First application of the dispersive optical model to ($p$,$2p$) reaction analysis within the distorted-wave impulse approximation framework

The independent particle picture provides an excellent first characterization of the structure of a nucleus. An important indicator of this picture is the spectroscopic factor for valence orbitals, which represents the removal probability for each nucleon orbital to a low-lying state of the system with one proton less. The nucleon knockout reaction has been one of the best tools to study this aspect of nuclei. The electron-induced proton knockout reaction, ($e$,$e^{\prime}p$) [1, 2, 3, 4, 5, 6, 7, 8, 9], has been considered the cleanest spectroscopic method for decades. Despite some concerns about the uncertainties associated with proton-induced proton knockout reactions, ($p$,$2p$) [10, 11, 12, 13, 14, 15, 16, 17], a recent review [16] established $(p,2p)$ as an complementary spectroscopic tool to ($e$,$e^{\prime}p$) with about 15% uncertainty for incident energy above 200 MeV.

As discussed in Ref. [16], the effect of nonlocality on the distorted waves and the bound-state wave function is considered to be a major source of the theoretical uncertainties in the description of the ($p$,$2p$) reactions. Usually, the effect is phenomenologically taken into account by including the Perey factor [18]; the Darwin factor is used when an optical potential based on the Dirac phenomenology [19, 20, 21] is adopted. However, the validity of this phenomenological treatment of nonlocality has not been estimated quantitatively. Recently, a fully nonlocal dispersive optical model (DOM) has been developed [22, 23] extending the original work by Mahaux and Sartor [24]. The DOM describes the nucleon scattering potential and the binding potential that gives single-particle levels on the same footing, making use of a subtracted dispersion relation. The single-particle wave function (SPWF) and its spectroscopic factor as well as the distorted waves obtained by the present DOM framework were applied to the nonrelativistic distorted wave impulse approximation (DWIA) analysis of ^40,48Ca($e$,$e^{\prime}p$)^39,47K reactions [8, 9] without any further adjustment. It was concluded that an accurate treatment of the nonlocality as practised in the DOM is necessary to generate spectroscopic factors that automatically describe the ($e$,$e^{\prime}p$) knockout cross sections after the DOM potential has been constrained by all available elastic scattering data (up to 200 MeV) and relevant ground-state information.

The finding in Refs. [8, 9] demonstrates the importance of the implementation of the DOM wave functions to the DWIA analysis of ($p$,$2p$) processes in which three proton distorted waves are present. Therefore, the use of the DOM wave functions is expected to clarify the role of the effective interaction that is involved in the extraction of spectroscopic factors from ($p$,$2p$) data. In this study, we report the first application of such a DWIA analysis with the DOM wave functions to ⁴⁰Ca($p$,$2p$)³⁹K data at 200 MeV [25].

The organization of this paper is as follows. In Sec. II, we briefly introduce the present DOM + DWIA framework. In Sec. III, a theoretical analysis of ⁴⁰Ca($p$,$2p$)³⁹K data and the consistency between DOM proton spectroscopic factors that automatically describe $(e,e^{\prime}p)$ cross sections and those obtained from ($p$,$2p$) analyses are discussed. We also compare the present framework with a conventional DWIA analysis employing a phenomenological SPWF and optical potentials. Finally, a summary is given in Sec. IV.

In this section, we discuss extracted spectroscopic factors from the ⁴⁰Ca$(p,2p)^{39}$K reaction in comparison with the DOM result that is consistent with the ⁴⁰Ca$(e,e^{\prime}p)^{39}$K data. We also address uncertainties arising from the choice of the optical potential and the effective $p$-$p$ interaction. The theoretical knockout cross section is calculated using the DWIA framework with the DOM SPWF and distorted waves. Results using phenomenological inputs are also discussed for comparison.

The spectroscopic factor of $0d_{3/2}$ is extracted from the ratio of the theoretical cross section and the experimental data of the ⁴⁰Ca($p$,$2p$)³⁹K reaction at 197 MeV. The reaction kinematics is in a coplanar kinematics and the opening angles of the emitted protons are fixed at the same angle: $\phi_{1}^{\mathrm{L}}=0^{\circ}$, $\phi_{2}^{\mathrm{L}}=180^{\circ}$, and $\theta_{1}^{\mathrm{L}}=\theta_{2}^{\mathrm{L}}=42.0^{\circ}$ in the Madison convention [31]. The kinematics of the three particles is then uniquely determined once $T_{1}^{\mathrm{L}}$ is given.

