Coulomb Effects in Few-Body Reactions

The method of screening and renormalization is used to include the Coulomb interaction between the charged particles in the momentum-space description of three- and four-body nuclear reactions. The necessity for the renormalization of the scattering amplitudes and the reliability of the method is demonstrated. The Coulomb effect on observables is discussed.


I. INTRODUCTION
The inclusion of the long-range Coulomb interaction in the description of the three-and four-particle scattering is a challenging task in theoretical few-body nuclear physics. The long range of the Coulomb potential prevents the direct application of the standard scattering theory. There is a number of suggestions how to overcome this difficulty; most of them are based on the configuration-space framework [1][2][3][4] and are limited to energies below three-body breakup threshold, while the others [5][6][7][8] have not matured yet into practical applications. Up to now only few approaches led to the results above threebody breakup threshold. Those are configuration-space calculations for proton-deuteron (p-d) elastic scattering using the Kohn variational principle [9] and the screening and renormalization method in the framework of momentum-space integral equations [10][11][12][13]; the latter approach will be discussed here in more details. Very recently p-d results above three-body breakup threshold were also obtained using modified Faddeev equation in configuration space together with the dumping of particular Coulomb contributions [14].

II. METHOD OF SCREENING AND RENORMALIZATION
In nature the Coulomb potential w C is always screened at large distances. The comparison of the data from typical nuclear physics experiments and theoretical predictions with full Coulomb is meaningfull only if the full and screened Coulomb become physically indistinguishable. This was proved in Refs. [15,16] where the screening and renormalization method for the scattering of two charged particles was proposed. We base our treatment of the Coulomb interaction on that idea.
The standard scattering theory is formally applicable to the screened Coulomb potential w R , i.e., the Lippmann-Schwinger equation yields the two-particle transition matrix where g 0 is the free resolvent. In the configuration-space representation we choose the screened Coulomb potential as w R (r) = w C (r) e −(r/R) n (2) with R being the screening radius and n controlling the smoothness of the screening. It was proven in Ref. [15] that in the limit of infinite screening radius R the on-shell screened Coulomb transition matrix (screened Coulomb scattering amplitude) p ′ |t R |p with p ′ = p, renormalized by the infinitely oscillating phase factor z −1 R (p) = e 2iφ R (p) , approaches the full Coulomb amplitude p ′ |t C |p in general as a distribution, i.e., for any test function ϕ(p) with the properties given in Ref. [15]; in particular, it must vanish for forward scattering, i.e., ϕ(p ′ ) = 0. Reference [15] uses partial wave expansion, which for the full Coulomb amplitude itself converges only as a distribution, and it therefore is unable to make any conclusions on the possible pointwise convergence. However, as argued already in Ref. [15], the convergence of the renormalized screened Coulomb scattering amplitude to the full Coulomb amplitude in the sense of distributions is sufficient for the description of physical observables. For a fixed final state observation directionp ′ the cross section is determined not directly by the scattering amplitude, but by the outgoing wave packet that is related to the initial wave packet ϕ i (p) by the S -matrix or the scattering amplitude. Thus, in the step from the scattering amplitude to the cross section one has to go through the conceptual exercise of averaging the scattering amplitude over the initial state physical wave packet being peaked around the experimental beam momentum p i . Thus, on the rhs of Eq. (3) p ′ |t C |p i itself is picked out in that average. In addition, the outgoing wave packet is never observed in the forward direction, i.e.,p ′ p i ; the necessary property ϕ i (p ′ ) = 0 can therefore always be fulfilled by the sharpness of the initial wave packet. In the practical calculations [11,12] the above averaging is carried out implicitly, replacing the renormalized screened Coulomb amplitude in the R → ∞ limit by the full one, i.e., Since z −1 R (p) is only a phase factor, the above relations indeed demonstrate that the physical observables become insensitive to screening provided it takes place at sufficiently large distances R and, in the R → ∞ limit, coincide with the corresponding quantities referring to the full Coulomb. Furthermore, renormalization by z − 1 2 R (p i ) in the R → ∞ limit relates also the screened and full Coulomb wave functions [17], i.e., A. Three-particle scattering The screening and renormalization method based on the above relations can be extended to more complicated systems, albeit with some limitations. The systems of two-and threeparticles interacting via pairwise strong short-range and screened Coulomb potentials, v α and w αR , α = 1, 2, 3, is considered in Refs. [12,13,18]. Here we extend our treatment to a more general case where an irreducible three-body force is present; it is decomposed into three terms u α . The full resolvent with H 0 being the three-particle kinetic energy operator and E the available energy may be decomposed into channel resolvents and the multichannel three-particle transition operator U (R) βα according to all operators depend parametrically on the Coulomb screening radius R. The full multichannel transition matrix U (R) βα for elastic and rearrangement scattering is calculated from the integral equation [19] that is a generalisation of the Alt, Grassberger, and Sandhas (AGS) equation [20] in the presence of the three-body force.
