# Sub-barrier d+208Pb scattering and sensitivity to nucleon-nucleon
interactions

Peng Yin  CAS Key Laboratory of High Precision Nuclear Spectroscopy, Institute
of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China
Department of Physics and Astronomy, Iowa State University, Ames, IA 50011,
USA  Weijie Du111Corresponding author<EMAIL_ADDRESS>Department of Physics
and Astronomy, Iowa State University, Ames, IA 50011, USA  Wei Zuo  CAS Key
Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern
Physics, Chinese Academy of Sciences, Lanzhou 730000, China  School of Nuclear
Science and Technology, University of Chinese Academy of Sciences, Beijing
100049, China  Xingbo Zhao  CAS Key Laboratory of High Precision Nuclear
Spectroscopy, Institute of Modern Physics, Chinese Academy of Sciences,
Lanzhou 730000, China  School of Nuclear Science and Technology, University of
Chinese Academy of Sciences, Beijing 100049, China  James P. Vary  Department
of Physics and Astronomy, Iowa State University, Ames, IA 50011, USA

###### Abstract

We employ the non-perturbative time-dependent basis function (tBF) approach to
investigate in detail the sub-Coulomb barrier scattering of the deuteron
projectile on the 208Pb target. Specifically, we treat the target as a
quiescent source of a strong external Coulomb field that induces higher-order
effects in electric-dipole excitations of the deuteron projectile including
the population of states not accessible through direct electric-dipole
transitions from the ground state of the deuteron. We calculate the electric-
dipole polarizability of the deuteron and elastic scattering observables for
comparison with experimental data. With no adjustable parameters, the tBF
approach provides good agreement with experimentally available differential
cross-section ratios. The dependence of these measured quantities on nucleon-
nucleon interactions is investigated. The effects of the polarization
potential on the scattering observables are also presented in this work. We
also investigate the detailed scattering dynamics and identify characteristics
of coherent and incoherent processes.

###### pacs:

13.75.Cs, 21.10.Ky, 21.60.De, 24.10.-i, 25.70.De.

##  I Introduction

As the existing and upcoming radioactive ion beam facilities provide unique
access to the exotic phenomena in nuclear physics, reliable reaction theory is
crucial for establishing a substantial connection between the measured
reaction observables and the underlying nuclear structure. While
phenomenological reaction theories and standard approximation methods are
playing a significant role in modeling nuclear reaction data, reaction
approaches, that start from the ab initio calculations of nuclear structure,
have been developed and turn out to be numerically tractable and successful in
describing reactions involving light nuclei. For few-body systems with nucleon
number $A\leq 4$, the Faddeev Witala:2000am , Faddeev-Yakubovsky
Lazauskas:2004hq ; Lazauskas:2009gv , hyperspherical harmonics (HH)
Marcucci:2009xf ; Marcucci:2019hml , the Alt, Grassberger, and Sandhas (AGS)
Deltuva:2006sz ; Deltuva:2007xv , resonating group method (RGM) Hofmann:2005iy
, etc., are applicable and successful. For systems with more than four
nucleons, additional approaches, such as the Lorentz integral transform (LIT)
methods Gazit:2005nh ; Quaglioni:2007eg ; Bacca:2008tb ; Leidemann:2012hr ,
the Green’s function Monte Carlo (GFMC) Nollett:2006su , the fermionic
molecular dynamics (FMD) Chernykh:2007zz , no-core shell model (NCSM) with RGM
Quaglioni:2008sm ; Quaglioni:2009mn ; Navratil:2009ut ; Navratil:2011zs  and
the single-state harmonic oscillator representation of scattering equations
(SS-HORSE) method Shirokov:2018nlj  have been proposed and applied. However,
these approaches may be challenged to retain the full, non-perturbative
quantal coherence of all the potentially relevant intermediate and final
states in nucleus-nucleus reactions, especially for reactions involving
unstable rare isotopes.

In Refs. Du:2018a ; Du:2018b , we proposed the theoretical framework of the
non-perturbative time-dependent basis function (tBF) approach. The tBF method
establishes a firm connection between ab initio nuclear structure approaches,
e.g., the NCSM Barrett:2013nh , and nuclear reaction theory. Its counterpart
in relativistic quantum field theory, the time-dependent basis light front
quantization approach, has been applied to investigate the non-linear Compton
scattering Zhao:2013cma ; Hu:2019hjx , the interaction of an electron with
intense electromagnetic field Chen:2017uuq  and the scattering of a quark on a
heavy nucleus Li:2020uhl ; Li:2021zaw  at high energies. The key point of the
tBF method is the construction of the basis representation for the scattering
problem, employing the eigenstates of the system being investigated (a rare
isotope beam in future applications) that are obtained with ab initio nuclear
structure approaches. Based on this basis representation, we solve the
equation of motion (EOM) of the scattering process numerically as an initial
value problem in a non-perturbative manner, where the quantal coherence is
fully retained during the reaction process. It is noteworthy that the tBF
approach appears to be well-suited as a forefront application of quantum
computing in nuclear scattering dynamics Du:2020glq .

In Ref. Yin:2019kqv , we improved the tBF method and employed it to
investigate the deuteron scattering on 208Pb well below the Coulomb barrier of
approximately $11$ MeV. By considering all the possible electric-dipole (E1)
transition paths among all the states in the deuteron projectile and the
polarization potential, we successfully described the experimental data of
elastic cross-section ratios with the tBF method employing a realistic
nucleon-nucleon (NN) interaction from the chiral effective field theory (EFT).
No adjustable parameters were introduced. In the present work, we study the
detailed dynamics for the scattering of d+208Pb with the deuteron incident
energy $E_{d}=7$ MeV and the scattering angle $\theta=150^{\circ}$ using the
tBF method. We also calculate the angular distribution of the elastic
differential cross-section at $E_{d}=5-7$ MeV and investigate the effect of
the polarization potential. In addition, we calculate the E1 polarizability of
the deuteron and ratios of elastic scattering cross sections (at $E_{d}=3-7$
MeV) to compare with experimental data and present their dependence on NN
interactions.

The paper is organized as follows. In the next section, we present the
theoretical framework of the tBF approach. We show the main results in Sec.
III. Finally we give a summary of our conclusions and an outlook in Sec. IV.

##  II Theoretical framework of the tBF approach

In this work we investigate the scattering of the deuteron projectile on the
208Pb target below the Coulomb barrier employing the tBF approach. When the
Sommerfeld parameter

$\eta=\frac{Ze^{2}}{v}$ (1)

is much larger than unity, one may reasonably adopt a semi-classical approach
for the scattering trajectory Alder:1956im ; Alder:1975 . Here $Z$ represents
the charge of our target ($Z=82$ for Pb), $e$ is the unit of electric charge
and $v$ represents the incident velocity of the deuteron. Note that we adopt
the natural unit and set $\hbar=c=1$ throughout the paper. The minimum value
of the Sommerfeld parameter in this work is $6.93$ (corresponding to the
maximum incident deuteron energy $7$ MeV). Therefore we assume, for the
applications in the present work, that the center of mass (COM) of the
deuteron projectile moves along a classical trajectory. We then construct the
time-dependent Schrödinger equation of the scattering process describing how
the external field provided by the 208Pb target induces the internal
transitions in the deuteron projectile during the scattering. The main idea of
the tBF approach is to solve the time-dependent Schrödinger equation within a
basis representation determined from ab initio nuclear structure calculations.

We do not consider the excitation of 208Pb since its spin-parity ($3^{-}$)
requires a high multipole (E3) transition, its excitation energy ($2.6$ MeV)
is also rather high Aoki:2000nim  and the deuteron provides an external
excitation potential acting on the lead target that is $82$ times weaker than
it experiences from the lead target. Hence, this $3^{-}$ excitation of 208Pb
is expected to be at least two orders of magnitude smaller than the Coulomb
dissociation of the deuteron for $E_{d}=3-7$ MeV, as previously noted
Rodning:1982zz . The influence of vacuum polarization, atomic screening and
relativistic corrections on the elastic cross-section ratios is found to be
small Moro:1999fcl ; Rodning:1982zz  and therefore these effects are also not
taken into account in this work. We found the effects of the magnetic dipole
(M1) transitions to be negligibly small compared to the effects of the E1
transitions below the Coulomb barrier and would not affect the conclusions of
this paper so we omit the M1 transitions at the present time Yin:2019kqv .

###  II.1 Determination of the trajectory

Figure 1: (Color online) A sketch for the scattering of the deuteron
projectile on the 208Pb target. See the text for details.

We show in Fig. 1 the sketch of the scattering setup. We set the scattering
plane to be the $xz$ plane. We assume that the 208Pb target is infinitely
massive and we then work in the lab frame. For the low incident energies
considered, we assume that the 208Pb target is a point-like nucleus located at
the origin during the scattering. The initial velocity of the deuteron
projectile is parallel to the $z$ axis. $E_{d}$ and $b$ denote the bombarding
energy of the deuteron and the impact parameter, respectively. $\theta$
represents the scattering angle. The time-dependent vector ${\bm{R}}_{d}(t)$
denotes the position of the COM of the neutron-proton (np) system with respect
to the origin during the scattering. We calculate the trajectory assuming that
the deuteron projectile is initially located at $(x,z)=(b,z_{0})$ at $t=0$ as
shown in Fig. 1. Specifically, we adopt $z_{0}=-300$ MeV-1 ($\approx-59200$
fm) for which the scattering observables investigated in this work are
converged to the order of $10^{-5}$ Yin:2019kqv .

