $D$-wave scattering extracted from discrete finite-volume spectra computed on lattices which have a value of the quark mass corresponding to $m_pisim391$ MeV. In the $J^P=0^+$ sector we find analogues of the experimental $sigma$ and $f_0(980)$ states, where the $sigma$ appears as a stable bound-state below $pipi$ threshold, and, similar to what is seen in experiment, the $f_0(980)$ manifests itself as a dip in the $pipi$ cross section in the vicinity of the $Koverline{K}$ threshold. For $J^P=2^+$ we find two states resembling the $f_2(1270)$ and $f_2(1525)$, observed as narrow peaks, with the lighter state dominantly decaying to $pipi$ and the heavier state to $Koverline{K}$. The presence of all these states is determined rigorously by finding the pole singularity content of scattering amplitudes, and their couplings to decay channels are established using the residues of the poles.
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We present for the first time a determination of the energy dependence of the isoscalar $pipi$ elastic scattering phase-shift within a first-principles numerical lattice approach to QCD. Hadronic correlation functions are computed including all required quark propagation diagrams, and from these the discrete spectrum of states in the finite volume defined by the lattice boundary is extracted. From the volume dependence of the spectrum we obtain the $S$-wave phase-shift up to the $Koverline{K}$ threshold. Calculations are performed at two values of the $u,d$ quark mass corresponding to $m_pi = 236, 391$ MeV and the resulting amplitudes are described in terms of a $sigma$ meson which evolves from a bound-state below $pipi$ threshold at the heavier quark mass, to a broad resonance at the lighter quark mass.
We determine elastic and coupled-channel amplitudes for isospin-1 meson-meson scattering in $P$-wave, by calculating correlation functions using lattice QCD with light quark masses such that $m_pi = 236$ MeV in a cubic volume of $sim (4 ,mathrm{fm})^3$. Variational analyses of large matrices of correlation functions computed using operator constructions resembling $pipi$, $Koverline{K}$ and $qbar{q}$, in several moving frames and several lattice irreducible representations, leads to discrete energy spectra from which scattering amplitudes are extracted. In the elastic $pipi$ scattering region we obtain a detailed energy-dependence for the phase-shift, corresponding to a $rho$ resonance, and we extend the analysis into the coupled-channel $Koverline{K}$ region for the first time, finding a small coupling between the channels.
We report a direct lattice calculation of the $K$ to $pipi$ decay matrix elements for both the $Delta I=1/2$ and 3/2 amplitudes $A_0$ and $A_2$ on 2+1 flavor, domain wall fermion, $16^3times32times16$ lattices. This is a complete calculation in which all contractions for the required ten, four-quark operators are evaluated, including the disconnected graphs in which no quark line connects the initial kaon and final two-pion states. These lattice operators are non-perturbatively renormalized using the Rome-Southampton method and the quadratic divergences are studied and removed. This is an important but notoriously difficult calculation, requiring high statistics on a large volume. In this paper we take a major step towards the computation of the physical $Ktopipi$ amplitudes by performing a complete calculation at unphysical kinematics with pions of mass 422,MeV at rest in the kaon rest frame. With this simplification we are able to resolve Re$(A_0)$ from zero for the first time, with a 25% statistical error and can develop and evaluate methods for computing the complete, complex amplitude $A_0$, a calculation central to understanding the $Delta =1/2$ rule and testing the standard model of CP violation in the kaon system.
We calculate the parameters describing elastic $I=1$, $P$-wave $pipi$ scattering using lattice QCD with $2+1$ flavors of clover fermions. Our calculation is performed with a pion mass of $m_pi approx 320::{rm MeV}$ and a lattice size of $Lapprox 3.6$ fm. We construct the two-point correlation matrices with both quark-antiquark and two-hadron interpolating fields using a combination of smeared forward, sequential and stochastic propagators. The spectra in all relevant irreducible representations for total momenta $|vec{P}| leq sqrt{3} frac{2pi}{L}$ are extracted with two alternative methods: a variational analysis as well as multi-exponential matrix fits. We perform an analysis using Luschers formalism for the energies below the inelastic thresholds, and investigate several phase shift models, including possible nonresonant contributions. We find that our data are well described by the minimal Breit-Wigner form, with no statistically significant nonresonant component. In determining the $rho$ resonance mass and coupling we compare two different approaches: fitting the individually extracted phase shifts versus fitting the $t$-matrix model directly to the energy spectrum. We find that both methods give consistent results, and at a pion mass of $am_{pi}=0.18295(36)_{stat}$ obtain $g_{rhopipi} = 5.69(13)_{stat}(16)_{sys}$, $am_rho = 0.4609(16)_{stat}(14)_{sys}$, and $am_{rho}/am_{N} = 0.7476(38)_{stat}(23)_{sys} $, where the first uncertainty is statistical and the second is the systematic uncertainty due to the choice of fit ranges.
The pi+pi+ s-wave scattering phase-shift is determined below the inelastic threshold using Lattice QCD. Calculations were performed at a pion mass of m_pi~390 MeV with an anisotropic n_f=2+1 clover fermion discretization in four lattice volumes, with spatial extent L~2.0, 2.5, 3.0 and 3.9 fm, and with a lattice spacing of b_s~0.123 fm in the spatial direction and b_t b_s/3.5 in the time direction. The phase-shift is determined from the energy-eigenvalues of pi+pi+ systems with both zero and non-zero total momentum in the lattice volume using Luschers method. Our calculations are precise enough to allow for a determination of the threshold scattering parameters, the scattering length a, the effective range r, and the shape-parameter P, in this channel and to examine the prediction of two-flavor chiral perturbation theory: m_pi^2 a r = 3+O(m_pi^2/Lambda_chi^2). Chiral perturbation theory is used, with the Lattice QCD results as input, to predict the scattering phase-shift (and threshold parameters) at the physical pion mass. Our results are consistent with determinations from the Roy equations and with the existing experimental phase shift data.
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