is the free resolvent,δ βα = 1 − δ βα , and the two-particle transition matrix is derived from the full channel interaction v α + w αR , i.e., The on-shell matrix elements b β q ′ |U (R) βα |b α q are amplitudes (up to a factor) for elastic (β = α) and rearrangement (β α) scattering. The channel states |b α q are the eigenstates of the corresponding channel Hamiltonian H α = H 0 + v α with the energy eigenvalue E. |b α q is a product of the bound state wave function |b α for the pair α and the plane wave with the relative particle-pair α momentum q; the dependence on the discrete quantum numbers is suppressed in our notation.
In order to isolate the screened Coulomb contributions to the transition amplitude that diverge in the infinite R limit, we use a decomposition of the full resolvent into alternative channel resolvents where W c.m. αR is the screened Coulomb potential between the spectator particle α and the center of mass (c.m.) of the remaining pair. The same screening function is used for both Coulomb potentials w αR and W c.m.
αR . The corresponding transition matrix is a two-body operator and therefore its on-shell and half-shell behaviour in the limit R → ∞ is given by Eqs. (3) and (7). It relates the channel resolvents as Thus, the full resolvent can alternatively be decomposed into where the reduced transition operatorŨ (R) βα (Z) may be calculated through the integral equatioñ (19) which is driven by the strong two-and three-body potentials v α and u α and the potential of three-body nature W αR = γ (δ αγ w γR − δ αγ W c.m. γR ). This potential W αR accounts for the difference between the direct Coulomb interaction w γR and the auxiliary one W c.m.
γR that takes place between the charged particle and the c.m. of the remaining pair. When calculated between on-shell screened Coulomb states,Ũ (R) βα is of shortrange, even in the infinite R limit. Equation (17), together with Eq. (11), gives a relation between full and reduced threeparticle transition operators, i.e., Thus, the three-particle transition operator U (R) βα has a longrange part δ βα T c.m.
αR whereas the remainder U (R) βα − δ βα T c.m. αR is a short-range operator that is externally distorted due to the screened Coulomb waves generated by [ Onshell, both parts do not have a proper limit as R → ∞ but the limit exists after renormalization by an appropriate phase factor, yielding the transition amplitude for full Coulomb The first term on the right-hand side of Eq. (22) is known analytically [15]; it corresponds to the particle-pair α full Coulomb transition amplitude that results from the implicit renormalization of T c.m. αR according to Eq. (6). The R → ∞ limit for the remaining part (U (R) βα − δ βα T c.m. αR ) of the multichannel transition matrix is performed numerically; due to the short-range nature of this term, as demonstrated in Eq. (20), the convergence with the increasing screening radius R is fast and the limit is reached with sufficient accuracy at finite R; furthermore, it can be calculated using the partial-wave expansion. We emphasize that Eq. (22) is by no means an approximation since it is based on the obviously exact identity (21) where the R → ∞ limit for each term exists and is calculated separately.
The renormalization factor for R → ∞ is a diverging phase factor where Φ αR (q), though independent of the particle-pair relative angular momentum l in the infinite R limit, may be realized by with the diverging screened Coulomb phase shift η lR (q) corresponding to standard boundary conditions and the proper Coulomb one σ l (q) referring to the logarithmically distorted proper Coulomb boundary conditions. For the screened Coulomb potential of Eq. (2) the infinite R limit of Φ αR (q) is known analytically, where C ≈ 0.5772156649 is the Euler number and K α (q) is the Coulomb parameter. The form of the renormalization phase Φ αR (q) to be used in the actual calculations with finite screening radii R is not unique, but the converged results show independence of the chosen form of Φ αR (q). For breakup observables we follow a very similar strategy where the starting point is the AGS-like equation for the breakup operator and its relation to the full resolvent, i.e., In the same spirit, we introduce auxiliary Coulomb resolvent that keeps only the screened Coulomb interaction. The proper three-body Coulomb wave function and its relation to the three-body screened Coulomb wave function generated by G R is, in general, unknown. This prevents the application of the screening and renormalization method to the reactions involving three free charged particles (nucleons or nuclei) in the final state. However, in the system of two charged particles and a neutral one, only the channel γ = ρ, corresponding to the correlated pair of charged particles, contributes to G R which simplifies to making channel ρ the most convenient choice for the description of the final breakup state. Thus, for the purpose of breakup, a decomposition of the full resolvent, alternative to Eq. (27) is where the reduced breakup operatorŨ (R) 0α may be calculated through quadraturẽ from the corresponding reduced operatorŨ (R) βα (Z) of elastic/rearrangement scattering. On-shell, the reduced operator U (R) 0α (Z) calculated between screened Coulomb distorted initial and final states is of finite range, though the two contributions in Eq. (33) have slightly different range properties as discussed in Ref. [13]. The relation between the full and reduced breakup operators is In the full breakup operator U (R) 0α (Z) the external distortions show up in screened Coulomb waves generated by ( αR ) in the initial state and by (1 + T ρR G 0 ) in the final state; both wave functions do not have proper limits as R → ∞. Therefore the full breakup transition amplitude in the case of the unscreened Coulomb potential is obtained via the renormalization of the on-shell breakup transition matrix where p ′ is the relative momentum between the charged particles in the final state, q ′ the corresponding particle-pair relative momentum, and the final-state renormalization factor. The limit in Eq. (35) has to be performed numerically, but, due to the short-range nature of the breakup operator, the convergence with the increasing screening radius R is fast and the limit is reached with sufficient accuracy at finite R.

