Two-body Bound State Research Articles

Three body asymptotic completeness forP(?)2 models

We consider weakly coupled even λP(Φ)2 models that do not have a two-body bound state, and prove asymptotic completeness on the subspace of states with mass between 3m+a(λ) and 4m−b(λ), wherea andb are positive functions tending to zero with λ. The analytic structure of the six point function, integrated over the three incoming momenta, shows only two Landau singular manifolds (plus normal thresholds) associated to three particle processes.

Relativistic bound states of fermions and antifermions. II

We determine the eigenstates of the bound-state Hamiltonian proposed in the preceding article using the covariant Bethe-Salpeter (BS) formalism. The interaction Hamiltonian is the invariant direct four-fermion interaction, and we use the invariant cutoff function. We evaluate the two-body irreducible kernel, considering a very wide class of Feynman diagrams including the ones with the infinite order, which we approximate and express in terms of parameters. This kernel is expected to be adequate for the $S$-wave bound states. We then solve the BS equation for the bound state exactly. We derive in this way a set of eigenvalue equations, which is formally identical to the set obtained in the preceding article. There appears no difficulty in explaining all the observed pseudoscalar and vector mesons by adjusting the parameters in the eigenvalue equations. These mesons include $\ensuremath{\pi}$, $\ensuremath{\rho}$, and all the way up to ${\ensuremath{\eta}}_{C}$ and $\ensuremath{\psi}(3100)$. In particular we can achieve this fitting independently of the cutoff parameters. Assuming that $\ensuremath{\Upsilon}$ is also a two-body bound state of $B\overline{B}$, where $B$ is the underlying bottom fermion, we determine not only the physical mass of $B$ but also the masses of all the expected pseudoscalar and vector bottom mesons. The confirmation of these masses is to be considered as a strong support to this approach.

Solution of a relativistic quasipotential wave equation for a two-body bound state

We give a short derivation of a relativistic two-body bound-state quasipotential integral equation the kernel of which is symmetric, fully continuous, and positive definite. We discuss a numerical solution of the eigenvalue problem based on the analytical structure of the kernel. We have found the first 50 approximate eigenvalues and the corresponding eigenfunctions. We have also estimated the error in the numerical calculation. We give a graphical representation of the first four eigenfunctions.NUCLEAR STRUCTURE Deuteron bound state. Numerical solution of the relativistic quasipotential equation for a scalar two-body bound state.

Efimov effect in the four-body case

Introducing a suitable cutoff parameter r/sub 0/ for the effective potential in the three-body subsystem, we show that the number of four-body bound states is roughly k ln(epsilona(r/sub 0/)), where epsilon is a momentum cutoff and 2/3..pi.. < or = k < or = 1/3, if the scattering length a(r/sub 0/) for a two-body bound state and a third particle tends to infinity. In a numerical calculation we found that the value r/sub 0/ approx. = 1.82 fm, which provides a sufficiently resonant effective interaction in the three-body subsystem, is independent of the applied two-body potentials within the numerical accuracy.

Sturmian expansion method for bound state problems

A novel precision method is developed to solve a two-body bound state problem where the interaction may be an arbitrary relativistic, complex, nonlocal operator and, in particular, may contain also the Coulomb potential. The method is based on the expansion of the Green's function (free or Coulomb modified) in terms of a discrete set of Coulomb Sturm functions which have equally simple forms in x or p space representation. The method should be particularly useful for hadronic atom problems. Other applications are also suggested.

Efimov effect in an analytically solvable model

The Efimov effect is demonstrated in a model consisting of two heavy particles and a light one when the light-heavy interaction leads to a zero-energy two-body bound state. The model is solved in the Born-Oppenheimer approximation with the light-heavy interaction taken to be a separable S-wave potential of Yamaguchi form. It is demonstrated that in the case of a- two-body zero-energy bound state the binding energy of the light particle in the two-center potential exactly yields an effective 1 r 2 potential for the relative motion of the heavies. If the light-heavy mass ratio is made small enough, infinitely many bound states (the Efimov effect) are obtained. The approach to this limit is studied and the nature of the potential for large scattering length is obtained. An upper bound for the number of bound states is calculated using a result of Bargmann and Calogero and Efimov's ln( a r 0) result is found. We note that the long-range effect arises from the large extent of the bound state, the pair wave function being essentially exp(− r a ) when the scattering length a is large.

Two-dimensional many-body problem I. the low-density hard-disk bose gas

The ground-state energy of a low-density hard-disk bose gas is analysed with a variational calculation. An upper bound, using a Jastrow trial function, and a lower bound are obtained which show the same leading functional dependence and hence the dependence of the lead term in the ground-state energy is firmly established. The upper-bound calculation is extended to provide estimates of succeeding terms. The calculation parallels calculations for the hard-sphere gas, but there is a marked difference in the treatment of the third term. Schick's results for the hard-disk gas are recovered. The upper bound for the leading term is generalized to pair potentials which do not support a two-body bound state and calculations for a two-dimensional Lennard-Jones fluid are discussed.

