arXiv subjects

J. Bijtebier

Publications and source records attributed to J. Bijtebier.

9 recordsLinked to original sources

arXiv2001

Bound state equation for 4 or more relativistic particles

We apply the 3D reduction method we recently proposed for the N-particle Bethe-Salpeter equation to the 4-particle case. We find that the writing of the Bethe-Salpeter equation is not a straightforward task when N is larger or equal to 4, owing to the presence of mutually unconnected interactions, which could lead to an overcounting of some diagrams in the resulting full propagator. We overcome this difficulty in the N=4 case by including three counterterms in the Bethe-Salpeter kernel. The application of our 3D reduction method to the resulting Bethe-Salpeter equation suggests us a modified 3D reduction method, which gives directly the 3D potential, without the need of writing the Bethe-Salpeter kernel explicitly. The modified reduction method is usable for all N.

hep-th↗

arXiv2001

3D reduction of the N-body Bethe-Salpeter equation

We perform a 3D reduction of the two-fermion Bethe-Salpeter equation, by series expansion around a positive-energy instantaneous approximation of the Bethe-Salpeter kernel, followed by another series expansion at the 3D level in order to get a manifestly hermitian 3D potential. It turns out that this potential does not depend on the choice of the starting approximation of the kernel anymore, and can be written in a very compact form. This result can also be obtained directly by starting with an approximation of the free propagator, based on integrals in the relative energies instead of the more usual delta-constraint. Furthermore, the method can be generalized to a system of N particles, consisting in any combination of bosons and fermions. As an example, we write the 3D equation for systems of two or three fermions exchanging photons, in Feynman or Coulomb's gauge.

hep-th↗

arXiv2000

Towards a 3D reduction of the N-body Bethe-Salpeter equation

Starting with the homogeneous Bethe-Salpeter equation for two fermions, we perform a 3D reduction using a series expansion around an unspecified positive-energy instantaneous approximation of the kernel. A second series expansion is made, at the 3D level, in order to get an "hermitian" potential. The combination of both series gives a very simple result, which does not depend of the initial approximation of the kernel anymore, and could be obtained directly by starting with an approximation of the free propagator. The generalisation of this result to a system of f (=0,...N) fermions and N-f bosons is easy.

hep-th↗

arXiv2000

A cluster-separable Born approximation for the 3D reduction of the three-fermion Bethe-Salpeter equation

We perform a 3D reduction of the two-fermion Bethe-Salpeter equation based on Sazdjian's explicitly covariant propagator, combined with a covariant substitute of the projector on the positive-energy free states. We use this combination in the two fermions in an external potential and in the three-fermion problems. The covariance of the two-fermion propagators insures the covariance of the two-body equations obtained by switching off the external potential, or by switching off all interactions between any pair of two fermions and the third one, even if the series giving the 3D potential is limited to the Born term or more generally truncated. The covariant substitute of the positive energy projector preserves the equations against continuum dissolution without breaking the covariance.

hep-th↗

arXiv1999

Two and three-fermion 3D equations deduced from Bethe-Salpeter equations

We write a 3D equation for three fermions by combining the three two-body potentials obtained in 3D reductions (based on a series expansion around a relative-energy fixing "approximation" of the free propagators) of the corresponding two-fermion Bethe-Salpeter equations to equivalent 3D equations, putting the third fermion on its positive-energy mass shell. In this way, the cluster-separated limits are exact, and the Lorentz invariance / cluster separability requirement is automatically satisfied, provided no supplementary approximation, like the Born approximation, is made. The use of positive free-energy projectors in the chosen reductions of the two-fermion Bethe-Salpeter equations prevents continuum dissolution in our 3D three-fermion equation. The potentials are hermitian below the inelastic threshold and depend only slowly on the total three-fermion energy. This "hand-made" three-fermion 3D equation is also obtained by starting with an approximation of the three-fermion Bethe-Salpeter equation, in which the three-body kernel is neglected and the two-body kernels approached by positive-energy instantaneous expressions, with the spectator fermion on the mass shell. The neglected terms are then transformed into corrections to the 3D equation, in three steps implying each a series expansion. The result is of course complicated, but the lowest-order contributions of these correction terms to the energy spectrum remain manageable.We also present some other 3D reduction procedures and compare them to our's: use of Sazdjian's covariant approximation of the free propagator, 3D reductions performed by a series expansion around instantaneous approximations of the kernels instead of "approximations" of the propagators, Gross' spectator model.

