# Constraining the in-medium nucleon-nucleon cross section from the width of
nuclear giant dipole resonance

Rui Wang<EMAIL_ADDRESS>Zhen Zhang<EMAIL_ADDRESS>Lie-Wen
Chen<EMAIL_ADDRESS>Che Ming Ko<EMAIL_ADDRESS>Yu-Gang Ma
<EMAIL_ADDRESS>Key Laboratory of Nuclear Physics and Ion-beam
Application (MOE), Institute of Modern Physics, Fudan University, Shanghai
$200433$, China  Shanghai Institute of Applied Physics, Chinese Academy of
Sciences, Shanghai $201800$, China  Sino-French Institute of Nuclear
Engineering and Technology, Sun Yat-Sen University, Zhuhai $519082$, China
School of Physics and Astronomy and Shanghai Key Laboratory for Particle
Physics and Cosmology, Shanghai Jiao Tong University, Shanghai $200240$, China
Cyclotron Institute and Department of Physics and Astronomy, Texas A$\&$M
University, College Station, Texas $77843$, USA

###### Abstract

We develop a new lattice Hamiltonian method for solving the Boltzmann-Uehling-
Uhlenbeck (BUU) equation. Adopting the stochastic approach to treat the
collision term and using the GPU parallel computing to carry out the
calculations allows for a rather high accuracy in evaluating the collision
term, especially its Pauli blocking, leading thus to a new level of precision
in solving the BUU equation. Applying this lattice BUU method to study the
width of giant dipole resonance (GDR) in nuclei, where the accurate treatment
of the collision term is crucial, we find that the obtained GDR width of
[208]Pb shows a strong dependence on the in-medium nucleon-nucleon cross
section $\sigma_{\rm NN}^{*}$. A very large medium reduction of $\sigma_{\rm
NN}^{*}$ is needed to reproduce the measured value of the GDR width of [208]Pb
at the Research Center for Nuclear Physics in Osaka, Japan.

###### keywords:

Heavy-ion collisions , Transport models , Nuclear giant dipole resonance width
, In-medium nucleon-nucleon cross section

_Introduction._ The in-medium nucleon-nucleon (NN) cross section $\sigma_{\rm
NN}^{*}$ has significant effects on the dynamics of heavy-ion collisions
(HICs), and it thus plays a crucial role in understanding the reaction
mechanisms as well as various phenomena and observables in these collisions
[1, 2, 3, 4, 5, 6]. The importance of $\sigma_{\rm NN}^{*}$ also lies in its
intimate relation to the transport properties of nuclear matter [7, 8] and the
nucleon effective interactions [9]. Since a major goal of studying HICs is to
extract the equation of state (EOS) of nuclear matter from experimental data
[10, 11, 12, 13, 14, 15], a thorough understanding of $\sigma_{\rm NN}^{*}$
helps reduce the uncertainties in transport models [16, 17] that are used for
describing these reactions. While the NN cross section in free space
$\sigma_{\rm NN}^{\rm free}$ can be directly measured in experiments, the
determination of the value of $\sigma_{\rm NN}^{*}$ in nuclear medium usually
relies on theoretical investigations. These include calculations based on
microscopic theories, such as the nonrelativistic and relativistic Brueckner
theories [18, 19, 20, 21, 22, 23, 9] and the closed time path Green’s function
approach [24, 25]. Also, there have been attempts to extract $\sigma_{\rm
NN}^{*}$ from experiments by comparing results from transport model
calculations, where $\sigma_{\rm NN}^{*}$ is a crucial input, with observables
measured in HICs that are sensitive to $\sigma_{\rm NN}^{*}$, e.g., the
collective flow and nuclear stopping [1, 26, 27, 28, 29, 8, 30]. Although
these studies have reached the consensus that the NN cross section is
suppressed in nuclear medium, the reduction factor is still far from
certainty.

