Solving the Bethe-Salpeter equation in Minkowski space for a fermion-scalar system

Abstract

The ladder Bethe-Salpeter Equation of a bound (1/2) + system, composed by a fermion and a scalar boson, is solved in Minkowski space, for the first time. The formal tools are the same already successfully adopted for two-scalar and two-fermion systems, namely the Nakanishi integral representation of the Bethe-Salpeter amplitude and the light-front projection of the fulfilled equation. Numerical results are presented and discussed for two interaction kernels: i) a massive scalar exchange and ii) a massive vector exchange, illustrating both the correlation between binding energies and the interaction coupling constants, as well as the valence content of the interacting state, through the valence probabilities and the light-front momentum distributions. In the case of the scalar exchange, an interesting side effect, to be ascribed to the repulsion generated by the small components of the Dirac spinor, is pointed out, while for the vector exchange the manifestation of the helicity conservation opens new interesting questions to be addressed within a fully non-perturbative framework, as well as the onset of a scale-invariant regime.

Fermion and photon gap-equations in Minkowski space within the Nakanishi integral representation method

“…Those steps have been explored for the BSEs in Refs. [64][65][66][67][68][69][70][71][72][73][74][75][76][77][78]. For the fermion self-energy, it is necessary to introduce two NWFs, since one has to deal with two scalar functions.…”

“…[44,[58][59][60][61][62][63]) and BSE (see, e.g., Refs. [64][65][66][67][68][69][70][71][72][73][74][75][76][77][78], where systems with and without spin degrees of freedom are investigated).…”

“…y − A 7 (s, s , ω, v, ξ, w) , (D.128)with Δ , C(i) AV , C(i) AS and C γ from Eqs (69),(70),(71),(73),(74). and(78), and the following equations that relate the NWFs to the KL weights (see Eq.…”

Extracting a model quark propagator's spectral density

“…Together with the practicity of our approach and the reduced error estimate, we believe that the algorithm discussed herein is also suitable to study other Green functions. Moreover, the SPM has been proven useful in connecting Euclidean and Minkowskian quantities [48][49][50], a key goal in modern hadron physics [51,52].…”

Nakanishi integral representation for the quark-photon vertex

“…The above properties of NIR were exploited to provide solutions of many particular quantum field toy model problems [9]. Especially, the NIR was used in two body bound state calculation in various models [10], [11], [12], [13], [14], [16], [17]. Encouraging results for the electromagnetic form factors were obtained [18] within the formalism as well.…”