Calculation of proton-He elastic scattering between 7 and 35 MeV
Theoretical calculations of the four-particle scattering above the four-cluster breakup threshold are technically very difficult due to nontrivial singularities or boundary conditions. Further complications arise when the long-range Coulomb force is present.
We aim at calculating proton-He elastic scattering observables above three- and four-cluster breakup threshold.
We employ Alt, Grassberger, and Sandhas (AGS) equations for the four-nucleon transition operators and solve them in the momentum-space framework using the complex-energy method whose accuracy and practical applicability is improved by a special integration method.
Using realistic nuclear interaction models we obtain fully converged results for the proton-He elastic scattering. The differential cross section, proton and He analyzing powers, spin correlation and spin transfer coefficients are calculated at proton energies ranging from 7 to 35 MeV. Effective three- and four-nucleon forces are included via the explicit excitation of a nucleon to a isobar.
Realistic proton-He scattering calculations above the four-nucleon breakup threshold are feasible. There is quite good agreement between the theoretical predictions and experimental data for the proton-He scattering in the considered energy regime. The most remarkable disagreements are the peak of the proton analyzing power at lower energies and the minimum of the differential cross section at higher energies. Inclusion of the isobar reduces the latter discrepancy.
pacs:21.45.-v, 25.10.+s, 21.30.-x, 24.70.+s
Proton-He () scattering is one of the most commonly used experiments to study the four-nucleon system tilley:92a : It involves two charged particles that are stable and easy to detect with acceptable precision, there are no competing channels until MeV proton lab energy, and beyond that only three- and four-cluster breakup takes place up to the pion-production threshold. Much like neutron-H () scattering, is dominated by isospin , but has the deciding experimental advantage of having a proton beam and a non-radioactive He target. On the contrary, from the theoretical point of view, is more difficult to calculate than due to the long-range Coulomb force between protons () that gives rise to complicated boundary conditions in the coordinate-space and non-compact kernel in the momentum-space representation. Nevertheless, these difficulties have been solved below the three-cluster breakup threshold using three different theoretical frameworks, namely, the hyperspherical harmonics (HH) expansion method viviani:01a ; kievsky:08a , the Faddeev-Yakubovsky (FY) equations yakubovsky:67 for the wave function components in coordinate space lazauskas:04a ; lazauskas:09a , and the Alt, Grassberger and Sandhas (AGS) equations grassberger:67 for transition operators that were solved in the momentum space deltuva:07a ; deltuva:07b . A good agreement between these methods has been demonstrated in a benchmark for and elastic scattering observables viviani:11a using realistic nucleon-nucleon (NN) potentials.
Recently we extended the AGS calculations to energies above three- and four-cluster breakup thresholds deltuva:12c ; deltuva:13a . The complex energy method kamada:03a ; uzu:03a was used to deal with the complicated singularities in the four-particle scattering equations; its accuracy and practical applicability was greatly improved by a special integration method deltuva:12c . This allowed us to achieve fully converged results for elastic scattering and neutron-neutron-deuteron recombination into using realistic NN interactions. We note that the FY calculations of elastic scattering have been recently extended as well to energies above the four-nucleon breakup threshold lazauskas:12a , however, using a semi-realistic NN potential limited to -waves.
In the present work we extend the method of Ref. deltuva:12c to calculate the elastic scattering above breakup threshold and compare with existing data for cross sections and spin observables over a wide range of proton beam energies up to MeV. The Coulomb interaction is treated as in Refs. deltuva:05a ; deltuva:07b using the method of screening of the Coulomb potential followed by the phase renormalization of transition amplitudes taylor:74a ; alt:80a . Thus, standard AGS scattering equations with short-range potentials are applicable. At energies bellow three-cluster threshold our results agree with those obtained by other methods as mentioned in Ref. viviani:11a . Compared to our previous scattering calculations above the breakup threshold deltuva:12c , the most serious complication for is the convergence of the partial-wave expansion that requires a larger number of states due to the longer range of the screened Coulomb potential.
Ii 4N scattering equations
We use the symmetrized AGS equations deltuva:07a as appropriate for the four-nucleon system in the isospin formalism. They are integral equations for the four-particle transition operators , i.e.,
Here, corresponds to the partition (12,3)4 whereas corresponds to the partition (12)(34); there are no other distinct two-cluster partitions in the system of four identical particles. The free resolvent at the complex energy is given by
with being the free Hamiltonian. The pair (12) transition matrix
is derived from the potential ; for the pair includes both the nuclear and the screened Coulomb potential . Our calculations are done in momentum space; however, we start with the configuration-space representation
where is the true Coulomb potential, is the fine structure constant, is the screening radius, and controls the smoothness of the screening. All transition operators acquire parametric dependence on but it is suppressed in our notation, except for the scattering amplitudes. The symmetrized 3+1 or 2+2 subsystem transition operators are obtained from the respective integral equations
The basis states are antisymmetric under exchange of two particles in the subsystem (12) for the partition and in (12) and (34) for the partition. The full antisymmetry of the four-nucleon system is ensured by the permutation operators of particles and with and .
