Three-body Amplitudes Research Articles

An Alternative Scheme for Pionless EFT: Neutron-Deuteron Scattering in the Doublet S-Wave

Using the 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 breakdown of unitarity in the three-body amplitude. We discuss the extension of our alternative subtraction scheme for including effective range corrections in pionless effective field theory for spinless bosons to nucleons. In particular, we consider the neutron-deuteron system in the doublet S-wave channel explicitly.

Spectroscopic amplitudes in microscopic three-cluster systems

I investigate three-body spectroscopic amplitudes in a microscopic three-cluster model. The total wave functions are described in the resonating group method (RGM) formalism based on cluster wave functions and on relative functions defined in the hyperspherical method. Core excitations are included. I develop a method aimed to determine three-body spectroscopic amplitudes and spectroscopic factors. This technique also provides three-body wave functions where the antisymmetrization is treated approximately. In this way, the various channels are orthogonal to each other and can be used without the antisymmetrization operator. This method is well known in two-cluster systems, and is based on the eigenvalues and the eigenvectors of the antisymmetrization operator. I illustrate the formalism with the $^{6}\mathrm{He}$, $^{11}\mathrm{Li}$, and $^{14}\mathrm{Be}$ nuclei where $^{9}\mathrm{Li}$ and $^{12}\mathrm{Be}$ excited states are taken into account.

Chiral dynamics and S-wave contributions in {bar{B}}^0/D^0 rightarrow pi ^{pm }eta l^{mp } nu decays

In this work, we analyze the semi-leptonic decays bar{B}^0/D^0 rightarrow (a_0(980)^{pm }rightarrow pi ^{pm }eta ) l^{mp } nu within light-cone sum rules. The two and three-body light-cone distribution amplitudes (LCDAs) of the B meson and the only available two-body LCDA of the D meson are used. To include the finite-width effect of the a_0(980), we use a scalar form factor to describe the final-state interaction between the pi eta mesons, which was previously calculated within unitarized Chiral Perturbation Theory. The result for the decay branching fraction of the D^0 decay is in good agreement with that measured by the BESIII Collaboration, while the branching fraction of the {bar{B}}^0 decay can be tested in future experiments.

Finite-width effects in three-body B decays

It is customary to apply the so-called narrow width approximation $\Gamma(B\to RP_3\to P_1P_2P_3)=\Gamma(B\to RP_3)Br(R\to P_1P_2)$ to extract the branching fraction of the quasi-two-body decay $B\to RP_3$, with $R$ and $P_3$ being an intermediate resonant state and a pseudoscalar meson, respectively. However, the above factorization is valid only in the zero width limit. We consider a correction parameter $\eta_R$ from finite width effects. Our main results are: (i) We present a general framework for computing $\eta_R$ and show that it can be expressed in terms of the normalized differential rate and determined by its value at the resonance. (ii) We introduce a form factor $F(s_{12},m_R)$ for the strong coupling involved in the $R(m_{12})\to P_1P_2$ decay when $m_{12}$ is away from $m_R$. We find that off-shell effects are small in vector meson productions, but prominent in the $K_2^*(1430)$, $\sigma/f_0(500)$ and $K_0^*(1430)$ resonances. (iii) We evaluate $\eta_R$ in the theoretical framework of QCD factorization (QCDF) and in the experimental parameterization (EXPP) for three-body decay amplitudes. In general, $\eta_R^{\rm QCDF}$ and $\eta_R^{\rm EXPP}$ are similar for vector mesons, but different for tensor and scalar resonances. A study of the differential rates enables us to understand the origin of their differences. (iv) Finite-width corrections to $Br(B^-\to RP)_{\rm NWA}$ obtained in the narrow width approximation are generally small, less than 10\%, but they are prominent in $B^-\to\sigma/f_0(500)\pi^-$ and $B^-\to \overline K_0^{*0}(1430)\pi^-$ decays. The EXPP of the normalized differential rates should be contrasted with the theoretical predictions from QCDF calculation as the latter properly takes into account the energy dependence in weak decay amplitudes.

