Implications for (d,p) reaction theory from nonlocal dispersive optical model analysis of {}^{40}Ca(d,p){}^{41}Ca.

# Implications for (d,p) reaction theory from nonlocal dispersive optical model analysis of 40Ca(d,p)41Ca.

S. J. Waldecker Department of Chemistry and Physics, University of Tennessee at Chattanooga, 615 McCallie Avenue, Chattanooga, TN 37403    N. K. Timofeyuk Department of Physics, Faculty of Engineering and Physical Sciences, University of Surrey, Guildford, Surrey GU2 7XH, UK
September 9, 2019
###### Abstract

The nonlocal dispersive optical model (NLDOM) nucleon potentials are used for the first time in the adiabatic analysis of a (d,p) reaction to generate distorted waves both in the entrance and exit channels. These potentials were designed and fitted by Mahzoon et al. [Phys. Rev. Lett. 112, 162502 (2014)] to constrain relevant single-particle physics in a consistent way by imposing the fundamental properties, such as nonlocality, energy-dependence and dispersive relations, that follow from the complex nature of nuclei. However, the NLDOM prediction for the Ca(d,p)Ca cross sections at low energy, typical for some modern radioactive beam ISOL facilities, is about 70% higher than the experimental data despite being reduced by the NLDOM spectroscopic factor of 0.73. This overestimation comes most likely either from insufficient absorption or due to constructive interference between ingoing and outgoing waves. This indicates strongly that additional physics arising from many-body effects is missing in the widely used current versions of (d,p) reaction theories.

###### pacs:
25.45.Hi, 21.10.Jx, 27.40.+z

September 9, 2019

## I Introduction

One nucleon transfer reactions have been a tool for nuclear spectroscopic studies for half of a century. Today, they are used in experiments with radioactive beams and among them the (d,p) reactions are perhaps the most popular choice. Analysis of these reactions relies on (d,p) reaction theory, which is traditionally either the distorted-wave Born approximation (DWBA) Satchler or adiabatic distorted wave approximation (ADWA) JT , the latter being a computationally inexpensive way of taking deuteron breakup into account. The deuteron breakup means that the (d,p) amplitude should contain the degrees of freedom explicitly, which requires solving the three-body Schrödinger equation. Several methods exist to solve this equation exactly, the CDCC Mor09 ; Del15 and the Faddeev approach Del09a being the most used. The usual assumption in these calculations is that the Hamiltonian contains the and potentials (often taken at half the deuteron incident energy) that describe nucleon elastic scattering. However, it has been shown in Joh14 that the and potentials for the problem are very complicated objects which depend on the position and the energy of the third nucleon and are not equal to optical potentials taken at half the deuteron energy. It was also shown in Joh14 that in the case of (d,p) reactions the averaging over the first Weinberg component (which is the same as making the adiabatic approximation) results in a simple prescription for choosing the and potentials appropriate for analysis of (d,p) reactions. This prescription is possible due to the main contribution to the (d,p) amplitude coming from small separations.

The prescription in Joh14 says that within the Feshbach formalism, the and potentials should be nonlocal energy-dependent potentials evaluated at half the deuteron incident energy plus half of the kinetic energy in deuteron averaged over the potential, which is about 57 MeV. After evaluation of these potentials, they should be treated as energy-independent and nonlocal. A simple recipe to include such potentials into the available (d,p) reaction scheme, based on the local energy approximation, is given in Tim13b .

At the time when Joh14 was written, only one energy-dependent nonlocal potential had been known GRZ , derived from Watson multiple scattering theory. It has an energy-independent real part and an energy-dependent imaginary potential. Soon after the publication of Joh14 , a nonlocal version of the dispersive optical model (NLDOM) became available for Ca NLDOM . The nonlocal structure of NLDOM is more complicated than that from previous nonlocal optical potentials in that it is described by seven different nonlocality parameters. Based on the nucleon self-energy from Green’s function many-body theory, the NLDOM potential contains both real and imaginary dynamic terms that are connected through a dispersion relation, which enforces causality and links the negative and positive energy regions. This dispersion relation is important for constraining the NLDOM parameters with both scattering and bound-state data while simultaneously providing a good description of these data. The potential from Ref. GRZ was also constrained with both scattering and bound-state data but without incorporating the dispersion relation.

In this paper, we analyze the Ca(d,p)Ca reaction at 11.8 MeV using NLDOM to generate the distorting potentials in both the entrance and exit channels. The NLDOM potential has already been used in Ross15 to calculate the Ca(p,d)Ca cross sections but only within the DWBA (which means neglecting deuteron breakup) and no comparison to the experimental data was made. Our choice of the reaction is due to the availability of the p-Ca and n-Ca optical potentials needed to construct the d-Ca potential. The choice of the deuteron energy is due to several radioactive beams facilities existing in the world that use this low-energy range. Also, it is this low-energy range where the dispersive relations cause the most prominent effects in the energy behavior of the optical potential. In addition, at these energies, spin-orbit effects and finite range effects can be neglected and the prescription from Joh14 should be valid.