The scattering amplitude with nuclear plus screened Coulomb interactions at available energy is obtained from the on-shell matrix element in the limit . Here is the Faddeev component of the asymptotic state in the channel , characterized by the bound state energy MeV and the relative momentum , being the average nucleon mass. Due to energy conservation .
The amplitude is decomposed into its long-range part , being the two-body on-shell transition matrix derived from the screened Coulomb potential of the form (4) between the proton and the center of mass (c.m.) of , and the remaining Coulomb-distorted short-range part. Renormalizing by the phase factor taylor:74a ; alt:80a ; deltuva:07b , in the limit, yields the full transition amplitude in the presence of Coulomb
where the first term is obtained from that converges, in general as a distribution, to the exact Coulomb amplitude between the proton and the c.m. of the nucleus taylor:74a . The renormalization factor is defined in Refs. deltuva:05a ; deltuva:07b . The second term in Eq. (6), after renormalization by , represents the Coulomb-modified nuclear short-range amplitude. It has to be calculated numerically, but, due to its short-range nature, the limit is reached with sufficient accuracy at finite screening radii . Since the convergence with is faster at higher energies, the required screening radii are smaller than in our low-energy calculations deltuva:07b . We found that ranging from 8 to 10 fm leads to well-converged results in the energy regime considered in the present paper. Furthermore, we take a sharper screening with such that at short distances the screened Coulomb approximates the full Coulomb better than with used in Ref. deltuva:07b and at the same time vanishes more rapidly at thereby accelerating the partial-wave convergence.
We solve the AGS equations (1) in the momentum-space partial-wave framework. The states of the total angular momentum with the projection are defined as for the configuration and for the . Here and are the four-particle Jacobi momenta in the convention of Ref. deltuva:12a , , , and are the associated orbital angular momenta, and are the total angular momenta of pairs (12) and (34), is the total angular momentum of the (123) subsystem, and are the spins of nucleons 3 and 4, and , , and are channel spins of two-, three-, and four-particle system. A similar coupling scheme is used for the isospin. We include a large number of four-nucleon partial waves, up to , , and , such that the results are well converged. In fact, lower cutoffs are sufficient for lower , e.g., and are sufficient for . Furthermore, for most observables or even are enough for the convergence; yields small but still visible effect only above MeV.
The numerical calculations are performed for complex energies, i.e., with finite . The limit needed for the calculation of the amplitude is obtained by the extrapolation of finite results as proposed in Ref. kamada:03a . A special integration method developed in Ref. deltuva:12c is used to treat the quasi-singularities of the AGS equations (1). We obtain accurate results by using ranging from 1 to 2 MeV at lowest considered energies and from 2 to 4 MeV at highest considered energies. Grid points for the discretization of each momentum variable range from 30 (at lower energies) to 35 (at higher energies). Further details on the other numerical techniques for solving the four-nucleon AGS equations can be found in Ref. deltuva:07a .
We study the scattering using realistic high-precision NN potentials, namely, the Argonne (AV18) potential wiringa:95a , the inside-nonlocal outside-Yukawa (INOY04) potential by Doleschall doleschall:04a ; lazauskas:04a , the charge-dependent Bonn potential (CD Bonn) machleidt:01a , and its extension CD Bonn + deltuva:03c allowing for an excitation of a nucleon to a isobar and thereby yielding effective three- and four-nucleon forces (3NF and 4NF). The binding energy calculated with AV18, CD Bonn, CD Bonn + , and INOY04 potentials is 6.92, 7.26, 7.54, and 7.73 MeV, respectively; the experimental value is 7.72 MeV. Therefore most of our predictions correspond to INOY04 as it is the only potential that nearly reproduces the experimental binding energy of . The calculations with other potentials are done at fewer selected energies.
In Figs. 1 - 2 we show the differential cross section for elastic scattering as a function of the c.m. scattering angle at a number of proton energies ranging from to 35.0 MeV. This observable decreases rapidly with the increasing energy and also changes the shape; the calculations describe the energy and angular dependence of the experimental data fairly well. Below MeV the experimental data are slightly underpredicted at forward angles as happens also at energies below the three-cluster breakup threshold viviani:11a ; deltuva:07b . At the minimum the predictions scale with the binding energy: the weaker the binding the lower the dip of that is located between and . The scaling is more pronounced at higher . For the INOY04 potential that fits the binding energy, one gets an good agreement in the whole angular region up to MeV but, as the energy increases, the calculated cross section underpredicts the data at the minimum much like what happens in nucleon-deuteron elastic scattering witala:98a ; nemoto:98c ; deltuva:05a but for nucleon energies above 60 MeV. In line with the conjectures that were made 15 years ago for the three-nucleon system witala:98a ; nemoto:98c , this underprediction of the data at the minimum of may be a sign for the need to include the 3NF.
