Coupled-channel Case Research Articles

Study on hadron-hadron interaction with femtoscopic technique

We study the method to study the exotic hadrons using the femtoscopic correlation function in the high-energy collisions. The formula for the correlation function extended to the coupled-channel case is employed. We see that how the coupled-channel effect works for the correlation function from the different hadron emitting sources with the case of the K−p correlation function. The method is applied to the prediction for the DD*/DD̅* correlation function for the study of the Tcc/X(3872) state.

Dalitz plot distributions in presence of triangle singularities

We discuss properties of three-particle Dalitz distributions in coupled channel systems in presence of triangle singularities. The single channel case was discussed long ago where it was found that as a consequence of unitarity, effects of a triangle singularity seen in the Dalitz plot are not seen in Dalitz plot projections. In the coupled channel case we find the same is true for the sum of intensities of all interacting channels. Unlike the single channel case, however, triangle singularities do remain visible in Dalitz plot projections of individual channels.

Unitarity, analyticity, dispersion relations, and resonances in strongly interactingWLWL,ZLZL, andhhscattering

If the Electroweak Symmetry Breaking Sector turns out to be strongly interacting, the actively investigated effective theory for longitudinal gauge bosons plus Higgs can be efficiently extended to cover the regime of saturation of unitarity (where the perturbative expansion breaks down). This is achieved by dispersion relations, whose subtraction constants and left cut contribution can be approximately obtained in different ways giving rise to different unitarization procedures. We illustrate the ideas with the Inverse Amplitude Method, one version of the N/D method and another improved version of the K-matrix. In the three cases we get partial waves which are unitary, analytical with the proper left and right cuts and in some cases poles in the second Riemann sheet that can be understood as dynamically generated resonances. In addition they reproduce at Next to Leading Order (NLO) the perturbative expansion for the five partial waves not vanishing (up to J=2) and they are renormalization scale ($\mu$) independent. Also the unitarization formalisms are extended to the coupled channel case. Then we apply the results to the elastic scattering amplitude for the longitudinal components of the gauge bosons $V=W, Z$ at high energy. We also compute $h h \rightarrow h h$ and the inelastic process $VV\rightarrow h h$ which are coupled to the elastic $VV$ channel for custodial isospin $I=0$. We numerically compare the three methods for various values of the low-energy couplings and explain the reasons for the differences found in the $I=J=1$ partial wave. Then we study the resonances appearing in the different elastic and coupled channels in terms of the effective Lagrangian parameters.

Coupled-channel scattering in1+1dimensional lattice model

Based on the Lippmann-Schwinger equation approach, a generalized L\"uscher's formula in $1+1$ dimensions for two particles scattering in both the elastic and coupled-channel cases in moving frames is derived. A two-dimensional coupled-channel scattering lattice model is presented, which represents a two-coupled-channel resonant scattering scalars system. The Monte Carlo simulation is performed on finite lattices and in various moving frames. The two-dimensional generalized L\"uscher's formula is used to extract the scattering amplitudes for the coupled-channel system from the discrete finite-volume spectrum.

Coupled-channel scattering on a torus

Based on the Hamiltonian formalism approach, a generalized L\"uscher's formula for two particle scattering in both the elastic and coupled-channel cases in moving frames is derived from a relativistic Lippmann-Schwinger equation. Some strategies for extracting scattering amplitudes for a coupled-channel system from the discrete finite-volume spectrum are discussed and illustrated with a toy model of two-channel resonant scattering. This formalism will, in the near future, be used to extract information about hadron scattering from lattice QCD computations.

Formula omitted] methods extended for the solution of coupled channel Schrödinger equations

The successful CPM { P , N } methods for the one-dimensional time-independent Schrödinger problem are generalized to the coupled channel case. The derivation of the formulae is discussed and a Maple program code is included which allows to determine the analytic expressions of the perturbation corrections needed to construct methods of different orders. Some numerical illustrations are given showing the accuracy, the robustness and the CPU-time gain of the proposed algorithms over the sixth order LILIX method which was presented in [L.Gr. Ixaru, LILIX—A package for the solution of the coupled channel Schrödinger equation, Comput. Phys. Comm. 147 (2002) 834–852].

Energy-dependent point interactions in one dimension

We consider a new type of point interaction in one-dimensional quantum mechanics. It is characterized by a boundary condition at the origin that involves the second and/or higher order derivatives of the wavefunction. The interaction is effectively energy dependent. It leads to a unitary S-matrix for the transmission–reflection problem. The energy dependence of the interaction can be chosen such that any given unitary S-matrix (or the transmission and reflection coefficients) can be reproduced at all energies. Generalization of the results to coupled-channel cases is discussed.

