Research

High-momentum Protons and the (e,e’p) Cross Section

The presence of high-momentum nucleons in the nuclear ground state remains an issue of great interest. The possible experimental demonstration of such high-momentum components has been explored in a theoretical calculation of the cross section for the (e,e’p) (pub 1, 2) by solving the Dyson equation in the finite nucleus using a realistic nucleon-nucleon (NN) interaction. A meaningful theoretical determination of the coincidence cross section for the (e,e’p) reaction involves both the nucleon spectral function and a realistic description of the wave function of the ultimately detected proton inside the nucleus. In order to proceed to a more complete theoretical treatment of the coincidence cross section and to establish the possible detectability of removal processes involving protons with high initial momentum, our group, in collaboration with the Pavia group, has used the theoretical spectral functions obtained (pub 1, 2) to determine the coincidence cross section for the (e,e’p) reaction on ¹⁶O. The results of this collaboration have been published (pub 3, 4). At low missing energy the removal of protons from p1/2 and p3/2 quasihole states has been studied. The comparison of the missing momentum dependence of these transitions with experimental data from NIKHEF ref.1 and Mainz ref.2 is very favorable. The slight difference between theory and experiment at high missing momenta can at most account for a very tiny fraction of the single-particle (sp) strength. This strength is predicted to be present at these momenta  (pub 1, 2) based on information of the momentum distribution. A comparison for the p3/2 quasihole state with the results of ref.3 using the Argonne v14 NN interaction ref.4 suggests that the Green’s function result yields a slightly better description of the momentum dependence of the cross section. While the shape of the cross section is nicely described by these new results, the associated spectroscopic factors are overestimated substantially. A substantial fraction of this discrepancy can be ascribed to the influence of long-range correlations (pub 5). A nonnegligible discrepancy remains, however, since a correct treatment of the center-of-mass motion cannot explain the remaining difference ref.5. Our results (pub 3, 4) clearly support the notion that the presence of high-momentum components in the nuclear ground state can only be explored by considering high missing energies in the (e,e’p) reaction. Although other processes may contribute at these energies, our results demonstrate that detectable cross sections can be expected, indicating the presence of high-momentum nucleons in the nucleus. On the other hand, we obtain negligible cross sections for high-momentum nucleons at low missing energy, further supporting the results obtained (pub 1, 2). Experimental work at JLab has recently confirmed the presence of high-momentum components at high missing energy ref.6.

PUBLICATIONS

1. Single-particle spectral function of O-16
   H Muther, WH Dickhoff, Phys. Rev. 49 (nucl-th/9307003), R17-R20
2. Momentum and energy distributions of nucleons in finite nuclei due to short-range correlations
   H Müther, A Polls, WH Dickhoff, Physical Review C 51 (6), 3040
3. High-momentum proton removal from ¹⁶O and the (e,e’p) cross section
   A Polls, M Radici, S Boffi, WH Dickhoff, H Müther, Physical Review C 55 (2), 810
4. Single-particle spectral function for ¹⁶O and the (e,e’p) cross section
   A Polls, M Radici, S Boffi, WH Dickhoff, H Muether, Proceedings of the Workshop on Electron-Nucleus Scattering, 350-362
5. Spectroscopic factors for nucleon knock-out from O 16 at small missing energy
   WJW Geurts, K Allaart, WH Dickhoff, H Müther, Physical Review C 53 (5), 2207

REFERENCES

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5. D. Van Neck et al., Phys. Rev. C57, 2308 (1998).
6. D. Rohe et al. Phys. Rev. Lett.  93, 182501 (2004).

Calculation of the (e,e’2p) Cross Section

The combined effects of the short-range NN interaction and of long-range correlations associated with the finite size of the nucleus have been investigated publ.70 for the two-proton removal amplitudes from ¹⁶O to discrete final states of the final nucleus ¹⁴C. As a first step, the sp spectral function has to be obtained by including the depletion of shell-model orbits by short-range correlations. This is accomplished by treating the energy dependence of the sp energies obtained from the G-matrix effective interaction. With the same interaction, the Tamm-Dancoff coupling of the sp motion to two-particle-one-hole and two-hole-one-particle states was included to describe the effect of long-range correlations publ.69. With this description of the sp propagator, the two-proton removal amplitudes were calculated using the dressed RPA method (DRPA). The effect of short-range correlations on this two-proton removal spectral function is then taken into account by the replacement of the relative uncorrelated ¹S₀ and ³P_J wave functions by correlated ones, using defect wave functions from the solution of the Bethe-Goldstone equation in ¹⁶O. For low-lying discrete final states, the resulting spectral functions display a considerable sensitivity to the chosen realistic NN interaction.