B. Practical realization
To calculate the short-range part of the elastic, rearrangement, and breakup scattering amplitudes (22) and (35) we solve standard scattering equations (12), (15), and (26) at finite Coulomb screening radius R using the momentum-space partial-wave representation as described in detail in Refs. [21][22][23]. We have to make sure that R is large enough to achieve (after renormalization) the R-independence of the results up to a desired accuracy. However, those R values are larger than the range of the nuclear interaction resulting in a slower convergence of the partial-wave expansion. As we found in Ref. [12], the practical success of the screening and renormalization method depends strongly on the choice of the screening function, i.e., on the choice of the exponent n in Eq. (2). One of the essential differences compared to previous works [10,11,24,25] is that we use a sharper screening than the Yukawa screening (n = 1). We want to ensure that the screened Coulomb potential w R approximates well the true Coulomb one w C for distances r < R and simultaneously vanishes rapidly for r > R, providing a comparatively fast convergence of the partial-wave expansion. However, the sharp cutoff (n → ∞) yields an unpleasant oscillatory behavior in the momentum-space representation, leading to convergence problems. We find values 3 ≤ n ≤ 8 to provide a sufficiently smooth, but at the same time a sufficiently rapid screening around r = R. The screening functions for different n values are compared in Fig. 1, showing that the present choice n = 4 includes much more of the exact Coulomb potential at short distances than the Yukawa screening used previously. For example, Yukawa screening requires a screening radius of R = 1280 fm in order to approximate true Coulomb at relative distance r = 5 fm as well as the present choice does with R = 20 fm. As shown in Fig. 2 for the proton-proton (pp) 1 S 0 phase shift, the convergence with the screening radius for n = 4 is much faster than for Yukawa screening or sharp cutoff. Furthermore, due to the much shorter range of the n = 4 screening compared to the Yukawa screening, the quasisingularities of the screened Coulomb potential are far less pronounced in the n = 4 case. With our optimal choice 3 ≤ n ≤ 8 the convergence of the partial-wave expansion, though being slower than for the nuclear interaction alone, can still be achieved; alternatively, a perturbative approach for higher two-particle partial waves [26] that is computationally less demanding but, nevertheless, highly reliable as proved in Ref. [27], can be used as well.