Bar-shape and eigenshape parameters from the zero-energy wave functions.

It is shown that the Biedenharn-Blatt formula for the shape parameter for the coupled-channel case as an integral involving the zero-energy wave function actually gives the bar-shape parameter ${\mathit{P}}_{\mathit{b}}$ and not the eigenshape parameter ${\mathit{P}}_{\mathit{e}}$ as stated by the authors. We show that the zero-energy bar-wave functions and the zero-energy eigenwave functions are identical, and thus we obtain, by two different methods, the formula ${\mathit{P}}_{\mathit{b}}$-${\mathit{P}}_{\mathit{e}}$=${\mathit{q}}^{2}$/(${\mathit{a}}_{\mathit{t}}$${\mathit{r}}_{0}^{3}$), where ${\mathit{a}}_{\mathit{t}}$ is the triplet scattering length, ${\mathit{r}}_{0}$ the effective range, and q the energy derivative of the eigenmixing parameter at zero energy. The term ${\mathit{q}}^{2}$/(${\mathit{a}}_{\mathit{t}}$${\mathit{r}}_{0}^{3}$) results from the ${\mathit{k}}^{2}$ behavior of the eigenmixing parameter at low energies, the bar-mixing parameter having a ${\mathit{k}}^{3}$ dependence, and the order in which two particular limits are taken. We verify our new formula by considering potentials derived from the Reid hard-core potential model for the $^{3}$${\mathit{S}}_{1\mathrm{\ensuremath{-}}}^{3}$${\mathit{D}}_{1}$ state of the neutron-proton system, with the tensor potential taken to have a variable strength.

Equivalent local potentials in a coupled-channel Dirac phenomenology.

We discuss the transformation from a scalar and vector Dirac equation to either a scalar and tensor or a vector and tensor equation in both the single-channel (static potential) and coupled-channel (nonstatic potential) cases. The relationship to the \ensuremath{\sigma}-\ensuremath{\omega} Lagrangian in the mean-field theory is pointed out. The failure to obtain such a transformation in the coupled-channel case, except when the states are degenerate, is explained simply and analyzed in some detail.

Coupled channel NN interaction in the Gamow separable approximation

Abstract The Gamow separable expansion for realistic nucleon-nucleon potentials is generalized in order to include the coupled channel case. The Gamow vectors are obtained according to the method proposed by Baldo, Ferreira and Streit. Excellent accuracy is obtained in the 3 S 1  3 D 1 channel for the Reid soft-core and Argonne V14 interactions.

Gamow vectors as solutions of a non-hermitian eigenvalue problem

Gamow vectors have been extensively used for describing resonance phenomena, but they do not belong to the usual Hilbert space. They can be obtained for a single channel potential by solving the Schrödinger equation with out-going boundary conditions. A new method for calculating Gamow vectors through the solution of a matrix eigenvalue problem is presented. The matrix to be diagonalized is real and non-symmetric, allowing the treatment of bound states and resonances on the same footing. The advantage of the method is to be equally applicable to the coupled channel problem when the standard procedure of solving the set of radial differential equations have the problem of denning the regular solution at the origin. Furthermore, the solution of the eigenvalue problem gives directly the Gamow momenta and vectors, thus avoiding the usual search in the complex momentum plane which can be rather cumbersome. Applications to single as well as coupled channel cases, both for model potentials and realistic nucleon-nucleon interaction, show great accuracy of the method.

On-shell equivalent transformations for eliminating energy dependence from coupled channel equations with nonlocal linearly energy-dependent potentials.

A general method is developed for constructing on-shell equivalent transformations which eliminate the energy dependence of coupled channel equations with nonlocal linearly energy-dependent potentials. Our derivation shows that such transformations are not unique because of ambiguities of the half-off-shell t matrix compatible with the original Hermitian on-shell equivalent Hamiltonian. A class of such transformations for single and coupled channel cases is explicitly constructed. Our method is applied to a constituent quark-cluster model for nucleon-nucleon scattering.