hep-th↗

arXiv1999

Higher-order corrections to the two and three-fermion Salpeter equations

We compare two opposite ways of performing a 3D reduction of the two-fermion Bethe-Salpeter equation beyond the instantaneous approximation and Salpeter's equation. The more usual method consists in performing an expansion around an instantaneous approximation of the product of the free propagators (propagator-approximated reduction). The second method starts with an instantaneous approximation of the Bethe-Salpeter kernel (kernel-approximated reduction). In both reductions the final 3D potential can be obtained by following simple modified Feynman rules. The kernel-approximated reduction, however, does not give the correct scattering amplitudes, and must thus be limited to the computation of bound states. Our 3D reduction of the three-fermion Bethe-Salpeter equation is inspired by these results. We expand this equation around positive-energy instantaneous approximations of the three two-body kernels, but these starting approximations are given by propagator-approximated reductions at the two-body level. The three-fermion Bethe-Salpeter equation is first transformed into a set of three coupled equations for three wave functions depending each on one two-fermion total energy, then into a set of three 3D equations and finally into a single 3D equation. This last equation is rather complicated, as it combines three series expansions, but we use it to write a manageable expression of the first-order corrections to the energy spectrum.

hep-th↗

arXiv1998

3D reduction of the three-fermion Bethe-Salpeter equation

We present a 3D approximation of the three-fermion Bethe-Salpeter equation. Our 3D equation is covariantly cluster separable and the two-fermion cluster separated limits are exact equivalents of the corresponding two-fermion Bethe-Salpeter equations. The potentials include positive free energy projectors in order to avoid continuum dissolution.

hep-th↗

arXiv1998

A three-fermion Salpeter equation

We write a 3D equation for three fermions by combining the three two-body potentials obtained by the reductions of the corresponding two-fermion Bethe-Salpeter equations to equivalent 3D equations, putting the spectator fermion on the mass shell. In this way, the cluster-separated limits are still exact, and the Lorentz invariance / cluster separability requirement is automatically satisfied, provided no supplementary approximation, like the Born approximation, is made. The use of positive free-energy projectors in the chosen reductions of the two-fermion Bethe-Salpeter equations prevents continuum dissolution in our 3D three-fermion equation. The potentials are hermitian and depend only slowly on the total three-fermion energy. The one high-mass limits are approximately exact. In view of a possible perturbation calculation, correcting the remaining discrepancies with the three-fermion Bethe-Salpeter equation, we succeeded in deriving our 3D equation from an approximation of the three-fermion Bethe-Salpeter equation, in which the three-body kernel is neglected and the two-body kernels approached by positive-energy instantaneous expressions, with the spectator fermion on the mass shell. The neglected terms are transformed into corrections to the 3D equation. A comparison is made with Gross' spectator model.

hep-th↗

arXiv1997

Bethe-Salpeter equation: 3D reductions, heavy mass limits and abnormal solutions

We show that the 3D reductions of the Bethe-Salpeter equation have the same bound state spectrum as the original equation, with the possible exception of some solutions for which the corresponding 3D wave function vanishes. The abnormal solutions of the Bethe-Salpeter equation (corresponding to excitations in the relative time-energy degree of freedom), when they exist, are recovered in the 3D reductions via a complicated dependence of the final potential on the total energy. We know however that the one-body (or one high mass) limit of some 3D reductions of the exact Bethe-Salpeter equation leads to a compact 3D equation (by a mutual cancellation of the ladder and crossed graph contributions), which does not exhibit this kind of dependence on the total energy anymore. We conclude that the exact Bethe-Salpeter equation has no abnormal solution at this limit, or has only solutions for which our 3D wave function vanishes. This is in contrast with the results of the ladder approximation, where no such cancellation occurs. We draw the same conclusions for the static model, which we obtain by letting the mass of the lighter particle go also to infinity. These results support Wick's conjecture that the abnormal solutions are a spurious consequence of the ladder approximation.

nucl-th↗