Faddeev Equations Research Articles

Lellouch-Lüscher factor for the K → 3π decays

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.

Impact of the N4LO\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {N}^4\\hbox {LO}$$\\end{document} Short-Range Three-Nucleon Force Components on the Nucleon-Deuteron Spin Correlation Coefficients

The spin correlation coefficients in the neutron-deuteron elastic scattering process at incoming neutron laboratory energies E=10\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {E}=10$$\\end{document}, 135, 190, and 250 MeV are determined by solving the momentum space three-nucleon (3N) Faddeev equations. The chiral two-nucleon (2N) interaction with momentum-space semi-local (SMS) regularization up to the fifth order of chiral expansion (N4LO\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {N}^4\\hbox {LO}$$\\end{document}), supplemented by the F-waves terms from the sixth order (N5LO\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {N}^5\\hbox {LO}$$\\end{document}), is used. Additionally, the consistent 3N force (3NF) at the third order of chiral expansion, supplemented by the short-range contributions from N4LO\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {N}^4\\hbox {LO}$$\\end{document} is applied. As a results, we give predictions for the complete set of spin correlation coefficients Cα,β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$C_{\\alpha ,\\beta }$$\\end{document}. We find that the effect of the investigated three-nucleon N4LO\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {N}^4\\hbox {LO}$$\\end{document} components amounts up to several dozen percent, depending on reaction energy, scattering angle and type of spin correlation coefficient itself. Our results can serve as a guide for future measurements of the spin correlation coefficients.

Space-like electromagnetic form factors of lambda- and sigma-baryons from quark-diquark Faddeev equations

An important goal of ongoing and future experiments is to explore spectra and transition form factors of baryons with non-zero strangeness. Of particular interest is the transition form factorγ(∗)Σ0→Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma ^{(*)} \\Sigma ^0 \\rightarrow \\Lambda $$\\end{document} in the time-like momentum region that can be extracted from Dalitz decays. On the road towards a theoretical description of these form factors we extend a covariant dynamical quark-diquark model for the baryon Faddeev equation to the strange-quark sector. Based on an excellent description of the mass spectrum of selected baryon octet and decuplet states and reasonable results for the nucleon form factors we determine the elastic electromagnetic form factors of Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Lambda $$\\end{document} and Σ+,Σ0,Σ-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Sigma ^+, \\Sigma ^0, \\Sigma ^-$$\\end{document} hyperons in the space-like region as well as the ones for the octet transition γ(∗)Σ0→Λ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma ^{(*)} \\Sigma ^0 \\rightarrow \\Lambda $$\\end{document}. We discuss qualitative and quantitative features of the diquark-quark picture and compare systematically with previous results from a three-body Faddeev approach and lattice data where available.

The charge and mass symmetry breaking in the KK K¯ system

In the framework of the Faddeev equations in configuration space, we investigate the K(1460) meson as a resonant state of the KKK¯ kaonic system. We perform calculations for the particle configurations K 0 K + K − and K0K+K0¯ within two models: the ABC model, in which all three particles are distinguishable, and the AAC model when two particles are identical. The models differ in their treatment of the kaon mass difference and the attractive Coulomb force between the K + K − pair. We found that the Coulomb shift adds over 1 MeV to the three-body binding energy. The expected correction to the binding energy due to mass redistribution from AA to AB is found to be negligible, up to a maximum of 6% of the relative mass correction. At the same time, the symmetry of the wave function is distorted depending on the mass ratio value. We found that the repulsive KK interaction plays an essential role in the binding energy of the KKK¯ system and reports the mass of 1461.8 or 1464.1 MeV for the neutral K 0(1460) and 1466.5 or 1468.8 MeV for the charged K +(1460) resonances, respectively, depending on the parameter sets for KK and KK¯ interactions.

Main properties of the Faddeev equation for 2 \times 2 operator matrices

In the present paper we consider a 2 \times 2 operator matrix H. We construct an analog of the well-known Faddeev equation for the eigenvectors of H and study some important properties of this equation, related with the number of eigenvalues. In particular, the Birman–Schwinger principle for H is proven.

