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

K. Yoshida<EMAIL_ADDRESS>Advanced Science Research Center, Japan
Atomic Energy Agency, Tokai, Ibaraki 319-1195, Japan  M. C. Atkinson  Theory
Group, TRIUMF, vancouver, british columbia V6T 2A3, Canada  K. Ogata  Research
Center for Nuclear Physics (RCNP), Osaka University, Ibaraki 567-0047, Japan
Department of Physics, Osaka City University, Osaka 558-8585, Japan  Nambu
Yoichiro Institute of Theoretical and Experimental Physics (NITEP), Osaka City
University, Osaka 558-8585, Japan  W. H. Dickhoff  Department of Physics,
Washington University, St. Louis, Missouri 63130, USA

###### Abstract

Background

Both ($e$,$e^{\prime}p$) and ($p$,$2p$) reactions have been performed to study
the proton single-particle character of nuclear states with its related
spectroscopic factor. Recently, the dispersive optical model (DOM) was applied
to the ($e$,$e^{\prime}p$) analysis revealing that the traditional treatment
of the single-particle overlap function, distorted waves, and nonlocality must
be further improved to achieve quantitative nuclear spectroscopy.

Purpose

We apply the DOM wave functions to the traditional ($p$,$2p$) analysis and
investigate the consistency of the DOM spectroscopic factor that describes the
($e$,$e^{\prime}p$) cross section with the result of the ($p$,$2p$) analysis.
Additionally, we make a comparison with a phenomenological single-particle
wave function and optical potential. Uncertainty arising from a choice of
$p$-$p$ interaction is also investigated.

Method

We implement the DOM wave functions to the nonrelativistic distorted wave
impulse approximation (DWIA) framework for ($p$,$2p$) reactions.

Results

DOM + DWIA analysis on 40Ca($p$,$2p$)39K data generates a proton $0d_{3/2}$
spectroscopic factor of 0.560, which is meaningfully smaller than the DOM
value of 0.71 shown to be consistent with the ($e$,$e^{\prime}p$) analysis.
Uncertainties arising from choices of single-particle wave function, optical
potential, and $p$-$p$ interaction do not explain this inconsistency.

Conclusions

The inconsistency in the spectroscopic factor suggests there is urgent need
for improving the description of $p$-$p$ scattering in a nucleus and the
resulting in-medium interaction with corresponding implications for the
analysis of this reaction in inverse kinematics.

††preprint: NITEP 119

##  I Introduction

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.

##  II Theoretical framework

###  II.1 Distorted wave impulse approximation

In the present study, the factorized form of the nonrelativistic DWIA with the
spin degrees of freedom is employed. The transition matrix $T$ within the
distorted wave impulse approximation framework is given by

$\displaystyle T_{\mu_{1}\mu_{2}\mu_{0}\mu_{j}}$
$\displaystyle=\sum_{\mu^{\prime}_{1}\mu^{\prime}_{2}\mu^{\prime}_{0}\mu_{p}}\tilde{t}_{\mu^{\prime}_{1}\mu^{\prime}_{2}\mu^{\prime}_{0}\mu_{p}}\int
d\bm{R}\,\chi_{1,\mu^{\prime}_{1}\mu_{1}}^{(-)*}(\bm{R})\chi_{2,\mu^{\prime}_{2}\mu_{2}}^{(-)*}(\bm{R})\chi_{0,\mu^{\prime}_{0}\mu_{0}}^{(+)}(\bm{R})e^{-i\alpha_{R}\bm{K}_{0}\cdot\bm{R}}\sum_{m}\left(lms_{p}\mu_{p}|j\mu_{j}\right)\psi^{n}_{ljm}(\bm{R}).$
(1)

