TRIANGULAR EXCHANGE DIAGRAM WITHIN THE FULL OFF-SHELL COULOMB T-MATRIX IN DEUTERON INDUCED REACTIONS

The amplitude of the triangular diagrams in the three-body models of deuteron induced reactions with charged particles contains the off-shell two-body T -matrix describing the intermediate-state Coulomb scattering of charged subsystems. Up to now the latter has usually been replaced by the Coulomb potential due to the computational reason. In this paper, we first investigate theoretically and numerically the validity of the mentioned Coulomb-Born approximation. The results show the important contribution of the higher order of the latter approximation for the (d,p) reactions off heavy target.


Introduction
Treatment the long-range character of the Coulomb interaction in low-energy deuteron inducedreactions off heavy targets using the 3-body Faddeev formalism [1] is a challenging task.In principle, scattering of deuteron off a nuclear target (an inert core A ) can be exactly described in the framework of the Faddeev integral equations written in the AGS form [2]. The latter equations are usually solved in momentum-space partial-wave basis where they become a system of integral equations with two continuous momenta variables.However, the Faddeev-AGS formalism is only designed for short-range potentials.In fact the long-range behavior of Coulomb interaction gives rise to severe singularities in the kernel of the integral equation that the latter may lack the compactness property known to exist in the case of purely short-range interactions.
One way of dealing with this problem is to use the screening and renormalization method [3,5] which has successfully described not only the 3-body but also 4-body nuclear systems [6][7][8][9][10].
Unfortunately, the technical difficulties arise in the renormalization procedure as the charge of the target increases making this approach is unreliable for targets with charge 28 Z [11].So after almost 50 years from the pioneer works of A. M. Veselova [4], the Coulomb potential remains a serious barrier for having an accurate description of low-energy nuclear reactions using the three-body Faddeev integral equations.New approach based on employing the non-screened Coulomb potential is derived [12].Instead of the usual plane-wave basis, the AGS equations are cast in a momentum-space Coulomb distorted partial-wave representation.Applying two potential formulas the AGS equations are converted to the form in which the matrix elements are sandwiched by the Coulomb distorted waves in the initial and final states.The obtained Faddeev-AGS equations couple all the neutron transfer, proton transfer, in(elastic) scattering, and triangular exchange diagrams (see Fig. 1).The channel Coulomb potentials [sub-diagram (e)] are subtracted from the in(elastic) diagram to remove the strongest singularity of the elastic scattering triangular diagram with Coulomb four-ray vertex which leads to a noncompact singularity of the Faddeev-AGS equations [3,5,11].These generalized Faddeev-AGS equations provide the most advanced and complete description of the deuteron induced reactions.Despite the core formalism having been developed, in order to make it practical an important additional analytical and computational work is required.
In this paper, we propose a new method to approximately take into account the triangular exchange diagrams instead of directly calculating it.Through this investigation we also show for the first time the role of the full off-shell Coulomb T-matrix in describing the intermediate-state Coulomb scattering of proton and target.In the previous works [13][14][15], the Coulomb T-matrix is usually replaced by the corresponding Coulomb potential.This approximation considerably reduces the analytical and numerical efforts due to the treatment of the full off-shell Coulomb T matrix.As has been shown in Refs.[16,17], the Coulomb-Born approximation is accurate to a few percent for all energies and scattering angles for the d(p,d)p reaction.A priori, the increase of the charge of the nuclear targets gives rise to the uncertainties of the Coulomb-Born approximation.To the best of our knowledge the validity of the latter approximation is still an open question in the region of medium and heavy targets.