Relativistic Faddeev 3D equations for three-body bound states without two-body t-matrices

Abstract This paper explores a novel revision of the Faddeev equation for three-body (3B) bound states, as initially proposed in Ref. [J. Golak, K. Topolnicki, R. Skibiński, W. Glöckle, H. Kamada, A. Nogga, Few Body Syst. 54, 2427 (2013)]. This innovative approach, referred to as t-matrix-free in this paper, directly incorporates two-body (2B) interactions and completely avoids the 2B transition matrices. We extend this formalism to relativistic 3B bound states using a three-dimensional (3D) approach without using partial wave decomposition. To validate the proposed formulation, we perform a numerical study using spin-independent Malfliet–Tjon and Yamaguchi interactions. Our results demonstrate that the relativistic t-matrix-free Faddeev equation, which directly implements boosted interactions, accurately reproduces the 3B mass eigenvalues obtained from the conventional form of the Faddeev equation, referred to as t-matrix-dependent in this paper, with boosted 2B t-matrices. Moreover, the proposed formulation provides a simpler alternative to the standard approach, avoiding the computational complexity of calculating boosted 2B t-matrices and leading to significant computational time savings.

Main Properties of the Faddeev Equation for 2 × 2 Operator Matrices

Main Properties of the Faddeev Equation for 2 × 2 Operator Matrices

The Faddeev and Schrödinger approaches to Efimov states—a numerical comparison

We compare effective hyperradial three-body potentials calculated using the S-wave part of the Faddeev equations to calculations using the full Schrödinger equation. As two-body model potential we test both a short-range potential and a Lennard-Jones potential with van der Waals tail. In the former case we find excellent agreement between the two methods for the lowest adiabatic state, indicating that the Faddeev method can be a useful tool also for numerical computations. For excited states the two methods show important differences, but agree for hyperradii larger than about five times the range of the potential (independent of the value of the scattering length and of the number of bound states). For the van der Waals potential, we focus on how well the Faddeev method reproduces the so-called van der Waals universality. We find that indeed the universality is manifest also using this method, but at a slightly different value of the universal parameter . We further derive an efficient method to solve the integro-differential equation arising in the Faddeev method.

Nonlocal optical potential in inelastic deuteron scattering off Mg24

Nuclear reactions are a powerful tool for obtaining information on yet unknown aspects of nuclear structure. This work combines an elaborate reaction framework, based on the Faddeev equations, with novel nonlocal nucleon-nucleus potentials including core excitations. The authors show that nonlocalities lead to an improved description of inelastic deuteron scattering data and may impact the extracted structure information. These results may serve as motivation for exploring a wider range of reaction frameworks.

Trion clustering structure and binding energy in two-dimensional semiconductor materials: Faddeev equations approach

In this work, we develop the basic formalism to study trions in semiconductor layered materials using the Faddeev equations in momentum space for three different particles lying in two dimensions. We solve the trion Faddeev coupled integral equations for both short-range one-term separable Yamaguchi potential and Rytova-Keldysh (RK) interaction applied to the MoS$_2$ layer. We devise two distinct regularization methods to overcome the challenge posed by the repulsive electron-electron RK potential in the numerical solution of the Faddeev equations in momentum space. The first method regulates the repulsive interaction in the infrared region, while the second regulates it in the ultraviolet region. By extrapolating the trion energy to the situation without screening, the two methods gave consistent results for the MoS$_2$ layer with a trion binding energy of $-49.5(1)$~meV for the exciton energy of $-753.3$~meV. We analyzed the trion structure for the RK and Yamaguchi potentials in detail, showing their overall similarities and the dominant cluster structure, where the strongly bound exciton is weakly bound to an electron. We found that this property is manifested in the dominance of two of the Faddeev components over the one where the hole is a spectator of the interacting electron pair.

Baryon and meson masses in the Nambu–Jona-Lasinio model: A Bayesian approach

We investigate the capabilities of the Nambu--Jona-Lasinio model to describe and reproduce fundamental vacuum properties of Quantum Chromodynamics, notably the hadronic spectrum. Mesons are described as quark-antiquark bound states at the level of the random phase approximation of the Bethe-Salpeter equation, while baryons are characterized as quark-diquark bound states within the static approximation of the Faddeev equation. Within a Bayesian framework, we constrain the model by phenomenologically known quantities and study the implications on its parameters and predictions in vacuum, as well as the correlations between the two. We find that within our framework, the vacuum masses of mesons and baryons can be reasonably well reproduced. Scalar diquarks need to be significantly bound in order to correctly reproduce the masses of the baryon octet, therefore enforcing values of the scalar diquark coupling larger than what is suggested by the canonical Fierz values. These findings could have important implications on the phenomenology of strongly-interacting matter at high temperature and density as well as of compact star physics.

