# Nuclear giant quadruple resonance within transport approach and its
constraint on nucleon effective mass

Yi-Dan Song(宋一丹)  Key Laboratory of Nuclear Physics and Ion-beam Application
(MOE), Fudan University, Shanghai $200433$, China  Rui Wang(王睿)
<EMAIL_ADDRESS>Shanghai Institute of Applied Physics, Chinese Academy of
Sciences, Shanghai $201800$, China  Zhen Zhang(张振)  Sino-French Institute of
Nuclear Engineering and Technology, Sun Yat-sen University, Zhuhai $519082$,
China  Yu-Gang Ma(马余刚<EMAIL_ADDRESS>Key Laboratory of Nuclear
Physics and Ion-beam Application (MOE), Fudan University, Shanghai $200433$,
China  Shanghai Institute of Applied Physics, Chinese Academy of Sciences,
Shanghai $201800$, China

###### Abstract

We study the nuclear iso-scalar giant quadruple resonance (ISGQR) based on the
Boltzmann-Uehling-Uhlenbeck (BUU) transport equation. The mean-field part of
the BUU equation is described by the Skyrme nucleon-nucleon effective
interaction, and its collision term, which embodies the two-particle-two-hole
($2$p-$2$h) correlation, is implemented through the stochastic approach. We
find that the width of ISGQR for heavy nuclei is exhausted dominated by
collisional damping, which is incorporated into the BUU equation through its
collision term, and it can be well reproduced through employing a proper in-
medium nucleon-nucleon cross section. Based on further Vlasov and BUU
calculations with a number of representative Skyrme interactions, the iso-
scalar nucleon effective mass at saturation density is extracted respectively
as $m^{*}_{s,0}/m$ $=$ $0.83\pm 0.04$ and $m^{*}_{s,0}/m$ $=$ $0.82\pm 0.03$
from the measured excitation energy $E_{x}$ of the ISGQR of
$\isotope[208]{Pb}$. The small discrepancy between the two constraints
indicates the negligible role of $2$p-$2$h correlation in constraining
$m_{s,0}^{*}$ with the ISGQR excitation energy.

##  I Introduction

Nuclear giant resonances are a global feature of nuclei that arises from the
collective motion of their constituent nucleons Bertsch et al. (1983); Pitthan
and Walcher (1971); Bertrand (1981); Youngblood et al. (1981); Knöpfle et al.
(1981); Buenerd (1984); Youngblood et al. (2004); Gupta et al. (2018);
Monrozeau et al. (2008); He et al. (2014); Huang and Ma (2021); Dang et al.
(2019a, b). Following the iso-vector giant dipole resonance (IVGDR) Bertsch et
al. (1983), the iso-scalar giant quadruple resonance (ISGQR), discovered in
the early $1970$s Pitthan and Walcher (1971), was the second fundamental mode
of nuclear giant resonances. Systematic studies on giant resonance for
different nuclear masses have been carried out by, ($d$, $d^{\prime}$) or
($\alpha$, $\alpha^{\prime}$) reactions Bertrand (1981); Youngblood et al.
(1981); Knöpfle et al. (1981); Buenerd (1984); Youngblood et al. (2004); Gupta
et al. (2018); Monrozeau et al. (2008), which preferentially excites iso-
scalar transitions due to their zero isospin nature. One of the important
features of the nuclear ISGQR is that its excitation energy $E_{x}$ is related
to the nucleon effective mass in symmetric nuclear matter, or iso-scalar
nucleon effective mass, $m_{s}^{*}(\rho)$ Blaizot (1980). It is well known
that the strong exchange effects of nuclear interaction lead to a momentum
dependence of nucleon potential in nuclear medium Wiringa (1988). The nucleon
effective mass is usually employed to characterize this momentum-dependence of
the single-nucleon potential, and it plays an important role in the dynamics
of heavy-ion collisions Li et al. (2008); Zhang and Su (2020); Li et al.
(2016); Wei et al. (2020). Apart from this, the isospin splitting of the
nucleon effective masses, which is related to the $m_{s}^{*}$ and the iso-
vector nucleon effective mass $m_{v}^{*}$, has strong influences on various
quantities or processes, e.g., the properties of mirror nuclei Nolen and
Schiffer (1969), transport properties of asymmetric nuclear matter Li and Chen
(2005); Li et al. (2004), neutrino emission in neutron stars Baldo et al.
(2014). Therefore, how to constrain accurately the nucleon effective mass has
been a hot topic in nuclear physics, and one may refer to Ref. Li et al.
(2018) for a recent review.

Since the excitation energy $E_{x}$ of the nuclear ISGQR is an effective probe
of the $m_{s}^{*}$, several methods have been employed to calculate the
$E_{x}$ of the ISGQR and consequently constrain the $m_{s}^{*}$, e.g., the
(quasi-)random-phase approximation (RPA) approach Bohigas et al. (1979); Roca-
Maza et al. (2013); Yoshida and Nakatsukasa (2013); Zhang and Chen (2016);
Yüksel et al. (2018); Bonasera et al. (2018); Zhang et al. (2021), the
macroscopic Langevin equation Su et al. (2020), and dynamical approaches like
transport models Kong et al. (2017); Xu and Qin (2020). One of the advantages
of the transport model is that the beyond mean-field effects can be easily
included via the collision term. The Boltzmann-Uehling-Uhlenbeck (BUU)
equation Bertsch (1988) is one of the main transport models that has been
widely applied to the study of heavy-ion collisions. On the one hand, with the
absence of the collision term, the BUU equation reduces to the Vlasov
equation, which can be regarded as a semi-classical approximation of the mean-
field level time-dependent Hartree-Fock (density functional) approach
Nakatsukasa et al. (2016); Simenel and Umar (2018); Stevenson and Barton
(2019). On the other hand, the nucleon-nucleon scatterings in the BUU equation
embodies beyond-mean-field correlations, or the two-particle-two-hole
($2$p-$2$h) correlation, and it incorporates two-body dissipation into the
evolution. It has been shown that the nucleon-nucleon scatterings are
essential to describe the experimental full width at half maximum $\Gamma$ of
IVGDR for heavy nuclei Wang et al. (2020a).