The DOM-DWIA result is compared with those of the phenomenological SPWF and the optical potential in Fig 1. For this comparison, the DOM-DWIA cross section is adjusted to the data and the DOM spectroscopic factor was not utilized. The phenomenological SPWF suggested by Kramer et al. [7], the Koning-Delaroche optical potential parameter set (KD) [32], and the Dirac phenomenology (DP) [19, 20, 21] are also considered. Calculated TDXs and the experimental data are shown in Fig. 1. Spectroscopic factors are therefore extracted from the ratio of the present calculations and the experimental data taken by the E258 experiment at RCNP [25] by minimizing

$\sigma_{i}^{\textrm{DWIA}}$ and $\sigma_{i}$ are theoretical and experimental cross sections at data points, respectively, and $\delta_{i}$ is associated error of the experimental data. Obtained spectroscopic factors are summarized in Table 1. Following Ref. [16], only the data points around the peak, larger than $25~{}\mu$b/(MeV sr²), are fitted to reduce the uncertainty.

The spectroscopic factors obtained from the phenomenological ($p$,$2p$) analysis are consistent with the phenomenological ($e$,$e^{\prime}p$) analysis which gave $0.65\pm 0.06$ [8]. On the other hand, the spectroscopic factor obtained using the DOM wave functions to reproduce the ($p$,$2p$) cross section is in disagreement with the DOM-calculated [using Eq. (6)] value of $0.71\pm 0.04$. One of the reasons for this inconsistency may lie in a difference in the peripherality of the reaction probes, but it is not yet well understood.

The importance of having to deal with three distorted proton waves in the ($p$,$2p$) reaction as compared to just one in the ($e$,$e^{\prime}p$) case remains an issue. There is an uncertainty associated with the DOM distorted waves due to the experimental data points used in the DOM fit. Considering the strong correlation between the proton reaction cross section and the ⁴⁸Ca$(e,e^{\prime}p)^{47}$K cross section demonstrated in Ref. [9], we look to uncertainties in the experimental proton reaction cross section data points in energy regions corresponding to those of the distorted proton waves to get a rough estimate of the uncertainty associated with the DOM distorted waves. The proton reaction cross-section data points from Refs. [33, 34] suggest an uncertainty in the corresponding DOM distorted waves around 3%. Furthermore, due to the kinematics of the reaction, one of the proton energies is as low as 36 MeV. In the DOM analysis of ⁴⁰Ca$(e,e^{\prime}p)^{39}$K, the description of the experimental cross section for outgoing proton energies of 70 MeV, the lowest of the considered proton energies, is unsatisfactory [8]. This indicates that the impulse approximation may not be applicable at proton energies of 70 MeV and below. Since one of the outgoing proton energies in this ⁴⁰Ca$(p,2p)^{39}$K reaction is even less than 70 MeV, it is reasonable to expect some discrepancy in the ⁴⁰Ca$(p,2p)^{39}$K TDX. This discrepancy may be reduced when higher proton beam energies are considered but this implies that the DOM analysis has to be extended to higher energies.

As a further step to clarify this issue, we also investigated the uncertainty arising from a different choice of the $p$-$p$ effective interactions. Three different types of the $p$-$p$ effective interactions, the Franey-Love effective interaction (FL) [35], the Melbourne $g$-matrix interaction at mean density (Mel) [36], and that at zero density (Mel free) were utilized. The mean density of the reaction is defined in Sec. 6.1. of Ref. [16]. The SPWF and the distorted waves from the DOM are adopted in this analysis. The choice of the $p$-$p$ effective interaction only changes the magnitude of the TDX, keeping their shapes unchanged as shown in Fig. 2.

Extracted spectroscopic factors are summarized in Table. 1. An uncertainty of the choice of the $p$-$p$ effective interaction results in $\mathcal{Z}_{0d_{3/2}}=0.489$–$0.560$ and does not explain the inconsistency between the DOM ($p$,$2p$) result and the successful DOM description of the ($e$,$e^{\prime}p$) cross section [8].

