Abstract
The challenge to obtain from the Euclidean Bethe–Salpeter amplitude the amplitude in Minkowski is solved by resorting to un-Wick rotating the Euclidean homogeneous integral equation. The results obtained with this new practical method for the amputated Bethe–Salpeter amplitude for a two-boson bound state reveals a rich analytic structure of this amplitude, which can be traced back to the Minkowski space Bethe–Salpeter equation using the Nakanishi integral representation. The method can be extended to small rotation angles bringing the Euclidean solution closer to the Minkowski one and could allow in principle the extraction of the longitudinal parton density functions and momentum distribution amplitude, for example.
Highlights
Techniques to solve the Bethe–Salpeter Equation (BSE) in Minkowski space have been developed for bound state of bosons [1, 2, 3, 4, 5] and fermions [6, 7, 8], at the expense of being algebraically quite involved, either by use of the Nakanishi integral representation (NIR) [9] or by direct integration
Calculations done in Euclidean space after performing the Wick rotation of the BSE are conceptually straigthforward [10, 11], but it is nontrivial to obtain structure observables that are defined on the light-front, such as e.g. parton distributions, from Euclidean solutions
Our goal here is to present solutions of the BSE for two-bosons close to the Minkowski space, by introducing a rotation into the complex plane of k0 → k0 exp(ıθ), where θ = π/2 is the standard Wick-rotation associated with the Euclidean space formulation, while the Minkowski space formulation corresponds to θ = 0
Summary
Techniques to solve the Bethe–Salpeter Equation (BSE) in Minkowski space have been developed for bound state of bosons [1, 2, 3, 4, 5] and fermions [6, 7, 8], at the expense of being algebraically quite involved, either by use of the Nakanishi integral representation (NIR) [9] or by direct integration. 1. Introduction Techniques to solve the Bethe–Salpeter Equation (BSE) in Minkowski space have been developed for bound state of bosons [1, 2, 3, 4, 5] and fermions [6, 7, 8], at the expense of being algebraically quite involved, either by use of the Nakanishi integral representation (NIR) [9] or by direct integration. Calculations done in Euclidean space after performing the Wick rotation of the BSE are conceptually straigthforward [10, 11], but it is nontrivial to obtain structure observables that are defined on the light-front, such as e.g. parton distributions, from Euclidean solutions.
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