In the non-relativistic limit, the trajectory for the scattering of two
electric point charges is given by the well known Rutherford trajectory. When
two nuclei approach to distances comparable to the sum of their radii, they
interact through the strong nuclear potential and the trajectory will deviate
from the Rutherford trajectory. Even for scatterings well below the Coulomb
barrier, where the nuclear effects become small, some other effects will also
lead to deviations from the Rutherford trajectory. One of the most important
effects is expected to arise from dipole polarization Moro:1999fcl .

During the deuteron scattering on 208Pb, the charge distribution in the
projectile is distorted by the electric field of the target. Then the proton
and the neutron in the deuteron are forced to larger amplitudes of relative
motion with the proton repelled away from the Pb target relative to the
neutron. This effect is usually referred to as the dipole polarization of the
projectile. In this work, the effect of the dipole polarization will be
represented by a polarization potential acting on the COM of the deuteron
during the scattering process. Simultaneously, we will treat the effects of
the external field on the relative motion inside the deuteron at the quantum
amplitude level in tBF.

Taking into account the dipole polarization effect of the deuteron projectile,
the trajectory of the projectile is determined by the following combined
potential

$\displaystyle V_{\rm pot}$ $\displaystyle=$ $\displaystyle V_{\rm c}+V_{\rm
pol},$ (2)

where $V_{\rm c}$ and $V_{\rm pol}$ denote the Coulomb potential and the
polarization potential, respectively, acting on the projectile COM. Omitting
the correction of the polarization potential, the trajectory takes the form of
the conventional hyperbolic Rutherford trajectory.

Applying second-order perturbation theory Rodning:1982zz ; Moro:1999fcl ;
Baur:1977nso ; Alder:1956im , the polarization potential is written as

$\displaystyle V_{\rm pol}=-\frac{1}{2}\alpha\frac{Z^{2}e^{2}}{R_{d}^{4}(t)}.$
(3)

$\alpha$ denotes the E1 polarizability of the deuteron which is defined as
Alder:1956im

$\displaystyle\alpha$ $\displaystyle=$ $\displaystyle\frac{8\pi}{9}\sum_{n\neq
0}\frac{B(E1;0\rightarrow n)}{(E_{n}-E_{0})},$ (4)

where the indexes $0$ and $n$ denote the ground state and the scattering
states of the deuteron, respectively. $B(E1;0\rightarrow n)$ represents the E1
strength for the coupling between the deuteron ground state and the scattering
state, which is calculated as follows

$B(E1;0\rightarrow
n)=\sum_{M_{n},\mu}\left|\left\langle\beta_{0},M_{0}|\mathcal{M}(E1,\mu)|\beta_{n},M_{n}\right\rangle\right|^{2}.$
(5)

$|\beta_{j}\rangle$ denotes the wave function of the deuteron ground state
($j=0$) or the scattering state ($j>0$), which is calculated by solving the
following eigenequation:

$\displaystyle H_{0}|\beta_{j}\rangle$ $\displaystyle=$ $\displaystyle E_{j}\
|\beta_{j}\rangle,$ (6)

with $E_{j}$ the eigenenergy of the state $|\beta_{j}\rangle$. $H_{0}$
represents the fully interacting Hamiltonian for the intrinsic motion of the
np system in the absence of any external fields:

$\displaystyle H_{0}$ $\displaystyle=$ $\displaystyle T_{\rm rel}+V_{\rm NN},$
(7)

with $T_{\rm rel}$ and $V_{\rm NN}$ being the relative kinetic energy and the
NN interaction, respectively. $M_{0}$ and $M_{n}$ in Eq. (5) represent the
total angular momentum projections of $|\beta_{0}\rangle$ and
$|\beta_{n}\rangle$, respectively, while $\mu$ is the angular momentum
projection of the E1 operator so that $M_{n}+\mu=M_{0}$. In practice we obtain
$|\beta_{j}\rangle$ by diagonalizing the Hamiltonian $H_{0}$ in the harmonic
oscillator (HO) basis, which is characterized by the basis strength $\omega$
and the basis truncation parameter $N_{\rm max}$ (defined as the maximum of
twice the radial quantum number plus the orbital angular momentum)
Barrett:2013nh ; Vary:2018jxg . In the HO basis, the scattering states are
discretized. In what follows, we therefore check for the convergence of the
scattering observables with increasing basis space cutoff $N_{\rm max}$ for a
range of values of $\omega$. $\mathcal{M}(E1,\mu)$ in Eq. (5) is the E1
operator which is expressed in the internal coordinates of the deuteron as

$\mathcal{M}(E1,\mu)=\sum_{k=n,p}e_{k}r_{k}Y_{1\mu}(\theta_{k},\phi_{k}).$ (8)

where $Y_{\lambda\mu}$ denotes the spherical harmonics ($\lambda=1$ denotes
the dipole component) in the Condon-Shortley convention. $e_{k}$ and
${\bm{r}}_{k}=(r_{k},\theta_{k},\phi_{k})$ are the charge and the position
vector of the proton ($k=p$) or neutron ($k=n$) in the deuteron projectile.

In this work, we adopt five NN interactions for our investigations: the JISP16
Shirokov:2005bk ; Shirokov:2014kqa , Daejeon16 Shirokov:2016ead  NN
interactions and three NN interactions from the chiral EFT. Specifically, we
select the Low Energy Nuclear Physics International Collaboration (LENPIC) NN
interactions Epelbaum:2014efa ; Epelbaum:2014sza ; Binder:2016 ; Binder:2018
for the chiral EFT interactions, which have been developed for each chiral
order up through N4LO. These LENPIC interactions employ a semilocal
coordinate-space regulator and we take the interactions with the regulator
range of $1.0$ fm. We take three LENPIC interactions at the chiral orders of
N2LO, N3LO and N4LO, which we refer to as “LENPIC-N2LO”, “LENPIC-N3LO” and
“LENPIC-N4LO”, respectively.

Figure 2: (Color online) Trajectory of the deuteron incident from the negative
$z$ direction and scattering on 208Pb at $E_{d}=7$ MeV to the final angle of
$\theta=150^{\circ}$. Distances are marked in units of MeV-1. Inset:
amplification of the trajectories near the closest approach. The dashed and
solid lines denote the results with and without the polarization potential
respectively.

As an example, we show in Fig. 2 the trajectories of the deuteron scattering
on 208Pb with the incident energy $E_{d}=7$ MeV and the scattering angle
$\theta=150^{\circ}$, determined by $V_{c}$ (solid line) or $V_{c}+V_{\rm
pol}$ [Eq. (2)] (dashed line). Note that the trajectory determined by $V_{c}$
corresponds to the Rutherford trajectory. We adopt the polarization potential
in Eq. (3) with $\alpha=0.635$ fm3 which was determined in Ref. Yin:2019kqv
for the LENPIC-N4LO interaction. We find that the effect of the polarization
potential on the trajectory is almost invisible in the large scale. However,
the difference between these two trajectories are visible near the closest
approach (see the inset). In particular, the correction of the attractive
polarization potential leads to a slightly smaller distance of closest
approach for this scattering angle compared to the result of the Rutherford
trajectory.

###  II.2 EOM for the scattering

The full Hamiltonian of the deuteron projectile moving in the time-dependent
background field provided by the 208Pb target can be written as

$\displaystyle H_{\rm full}(t)$ $\displaystyle=$ $\displaystyle H_{0}+V_{\rm
int}(t),$ (9)

where $V_{\rm int}(t)$ denotes the time-dependent interaction between the
projectile and the target during the scattering. The EOM of the projectile
during the scattering, in the interaction picture, can be written as

$\displaystyle i\frac{\partial}{\partial t}|\psi;t\rangle_{I}$
$\displaystyle=$ $\displaystyle e^{i{H_{0}t}}\ V_{\rm int}(t)\ e^{-i{H_{0}}t}\
|\psi;t\rangle_{I}\ \equiv\ V_{I}(t)\ |\psi;t\rangle_{I}\ ,$ (10)

where $V_{I}(t)$ denotes the time-dependent interaction between the projectile
and the target in the interaction picture. The subscript “I” specifies the
interaction picture. The wave function of the projectile at $t\geq t_{0}$ can
be solved as

$\displaystyle|\psi;t\rangle_{I}$ $\displaystyle=$ $\displaystyle
U_{I}(t;t_{0})|\psi;t_{0}\rangle_{I},$ (11)

where $t_{0}$ denotes the time when the deuteron projectile is in its initial
state. $U_{I}(t;t_{0})$ is the unitary operator for the time-evolution

$\displaystyle U_{I}(t;t_{0})$ $\displaystyle=$
$\displaystyle\hat{T}e^{-i\int_{t_{0}}^{t}V_{I}(t^{\prime})dt^{\prime}},$ (12)

with $\hat{T}$ the time-ordering operator towards the future.

###  II.3 Numerical solution of the EOM

The internal transitions in the deuteron projectile are negligible for large
separation of the projectile and the target $R_{d}(t)$. To improve the
computational efficiency, we do not consider the internal transitions in the
deuteron projectile before $t=t_{0}$ where $R_{d}(t<t_{0})$ is larger than
$R_{d}^{c}$, which will be determined in the following. We initiate the
internal transitions from the position $R_{d}(t_{0})=R_{d}^{c}$.