Although there already exist several works that aim to study the ISGQR and its
constraint on the $m_{s}^{*}$ using the BUU transport equation Kong et al.
(2017); Xu and Qin (2020), they suffer from the problems of unstable nuclear
ground-state evolution and inaccurate treatment of the collision term,
especially the Pauli blocking, which has been shown to have significant effect
on the width $\Gamma$ of nuclear giant resonances Wang et al. (2020a, b). In
the present work, to study the ISGQR of finite nuclei, we employ a lattice
Hamiltonian method Lenk and Pandharipande (1989) to solve the BUU equation,
and its nucleon-nucleon collision term is implemented through a full-ensemble
stochastic collision approach Wang et al. (2019, 2020a, 2020b). The ground-
state of a nucleus is obtained by varying the total energy with respect to the
nucleon density distribution based on the same Hamiltonian that governs the
equations of motion of nucleons. Besides that, the present framework of
solving the BUU equation is implemented numerically through GPU (Graphics
Processing Unit) high-performance parallel computing Ruetsch and Fatica
(2013), which enables us to employ huge amount of test particles when solving
the BUU equation, and increases the numerical efficiency and accuracy
profoundly. The above features help to improve the ground-state stability and
the accuracy of the collision term significantly, and make it possible to
solve the BUU equation precisely. Therefore, one might describe through the
BUU equation the time evolution of the ISGQR and obtain its strength function
$S(E)$, and subsequently its excitation energy $E_{x}$ and width $\Gamma$.

The present article is organized as follows. In Sec. II, we introduce the
basic concept of solving the BUU equation, and the necessary formalism of
nucleon effective masses in Skyrme-Hartree-Fock (SHF) approach, as well as how
to study nuclear collective motions in the transport approach. In Sec. III, we
study the ISGQR of [208]Pb and its strength function within the BUU equation,
and present a correlation between its excitation energy $E_{x}$ and iso-scalar
nucleon effective mass $m_{s}^{*}$ through the BUU calculation from various
Skyrme interactions, based on which we constrain the $m_{s}^{*}$. Finally, we
summarize the present work in Sec. IV.

##  II Method

In the present study, the ISGQR of finite nuclei is studied within the
framework of the BUU transport equation. In this section, we briefly present
how we solve the BUU equation effectively, i.e., the lattice Hamiltonian
method adopted for the mean-field evolution and the stochastic method employed
for the collision term, and how to describe the nuclear collective motions
with Wigner function $f(\vec{r},\vec{p})$. Since we employ the Skyrme
effective nucleon-nucleon interaction to describe the mean-field part of the
BUU equation, we also introduce several basic features of the Skyrme
interaction and nucleon effective masses within the Skyrme interaction.

###  II.1 Boltzmann-Uehling-Uhlenbeck transport equation

The BUU transport equation describes the time evolution of the Wigner function
$f(\vec{r},\vec{p})$. If we consider a momentum-dependent potential
$U(\vec{r},\vec{p})$, the BUU equation can be written as,

$\frac{\partial{f}}{\partial{t}}+\frac{\vec{p}}{E}\cdot\triangledown_{\vec{r}}f+\triangledown_{\vec{p}}U(\vec{r},\vec{p})\cdot\triangledown_{\vec{r}}f-\triangledown_{\vec{r}}U(\vec{r},\vec{p})\cdot\triangledown_{\vec{p}}f=I_{c}.$
(1)

In the above equation, the left hand side describes the time evolution of the
$f(\vec{r},\vec{p})$ in the mean-field $U(\vec{r},\vec{p})$, while the
collision integral $I_{c}$ is responsible for part of the beyond mean-field
many-body correlation. The general form of $I_{c}$ is

$\begin{split}I_{c}=&-g\int\frac{d^{3}p_{2}}{(2\pi\hbar)^{3}}\frac{d^{3}p_{3}}{(2\pi\hbar)^{3}}\frac{d^{3}p_{4}}{(2\pi\hbar)^{3}}\\\
&\times|\mathcal{M}_{12\to
34}|^{2}(2\pi)^{4}\delta^{4}(p_{1}+p_{2}-p_{3}-p_{4})\\\
&\times\big{[}ff_{2}(1-f_{3})(1-f_{4})-f_{3}f_{4}(1-f)(1-f_{2})\big{]},\end{split}$
(2)

where $f_{i}$ is short for $f(\vec{r},\vec{p}_{i})$, $g$ is a degeneracy
factor, and $\mathcal{M}_{12\to 34}$ is the in-medium transition matrix
element. The $1-f$ in $I_{c}$ is added to take the Pauli principle into
account due to nucleons’ Fermion statistics.

The lattice Hamiltonian method Lenk and Pandharipande (1989) is adopted to
solve the mean-field evolution of the BUU equation. We first mimic $f$ by a
large number of test nucleons Wong (1982),

$f(\vec{r},\vec{p},t)=\frac{1}{g}\frac{(2\pi\hbar)^{3}}{N_{E}}\sum^{AN_{E}}_{i}S[\vec{r}_{i}(t)-\vec{r}~{}]\delta[\vec{p}_{i}(t)-\vec{p}~{}],$
(3)

with $A$ to be the mass number of the system and $N_{E}$ the number of
parallel ensembles (or the number of test particles in some literature). A
form factor $S$ in coordinate space is introduced to modify the relation
between test nucleons and $f$. Based on the $f$ in Eq. (3), we then calculate
the total Hamiltonian $H$ approximated by the lattice Hamiltonian $H_{L}$,

$H=\int\mathcal{H}(\vec{r})d\vec{r}\thickapprox
l_{x}l_{y}l_{z}\sum_{\alpha}\mathcal{H}(\vec{r}_{\alpha})\equiv H_{L},$ (4)

to obtain the equations of motion of test nucleons and subsequently other
physical quantities. In the above formula, $\vec{r}_{\alpha}$ represents
certain lattice site, and $l_{x}$, $l_{y}$, $l_{z}$ are lattice spacing.