The transport model used in describing HICs is a straightforward tool for
studying $\sigma_{\rm NN}^{*}$ because one of its main ingredients, the NN
collision term, embodies the information of $\sigma_{\rm NN}^{*}$. Since the
mean field or the EOS of nuclear matter is another major ingredient of one-
body transport models, finding the proper observables that depend on
$\sigma_{\rm NN}^{*}$ rather than the nuclear EOS is essential for studying
$\sigma_{\rm NN}^{*}$. One such observable is the width of nuclear giant
dipole resonance (GDR) as it is naturally related to $\sigma_{\rm NN}^{*}$
through the NN collision term in transport models. In general, the damping
width of nuclear collective motion originates from three sources: 1) the
escape width associated with particle emissions; 2) the fragmentation or the
Landau damping width due to couplings between single particle states and the
mean field; 3) the spreading or collisional damping width caused by the
coupling to more complex states like the two-particle-two-hole ($2p$-$2h$),
$3p$-$3h$, etc. For a heavy nucleus at zero temperature, the width of its GDR
is mainly exhausted by collisional damping [31, 32, 33] before the
contribution from deformation fluctuations appears as a result of the finite
temperature effect [34]. In the transport model, the collisional damping is
incorporated in the binary collisions of nucleons and thus depends directly on
$\sigma_{\rm NN}^{*}$. It is therefore expected that the GDR width of a heavy
nucleus in studies based on the transport model depends strongly on
$\sigma_{\rm NN}^{*}$ and weakly on the nuclear EOS.

The major obstacle that has so far prevented the use of transport models to
accurately calculate the spreading width of GDR is due to the fermionic nature
of nucleons. Specifically, the accurate treatment of Pauli blocking in
transport models is challenging [16, 17], especially for small amplitude
nuclear collective motions. Both subtle implementations and advanced computing
techniques are required for overcoming this difficulty. In the present work,
we extend the previous study using the lattice Hamiltonian Vlasov method based
on the next-to-next-to-next-to leading order (N$3$LO) Skyrme pseudopotential
[35] to include a stochastic elastic NN collision term. Solving the resulting
Boltzmann-Uehling-Uhlenbeck (BUU)-type one-body transport model with the high
computation efficiency provided by GPU parallel computing [36], which enables
the accurate treatment of Pauli blocking in the collision term of the BUU
equation, allows us to calculate precisely the spreading width of the GDR in
nuclei. We then obtain a stringent constraint on the in-medium NN cross
section $\sigma_{\rm NN}^{*}$ by comparing the GDR width of [208]Pb from the
present lattice BUU (LBUU) method with that measured from
$\isotope[208]{Pb}(\vec{p},\vec{p}^{\prime})$ reaction with polarized protons
at the Research Center for Nuclear Physics (RCNP) in Osaka, Japan [37].

_Model description._ The BUU equation is a semi-classical approximation to the
quantum transport equation [38, 39]. For a momentum-dependent mean-field
potential $U(\vec{r},\vec{p})$, it reads as

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

where $f=f(\vec{r},\vec{p})$ is the one-body phase-space distribution function
of nucleons or their Wigner function. The r.h.s of Eq. (1) is the NN collision
term including the Pauli blocking effect due to the Fermi statistics of
nucleons, i.e.,

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

where $g$ is the degeneracy, ${\cal M}_{12\rightarrow 34}$ is the in-medium
transition matrix element, and $(1-f_{i})$ is the Pauli suppression factor. It
is worth mentioning that higher-order quantum corrections to Eq. (1) can be
added perturbatively [40].

In the present work, we solve the BUU equation by the lattice Hamiltonian (LH)
method [41, 42, 43], which is a variant of the usual test particle method
[44]. In the LH method, the total Hamiltonian $H$ of the system is
approximated by the lattice Hamiltonian $H_{L}$, i.e.,

$H=\int{\cal H}(\vec{r}){\rm d}\vec{r}\approx l^{3}\sum_{\alpha}{\cal
H}(\vec{r}_{\alpha})\equiv H_{L},$ (3)

where $\mathcal{H}$ is the Hamiltonian density, $\vec{r}_{\alpha}$ represents
the coordinate of certain lattice site $\alpha$, and $l$ is the lattice
spacing. For the nucleon one-body phase-space distribution function
$f_{\tau}(\vec{r}_{\alpha},\vec{p})$, it is expressed as