To solve the EOM numerically, we divide the interval $[t_{0},t]$ into $n$
segments with the equal time step $\delta t=(t-t_{0})/n$, which must be
sufficiently small to attain numerically stable results. We then replace the
integration in the time-evolution operator $U_{I}(t;t_{0})$ by additive
increments and keep only terms up to the order of $\delta t$ in the following
Taylor expansion:

$\displaystyle U_{I}(t;t_{0})\longrightarrow[1-iV_{I}(t)\delta
t][1-iV_{I}(t_{n-1})\delta t]\cdots[1-iV_{I}(t_{1})\delta t].$ (13)

The numerical solution of the EOM according to Eq. (13) is called the Euler
scheme, which was found numerically unstable during the time-evolution
Askar:1978 . We therefore employ the non-perturbative multistep differencing
scheme up to the second-order (MSD2 scheme) Iitaka:1994  to obtain the time-
dependent wave function of the deuteron projectile in our tBF method. Using
the MSD2 scheme, we evaluate the state vector at the first time step
($t_{1}=t_{0}+\delta t$) still via the Euler Scheme:

$\displaystyle|\psi;t_{1}\rangle_{I}=[1-iV_{I}(t_{0})\delta
t]|\psi;t_{0}\rangle_{I}.$ (14)

However, we calculate the state vector at
$t^{\prime}=t_{2},t_{3},\cdots,t_{n-1}$ with

$\displaystyle|\psi;t^{\prime}+\delta
t\rangle_{I}=|\psi;t^{\prime}\rangle_{I}-2iV_{I}(t^{\prime})\delta
t|\psi;t^{\prime}\rangle_{I}.$ (15)

In this work, we take $R_{d}^{c}=5$ MeV-1 ($\approx 987$ fm) and $\delta
t=0.0001$ MeV-1 under which the calculated observables are converged up to the
fifth significant figure as illustrated in Ref. Yin:2019kqv .

###  II.4 State vector and interaction in the basis representation

In the numerical implementation of the MSD2 scheme [Eq. (14) and Eq. (15)], we
calculate the time-dependent state vector of the deuteron projectile in a
basis representation formed by the deuteron bound and scattering states, i.e.,
$\\{|\beta_{j},M_{j}\rangle\\}$. In this basis representation, the state
vector of the deuteron projectile at any moment $t$ is written as

$\displaystyle|\psi;t\rangle_{I}=\sum_{j}A_{j}^{I}(t)|\beta_{j},M_{j}\rangle,$
(16)

where $A_{j}^{I}(t)$ is the amplitude corresponding to the basis
$|\beta_{j},M_{j}\rangle$. The time-dependent interaction $V_{I}(t)$ becomes a
matrix in this basis representation. We consider only the E1 component of the
Coulomb interaction between the projectile and the target since that is known
to be the dominant deuteron excitation mode for sub-barrier scatterings
Moro:1999fcl . We calculate the matrix elements of the E1 interaction as
follows:

$\displaystyle\langle\beta_{j},M_{j}|V_{I}(t)|\beta_{k},M_{k}\rangle=-\frac{4\pi}{3}Ze^{2}e^{i(E_{j}-E_{k})t}\sum_{\mu}\frac{Y_{1\mu}^{*}(\Omega_{R})}{|R_{d}(t)|^{2}}\int
d\bm{r}\langle\beta_{j},M_{j}|\bm{r}\rangle\frac{r}{2}Y_{1\mu}(\Omega_{r})\langle\bm{r}|\beta_{k},M_{k}\rangle,$
(17)

where $\bm{r}=\bm{r}_{p}-\bm{r}_{n}$ denotes the relative coordinates of the
deuteron projectile. After adopting an initial state
$|\psi;t_{0}\rangle_{I}=\sum_{j}A_{j}^{I}(t_{0})|\beta_{j},M_{j}\rangle$ in
the above basis representation, we can calculate the state vector of the
deuteron projectile at $t>t_{0}$ with Eq. (14) and Eq. (15) straightforwardly.
It is noteworthy that the time-varying $R_{d}(t)$ (separation between the COMs
of the projectile and the target) in Eq. (17) provides the only source of the
time-dependence in the present scattering problem.

###  II.5 Observables

In this work, we calculate the differential cross-section for elastic
scattering by

$\left(\frac{d\sigma}{d\Omega}\right)_{\rm el}=P_{\rm
el}\left(\frac{d\sigma}{d\Omega}\right)_{\rm class},$ (18)

where the elastic scattering probability $P_{\rm el}$ is obtained by summing
over the probabilities in the three orientations of the deuteron ground state
after the time-evolution with the tBF method. That is, we take into account
the spin flips of the deuteron ground state, a higher-order process, in the
elastic channel which would be appropriate for experiments that do not measure
the spin of the deuteron in the final state.
$\left(\frac{d\sigma}{d\Omega}\right)_{\rm class}$ denotes the classical
differential cross-section, which is calculated using a trajectory determined
by the adopted potential acting on the COM of the deuteron (either $V_{\rm c}$
or $V_{\rm pot}$). Specifically, we calculate
$\left(\frac{d\sigma}{d\Omega}\right)_{\rm class}$ with

$\displaystyle\left(\frac{d\sigma}{d\Omega}\right)_{\rm class}$
$\displaystyle=$
$\displaystyle\frac{b}{\sin\theta}\left|\frac{db}{d\theta}\right|,$ (19)

where $b$ and $\theta$ denote the impact parameter and the scattering angle,
respectively. For reference, in the case where $V_{\rm c}$ alone is used, the
Rutherford cross-section would emerge since
$b=\frac{Ze^{2}}{2E_{d}}\cot\left(\frac{\theta}{2}\right)$.

In order to compare with available experimental data Rodning:1982zz , we
calculate the following quantity

$\displaystyle R(E_{d})=\frac{\sigma(3\ \rm{MeV},\theta_{1})}{\sigma(3\
\rm{MeV},\theta_{2})}\frac{\sigma(E_{d},\theta_{2})}{\sigma(E_{d},\theta_{1})},$
(20)

where $\sigma(E_{d},\theta)=2\pi\left(\frac{d\sigma}{d\Omega}\right)_{\rm el}$
denotes the differential cross-section of the elastically scattered deuterons
at the angle $\theta$ with the bombarding energy $E_{d}$.

##  III Results and Discussions

In this work we set the initial state of the deuteron projectile to be its
ground state (${}^{3}S_{1}-{{}^{3}D_{1}}$ channel). The polarization will be
defined for each of the specific applications below. Since E1 transitions
respect the conservation of the total spin $S$ of the np system, we take only
channels with $S=1$ into account. We restrict the total angular momentum $J$
to be $J\leq 2$ though higher angular momentum states could, in principle, be
populated through higher-order transitions. We introduce a quantity $E_{\rm
cut}$ to represent the upper energy limit of the retained scattering states of
the np system. To be specific, we adopt the eigenstates of the np system with
eigenenergies below $E_{\rm cut}$ in ${}^{3}S_{1}-{{}^{3}D_{1}}$,
${}^{3}P_{0}$, ${}^{3}P_{1}$, ${}^{3}D_{2}$ and ${}^{3}P_{2}-{{}^{3}F_{2}}$
channels to form the basis representation of the tBF approach in this work.

Figure 3: (Color online) The deuteron spectrum, up to $E_{\rm cut}=14$ MeV for
$S=1$ and $J\leq 2$, as a function of $N_{\rm max}$ obtained with the
LENPIC-N4LO interaction with $\omega=20$ MeV. The horizontal dashed line
denotes the boundary between the bound and scattering states. We indicate the
energy of the deuteron ground state below the lowest level of each spectrum.

In Fig. 3, we show the spectra of the deuteron in five channels
(${}^{3}S_{1}-{{}^{3}D_{1}}$, ${}^{3}P_{0}$, ${}^{3}P_{1}$,
${}^{3}P_{2}-{{}^{3}F_{2}}$, and ${}^{3}D_{2}$) below $14$ MeV calculated with
$\omega=20$ MeV and a sequence of $N_{\rm max}$ truncations by adopting the
LENPIC-N4LO interaction. We also show the number of states in these five
channels which takes into account the degeneracy with respect to the magnetic
quantum number of each state. The set of states for a given truncation forms
the basis representation of the tBF approach as in Eq. (16) which is used to
solve the time-dependent Schrödinger equation. The deuteron bound state energy
becomes well converged with increasing $N_{\rm max}$ and tends to the
experimental value. As the basis truncation $N_{\rm max}$ increases, we find
that the discretized scattering states become increasingly dense and flow
towards the breakup threshold as expected. The ultimate test of sufficiency of
the continuum discretization will be the convergence rate of specific
experimental observables with increasing $N_{\rm max}$ as we will address
below.

Figure 4: (Color online) The E1 polarizability of the deuteron as a function
$N_{\rm max}$. The results in panel (a) are obtained using five different NN
interactions with $\omega=20$ MeV. The results in panel (b) are calculated
employing the LENPIC-N4LO interaction with various $\omega$ values. Two
experimental results and their uncertainty bands from Ref. Rodning:1982zz
(blue region) and Ref. Friar:1983zza  (green region) are presented for
comparison.

In Fig. 4 (a), we show the E1 polarizability of the deuteron, $\alpha$, as a
function of the truncation parameter of the HO basis, $N_{\rm max}$, for the
LENPIC (-N2LO, -N3LO, -N4LO), JISP16 and Daejeon16 NN interactions with the
identical strength of the HO basis $\omega=20$ MeV. We also show two sets of
results extracted from experiments along with their quoted uncertainties for
comparison. We find in Fig. 4 (a) that the E1 polarizabilities of the deuteron
calculated with these five NN interactions converge to different values at
sufficiently large $N_{\rm max}$, which are all consistent with the
experimental data.