We adopt the stochastic collision approach Danielewicz and Bertsch (1991); Xu
and Greiner (2005) to deal with the collision term $I_{c}$. The scattering
probability of a scattering event, involving test nucleons $i$ and $j$ in a
time interval $\Delta t$, can be calculated directly through the collision
term by substituting the $f$ in Eq. (3) into Eq. (2). It reads

$P_{ij}=\upsilon_{rel}\sigma^{*}_{\rm
NN}S(\vec{r}_{i}-\vec{r}_{\alpha})S(\vec{r}_{j}-\vec{r}_{\alpha})l_{x}l_{y}l_{z}\Delta
t,$ (5)

where $\upsilon_{rel}$ is the relative velocity between the two test nucleons,
and $\sigma^{*}_{\rm NN}$ is the in-medium nucleon-nucleon cross section. Note
its difference with the probability in normal stochastic approach Xu and
Greiner (2005), due to the existence of the form factor $S$ in the present
study. If the collision of the $i$th and $j$th test nucleons happens, we
calculate a Pauli blocking factor
$\big{[}1-f(\vec{r}_{\alpha},\vec{p}_{i}^{\prime})\big{]}\big{[}1-f(\vec{r}_{\alpha},\vec{p}_{j}^{\prime})\big{]}$
according to their final state momentum $\vec{p}_{i}^{\prime}$ and
$\vec{p}_{j}^{\prime}$, to determine whether the collision is blocked by the
Pauli principle.

We have omitted the isospin degree of freedom in the above context for
clarity. The present lattice BUU framework of solving the BUU equation has
been applied successfully to the giant resonances of heavy nuclei Wang et al.
(2019, 2020a). One can find more detailed descriptions about the present
framework, e.g., the equations of motion of test nucleons, in Ref. Wang et al.
(2020b).

###  II.2 Nucleon effective mass in Skyrme-Hartree-Fock approach

As mentioned above, we employ the SHF approach Vautherin and Brink (1972);
Chabanat et al. (1997, 1998) to describe the mean-field in the BUU equation.
The standard parametrization form of the Skyrme interaction Chabanat et al.
(1998, 1997) is,

$\begin{split}\upsilon_{12}&=t_{0}(1+x_{0}{\hat{P}_{\sigma}})\delta(\vec{r}_{1}-{\vec{r}_{2}})\\\
&+\frac{1}{6}t_{3}(1+x_{3}{\hat{P}_{\sigma}})\rho^{\alpha}(\vec{R})\delta(\vec{r}_{1}-{\vec{r}_{2}})\\\
&+t_{1}(1+x_{1}{\hat{P}_{\sigma}})\frac{1}{2}\Big{[}\delta(\vec{r}_{1}-{\vec{r}_{2}}){\hat{\vec{k}}^{{}^{\prime}2}}+\hat{\vec{k}}^{2}\delta(\vec{r}_{1}-{\vec{r}_{2}})\Big{]}\\\
&+t_{2}(1+x_{2}{\hat{P}_{\sigma}})\hat{\vec{k}}^{{}^{\prime}}\cdot\delta(\vec{r}_{1}-{\vec{r}_{2}})\hat{\vec{k}}\\\
&+iW_{0}(\hat{\vec{\sigma}}_{1}+\hat{\vec{\sigma}}_{2})\cdot\hat{\vec{k}}^{{}^{\prime}}\times\delta(\vec{r}_{1}-{\vec{r}_{2}})\hat{\vec{k}},\end{split}$
(6)

with the $\vec{R}=\frac{1}{2}(\vec{r}_{1}+{\vec{r}_{2}})$. The relative
momentum operators
$\hat{\vec{k}}=(\hat{\vec{\nabla}}_{1}-\hat{\vec{\nabla}}_{2})/2i$ and
$\hat{\vec{k}}^{{}^{\prime}}=-(\hat{\vec{\nabla}}_{1}-\hat{\vec{\nabla}}_{2})/2i$
act on the right and left, respectively. The $\hat{\vec{\sigma}}_{i}$ and
$\hat{\vec{P}}_{\sigma}$ are the Pauli spin matrices and the spin exchange
operator, respectively. The $t_{0}$ $-$ $t_{3}$, $x_{0}$ $-$ $x_{3}$, $\alpha$
are Skyrme parameters. In Eq. (6), the first term is the local density term,
and the second term is the density-dependent term, which is used to describe
effectively the three-body interaction between nucleons. The third and fourth
terms are momentum dependent terms, and the fifth term is spin-orbit coupling
term which is omitted in the present study due to the assumption of spin
saturation.

We can include in Eq. (6) additional high-order momentum dependent terms, to
form the Skyrme pseudopotential Carlsson et al. (2008); Raimondi et al.
(2011). The Skyrme pseudopotential with next-to-next-to-next-to-leading order
(N$3$LO) momentum dependent terms can reproduce the empirical nuclear optical
potential up to about 1 GeV in kinetic energy Wang et al. (2018).