Calculations of the triple-coincidence cross section have been completed using the reaction code of the Pavia group ref.1, employing as input the two-nucleon removal amplitudes described above. This code includes the effects of two-body currents due to the excitation of the D resonance as well as a treatment of FSI by means of spin-dependent optical potentials. The results of this study publ.73 suggest that the (e,e’2p) reaction may be highly selective since the calculations indicate the exclusive population of the lowest 0⁺ and 2⁺ states by the removal of a ¹S₀ proton pair, whereas the 1⁺ state is dominated by ³P_J removal. The calculations reported in publ.73 were performed for kinematical settings relevant to recent experiments at NIKHEF and Mainz. Under these conditions, the knockout of a ³P_J proton pair is largely due to the (two-body) D current. The ¹S₀ pair knockout, on the other hand, is dominated by the one-body current and therefore sensitive to two-body short-range correlations. Accordingly, good prospects for the study of these long-sought correlations exist in the ¹⁶O(e,e’2p) reaction involving the lowest states of ¹⁴C. Our group has collaborated on a publication of the analysis of the NIKHEF data publ.75 in which actual triple-coincidence cross sections are reported for the ¹⁶O(e,e’2p)¹⁴C reaction. The comparison of the data with our calculations is also published in this letter and shows clear signatures of short-range correlations in the ¹⁶O ground state. Further data in different kinematical conditions confirm this conclusion and were published in publ.79.

REFERENCES

1. C. Giusti and F. D. Pacati, Nucl. Phys. A615, 373 (1997).

Scattering of Correlated Nucleons in Nuclear Matter

Initial results on the scattering of dressed nucleons in the nuclear medium have been reported in publ.57, publ.67, and publ.76. Since little or no previous experience with this type of scattering process is available, the scattering of mean-field nucleons in nuclear matter has been studied to facilitate a comparison with studies using Brueckner theory ref.1 or a finite temperature Green’s function formalism ref.2, ref.3. For the case of liquid ³He, the relevant description of the scattering process in terms of phase shifts and their behavior near threshold has been developed for mean-field fermions in ref.4. In this context it is important to emphasize the role of hole-hole propagation, which leads to a considerable enhancement of pairing correlations. As discussed in publ.24 and publ.42, a ladder summation is able to identify a pairing instability around twice the Fermi energy ε_F, for a sufficiently attractive interaction. We have shown that the solution of the scattering equation near twice the Fermi energy indicates this pairing feature by yielding phase shifts that tend to p when the continuum energy approaches 2ε_F on either side. This result mirrors Levinson’s theorem for free particles. Both the ³S₁–³D₁ and ¹S₀ channels yield phase shifts that tend to π when the energy approaches 2ε_F in a wide range of densities. This feature of the effective interaction is responsible for the observed enhancement of the NN cross section at finite temperature as calculated in ref.3.

In order to treat the scattering of dressed particles in the nuclear medium, it is necessary to develop an appropriate scattering theory to deal with the new features that arise from the dressing of the participating nucleons. By casting the conventional asymptotic analysis of scattering in free space in the language of the two-body propagator, it becomes possible to develop modifications of this analysis due to the dressing of the nucleons in the medium. While the scattering energy singles out a unique (on-shell) momentum characterizing the relative wave function of free or mean-field nucleons, this uniqueness is no longer maintained for dressed nucleons. The resulting distribution of momenta in the relative wave function leads to a localization of the scattering process in coordinate space which can be expressed as a healing of the correlated wave function to the noninteracting one. This property has been considered the physical justification of the mean-field-like properties observed in the presence of strong short-range correlations. Our present analysis publ.74, reconciles this healing concept for dressed nucleons with the substantial fragmentation of the nucleon single-particle strength observed in nuclei. The scattering process in nuclear matter involving mean-field nucleons always yields a phase shift and hence no healing. A realistic description of scattering processes in nuclear matter therefore requires the dressing of the nucleons in order bring into play this healing property of the relative wave function. The localization of the scattered wave implies that the particles no longer remember a scattering event beyond some finite distance. This feature suggests that the notion of a cross section in the medium is a tenuous one.

