Two-body Propagators Research Articles

Revisiting Universal Variables for Robust, Analytical Orbit Propagation Under the Vinti Potential

To meet the growing complexity and demands of modern spacecraft missions, analytical solutions to initial value problems see continued use, typically supporting global searches of large trajectory design spaces. These efforts often employ universal two-body orbit propagators for their recognized speed and robustness, but many applications, like active space debris removal, would benefit from a comparable propagator with greater accuracy. Vinti propagators, which consider planetary oblateness, may serve this purpose, but existing Vinti solutions possess computational difficulties in certain orbital regimes. To mitigate these deficiencies, the present study develops and validates an analytical, third-order universal Vinti propagator free of computational difficulties by leveraging standard, oblate spheroidal (OS) universal variables and OS equinoctial orbital elements. Accuracy of the third-order approximation is assessed for multiple examples across an array of orbital regimes. Computational runtime is also evaluated, and performance is directly compared to the benchmark Vinti6 algorithm. On average, the Vinti propagator implemented in this work is only slower than a typical universal Kepler propagator by a factor of 4.0 and slower than Vinti6 by a factor of 1.8, but with greater robustness than the benchmark. The new form of the equations of motion also has favorable implications for the associated boundary value problem.

Incorporating exact two-body propagators for zero-range interactions intoN-body Monte Carlo simulations

Ultracold atomic gases are, to a very good approximation, described by pairwise zero-range interactions. This paper demonstrates that $N$-body systems with two-body zero-range interactions can be treated reliably and efficiently by the finite temperature and ground state path integral Monte Carlo approaches, using the exact two-body propagator for zero-range interactions in the pair product approximation. Harmonically trapped one- and three-dimensional systems are considered. A new propagator for the harmonically trapped two-body system with infinitely strong zero-range interaction, which may also have applications in real time evolution schemes, is presented.

Quasiparticle properties of an impurity in a Fermi gas

We report on a study of a spin-down impurity strongly coupled to a spin-up Fermi sea (a so-called Fermi polaron) with the diagrammatic Monte-Carlo (DiagMC) technique. Conditions of zero temperature and three dimensions are considered for an ultracold atomic gas with resonant interactions in the zero-range limit. A Feynman diagrammatic series is developed for the one-body and two-body propagators providing information about the polaron and molecule channel respectively. The DiagMC technique allows us to reach diagram orders that are high enough for extrapolation to infinite order. The robustness of the extracted results is examined by checking various resummation techniques and by running the simulations with various choices for the propagators and vertex functions. It turns out that dressing the lines in the diagrams as much as possible is not always the optimal choice. We also identify classes of dominant diagrams for the one-body and two-body self-energy in the region of strong interaction. These dominant diagrams turn out to be the leading processes of the strong-coupling limit. The quasiparticle energies and $Z$-factor are obtained as a function of the interaction strength. We find that the DiagMC results for the molecule and polaron properties are very similar to those obtained with a variational ansatz. Surprisingly, this variational ansatz gives very good predictions for the quasiparticle residue even when this residue is significantly smaller than one.

Obtaining the one-body limit from the relativistic two-body equation

We compare several relativistic two-body propagators by writing them in a Lorentz invariant but non-spectral form using relativistic Jacobi coordinates. By doing so we are able, for spin-zero particles, to identify a factor, R, in several of the propagators which is responsible for the incorrect behaviour as one of the particles becomes very heavy (the one-body limit). Since the factor is one for equal-mass particles, it is possible to obtain propagators that agree with the well-known propagators for the equal-mass case but give the correct one-body limit. For Dirac particles the above procedure can still leave positive projection operators which destroy the one-body limit and are inconsistent with Dirac phenomenology. A small additional modification can cure this problem.

Smooth relativistic two-body propagators

Abstract In this paper we develop two-body propagators for use in relativistic systems. The starting point is the observation of Thies that the success of the Dirac approach may be attributed to the cancellation of short-range structure generated by the iteration of a momentum dependent potential in the Lippmann-Schwinger equation. We elevate this observation to a principle and look for two-body propagators that have as little short-range structure as possible. The propagators so found treat the two particles symmetrically, reduce to the correct relativistic equation when either one of the two particles becomes very massive and have the correct contribution from the two-body cut. For the pi-nucleon system we find that both the soft-pion theorems and unitarity can be easily satisfied. For proton-nucleus or electron-nucleus scattering we get a simple method of including nuclear recoil. The present approach generates propagators similar to those of Todorov and of Crater and Van Alstine.