In the SHF approach Vautherin and Brink (1972); Chabanat et al. (1997, 1998),
the nucleon effective mass $m^{*}_{q}/m$ for a nucleon $q$ ($n$ or $p$) in
asymmetric nuclear matter with density $\rho$ can be obtained as follow,

$\begin{split}\frac{\hbar^{2}}{2m^{*}_{q}(\rho,\delta)}&=\frac{\hbar^{2}}{2m}+\frac{1}{4}t_{1}\Big{[}(1+\frac{1}{2}x_{1})\rho-(\frac{1}{2}+x_{1})\rho_{q}\Big{]}\\\
&+\frac{1}{4}t_{2}\Big{[}(1+\frac{1}{2}x_{2})\rho+(\frac{1}{2}+x_{2})\rho_{q}\Big{]},\end{split}$
(7)

with isospin asymmetry $\delta=(\rho_{n}-\rho_{p})/(\rho_{n}+\rho_{p})$. When
setting $\rho_{q}=\rho/2$, we obtain the iso-scalar nucleon effective mass

$\frac{\hbar^{2}}{2m^{*}_{s}(\rho)}=\frac{\hbar^{2}}{2m}+\frac{3}{16}t_{1}\rho+\frac{1}{16}t_{2}(5+4x_{2})\rho.$
(8)

It’s expression in the Skyrme pseudopotential can be obtained
straightforwardly.

###  II.3 Nuclear collective motions in the Wigner representation

One of the key quantities when studying the collective motions is the
expectation value of the excitation operator $\hat{Q}$. In the Wigner
representation Carruthers and Zachariasen (1983); Bonasera et al. (1993),
i.e., the variable in the equation of motion has been transformed from wave
function or density matrix to $f(\vec{r},\vec{p})$, we should express the
expectation $\langle{\hat{Q}}\rangle$ in terms of $f(\vec{r},\vec{p})$. The
$\hat{Q}$ is usually considered as a one-body operator, which means that it
can be expressed as the sum of single-nucleon operator ${\hat{q}_{i}}$ acting
on each nucleon. The expectation of $\hat{Q}$ can then be expressed as

$\displaystyle\langle{\hat{Q}}\rangle\equiv$
$\displaystyle\langle\phi|{\hat{Q}}|\phi\rangle={\sum^{A}_{i}\langle\phi|{\hat{q}_{i}}|\phi\rangle}$
$\displaystyle=$
$\displaystyle\sum^{A}_{i}\int{\langle\phi|\vec{r}_{1}...\vec{r}_{N}\rangle\langle\vec{r}_{1}...\vec{r}_{N}|{\hat{q}_{i}}|{\vec{r}_{1}}^{\prime}...\vec{r}_{N}^{\prime}}\rangle$
$\displaystyle\times\langle{\vec{r}_{1}^{\prime}}...{\vec{r}_{N}^{\prime}}|\phi\rangle{d^{3}\vec{r}_{1}}...{d^{3}\vec{r}_{N}}{d^{3}\vec{r}_{1}^{\prime}}...{d^{3}\vec{r}_{N}^{\prime}}$
$\displaystyle=$
$\displaystyle\int{\rho(\vec{r}_{i}^{\prime},\vec{r}_{i})\langle\vec{r}_{i}|{\hat{q}_{i}}|{\vec{r}_{i}^{\prime}}\rangle
d^{3}\vec{r}_{i}{d^{3}\vec{r}_{i}^{\prime}}},$ (9)

where $|\vec{r}_{i}\rangle$ and $|\vec{r}^{\prime}_{i}\rangle$ are coordinate
eigen-states, and $\rho({\vec{r}_{i}^{\prime}},\vec{r}_{i})$ is the one-body
density matrix. By transforming the relative coordinate
$\vec{r}_{i}-\vec{r}_{i}^{\prime}$ in Eq. (II.3) to momentum space, we obtain
$\langle{\hat{Q}}\rangle$ in terms of the Wigner function
$f(\vec{r},\vec{p})$, i.e.,

$\langle{\hat{Q}}\rangle=\int{f(\vec{r},\vec{p})q(\vec{r},\vec{p})d^{3}\vec{r}d^{3}\vec{p}},$
(10)

where the $q(\vec{r},\vec{p})$ is the Wigner transform of
$\langle\vec{r}_{i}|{\hat{q}_{i}}|{\vec{r}_{i}^{\prime}}\rangle$ and $\vec{r}$
$=$ $(\vec{r}_{i}-\vec{r}_{i}^{\prime})/2$. Therefore, the expectation of
$\hat{Q}$ can be calculated within the framework of BUU equation. The specific
form of the excitation operator $\hat{Q}$ for iso-scalar quadruple mode will
be given in Sec. III.1.

##  III Results and discussion

###  III.1 Strength function of ISGQR

In transport approach, an excited nucleus can be generated by changing the
$f(\vec{r},\vec{p})$ of a ground state nucleus according to the form of the
excitation operator. This is equivalent to changing the initial coordinates
and momenta of test nucleons if we adopt the test particle ansatz. For the
coordinates $\vec{r}_{i}$ and momenta $\vec{p}_{i}$ of test nucleons of a
ground state nucleus, since we deals with a semi-classical transport equation,
they are obtained based on the Thomas-Fermi approach. In the Thomas-Fermi
approach, the radial density distribution $\rho_{\tau}(r)$ of a ground state
nucleus is obtained by varying the total Hamiltonian with respect to it. Since
the same Hamiltonian also governs the evolution of the test nucleons, a very
stable ground state evolution can be obtained in the transport calculation.
The initial coordinates $\vec{r}_{i}$ of test nucleons are generated according
to the obtained radial density distribution, while their initial momenta
$\vec{p}_{i}$ are generated from a zero-temperature Fermi distribution with
the Fermi momentum given as $p^{\rm F}_{\tau}$ $=$
$\hbar\big{[}3\pi^{2}\rho_{\tau}(r)\big{]}^{1/3}$.

