Weak sensitivity of threebody () reactions to force models
Abstract
 Background

Adiabatic distortedwave approximation (ADWA) study of threebody transfer reactions [G.W. Bailey, N.K. Timofeyuk, and J.A. Tostevin, Phys. Rev. Lett. 117, 162502 (2016)] reported strong sensitivity of cross sections to the neutronproton interaction model when the nucleonnucleus optical potential is nonlocal.
 Purpose

The verification of this unusual finding using more reliable methods is aimed for in the present work.
 Methods

A rigorous Faddeevtype threebody scattering theory is applied to the study of transfer reactions. The equations for transition operators are solved in the momentumspace partialwave framework.
 Results

Differential cross sections for reactions are calculated using nonlocal nuclear optical potentials and a number of realistic potentials. Only a weak dependence on the force model is observed, typically one order of magnitude lower than in the previous ADWA study. The shape of the angular distribution of the experimental data is well reproduced.
 Conclusions

Cross sections of transfer reactions calculated using a rigorous threebody method show little sensitivity to the interaction model. This indicates a failure of the ADWA in the context of nonlocal potentials. Some evident shortcomings of the ADWA are pointed out.
I Introduction
Threebody transfer reactions with nucleonnucleus nonlocal optical potentials (NLOP) and a number of neutronproton potentials have been studied in Ref. PhysRevLett.117.162502 () using the adiabatic distortedwave approximation (ADWA). The authors of Ref. PhysRevLett.117.162502 () claimed that NLOP leads to a substantial increase of the sensitivity of lowenergy () cross sections to highmomentum components in the deuteron wave function and interaction. This sensitivity manifested itself as a strong dependence of cross sections on the potential model, even exceeding a factor of 2 in particular cases such as the (g.s.) reaction. Such sensitivity contrasts with the usual behavior in the lowenergy nuclear physics and calls for further studies using alternative methods. The authors of Ref. PhysRevLett.117.162502 () suggested that the Weinberg state expansion, the basis of ADWA, should be revisited in the presence of nonlocality. The present work, being an extended version of the pioneering study deltuva:commdp (), goes well beyond the suggestion of Ref. PhysRevLett.117.162502 (). I use a rigorous treatment of threebody reactions with NLOP based on the Faddeev theory faddeev:60a (). Although much more complicated than ADWA in terms of practical implementation deltuva:09b (), this treatment has the advantage of providing an exact solution of the threebody scattering problem. It does not rely on the Weinberg state expansion, thus, implicitly includes all Weinberg states up to infinity. Furthermore, it includes also the so called ”remnant term”, neglected in ADWA of Ref. PhysRevLett.117.162502 (), all relevant waves beside the deuteron wave, and the proper Coulomb force acting between the proton and nucleus, not between the centerofmass (c.m.) of the deuteron and nucleus as in the initial channel treatment by ADWA.
Section II recalls the threebody Faddeev formalism for transition operators. The results for reactions, the study of the force model sensitivity, and the comparison with the experimental data are presented in Sec. III. Section IV contains the conclusions and discussion, also pointing out some inadequacies of the ADWA.
Ii Theory
A rigorous threebody treatment of nuclear reactions with NLOP based on the Faddeev equations in the AltGrassbergerSandhas (AGS) form alt:67a () was first implemented in Refs. deltuva:09b (); deltuva:09d (). At the given threebody energy in the c.m. frame the transition operators obey the system of integral equations
(1) 
with , the free resolvent , and the free Hamiltonian for the relative motion . The twoparticle transition matrices in threeparticle space
(2) 
are obtained from the corresponding twoparticle potentials . Oddmanout notation is used to label pairs of particles and twocluster channels with asymptotic states , while onshell matrix elements yield reaction amplitudes. The integral equations (1) are solved in the momentumspace partialwave framework, leading to wellconverged results for differential cross sections, with the protonnucleus Coulomb force included via the screening and renormalization method. Further technical details on the solution can be found in Refs. deltuva:09b (); deltuva:09d ().
Iii Results
I study reactions using a number of realistic highprecision potentials: Argonne V18 (AV18) wiringa:95a (), chargedependent Bonn (CD Bonn) machleidt:01a (), Reid93 stoks:94a (), chiral effective field theory (EFT) potentials at nexttonexttonexttonexttoleading order (N4LO) PhysRevLett.115.122301 () with regulators of 0.8 and 1.2 fm, and the AV18 potential softened by the similarity renormalization group (SRG) transformation bogner:07b (); bogner:07c () with the flow parameter . The deuteron state probability, characterizing the strength of the tensor force and highmomentum components, acquires values in a broad range from 2.53% (SRG) to 5.76% (AV18), even broader than in Ref. PhysRevLett.117.162502 (). To compare with ADWA results of Ref. PhysRevLett.117.162502 (), I take the same NLOP for  and , i.e., the parametrization of Giannini and Ricco giannini () without the spinorbit part. Binding potentials for are also chosen as in Ref. PhysRevLett.117.162502 (), i.e., they have local WoodsSaxon form with the radius fm, diffuseness fm, and spinorbit strength MeV, whereas the strength of the central potential is adjusted to the neutron separation energy for the given state.
The results at the deuteron beam energy MeV for differential cross sections with ground and excited final states are presented in Fig. 1. Both in the present work and in Ref. PhysRevLett.117.162502 () the sensitivity to the potential reaches its maximum at forward angles, therefore the following comparison refers to the c.m. scattering angle deg. The largest sensitivity to the potential using ADWA in Ref. PhysRevLett.117.162502 () was found for the transfer to the ground state with spin/parity and MeV. At deg the spread of ADWA results with realistic potentials in Ref. PhysRevLett.117.162502 () is roughly a factor of 2.5, or about % when measured from the central value. In contrast, rigorous Faddeev calculations, as shown in the top panel of Fig. 1, exhibit much weaker sensitivity to the force model, about % when measured from the respective central value. Furthermore, the magnitude of the differential cross section predicted using the Faddeev framework at deg is lower by a factor of 2 (CD Bonn) to 4 (AV18) as compared to ADWA; in the latter case even the shape is quite different.