C. Renormalization in proton-deuteron scattering
For the nucleon-deuteron scattering it is convenient to consider nucleons as identical particles and use the isospin formalism. Since the isospin conservation is violated by the Coulomb force, both total 3N isospin T = 1 2 and 3 2 states have to be included. The symmetrized transition operator for elastic scattering is the solution of the symmetrized AGS integral equation [19] We omit the spectator index that is not needed anymore. The basis states are antisymmetric in the pair only, the full antisymmetry is ensured by P = P 12 P 23 + P 13 P 23 where P αβ is the permutation operator of particles α and β. operator is then obtained from the quadrature The amplitudes for p-d elastic scattering and breakup referring to unscreened Coulomb are obtained after the renormalization of the corresponding on-shell matrix elements of the symmetrized transition operators, i.e., However, a recent work [28] on an alternative Coulomb treatment in pd scattering proposed a different renormalization prescription: p-d elastic scattering amplitude calculated with screened Coulomb does not need renormalization at all, i.e., the limit lim R→∞ q ′ |U (R) |q should exist; the renormal-ization for the breakup amplitude is needed but it is different from ours given in Eq. (40). However, numerical results of Ref. [28] involve approximations that are not well under control. First, the screened Coulomb proton-proton transition matrix is approximated by the screened Coulomb potential thereby replacing a complex quantity by a real one. Second, the contributions q ′ |PT R PG 0 T (R) G 0 U (R) |q and p ′ q ′ |(1 + P)T R PG 0 T (R) G 0 U (R) |q to the elastic scattering and breakup amplitudes, respectively, are neglected; this may have even more serious consequences since the above terms among others include screened Coulomb contributions that are of the leading (first) order. Indeed, under the approximations of Ref. [28] the R-dependence of the resulting screened Coulomb amplitude for p-d elastic scattering is weak and the R → ∞ limit of renormalized amplitude in Eq. (39) does not exists. However, the situation changes completely when the scattering amplitudes are calculated exactly as in our work. In Figs. 3 and 4 we study the dependence on the screening ra- dius R for selected components of the nonrenormalized and renormalized p-d elastic scattering amplitudes q ′ |U (R) |q and q ′ |U (C) |q . The CD Bonn + ∆ two-baryon potential [29] is taken as the hadronic interaction and the proton lab energy E p = 10 MeV. In Fig. 3 we show spin-nondiagonal amplitudes that have only the Coulomb-distorted short-range part. The nonrenormalized amplitude q ′ |U (R) |q shows a clear R-dependence; in fact, its absolute value becomes Rindependent but the phase changes like ln R. In contrast, the renormalized amplitude q ′ |U (C) |q , within the accuracy of the plot, becomes independent of R for R ≥ 10 fm. The spindiagonal amplitude having also a long-range part shows, as expected, very strong R-dependence at small scattering angles before renormalization but becomes R-independent after renormalization as well as Fig. 4 demonstrates. In Fig. 5 we show the corresponding results for p-d breakup at E p = 13 MeV in the space star configuration. For breakup reaction the renormalized amplitude p ′ q ′ |U (C) 0 |q converges with the screening radius somehow slower than for elastic scatter-ing, it becomes independent of R for R ≥ 20 fm whereas the nonrenormalized amplitude p ′ q ′ |U (R) 0 |q shows a clear R-dependence. Furthermore, although the breakup amplitude (without the three-nucleon force) can be decomposed into three terms T (R) γ G 0 U (R) γα , separately none of them has an R → ∞ limit with or without renormalization in contrast to the conjecture of Ref. [28]; infinite R limit exists only for renormalized full breakup amplitude. Thus, fully converged numerical results without uncontrolled approximations clearly support the standard screening and renormalization theory as given in Eqs. (39) and (40) and not the one of Ref. [28].
The convergence of the observables with the screening radius R used to calculate the Coulomb-distorted short-range part of the amplitudes is the internal criterion for the reliability of our method. Numerous examples can be found in Refs. [12,13,30,31]. In most cases the convergence is impressively fast; the screening radius R = 10 to 30 fm is sufficient. The exceptions requiring larger screening radii are the observables at very low energies and the breakup differ- ential cross section in kinematical situations characterized by very low relative energy E rel between the two charged particles, e.g., p-d breakup or photodisintegration of 3 He close to the pp final-state interaction (pp-FSI) regime [13]. The slow convergence under those conditions is not surprising, since the renormalization factor itself as well as the Coulomb parameter become ill-defined, indicating that the screening and renormalization procedure cannot be applied at E rel = 0. Therefore an extrapolation has to be used to calculate the observables at E rel = 0, which works pretty well when the observables vary smoothly with E rel which is the case in three-body breakup reactions. Furthermore, the results for p-d elastic scattering obtained by the present technique were compared [32] with those of Ref. [9] obtained from the variational solution of the three-nucleon Schrödinger equation in configuration space with the inclusion of an unscreened Coulomb potential between the protons and imposing the proper Coulomb boundary conditions explicitly. Good agreement over a wide energy range was found indicating that both techniques for including the Coulomb interaction are reliable. At very low energies the coordinate-space treatments remain favored since there the method of screening and renormalization converges slowly and therefore becomes technically too demanding, but at higher energies and for three-body breakup reactions it is more efficient.