For the ISGQR, the excitation operator ${{\hat{Q}}_{\rm ISQ}}$ can be written
as the sum of single-nucleon operator ${{\hat{q}}_{\rm ISQ}}$,

${\hat{Q}_{\rm ISQ}}=\sum^{A}_{\rm i=1}\hat{q}_{\rm
ISQ}={\sum^{A}_{i=1}}\frac{1}{A}\sqrt{\frac{5}{16\pi}}(3{\hat{z}_{i}}^{2}-\hat{r}_{i}^{2}),$
(11)

where $\hat{r}_{i}$ and $\hat{z}_{i}$ are the coordinate operator of the
$i$-th nucleon and its $3$-rd component, respectively, and $A$ is the mass
number of the nucleus. For ${{\hat{q}}_{\rm ISQ}}$, its corresponding ${q_{\rm
ISQ}}(\vec{r},\vec{p})$ in Eq. (10) is expressed as

${q_{\rm
ISQ}}(\vec{r},\vec{p})=\frac{1}{A}\sqrt{\frac{5}{16\pi}}(3z^{2}-\vec{r}\,^{2}).$
(12)

Note here the coordinate operators have been replaced by their eigenvalues.

To obtain the iso-scalar quadruple mode, we can either change the initial
$\vec{r}_{i}$ and $\vec{p}_{i}$ of test nucleons according to

$\vec{p}_{i}\rightarrow\left\\{\begin{aligned}
{p_{xi}}+2{\lambda}\sqrt{\frac{5}{16\pi}}\frac{x_{i}}{A}\\\
{p_{yi}}+2{\lambda}\sqrt{\frac{5}{16\pi}}\frac{y_{i}}{A}\\\
{p_{zi}}-4{\lambda}\sqrt{\frac{5}{16\pi}}\frac{z_{i}}{A},\\\
\end{aligned}\right.$ (13)

or according to

$\vec{r}_{i}\rightarrow\left\\{\begin{aligned} x_{i}-x_{i}\lambda\\\
y_{i}-y_{i}\lambda\\\
z_{i}+2z_{i}\lambda\end{aligned}\right.\qquad\vec{p}_{i}\rightarrow\left\\{\begin{aligned}
p_{xi}+2p_{xi}\lambda\\\ p_{yi}+2p_{yi}\lambda\\\
p_{zi}-p_{zi}\lambda,\end{aligned}\right.$ (14)

with $\lambda$ to be a small quantity. The former scenario corresponds to
exciting the nucleus at $t$ $=$ $t_{0}$, i.e., $\hat{H}_{\rm ex}$ $\propto$
$\lambda\hat{Q}_{\rm ISQ}\delta(t-t_{0})$, which is a common practice within
the linear response theory Fetter and Walecka (1971). The latter scenario
corresponds to starting the nucleus at the ground state with the presence of
the excitation operator Patra et al. (2002), and then remove the excitation at
$t$ $=$ $t_{0}$, i.e., $\hat{H}_{\rm ex}$ $\propto$ $\lambda\hat{Q}_{\rm
ISQ}\theta(t_{0}-t)$. We will examine both of these two scenarios when
calculating the ISGQR.

After exciting the nucleus to a certain mode, we calculate the time evolution
of the expectation of the excitation operator $\langle\hat{Q}\rangle(t)$
within the BUU equation through Eq. (10), and subsequently obtain the strength
function $S(E)$ for that mode. In the following we calculate the ISGQR of
[208]Pb with the Skyrme interaction SkM* Bartel et al. (1982). We consider two
cases, i.e., the pure Vlasov calculation, in which only the mean-field, i.e.,
one-particle-one-hole ($1$p-$1$h) correlation is taken into account, and the
full BUU calculation, with the nucleon-nucleon collision term reflects
effectively additional $2$p-$2$h correlation. This nucleon-nucleon collision
term is responsible for the collisional damping of the nuclear collective
motions, which is indispensable for describing the width of the latter Wang et
al. (2020a). It has been shown that a large in-medium reduction of nucleon-
nucleon cross section is essential to reproduce the experimental width of
IVGDR of [208]Pb Wang et al. (2020a). In the full BUU calculation, for
simplicity, we employ a constant in-medium correction, i.e., $\sigma_{\rm
NN}^{*}$ $=$ $0.60\sigma_{\rm NN}^{\rm free}$. The free nucleon-nucleon cross
section $\sigma_{\rm NN}^{\rm free}$ is parameterized based on the
experimental nucleon-nucleon scattering data as in Ref. Cugnon et al. (1996).
This in-medium nucleon-nucleon cross section can reproduce the weighted
average of the experimental width $\Gamma$ $=$ $3.0\pm 0.1$ Roca-Maza et al.
(2013) of ISGQR of [208]Pb. In the present study, we choose lattice spacing
$l_{x}$ $=$ $l_{y}$ $=$ $l_{z}$ $=$ $0.5~{}\rm fm$, time step $\Delta{t}=\
0.5\,\text{fm}/c$, and $N_{E}$ $=$ $10000$, to ensure the convergence of the
numerical results.

Figure 1: The time evolution of the $\Delta\langle\hat{Q}_{\rm ISQ}\rangle$ of
$\isotope[208]{Pb}$ in the ground-state and after a perturbation excitation of
${\hat{H}}_{ex}(t)$ $\propto$ ${\lambda}{\hat{Q}}_{\rm ISQ}{\delta(t-t_{0})}$
with $\lambda=0.1\ {\rm GeV}\cdot fm^{-1}/c$ based on Eq. (13). The results
correspond to the ground-state, the pure Vlasov and the full BUU calculation,
respectively, which are calculated with the Skyrme interaction SkM*.

