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Using non-relativistic effective field theory, we derive a three-particle analog of the Lellouch-Lüscher formula at the leading order. This formula relates the three-particle decay amplitudes in a finite volume with their infinite-volume counterparts and, hence, can be used to study the three-particle decays on the lattice. The generalization of the approach to higher orders is briefly discussed. [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 consider two ways of the experimental determination of the S-wave ππ scattering lengths: from K → 3π decays and from Ke4 decays. In both cases the unitary cusps, which emerge due to the difference in the charged and neutral pion masses, play a crucial role. We construct a systematic effective field theory framework which can be used to describe decay amplitudes in the cusp region. [ABSTRACT FROM AUTHOR]

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]

We present a general method for deriving the energy shift of an interacting system of N spinless particles in a finite volume. To this end, we use the nonrelativistic effective field theory (NREFT), and match the pertinent low-energy constants to the scattering amplitudes. Relativistic corrections are explicitly included up to a given order in the 1/L expansion. We apply this method to obtain the ground state of N particles, and the first excited state of two and three particles to order L−6 in terms of the threshold parameters of the two- and three-particle relativistic scattering amplitudes. We use these expressions to analyze the N-particle ground state energy shift in the complex φ4 theory. [ABSTRACT FROM AUTHOR]

A three-particle quantization condition on the lattice is written down in a manifestly relativistic-invariant form by using a generalization of the non-relativistic effective field theory (NREFT) approach. Inclusion of the higher partial waves is explicitly addressed. A partial diagonalization of the quantization condition into the various irreducible representations of the (little groups of the) octahedral group has been carried out both in the center-of-mass frame and in moving frames. Furthermore, producing synthetic data in a toy model, the relativistic invariance is explicitly demonstrated for the three-body bound state spectrum. [ABSTRACT FROM AUTHOR]

We test an alternative proposal by Bruno and Hansen (J High Energy Phys 2021(6), https://doi.org/10.1007/JHEP06(2021)043, 2021) to extract the scattering length from lattice simulations in a finite volume. For this, we use a scalar ϕ 4 theory with two mass nondegenerate particles and explore various strategies to implement this new method. We find that the results are comparable to those obtained from the Lüscher method, with somewhat smaller statistical uncertainties at larger volumes. [ABSTRACT FROM AUTHOR]

The extraction of two- and three-body hadronic scattering amplitudes and the properties of the low-lying hadronic resonances from the finite-volume energy levels in lattice QCD represents a rapidly developing field of research. The use of various modifications of the Lüscher finite-volume method has opened a path to calculate infinite-volume scattering amplitudes on the lattice. Many new results have been obtained recently for different two- and three-body scattering processes, including the extraction of resonance poles and their properties from lattice data. Such studies, however, require robust parametrizations of the infinite-volume scattering amplitudes, which rely on basic properties of S-matrix theory and—preferably—encompass systems with quark masses at and away from the physical point. Parametrizations of this kind, provided by unitarized Chiral Perturbation Theory, are discussed in this review. Special attention is paid to three-body systems on the lattice, owing to the rapidly growing interest in the field. Here, we briefly survey the formalism, chiral extrapolation, as well as finite-volume analyses of lattice data. [ABSTRACT FROM AUTHOR]

A dispersive analysis of η→3π decays has been performed in the past by many authors. The numerical analysis of the pertinent integral equations is hampered by two technical difficulties: (i) The angular averages of the amplitudes need to be performed along a complicated path in the complex plane. (ii) The averaged amplitudes develop singularities along the path of integration in the dispersive representation of the full amplitudes. It is a delicate affair to handle these singularities properly, and independent checks of the obtained solutions are demanding and time consuming. In the present article, we propose a solution method that avoids these difficulties. It is based on a simple deformation of the path of integration in the dispersive representation (not in the angular average). Numerical solutions are then obtained rather straightforwardly. We expect that the method also works for ω→3π. [ABSTRACT FROM AUTHOR]

Using non-relativistic effective Lagrangians in the particle-dimer picture, we rederive the expression for the energy shift of a loosely bound three-particle bound state of identical bosons in the unitary limit. The effective field theory formalism allows us to explicitly investigate the role of the three-particle force. Moreover, we discuss relaxing the unitary limit of infinite scattering length and demonstrate a smooth transition from the weakly bound three-particle state to a two-particle bound state of a particle and a deeply bound dimer. [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]