In Sec. II, we review the NLDOM and show that, similar to the standard Perey-Buck case, a local-equivalent potential exists for NLDOM and a generalization of the Perey factor can be introduced. In a similar fashion, we show in Sec. III that the d-Ca distorting potential can be constructed by extending the local scheme proposed in Tim13b to the case with several nonlocality parameters. We summarize the adiabatic approximation in lowest order and introduce first order corrections. In Sec. IV we calculate the cross section of the Ca(d,p)Ca reaction at 11.8 MeV and show that, using the prescription from  Joh14 , the NLDOM strongly overestimates the experimental data. In Sec. V, we discuss the implications for the (d,p) reaction theory following from our analysis.

## Ii Nonlocal DOM potential and nucleon scattering

The NLDOM potential from Ref. NLDOM models the irreducible nucleon self-energy with real and imaginary parts that are both explicitly nonlocal. The potential contains eight terms, which were constrained with both bound-state and scattering data associated with Ca. It is written in the form

 Σ(\boldmathr,\boldmathr′;E) = Uvol1HF(~r)H(\boldmathx;βvol1) (1) + Uvol2HF(~r)H(\boldmathx;βvol2) + UwbHF(~r)H(\boldmathx;βwb) + Usur+dy(~r;E)H(\boldmathx;βsur+) + Usur−dy(~r;E)H(\boldmathx;βsur−) + Uvol+dy(~r;E)H(\boldmathx;βvol+) + Uvol−dy(~r;E)H(\boldmathx;βvol−) + Uso(r;E)

where and . Following Perey and Buck PB , the nonlocality function is assumed to be of the form

 H(\boldmathx;β)=exp(−\boldmathx2/β2)/(π3/2β3), (2)

where is the nonlocality range.

In Eq. (1) the terms represent the static part of the self-energy and are purely real. For this reason, these terms are referred to as parts of the Hartree-Fock (HF) potential, but technically they do not form the true HF potential, because in practice a subtracted dispersion relationship is used NLDOM . The terms represent the dynamic part of the self-energy and are complex. The real part of the dynamic self-energy is determined completely from the dispersion integral of the imaginary part. The dynamic self-energy consists of surface and volume terms, and these have different nonlocalities for energies above the Fermi energy (denoted with a ’+’ sign) and energies below (denoted by a ’-’ sign). The inclusion of several nonlocality parameters was based on the microscopic calculation in Ref. Dussan11 , which indicated different degrees of nonlocality in different energy regions. Table 1 shows the value of each nonlocality parameter. Note that some of these parameters are about twice as large as those known from traditional Perey-Buck potentials. The term is the spin orbit potential, which was assumed to be local. It has a weak energy dependence that only becomes important at high energies.

Overlap functions can also be generated with the dispersive optical model. For discrete states in the and systems, one can show that these overlap functions obey a Schrödinger-like equation Dickhoff08 with the nucleon self-energy taking the role of an effective potential. In order to use the NLDOM overlap function in the analysis of Ca (d,p)Ca reactions, the calculated binding energy of the neutron level in Ca must match the experimental one of MeV. However, in NLDOM , such a constraint was not employed. For the purposes of this study, some of the parameters were refit in order to reproduce the experimental binding energy of the neutron level. Only the parameters associated with the terms were refit, and they were constrained by elastic scattering, charge density, and energy level data. All other parameters were unchanged. The new parameters are shown in Table 2 and compared with those from the analysis in NLDOM . The quality of the new fit is comparable to that obtained in NLDOM .

The nonlocal terms in Eq. (1) can be written more succinctly as

 ΣN(\boldmathr,\boldmathr′;E)=∑iUNA,i(~r)Hi(x), (3)

where the energy dependence of the dynamic terms is implied, and for proton and neutron potentials, respectively.

In this paper we will construct an effective local model for the deuteron-target adiabatic potential using NLDOM. Therefore, we first evaluate the effective local potential model for nucleon scattering. Following the procedure in PB for transforming a nonlocal potential to a local equivalent, one finds that for a nonlocal potential with multiple nonlocalities of the Perey-Buck type the local-equivalent potential can be found by solving the transcendental equation

 Uloc(r)=∑iUNA,i(r)exp[−μNβ2i2ℏ2(E−Uloc(r))] (4)

where is the reduced mass of the system. This equation is obtained in the lowest order of the expansion of over and corrections to any order can be built systematically using developments from Fid . In particular, the first order correction is

 ΔUN=ℏ2μN[(∇ff)2−12∇2ff], (5)

where the function is the Perey factor explained below. For proton scattering, equation (4) must be corrected by reducing the centre-of-mass energy in the r.h.s. of Eq. (4) by the local Coulomb interaction .

According to Fid , the local spin-orbit term must also be corrected when transforming to a local-equivalent potential. For the present case of a potential with multiple nonlocality parameters, the new spin-orbit term, , of the local-equivalent potential is

 Uleso=Uso/(1−U1) (6)

where

 U1=(1−∑iμNβ2i2ℏ2UNA,i(r)Gi(r,E)) (7)

and

 Gi(r,E)=exp[−μNβ2i2ℏ2(E−Uloc(r))]. (8)

Figure 1 compares proton scattering differential cross sections (normalized by the Rutherford cross section) determined from solving the NLDOM scattering problem exactly using the iterative procedure outlined in Ref.  Michel09 and from solving the local-equivalent problem. Both the Coulomb and spin-orbit corrections are included. The experimental data are from Refs. Dicello ; McCamis ; Blumberg ; Noro . Aside from small deviations at large angles, the results from using Eq. (4) are very similar to the exact solutions. The results from including are also shown. Overall, this correction improves the correspondence between the exact and approximate solutions of the nonlocal problem for angles and above.

