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We derive an explicit expression for the Lellouch-Lüscher (LL) factor in the K → 3π decays at leading order (without derivative couplings). Several important technical details are addressed, like a proper decomposition into the isospin amplitudes, the choice of a minimal set of effective couplings and the renormalization, as well as the algorithm for the solution of the pertinent Faddeev equations in the infinite volume which is based on the contour deformation method. Most importantly, our numerical results demonstrate that the three-body force contributes very little to the LL factor. This result paves the way for the study of the K → 3π decays on the lattice. [ABSTRACT FROM AUTHOR]

A manifestly relativistic-invariant Lellouch-Lüscher formalism for the decays into three identical particles with no two-to-three transitions is proposed. Similarly to [1], the formalism is based on the use of the non-relativistic effective Lagrangians. Manifest Lorentz invariance is guaranteed, as in [2], by choosing the quantization axis along the total four-momentum of the three-particle system. A systematic inclusion of the higher-order derivative couplings, as well as higher partial waves is addressed. [ABSTRACT FROM AUTHOR]

We study the properties of three-body resonances using a lattice complex scalar φ4 theory with two scalars, with parameters chosen such that one heavy particle can decay into three light ones. We determine the two- and three-body spectra for several lattice volumes using variational techniques, and then analyze them with two versions of the three-particle finite-volume formalism: the Relativistic Field Theory approach and the Finite-Volume Unitarity approach. We find that both methods provide an equivalent description of the energy levels, and we are able to fit the spectra using simple parametrizations of the scattering quantities. By solving the integral equations of the corresponding three-particle formalisms, we determine the pole position of the resonance in the complex energy plane and thereby its mass and width. We find very good agreement between the two methods at different values of the coupling of the theory. [ABSTRACT FROM AUTHOR]

Using effective-range expansion for the two-body amplitudes may generate spurious sub-threshold poles outside of the convergence range of the expansion. In the infinite volume, the emergence of such poles leads to the inconsistencies in the three-body equations, e.g., to the breakdown of unitarity. We investigate the effect of the spurious poles on the three-body quantization condition in a finite volume and show that it leads to a peculiar dependence of the energy levels on the box size L. Furthermore, within a simple model, it is demonstrated that the procedure for the removal of these poles, which was recently proposed in ref. [1] in the infinite volume, can be adapted to the finite-volume calculations. The structure of the exact energy levels is reproduced with an accuracy that systematically improves order by order in the EFT expansion. [ABSTRACT FROM AUTHOR]