The LENPIC-N3LO and LENPIC-N4LO interactions are known to provide challenges
for obtaining converged spectra in ab initio calculations in light nuclei
Binder:2018 . Since the HO basis space parameters can be characterized by an
ultraviolet (UV) cutoff $\Lambda=\sqrt{m_{N}(N_{\rm max}+3/2)\omega}$ ($m_{N}$
is the nucleon mass) Stetcu:2006ey ; Coon:2012ab , we can deduce from Fig. 4
that the E1 polarizability of the deuteron for fixed $\omega$ requires a
sufficiently large $N_{\rm max}$ for these interactions to access their
higher-momentum contributions. One may interpret the erratic oscillations at
small $N_{\rm max}$ for these two interactions in Fig. 4 (a) as arising from
basis spaces that are deficient in the needed high-momentum contributions.
However, the results calculated with the LENPIC-N2LO, JISP16 and Daejeon16
show smooth trends at low to moderate $N_{\rm max}$ indicating that these
three interactions are softer in the UV region than the LENPIC-N3LO and
LENPIC-N4LO interactions. Above about $N_{\rm max}=40$ at $\omega=20$ MeV, we
observe that all these five NN interactions produce a similar trend reflecting
similar accumulations of the long-range (infra-red region) components of the
nuclear wave function that contribute to the E1 polarizability.

In Fig. 4 (b), we present the E1 polarizability of the deuteron as a function
of $N_{\rm max}$, calculated with the LENPIC-N4LO interaction with various
$\omega$ values. The results with different $\omega$ values converge to the
same asymptotic value at sufficiently large $N_{\rm max}$. However, different
$\omega$ values result in different convergence patterns at small $N_{\rm
max}$. In particular, we observe erratic fluctuations at small $N_{\rm max}$
for small $\omega$ values. However, these fluctuations vanish at large
$\omega$ since the UV cutoff $\Lambda$ increases with $\omega$ and the physics
at high-momentum is more adequately accommodated at larger $\omega$ and small
to moderate $N_{\rm max}$.

Table 1:  The E1 polarizability of the deuteron, $\alpha$, for $11$ NN interactions and two results extracted from experiments. |  $\alpha$ (fm3) |  Refs.
---|---|---
LENPIC-N2LO |  $0.6292(1)$ |  this work
LENPIC-N3LO |  $0.6352(1)$ |  this work
LENPIC-N4LO |  $0.6349(1)$ |  this work
JISP16 |  $0.6164(1)$ |  this work
Daejeon16 |  $0.6658(1)$ |  this work
Reid soft core (68) |  $0.6237$ |  Friar:1997fp
Reid soft core (93) |  $0.6345$ |  Friar:1997fp
Argonne V14 |  $0.6419$ |  Friar:1997fp
Argonne V18 |  $0.6343$ |  Friar:1997fp
Bonn (CS) |  $0.6336$ |  Friar:1997fp
Paris |  $0.6352$ |  Friar:1997fp
Exp. |  $0.70(5)$ |  Rodning:1982zz
Exp. |  $0.61(4)$ |  Friar:1983zza

We summarize in Table 1 the E1 polarizabilities of the deuteron at $N_{\rm
max}=240$ obtained with the LENPIC (-N2LO, -N3LO, -N4LO), JISP16 and Daejeon16
interactions with $\omega=20$ MeV. The uncertainty for each interaction
denotes the absolute difference between the result at $N_{\rm max}=240$ and
the result at $N_{\rm max}=280$. The results calculated with the three LENPIC
interactions differ from each other by less than $1\%$. The JISP16 and
Daejeon16 interactions predict the smallest and the largest E1
polarizabilities of the deuteron, respectively. The spread of the E1
polarizability, induced by these five interactions, is about $8\%$. For
comparison, we also show in Table 1 the E1 polarizabilities of the deuteron
extracted from experiments in Refs. Rodning:1982zz ; Friar:1983zza  and those
obtained with the Reid soft core (68) Reid:1968sq , Reid soft core (93)
Stoks:1994wp , Argonne V14 Wiringa:1984tg , Argonne V18 Wiringa:1994wb  and
Bonn (CS) Machleidt:1987hj  interactions in Ref. Friar:1997fp . The results
obtained in our work and prior theoretical results Friar:1997fp , all shown in
Table 1, are consistent with experiment Rodning:1982zz ; Friar:1983zza .

Figure 5: (Color online) Probability of the initial state as a function of the
evolution time at $E_{d}=7$ MeV and $\theta=150^{\circ}$ for various $N_{\rm
max}$ [panel (a)] and $E_{\rm cut}$ [panel (b)].

In Fig. 5, we display the probability of the deuteron remaining in its initial
state, which we refer to as $P_{0}$, as a function of the evolution time
during the deuteron scattering on 208Pb at $E_{d}=7$ MeV and
$\theta=150^{\circ}$, calculated with the tBF method employing the LENPIC-N4LO
interaction with $\omega=20$ MeV at various $N_{\rm max}$ values [panel (a)]
and various $E_{\rm cut}$ values [panel (b)]. We adopt the trajectory,
determined by $V_{c}+V_{\rm pol}$ [see Eq. (2)] with $\alpha=0.635$ fm3, in
the tBF method as in Fig. 2. In the following we will use the same $\it{NN}$
interaction and the polarization potential as in Fig. 5 unless explicitly
stated otherwise. We take a polarized deuteron with the total angular momentum
projection $M_{0}=-1$ in its ground state for the initial state. We show the
results for different $N_{\rm max}$ and $E_{\rm cut}$ to investigate the
convergence of $P_{0}$ with respect to these two truncation parameters.

In the early stage of the time-evolution, the probability $P_{0}$ decreases
with time but is still very close to $1$ since the interaction between the
projectile and the target is very weak. This deviation from unity is not
visible in Fig. 5. When the deuteron projectile is sufficiently close to the
208Pb target, transitions become significant and therefore the probability
$P_{0}$ drops sharply below unity before partially recovering, with an
oscillatory fine structure, towards the final value at longer times.

Eventually the probability $P_{0}$ relaxes to its asymptotic value as the
Coulomb field fades away. We notice in Fig. 5 that the asymptotic value of the
probability $P_{0}$ is well converged with respect to the truncation
parameters $N_{\rm max}$ and $E_{\rm cut}$. The uncertainty of $P_{0}$ induced
by $N_{\rm max}$ and $E_{\rm cut}$ is on the order of $10^{-4}$. In all
remaining calculations, we take $N_{\rm max}=200$, $\omega=20$ MeV and $E_{\rm
cut}=14$ MeV, which provides $165$ states of the np system. We deem this to be
sufficient since we find the observables that we investigate are converged to
the order of $10^{-4}$ under this choice.

Figure 6: (Color online) Probabilities, on a semi-log scale, of the deuteron
ground state with $M_{0}=0$ and $M_{0}=1$ and the deuteron scattering states
as functions of the evolution time for the deuteron scattering on 208Pb at
$E_{d}=7$ MeV and $\theta=150^{\circ}$. The solid and the dashed curves
represent the states allowed and forbidden respectively by the E1 transition
operator acting on the initial state with $M_{0}=-1$ in first-order
perturbation theory. We show in the legend the six states (e.g., “1st,
3P2-3F2” denotes the lowest state in the 3P2-3F2 channel) with the largest
probabilities before the crest. These states are labeled in ascending order of
excitation energy in the spectrum of the deuteron.

In Fig. 6, we show the time-evolution of the probabilities of all the states,
except for the initial state, for the deuteron scattering on 208Pb at
$E_{d}=7$ MeV and $\theta=150^{\circ}$, calculated by the tBF method. We take
the same initial state (i.e., a polarized deuteron ground state with
$M_{0}=-1$) as in Fig. 5. We signify the states, allowed and forbidden by the
E1 transition operator acting on our initial state in first-order perturbation
theory, by the solid and the dashed curves in Fig. 6, respectively. For
simplicity, we will refer to these states as “E1 allowed” and “E1 forbidden”,
respectively.

In the early stage of the time-evolution in Fig. 6, only E1 allowed states
populate significantly. We note that six of them (see the legend of Fig. 6)
populate well above the others before reaching the crest. The wave functions
of the first, third and fifth states in the ${}^{3}P_{2}-^{3}F_{2}$ channel
are dominated by the ${}^{3}P_{2}$ component and therefore obtain significant
populations from the ${}^{3}S_{1}$ component of the deuteron ground state in
this stage, according to the E1 selection rules. However, the second and
fourth states in the ${}^{3}P_{2}-^{3}F_{2}$ channel, which are dominated by
the ${}^{3}F_{2}$ channel, are not able to populate via direct transitions
from the ${}^{3}S_{1}$ component of the deuteron ground state and therefore
are less populated than the first, third and fifth states in this channel. The
populations of the E1 forbidden states are very small, but still non-zero,
during this early stage. Of course, E1 forbidden states populate via
transitions from the E1 allowed states according to the E1 selection rule.
Therefore, as expected, the E1 forbidden states do not populate significantly
until the E1 allowed states accumulate appreciable populations. Note that the
de-excitations among all the states are also taken into account in the tBF
method. One may be able to visualize that an entire complex web of quantum
excitations and de-excitations is evolving with time and Fig. 6 attempts to
portray some aspects of the probability flow among these states.

Figure 7: (Color online) Time-evolution for the probability of the deuteron
remaining in the initial state for three different sets of transition
couplings. See the text for details.

