Abstract
The structure of the three-boson bound state in Minkowski space is studied for a model with contact interaction. The Faddeev-Bethe-Salpeter equation is solved both in Minkowski and Euclidean spaces. The results are in fair agreement for comparable quantities, like the transverse amplitude obtained when the longitudinal constituent momenta of the light-front valence wave function are integrated out. The Minkowski space solution is obtained numerically by using a recently proposed method based on the direct integration over the singularities of the propagators and interaction kernel of the four-dimensional integral equation. The complex singular structure of the Faddeev components of the Bethe-Salpeter vertex function for space and time-like momenta in an example of a Borromean system is investigated in detail. Furthermore, the transverse amplitude is studied as a mean to access the double-parton transverse momentum distribution. Following that, we show that the two-body short-range correlation contained in the valence wave function is evidenced when the pair has a large relative momentum in a back-to-back configuration, where one of the Faddeev components of the Bethe-Salpeter amplitude dominates over the others. In this situation a power-law behavior is derived and confirmed numerically.
Highlights
The Bethe-Salpeter (BS) approach is an important and efficient tool to investigate relativistic few-body systems
Concerning the Euclidean space solution, the reasons of this time lag was due to the fact that, though the BS equation was given in Ref. [1] in a simple and transparent form, as it was there presented the equation did not allow to make the Wick rotation directly
The Minkowski space solution is obtained by the direct integration of the singularities of the propagators and interaction kernel [3], what allows us to explore in the space and timelike momenta regions the complex singular structure of the Faddeev components of the BS vertex function of such a Borromean state
Summary
The Bethe-Salpeter (BS) approach is an important and efficient tool to investigate relativistic few-body systems. [2], the effect coming from higher Fock components on the binding energy and transverse amplitude is huge, even for weakly bound states This is different from the two-body case, where the truncation at the valence state does not present such a dramatic effect As the goal is to address the zero-range interaction case, a major point is the influence of relativistic effects on the stability of the three-body system and the impact on its structure To accomplish such a goal we focus on Borromean systems and the Faddeev-BS equation is solved both in Minkowski and Euclidean spaces. The Minkowski space solution is obtained by the direct integration of the singularities of the propagators and interaction kernel [3], what allows us to explore in the space and timelike momenta regions the complex singular structure of the Faddeev components of the BS vertex function of such a Borromean state. Some of the more lengthy derivations, and a brief summary of the numerical methods, are available in appendixes
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