Multiple scattering of charged particles byNatoms

The scattering of charged particles by $N$ atoms is investigated theoretically. We derive the scattering equations by applying the approach of Faddeev that was originally developed to study quantum mechanically the scattering of three particles. We focus on the case in which the distance between the $N$ target atoms is sufficiently large to allow the interactions between the atoms to be neglected. The Faddeev equations can then be reduced to a linear system whose solution is valid from the low-energy diffusive scattering regime, where the Foldy-Lax approach is often applied, to the ballistic regime, where multiple-scattering effects are small. To illustrate the method, results of calculations of the scattering of electrons by random configurations of 250 atoms, modeled by spherical potential wells, are presented. As an additional application, the solution obtained by Mott for the inelastic scattering of $\ensuremath{\alpha}$ particles by two atoms is extended to $N$ atoms.

Predictions of superexotic heavy mesons from K*B(*)B* interactions

We make a theoretical study of the three-body system composed of ${\bar K^*}{\bar B^*}{\bar B^*}$ to look for possible bound states, which could be associated to mesonic resonances of very exotic nature, containing open strange and double-bottom flavours. The three-body interaction is evaluated by using the fixed center approach to the Faddeev equations where the ${\bar B^*}{\bar B^*}$ is bound forming an $I(J^P)=0(1^+)$ state, as it was found in previous works, and the third particle, the ${\bar B^*}$, of much smaller mass, interacts with the components of the cluster. We obtain bound states for all the channels considered: spin $J=0$, 1 and 2, all of them with isospin $I=1/2$ and negative parity.

$$^{3}\hbox{He} - \alpha $$ Radiative Capture Reaction by EFT

We have investigated the $$^3\hbox{He} (\alpha ,\gamma ) ^7\hbox{Be}$$ radiative capture reaction using the pionless Effective Field Theory (EFT). The amplitude, cross-section, and S-factor for the $$E_1$$ and $$E_2$$ transitions to the $$^{7}\hbox{Be}$$ ground state have been computed using the Faddeev equation approach up to Next-to Leading Order in the astrophysical energy range. Our aim has been the study of Coulomb interaction effects and the contribution of EFT higher-order calculations in optimizing the results. The computed S-factor for the reaction agrees with experimental data and previous theoretical calculations.

Geometrical properties of ΩNN three-body states by realistic NN and first principles Lattice QCD ΩN potentials

The Faddeev equations in coordinate space are solved to study the ΩNN and ΩΩN three-body systems using the latest ΩNS25 and ΩΩ S01 interactions developed by the HAL QCD Collaboration. We recalculate the binding energy of the ΩNN system by examining three NN potentials, i.e., modern realistic AV18 potential, Yukawa-type Malfliet-Tjon (MT) interaction, and Gogny-Pires-Tourreil (GPT) soft and local potential. We take into account the contribution of the Coulomb potential. Our numerical calculations for Ωd(T=0) in maximum spin 5/2+ confirm ground state binding energy of 20.953, 19.368, and 20.439 MeV and a matter radius of 1.097, 1.373, and 1.309 fm using MT, GPT, and AV18 NN potentials, respectively. In the case of Ωd(0)5/2+ system, our numerical analysis shows that considering higher partial waves than s wave in NN interactions leads to an increase of about 0.2 MeV using GPT and about 0.1 MeV reduction with AV18 potentials. We study the convergence of three-body binding energies in a cluster model using the hyperspherical harmonics method and investigate the geometrical properties of Ωd(0)5/2+ ground states.

Faddeev fixed-center approximation to the ηK*K¯* , πK*K¯* , and KK*K¯* systems

The three-body $\eta K^*\bar{K}^*$, $\pi K^*\bar{K}^*$ and $KK^*\bar{K}^*$ systems are investigated within the framework of fixed-center approximation to the Faddeev equations, where $K^*\bar{K}^*$ is treated as the scalar meson $f_0(1710)$. The interactions between $\pi$, $\eta$, $K$ and $K^*$ are taking from the chiral unitary approach. By scattering the $\eta$ meson on the clusterized $(K^*\bar{K}^*)_{f_0(1710)}$ system, we find a peak in the modulus squared of the three-body scattering amplitude and it can be associated as a bound state with quantum numbers $I^G(J^{PC})=0^+(0^{-+})$. Its mass and width are around 2054 MeV and 60 MeV, respectively. This state could be associated to the $\eta(2100)$ meson. For the $\pi (K^*\bar{K}^*)_{f_0(1710)}$ scattering, we find a bump structure around 1900-2000 MeV with quantum numbers $1^-({0^{-+}})$. While for the $K K^* \bar{K}^*)_{f_0(1710)}$ system, there are three structures. One of them is much stable and its mass is about 2130 MeV. It is expected that these theoretical predictions here could be tested by future experimental measurements, such as by the BESIII, BelleII and LHCb collaborations.

