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

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. $k^{\pm }=k^0\pm k^3$. 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.

Figure 1

The binding energy $B/m$ vs $g^2$ for a massless vector exchange. Solid line: our calculations, with the exact analytical treatment of the endpoint singularities. Full dots: $g^2$ from Ref. [8], with a numerical treatment of the singularities. A critical value $g_{\mathrm{crit}}$ is clearly approached for $B/m\to 2$.

Figure 2

The light-front amplitude ${\psi }_i$ times $\gamma /m^2$ [cf. Eq. (7)] vs $\gamma /m^2$ at fixed $z=0$, for the vector coupling and $B/m=0.1$ (thin lines) and 1.0 (thick lines). Solid line: $(\gamma /m^2){\psi }_1$. Dashed line: $(\gamma /m^2){\psi }_2$. Dot-dashed line: $(\gamma /m^2){\psi }_4$. Notice that ${\psi }_3=0$ for $z=0$ (see text). The lines for $B/m=1.0$ have been divide by 10.