In Fig. 1, we show the time evolution of the $\Delta\langle{\hat{Q}}_{\rm
ISQ}\rangle$ through Eq. (13) with the Skyrme interaction SKM∗. The black and
red solid lines represent the $\Delta\langle{\hat{Q}}_{\rm ISQ}\rangle$ for
pure Vlasov and full BUU calculations, respectively. In both cases, the
expectation of the excited nucleus $\langle\phi|{\hat{Q}}_{\rm
ISQ}|\phi\rangle$ has been subtracted by the that of the ground state nucleus
$\langle 0|{\hat{Q}}_{\rm ISQ}|0\rangle$. The latter is expected to be zero,
and it is shown in Fig. 1 with the blue dash line. We notice from the figure
that in the Vlasov calculation where only the mean-field ($1$p-$1$h) is
included, the $\Delta\langle{\hat{Q}}_{\rm ISQ}\rangle$ shows a regular
oscillation, with the amplitude of the oscillation almost unchanged.
Meanwhile, in the full BUU calculation where nucleon-nucleon scatterings are
also included, the oscillation of the $\Delta\langle{\hat{Q}}_{\rm
ISQ}\rangle$ damps very quickly and return to zero eventually. The above
feature indicates that the collisional damping, which is a two-body
dissipation, dominates the ISGQR width of heavy nuclei, while the Landau
damping, the one-body dissipation that relates only to the mean-field, has
negligible effect on it. This is different from the IVGDR case where the
Landau damping is responsible for almost half of the width Wang et al.
(2020a). Such a feature suggests the ISGQR to be an ideal site for
constraining the in-medium nucleon-nucleon cross section.

Figure 2:  Same as Fig. 1 but for perturbation excitation of
${\hat{H}}_{ex}(t)$ $\propto$ ${\lambda}{\hat{Q}_{\rm ISQ}}{\theta(t_{0}-t)}$
with $\lambda=0.01\ {\rm GeV}\cdot fm^{-1}/c$.

Similarly, we show the time evolution of the $\Delta\langle{\hat{Q}}_{\rm
ISQ}\rangle$ of the $\isotope[208]{Pb}$ after a perturbative iso-scalar
quadruple excitation through Eq. (14). Note that since in this case, the
nucleus starts at the ground state with the presence of the excitation
operator $\hat{Q}_{\rm ISQ}$, the initial $\Delta\langle\hat{Q}_{\rm
ISQ}\rangle$ for the pure Vlasov and full BUU calculations, shown as solid
line in the figure, have finite values. Similar features with Fig. 1 about the
damping of the $\Delta\langle{\hat{Q}}_{\rm ISQ}\rangle(t)$ for Vlasov and BUU
calculations can be observed in Fig. 2. It is obviously seen that the result
of the Vlasov calculation has similar oscillation with Fig. 1, but more
regular. One explanation to this phenomenon is that if one excites the nucleus
through Eq. (13), it is easier for the test nucleons at the surface of the
nucleus along $x$ and $y$ axis to escape, whereas in the excitation scenario
in Eq. (14) this effect is suppressed due to the deformation in the initial
coordinate space. However, by adding the number of $N_{E}$, the escape of the
test nucleons for the first excitation scenario can be diminished.

Based on the obtained $\Delta\langle{\hat{Q}}_{\rm ISQ}\rangle(t)$, the
strength function $S(E)$ for the iso-scalar quadruple mode can be calculated
through the Fourier transform. If one excites the nucleus through Eq. (13),
the $S(E)$ can be obtained from

$S(E)=-\frac{1}{\pi\lambda}{\int^{\infty}_{0}}{dt\Delta\langle{\hat{Q}}\rangle(t){\rm
sin}\frac{Et}{\hbar}}.$ (15)

For the excitation scenario in Eq. (14), analogous expression can be obtained
by replacing the sine in the above equation with the cosine.

We exhibit in Fig. 3 the full BUU calculations of the strength function $S(E)$
for the iso-scalar quadruple excitation of [208]Pb with two standard Skyrme
interactions, SkM*, and KDE Agrawal et al. (2005) and one Skyrme
pseudopotential SP$6$h Wang et al. (2018). For KDE and SP$6$h, we excite the
nucleus through Eq. (14), shown as black and blue dashed lines, respectively.
For SkM*, apart from the result through Eq. (14), represented by the red
dashed line, the result through Eq. (13) is also adopted and the result is
shown as the red solid line. The results with SkM* indicate that the two
different excitation scenarios expressed in Eq. (13) and Eq. (14) give similar
peak energies and width, as well as the shape of the strength function. In the
following, we employ the excitation scenario in Eq. (14) to generate the iso-
scalar quadruple mode of the nucleus, since it is easier to obtain a regular
oscillation of $\Delta\langle{\hat{Q}}_{\rm ISQ}\rangle$. The gray hatched
band in Fig. 3 at $E$ $=$ $10.9\pm 0.1~{}\rm MeV$ represents the weighted
average of different experimental values of the excitation energy of ISGQR of
[208]Pb Roca-Maza et al. (2013). This value can be reproduced by SkM*, whose
$E_{x}$ $=$ $10.86~{}\rm MeV$. Different peak energies of different Skyrme
interactions actually reflect their different iso-scalar single particle
behavior, i.e., $m_{s,0}^{*}$. This correlation can be used to constrain the
$m_{s,0}^{*}$, which is the main topic in the next subsection. The
experimental E$2$ energy-weighted sum rule (EWSR) measured by Youngblood et
al. Youngblood et al. (2004) is also included for comparison, i.e., the green
symbols in Fig. 3. Since the original data in Ref. Youngblood et al. (2004)
are only shown in fraction, the data point are scaled to the peak value of the
$S(E)$ obtained with SkM*. We notice from the figure that its overall shape
can be well reproduced within the framework of the BUU equation.