A novel method for the extraction of form factors of unstable particles on the lattice is proposed. The approach is based on the study of two-particle scattering in a static, spatially periodic external field by using a generalization of the Lüscher method in the presence of such a field. It is shown that the resonance form factor is given by the derivative of the resonance pole position in the complex plane with respect to the coupling constant to the external field. Unlike the standard approach, this proposal does not suffer from problems caused by the presence of the triangle diagram. [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]

The spectrum of a bound state of three identical particles with a mass m in a finite cubic box is studied. It is shown that in the unitary limit, the energy shift of a shallow bound state is given by ΔE = c(k2/m)(xL)-3/2∣A∣2 exp(-2xL/√3), where k is the bound-state momentum, L is the box size, ∣∣A2 denotes the three-body analog of the asymptotic normalization coefficient of the bound state wave function, and c is a numerical constant. The formula is valid for kL ⪢ 1. [ABSTRACT FROM AUTHOR]

Abstract: Based on the analysis of decays, the NA48/2 Collaboration has recently determined the S-wave ππ scattering lengths with high precision. In addition, the scattering length has been independently measured, although less precisely so. The present article discusses in detail one of the theoretical frameworks used in the data analysis. [Copyright &y& Elsevier]

Abstract: In this paper we propose a model-independent method to extract the resonance parameters on the lattice directly from the Euclidean 2-point correlation functions of the field operators at finite times. The method is tested in case of the two-point function of the Δ-resonance, calculated at one loop in Small Scale Expansion. Further, the method is applied to a -dimensional model with two coupled Ising spins and the results are compared with earlier ones obtained by using Lüscher''s approach. [Copyright &y& Elsevier]

Abstract: We propose a method to determine the isoscalar scattering length on the lattice. Our method represents the generalization of Lüscher''s approach in the presence of inelastic channels (complex scattering length). In addition, the proposed approach allows one to find the position of the S-matrix pole corresponding the resonance. [Copyright &y& Elsevier]

A brief review of the experimental and theoretical attempts to determine $\bar KN$ scattering lengths is given. A proposal is formulated to calculate these (complex) scattering lengths in lattice QCD, using an appropriate modification of Lüscher’s formula. [ABSTRACT FROM AUTHOR]

Abstract: We study the extraction of Δ-resonance parameters from lattice data for small quark masses, corresponding to the case of an unstable Δ. To this end, we calculate the spectrum of the correlator of two Δ-fields in a finite Euclidean box up-to-and-including in the small scale expansion using infrared regularization. On the basis of our numerical study, we argue that the extraction of the parameters of the Δ-resonance (in particular, of the mass and the pion–nucleon–delta coupling constant) from the measured volume dependence of the lowest energy levels should be feasible. [Copyright &y& Elsevier]

The extraction of the B → K ⁎ transition form factors from lattice data is studied, applying non-relativistic effective field theory in a finite volume. The possible mixing of πK and ηK states is taken into account. The two-channel analogue of the Lellouch–Lüscher formula is reproduced. Due to the resonance nature of the K ⁎ , an equation is derived, which allows to determine the form factors at the pole position in a process-independent manner. The infinitely-narrow width approximation of the results is discussed. [ABSTRACT FROM AUTHOR]

Using the non-relativistic effective field theory framework in a finite volume, we discuss the extraction of the ΔNγ* transition form factors from lattice data. A counterpart of the Lüscher approach for the matrix elements of unstable states is formulated. In particular, we thoroughly discuss various kinematic settings, which are used in the calculation of the above matrix element on the lattice. The emerging Lüscher-Lellouch factor and the analytic continuation of the matrix elements into the complex plane are also considered in detail. A full group-theoretical analysis of the problem is made, including the partial-wave mixing and projecting out the invariant form factors from data. [ABSTRACT FROM AUTHOR]

The divergent part of the one-loop effective action in Chiral Perturbation Theory with virtual photons has been evaluated in an arbitrary covariant gauge. The differential operator that emerges in the functional determinant is of non-minimal type, for which the standard heat kernel methods are not directly applicable. Both the SU(2) and SU(3) cases have been worked out. A comparison with existing results in the literature is given. [ABSTRACT FROM AUTHOR]