III. RESULTS
In this section we present selected results for reactions in various three-and four-body nuclear systems.

A. Proton-deuteron scattering
Realistic models for the nuclear interaction are used, e.g., CD Bonn +∆ two-baryon potential [29] yielding an effective three-nucleon force or the AV18 two-nucleon potential Results including the Coulomb interaction (red curves) are compared to results without Coulomb (blue curves). Hadronic interaction model is AV18 + UIX. The experimental data are from Ref. [34] (full circles). For the space star configuration also the nd data from Refs. [35,36] (open and full squares) are shown. [37] with the irreducible Urbana IX (UIX) three-nucleon force [38]. The Coulomb effect is important in low-energy p-d elastic scattering as Fig. 6 demonstrates; the description of the experimental data is quite satisfactory except for the proton and deuteron vector analyzing powers. In p-d elastic scattering at higher energies the Coulomb effect gets confined to forward angles.
In Fig. 7 we show results for p-d breakup at 13 MeV pro-ton lab energy. Although the inclusion of the Coulomb force slightly improves the agreement with data in the space star configuration, the Coulomb effect is far too small to reproduce the difference between the experimental p-d and n-d data and to resolve the so-called space star anomaly. Slightly larger and beneficial Coulomb effects are seen in quasifree scattering (QFS) and collinear configurations. The Coulomb effect in may become large even at higher energies when the relative pp energy in the final state is very low as demonstrated in Fig. 8 for p-d breakup at 130 MeV deuteron lab energy. In there, the Coulomb repulsion is responsible for decreasing the cross section; the pp-FSI peak obtained in the absence of Coulomb may even be converted into a minimum. However, in some other configurations Coulomb may lead to a moderate increase of the differential cross section as shown in Fig. 9.

B. Photodisintegration of 3 He
The results are obtained with the CD Bonn +∆ two-baryon potential and include effective two-and three-nucleon electromagnetic currents mediated by the ∆ isobar [41,42]. As in p-d breakup, the Coulomb effect may be very strong in the three-body photodisintegration of 3 He close to the pp-FSI kinematics. The effect remains also after partial integration when calculating the semi-inclusive observables as shown in Figs. 10 and 11. Not only the differential cross section gets affected but also the spin observables like the linear photon asymmetry for 3 He(γ, p)pn reaction.

