O(4) Expansion of the ladder Bethe-Salpeter equation

The Bethe-Salpeter amplitude is expanded on a hyperspherical basis, thereby reducing the original 4-dimensional integral equation into an infinite set of coupled 1-dimensional ones. It is shown that this representation offers a highly accurate method to determine numerically the bound state solutions. For generic cases only a few hyperspherical waves are needed to achieve convergence, both for the ground state as well as for radially or orbitally excited states. The wave function is reconstructed for several cases and in particular it is shown that it becomes independent of the relative time in the nonrelativistic regime.Comment: 21 pages, revte

Nonperturbative study of generalized ladder graphs in a \phi^2\chi theory

The Feynman-Schwinger representation is used to construct scalar-scalar bound states for the set of all ladder and crossed-ladder graphs in a \phi^2\chi theory in (3+1) dimensions. The results are compared to those of the usual Bethe-Salpeter equation in the ladder approximation and of several quasi-potential equations. Particularly for large couplings, the ladder predictions are seen to underestimate the binding energy significantly as compared to the generalized ladder case, whereas the solutions of the quasi-potential equations provide a better correspondence. Results for the calculated bound state wave functions are also presented.Comment: 5 pages revtex, 3 Postscripts figures, uses epsf.sty, accepted for publication in Physical Review Letter

Nonperturbative dynamics of scalar field theories through the Feynman-Schwinger representation

In this paper we present a summary of results obtained for scalar field theories using the Feynman-Schwinger (FSR) approach. Specifically, scalar QED and chi^2phi theories are considered. The motivation behind the applications discussed in this paper is to use the FSR method as a rigorous tool for testing the quality of commonly used approximations in field theory. Exact calculations in a quenched theory are presented for one-, two-, and three-body bound states. Results obtained indicate that some of the commonly used approximations, such as Bethe-Salpeter ladder summation for bound states and the rainbow summation for one body problems, produce significantly different results from those obtained from the FSR approach. We find that more accurate results can be obtained using other, simpler, approximation schemes.Comment: 25 pags, 19 figures, prepared for the volume celebrating the 70th birthday of Yuri Simono

Relativistic bound-state equations in three dimensions

Firstly, a systematic procedure is derived for obtaining three-dimensional bound-state equations from four-dimensional ones. Unlike ``quasi-potential approaches'' this procedure does not involve the use of delta-function constraints on the relative four-momentum. In the absence of negative-energy states, the kernels of the three-dimensional equations derived by this technique may be represented as sums of time-ordered perturbation theory diagrams. Consequently, such equations have two major advantages over quasi-potential equations: they may easily be written down in any Lorentz frame, and they include the meson-retardation effects present in the original four-dimensional equation. Secondly, a simple four-dimensional equation with the correct one-body limit is obtained by a reorganization of the generalized ladder Bethe-Salpeter kernel. Thirdly, our approach to deriving three-dimensional equations is applied to this four-dimensional equation, thus yielding a retarded interaction for use in the three-dimensional bound-state equation of Wallace and Mandelzweig. The resulting three-dimensional equation has the correct one-body limit and may be systematically improved upon. The quality of the three-dimensional equation, and our general technique for deriving such equations, is then tested by calculating bound-state properties in a scalar field theory using six different bound-state equations. It is found that equations obtained using the method espoused here approximate the wave functions obtained from their parent four-dimensional equations significantly better than the corresponding quasi-potential equations do.Comment: 28 pages, RevTeX, 6 figures attached as postscript files. Accepted for publication in Phys. Rev. C. Minor changes from original version do not affect argument or conclusion

Cross-ladder effects in Bethe-Salpeter and Light-Front equations

Bethe-Salpeter (BS) equation in Minkowski space for scalar particles is solved for a kernel given by a sum of ladder and cross-ladder exchanges. The solution of corresponding Light-Front (LF) equation, where we add the time-ordered stretched boxes, is also obtained. Cross-ladder contributions are found to be very large and attractive, whereas the influence of stretched boxes is negligible. Both approaches -- BS and LF -- give very close results.Comment: 11 pages, 7 figure

Solving Bethe-Salpeter equation in Minkowski space

We develop a new method of solving Bethe-Salpeter (BS) equation in Minkowski space. It is based on projecting the BS equation on the light-front (LF) plane and on the Nakanishi integral representation of the BS amplitude. This method is valid for any kernel given by the irreducible Feynman graphs. For massless ladder exchange, our approach reproduces analytically the Wick-Cutkosky equation. For massive ladder exchange, the numerical results coincide with the ones obtained by Wick rotation.Comment: 10 pages, 4 figure