We show in Fig. 7 the probability of the deuteron remaining in its initial
state (the deuteron ground state with $M_{0}=-1$) as a function of the
evolution time during the deuteron scattering on 208Pb at $E_{d}=7$ MeV and
$\theta=150^{\circ}$ with three different sets of transition couplings. In
other words, the deviation of the asymptotic value of $P_{0}$ from unity
measures the inelastic effects. The black solid line denotes the tBF results
with all the possible couplings among all the $165$ states included. The red
dashed line, signified as “Type I”, represents the case where we retain only
the couplings between the initial state and its E1 allowed states ($60$ states
in total). The blue dotted line, signified as “Type II”, represents the case
where we retain the couplings between the initial state and the six states,
highlighted in Fig. 6, as well as the couplings between these six states and
their corresponding E1 allowed states ($46$ states in total).

The E1 transitions between the initial state and the six E1 allowed scattering
states, which are highlighted in Fig. 6, are dominant in the early stage of
the time-evolution and all the three curves in Fig. 7 take into account these
transitions. Therefore these three curves in Fig. 7 are almost identical in
the early stage of the time-evolution. As the projectile approaches the target
and the Coulomb interaction between them increases accordingly, we notice that
differences among these three curves emerge. After the virtual excitation
denoted by the dip in the red dashed line, the de-excitation processes bring
almost all the populations back to values near zero for the Type I coupling.
This reflects an approximate time-reversal symmetry which is effectively
achieved in this restricted inelastic scattering framework. However, the
initial state retains appreciable transitions to the excited states later in
the time-evolution with the Type II coupling (blue dotted line) and with all
the couplings (black solid line). Comparing the asymptotic values of the black
and the red lines ($0.9784$ and $0.9998$, respectively), we find that
retaining only the coupling to E1 allowed states (Type I coupling)
dramatically underestimates the inelastic effects which clearly demonstrates
that the higher-order effects play a major role in the scattering process.
Comparing the asymptotic values of the black and the blue curves ($0.9784$ and
$0.9823$, respectively), we find that the Type II coupling accounts for about
$82\%$ of the inelastic effects. The remaining $18\%$ represents additional
transitions to E1 allowed states and additional non-linearities included
coherently with the dominant processes.

Figure 8: (Color online) Probability of the initial state as a function of the
evolution time evaluated by the tBF method (solid red lines) and the time-
reversed calculations (dashed blue lines). Panels (a) and (b) denote the
results with the Type I coupling and all the couplings, respectively.

In order to check our numerical accuracy, we can exploit the time-reversal
symmetry of the tBF approach. We show in Fig. 8 the time-evolution of the
probability of the deuteron remaining in its initial state (the deuteron
ground state with $M_{0}=-1$), predicted by the standard (forward time-
propagation) tBF method (red solid curves) as well as the corresponding time-
reversed calculations (blue dashed curves) considering two different couplings
[Type I coupling in panel (a) and all the possible couplings in panel (b)],
for the d+208Pb scattering at $E_{d}=7$ MeV and $\theta=150^{\circ}$. We take
the time step $\delta t=0.0001$ MeV-1 for the standard tBF calculations while
$\delta t=-0.0001$ MeV-1 for the time-reversed calculations. The time at which
we switch to the time-reversed calculation corresponds to the closest approach
of the projectile and the target, which is at about $t=3476.12$ MeV-1.

We find in Fig. 8 that the standard tBF result and its corresponding time-
reversed result are coincident, to within five significant figures in these
results, for both of the two choices of E1 couplings [panel (a) and panel
(b)]. It is important to recall that the full wave function of the state
(i.e., not the probabilities) is evolved first forward and then backwards in
time so that the coherent complex amplitude is being verified. This
reproduction of time-reversal symmetry to high accuracy supports the numerical
precision of the tBF approach in the current application.

Figure 9: (Color online) Time-evolution for the probabilities of six selected
states in the incoming [panel (a)] and outgoing [panel (b)] segments of the
d+208Pb scattering at $E_{d}=7$ MeV and $\theta=150^{\circ}$, which are
calculated with all the couplings (solid curves) or Type I coupling (solid
squares). These states are labeled in ascending order of excitation energy in
the spectrum of the deuteron. The horizontal arrows delimit a cycle that is
discussed in the text.

We note that the probabilities of all the states in Fig. 6 exhibit
oscillations. To shed more light on these oscillating patterns, we show in
Fig. 9 the probabilities of the six states, highlighted in Fig. 6, as an
example. In panel (a) we show the results in the early stage of the time-
evolution, which we refer to as the “incoming segment”, in an expanded view.
In panel (b) we show the results after the closest approach, which we refer to
as the “outgoing segment”, also in an expanded view. For both the incoming and
outgoing segments, we present in Fig. 9 the results calculated by the tBF
method with all the couplings and with Type I coupling. In the incoming
segment, we notice that the results for both of the two sets of couplings are
coincident since these are six states that populate dominantly in the incoming
segment as shown in Fig. 6. As the deuteron projectile approaches the 208Pb
target, the higher-order effects become significant. As a consequence, the
results from these two couplings become significantly different in the
outgoing segment.

Table 2:  $E_{x}$, $\Gamma$ and $\kappa$ for the six scattering states of the
deuteron which are shown in Fig. 9. $\Gamma$ and $\kappa$ in the incoming
segment are distinguished from those in the outgoing segment for the tBF
calculations with either the Type I coupling or all the couplings. See the
text for additional details.

| $E_{\rm x}$ (MeV) |  $\Gamma$(MeV-1) | $\kappa$
---|---|---|---
Incoming | Outgoing | Incoming | Outgoing
Type I | All | Type I | All | Type I | All | Type I | All
$1^{\rm st}$ 3P2-3F2 | $2.72$ | $2.31$ | $2.31$ | $2.30$ | $2.10$ | $6.28$ | $6.28$ | $6.26$ | $5.71$
$1^{\rm st}$ 3P1 | $2.72$ | $2.31$ | $2.31$ | $2.30$ | $2.13$ | $6.28$ | $6.28$ | $6.26$ | $5.79$
$3^{\rm rd}$ 3P2-3F2 | $3.69$ | $1.70$ | $1.70$ | $1.69$ | $2.16$ | $6.27$ | $6.27$ | $6.24$ | $7.97$
$2^{\rm nd}$ 3P1 | $3.70$ | $1.70$ | $1.70$ | $1.69$ | $2.12$ | $6.29$ | $6.29$ | $6.25$ | $7.84$
$5^{\rm th}$ 3P2-3F2 | $5.15$ | $1.22$ | $1.22$ | $1.17$ | $1.32$ | $6.28$ | $6.28$ | $6.03$ | $6.80$
$3^{\rm rd}$ 3P1 | $5.17$ | $1.22$ | $1.22$ | $1.17$ | $1.31$ | $6.31$ | $6.31$ | $6.05$ | $6.77$

We find in Fig. 9 that the probabilities of all these six states oscillate in
approximately periodic patterns in each segment for both of the two sets of
couplings. In Fig. 9 we define a quantity $\Gamma$, for every state in a given
segment, as the time difference between two adjacent troughs, within the time
range of the same segment. We show in Table 2 the quantity $\Gamma$ of these
six states in both the incoming (Columns 3 and 4) and the outgoing segments
(Columns 5 and 6). We also distinguish in Table 2 the quantity $\Gamma$
calculated with the Type I coupling (Columns 3 and 5) from the one with all
the couplings (Columns 4 and 6). We then calculate the products of the
excitation energy $E_{x}$ (Column 2) and $\Gamma$, denoted as $\kappa$, and
present the results in Table 2 (Columns 7-10). We find in Table 2 that the
$\kappa$ values calculated with these two couplings (Columns 7 and 8) are the
same up to the third significant figure for each state in the incoming
segment. This is readily understood since the higher-order effects are
negligible in this segment. We also notice in Table 2 that the $\kappa$ values
of these six states are all approximately $2\pi$. We recognize this as a
“phase coherence” representing periodicity in the entanglement of the ground
state with these particular excited states induced by the E1 operator.

In the outgoing segment, we find that the $\kappa$ values of these six states,
calculated with the Type I coupling (Column 9), are also around $2\pi$, which
means the phase coherence phenomenon remains valid for this case. However, we
observe noticeable differences for the $\kappa$ values of these six states,
calculated with all the couplings (Column 10), and most of the $\kappa$ values
deviate significantly from $2\pi$. We refer to this as a “phase decoherence”.
Thus, the maintenance of the phase coherence phenomenon depends on the absence
of the couplings between the excited states. Such interference effect at the
amplitude level due to the time-varying interaction between the particle and
the external field is reminiscent of the Landau-Pomeranchuk-Migdal (LPM)
effect known in relativistic inelastic collisions Landau:1953um ;
Migdal:1956tc ; Yao:2022eqm .

Figure 10: (Color online) Angular distribution of the elastic differential
cross-section (left panel) and the quantity $T$ (right panel) after the
deuteron scatterings on 208Pb at $E_{d}=5,6$ and $7$ MeV. The solid and open
symbols in panel (a) are calculated by employing the classical trajectories
determined by the Coulomb potential with and without the inclusion of the
polarization potential, respectively. See the text for the details.

In panel (a) of Fig. 10, we show the angular distribution of the elastic
differential cross-section $\left(\frac{d\sigma}{d\Omega}\right)_{\rm el}$
divided by the Rutherford cross-section
$\left(\frac{d\sigma}{d\Omega}\right)_{\rm R}$ for the deuteron scattering on
208Pb at $E_{d}=5,6$ and $7$ MeV calculated by the tBF method. We take the
evenly weighted coherent sum of the magnetic substates of the deuteron ground
state as the initial state. We denote the elastic differential cross-section
calculated with and without the polarization potential by
$\left[\frac{\left(d\sigma/d\Omega\right)_{\rm
el}}{\left(d\sigma/d\Omega\right)_{\rm R}}\right]_{a}$ and
$\left[\frac{\left(d\sigma/d\Omega\right)_{\rm
el}}{\left(d\sigma/d\Omega\right)_{\rm R}}\right]_{b}$, respectively, for
convenience.