C. Three-body nuclear reactions
The screening and renormalization method in the framework of momentum-space AGS equations can be applied to the description of three-body-like nuclear reactions involving heavier nuclei whose interaction with nucleons is described by the optical potentials. Examples are deuteron scattering on a stable nucleus A or proton scattering on a weakly bound twobody system (An) consisting of the core A and the neutron n; all elastic, transfer, charge-exchange, and breakup reactions, allowed by the chosen Hamiltonian, can be calculated on the same footing. The calculations have been performed for a number of reactions with the core A ranging from the α particle to 58 Ni [30,45,46].
In Fig. 12 we show the differential cross section for α-d breakup reaction at 15 MeV α lab energy that is calculated using three-body (α, p, n) model. The most important Coulomb effect is the shift of the αp P-wave resonance position that leads to the corresponding changes in the structure of the observables. The predictions without Coulomb fail completely in accounting for the experimental data, while inclusion of the Coulomb moves the peaks of the differential cross section to the right positions, although the height of those peaks is not always reproduced, probably due to deficiencies of the nucleon-α potentials.
The developed technique allowed to test the accuracy of traditional approximate nuclear reaction approaches like Continuum Discretized Coupled Channels (CDCC) method [47] as has been done in Ref. [48] for d + 12 C, d + 58 Ni, and p + 11 Be reactions. It was found that CDCC is indeed a reliable method to calculate d + A elastic and breakup cross sections but may lack accuracy for transfer reactions such as p + 11 Be → d + 10 Be and for breakup of one-neutron halo nuclei p + 11 Be → p + n + 10 Be. Furthermore, novel dynamical input like energy-dependent [49] or nonlocal optical potentials [50,51] could be included for the first time due to the use of the momentum-space framework. Especially important nonlocality effects were found for (d, p) and (p, d) transfer reactions involving stable [52] as well as exotic nuclei [53].

D. Four-nucleon scattering
Exact description of the four-nucleon scattering is given by the Faddeev-Yakubovsky equations [54] for the wave-function components or by the equivalent AGS equations [55]. We use isospin formalism and solve the symmetrized AGS equations [56] for the transition operators. Coulomb interaction is included using the same idea of screening and renormalization. Long-and Coulomb-distorted short-range parts in the scattering amplitudes are separated [57,58]. The former is of twobody nature and its R → ∞ limit is known analytically. The Coulomb-distorted short-range part is calculated by solving symmetrized AGS equations numerically where the screened Coulomb potential is added to the nuclear proton-proton potential; the R → ∞ limit is reached with sufficient accuracy at finite screening radii R as demonstrated in Ref. [57].
The four-nucleon scattering calculations have been performed so far only below three-body breakup threshold where the Coulomb effect is extremely important. This is demonstrated in Fig. 14 for proton-3 He elastic scattering at 4 MeV proton lab energy. As in p-d scattering, the proton analyzing power is underpredicted by the theory [57,59].
In Figs. 15 and 16 we show examples for transfer and charge-exchange reactions in the four-nucleon system. The two-nucleon interactions we use are AV18 [37], the one derived from chiral perturbation theory at next-to-next-to-nextto-leading order (N3LO) [62], CD Bonn [63], and inside nonlocal outside Yukawa (INOY04) potential by Doleschall [64]. Although here we do not include a three-nucleon force, its presence is simulated by using the potential INOY04 that fits both 3 He and 3 H experimental binding energies (7.72 MeV and 8.48 MeV, respectively). The results for the two-baryon potential CD Bonn + ∆ [29] allowing for a virtual excitation of a nucleon to a ∆-isobar and thereby yielding consistent effective three-and four-nucleon forces are qualitatively similar and can be found in Ref. [65]. Most of the experimental data are quite well described [57][58][59]66] at least by some of the used two-nucleon force models, but there exist also several discrepancies, e.g., for the neutron-3 H total cross section [56,67] and for the proton analyzing power in the proton-3 He elastic scattering [57,59,60] and in the p + 3 H → n + 3 He charge-exchange reaction [58].  Fig. 7 and the experimental data from Ref. [39].

IV. SUMMARY
We used the method of screening and renormalization to include the Coulomb interaction between the charged particles in few-body nuclear reactions. We demonstrated analytically and numerically that the limit of infinite screening radius exists only for renormalized amplitudes. The short range part of the scattering amplitudes were obtained from the exact few- The experimental data are from Ref. [43].
body scattering equations that were solved in the momentumspace framework. We obtained fully converged results for three-and four-nucleon scattering and for three-body-like nuclear reactions. The cross section data are from Refs. [73] (squares) and [74] (circles) and the analyzing power data are from Refs. [74,75].