While the cross section of dressed particles cannot be written down formally, it is still possible to generate approximate expressions characterizing the strength of the interaction in the medium in terms of phase shifts and cross sections, which can be fruitfully compared to calculations involving mean-field nucleons publ.76. Results of calculations involving dressed nucleons generate phase shifts and cross sections which deviate substantially from the results for mean-field nucleons alluded to above. A detailed paper containing this information for a realistic NN interaction can be found in publ.78.

REFERENCES

1. G. Giansiracusa, U. Lombardo, and N. Sandalescu, Phys. Rev. C53, R1478 (1996).
2. M. Schmidt, G. Röpke, and H. Schulz, Ann. Phys. 202, 57 (1990).
3. T. Alm, G. Röpke, and M. Schmidt, Phys. Rev. C50, 31 (1994).
4. R. F. Bishop, M. R. Strayer, and J. M. Irvine, Phys. Rev. A10, 2423 (1974).

Λ-Propagation and Weak Decay in Nuclear Matter

A Λ-hyperon, when placed in the nuclear medium, will be modified from its noninteracting form in much the same way as the nucleons themselves are altered by their mutual interactions. The same coupling between sp and two-particle-one-hole (2p1h) states that alters the Λ sp spectrum serves to move particle strength to high energy for a given momentum. The resulting spectral function calculated for a realistic hyperon-nucleon (YN) interaction ref.1 indicates that about 15% of the Λ strength is removed from the lowest states, compared to about 30% for a nucleon at the Fermi momentum in a similar calculation. A complete study of the Λ self-energy and spectral function in nuclear matter can be found in publ.98. The primary decay mode for the Λ particle in a dense nuclear medium is a non-mesonic ΛN-NN decay ref.2. Details of the YN weak interaction aside, this decay mechanism is sensitive to correlations in the initial YN state, to how the nucleon which “catalyzes” the decay is described, and to how the final-state nucleons are dressed in the medium and correlated with each other. Correlations between the YN pair differ among the important partial wave channels, and are not as strong as typically assumed. Our calculations suggest that the nonmesonic decay width increases accordingly with a more realistic treatment of correlations. We have also observed that the ΣN-NN decay channel is made important by the strong coupling between ΛN and ΣN states ref.3. As a result, the contribution to the decay from the strong conversion to the ΣN state prior to subsequent weak decay of the Σ increases the decay width substantially. A paper on this work has been published in publ.100. For even more details see the thesis of Neil Robertson that is available from here.

REFERENCES

1. P. M. M. Maessen, T. A. Rijken, and J. J. de Swart, Phys. Rev. C40, 2226 (1989).
2. B. F. Gibson and E. V. Hungerford III, Phys. Reports 257, 349 (1995).
3. H. Bando, Y. Shono, and H. Takaki, Int. J. of Modern Phys. A3, 1581 (1988).

Dispersive Optical Model (DOM) and Exotic Nuclei

A long-standing collaboration between the experimental faculty in radiochemistry (Bob Charity and Lee Sobotka) and our group aims to investigate the role of correlations in nuclei beyond the mean field as a function of nucleon asymmetry (δ=N-Z/A). These efforts were initiated in Phys. Rev. Lett. 97, 162503 (2006) (see Publ.105), Phys. Rev. C76, 044314 (2007) (see Publ.108), and were continued in Phys. Rev. C83, 064605 (2011) (see Publ.119). The dispersive optical model (DOM) was originally developed by Mahaux and Sartor and provides an excellent framework to connect and analyze elastic nuclear reactions and nuclear structure data that can be represented by the single-particle Green’s function. The DOM utilizes a subtracted dispersion relation linking negative and positive energy domains with emphasis on the physics near the Fermi energy (the subtraction point) but extending to 200 MeV scattering energies above and the complete domain of negative energies.