$f_{\tau}(\vec{r}_{\alpha},\vec{p},t)=\frac{(2\pi\hbar)^{3}}{gN_{\rm
E}}\sum_{i}^{\alpha,\tau}S\big{[}\vec{r}_{i}(t)-\vec{r}_{\alpha}\big{]}\delta\big{[}\vec{p}_{i}(t)-\vec{p}\big{]},$
(4)

where $S$ is the form factor and $N_{\rm E}$ is the number of ensembles or
test particles used in the calculation. The sum in Eq.(4) runs over all test
nucleons of isospin state $\tau$ that contribute to the lattice site $\alpha$.
In the present work, we adopt a triangular form factor $S$ with the size of
$4l$, and its detail can be found in Ref. [35]. The Hamiltonian in Eq. (3)
contains both the Coulomb and the nuclear part [35] with the latter obtained
from the N$3$LO Skyrme pseudopotential [45] SP$6$h, whose details can be found
in Ref. [46].

In the present LBUU method, the ground state of a spherical nucleus at zero-
temperature is obtained from the Thomas-Fermi approach [41, 47, 48, 49] via
the variation of the Hamiltonian with respect to the radial nucleon density
$\rho_{\tau}(r)$. The obtained $\rho_{\tau}(r)$ is then used to determine the
initial coordinates of test nucleons, while their initial momenta are
generated according to zero-temperature Fermi distribution with local Fermi
momentum given by $p_{\tau}^{F}(r)$ $=$
$\hbar\big{[}3\pi^{2}\rho_{\tau}(r)\big{]}^{1/3}$. This method for
initialization ensures the stability of ground-state nuclei in BUU-like
transport models [49, 35].

For the collision term in the BUU equation, we implement it using the
stochastic approach [50], which is more reliable than the commonly used
geometric method when the mean free path $\lambda_{\rm MFP}$ of a test nucleon
is not much larger than the interaction length between two test nucleons [51]
or when the NN scattering cross section is very large. The collision
probability $P_{ij}$ of two test nucleons in the stochastic approach is
determined from the NN collision term in Eq. (2), which is

$P_{ij}=v_{\rm rel}\sigma_{\rm
NN}^{*}S(\vec{r}_{i}-\vec{r}_{\alpha})S(\vec{r}_{j}-\vec{r}_{\alpha})l^{3}\Delta
t.$ (5)

To reduce the statistical fluctuations of collision events and better reflect
the nature of the BUU equation, we include collisions of test nucleons from
different ensembles. In this case, the collision probability is reduced to
$P_{ij}/N_{\rm E}$, because of the scaling $\sigma_{\rm NN}^{*}$ $\rightarrow$
$\sigma_{\rm NN}^{*}/N_{\rm E}$ of the in-medium NN cross section between test
nucleons. Under such a scaling, the diluteness of the system, which is
characterized by $\sqrt{\sigma_{\rm NN}^{*}}/\lambda_{\rm MFP}$, is reduced by
the factor $\sqrt{N_{\rm E}}$, and this makes it possible to solve the BUU
equation almost exactly with a sufficiently large $N_{\rm E}$ achieved by
adopting the GPU parallel computing.