For Perey-Buck potentials with one nonlocality parameter the wave function obtained from the phase-equivalent local model defined by a potential differs in the nuclear interior from the exact wave function by the Perey factor Fid

 f(r)=exp[μNβ24ℏ2Uloc(r)]. (9)

Elastic scattering observables do not depend on the Perey factor. Transfer cross sections may depend on it if they are not peripheral. In the particular case of Ca(d,p)Ca, the internal part contributes up to 20 Pan07 for the energy being considered, and this contribution is more important in the DWBA than in the ADWA. One can show it is also possible to derive the Perey factor for optical potentials with multiple nonlocalities such as in NLDOM. In this case the Perey factor is

 f(r)=exp[μNβ2eff(r)4ℏ2Uloc], (10)

where

 β2eff(r)=−U−1loc(r)∫∞rdr1∑iβ2iU′NA,i(r1)Gi(r1,E)1−U1(r1,E).

This Perey factor has some effective -dependent range , which can be complex. The real and imaginary parts are shown in Fig. 2 for the case of p - Ca elastic scattering at several proton energies. The imaginary part is small and has a negligible effect on the (d,p) cross sections. We note that decreases with energy. This decrease reflects the fact that is larger than . Since the volume imaginary potential dominates at higher energies, the term in Eq. (4) with becomes more important with increasing energy. This term also seems to dominate at large , as Re converges to fm for all energies.

The Perey factor for NLDOM is shown in Fig. 3, evaluated at the energy MeV, which is the center of mass energy for the outgoing proton in the Ca(d,p)Ca reaction with MeV (in the lab frame). It is compared to the Perey factor of an earlier version of the DOM Mueller11 that is purely local (LDOM). This Perey factor will be used in Sec. IV. It is given by Mahaux87 ; Mahaux91

 f(r,E)=√~m(r,E)/m, (12)

where is the so-called momentum-dependent effective mass and is related to the LDOM Hartree-Fock potential as

 ~m(r,E)m=1−dVHF(r,E)dE. (13)

Figure 3 also shows the Perey factor calculated with the widely used CH89 potential using Eq.( 9) and assuming fm. The Perey factors from LDOM and CH89 both have less effect in the surface region than the one calculated with NLDOM.

To calculate the Ca(d,p)Ca cross sections a choice needs to be made for the optical potential in the exit channel. In principle, this potential is auxiliary, and it is believed that choosing that describes proton elastic scattering in the exit channel makes the remnant term in the transfer amplitude to disappear Satchler . Since NLDOM was not fit to Ca scattering data, one choice for the auxiliary Ca potential is to use NLDOM but evaluated with instead of . An alternative, originally proposed in GW and then further explored in Tim99 , stems from the argument that the remnant term can be removed from the transition operator exactly, leading to a different model for the exit state wave function. In this model, the three-body Hamiltonian associated with the exit channel contains the Ca optical potential, the Ca bound-state potential and no interaction. In the limit of infinitely large core and in the zero-range approximation, the corresponding three-body wave function contains the Ca distorted wave function calculated with the Ca optical potential. Corrections due to recoil excitation and breakup are considered in Tim99 . For light nuclei the validity of the transfer amplitude with no remnant has also been confirmed by Mor09 . We analyzed the (d,p) reaction with both choices for , and the resulting cross sections were found to differ at the peak by about 1%. For the purposes of this study, both choices give practically the same result. Below, we choose to use NLDOM evaluated with for .

## Iii The deuteron-target potential for (d,p) reactions in the adiabatic approximation

Following Johnson-Tandy JT we retain only the first Weinberg component of the system in the transfer amplitude since this amplitude is sensitive only to those parts of this wave function in which the neutron is close to the proton . Recently, exact continuum-discretized coupled channel calculations confirmed that this component indeed dominates Pan13 . The first Weinberg component is a product of the deuteron wave function times the relative motion wave function which is the solution of the two-body Schrödinger equation with an adiabatic potential constructed from and optical potentials. The generalization of the deuteron adiabatic potential for the case of nonlocal, energy-independent optical potentials of the Perey-Buck type is given in Tim13b . If nonlocal potentials (such as NLDOM) explicitly depend on energy then they should be evaluated at the energy and then treated as nonlocal and energy-independent Joh14 . The term is half the kinetic energy in deuteron averaged over the short-ranged potential . The value of this term is about 57 MeV Joh14 , so for MeV, the NLDOM potential should be evaluated at about MeV. This at first sight seems counterintuitive. However, keeping in mind that neutron transfer takes place when proton and neutron in deuteron are at very short separations where the Heisenberg principle dictates high relative momentum, it becomes clear that there is an additional kinetic energy in the system which should be taken into account when choosing the energy at which the potential should be evaluated.