Three-body non-leptonic heavy-to-heavy B decays at NNLO in QCD

Exclusive non-leptonic two-body decays of B mesons have been studied extensively in the past two decades within the framework of factorization. However, the exploration of the corresponding three-body case has only started recently, in part motivated by new data. We consider here the simplest non-leptonic three-body B decays from the point of view of factorization, namely heavy-to-heavy transitions. We provide a careful derivation of the SCET/QCDF factorized amplitudes to NNLO in αs, and discuss the numerical impact of NLO and NNLO corrections. We then study the narrow-width limit, showing that the three-body amplitude reproduces analytically the known quasi-two-body decay amplitudes, and compute finite-width corrections. Finally, we discuss certain observables that are sensitive to perturbative NLO and NNLO corrections and to higher Gegenbauer moments of the dimeson LCDAs. This is the first study of non-leptonic three-body B decays to NNLO in QCD.

Finite-Volume Spectrum of π^{+}π^{+} and π^{+}π^{+}π^{+} Systems.

The abinitio understanding of hadronic three-body systems above threshold, such as exotic resonances or the baryon spectrum, requires the mapping of the finite-volume eigenvalue spectrum, produced in lattice QCD calculations, to the infinite volume. We present the first application of such a formalism to a physical system in form of three interacting positively charged pions. The results for the ground state energies agree with the available lattice QCD results by the NPLQCD collaboration at unphysical pion masses. Extrapolations to physical pion masses are performed using input from effective field theory. The excited energy spectrum is predicted. This demonstrates the feasibility to determine three-body amplitudes above threshold from lattice QCD, including resonance properties of axial mesons, exotics, and excited baryons.

What is the right formalism to search for resonances? II. The pentaquark chain

We discuss the differences between several partial-wave analysis formalisms used in the construction of three-body decay amplitudes involving fermions. Specifically, we consider the decay Lambda _b rightarrow psi ,p K^-, where the hidden charm pentaquark signal has been reported. We analyze the analytical properties of the amplitudes and separate kinematical and dynamical singularities. The result is an amplitude with the minimal energy dependence compatible with the S-matrix principles.

Description of ρ(1700) as a $$\rho K\bar K$$ system with the fixed-center approximation

We study the $\rho K\bar{K}$ system with an aim to describe the $\rho (1700)$ resonance. The chiral unitary approach has achieved success in a description of systems of the light hadron sector. With this method, the $K \bar{K}$ system in the isospin sector $I=0$, is found to be a dominant component of the $f_0 (980)$ resonance. Therefore, by regarding the $K\bar{K}$ system as a cluster, the $f_0 (980)$ resonance, we evaluate the $\rho K\bar{K}$ system applying the fixed center approximation to the Faddeev equations. We construct the $\rho K$ unitarized amplitude using the chiral unitary approach. As a result, we find a peak in the three-body amplitude around 1739 MeV and a width of about 227 MeV. The effect of the width of $\rho$ and $f_0 (980)$ is also discussed. We associate this peak to the $\rho (1700)$ which has a mass of $1720 \pm 20$ MeV and a width of $250 \pm 100$ MeV.

Low Energy D(p, p)D Elastic Scattering Including Coulomb Force

The motivation of this paper is to obtain the three-body amplitudes for the Coulomb potential plus a nuclear force in momentum space. Not only the two-body off-shell nuclear amplitude but also the two-body off-shell Coulomb amplitude is important in the three-body calculation. For calculating the Coulomb amplitude, the modified Coulomb potential whose Fourier transformation is analytically equivalent to the pure configuration space Coulomb potential, is introduced. In addition, the decisive screening range parameter is also utilized instead of the screening range. The modified Coulomb potential plus the parameter is called decisive modified Coulomb potential. The three-body proton-deuteron elastic scattering is calculated by using the proper two-body off-shell amplitude for the decisive modified Coulomb potential.

Three-body breakup within the fully discretized Faddeev equations

A novel approach is developed to find the three-body breakup amplitudes and cross sections within the modified Faddeev equation framework. The method is based on the lattice-like discretization of the three-body continuum with a three-body stationary wave-packet basis in momentum space. The approach makes it possible to simplify drastically all the three- and few-body breakup calculations due to discrete wave-packet representations for the few-body continuum and simultaneous lattice representation for all the scattering operators entering the integral equation kernels. As a result, the few-body breakup can be treated as a particular case of multi-channel scattering in which part of the channels represents the true few-body continuum states. As an illustration for the novel approach, an accurate calculations for the three-body breakup process $n+d\to n+n+p$ with non-local and local $NN$ interactions are calculated. The results obtained reproduce nicely the benchmark calculation results using the traditional Faddeev scheme which requires much more tedious and time-consuming calculations.