Comparing the two results with and without the polarization potential at the
same incident energy, we find that the polarization potential reduces the
elastic cross-section for all the incident energies and scattering angles
presented in Fig. 10 (a). The difference between
$\left[\frac{\left(d\sigma/d\Omega\right)_{\rm
el}}{\left(d\sigma/d\Omega\right)_{\rm R}}\right]_{a}$ and
$\left[\frac{\left(d\sigma/d\Omega\right)_{\rm
el}}{\left(d\sigma/d\Omega\right)_{\rm R}}\right]_{b}$ is rather small at
small scattering angles since the polarization potential is very weak. As the
scattering angle increases, the polarization effect grows in importance since
the deuteron projectile approaches closer to the target. For similar reasons,
we observe in Fig. 10 (a) that the difference between
$\left[\frac{\left(d\sigma/d\Omega\right)_{\rm
el}}{\left(d\sigma/d\Omega\right)_{\rm R}}\right]_{a}$ and
$\left[\frac{\left(d\sigma/d\Omega\right)_{\rm
el}}{\left(d\sigma/d\Omega\right)_{\rm R}}\right]_{b}$ at the same scattering
angle increases with the deuteron incident energy.

We do not anticipate diffractive effects in our treatment since that would
require a consideration of the strong interaction between the projectile and
the target and a quantum treatment of the COM motion of the projectile during
the scattering. Our approximations of an impact parameter treatment seem
reasonable since the Sommerfeld parameter is larger than unity for our current
application as discussed above. In addition, it has been noted that
diffractive scattering effects are expected to be negligible below $E_{d}=12$
MeV Moro:1999fcl . We therefore find in Fig. 10 (a) that the elastic cross-
section predicted by the tBF method decreases monotonically with the
scattering angle independent of whether the polarization potential is included
or not.

For classical Rutherford scattering, we have
$\left(\frac{d\sigma}{d\Omega}\right)_{\rm
el}=\left(\frac{d\sigma}{d\Omega}\right)_{\rm R}$ since $P_{\rm el}=1$ and
$\left(\frac{d\sigma}{d\Omega}\right)_{\rm
class}=\left(\frac{d\sigma}{d\Omega}\right)_{\rm R}$ [Eq. (18)]. Therefore the
deviation of the ratio of angular distributions from unity as measured by
$1-\left[\frac{\left(d\sigma/d\Omega\right)_{\rm
el}}{\left(d\sigma/d\Omega\right)_{\rm R}}\right]$, indicates the deviation of
a scattering from the classical Rutherford scattering.

For the tBF results with the correction of the polarization potential to the
Rutherford trajectories, $1-\left[\frac{\left(d\sigma/d\Omega\right)_{\rm
el}}{\left(d\sigma/d\Omega\right)_{\rm R}}\right]_{a}$ stems from the
following two effects:

  1. 1.

internal transitions of the projectile induced by the E1 interaction between
the projectile and the target during the scattering which lead to $P_{\rm
el}<1$, and

  2. 2.

the correction of the polarization potential to the classical Rutherford
trajectories which gives rise to $\left(\frac{d\sigma}{d\Omega}\right)_{\rm
class}<\left(\frac{d\sigma}{d\Omega}\right)_{\rm R}$.

We incorporate these two effects into the tBF method using the quantum and
classical method, respectively, as mentioned in Sec. II. However,
$\left[\frac{\left(d\sigma/d\Omega\right)_{\rm
el}}{\left(d\sigma/d\Omega\right)_{\rm R}}\right]_{b}$, calculated by the tBF
method without the correction of the polarization potential, deviates from
unity solely due to the internal E1 transitions in the projectile during the
scattering. Therefore, we estimate the significance of the internal E1
transitions in generating $1-\left[\frac{\left(d\sigma/d\Omega\right)_{\rm
el}}{\left(d\sigma/d\Omega\right)_{\rm R}}\right]_{a}$ with the following
quantity

$\displaystyle T(E_{d},\theta)$ $\displaystyle=$
$\displaystyle\frac{1-\left[\frac{\left(d\sigma/d\Omega\right)_{\rm
el}(E_{d},\theta)}{\left(d\sigma/d\Omega\right)_{\rm
R}(E_{d},\theta)}\right]_{b}}{1-\left[\frac{\left(d\sigma/d\Omega\right)_{\rm
el}(E_{d},\theta)}{\left(d\sigma/d\Omega\right)_{\rm
R}(E_{d},\theta)}\right]_{a}}.$ (21)

We show the corresponding results in Fig. 10 (b). The larger the quantity $T$
is, the more the internal E1 transitions in the deuteron projectile contribute
to $1-\left[\frac{\left(d\sigma/d\Omega\right)_{\rm
el}}{\left(d\sigma/d\Omega\right)_{\rm R}}\right]_{a}$ compared to the
polarization potential contributions. We find in Fig. 10 (b) that the effect
of the internal E1 transitions in the deuteron projectile on
$1-\left[\frac{\left(d\sigma/d\Omega\right)_{\rm
el}}{\left(d\sigma/d\Omega\right)_{\rm R}}\right]_{a}$ increases monotonically
with the scattering angle at $E_{d}=5,6$ and $7$ MeV. The deviation of
$\left[\frac{\left(d\sigma/d\Omega\right)_{\rm
el}}{\left(d\sigma/d\Omega\right)_{\rm R}}\right]_{a}$ from unity is always
dominated by the polarization potential for all the scattering angles
presented in Fig. 10 at $E_{d}=5$ and $6$ MeV. However, the contribution of
the internal E1 transitions in the deuteron projectile to
$1-\left[\frac{\left(d\sigma/d\Omega\right)_{\rm
el}}{\left(d\sigma/d\Omega\right)_{\rm R}}\right]_{a}$ is less significant
than the polarization potential below $140^{\circ}$ while it becomes dominant
above $140^{\circ}$ at $E_{d}=7$ MeV.

Figure 11: (Color online) $R(E_{d})$ with
$(\theta_{1},\theta_{2})=(60^{\circ},150^{\circ})$ as a function of the
bombarding energy $E_{d}$ for the scattering of d+208Pb. The dashed lines
denote the results calculated by the tBF method employing five NN interactions
with no polarization potential. The solid lines represent the results
calculated with the same NN interactions including the polarization potential.
The experimental data (black solid squares with error bars) are also shown for
comparison.

In Fig. 11 we show the quantity $R(E_{d})$ with
$(\theta_{1},\theta_{2})=(60^{\circ},150^{\circ})$ in Eq. (20) for the
scattering of the deuteron on 208Pb at $E_{d}=3-7$ MeV. We again adopt the
initial state with equal amplitudes for the three magnetic substates as used
for the results in Fig. 10. We also present the experimental data in Fig. 11
for comparison Rodning:1982zz . To investigate the sensitivity of $R(E_{d})$
to the NN interactions, we show in Fig. 11 the tBF results calculated with the
LENPIC (-N2LO, -N3LO, -N4LO), JISP16 and Daejeon16 NN interactions. For pure
Rutherford scattering $\sigma(E_{d},\theta_{1})/\sigma(E_{d},\theta_{2})$ in
Eq. (20) is independent of $E_{d}$ and therefore $R(E_{d})=1$ in this case
Rodning:1982zz ; Moro:1999fcl . Therefore the deviation of the quantity
$R(E_{d})$ from unity, i.e., $1-R(E_{d})$, reveals the deviation of a
scattering from the classical Rutherford scattering.

For the tBF results without the polarization potential (dashed lines) in Fig.
11, we use the Rutherford trajectories determined by the Coulomb potential.
$1-R(E_{d})$ is entirely induced by the internal transitions in the deuteron
projectile in this case as we pointed out in Ref. Yin:2019kqv . We notice in
Fig. 11 that the tBF calculations with no polarization potential are not able
to reproduce the experimental data. By taking the polarization potential into
account (solid lines), our tBF approach reproduces the experimental data at
$E_{d}=3-6$ MeV while falling below experiment at $E_{d}=7$ MeV for all the
five adopted NN interactions, which indicates that the polarization potential
is significant in the current application. In our calculations we take
$\alpha$ from Table 1 and retain three significant figures. In Ref.
Yin:2019kqv , we have compared the tBF results of the LENPIC-N4LO interaction
with the optical model calculations in Refs. Rodning:1982zz ; Moro:1999fcl .
We will concentrate mainly on the sensitivity of $R(E_{d})$ to the NN
interactions in the following.

It is noteworthy that the semiclassical tBF approach neglects the effect of
the energy loss on the center of mass motion of the projectile. The energy
transferred to the intrinsic degree of freedom of the projectile, approximated
by the average excitation energy of the projectile, should be compensated by a
reduction of the energy in its COM degree of freedom. We have shown in Ref.
Yin:2019kqv  that the energy loss effect could enhance $R(E_{d}=7$ MeV$)$ by
about $1\%$ using the LENPIC-N4LO interaction, which would lead to an improved
tBF description of the experimental data at $E_{d}=7$ MeV. The effect of the
energy loss on $R(E_{d})$ is insignificant at $E_{d}=3-6$ MeV and therefore
the current calculation is able to reproduce those experimental data.