### iii.1 Lowest order equivalent local model

The nonlocal Schrödinger equation for from Tim13b can easily be generalized for the case of nonlocal optical potentials with multiple nonlocalities:

 (TR+UC(R)−Ed)χ(\boldmathR)=−∑N=n,p∑i∫d\boldmathsd\boldmathx ×ϕ1(\boldmathx+α1\boldmaths)UNA,i(\boldmathx2−\boldmathR)Hi(s)ϕ0(\boldmathx)χ(α2\boldmaths2+% \boldmathR)

where is the radius-vector between and , is the kinetic energy operator associated with , , , is the deuteron ground state wave function and

 ϕ1(\boldmathr)=Vnp(\boldmathr)ϕ0(\boldmathr)⟨ϕ0|Vnp|ϕ0⟩. (15)

Solving the nonlocal problem (LABEL:NLeq) directly is certainly possible, and has recently been done in Ref. Titus16 . However, in this paper, we construct the local-equivalent model, as simplified local-equivalent models can provide useful insight into the physics of a problem and make available transfer reaction codes applicable to nonlocal problems. The local-equivalent approximation of (LABEL:NLeq) can be obtained by expanding both and into Taylor series. In the lowest order approximation, using we get

 (TR+UC(R)−Ed)χ(\boldmathR)=−∑iUdA,i(R)~H(0)i(TR)χ(\boldmathR), (16)

where ,

 ~H(0)i(TR)=M(0)0,i∞∑n=0(−)nn!(μdα222ℏ2(β(0)n,i)2TR)n, (17)

the coefficients are defined by

 β(0)n,i=1√2⎡⎢⎣M(0)2n,i(2n+1)!!M(0)0,i⎤⎥⎦12n, (18)

and the moments are defined by

 M(0)2n,i=∫d\boldmathsd\boldmathxs2nHi(s)ϕ1(\boldmathx−α1\boldmaths)ϕ0(\boldmathx). (19)

Eq. (16) is further simplified by introducing the local-energy approximation Satchler ,

 TR≈T0(R)=Ed−U0loc(R)−UC(R), (20)

where the local potential is defined as

 U0loc(R)=∑iUdA,i(R)~H(0)i(T0(R)). (21)

This approximation works very well for nucleon optical potentials with one nonlocality parameter Tim13b . We will show that it remains good for NLDOM with its multiple nonlocalities, but we first present the results of calculations of for = 11.8 MeV using the deuteron wave function from the Hultèn model, the same as in Tim13b . It is pointed out in Tim13b that a realistic deuteron wave function gives the moment which is very similar to that obtained with the Hultén wave function. In Table 3 we show the calculated and terms. For very small nonlocalities, should be approximately equal to Tim13a ; Tim13b . For typical nonlocalities of fm they are smaller than the value but are essentially independent of , allowing one to replace the coefficients with a constant. We define this constant to be

 βd,i=β(0)1,i. (22)

The summation in (17) then gives an exponential form

 ~H(0)i(TR)≈H(0)i(γiTR)=M(0)0,iexp(−γiTR), (23)

where

 γi=μdα22β2d,i2ℏ2. (24)

As increases beyond 1.0 fm, the approximation in Eq. (22) becomes less valid and the coefficients deviate from even more. This trend can be seen in Table 3 and is especially apparent for fm, which is the largest nonlocality parameter used in NLDOM. Thus, we solve Eqs. (17), (20), (21) by restricting the sum over in (17) by some . We found that is sufficient to obtain a converged solution for . However, we also found that using Eq. (22) leads to practically the same result because the real and imaginary parts of the NLDOM terms associated with fm are small at the energies being considered. The potentials obtained from solution of (21) with (17) and (23) are shown in Fig. 4. Using Eq. (22) for all nonlocality parameters, one obtains from the transcendental equation

 U0loc = ∑iM(0)0,iUdA,i× (25) exp[−μdα22β2d,i2ℏ2(Ed−U0loc−UC)].

The Coulomb potential is approximated by a constant, given by

 UC=−1.08+1.35ZA1/3, (26)

which was used in Refs. Gia76 ; GRZ . The difference between using this approximation and a more realistic potential is only about 1%, in terms of the peak cross section of the proton angular distribution for the Ca(d,p)Ca reaction at MeV.

### iii.2 Correction to the local-energy approximation in the lowest order

It was shown in Tim13b that corrections to the lowest order local model beyond the local-energy approximation are small because they are the fourth-order effect of the nucleon nonlocality , as the second-order terms cancel each other for Perey-Buck potentials with one nonlocality parameter. The NLDOM from NLDOM contains several nonlocalities, and second-order contributions may not cancel. Moreover, some of these nonlocalities are large so that the contributions beyond the local-energy approximation are expected to be larger than those in Tim13b . In this section we study these corrections using results from sections IV.C and A.4 of Tim13b . Including leading correction term, linear in the kinetic energy operator , and using the exponential form (23) for , the right-hand side of Eq. (16) becomes

 ∑iUdA,iH(0)i(γiTR)χ(% \boldmathR)≈∑iUdA,iH(0)i(γiT0) ×[1−γi(TR−T0+Δi)−ℏ2γ2i2μd\boldmath∇T0⋅\boldmath∇R]χ(\boldmathR), (27)

where the energy correction , arising because and do not commute, is given by

 Δi=ℏ2γi2μd(T′′02+T′0R−γi3T′20). (28)