Figure 3:  The $S(E)$ of the ISGQR for $\isotope[208]{Pb}$ obtained based on
the BUU equation. The solid line represents the result obtained through the
first excitation scenario [Eq. (13)], while the dashed lines correspond to
that through the second excitation scenario [Eq. (14)]. The gray hatched band
represents the weighted average of the experimental $E_{x}$ for
$\isotope[208]{Pb}$ at $E_{x}$ $=$ $10.9\pm 0.1~{}\rm MeV$. The green symbols
are the experimental E$2$ EWSR by Youngblood et al. Youngblood et al. (2004).

###  III.2 Correlation between $E_{x}$ and $m_{s,0}^{*}$

Based on the quantal harmonic oscillator approach Bohr and Mottelson (1975),
one can obtain a direct relation between the excitation energy $E_{x}$ of the
ISGQR and the iso-scalar nucleon effective mass,

$E_{x}=\sqrt{\frac{2m}{m^{*}_{s,0}}}\hbar\omega_{0},$ (16)

where $\hbar\omega_{0}$ is the frequency of the harmonic oscillator, which is
related to the restoring force of the ISGQR, and $m^{*}_{s,0}$ is the iso-
scalar effective mass at the nuclear saturation density $\rho_{0}$. By this
semiempirical equation, we obtain that a strong linear correlation exists in
between $10^{3}/{E_{x}}^{2}$ and ${m^{*}_{s,0}}/{m}$,

$\frac{10^{3}}{{E_{x}}^{2}}=\frac{m^{*}_{s,0}}{m}k+b,$ (17)

where the slope $k$ is approximately equal to
${10^{3}}/{2(\hbar\omega_{0})^{2}}$ and $b$ is the intercept. Based on the BUU
equation, we can calculate the $E_{x}$ of the ISGQR for certain nucleus
employing various Skyrme interactions with different $m^{*}_{s,0}$.

Figure 4:  The correlation between the $m^{*}_{s,0}/m$ and the $E_{x}$ of the
ISGQR for $\isotope[208]{Pb}$, $\isotope[144]{Sm}$, $\isotope[116]{Sn}$ and
$\isotope[90]{Zr}$ obtained by pure Vlasov calculation with 65 Skyrme
interactions. Their linear fittings are shown in solid lines. The hatched
bands correspond to the experimental $E_{x}$ of the ISGQR these nuclei.

In order to obtain the correlation between the $E_{x}$ and the $m^{*}_{s,0}$,
we select $65$ representative Skyrme interactions or Skyrme pseudopotentials,
to calculate the strength function of the ISGQR of various heavy nuclei based
on the Eq. (14). All of these interactions (SV-m56-O, SkI5, SkI3, SGI, SkA,
SV-m64-O, SkI6, SkMP, SLy9, SAMi, SLy6, NRAPR, SkI1, SLy4, SLy0, SLy1, SLy3,
SV-mas07, KDE0v, KDE0v1, SKRA, KDE, SK272, Rs, Gs, SGII, SkM, SkM*, SK255, SV-
mas08, MSL0, MSL1, SkT7, SkT7a, SkT8, SkT8a, SkT9, SkT9a, SkS3, SkS1, SkS4,
SV-sym32, SkO’, SV-bas, SV-K241, BSk13, BSk5, SV-min, Skxs20, SKXm, Skxs15,
Ska25s20, Ska35s20, SKX, SkSC15, SkT1, SkT2, SkT3, SKXce, Ska35s15, Ska45s20,
BSk1, MSk7, details about these standard Skyrme interactions can be found in
Ref. Dutra et al. (2012) and references therein, Sk$\chi$m* Zhang et al.
(2018), SP$6$h Wang et al. (2018)) can roughly reproduce the empirical values
of several characteristic quantities of the nuclear equation of state Roca-
Maza and Paar (2018), i.e., nuclear saturation density $\rho_{0}$, energy per
nucleon for symmetric nuclear matter at saturation density $E_{0}$, the
nuclear incompressibility at saturation density $K_{0}$, and the symmetry
energy at saturation density $E_{\rm sym}(\rho_{0})$. The $m^{*}_{s,0}/m$ in
these Skyrme interactions range from $0.56$ to $1.05$, which can be calculated
through Eq. (8).

Figure 4 displays the pure Vlasov results of the inverse squared excitation
energy $1/E_{x}^{2}$ of ISGQR versus $m^{*}_{s,0}/m$ with the above $65$
Skyrme interactions for $\isotope[208]{Pb}$, $\isotope[144]{Sm}$,
$\isotope[116]{Sn}$ and $\isotope[90]{Zr}$. The obtained $E_{x}$ of the ISGQR
decreases from light to heavy nuclei, which follows the general trend. As can
be noticed in the figure, the $1/E_{x}^{2}$ and $m^{*}_{s,0}/m$ for these
nuclei exhibit nice linear correlations (their linear fits are shown as solid
lines), which indicates that Eq. (17) holds for a large nuclear mass range.
The experimental $E_{x}$ of the ISGQR for different nuclei extracted in Ref.
Youngblood et al. (2004); Gupta et al. (2018) are also included in Fig. 4 with
different hatched bands, i.e., $10.89\pm 0.30~{}\rm MeV$, $12.78\pm 0.20~{}\rm
MeV$, $13.50\pm 0.35~{}\rm MeV$ and $14.56\pm 0.20~{}\rm MeV$ for
$\isotope[208]{Pb}$, $\isotope[144]{Sm}$, $\isotope[116]{Sn}$ and
$\isotope[90]{Zr}$, respectively. Due to the limitation of the density
functional, the Skyrme interactions which reproduce the experimental $E_{x}$
of heavier nuclei give slightly lower $E_{x}$ of lighter nuclei compared with
their experimental value, and vice versa, though the deviations are very
small. This result is consistent with those from similar systematic analyses
with Skyrme interactions based on the RPA Bonasera et al. (2018) and the
macroscopic Langevin equation Su et al. (2020).