Molecular states of D*D*K¯* nature

We study the interaction of two ${D}^{*}$ and a ${\overline{K}}^{*}$ by using the fixed center approximation to the Faddeev equations to search for bound states of the three-body system. Since the ${D}^{*}{D}^{*}$ interaction is attractive and gives a bound state, and so is the case of the ${D}^{*}{\overline{K}}^{*}$ interaction, where the ${J}^{P}={0}^{+}$ bound state is identified with the ${X}_{0}(2900)$, the ${D}^{*}{D}^{*}{\overline{K}}^{*}$ system leads to manifestly exotic bound states with $ccs$ open quarks. We obtain bound states of isospin $I=1/2$, negative parity and total spin $J=0$, 1, 2. For $J=0$ we obtain one state, and for $J=1$, 2 we obtain two states in each case. The binding energies range from 56 to 152 MeV and the widths from 80 to 100 MeV.

Posterior predictive distributions of neutron-deuteron cross sections

We quantify the posterior predictive distributions (PPDs) of elastic neutron-deuteron ($nd$) scattering cross sections using nucleon-nucleon ($NN$) interactions from chiral effective field theory ($\ensuremath{\chi}\mathrm{EFT}$) up to and including next-to-next-to-next-to-leading order ($\mathrm{N}^{3}\mathrm{LO}$). These PPDs quantify the spread in $nd$ predictions due to the variability of the low-energy constants (LECs) inferred from $NN$ scattering data. We use the wave-packet continuum discretization method to solve the Alt-Grassberger-Sandhas form of the Faddeev equations for elastic scattering. We draw 100 samples from the PPDs of $nd$ cross sections up to 67 MeV in scattering energy, i.e., in the energy region where the effects of three-nucleon forces are expected to be small. We find that the uncertainty about $NN$ LECs inferred from $NN$ scattering data, when assuming uncorrelated errors, does not translate to significant uncertainty in the low-energy $nd$ continuum. Based on our estimates, the uncertainty of $nd$ predictions are dominated by the $\ensuremath{\chi}\mathrm{EFT}$ truncation error, at least below $\mathrm{N}^{3}\mathrm{LO}$. At this order, the 90% credible interval of the PPD and the truncation error are comparable, although both are very small on an absolute scale.

Exotic states with triple charm

In this work we investigate the possibility of the formation of states from the dynamics involved in the D^*D^*D^* system by considering that two D^*’s generate a J^P=1^+ bound state, with isospin 0, which has been predicted in an earlier theoretical work. We solve the Faddeev equations for this system within the fixed center approximation and find the existence of J^P=0^-, 1^- and 2^- states with charm 3, isospin 1/2, masses sim 6000 MeV, which are manifestly exotic hadrons, i.e., with a multiquark inner structure.

Wave functions of (I,JP)=(12,32∓) baryons

Using a Poincar\'e-covariant $\mathrm{quark}+\mathrm{diquark}$ Faddeev equation, we provide structural information on the four lightest $(I,{J}^{P})=(\frac{1}{2},{\frac{3}{2}}^{\ensuremath{\mp}})$ baryon multiplets. These systems may contain five distinct types of diquarks; but in order to obtain reliable results, it is sufficient to retain only isoscalar-scalar and isovector-axialvector correlations, with the latter being especially important. Viewed with low resolution, the Faddeev equation description of these states bears some resemblance to the associated quark model pictures; namely, they form a set of states related via orbital angular momentum excitation: the negative parity states are primarily $\mathsf{P}$-wave in character, whereas the positive parity states are $\mathsf{D}$ wave. However, a closer look reveals far greater structural complexity than is typical of quark model descriptions, with $\mathsf{P}$, $\mathsf{D}$, $\mathsf{S}$, $\mathsf{F}$ waves and interferences between them all playing a large role in forming observables. Large momentum transfer resonance electroexcitation measurements can be used to test these predictions and may thereby provide insights into the nature of emergent hadron mass.