The solution of Eq. (16) in this approximation is the product

 χ(\boldmathR)=f0(\boldmathR)φ(%\boldmath$R$), (29)

where is the scattering wave of the local model

 (TR+UC−Ed)φ=−(U0loc+ΔU0)φ. (30)

The term is discussed in the previous section, and the correction term is

 ΔU0=TRf0f0+ℏ24μd(U2T′01−U1)2−UΔ1−U1 (31)

with

 Un(R)=∑iUdA,iH(0)i(γiT0)γni, (32) UΔ(R)=∑iUdA,iH(0)i(γiT0)γiΔi. (33)

The function is the analog of the Perey factor discussed in Sec. II. It modifies the scattering wave function in the nuclear interior and satisfies the first order differential equation

 \boldmath∇f0f0=−12U2(R)1−U1(R)\boldmath∇T0, (34)

with the boundary condition at . The solution of this equation is

 f0(R)=exp(12∫∞RdR1U2(R1)1−U1(R1)T′0(R1)). (35)

Because of multiple nonlocalities, the analytical integration in (35) cannot be done. So, it is difficult to see if the contributions to from second-order terms on cancel. Most likely, they do not.

The Perey factor and the correction to the equivalent local potential are shown in Fig. 5 and Fig. 6, respectively, for Ca at a deuteron incident energy of 11.8 MeV. The Perey factor increases the scattering wave in the nuclear interior by about 6, which is a couple of percent higher than the result in Tim13b . The correction to , however, remains small. Its real part is very close to the one obtained in Tim13b in the maximum, being about 150 keV, while the imaginary part is much smaller. Thus, for NLDOM the second order corrections most likely remain small and the local-energy approximation remains good.

### iii.3 First order corrections

The first order correction to the local-equivalent lowest-order model of Sec. III.1 is obtained by retaining two terms in the Taylor series expansion of the central potential :

 U(±\boldmathx2−\boldmathR)≈U(\boldmathR)∓12\boldmathx⋅\boldmath∇U(\boldmathR). (36)

In this case, using techniques of Tim13b , we obtain the following:

 (TR+UC(R)−Ed)χ(\boldmathR)= −∑iUdA,i(R)H(0)i(γiTR)χ(%\boldmath$R$) −∑i\boldmath∇[UdA,i(R)]H(1)i(~γiTR)\boldmath∇χ(\boldmathR), (37)

where

 H(1)i(~γiTR)=M(1)0,i3M(0)0,iH(0)i(~γiTR) (38)

and the moments are defined as

 M(1)2n,i=∫d\boldmathsd\boldmathxs2nHi(s)ϕ1(\boldmathx−α1\boldmaths)ϕ0(\boldmathx)α2\boldmaths⋅\boldmathx4. (39)

The new factor arises from the fact that the moments lead to new coefficients that are also practically independent of (see Table 3). Introducing a new constant

 ~βd,i=β(1)1,i (40)

the factor can be written as

 ~γi=μdα22~β2d,i2ℏ2. (41)

The coefficients are defined as

 β(1)n,i=1√2⎡⎢⎣3M(1)2n,i(2n+3)!!M(1)0,i⎤⎥⎦12n. (42)

At this point we make the local-energy approximation (20) but we also include the correction to this approximation similar to that considered above. We expand in the first term of r.h.s of Eq. (37) as in Eq. (27), but we use a simpler expansion for in the second term,

 H(1)i(~γiTR)=H(1)i(~γiT0)(1−~γi(TR−T0)), (43)

because is an order of magnitude smaller than so that the higher order terms on in terms with and will be small. For a similar reason we keep only the leading correction to :

 (TR−T0)\boldmath∇χ(R)≈% \boldmath∇T0χ(R). (44)

With these approximations we can solve Eq. (37) by introducing the same representation used both for proton scattering in Sec. II and for correction to local-energy-approximation above. The scattering wave is found from the local equation

 (TR+UC−Ed)φ=−(U0loc+ΔU)φ, (45)

with the same as before but corrected by

 ΔU=ΔU0+ΔU1, (46)

in which the first term is the same as in Eq. (31), and the second term is given by

 ΔU1=TRf1f1+μdℏ2U20(1−U1)2−U1T′01−U1−12U0U2T′0(1−U1)2,

where

 Un(R) = ∑i\boldmath∇[UdA,i(R)]H(1)i(~γiT0)~γni. (48)

The Perey factor is the solution of the first order differential equation

 \boldmath∇ff=μdℏ2U01−U1−12U2T′01−U1 (49)

with the boundary condition at . The solution to this equation can be written as

 f(R)=f0(R)f1(R) (50)

where is given by Eq. (35) and is given by

 f1(R)=exp(−μdℏ2∫∞RdR′U0(R′)1−U1(R′)). (51)

The correction and the four terms on the right-hand side of Eq. (47) are shown in Fig. 7. The Perey factors and are plotted in Fig. 5. The correction and the Perey factor are comparable to the corresponding quantities obtained in [7] for a Perey-Buck potential with a single nonlocality. We can rewrite the Perey factor in the form of Eq. (10). The corresponding effective deuteron nonlocality, is plotted in Fig. 8 as a function of for MeV. Note that is complex, but, as in the case for protons, the imaginary part is small and changes the cross section of the proton angular distribution by less than 0.5%. The real part lies between 0.50 and 0.60 fm, and these values are very similar to 0.56 fm, which is the value used for deuteron elastic scattering.

