Advances in solving the two-fermion homogeneous Bethe-Salpeter equation in Minkowski space

Abstract

Actual solutions of the Bethe-Salpeter equation for a two-fermion bound system are becoming available directly in Minkowski space, by virtue of a novel technique, based on the so-called Nakanishi integral representation of the Bethe-Salpeter amplitude and improved by expressing the relevant momenta through light-front components, i.e. k ± = k 0 ± k 3 . We solve a crucial problem that widens the applicability of the method to real situations by providing an analytically exact treatment of the singularities plaguing the two-fermion problem in Minkowski space, irrespective of the complexity of the irreducible Bethe-Salpeter kernel. This paves the way for feasible numerical investigations of relativistic composite systems, with any spin degrees of freedom. We present a thorough comparison with existing numerical results, evaluated in both Minkowski and Euclidean space, fully corroborating our analytical treatment, as well as fresh light-front amplitudes illustrating the potentiality of non perturbative calculations performed directly in Minkowski space.

Minkowski space approach for the Bethe-Salpeter equation

“…The success in applying the NIR to the BS formalism for the spectrum of bosonic systems, as outlined here and detailed in [13,14], provides enough motivation for possible extensions, as to consider these studies beyond the ladder approximation [9]. Following previous investigations on two-fermion systems within the NIR [11,12], it is quite relevant also to look for the corresponding excited bound-state spectrum.…”

“…Moreover, in [10], within the NIR the effect of the self-energy contribution to the binding energy of the ground state has been explored in the scalar theory. It is important to point out that the very relevant problem of two-fermion bound state has been investigated within the NIR framework in [11,12], for the case J π = 0 + , where J is the total spin and π the parity of the bound state.…”

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

“…its restriction to a vanishing relative LF time (an introduction to this technique and its application to the BSE is given in [15]). This approach, together with the analogous one developed by Carbonell and Karmanov (see, e.g., [16][17][18]), has already achieved relevant outcomes, addressing: i) two-scalar systems both in bound states and at the zero-energy limit [15,[19][20][21]; the two-fermion bound state in a 0 + channel [22,23]. To summarize the results of the previous studies and of the present one, we can state that such an approach is able to yield actual solutions of the BSE, directly in Minkowski space.…”

Residues of ΛQ -type and ΣQ -type baryons in the Bethe-Salpeter equation approach

“…To solve Eq. (18) and the two groups of coupled integral equations (25), (26), and (33)- (35) numerically, we discretize the integration regions into n pieces with n being sufficiently large. Then the equations become eigenvalue equations and can be solved with the Gaussian integration method.…”

“…Thus we employ this approximation in this work for the purpose of simplifying our calculations. It should be noted that solving the BS equation in the Minkowski space has been done before by other researchers [25,26], and we will also try to accomplish that by modifying our model in our future work. Finally, the obtained BS wave functions will be applied to calculate the residues of Λ Q , Ξ A Q , Σ ðÃÞ Q , Ξ SðÃÞ Q and Ω ðÃÞ Q .…”