Theoretical Formalism
We consider the system consisting of two nucleons ( 1 proton and 1 neutron) and a nucleus consisting of A nucleons (no core excitation) to the three-body system.Denote the masses of the    ; p is the momentum of the transfer particle.The standard expression for the off-shell Coulomb scattering amplitude is [18][19][20].
where k and k' are the relative off-shell momenta of charged particles and before and after scattering moving with the relative kinetic energy ẑ , Then the amplitude of triangular exchange diagram simplifies to where we used the substitution p = p p' .Also k = p' ( / )p mm is the relative momentum of particles and in the three-ray vertex () .Similarly is the relative momentum of particles and in the three-ray vertex () of the same diagram.Now we rewrite where we introduced 22 =k k and used the substitution Similarly Here we took into account that from the energy-momentum conservation in both three-ray vertices of the diagram (b) in Fig. 1 follows =. ( Because we consider the singularity of the triangular exchange diagram generated by the coincidence of zeroes of three denominators (pinch-point singularity) in Eq. ( 4), we use the substitution (6) obtaining in the leading order Thus we have shown that the strongest singularity of the amplitude of the triangular exchange diagram is a pole singularity at The same singularity has the pole diagram in sub-figure (a) in Fig. 1.Moreover (a) 1 (p' , p ) M (11) in Eq. ( 9) is the amplitude of the pole diagram.We can conclude from the simple consideration presented here that near the singularity (10)  T Coulomb scattering amplitude rather than the Coulomb-Born amplitude was obtained in [21][22][23].The renormalization factor, which determines the strength of the pole singularity of the triangular exchange diagram, is

Conclusions
The analytical results have been demonstrated by the numerical calculations presented in Figs. 2, 3, and 4. The magnitude of the pole and triangular exchange diagrams have been calculated for shows the quantity R for the (d, p) reactions off 2 H, 48 Ca and 208 Pb targets at different energies below 50 MeV.In the light-target region, the Coulomb-Born approximation works very well even at very low energy around 1 MeV.The obtained results show that the Coulomb-Born approximation is more accurate at small scattering angles less than 80 degree.It fails rather dramatically in the medium and completely collapses in heavy regions at any energies below 50 MeV.We could conclude that the full off-shell Coulomb T matrix must be taken into account for the (d,p) reactions off the medium and heavy targets.

Fig. 1 .Fig. 2 .Fig. 3 .
Fig.1shows the exchange process from the initial channel () to the final channel () where particle having a center-of-mass (c.m.) momentum p impinges on the bound state of particles and and in the final state particles and are bound in a state and particle is free with a c.m. momentum p .This triangular exchange diagram is nothing else the neutron transfer diagram taking into account the Coulomb scattering between the proton and the target.First, we analytically demonstrate that the inclusion of a four-ray vertex corresponding to the offenergy shell Coulomb scattering in the triangular exchange diagram will affect neither the position nor the character of the proper singularity in cos( ) ( is the scattering angle in c.m.) of

Fig. 4 .
Fig. 4. The dot-dashed lines show the magnitude of the neutron transfer diagram whereas the solid (dashed) lines present the triangular exchange amplitude with (without) Coulomb-Born approximation for the 208 Pb(p,d) 209 Pb reactions at 1, 15, 30, and 50 MeV

C
the physical region and the strongest singularity of the triangular exchange diagram is the one generated by the coincidence of the singularities of the propagators = 0 of the Coulomb scattering amplitude.To show how these singularity of the triangular exchange diagram appears and to simplify consideration we replace the off-shell Coulomb scattering amplitude (Tz of particles and by the Coulomb-Born amplitude, which is the Fourier transform of the Coulomb potential the amplitude of the triangular exchange diagram behaves as renormalized amplitude of the pole diagram in sub-figure (a) of Fig. 1: D is the renormalization factor determining the strength of the pole singularity.The additional term =0.A general expression for the renormalization factor for the triangular exchange diagram containing the full C

2 H, 48 1 D and 2 D
Ca and 208 Pb targets at different energies below 50 MeV.The proper singularity of triangular exchange diagram (arising as result of simultaneous vanishing of in the denominator of Eq. (9) appears for d(p,d)p reaction at energy higher than 30 MeV, and this singularity located farther from the physical domain than the pole singularity.Based on the obtained results, we can considerably simplify the calculations of the effective potentials of Faddeev-AGS equations by combining the neutron transfer pole amplitudes with the Coulomb exchange triangular diagram amplitudes taking into account the fact that near the pole the exchange triangular diagram has also a pole singularity.To investigate the validity of the Coulomb-Born approximation, it is useful to define the ratio of the full off-shell Coulomb T-matrix to the Coulomb potential scattering magnitude quality of the Coulomb-Born approximation.Fig.5

Fig. 5 .
Fig. 5.The ratio of the full off-shell Coulomb T-matrix to the Coulomb potential scattering magnitudes in function of mass of the target at 1, 15, 30, and 50 MeV