## Iv Transfer reaction 40Ca(d,p)41Ca at 11.8 MeV

We calculated the proton angular distributions for the Ca(d,p)Ca reaction at MeV in the adiabatic zero-range approximation. The finite range effects at these energies are expected to be small FR . The standard value for , given by MeVfm, was used. The distorted potentials both in the deuteron and proton channels, generated with NLDOM, were read into the TWOFNR code 2FNR . There is no option in TWOFNR for incorporating complex -dependent effective nonlocalities . Therefore, in order to reduce the corresponding distorted waves in the nuclear interior, we multiplied the NLDOM CaCa overlap function (also read into the TWOFNR code) by the Perey factors of the proton and deuteron channels, given by Eqs. (10) and (50), respectively. This is legitimate in the zero-range approximation, where the integrand of the (d,p) reaction amplitude is a function of only one vector variable. In this case, the Perey factor for the proton channel had to be calculated on a different grid.

The overlap function , generated by NLDOM and read into TWOFNR, is compared in Fig. 9 to (i) the overlap function obtained from a Woods-Saxon potential with standard geometry ( fm, fm); (ii) the overlap function generated with LDOM and (iii) the overlap function , calculated in a standard Woods-Saxon model employing a nonlocality correction via the Perey factor with fm. All these overlap functions are normalized to 0.73, which is the spectroscopic factor calculated from NLDOM.

We have found that can be described very well (with about 1% accuracy) by a local two-body Woods-Saxon potential model that has the radius 1.252 fm, diffuseness fm and the spin-orbit strength MeV. These parameters are very close to the standard values of = 1.25 fm and 0.65 fm used to generate . However, relative to , using as the overlap function increases the transfer cross section at the peak, , by about 15% (for the reaction at MeV).

The NLDOM and LDOM overlap functions have similar shapes, but their r.m.s. radii and asymptotic normalization coefficients (ANC) somewhat differ. The radius of 4.030 fm is slightly larger than that of , which is 3.965 fm. Also, the single-particle ANC for is fm, which is about 10% larger than that of . The spectroscopic factors for these two overlaps are practically the same. As a result, produces a larger many-body ANC squared, , equal to fm, whereas has fm. Interestingly, the NLDOM value of is very close to the prediction of fm of the source term approach Tim11 , which is based on the independent-particle-model for Ca and Ca. This approach accounts for correlations between nucleons via an effective interaction potential of the removed nucleon with nucleons in the core Tim09 .

The standard overlap is very close to (see Fig. 9). The overlap , which is sometimes used in (d,p) calculations, has a distinctly larger radius and is not consistent with NLDOM. Below, in all our (d,p) calculations we use only the NLDOM overlap with its own normalization of 0.73, which allows for making conclusions from comparison between theoretical and experimental cross sections without any further renormalization.

Proton angular distributions calculated using the NLDOM potentials are presented in Fig. 10. The solid curve corresponds to the lowest-order result. The long-dashed curve shows that incorporating the first-order corrections for the proton channel, via Eqs. (5) and (10), reduces the lowest-order (d,p) peak cross sections by 3 %. Further first-order corrections, coming from Eqs. (46) and (50) for the deuteron channel and shown by the short-dashed curve, reduces by another 5 %.

Finally, including the spin-orbit potential raises and makes it comparable to the cross sections obtained with no corrections in the deuteron channel (the dot-dashed curve). The experimental data are from ca40 . The spread between all these calculations does not exceed 12% and all of them considerably overestimate the experimental data, shown in Fig. 10 as well. This overestimation (by about 70% after normalizing overlap function to 0.73) cannot come from the local approximations we have made to solve the nonlocal problem. The first-order corrections of about 12% mean that the second-order corrections would most likely be around 1% or less. Thus, other reasons for this overestimation should be investigated.

It was already noted in Tim13a ; Tim13b that the adiabatic model with nonlocal energy-independent potentials gives higher cross sections, as compared to the standard adiabatic model, due to a weaker attraction in the deuteron channel. The higher cross sections are confirmed in other (d,p) studies with such potentials Titus16 . Here, the overestimation of the (d,p) cross sections using the NLDOM is stronger than in the case of energy-independent potentials. This can be seen in Fig. 11, which compares the NLDOM angular distribution with the angular distributions from two nonlocal, energy-independent potentials, referred to as GR Gia76 and TPM TPM . Figure 11 also shows the angular distribution from another nonlocal, energy-dependent potential, referred to as GRZ GRZ and used previously in Joh14 . This potential generates an angular distribution very similar to the one generated with NLDOM. The structure of the GRZ potential is not as complicated as the NLDOM potential, but it has a typical low-energy behaviour of the imaginary part, vanishing at (for N = Z). For all distributions, the corresponding nucleon optical model parameterizations were consistently used to calculate the exit and entrance channel potentials, taking into account the first-order corrections and the Perey effect.