In principle, mean-field calculations are more suitable for heavier nuclei,
therefore we choose [208]Pb to constrain the $m^{*}_{s,0}$. In Fig. 5, in
addition to the pure Vlasov result of the $E_{x}$ \- $m^{*}_{s,0}/m$
correlation, the result from full BUU calculation with the nucleon-nucleon
scatterings is also included. Through a linear fit, shown as solid lines in
Fig. 5, we obtain the slope $k$ and intercept $b$ in the relation expressed by
Eq. (17), and they are also given in Fig. 5 along with the Pearson coefficient
of the fittings. As shown in the figure, the results of Vlasov and BUU
calculation are quite similar except a small deviation. Due to the damping
provided by the nucleon-nucleon scatterings, the slope $k$ obtained from BUU
calculations is a little bit larger then that in the Vlasov calculation. The
constrains on the ${m^{*}_{s,0}}/{m}$ can be obtained through
${m^{*}_{s,0}}/{m}$ $=$ $(10^{3}/E_{x}^{2}-b)/k$. Here a weighted average
value of $E_{x}$ $=$ $10.9\pm 0.1~{}\rm MeV$ over several experiment
measurements Roca-Maza et al. (2013) is adopted, and represented by pink
hatched band in Fig. 5. For the Vlasov calculation, we have

$\frac{m^{*}_{s,0}}{m}=0.83\pm 0.04,$ (18)

while for the BUU calculation

$\frac{m^{*}_{s,0}}{m}=0.82\pm 0.03.$ (19)

The uncertainties in the above constraints on $m_{s,0}^{*}/m$ are propagated
from the uncertainties of $k$, $b$ and $E_{x}$. The central value of the
constraint from the BUU calculation is slightly lower than that from the
Vlasov calculation, whereas they are within the errors of each other. This
indicates the inclusion of the nucleon-nucleon scatterings ($2$p-$2$h
correlation) has only negligible effect on the constraint of the
$m_{s,0}^{*}/m$. These effective mass constraints are consistent with the
$m^{*}_{s,0}/m\approx 0.8-0.9$ obtained from the ISGQR of the spherical
nucleus $\isotope[144]{Sm}$ and the deformed nucleus $\isotope[154]{Sm}$
Yoshida and Nakatsukasa (2013). The constraint on $m_{s,0}^{*}$ from the
Vlasov calculation is also in agreement with the constraint $m_{s,0}^{*}/m$
$=$ $0.91\pm 0.05$ obtained from a conventional analysis Zhang and Chen (2016)
and $m_{s_{0}}^{*}/m=0.87\pm 0.04$ from a recent Bayesian analysis Zhang et
al. (2021) of ISGQR of [208]Pb based on the Skyrme-RPA calculations, where the
$2$p-$2$h correlation is also missing.

Figure 5:  The correlation between the $m^{*}_{s,0}/m$ and the $E_{x}$ of the
ISGQR for $\isotope[208]{Pb}$. The green circles and red triangles are results
from the pure Vlasov and the full BUU calculations, respectively, for
different Skyrme interactions. Their linear fittings are shown in solid lines,
whose slope $k$, intercept $b$ and Pearson coefficient $r$ are also listed.
The pink hatched band corresponds to the weighted average of the experimental
$E_{x}$ $=$ $10.9\pm 0.1~{}\rm MeV$.

##  IV Summary

In the present study, the nuclear ISGQR has been investigated based on the BUU
transport equation. The mean-field part of the BUU equation is described by
the Skyrme nucleon-nucleon effective interaction, and numerically solved
through a lattice Hamiltonian method. We employ the stochastic approach to
deal with the collision term of the BUU equation, which provides a two-body
dissipation and is responsible for the collisional damping of the nuclear
collective motions. The measured ISGQR width and the overall shape have been
reproduced based on the BUU equation with a proper in-medium nucleon-nucleon
cross section. Unlike in the case of the IVGDR, where collisional damping is
only responsible for about half of the total width, we find that the width of
ISGQR of heavy nuclei ([208]Pb) is dominantly exhausted by the collisional
damping, which suggests the ISGQR to be an ideal site for constraining the in-
medium nucleon-nucleon cross section. A strong linear correlation between the
iso-scalar nucleon effective mass $m^{*}_{s,0}$ and $1/E_{x}^{2}$, the inverse
of the squared excitation energy of ISGQR, has been obtained for [90]Zr,
[116]Sn, [144]Sm and [208]Pb based on the collisionless BUU equation, i.e.,
the Vlasov equation. Through the weighted average of the experimental
excitation energy $E_{x}$ of [208]Pb, we have extracted a constraint on the
iso-scalar nucleon effective mass at $m^{*}_{s,0}/m$ $=$ $0.83\pm 0.04$. If we
include the collision term of the BUU equation, which embodies the beyond
mean-field $2$p-$2$h correlation, the constraint on $m^{*}_{s,0}/m$ only
decreases slightly to $0.82\pm 0.03$, which indicates that the $2$p-$2$h
correlation has negligible effect on the excitation energy, and subsequently
on the constraint of the iso-scalar nucleon effective mass.

###### Acknowledgements.

We thank Chen Zhong for setting up and maintaining the GPU server. This work
is partially supported by the National Natural Science Foundation of China
under Contracts No. $11890714$, No. $11891070$ and No. $11905302$, the
Guangdong Major Project of Basic and Applied Basic Research No.
$2020$B$0301030008$, and the Key Research Program of the CAS under Grant No.
No. XDB34000000.

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