In the original version with local potentials, it requires functional forms for local absorptive potentials at positive and negative energies, as well as a real potential at the Fermi energy referred to by Mahaux and Sartor as the Hartree-Fock (HF) potential because its functional form can be linked to results of empirical HF calculations. Since the HF potential is intrinsically nonlocal, it has in the past been transformed into a local but energy-dependent potential to facilitate the numerical effort and as such has been implemented in most DOM applications. Unfortunately, such a procedure compromises the normalization of the solutions of the Dyson equation which acts as the Schrödinger equation of a particle or hole in the medium under the influence of the DOM potential. We resolved this problem by replacing this normalization-distorting energy dependence of the local HF potential by a nonlocal and energy-indepent HF potential. The corresponding analysis was published earlier in Phys. Rev. C82, 054306 (2010) (see Publ.118). With this restoration it is possible to describe correlations beyond the mean-field in the DOM framework without resorting to the approximate expressions developed by Mahaux and Sartor. The corresponding quantities are the nucleon spectral function, one-body density matrix, natural orbits, momentum distribution, etc. The DOM can therefore properly describe ground-state properties of nuclei as a function of nucleon asymmetry in addition to standard ingredients like elastic nucleon scattering data and level structure. Predictions of nucleon correlations at larger nucleon asymmetries can then be made after data at smaller asymmetries constrain the potentials that represent the nucleon self-energy. An example is provided by the constraints provided by elastic scattering data on stable Sn nuclei. These exhibit an increasing surface absorption of protons when neutrons are added to the system (see Publ.119). A simple extrapolation for Sn isotopes beyond stability then generates predictions for increasing correlations of minority protons with increasing neutron number. Such predictions can be investigated by performing experiments with exotic beams. The predicted neutron properties for the double closed-shell ¹³²Sn nucleus exhibit similar correlations as those in ²⁰⁸Pb as expressed in terms of spectroscopic factors. Further exploration of the neutron drip line in Sn nuclei clarifies that the proton spectral strength of the last occupied g_9/2 orbit exhibits a sharp decline of the spectroscopic factor when the neutron drip line and the corresponding continuum is near in energy. We demonstrate however that the loss of strength in the main peak is accompanied by a shift of this strength into the nearby continuum which may be accessible experimentally. The resulting paper summarizing these results has been published in Eur. Phys. J. A50:23 (2014) (see Publ.124).

DOM potentials can also be utilized for the description of transfer reactions. The adiabatic wave approximation (ADWA) developed by Johnson and Soper in the seventies can be employed for the description of the (d,p) reaction and has the advantage that it includes the deuteron breakup channel. Furthermore, the deuteron optical potential is described by the sum of the neutron and proton optical potentials at half the incident deuteron energy. The DOM also supplies overlap functions to discrete final states with one neutron added. A collaboration between the reaction group of Filomena Nunes at NSCL/MSU and our DOM effort in St. Louis was therefore a natural development and resulted in a recent publication Phys. Rev. C84, 044611 (2011) (see Publ.121). These results show great promise for the unambiguous extraction of spectroscopic information for transfer reactions in inverse kinematics but require the consideration of nonlocal potentials as discussed below.

The properties of a nucleon that is strongly influenced by the presence of other nucleons have traditionally been studied in separate energy domains. Positive energy nucleons are described by fitted optical potentials mostly in local form. Bound nucleons have been analyzed in static potentials that lead to an independent-particle model modified by the interaction between valence nucleons as in traditional shell-model calculations. The link between nuclear reactions and nuclear structure is provided by considering these potentials as representing  different energy domains of one underlying nucleon self-energy as implemented in the DOM. So far the main stumbling block to describe ground-state properties pertaining to nuclear structure has been the need to utilize the approximate expressions for the properties of nucleons below the Fermi developed by Mahaux and Sartor. These expressions correct for the normalization-distorting energy dependence of the HF potential but do not adequately deal with the spectral properties when the absorptive potentials are substantial. As discussed above, it is possible to restore the proper treatment of nonlocality in the HF contribution, to overcome the problem of the normalization distortion. When the traditional local form of the absorptive potentials are maintained, it is however impossible to generate a good description of the nuclear charge density or even particle number. Recent work in our group established the microscopic content of the self-energy due to long-range (see Phys. Rev. C84, 034616 (2011) Publ.120) and short-range correlations (see Phys. Rev. C84, 044319 (2011) Publ.122) demonstrating that  nonlocal absorptive potentials are theoretically well-founded if not unavoidable.

