$D\to (K\pi {)}_{27}$ at the SU(3)-flavour-symmetric point I: Methodology and strong phase determination

Abstract

We present part one of an SU(3)-flavour-symmetric lattice QCD calculation of the amplitude for a $D$-meson decaying to a $K\pi$ final state in the 27-dimensional irreducible representation of the flavour symmetry group, denoted $(K\pi {)}_{27}$. The Wilson--clover gauge ensembles used in this work, generated by the OpenLat collaboration, are tuned such that $M_{\pi }=M_K\approx 410\mathrm{MeV}$. Using the distillation framework, we construct a matrix of Euclidean correlation functions from pairs of single-hadron operators projected to definite spatial momentum. Solving a generalised eigenvalue problem yields the finite-volume energy spectrum that is used to determine the scattering phase shift from threshold up to $4M_{\pi }\approx 1640\mathrm{MeV}$, which sits below but plausibly within reach of $M_D^{\mathrm{SU}(3)}\simeq 1900\mathrm{MeV}$. The calculation is performed across three lattice spacings, and we apply two strategies in which the continuum limit is taken at different stages of the computation: (i) on the extracted scattering parameters and (ii) on the finite-volume energies at fixed physical volume before extracting the scattering parameters. We find consistent results across these methods for the scattering phase shift as a function of the centre-of-mass energy, ${\delta }_{27}(E_{\mathsf{cm}})$. Taking a scattering-length-only parametrisation, we infer a value for the strong phase of the weak decay, ${\delta }_{27}(M_D^{\mathrm{SU}(3)})=-38.4(2.4{)}^{\circ }$. We further describe the methodology for using the same operator basis to compute three-point correlation functions to extract $⟨(K\pi {)}_{27}|H_W|D⟩$, for the tree-level effective weak Hamiltonian $H_W$, and for relating such finite-volume matrix elements to the full decay amplitude. The complete analysis leading to the latter will be presented in a forthcoming manuscript.

Comments: 41 pages, 15 figures, 6 tables