Channel-interaction effects on the3p-subshell photoionization of chlorine

Photoionization cross sections and photoelectron angular distributions (asymmetry parameters) are obtained for $3p$-subshell ionization of the Cl ground state. A substantial amount of bound-state electron correlation is taken into account through use of a multiconfiguration description of the ground state and residual ionic core. Both independent-channel and coupled-channel final states are used in evaluating the length and velocity forms of the dipole transition amplitudes. In the coupled-channel case, channel interaction is included via the reaction-matrix ($K$-matrix) method which diagonalizes the total $N$-electron atomic Hamiltonian in the manifold of single-channel basis states of the model Hamiltonian. In all of the $^{2}D$ and $^{2}P$ final-state channels, the photoabsorption transition probabilities exhibit significant contributions from interchannel interaction. In determining the cross sections for any of the $^{2}S$ channels, it was necessary to include the low-lying $3s3{p}^{6}$; $^{2}S$ bound state because of its large Coulomb interaction with the continuum of the $3{p}^{4}^{1}D$, $\ensuremath{\epsilon}d$; $^{2}S$ channel. The addition of the channels which originate from a $3s\ensuremath{\rightarrow}\ensuremath{\epsilon}p$ transition into the set of interacting $^{2}S$ channels nearly cancels the effect of the $3s3{p}^{6}$; $^{2}S$ discrete state on the cross sections of both the $3{p}^{4}^{1}D$,$\ensuremath{\epsilon}d$; $^{2}S$ and $3{p}^{4}^{1}S$,$\ensuremath{\epsilon}s$; $^{2}S$ channels. The asymmetry parameters for the three ionic multiplets $^{3}P$, $^{1}D$, and $^{1}S$ all show strong channel-interaction effects. The effects of the $3s3{p}^{6}$; $^{2}S$ bound state and the $3s3{p}^{5}$; $^{3,1}P$,$\ensuremath{\epsilon}p$; $^{2}S$ channels on the $^{1}D$ and $^{1}S$ $\ensuremath{\beta}$ parameters are discussed.

Muskhelishvili-Omnès integral equation with inelastic unitarity: Single- and coupled-channel equations

Solutions for the Muskhelishvili-Omnes integral equation with inelastic unitarity are given for the single- and coupled-channel cases. Physical applications are discussed.

Connection Between Regge Behavior and Fixed-Angle Scattering

The physical origins and dynamical details of Regge behavior are studied by examining a general class of theories which are capable of describing the observed large-momentum-transfer data and extending them into the intermediate-t range. The mechanism responsible for the smooth connection between the deep-scattering region and the Regge region is discussed in detail. We derive a convenient new form of the exact integral equation which describes generalized ladder graphs containing irreducible scattering amplitudes as the rungs. The limiting behavior of the Regge trajectories and residues are then calculated in both the single-channel and the more realistic coupled-channel cases. The trajectories are found to approach negative constants for large negative momentum transfer and the residues fall as powers in the same limit. Furthermore, since the forward and backward Regge regimes must join smoothly onto the same fixed-angle behavior, there are relations between a priori unrelated trajectory functions and residues. The standard properties of Regge poles, in particular factorization and signature, are shown to be present even though the basic, fixed-angle interaction possesses neither of these properties. These general considerations are then applied to the more specific constituent-interchange model. Here we find that the asymptotic behavior of the trajectories and residues are controlled by the form factors of the particles involved in the scattering. Finally, we elucidate the relationship between the constituent-interchange diagrams and the Harari-Rosner duality diagrams.

Self-Consistent High-Energy Scattering of Zero-Mass Leptons: The Coupled-Channel Case

The method devised in a preceding article for calculating a consistent high-energy behavior of neutrino scattering due to neutrino exchange is extended to the case of two coupled massless fermions. It is argued that the two fermions either could be the two types of neutrinos, ${\ensuremath{\nu}}_{e}$ and ${\ensuremath{\nu}}_{\ensuremath{\mu}}$ or could even be massive leptons like the electron, as long as the mass is small compared to $\frac{1}{\sqrt{G}}$. If we call the two particles $a$ and $b$, it is shown that the $a\ensuremath{-}a$, $a\ensuremath{-}\overline{a}$, $b\ensuremath{-}b$, and $b\ensuremath{-}\overline{b}$ total cross sections, due to the exchange of $a\overline{a}$ and $b\overline{b}$ pairs, approach constant values as the energy tends to infinity. It is further shown that the $a\ensuremath{-}a$ and $a\ensuremath{-}\overline{a}$ cross sections approach one another, as do the $b\ensuremath{-}b$ and $b\ensuremath{-}\overline{b}$ cross sections. On the other hand, the $a\ensuremath{-}b$ and $a\ensuremath{-}\overline{b}$ cross sections vanish at high energies.