The incident and two emitted protons are labeled as particle $0$–$2$, while
the bound proton in the initial state is labeled as $p$.
$\chi_{i,\mu^{\prime}_{i}\mu_{i}}$ is a distorted wave of particle $i=0,1,2$
having the asymptotic (local) third component $\mu_{i}$ ($\mu^{\prime}_{i}$)
of its spin $s_{i}=1/2$. The outgoing and incoming boundary conditions of the
distorted waves are denoted by superscripts $(+)$ and $(-)$, respectively.
$\bm{K}_{0}$ is the momentum (wavenumber) of the incident proton and
$\alpha_{R}$ is the mass ratio of the struck particle and the target, $1/40$
in this study. $n$ is the radial quantum number, and $l,j,m$ are the single-
particle orbital angular momentum, total angular momentum, and third component
of $l$, respectively. $\psi^{n}_{ljm}$ is the SPWF normalized to unity.
$\tilde{t}_{\mu^{\prime}_{1}\mu^{\prime}_{2}\mu^{\prime}_{0}\mu_{p}}$ is the
matrix element of the $p$-$p$ effective interaction $t_{pp}$,

$\displaystyle\tilde{t}_{\mu^{\prime}_{1}\mu^{\prime}_{2}\mu^{\prime}_{0}\mu_{p}}$
$\displaystyle=\Braket{\bm{\kappa}^{\prime},\mu^{\prime}_{1}\mu^{\prime}_{2}}{t_{pp}}{\bm{\kappa},\mu^{\prime}_{0}\mu_{p}},$
(2)

where $\bm{\kappa}$ and $\bm{\kappa}^{\prime}$ are relative momenta of two
protons in the initial and the final states, respectively. The factorization
procedure of $t_{pp}$ is explained using the local semiclassical approximation
(LSCA) and the asymptotic momentum approximation (AMA) in the appendix. It
should be noted that the factorized DWIA is often regarded as a result of the
zero-range approximation but $t_{pp}$ is finite-range interaction in the
present study.

The triple differential cross section (TDX) with respect to the emitted proton
energy $T_{1}^{\mathrm{L}}$ and emission angles $\Omega_{1}^{\mathrm{L}}$ and
$\Omega_{2}^{\mathrm{L}}$ is given as

$\displaystyle\frac{d^{3}\sigma^{\mathrm{L}}}{dT_{1}^{\mathrm{L}}d\Omega_{1}^{\mathrm{L}}d\Omega_{2}^{\mathrm{L}}}$
$\displaystyle=\mathcal{Z}^{n}_{lj}\mathcal{J}_{\mathrm{LG}}F_{\mathrm{kin}}\frac{(2\pi)^{4}}{\hbar
v_{\alpha}}\frac{1}{(2s_{0}+1)(2j+1)}$
$\displaystyle\times\sum_{\mu_{1}\mu_{2}\mu_{0}\mu_{j}}\left|T_{\mu_{1}\mu_{2}\mu_{0}\mu_{j}}\right|^{2},$
(3)

with $\mathcal{Z}^{n}_{lj}$, $\mathcal{J}_{\mathrm{LG}}$, $F_{\mathrm{kin}}$,
$v_{\alpha}$ being the spectroscopic factor, the Jacobian from the center-of-
mass frame to the Laboratory frame, kinetic factor, and the relative velocity
of the incident proton and the target, respectively. Quantities with
superscript $\mathrm{L}$ are evaluated in the laboratory frame while the
others are in the center-of-mass frame. See Sec. 3.1 of Ref. [16] for details.

###  II.2 Dispersive optical model

The distorted waves and SPWF in Eq. (1) as well as the spectroscopic factor in
Eq. (3) are calculated using the DOM. The nonlocal DOM uses both bound and
scattering data to constrain the nucleon self-energy $\Sigma^{*}$ for a given
nucleus. This self-energy is a complex and nonlocal potential that unites the
nuclear structure and reaction domains [24, 22]. The DOM was originally
developed by Mahaux and Sartor [24], employing local real and imaginary
potentials connected through dispersion relations. However, only with the
introduction of nonlocality can realistic self-energies be obtained [22, 23,
26, 27, 28]. The Dyson equation then determines the single-particle
propagator, or Green’s function, from which bound-state and scattering
observables can be deduced.