To understand the difference in shown in Fig. 11, we compare the entrance and exit channel potentials generated from the four nonlocal parameterizations and present them in Fig. 12. The NLDOM, GRZ and GR generate real parts of similar depths and sizes both in entrance and exit channels while the TPM produces a real part of a moderately larger radial extent. The imaginary parts, however, show a marked difference, in both the entrance and exit channels. The energy-independent parameterizations GR and TPM produce a much larger imaginary part in the surface region than the energy-dependent parameterizations NLDOM and GRZ. The smaller imaginary parts produce less absorption thus increasing . This is even better seen in Fig. 13, which shows the (d,p) angular distributions calculated with four different parameterizations for the exit proton channel and using NLDOM for the deuteron channel. In this figure, we have also added the calculations with LDOM Mueller11 and the widely used local CH89 CH89 parameterizations in the proton channel while keeping the NLDOM in the deuteron channel. The LDOM and CH89 proton potentials are shown in panels c) and d) of Fig. 12.

Figure 13 shows that predictions with NLDOM, LDOM and GRZ form a different class from those obtained with GR, TPM and CH89. All the potentials from the first class have smaller imaginary parts and/or volume integrals than the potentials from the second class. Thus, overestimation of the cross section calculated with NLDOM seems to be at least partly due to a weaker absorption in the exit channel potential.

The weaker absorption may not be the only reason for large cross sections obtained with NLDOM. It was discussed in detail in Har71 that a particular relation between optical potentials in entrance and exit channels results in destructive interference between the ingoing and outgoing partial waves leading to -localization of radial (d,p) amplitudes in the adiabatic model. A similar situation may occur here. Indeed, the standard adiabatic Johnson-Soper (JS) JS calculations using both the local-equivalent NLDOM and LDOM potentials, taken at , predict much lower cross sections (see Fig. 14) while the imaginary parts of the JS potentials, shown in Fig. 15, are much smaller than those of local-equivalent deuteron potentials obtained in this work (TJ). This could be an indication of constructive interference between the ingoing and outgoing partial waves generated with NLDOM potentials. A new procedure was proposed in Sec. VI.B of Ref.  Joh14 , explaining how phenomenological local energy-dependent optical potentials can be used in (d,p) calculations if they represent local equivalents of nonlocal potentials. Using the LDOM potential within this procedure and assuming a hidden nonlocality of 0.85 fm gives very similar results to NLDOM both for the real part of deuteron distorting potential (Fig. 15) and the (d,p) cross sections (Fig. 14) despite stronger imaginary part in the deuteron channel. Although the JS cross sections are close to experimental data, in light of recent findings Tim13a ; Tim13b ; Joh14 ; Titus16 , constructing the adiabatic potentials from nucleon optical potentials taken at half the deuteron incident energy does not seem to be justified anymore.

## V Conclusion

We presented the first adiabatic (d,p) calculations with the NLDOM potential, which has been designed with the aim of forging the link between nuclear structure and nuclear reactions in a consistent way. It has its roots in the underlying self-consistent Green’s functions theory and possesses the fundamental properties - nonlocality, energy-dependence and dispersion relations - that arise from the complex structure of the target. The NLDOM explicitly takes into account a number of components of nuclear many-body theory that many other optical models do not.

One could expect that using an advanced optical potential parametrization such as NLDOM would result in properly fixed single-nucleon properties both below and above the Fermi surface crucial for agreement between predictions of (d,p) reaction theory and experimental data. However, we have shown that using the NLDOM to generate the distorting potentials entering the (d,p) amplitude strongly overestimates the (d,p) cross sections despite the reduced strength of the NLDOM one-neutron overlap function employed in the calculations. Moreover, the NLDOM predictions are very similar to those made with a much simpler nonlocal potential GRZ derived within Watson multiple scattering theory and Wolfenstein’s parameterization of the nucleon-nucleon scattering amplitude Gia76 ; GRZ . The energy dependence is presented in GRZ only in the imaginary part.

Since we do not have strong reason to doubt the quality of the NLDOM parameterization the main assumptions of the (d,p) theory used in the present calculations should be reviewed. We list them below:

• The (d,p) amplitude contains a projection of the total many-body wave function into the three-body channel only. Projections onto all excited states of A are neglected.

• Only and potentials are used to calculate the projection. According to Joh14 there are also multiple scattering terms playing the role of a three-body force. These are neglected.

• Averaged and potentials were obtained using the procedure from Ref. Joh14 , which uses the adiabatic approximation. Corrections to this approximation may change the energy value at which these potentials should be evaluated.

• It was assumed that the (d,p) transition operator contains the term only. Any other terms present in this amplitude Satchler are neglected.

• It was shown in Tim99 that keeping only in the (d,p) transition operator modifies the proton channel wave function. In our particular case, this would result in using p-Ca optical potential in the p+Ca channel. We have not seen any difference in (d,p) cross sections when replacing Ca by Ca and this could mean that the averaging procedure, introduced in Joh14 , when applied to the special three-body model, that does not have and has different asymptotic conditions Tim99 , may result in completely different requirements to the proton distorting potential in the exit channel. Using proton optical potentials may not be justified anymore.

• We used the adiabatic approximation to solve the three-body Schrödinger equation.