Two- and Three-Body Coulomb Solution Method in a Momentum Space

We present a method to calculate two- and three-body amplitudes including the Coulomb potential in a momentum space. Our aim is to obtain the exact two-body Coulomb amplitudes used in three-body calculations, which reproduce the analytic phase shifts. For the purpose, our theory is based on the modified Coulomb potential (MCP) whose Fourier transformation is equivalent to the pure Coulomb potential in a configuration space, and the two-potential theory with an auxiliary potential. Moreover, one can analytically determine a decisive range R dec in the MCP. By using the MCP, we obtain the two-body Coulomb modified nuclear amplitude as well as the pure Coulomb amplitude. The calculated phase shift are very good fitting with the experimental data.

Surface-integral formulation of scattering theory

We formulate scattering theory in the framework of a surface-integral approach utilizing analytically known asymptotic forms of the two-body and three-body scattering wavefunctions. This formulation is valid for both short-range and long-range Coulombic interactions. New general definitions for the potential scattering amplitude are presented. For the Coulombic potentials, the generalized amplitude gives the physical on-shell amplitude without recourse to a renormalization procedure. New post and prior forms for the Coulomb three-body breakup amplitude are derived. This resolves the problem of the inability of the conventional scattering theory to define the post form of the breakup amplitude for charged particles. The new definitions can be written as surface-integrals convenient for practical calculations. The surface-integral representations are extended to amplitudes of direct and rearrangement scattering processes taking place in an arbitrary three-body system. General definitions for the wave operators are given that unify the currently used channel-dependent definitions.

Calculation of the three-body decay amplitude with high-order Fierz transformation

We present a scheme to make use of the permutation operators of the SU(4)-group to extend the Fierz identity from the usual two-bilinears case to the three-bilinears one. Using this technique, we investigate the non-resonance contributions to the charmonium decays into three pseudoscalar mesons, e.g. and so on. The result shows that the branching fractions for the direct three-body decays of ψ(2S) and J/ψ are less than 10−6. This indicates that the direct three-body decays of J/ψ and ψ(2S) are highly suppressed.

Variational principles for breakup amplitudes: Three charged clusters

Variational principles for three-body reaction amplitudes are derived which allow for colliding systems that are charged and composite and are applicable for energies lying below the threshold for breakup into four subsystems. The starting point of the analysis is a formulation of the collision dynamics based on coupled integral equations of the Faddeev type that are applicable in the presence of long-range Coulomb interactions. A variational identity (which becomes a variational approximation with the introduction of trial functions) for the amplitude for the breakup of a bound pair by a third particle is derived within the integral equation formulation. The expression is then converted to a differential form of the Kohn type involving wave functions in configuration space. Knowledge of the asymptotic form of the wave function representing the time-reversed final state, in which three unbound particles are incident, is not required in performing this derivation. The variational principle is enhanced by the existence of a subsidiary minimum principle satisfied by that component of the wave function describing virtual excitations of one or more of the three clusters that make up the scattering system.

Differential cross sections for muonic atom scattering from hydrogenic molecules

The differential cross sections for low-energy muonic hydrogen atom scattering from hydrogenic molecules are directly expressed by the corresponding amplitudes for muonic atom scattering from hydrogen-isotope nuclei. The energy and angular dependence of these three-body amplitudes is thus taken naturally into account in scattering from molecules, without involving any pseudopotentials. Effects of the internal motion of nuclei inside the target molecules are included for every initial rotational-vibrational state. These effects are very significant as the considered three-body amplitudes often vary strongly within the energy interval $\lesssim{}0.1$ eV. The differential cross sections, calculated using the presented method, have been successfully used for planning and interpreting many experiments in low-energy muon physics. Studies of $\mu^{-}$ nuclear capture in $p\mu$ and the measurement of the Lamb shift in $p\mu$ atoms created in H$_2$ gaseous targets are recent examples.

