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SOLVING BETHE-SALPETER EQUATION FOR THE SCATTERING
STATES
V.A. Karmanov 1† and J. Carbonell 2
(1) Lebedev Physical Institute, Moscow, Russia
(2) CEA, Saclay, France
† E-mail: karmanov@sci.lebedev.ru
We present the first solution of the Bethe-Salpeter (BS) equation [1]:
∫ d
F(p,p ′′ ,P) = K(p,p ′′ 4 p ′
,P)−i ,P)
(2π) 4K(p,p′
×
1 1
[ (1
[ P (1
2 +p′)2 −m +iǫ]
2 P 2 −p′)2 −m +iǫ]F(p ′ ,p ′′ ,P),
2
in Minkowski space for the scattering states, for the off-mass-shell amplitude, for the
one-boson exchange (OBE) kernel K. Solving this equation is a difficult task because
of the existence of many singularities in the integrand. The singularities, of course, are
integrable in a mathematical sense (due to iǫ in the propagators), but cannot be handled
in a direct numerical way. Because of that, the BS equation for the scattering states was
previously solved for the separable kernels (see for review [2]). For the OBE kernel, only
the on-shell amplitude, determining the phase shifts, was found.
Our method (valid both for the bound and for the scattering states) is based on an
explicit analytical treatment of all the singularities. It can be applied both to equation
without the partial wave decomposition and to the partial wave one. For the present, we
consider the case of spinless particles.
Using this method we first solve the bound state equation and reproduce our previous
results, found in Minkowski space by another our method (see for review [3]).
We find that the BS results for the phase shifts and for the scattering length considerably
differ from the ones given by the Schrödinger equation. The relativistic effects are
significant even at small incident energy.
Above the meson creation threshold the phase shifts become imaginary. Then we
calculate inelasticity.
The off-shell BS amplitude found in our work, after its generalization to the NN
system, can be used for relativistic calculation of the deuteron electrodisintegration ed →
enp, which is an inelastic analogue of the deuteron e.m. form factors.
References
[1] E.E. Salpeter and H.A. Bethe, Phys. Rev. 84, 1232 (1951).
[2] S.G. Bondarenko, V.V. Burov and E.P. Rogochaya, Few Body Syst. 49, 121, (2011).
[3] J. Carbonell and V.A. Karmanov, Few-Body Syst. 49, 205 (2011).
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