Two-body propagators with dynamic effective interactions in a many-fermion medium: I. General structure

Excitations of the many-body medium cause dynamic effective interactions between fermions. In actual calculations these interactions are almost always replaced by static approximations. In this paper a Green function approach is outlined which preserves the dynamic structure of the interactions. The central quantity is the four-vertex or two-fermion scattering amplitude Γ in the medium, which depends on three energy variables s, t, and u. Integral equations connect Γ to irreducible effective interactions in the three channels. With static interactions the s- and t-channels correspond to particle-particle and particle-hole configurations, respectively. Dynamic interactions mix these configurations. They also introduce poles in the crossed channels: the exchange of a phonon is a t-pole in the s-channel while pairons induce s-poles in the t-channel. Poles of the one-body propagator G( ω) and the two-body propagators K( s) and K( t) correspond to states of the A ± 1, A ± 2 and A-particle systems. The dynamic coupling between channels results in an intricate connection between various aspects of the many-fermion problem.

Two-body propagators with dynamic effective interactions in a many-fermion medium: II. s-Channel

In open-shell nuclear structure calculations collective excitations of the closed-shell core play an important role. When the energies of these excitations are comparable to the spread of energies in the model space a dynamic treatment of the core states is essential. In a Green function formulation dynamic core states, represented by particle-hole phonons, are included in the effective interaction of the s-channel integral equation for the fourfermion vertex function Γ. The solution of the equation is obtained analytically and from it the s-channel two-body propagator K is calculated. The structure of the propagator K reveals that, in addition to poles corresponding to the normal shell model states, many new poles of K are obtained. These correspond to two-particle- n-phonon configurations. Furthermore, it is shown that, due to the phonons, correlations of the ground state are induced in the calculation of the propagator K. The difference between these correlations and the well-known particle-particle-RPA correlations also included in the formalism is pointed out. The expressions derived can easily be handled numerically. In the calculations the physical states of single-particle and single-hole nuclei can be incorporated by an appropriate dressing of the single-fermion Green function.

Two-body propagators with dynamic effective interactions in a many-fermion medium: III. t-Channel

The integral equation for the four-fermion vertex function Γ in a many-body medium is solved in the t-channel. The usual RPA is extended by taking into account retarded and advanced effects of the particle-hole interaction. Specific attention is paid to the exchange of pp- and hh-vibrations between the ph-pair. Other types of dynamic interaction such as ph-phonons propagating in time are also discussed. The physical relevance of dynamic interactions is pointed out by comparing with other extensions of the RPA. Situations in which dynamic interactions are expected to be important are discussed.

Two-body propagators with dynamic effective interactions in a many-fermion medium. I. General structure: C. A. Engelbrecht, F. J. W. Hahne, and W. D. Heiss. Atomic Energy Board, Pelindaba, Transvaal, South Africa

Two-body propagators with dynamic effective interactions in a many-fermion medium. I. General structure: C. A. Engelbrecht, F. J. W. Hahne, and W. D. Heiss. Atomic Energy Board, Pelindaba, Transvaal, South Africa

Two-body propagators with dynamic effective interactions in a many-fermion medium. II. s-Channel : F. J. W. Hahne, W. D. Heiss, and C. A. Engelbrecht. Atomic Energy Board, Pelindaba, Transvaal, South Africa

Two-body propagators with dynamic effective interactions in a many-fermion medium. II. s-Channel : F. J. W. Hahne, W. D. Heiss, and C. A. Engelbrecht. Atomic Energy Board, Pelindaba, Transvaal, South Africa

Two-body propagators with dynamic effective interactions in a many-fermion medium. III. t-Channel : W. D. Heiss, C. A. Engelbrecht, and F. J. W. Hahne. Atomic Energy Board, Pelindaba, Transvaal, South Africa

Two-body propagators with dynamic effective interactions in a many-fermion medium. III. t-Channel : W. D. Heiss, C. A. Engelbrecht, and F. J. W. Hahne. Atomic Energy Board, Pelindaba, Transvaal, South Africa

Propagators of a self-coupled spinor field in Edwards and Lieb’s approximation

The one- and two-body propagators of the scalar Fermi self-coupling are investigated by using their continuous integral representatinns. The Edwards- Lieb nonperturbative method is used to find the propagators in the-first approximation. The existence of ghost statss in this method is discussed. (T.F.H.)