In Fig. 3 we show the proton analyzing power for elastic scattering at proton energies ranging from 7.0 to 35.0 MeV. We observe that the sensitivity to the nuclear force model and energy is considerably weaker as compared to the regime below three-cluster threshold deltuva:07a ; deltuva:07b . Most remarkably, in contrast to low energies where the famous - -puzzle exists viviani:01a ; fisher:06 ; deltuva:07b , the peak of around 120 degrees is described fairly well but there is a discrepancy in the minimum region. This is similar to the energy evolution of the -puzzle in the three-nucleon system gloeckle:96a .
In Fig. 4 we show the analyzing power for elastic scattering at proton energies ranging from 7.0 to 35.0 MeV. varies slowly with energy but is slightly more sensitive to the NN potential. Contrary to , calculated is in better agreement with data over the whole energy range, particularly when the INOY04 potential is used.
The experimental data are scarcer for double polarization observables. In Fig. 5 we show the spin correlation coefficient for elastic scattering at 7.03, 8.52, 10.03, and 19.4 MeV, and in Fig. 6 we show for elastic scattering at 8.52 and 19.4 MeV proton energy. Calculated exhibits some sensitivity to the NN potential model and describes the data reasonably well; the agreement with data is the best when the INOY04 interaction is used. The same happens for but for the single data set we know of.
Finally in Fig. 7 we show the proton spin transfer coefficients , , and for elastic scattering at 8.52, 10.77, and 16.23 MeV proton energy. Note that 8.52-MeV predictions are compared to experimental data taken at 8.82 MeV but, given the weak energy dependence of these observables, the comparison is appropriate. The calculated spin transfer coefficients show a rich angular structure and follow the data reasonably well but cannot be fully tested by the available data confined to the angular region below . In contrast to other shown spin observables, the spin transfer coefficient around , i.e., in the region of the differential cross section minimum, varies quite rapidly with the energy. There is little sensitivity to NN interaction model, except for around at MeV. Theoretical results and experimental data for are close to 1 up to , but data sets at different energies seem to be inconsistent as they show different angular dependence. In contrast, theoretical predictions at the three considered energies show nearly the same angular dependence for .
As already mentioned, above MeV the minimum of the elastic differential cross section is underpredicted. In order to establish the importance of the 3NF as a means to cure this discrepancy we study the effect of the -isobar excitation on both the differential cross section and proton analyzing power at 30 MeV proton energy. This has been done before at energies below three-cluster breakup threshold deltuva:08a and we follow here the same procedure. The results are shown in Fig. 8 in a way that one can single out the -isobar effects of 2N nature, the so-called 2N dispersion, and of 3N and 4N nature, the 3NF and 4NF. The competition between 2N dispersion and 3NF, often found in the 3N system, is well seen also here for the differential cross section. As shown in Fig. 8 dispersive effect increases the discrepancy with data (dashed-double dotted curves) while 3NF and 4NF effects reverse that trend for the differential cross section (dotted curves). Nevertheless, when the two effects are put together the net result is an improvement towards the data (solid curves) but not quite enough to bridge the original gap. For only the dispersive effect around the minimum is visible; it moves the predictions away from data.
In summary, we performed fully converged proton- elastic scattering calculations with realistic potentials above the three- and four-cluster breakup threshold. The symmetrized Alt, Grassberger, and Sandhas four-particle equations were solved in the momentum-space framework. We used the complex energy method whose accuracy and the efficiency is greatly improved by the numerical integration technique with special weights. The Coulomb interaction was included rigorously using the method of screening and renormalization.
The differential cross section exhibits rapid energy dependence and, in the minimum region around , also sensitivity to the NN interaction model. The calculations using the INOY04 potential describe the experimental data well up to MeV but underpredict the differential cross section in the minimum at higher energies; other potential models fail even more. In contrast, most of the calculated spin observables show little sensitivity to the interaction model, and also the dependence on the beam energy is weaker than below the three-cluster breakup threshold. The overall agreement with the experimental data for the spin observables is quite good, considerably better than in the low-energy - scattering which is affected by -wave resonances. In particular, the peak of the proton analyzing power that is strongly underpredicted at low energies, is reproduced fairly well above MeV but there is discrepancy in the minimum. The observed sensitivity to the NN interaction model seems to be mostly due to different predictions of the binding energy; the calculations using the INOY04 potential with correct binding provide the best description of the experimental data.
We also studied the effect of three- and four-nucleon forces through the explicit inclusion of the -isobar excitation. We found that -generated many-nucleon forces significantly improve the description of the differential cross section but have almost no effect for the proton analyzing power. However, there are also quite strong dispersive -isobar effects that often reduce or even reverse the effect of 3NF. Therefore the total -isobar effect, although beneficial, is not large enough to bridge the gap between the differential cross section data and calculations. It even increases the discrepancy in the minimum of . It might be possible that using the standard approach of including static 3NF one might be able to explain the data at higher energies, particularly using Effective Field Theory generated interactions epelbaum:00a ; machleidt:11a . Extension of the method to other reactions in the four-nucleon system is in progress.
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