Results for transfer reactions leading to excited states of are also presented in Fig. 1, this time using AV18, CD Bonn, and SRG potentials only; predictions of other potentials lie between those of AV18 and SRG and are therefore not shown. In all cases the sensitivity to the force model is significantly weaker than in ADWA predictions of Ref. PhysRevLett.117.162502 (). For example, for the reaction leading to the , MeV state at deg the spread of Faddeevtype results, measured from their central values, is % and % for the angular momentum transfer and , respectively, while the spread of the corresponding ADWA results is about % and %. Furthermore, the magnitude of Faddeevtype predictions for all excited states shown in Fig. 1 is lower than ADWA by a factor of roughly 1.5, while the shape is qualitatively similar. Note that partial waves other than the deuteron wave contribute to the cross section up to 10%; those waves are neglected in ADWA PhysRevLett.117.162502 ().
Although the main goal of the present work is the study of cross section sensitivity to the force model, the comparison of angular distributions with the experimental data is also of some interest, even if the calculations neglect the core excitation that is an important dynamic ingredient. To account roughly for this shortcoming, the theoretical predictions are rescaled by factors , such that the differential cross section for the given state with the excitation energy becomes
(3) 
I emphasize that are not spectroscopic factors (SF), but, given the low reaction energy MeV, may be quite close to the SF as the rigorous study including the core excitation indicates deltuva:17b (). The values, obtained by adjusting the CD Bonn results from Fig. 1 to the experimental data from Ref. PhysRevLett.115.062701 (), are , , , and , where the excitation energies are given in MeV. The comparison is presented in Fig. 2. One may conclude that theoretical calculations describe the shape of the experimental data PhysRevLett.115.062701 () quite well.
Iv Discussion and conclusions
The present study of reactions, performed in the rigorous momentumspace threebody framework, indicates low sensitivity to the potential model, and thereby a failure of the ADWA with NLOP PhysRevLett.117.162502 (). This may appear quite unexpected, since a decent accuracy of the ADWA, 20% or better nunes:11b (); PhysRevC.95.064608 (), was found when using local optical potentials. However, given that no sensitivity to the model was observed for local optical potentials PhysRevC.95.024603 (), much larger disagreement in the case of NLOP probably indicates the inadequacy of ADWA specifically for treating NLOP. In fact, some inadequacy can easily be seen when confronting the leadingorder NLOP effect in ADWA PhysRevC.95.024603 () to the rigorous Faddeev scattering framework faddeev:60a (). This comparison is justified, since the leadingorder yields a good approximation of the complete ADWA PhysRevC.95.024603 (), and therefore shortcomings characteristic to the leadingorder ADWA should be valid also in the case of the complete ADWA. The leadingorder NLOP effect in ADWA is simulated replacing the energyindependent NLOP in the initial deuteron channel by the equivalent local optical potential (ELOP) PhysRevC.95.024603 (). Energydependent ELOP has to be evaluated at the energy where, depending on the underlying potential, typically acquires values from to 70 MeV. E.g., MeV (72 MeV) for CD Bonn (AV18) models in the considered reactions at MeV. However, an inspection of the interaction terms on the r.h.s. of the Faddeev equation (1), i.e.,
(4) 
reveals that the energy of any interacting twoparticle subsystem with the intermediate relative momentum is not fixed but depends on the respective spectator momentum that is an integration variable, formally ranging from zero to infinity; and are the corresponding reduced masses. The energy of the nucleonnucleus subsystem therefore formally acquires values from to , i.e., its upper limit is MeV, amounting to MeV in the present study. Thus, of the order 50 MeV as required in the ADWA appears to be a very unnatural value from the Faddeev formalism point of view, where the threebody amplitudes depend on fully offshell nucleonnucleus transition operators but with the relative twobody energy below . Note that there is no such a clear contradiction in the case of ADWA with local potentials that are taken at the energy . Furthermore, the ELOP that is used in the leadingorder ADWA is onshell equivalent to the original NLOP only at but deviates from it otherwise. As a consequence, ELOP and NLOP do not provide an equivalent description of the initial scattering state in collisions. Most importantly, the ability of the energyindependent NLOP to describe the nucleonnucleus subsystem over a broader energy range is lost when using ELOP with fixed parameters giannini ().
In conclusion, ADWA appears to be inadequate for threebody reactions with NLOP, where low sensitivity to the interaction models is found by rigorous Faddeevtype calculations. A similar conclusion is drawn also by the alternative study using the approximate continuumdiscretized coupledchannel method gomez:nlop (). However, more significant sensitivity to nucleonnucleon force models can be expected in truly ab initio calculations without any use of optical potentials, where even the threshold positions may depend on the force model. Rigorous fourparticle calculations of and reactions deltuva:17a () provide an example for such dependence.
Acknowledgements.
I thank E. Epelbaum for providing the codes for EFT potentials. I acknowledge the support by the Alexander von Humboldt Foundation under Grant No. LTU1185721HFSTE, and the hospitality of the RuhrUniversität Bochum where a part of this work was performed.References
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