Using the DOM self-energy, a Schrödinger-like equation can be generated to
calculate the SPWF in Eq. (1),

$\displaystyle\sum_{\gamma}\bra{\alpha}T_{\ell}+\Sigma^{*}_{\ell
j}(E)\ket{\gamma}\psi_{\ell j}^{n}(\gamma)=\varepsilon_{n}^{-}\psi_{\ell
j}^{n}(\alpha),$ (4)

where $\alpha$ and $\gamma$ are arbitrary basis variables, $\ell j$ correspond
to the orbital and total angular momentum of the orbital, $n$ is the principle
quantum number of the orbital, and $\bra{\alpha}T_{\ell}\ket{\gamma}$ is the
kinetic-energy matrix element, including the centrifugal term [8]. These wave
functions correspond to overlap functions

$\psi^{n}_{\ell j}(\alpha)=\bra{\Psi_{n}^{A-1}}a_{\alpha\ell
j}\ket{\Psi_{0}^{A}},\qquad\varepsilon_{n}^{-}=E_{0}^{A}-E_{n}^{A-1}.$ (5)

Such discrete solutions to Eq. (4) exist where there is no imaginary part of
the self-energy, so near the Fermi energy. The normalization for these wave
functions corresponds to the same spectroscopic factor of Eq. (3), which is
given by [29]

$\mathcal{Z}^{n}_{\ell j}=\bigg{(}1-\frac{\partial\Sigma_{\ell
j}^{*}(\alpha_{qh},\alpha_{qh};E)}{\partial
E}\bigg{|}_{\varepsilon_{n}^{-}}\bigg{)}^{-1},$ (6)

where $\alpha_{qh}$ represents the quasihole state that solves Eq. (4). Note
that because of the presence of imaginary parts of the self-energy at other
energies, there is also strength located there, and thus the spectroscopic
factor will be less than 1 and also less than the occupation probability. The
DOM self-energies of 40Ca and 48Ca were used in Refs. [8, 30, 9] to reproduce
40Ca$(e,e^{\prime}p)^{39}$K and 48Ca$(e,e^{\prime}p)^{47}$K momentum
distributions, respectively. The corresponding 40Ca spectroscopic factor of
$\mathcal{Z}_{0d_{3/2}}^{\mathrm{DOM}}=0.71\pm 0.04$, which is consistent with
40Ca$(e,e^{\prime}p)^{39}$K data, will now be used alongside the DOM SPWF and
distorted waves to analyze the 40Ca$(p,2p)^{39}$K knockout reaction.

##  III Results and discussion

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

$\displaystyle\chi^{2}(\mathcal{Z}_{0d_{3/2}})$
$\displaystyle=\sum_{i}\frac{\left(\mathcal{Z}_{0d_{3/2}}\sigma_{i}^{\textrm{DWIA}}-\sigma_{i}\right)^{2}}{\delta_{i}^{2}}.$
(7)

$\sigma_{i}^{\textrm{DWIA}}$ 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.

Figure 1:  TDX with different optical potentials. The solid, dashed, and dotted lines are TDXs with the Koning-Delaroche optical potential (KD) and Dirac phenomenology (DP), respectively. The result with DOM ingredients is also shown as the dot-dashed line. All results reflect cross sections that are normalized with the spectroscopic factors shown in Table 1. The experimental data taken by the E258 experiment at RCNP [25] are also shown.  Table 1:  Setup and resulting spectroscopic factors.  SPWF | Optical pot. |  $p$-$p$ int. | $\mathcal{Z}_{0d_{3/2}}$
---|---|---|---
Kramer | KD | FL | $0.623\pm 0.006$
Kramer | Dirac | FL | $0.672\pm 0.006$
DOM | DOM | FL | $0.560\pm 0.005$
DOM | DOM | Mel | $0.489\pm 0.005$
DOM | DOM | Mel (free) | $0.515\pm 0.005$

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.

Figure 2:  TDX with different $p$-$p$ effective interactions. The solid,
dashed, and dotted lines are TDXs with the Franey-Love effective interaction
[35] (FL), Melbourne $g$-matrix interaction at mean density [36] (Mel), and
that at zero density (Mel free), respectively. The experimental data are the
same as in Fig. 1.

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.

##  IV Summary

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.

## ACKNOWLEDGMENTS

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.