For the $i$-th and $j$-th test nucleons colliding at the lattice site
$\vec{r}_{\alpha}$, the direction of their final momenta $\vec{p}_{3}$ and
$\vec{p}_{4}$ are sampled according to the differential cross-section given in
Ref. [52]. However, this collision can only happen if it is allowed by the
Pauli principle via the factor
$[1-f(\vec{r}_{\alpha},\vec{p}_{3})]\times[1-f(\vec{r}_{\alpha},\vec{p}_{4})]$.
In the present LBUU method, the distribution function
$f_{\tau}(\vec{r}_{\alpha},\vec{p})$ is calculated from averaging its value in
Eq. (4) over a given momentum-space sphere centered at $\vec{p}$ with radius
$R_{\tau}^{p}(\vec{r}_{\alpha},\vec{p})$. In typical transport model
calculations, $R_{\tau}^{p}(\vec{r}_{\alpha},\vec{p})$ is taken to have a
constant value of about one hundred MeV. In the present work, we use an
improved form for $R_{\tau}^{p}(\vec{r}_{\alpha},\vec{p})$ that is
specifically proposed for small-amplitude nuclear collective dynamics near
ground state [48], i.e., $R_{\tau}^{p}(\vec{r}_{\alpha},\vec{p})$ $=$ ${\rm
max}[\Delta p,p_{\tau}^{F}(\vec{r}_{\alpha})-|\vec{p}|]$, where
$p_{\tau}^{F}=\hbar(3\pi^{2}\rho_{\tau})^{1/3}$ is the local nucleon Fermi
momentum and $\Delta p$ is a constant with the dimension of momentum that
needs to be taken to be sufficiently small.

The treatment of Pauli blocking in transport models is crucial in calculating
the width of nuclear collective excitations. At low incident energy or
temperature, the Pauli blocking is notoriously difficult to handle in
transport models [16, 17]. This is mainly caused by the inaccuracy in
calculating the local momentum distribution
$f_{\tau}(\vec{r}_{\alpha},\vec{p})$, which then leads to numerically spurious
collisions and thus an overestimated GDR width as a result of the enhanced
collisional damping. There are three main origins for the numerically spurious
collisions in transport models: 1) fluctuations in calculating
$f_{\tau}(\vec{r}_{\alpha},\vec{p})$ caused by insufficiently large $N_{\rm
E}$; 2) spurious thermal excitation caused by finite $\Delta p$ in calculating
$f_{\tau}(\vec{r}_{\alpha},\vec{p})$ (also see Ref. [48]); and 3) diffusion in
local momentum caused by finite lattice spacing $l$ when averaging over
different local densities on the nuclear surface.

In choosing the parameter values in the LBUU calculations, we use the
following criteria. For a given $l$ and $\Delta p$, $N_{\rm E}$ should be
large enough to eliminate the overwhelming majority of the spurious collisions
caused by the first origin mentioned above, and at the same time $l$ and
$\Delta p$ should be chosen to be sufficiently small to suppress the effects
due to the second and third origins on the GDR width. After careful tests
based on considerations of numerical accuracy and computation efficiency, we
find the optional values of $l$ $=$ $0.5~{}\rm fm$, $\Delta p$ $=$ $0.05~{}\rm
GeV$ and $N_{\rm E}$ $=$ $30000$. It is worth to mention that with the
adoption of GPU parallel computing, it is possible to use a value for $N_{\rm
E}$ that far exceeds those used in all previous calculations based on the BUU
transport equation. Further reducing $\Delta p$ and $l$ and increasing $N_{\rm
E}$ only leads to a negligible variation in the calculated GDR width.

We note that for the case of free NN cross section, an average of $97.93\%$ of
the attempted collisions in the ground state of [208]Pb are blocked by the
Pauli principle, resulting in an average of $1.30$ successful collisions of
physical nucleons per ${\rm fm}/c$ during the time evolution of $0-500$ ${\rm
fm}/c$. Also, the root mean square (rms) radius and the ground-state energy of
[208]Pb vary by less than $3.6\%$ ($0.2~{}\rm fm$) and $3.2\%$ ($50~{}\rm
MeV$), respectively, during this time evolution. With a reduced in-medium NN
cross section, both the number of successful collisions and the change in the
radius and binding energy are even smaller. Since the binding energy decreases
monotonically with time without oscillations, it is not expected to have much
effect on the calculated excitation energy of GDR. The energy violation, which
is caused by our use of in-vacuum energy conservation in NN scatterings,
instead of the in-medium energy conservation in the presence of the momentum-
dependent potential, is not expected to affect the calculated width of GDR
either. This is because the latter is controlled by the NN scattering rate,
which depends on the NN scattering cross section and the Pauli blocking
factor. We also note that although the radius and binding energy variations in
LBUU are larger than those in the lattice Hamiltonian Vlasov approach of Ref.
[35], where the rms radius and the binding energy almost do not change, and
the radial density profile only changes slightly during the time evolution of
0-1000 fm/c, the LBUU method used in the present study is sufficiently
accurate for investigating the GDR width.