Figure 12: (Color online) elastic cross section ratio $R(E_{d})$ with both
$(\theta_{1},\theta_{2})=(60^{\circ},140^{\circ})$ [panel (a)] and
$(\theta_{1},\theta_{2})=(60^{\circ},160^{\circ})$ [panel (b)] as functions of
the bombarding energy $E_{d}$ for the scattering of d+208Pb. The solid lines
represent the tBF results employing five NN interactions including their
associated polarization potentials. The experimental data (black solid squares
with error bars) are also shown for comparison.

Similarly, in Fig. 12 we show the calculated $R(E_{d})$ with both
$(\theta_{1},\theta_{2})=(60^{\circ},140^{\circ})$ and
$(60^{\circ},160^{\circ})$ [see Eq. 20], and compare with experiment. In the
tBF calculations we include the corrections from their associated polarization
potentials to the Rutherford trajectory, where we use the same $\alpha$ values
as in Fig. 11. We omit $\theta_{1}$ in the following since we use the same
$\theta_{1}=60^{\circ}$. In Fig. 12 we reproduce well the experimental
$R(E_{d})$ with $\theta_{2}=140^{\circ}$ and $160^{\circ}$ at $E_{d}=3-6$ MeV
for all the five interactions. We find in panel (b) that the tBF calculations
fall below the experimental data at $E_{d}=7$ MeV for $\theta_{2}=160^{\circ}$
which we attribute to the omission of the correction of the energy loss to the
COM motion of the projectile, as we observed in Fig. 11. However, the tBF
calculation is close to the experimental data at $E_{d}=7$ MeV for
$\theta_{2}=140^{\circ}$ (see panel (a) of Fig. 12) where the energy loss is
less significant compared to $\theta_{2}=150^{\circ}$ and
$\theta_{2}=160^{\circ}$.

Figure 13: (Color online) Contributions to the spread of $R(E_{d})$ arising
from the different NN interactions employed in this work. The total spread
(solid symbols and solid lines) is depicted for three scattering angles
($\theta_{2}=140^{\circ},150^{\circ}$ and $160^{\circ}$). In particular, we
show the contributions to the total spread at each angle arising from the E1
transition matrix elements (dashed line+symbol embedded with ‘+’) and from the
E1 polarizability of the deuteron (dotted line+open symbol). The black, red
and blue colors represent the results for $\theta_{2}=140^{\circ},150^{\circ}$
and $160^{\circ}$, respectively. The green symbols ‘$-$ $|$ $\times$’ denote
the addition of these two contributions for
$\theta_{2}=140^{\circ},150^{\circ}$ and $160^{\circ}$, respectively. The
deviation of the green symbols from their solid symbol counterparts is small
and hardly visible on this scale (see the text for details).

We find in Fig. 11 and Fig. 12 that $R(E_{d})$ calculated with the five
interactions are almost indistinguishable at low incident energies and
separate with increasing incident energy. For deeper insights into the
sensitivity of $R(E_{d})$ to NN interactions, we present in Fig. 13 the spread
of $R(E_{d})$, $\Delta R(E_{d})$, evaluated by the absolute difference between
the upper and lower edges of the solid lines in Fig. 11 and Fig. 12, as a
function of the deuteron incident energy (denoted as “Total” in Fig. 13). We
notice in Fig. 13 that the spread $\Delta R(E_{d})$ increases with the
deuteron incident energy as well as the scattering angle, indicating that
$R(E_{d})$ is more sensitive to NN interaction at higher incident energies and
larger scattering angles.

In this work, $\Delta R(E_{d})$ stems from the following two effects. The
first effect is the dependence of the E1 transition matrix elements [Eq. (17)]
on NN interactions, which contribute to $\Delta R(E_{d})$ via the time-
dependent Schrödinger equation [Eq. (10)]. The second effect is the dependence
of the E1 polarizability of the deuteron on the NN interactions, which
contributes to $\Delta R(E_{d})$ via the effect of the polarization potential
on the scattering trajectory. Note that $\Delta R(E_{d})$ is not simply a
linear superposition of the above two effects. We therefore perform the
following two sets of independent tBF calculations to investigate their
individual contributions to the total $\Delta R(E_{d})$ in Fig. 13. To
evaluate the contribution of the first effect, we solve the time-dependent
Schrödinger equation using different E1 transition matrix elements calculated
with the five interactions, where the trajectories are determined by the
Coulomb potential and the polarization potential with the same E1
polarizability $\alpha=0.635$ fm3, for each $\theta_{2}$. We present in Fig.
13 the corresponding spread of $R(E_{d})$, which is the absolute difference
between the two extreme values of $R(E_{d})$ obtained with these five
interactions at each $E_{d}$ and $\theta_{2}$, denoted as “1st-contr.”. To
estimate the contribution of the second effect, we solve the time-dependent
Schrödinger equation using the same E1 transition matrix elements of the
LENPIC-N4LO interaction, while we use the trajectories, determined by the
Coulomb potential and the polarization potential with the two extreme values
of the five E1 polarizabilities obtained in this work (Table 1), i.e.,
$\alpha=0.616$ fm3 and $\alpha=0.666$ fm3, for each $\theta_{2}$. We show in
Fig. 13 the corresponding spread of $R(E_{d})$, which stems entirely from
using these two extreme E1 polarizabilities, denoted as “2nd-contr.”. We find
in Fig. 13 that the addition of these two contributions, denoted as “1st+2nd”,
can almost reproduce the total $\Delta R(E_{d})$, which indicates that the
above evaluations to the individual contributions of these two effects appear
to be reasonable.

At low incident energies, since the internal transitions in the deuteron
projectile are insignificant and the polarization potential is weak, we find
in Fig. 13 that both the total spread of $R(E_{d})$ and its two contributions
are negligibly small. Both of these two effects grow as the incident energy
increases. Therefore the spreads of $R(E_{d})$ induced by these two effects
and the total spread become more noticeable at higher incident energies. We
find the spread of $R(E_{d})$ stems almost completely from the second effect
at low incident energies. As the incident energy increases, the second effect
remains dominant, though the first effect is growing in the direction of
becoming comparable.

Table 3:  $\chi^{2}/{\rm data}(\theta_{2})$ and $\chi^{2}_{t}/{\rm data}$ for the five NN interactions presented in Fig. 11 and Fig. 12. See the text for details. |  $\chi^{2}$/data ($\theta_{2}=140^{\circ}$) |  $\chi^{2}$/data ($\theta_{2}=150^{\circ}$) |  $\chi^{2}$/data ($\theta_{2}=160^{\circ}$) |  $\chi_{t}^{2}$/data
---|---|---|---|---
LENPIC-N2LO | $2.703$ | $3.105$ | $1.602$ | $2.470$
LENPIC-N3LO | $2.778$ | $3.619$ | $2.053$ | $2.817$
LENPIC-N4LO | $2.789$ | $3.593$ | $2.036$ | $2.806$
JISP16 | $2.725$ | $2.657$ | $1.210$ | $2.197$
Daejeon16 | $2.837$ | $4.517$ | $2.829$ | $3.395$

To quantify the deviation of our tBF results of $R(E_{d})$ from the
experimental data for $\theta_{2}=140^{\circ},150^{\circ}$ and $160^{\circ}$,
we present in Table 3 (Column $2-4$) the quantity $\chi^{2}/{\rm
data}(\theta_{2})$, which we define as follows

$\displaystyle\chi^{2}/{\rm
data}(\theta_{2})=\frac{1}{7}\sum_{i=1}^{7}\left[\frac{R^{\rm
tBF}_{i}(E_{d})-R^{\rm Exp}_{i}(E_{d})}{\delta R^{\rm
Exp}_{i}(E_{d})}\right]^{2},$ (22)

where $R^{\rm Exp}_{i}(E_{d})$ denotes the central value of the experimental
$R(E_{d})$ in Fig. 11 and Fig. 12. $R^{\rm tBF}_{i}(E_{d})$ represents the tBF
results for each interaction. $\delta R^{\rm Exp}_{i}(E_{d})$ corresponds to
the experimental error in Fig. 11 and Fig. 12. The subscript $i$ runs over the
seven points experimentally available for each $\theta_{2}$ in Fig. 11 and
Fig. 12. To evaluate the entire deviation of our results from the experimental
data, we also present in Table 3 the quantity $\chi_{t}^{2}/{\rm data}$
(Column $5$), defined by

$\displaystyle\chi_{t}^{2}/{\rm
data}=\frac{1}{21}\sum_{j=1}^{21}\left[\frac{R^{\rm tBF}_{j}(E_{d})-R^{\rm
Exp}_{j}(E_{d})}{\delta R^{\rm Exp}_{j}(E_{d})}\right]^{2},$ (23)

where the subscript $j$ runs over all the $21$ points experimentally available
in Fig. 11 and Fig. 12.

We find in Table 3 that $\chi^{2}/{\rm data}(\theta_{2})$ becomes more
sensitive to NN interaction as $\theta_{2}$ increases, which is consistent
with the results in Fig. 13. All the five interactions adopted in this work
provide rather small $\chi_{t}^{2}/{\rm data}$, indicating that the tBF
approach is able to describe reasonably the deuteron scattering on 208Pb at
$E_{d}=3-7$ MeV with these interactions. We expect to improve
$\chi_{t}^{2}/{\rm data}$ by taking into account the correction of the energy
loss to the center of mass motion of the projectile in the future.