*

##  Appendix A Factorization of $T$ matrix

In this appendix, the factorization procedure of the $T$ matrix is explained
by means of the LSCA, following previous research [46, 47, 48, 49]. Equation
(1) is originally an integral over two coordinates $\bm{R}$ and $\bm{s}$:

$\displaystyle T_{\mu_{1}\mu_{2}\mu_{0}\mu_{j}}$
$\displaystyle=\sum_{\mu^{\prime}_{1}\mu^{\prime}_{2}\mu^{\prime}_{0}\mu_{p}}\int
d\bm{R}\,d\bm{s}\,\chi_{1,\mu^{\prime}_{1}\mu_{1}}^{(-)*}(\bm{R}_{1})\chi_{2,\mu^{\prime}_{2}\mu_{2}}^{(-)*}(\bm{R}_{2})t_{pp}(\bm{s})[1-P]\chi_{0,\mu^{\prime}_{0}\mu_{0}}^{(+)}(\bm{R}_{0})\sum_{m}\left(lms_{p}\mu_{p}|j\mu_{j}\right)\psi^{n}_{ljm}(\bm{R}_{2}).$
(8)

$\bm{R}$, $\bm{s}$, $\bm{R}_{0}$, $\bm{R}_{1}$, and $\bm{R}_{2}$ are defined
as shown in Fig. 3. $t_{pp}(\bm{s})$ is a $p$-$p$ effective interaction and
$P$ is the exchange operator for the colliding two protons; thus, as usual,
both the direct and exchange terms are considered. $\bm{R}$, $\bm{s}$,
$\bm{R}_{0}$, $\bm{R}_{1}$, and $\bm{R}_{2}$ are rewritten in terms of
$\bm{R}$ and $\bm{s}$ as

$\displaystyle\bm{R}_{0}$
$\displaystyle=\left(1-\alpha_{R}\right)\bm{R}+\frac{A+1}{2A}\bm{s},$ (9)
$\displaystyle\bm{R}_{1}$ $\displaystyle=\bm{R}+\frac{1}{2}\bm{s},$ (10)
$\displaystyle\bm{R}_{2}$ $\displaystyle=\bm{R}-\frac{1}{2}\bm{s}.$ (11)

The nucleon and the target mass number are 1 and $A$, respectively.

Figure 3:  Coordinate of the ($p$,$2p$) three-body system.

The factorization of the $p$-$p$ $t$-matrix is done by considering three
approximations as follows. First, for a short distance $\Delta\bm{R}$, the
local semiclassical approximation (LSCA) [46, 47, 48, 49] can be applied to
the distorted waves:

$\displaystyle\chi(\bm{R}+\Delta\bm{R})$
$\displaystyle\approx\chi(\bm{R})e^{i\bm{K}(\bm{R})\cdot\Delta\bm{R}}.$ (12)

$\bm{K}(\bm{R})$ is the local momentum of the scattering particle at $\bm{R}$.
Further approximation is made by replacing the local momentum $\bm{K}(\bm{R})$
with its asymptotic one $\bm{K}$. This is called the asymptotic momentum
approximation (AMA) In Ref. [49], in which the AMA was referred to as LSCA-A,
the AMA was shown to work with almost the same accuracy as of LSCA for a short
propagation of the distorted waves of about 1.5 fm. It should be noted that
for the factorization of the $p$-$p$ transition matrix, the propagation range
of about the half the range of $t_{pp}(\bm{s})$ needs to be approximated.
According to the result of Ref. [49], both the LSCA and AMA are expected to be
sufficiently accurate for this purpose. Furthermore, it is reported in Ref.
[41] that the knockout cross section with AMA differs only $\sim 6\%$ from
that with LSCA in 120Sn($p$,$p\alpha$)116Cd reaction at 392 MeV.

Thus, the AMA (and the LSCA as well) can safely be applied to the distorted
waves in Eq. (8), which reads

$\displaystyle\chi_{0,\mu^{\prime}_{0}\mu_{0}}^{(+)}(\bm{R}_{0})\approx$
$\displaystyle\chi_{0,\mu^{\prime}_{0}\mu_{0}}^{(+)}(\bm{R})e^{-\alpha_{R}\bm{K}_{0}\cdot\bm{R}}$
$\displaystyle\times e^{i(A+1)\bm{K}_{0}\cdot\bm{s}/2A},$ (13)
$\displaystyle\chi_{1,\mu^{\prime}_{1}\mu_{1}}^{(-)*}(\bm{R}_{1})\approx$
$\displaystyle\chi_{1,\mu^{\prime}_{1}\mu_{1}}^{(-)*}(\bm{R})e^{i\bm{K}_{1}\cdot\bm{s}/2},$
(14) $\displaystyle\chi_{2,\mu^{\prime}_{2}\mu_{2}}^{(-)*}(\bm{R}_{2})\approx$
$\displaystyle\chi_{2,\mu^{\prime}_{2}\mu_{2}}^{(-)*}(\bm{R})e^{-i\bm{K}_{2}\cdot\bm{s}/2},$
(15)