Validity of the finite-range treatment of $t_{pp}$ and factorization approximation as well as the nonlocality correction and the relativistic correction in nonrelativistic DWIA have been discussed for decades [37, 38, 39, 40]. They make certain change in the cross section, but it is still not quantitatively conclusive. In Ref. [41], it is reported that TDX with AMA only differs 6% of that with LSCA in ¹²⁰Sn($p$,$p\alpha$)¹¹⁶Cd reaction case. Considering this situation, it is very important that we obtained good agreement in spectroscopic factors deduced from phenomenological ($e$,$e^{\prime}p$) and ($p$,$2p$) analyses before discussing DOM + DWIA analysis. It should be noted that the finite-range treatment of $t_{pp}$ is more crucial in the relativistic DWIA framework, as pointed out in Ref. [42].

Ideally, the spectroscopic factor should not depend on the reactions to be adopted. As shown in Ref. [16], the spectroscopic factors for the SP levels near the Fermi energies of stable nuclei extracted from ($p,2p$) reactions above 200 MeV are consistent with those from ($e,e^{\prime}p$) with uncertainties ranging from 10% to 15%. It should be noted that the nonlocality correction to the SPWF and distorted waves are considered to be a primary source of the uncertainties of the spectroscopic factor [16]. As mentioned above, $\mathcal{Z}_{0d_{3/2}}$ obtained with the DOM-DWIA analysis of the ⁴⁰Ca($p,2p$) data at 200 MeV, in which the nonlocality is treated in a sophisticated manner, differs by at least 21% from the value used to reproduce $(e,e^{\prime}p)$ data using the same SPWF and proton distorted wave calculated with the DOM. This implies the necessity of improving the treatment of $p$-$p$ scattering inside a nucleus beyond the standard $t$- or $g$-matrix approach. One immediate concern is that present treatments of the effective interaction do not allow for energy transfer in the elementary process. Since in the ($p,2p$) reaction a substantial excitation energy is involved, it implies that the mediators of the strong interaction, in particular the pion, must be allowed to propagate in the system and are certainly not static [43].

The ability of the DOM to provide both bound and scattering nucleon wave functions is opening a door to a new research opportunity for the nucleon-nucleon scattering process in many-body systems. This is of particular importance as the ($p,2p$) reaction can be employed in inverse kinematics [44, 45]. There is therefore a clear need to pursue an improved description of the effective interaction in the medium which will also depend on the nucleon asymmetry that is studied in exotic systems.

The DOM SPWF and distorted waves have been applied to the DWIA analysis of ($p$,$2p$) reaction for the first time. The proton $0d_{3/2}$ spectroscopic factor $\mathcal{Z}_{0d_{3/2}}=0.56\pm 0.005$ of ⁴⁰Ca was extracted from ⁴⁰Ca($p$,$2p$)³⁹K analysis with the present framework. The obtained spectroscopic factor is inconsistent with the DOM supplied one that reproduces ⁴⁰Ca($e$,$e^{\prime}p$)³⁹K data, $0.71\pm 0.04$. We tested several types of input for the $p$-$p$ effective interaction: the Franey-Love interaction, the Melbourne $g$-matrix at mean density, and that at zero density. In addition to using the DOM wave functions, Kramer’s SPWF and distorted waves obtained by the Koning-Delaroche and Dirac phenomenology optical potentials were considered. We conclude from the present analysis that the gap between the above-mentioned ($p$,$2p$) and ($e$,$e^{\prime}p$) spectroscopic factors cannot be explained by the uncertainties arising from these inputs. This result implies there is room for improving the treatment of the $p$-$p$ binary scattering beyond the traditional $t$- and $g$-matrix approach combined with the impulse approximation.

The authors thank T. Noro for providing them with updated experimental data before publication. This work was supported by JSPS KAKENHI Grant No. JP20K14475. This work was also supported by the U.S. National Science Foundation under Grant No. PHY-1912643. TRIUMF receives federal funding via a contribution agreement with the National Research Council of Canada.

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