Relativistic two-body bound state

We argue that the relativistic bound state of a fermion and an antifermion can be studied based upon an effective Hamiltonian which consists of only four fermion fields and is invariant and local in space and time. This four-fermion interaction has to be invariant under the interchange of two fermion fields which are the same. As a consequence, this effective interaction correlates various bound states which have different spin and orbital configurations, and also belong to different isospin or unitary multiplets. It also determines the bound states of two fermions. This correlation among all these bound states turns out to be quite stringent. Thus, if the effective interaction is SU(3) invariant and required to yield $\ensuremath{\pi}$ and $\ensuremath{\rho}$ as bound states, it necessarily implies a bound state which does not seem to exist at all, if the underlying fermions are assumed to be the usual quarks with or without color. We manage to maintain consistency with experiment if these fermions are assumed to be the usual octet of baryons, primarily because there are two kinds of octet available in this case. The mass difference between $\ensuremath{\pi}$ and $\ensuremath{\rho}$ is easy to explain in this study as a relativistic kinematical effect.

Hard-core potentials in quantum mechanics

The treatment of hard-core potentials in quantum mechanics is described. Known techniques and results are collected and discussed here so that they may be readily available. Two-body bound-state and scattering problems are considered in some detail. The use of hard-core potentials in analyses of more complicated systems is indicated.

Semiclassical bound-state methods in four-dimensional field theory: Trace identities, mode sums, and renormalization for scalar theories

Semiclassical and inverse scattering methods, previously restricted to two-dimensional models, are extended to four-dimensional scalar field theories. The necessary trace identities are derived, mode sums are carried out, and renormalization is explicitly demonstrated. The four-dimensional $\mathrm{O}(N)$ model is used to illustrate these techniques. A renormalized expression for bound-state energies of this model, parametrized by the scattering data of a three-dimensional Schr\"odinger equation, is obtained. This expression contains nontrivial quantum fluctuation effects, and corresponds to the leading-order term of a $\frac{1}{N}$ expansion. In this model, stable $n$-body bound states do not seem to appear, even though a two-body bound state has been found in a $\frac{1}{N}$ expansion of the Green's functions. Possible reasons for this failure to bind are discussed.

Super-soft-core nucleon-nucleon interaction with π-, ρ- and gw-exchange contributions

An improved version of the super-soft-core interaction is presented. The known π-, ρ- and gw-exchange contributions are incorporated, while the core and the remainder of the intermediate range are treated phenomenologically. The potential gives a high-quality fit to the two-body bound-state and scattering data up to 350 MeV. The amplitudes are weak in the sense that in nuclear matter, the ratio of second-order to first-order energy perturbations (for the scalar components) are in the region 20–30%. The wound integral κ = 0.077 at k F = 1.4 fm −1. The triplet-even tensor component is weaker than in other phenomenological potentials, but the deuteron form factor is well reproduced. The percentage D-state is 5.92%.

Two-particle transfer reactions: A four-body treatment I

A method is proposed through which interaction between transferred nucleons can be introduced to all orders. The transition amplitude for the two-nucleon transfer reaction is thus obtained showing the relevant two-particle transition amplitude that describes the interaction between the transferred nucleons. In the presence of two-body bound state in the transferred pair the total cross section is split into two parts one describing the transfer of a bound system of two nucleons, the other part corresponds to corrections due to the continuum. General remarks about the application of these ideas to heavy-ion scattering are made.

T-Matrix Perturbation Theory for the Three-Nucleon System with the Reid Potential

The binding energy of $^{3}\mathrm{H}$ for the Reid soft-core $^{1}S_{0}$ and $^{3}S_{1}\ensuremath{-}^{3}D_{1}$ potentials was determined, using $T$-matrix perturbation theory, to be 7.02 MeV. The one-term separable approximation to the Reid potential, constructed using the deuteron wave function in the $^{3}S_{1}\ensuremath{-}^{3}D_{1}$ channel and the form factor associated with the antibound state in the $^{1}S_{0}$ channel, gave a binding energy of 7.15 MeV. This is close enough to the exact answer to be very useful in realistic three-nucleon calculations. It also demonstrates the important contribution that the two-body bound state wave functions make to the triton binding energy.

Construction d'un potentiel nucléon-nucléon à cȩur très mou (ssc)

A phenomenologieal local nucleon-nucleon potential has been constructed satisfying the following criteria: a high-quality fit to two-body bound-state and scattering data up to 350 MeV; the amplitudes of all components are weak in the sense that perturbation theory in nuclear matter converges rapidly; the tensor force in the triplet even state is much weaker than in other potential models. Close agreement is found between Brueckner-theory and perturbation-theory calculations in nuclear matter. The deuteron form factor is well reproduced with P D = 5.5%.