with $\bm{K}_{i}$ ($i=0,1,2$) being the asymptotic momenta of particle $i$.
Note that in Eq. (A), we have also used the fact that $\alpha_{R}$ is small.
As for the single-particle wave function, the Fourier transformation is
applied to factorize out the $\bm{s}$ dependence

$\displaystyle\psi^{n}_{ljm}(\bm{R}_{2})$
$\displaystyle=\frac{1}{(2\pi)^{3}}\int
d\bm{K}_{N}\,\tilde{\psi}^{n}_{ljm}(\bm{K}_{N})e^{i\bm{K}_{N}\cdot\bm{R}_{2}}$
$\displaystyle=\frac{1}{(2\pi)^{3}}\int
d\bm{K}_{N}\,\tilde{\psi}^{n}_{ljm}(\bm{K}_{N})e^{i\bm{K}_{N}\cdot(\bm{R}-\bm{s}/2)}.$
(16)

The second approximation is to assume the quasi-free $p$-$p$ collision in the
($p$,$2p$) reaction. Although the distorted waves are not eigenstates of
momenta, the initial (final) $p$-$p$ relative momentum $\bm{\kappa}$
($\bm{\kappa}^{\prime}$) relevant to the ($p$,$2p$) reaction are approximately
given by the asymptotic momenta $\bm{K}_{i}$ as

$\displaystyle\bm{\kappa}$
$\displaystyle\approx\frac{A+1}{2A}\bm{K}_{0}-\frac{1}{2}\bm{K}_{N}$ (17)
$\displaystyle\bm{\kappa}^{\prime}$
$\displaystyle\approx\frac{1}{2}\bm{K}_{1}-\frac{1}{2}\bm{K}_{2}.$ (18)

It should be noted that this is a natural outcome of the use of the AMA
mentioned above. Similarly, as the last approximation, the momentum
conservation of the $p$-$p$ collision is assumed:

$\displaystyle\frac{A+1}{A}\bm{K}_{0}+\bm{K}_{N}\approx\bm{K}_{1}+\bm{K}_{2}.$
(19)

Again, the appearance of the asymptotic momenta in Eq. (19) is due to the use
of the AMA.

Under these approximations, once the ($p$,$2p$) kinematics is fixed,
$\bm{K}_{N}$ and $\bm{\kappa}$ are uniquely determined by $\bm{K}_{0}$,
$\bm{K}_{1}$ and $\bm{K}_{2}$. Inserting Eqs. (A)–(16) into Eq. (8) and
applying the inverse Fourier transform of
$\tilde{\psi}_{nlm}^{n}(\bm{K}_{N})$, one obtains the factorized form of the
$T$ matrix

$\displaystyle T_{\mu_{1}\mu_{2}\mu_{0}\mu_{j}}=$
$\displaystyle\sum_{\mu^{\prime}_{1}\mu^{\prime}_{2}\mu^{\prime}_{0}\mu_{p}}\int
d\bm{s}\,e^{-i\bm{\kappa}^{\prime}\cdot\bm{s}}t_{pp}(\bm{s})[1-P]e^{i\bm{\kappa}\cdot\bm{s}}\int
d\bm{R}\,\chi_{1,\mu^{\prime}_{1}\mu_{1}}^{(-)*}(\bm{R})\chi_{2,\mu^{\prime}_{2}\mu_{2}}^{(-)*}(\bm{R})\chi_{0,\mu^{\prime}_{0}\mu_{0}}^{(+)}(\bm{R})e^{-i\alpha_{\mathrm{R}}\bm{K}_{0}\cdot\bm{R}}\sum_{m}\left(lms_{p}\mu_{p}|j\mu_{j}\right)\psi^{n}_{ljm}(\bm{R}).$
(20)

This is equivalent to Eqs. (1) and (2), if one takes account of the spin
degrees of freedom in the $p$-$p$ $t$ matrix.

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