Abstract
The scalar three-body Bethe-Salpeter equation, with zero-range interaction, is solved in Minkowski space by direct integration of the four-dimensional integral equation. The singularities appearing in the propagators are treated properly by standard analytical and numerical methods, without relying on any ansatz or assumption. The results for the binding energies and transverse amplitudes are compared with the results computed in Euclidean space. A fair agreement between the calculations is found.
Highlights
The Bethe-Salpeter (BS) equation [1], formally defined in the Minkowski space, is an efficient tool to study relativistic systems in the non-perturbative regime
One successful way of solving the BS equation fully in Minkowski space is by looking for the solution in the form of the Nakanishi integral representation [4] combined with the light
For the first time, directly in Minkowski space, the three-body BS equation derived in Ref. [8] for scalar constituents interacting by the two-body contact interaction
Summary
The Bethe-Salpeter (BS) equation [1], formally defined in the Minkowski space, is an efficient tool to study relativistic systems in the non-perturbative regime. One of the commonest methods to solve the BS equation numerically is to perform an analytic continuation to the complex plane, through the Wick rotation [2], into the Euclidean space. After this transformation, the equation turns to be non-singular as the singularities move from the integration line (real axis) to the complex plane. The direct integration method is practicable, it is much more demanding numerically than solving the problem in the Euclidean space All these methods were successfully applied to two-body systems. In order to obtain observables, considering the many-body components beyond the valence consistently is critical to solve the four-dimensional equation fully in Minkowski space.
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