We have therefore for the first time treated the nonlocality of the absorptive potentials for the nucleus ⁴⁰Ca with the aim to include all available data below the Fermi energy that can be linked to the nucleon single-particle propagator while maintaining a correct description of the elastic-scattering data. The result is a DOM potential that can be interpreted as the nucleon self-energy constrained by all available experimental data up to 200 MeV. Such a self-energy allows for a consistent treatment of nuclear reactions that depend on distorted waves generated by optical potentials as well as overlap functions and their normalization for the addition and removal of nucleons to discrete final states. The re-analysis of such reactions may further improve the consistency of the extracted structure information. Extending this version of the DOM to N different from Z will allow for predictions of properties that require the simultaneous knowledge of both reaction and structure information since at present few weakly-interacting probes are available for exotic nuclei. It is in this sense that we aim at continuing to establish detailed links between the physics of the continuum and structure information below the Fermi energy because these domains are even more strongly coupled for exotic nuclei than for stable systems.

The chosen form of the nonlocal absorption adheres to the traditional treatment, i.e. a Gaussian form is chosen as suggested long ago by Perey and Buck. The description of elastic scattering data for both protons and neutrons is of the same quality as our earlier work with local potentialsdiscussed above. We also find the same quality description of total and reaction cross section for neutrons and protons, respectively. Nevertheless, the introduction of nonlocality has important consequences since it introduces an explicit orbital-angular-momentum dependence that generates very different distorted waves and therefore may generate different results when nuclear reactions are analyzed that require their knowledge. This feature already applies to the (e,e’p) reaction which has been employed to extract spectroscopic factors for the removal of valence protons. In the analysis of such reactions  the NIKHEF group has always utilized nonrelativistic local optical potentials. The Madrid group has shown that a relativistic optical potential generates spectroscopic factors that are 10-15% larger than those obtained by the NIKHEF group. Furthermore their work suggests that this is mainly due to the different treatment of nonlocality. We view this discrepancy as an essential future test of the DOM in which we plan to reanalyze these data with our nonlocal  DOM potentials. Meanwhile our results for the spectroscopic factors are consistent with those obtained with relativistic optical potentials since we obtain 0.78 for the 1s_1/2 and 0.76 for the 0d_3/2 protons, respectively.

The introduction of nonlocality in the absorptive potential has essential benefits for the convergence of the particle number as a function of the orbital angular momentum and brings it in line with ab initio results obtained in Phys. Rev. C84, 044319 (2011) allowing about 1-2% of the particles with orbital angular momentum larger than 5. More importantly, it is now possible to generate an accurate fit to the nuclear charge density. We have found it easier to obtain this result when we replaced the surface contribution of the HF potential by a wine bottle generating Gaussian centered at the origin in accord with similar results obtained with Green’s function Monte Carlo studies of overlap functions by the Argonne group. A new constraint was introduced in the fit of properties below the Fermi energy by considering the spectral function for the removal of high-momentum protons as obtained at Jefferson Lab for Al and Fe nuclei. As these data per proton are essentially identical, we have employed them to constrain the distribution of high-momentum protons. While generating a reasonable description of these data, we obtain a modest 10.6% of the protons occupying momenta above 1.4 fm^-1 in the ⁴⁰Ca ground state. The presence of about 10% of high-momentum nucleons in the ground state is quite consistent with earlier ab initio work of our group and others employing different methods provided the underlying nucleon-nucleon interaction was not too soft.

Employing the energy or Koltun sum rule in the form given by Dieperink and DeForest, then yields a binding energy of 7.91 MeV/A much closer to the experimental 8.55 MeV/A than found in our earlier work  published in Phys. Rev. C82, 054306 (2010). The constrained presence of the high-momentum nucleons is responsible for this change. The 7.91 MeV/A binding obtained here represents the contribution to the ground-state energy from two-body interactions including a kinetic energy of 22.64 MeV/A and was not part of the fit. This empirical approach therefore leaves about 0.64 MeV/A attraction for higher-body interactions about 1 MeV/A  less than the Green’s function Monte Carlo results of the Argonne group for light nuclei. We have published these results in Phys. Rev. Lett. 112, 162503 (2014) and posted the paper on the arXiv (1312:5209) (see Publ.125). In addition we have deposited supplementary material there to provide the detailed parameters that are contained in the present fit (see arXiv:1312.4886).