Publisher’s Note: Model-independent measurement ofS-waveK−π+systems usingD+→Kππdecays from Fermilab E791 [Phys. Rev. D 73, 032004 (2006)

A model-independent partial-wave analysis of the \swave component of the $K\pi$ system from decays of $D^{+}$ mesons to the three-body $\Km\pip\pip$ final state is described. Data come from the Fermilab E791 experiment. Amplitude measurements are made independently for ranges of $\Km\pip$ invariant mass, and results are obtained below 825 \MeVcc, where previous measurements exist only in two mass bins. This method of parametrizing a three-body decay amplitude represents a new approach to analysing such decays. Though no model is required for the \swave, a parametrization of the relatively well-known reference \pdash and \dwave s, optimized to describe the data used, is required. In this paper, a Breit-Wigner model is adopted to describe the resonances in these waves. The observed phase variation for the \sdash, \pdash and \dwave s do not match existing measurements of $I=\half$ $\Km\pip$ scattering in the invariant mass range in which scattering is predominantly elastic. If the data are mostly $I=\half$, this observation indicates that the Watson theorem, which requires these phases to have the same dependence on invariant mass, may not apply to these decays without allowing for some interaction with the other pion. The production rate of $\Km\pip$ from these decays, if assumed to be predominantly $I=\half$, is also found to have a significant dependence on invariant mass in the region above 1.25 \GeVcc. These measurements can provide a relatively model-free basis for future attempts to determine which strange scalar amplitudes contribute to the decays.

Model-independent measurement ofS-waveK−π+systems usingD+→Kππdecays from Fermilab E791

A model-independent partial-wave analysis of the S-wave component of the K{pi} system from decays of D{sup +} mesons to the three-body K{sup -}{pi}{sup +}{pi}{sup +} final state is described. Data come from the Fermilab E791 experiment. Amplitude measurements are made independently for ranges of K{sup -}{pi}{sup +} invariant mass, and results are obtained below 825 MeV/c{sup 2}, where previous measurements exist only in two mass bins. This method of parametrizing a three-body decay amplitude represents a new approach to analyzing such decays. Though no model is required for the S-wave, a parametrization of the relatively well-known reference P- and D-waves, optimized to describe the data used, is required. In this paper, a Breit-Wigner model is adopted to describe the resonances in these waves. The observed phase variation for the S-, P-, and D-waves do not match existing measurements of I=(1/2) K{sup -}{pi}{sup +} scattering in the invariant mass range in which scattering is predominantly elastic. If the data are mostly I=(1/2), this observation indicates that the Watson theorem, which requires these phases to have the same dependence on invariant mass, may not apply to these decays without allowing for some interaction with the other pion. The production rate of K{supmore » -}{pi}{sup +} from these decays, if assumed to be predominantly I=(1/2), is also found to have a significant dependence on invariant mass in the region above 1.25 GeV/c{sup 2}. These measurements can provide a relatively model-free basis for future attempts to determine which strange scalar amplitudes contribute to the decays.« less

Dalitz plot analysis in the FOCUS experiment

The K-matrix formalism has been applied to the charm sector for the .rst time in the FOCUS Dalitz plot analyses of the Ds+ and D+ → π+π−π+ final states. The results, presented here, are extremely encouraging since the same K-matrix description provides a coherent picture of both two-body scattering measurements in light-quark experiments as well as charm meson decay. Such a result was not obvious beforehand. Moreover, the non-resonant component of each decay appears to be described by known two-body S-wave dynamics, without the need to include any three-body constant amplitude contribution.

Tau polarizations in the three-body slepton decays with stau as the next lightest supersymmetric particle

In the gauge-mediated supersymmetry breaking models with scalar tau as the next-to-lightest supersymmetric particle, a scalar lepton may decay dominantly into its superpartner, tau lepton, and the lightest scalar tau particle. We give detailed formulas for the three-body decay amplitudes and the polarization asymmetry of the outgoing tau lepton . We find that the tau polarizations are sensitive to the model parameters such as the stau mixing angle, the neutralino to slepton mass ratio and the neutralino mixing effect.

Momentum space integral equations for three charged particles: Nondiagonal kernels

Standard solution methods are known to be applicable to Faddeev-type momentum space integral equations for three-body transition amplitudes, not only for purely short-range interactions but also, after suitable modifications, for potentials possessing Coulomb tails provided the total energy is below the three-body threshold. For energies above that threshold, however, long-range Coulomb forces have been suspected to give rise to such severe singularities in the kernels, even of the modified equations, that their compactness properties are lost. Using the rigorously equivalent formulation in terms of an effective-two-body theory we prove that, for all energies, the nondiagonal kernels occurring in the integral equations which determine the transition amplitudes for all binary collision processes, possess on and off the energy shell only integrable singularities, provided all three particles have charges of the same sign, i.e., all Coulomb interactions are purely repulsive. Hence, after a few iterations these kernels become compact. The case of the diagonal kernels is dealt with in a subsequent paper.