##  IV Conclusions and outlook

In this work, we investigated the scattering of the deuteron projectile on the
208Pb target below the Coulomb barrier with the non-perturbative time-
dependent basis function (tBF) approach, employing the LENPIC (-N2LO, -N3LO,
-N4LO), JISP16 and Daejeon16 nucleon-nucleon (NN) interactions. We constructed
the basis representation of the tBF method with the deuteron ground and
discretized scattering states, which are obtained by diagonalizing the
realistic Hamiltonian based on these interactions in a large harmonic
oscillator (HO) basis sufficient to obtain the observables presented here. We
considered all the possible electric-dipole (E1) transitions among these
states during the scattering. We employed the classical trajectory determined
by either the Coulomb potential or the Coulomb potential supplemented with the
polarization potential. For each NN interaction that we employed, we adopted
the E1 polarizability of the deuteron $\alpha$ (in the polarization potential)
obtained with the same interaction. For each of our chosen NN interactions,
the tBF calculations have no adjustable parameters.

Before performing the tBF calculations, we studied the spectra and E1
polarizability of the deuteron projectile, which are both time-independent.
Taking the LENPIC-N4LO interaction with $\omega=20$ MeV as an example, we
found that the deuteron ground state becomes well converged and tends to the
experimental value with increasing $N_{\rm max}$. By comparing the E1
polarizabilities of the deuteron, calculated by the LENPIC (-N2LO, -N3LO,
-N4LO), JISP16 and Daejeon16 interactions with $\omega=20$ MeV, we found that
all these five interactions produce converged results at sufficiently large
$N_{\rm max}$, which are all consistent with the data extracted from
experiments. The spread in the theoretical E1 polarizabilities is about $8\%$.
We also investigated the convergence pattern of the E1 polarizability of the
deuteron with respect to $\omega$ for the LENPIC-N4LO interaction. We found
that results are sensitive to $\omega$ at lower $N_{\rm max}$ though the
converged values for different $\omega$ are the same at large $N_{\rm max}$.

We then investigated the scattering dynamics during the scattering of d+208Pb
at $E_{d}=7$ MeV and $\theta=150^{\circ}$ with the tBF method employing the
LENPIC-N4LO interaction with $\omega=20$ MeV. We found that the asymptotic
value of the probability for the deuteron remaining in its initial state is
well converged for our chosen values of the two truncation parameters $N_{\rm
max}$ and $E_{\rm cut}$. We also found that the E1 allowed states, especially
six of the scattering states, populates dominantly in the early stage of the
scattering and the E1 forbidden states populates significantly in the later
scattering process via the higher-order effects. By considering three
different couplings (Type I, Type II and all the possible couplings in Fig.
7), we showed quantitatively that the Type I coupling, which includes only
transitions between the initial state and its E1 allowed states with six of
them found to dominate the early scattering, remarkably underestimates the
inelastic effects when all couplings were included. On the other hand, the
Type II coupling, which includes the E1 transitions between the initial state
and these six states as well as transitions between these six states and their
E1 allowed states (but none beyond), accounts for around $82\%$ of the
inelastic effects when all couplings are included.

Comparing the time-evolution of the initial state calculated by the standard
tBF method with that of the time-reversed calculation, we illustrated that the
time-reversal symmetry of the theoretical Hamiltonian dynamics in our
semiclassical approximation is well respected in our numerical implementation
of the tBF method. By investigating the oscillations of the probabilities of
these six scattering states mentioned above with the Type I coupling as well
as with all the possible couplings, we established a phase coherence in the
incoming segment representing E1-induced entanglement of each of these six
states with the initial state along with probability oscillations commensurate
with their energy differences. However, we observed phase decoherence for the
results with all the possible couplings, due to the higher-order effects in
the outgoing segment. This was confirmed by showing that phase coherence is
retained in the outgoing segment with the Type I coupling.

Finally, we investigated two additional observables with the tBF method. We
calculated the angular distribution of the elastic cross-section for the
d+208Pb scattering at $E_{d}=5,6$ and $7$ MeV with the LENPIC-N4LO
interaction. We found that the elastic cross-section ratio to the Rutherford
cross-section decreases with increasing incident energy and increasing
scattering angle both with and without the polarization potential. The
polarization potential reduces the elastic cross-section and this effect
increases with the bombarding energy and scattering angle. The reduction of
the elastic cross-section stems dominantly from the polarization potential at
$E_{d}=5$ and $6$ MeV. However, at $E_{d}=7$ MeV, the polarization potential
contributes dominantly to the reduction of the elastic cross-section for
scattering angles below $140^{\circ}$ while the internal E1 transitions in the
deuteron projectile become dominant above $140^{\circ}$.

We also calculated the elastic cross-section ratios $R(E_{d})$ for
$(\theta_{1},\theta_{2})=(60^{\circ},140^{\circ})$, $(60^{\circ},150^{\circ})$
and $(60^{\circ},160^{\circ})$ with the tBF method employing the LENPIC
(-N2LO, -N3LO, -N4LO), JISP16 and Daejeon16 interactions. By considering the
internal E1 transitions in the deuteron projectile and the polarization
potential during the scattering, each of these five interactions is able to
reproduce the experimental $R(E_{d})$ at $E_{d}=3-6$ MeV. We found that
$R(E_{d})$ falls below the experimental data at $E_{d}=7$ MeV for
$\theta_{2}=150^{\circ}$ and $160^{\circ}$ which we attributed to neglecting
the effect of the energy loss (due to the internal excitations) of the
projectile on its center of mass motion. The tBF calculation of $R(E_{d})$
with $\theta_{2}=140^{\circ}$ at $E_{d}=7$ MeV is close to experiment since
the energy loss is less significant compared to $\theta_{2}=150^{\circ}$ and
$160^{\circ}$. By analysing the spread of $R(E_{d})$ induced by these five
interactions, we found that $R(E_{d})$ becomes more sensitive to NN
interactions with increasing either the deuteron incident energy or the
scattering angle $\theta_{2}$. In particular, the sensitivity of $R(E_{d})$ to
NN interactions stems almost completely from the dependence of the E1
polarizability on the NN interactions at low incident energies. Its
contribution to the spread remains dominant in contrast to the contribution of
the dependence of the E1 transition matrix elements on the NN interactions,
though both of these two contributions grow with increasing incident energy.

By analysing $\chi^{2}/{\rm data}(\theta_{2})$ and $\chi^{2}_{t}/{\rm data}$,
we found the interactions adopted in this work provide reasonable descriptions
to the deuteron scattering on 208Pb at $E_{d}=3-7$ MeV without any adjustable
parameters. We anticipate improving the $\chi^{2}_{t}/{\rm data}$ by taking
into account the correction of the energy loss to the center of mass motion of
the projectile in future improvements to the tBF approach.

It should be noted that we adopted a semiclassical approximation in the
current tBF method, where the center of mass motion of the projectile was
calculated via classical mechanics. We considered the contributions of the E1
effects for the deuteron’s internal and center of mass degrees of freedom
separately. We took into account the E1 transitions in the internal degree of
freedom of the deuteron projectile by solving the time-dependent Schrödinger
equation. We approximated the effect of the E1 polarization on the center of
mass motion of the projectile by a polarization potential.

In the future, we will improve the tBF method to extend the range of its
applications. (1) We will extend the tBF method to a fully quantum mechanical
framework, within which the center of mass motion and the internal motion of
the projectile will be considered coherently. The energy loss omitted in this
work will then be resolved naturally. (2) We are extending the tBF method to
investigate nuclear reactions involving larger projectile nuclei, where we
will obtain the nuclear structure information with ab initio approaches, e.g.,
no-core shell model. (3) We will take into account the vacuum polarization
potential in the tBF method, which is expected to enhance the elastic cross-
section Baur:1977nso  while it has no appreciable contribution to $R(E_{d})$
Rodning:1982zz ; Moro:1999fcl . (4) In the tBF method we will introduce the
chiral two-body charge operator, which first appears at N3LO, as a higher-
order correction to the present one-body E1 operator Filin:2019eoe . We will
also investigate the contribution of other electromagnetic transitions, such
as E2 and M1, to scattering observables. (5) We will extend the tBF method to
higher bombarding energies by considering the strong interaction between the
projectile and the target Satchler:1979ni ; Moffa:1977zz .

## Acknowledgments

We acknowledge helpful discussions with Andrey Shirokov, Pieter Maris, Shiplu
Sarker, Robert A. M. Basili, Antonio M. Moro, Gerhard Baur, Zhigang Xiao, Li
Ou, William Lynch and Betty Tsang. This work is supported in part by the US
Department of Energy (DOE) under Grant Nos. DE-FG02-87ER40371 and DE-
SC00018223. A portion of the computational resources are provided by the
National Energy Research Scientific Computing Center (NERSC), which is
supported by the US DOE Office of Science. Xingbo Zhao is supported by new
faculty startup funding by the Institute of Modern Physics, Chinese Academy of
Sciences, by Key Research Program of Frontier Sciences, Chinese Academy of
Sciences, Grant No. ZDB-SLY-7020, by the Natural Science Foundation of Gansu
Province, China, Grant No. 20JR10RA067, by the Foundation for Key Talents of
Gansu Province and by the Central Funds Guiding the Local Science and
Technology Development of Gansu Province. Peng Yin and Wei Zuo are supported
by the National Natural Science Foundation of China (Grant Nos. 11975282,
11705240, 11435014), the Strategic Priority Research Program of Chinese
Academy of Sciences, Grant No. XDB34000000 and the Key Research Program of the
Chinese Academy of Sciences under Grant No. XDPB15. This work is also
partially supported by the CUSTIPEN (China-U.S. Theory Institute for Physics
with Exotic Nuclei) funded by the U.S. Department of Energy, office of Science
under Grant No. DE-SC0009971.

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