Partly Nonlocal Potentials in Nuclear Matter

A set of phase-equivalent partly nonlocal two-nucleon potentials in the s state are constructed, with short range separable-type repulsion superimposed on a local attractive potential of longer range. Nuclear matter calculations with potentials of this type reveal qualitative differences when the results are compared with those obtained from phase-equivalent rank-two separable potentials. These differences are linked to the different behaviors of the two-body bound state wave functions generated by an attractive local potential and its phase-equivalent separable counterpart.

K-matrix theory of three-body problems

The two-body Green function in the three-body system is decomposed into open channels and closed channels. The three-body wave function is decomposed in a manner that the scattering amplitudes satisfy the reciprocity relations. On the basis of this wave function, the relations between the S-, T- and K-matrices are obtained just in the same form as in the two-body problem. The integral kernel of these matrices do not contain any δ-function-type divergence. The eigenstates of the closed three-body channels are the bound states if the total energy of the system is negative. Below the threshold of the break-up channel, the three-body bound state appears as a sharp resonance if the closed channels are coupled to the open channel. The eigenstate above the threshold of the break-up channel is the three-body resonance. The relation between the two-body bound state and the three-body bound state is discussed for a simple example. The existence of the three-body resonance above the threshold of the break-up channel is demonstrated also for a simple example. The Breit-Wigner formula is derived in a unified way for below and above the threshold of the break-up channel.

SeparabletMatrices and the Three-Body Problem

A new, separable expansion of the two-body $t$ matrix is presented. Such an expansion reduces the Faddeev equations to coupled equations in one continuous variable. The leading term of the expansion is the separable approximation suggested by Kowalski and Noyes. Any separable approximation to the $t$ matrix obtained by truncating the expansion is exact half-off the energy shell, exactly satisfies the off-shell unitarity relation, and duplicates the exact $t$ matrix in the neighborhood of two-body bound-state and resonance energies. The rate of convergence of the expansion is tested by means of examples. Two terms give a good approximation to the $S$-wave part of the $t$ matrix arising from a square-well potential, which fits the low-energy two-nucleon scattering data in an average way. It is also shown that the first term of the expansion gives a very good approximation to the $t$ matrix arising from a pure hard-core potential. Results are given for the binding energy of a system of three identical spinless particles interacting via square-well potentials with and without hard cores. The two potentials have the same scattering length and effective range. The potential without a core produces a three-body binding energy of 10.1 MeV; the potential with a core produces a three-body binding energy of 8.00 MeV.

Separable Expansions of the Two-BodyTMatrix for Local Potentials and Their Use in the Faddeev Equation

An expansion of the off-shell two-body $T$ matrix is introduced in the form of a sum of terms separable in the initial- and the final-momentum variables. The convergence of this expansion is tested against exact solutions of several local potential problems. The substitution of the expansion into the Faddeev equations yields a set of coupled integral equations in one variable. As an example, the binding energy of three identical bosons is calculated using an $S$-wave Yukawa potential for the two-body interaction. It is found that a stationary state exists for the potential strength $G\ensuremath{\ge}1.4$, whereas a two-body bound state requires $G\ensuremath{\ge}1.8$. An off-shell effective-range formula is introduced for problems in which the shape of the two-body potential is not well known. It is shown that the difference between the solutions of the Yukawa potential and the exponential potential with the same scattering length and effective range is comparable to the deviation of the off-shell effective-range formula from either solution.

Proton-Deuteron Elastic Scattering at High Momentum Transfers

Abstract : An experimental and theoretical investigation of proton-deuteron elastic scattering at high momentum transfers is presented. The differential cross sections for backward elastic scattering at incident proton kinetic energies of 1.0, 1.3, and 1.5 GeV were measured for four-momentum transfer squared (-t) from 2.6 to 5.0 (GeV/c)sq, which corresponds to cosine of center-of-mass proton scattering angles from -0.5 to -0.9. A backward peak is observed, and the slope and magnitude of the peak were determined. At 2.0 GeV for forward elastic differential cross section was measured for -t from 0.44 to 1.54 (GeV/c)sq. A shoulderlike departure from the forward diffraction peak was observed. The one-nucleon exchange peripheral model was successful in interpreting the backward peak. Calculations based on modern three-body quantum mechanical formalisms for the three-nucleon system supporting a two-body bound state also suggest the one-nucleon exchange process as the dominant mechanism. The measured forward differential cross section has been explained by the importance of double-scattering of the incident proton at higher momentum transfers. A negative value for the ratio of the real part to the imaginary part of the neutron-proton elastic scattering amplitude at 2.0 GeV is shown to yield maximum agreement with the experimental data. (Author)

Two-body Bound State and Scattering with Non-central Forces

A class of local potentials containing central, tensor, and L.S parts is con� sidered, for which the n-p coupled radial equations for the states of any J reduce to uncoupled equations over a region R of r. If R is finite all potentials may be of short range. The possibilities of describing the deuteron and n-p scattering by this model are considered. It is shown that the observed values of the high.-energy mixing parameters �4 may be fitted without solving the wave equations.