The deviation from the adiabatic approximation for solving the Schrödinger equation has been studied both within the continuum-discretized coupled channel method Mor09 and using Faddeev equations Nunes11 . Although these corrections can be non-negligible, they cannot be responsible for 70% overestimation of (d,p) cross sections obtained in this work. At MeV these deviations were no more than 4, while at a larger energy range, MeV they could be up to 23. The unknown non-adiabatic corrections to optical and potentials entering the Schrödinger equation for the model Joh14 can change both real and imaginary parts of these effective potentials, which could affect the (d,p) cross sections. But given that the adiabatic approximation is a good first choice for the (d,p) reactions, most likely, they will not explain the 70% difference between the NLDOM predictions and experiment.

The contributions from the remnant term in the (d,p) amplitude (all other terms that are not ) have been studied in an inert core model, where they were found to be small Smith69 . We estimated the effect of the remnant term for the Ca(d,p)Ca reaction at MeV using FRESCO FRESCO , which employs the inert core approximation and requires the use of local-equivalent potentials. We also found the effect of the remnant term to be small, decreasing the cross-section by about 3. A more recent study Ramos15 showed that the remnant term contributions remain small even when incorporating core-excitation effects, although they can become more important for nuclei in which the core has a low excitation energy. Whether these contributions remain small for a nonlocal is not known.

The strong overestimation of the Ca(d,p)Ca cross sections at 11.8 MeV implies that neglected parts of the (d,p) amplitude and/or its constituents are much more important than was thought before. Given that the deuteron energy, chosen for this work, is often used in modern experiments with radioactive beams for spectroscopic and astrophysical reasons, and that the dispersive effects are strong at this energy, further development of direct reaction theories is crucial to understand transfer experiments performed either recently or in the past and planned for the future.

## Acknowledgement

N.K.T. gratefully acknowledges support from the UK STFC through Grant ST/L005743/1. We are grateful to R.C. Johnson for fruitful discussions. S. J. Waldecker would also like to thank W. H. Dickhoff and M. H. Mahzoon for fruitful discussions.

## References

• (1) G.R. Satchler, Direct Nuclear Reactions (Oxford Press, N.Y., 1983) pp.819-824.
• (2) R. C. Johnson and Tandy, Nucl. Phys. A 235, 56 (1974).
• (3) A. M. Moro, F.M.Nunes, R.C.Johnson, Phys.Rev. C 80, 064606 (2009).
• (4) A. Deltuva, Phys. Rev. C 91, 024607 (2015).
• (5) A. Deltuva and A.C. Fonseca, Phys. Rev. C 79, 014606 (2009).
• (6) R.C. Johnson and N.K. Timofeyuk, Phys. Rev. C 89, 024605 (2014).
• (7) N.K. Timofeyuk and R.C. Johnson, Phys. Rev. C 87, 064610 (2013).
• (8) M.M. Giannini, G Ricco, A Zucchiatti, Ann. Phys. 124, 208 (1980).
• (9) M. H. Mahzoon, R. J. Charity, W. H. Dickhoff, H. Dussan, and S. J. Waldecker, Phys. Rev. Lett. 112, 162503 (2014).
• (10) A. Ross, L. J. Titus, F. M. Nunes, M. H. Mahzoon, W. H. Dickhoff, and R. J. Charity, Phys. Rev. C 92, 044607 (2015).
• (11) H. Dussan, S. J. Waldecker, W. H. Dickhoff, H. Müther, and A. Polls, Phys. Rev. C 84, 044319 (2011).
• (12) W. H. Dickhoff and D. Van Neck, Many-Body Theory Exposed!, (World Scientific, Singapore, 2008), 2nd ed.
• (13) F. Perey and B. Buck, Nucl. Phys. 32, 353 (1962).
• (14) H. Fiedeldey, Nucl. Phys. 77, 149 (1966).
• (15) N. Michel, Eur. Phys. J. A 42, 523 (2009).
• (16) J. F. Dicello, G. Igo, W.T. Leland, and F. G. Perey, Phys. Rev. C 4, 1130 (1971).
• (17) R. H. McCamis et al., Phys. Rev. C 33, 1624 (1986).
• (18) L. N. Blumberg, E. E. G. Gross, A. van der Woude, A. Zucker, and R. H. Bassel, Phys. Rev. 147, 812 (1966).
• (19) T. Noro, H. Sakaguchi, M. Nakamura, K. Hatanaka, F. Ohtani, H. Sakamoto, and S. Kobayashi, Nucl. Phys. A 366, 189 (1981).
• (20) D.-Y. Pang, F. M. Nunes, A. M. Mukhamedzhanov, Phys.Rev. C 75, 024601 (2007).
• (21) J.M. Mueller et al., Phys. Rev. C 83, 064605 (2011).
• (22) C. H. Johnson, D. J. Horen, and C. Mahaux, Phys. Rev. C 36, 2252 (1987).
• (23) C. Mahaux and R. Sartor, Adv. Nucl. Phys. 20, 1 (1991).
• (24) M. L. Goldberger and K. M. Watson, Collision Theory (Wiley, New York, 1964).
• (25) N.K. Timofeyuk and R.C. Johnson, Phys. Rev. C 59, 1545 (1999).
• (26) D.-Y. Pang, N. K. Timofeyuk, R. C. Johnson, and J. A. Tostevin, Phys. Rev. C 87, 064613 (2013).
• (27) L. J. Titus, F. M. Nunes, and G. Potel, Phys. Rev. C <