_Results and discussions._ The collective excitation of a nucleus consisting
of $A$ nucleons can be induced by adding a perturbation to its Hamiltonian at
the initial time $t_{0}$, i.e., $\hat{H}_{ex}(t)$ $=$
$\lambda\hat{Q}\delta(t-t_{0})$, where $\hat{Q}$ is an appropriate excitation
operator and $\lambda$ is a small parameter. The width of a collective
excitation is defined by the full width at half maximum (FWHM) of its strength
function $S(E)$ as a function of the excitation energy $E$. In the linear
response theory [53], the $S(E)$ is obtained from the Fourier integral

$S(E)=-\frac{1}{\pi\lambda}\int_{0}^{\infty}{\rm
d}t\Delta\langle\hat{Q}\rangle(t){\rm sin}\frac{Et}{\hbar},$ (6)

where $\Delta\langle\hat{Q}\rangle(t)$ $=$ $\langle
0^{\prime}|\hat{Q}|0^{\prime}\rangle$ $-$ $\langle 0|\hat{Q}|0\rangle$ is the
time evolution of the response function of the nucleus to the excitation
operator $\hat{Q}$ with $|0\rangle$ and $|0^{\prime}\rangle$ denoting the
nuclear states before and after the perturbation, respectively. In terms of
the Wigner transform $q(\vec{r},\vec{p})$ of the one-body excitation operator
$\hat{q}$, which is related to $\hat{Q}$ by $\hat{Q}$ $=$
$\sum_{i}^{A}\hat{q}$, the expectation values in the above can be evaluated
according to $\langle\hat{Q}\rangle(t)=\int
f(\vec{r},\vec{p},t)q(\vec{r},\vec{p}){\rm d}\vec{r}{\rm d}\vec{p}$ using the
nucleon phase-space distribution function $f(\vec{r},\vec{p},t)$. Details on
the single-particle operator used in exciting a ground state nucleus in
transport models can be found in Ref. [35].

Figure 1: Time evolution of the isovector dipole response function
$\Delta\langle\hat{Q}_{\rm IVD}\rangle$ (left) and strength function $S(E)$
(right) of [208]Pb due to the perturbation of $\hat{H}_{ex}$ $=$
$\lambda\hat{Q}_{\rm IVD}\delta(t-t_{0})$ with $\lambda$ $=$ $15~{}{\rm
MeV}/c$ from the Vlasov calculation (solid lines) and the LBUU calculation
(dashed lines) with $\sigma_{\rm NN}^{\rm free}$. The dotted cyan curve in the
left window represents the expectation value of the $\hat{Q}_{\rm IVD}$ in the
ground state of [208]Pb from the LBUU calculation with $\sigma_{\rm NN}^{\rm
free}$.

We first employ the present LBUU method to study the effect of NN scatterings
on the isovector dipole response of [208]Pb using the excitation operator
$\hat{Q}_{\rm IVD}$ $=$ $\frac{N}{A}\sum_{i}^{Z}\hat{z}_{i}$ $-$
$\frac{Z}{A}\sum_{i}^{N}\hat{z}_{i}$. In Fig. 1, we show the results obtained
by using the free NN elastic scattering cross section taken from Ref. [52]
with $\sigma_{\rm NN}^{\rm free}(p_{\rm lab})=\sigma_{\rm NN}^{\rm
free}(0.1~{}{\rm GeV}/c)$ for neutron-neutron ($nn$) or proton-proton ($pp$)
collisions at $p_{\rm lab}\leq 0.1~{}{\rm GeV}/c$ and $\sigma_{\rm NN}^{\rm
free}(p_{\rm lab})=\sigma_{\rm NN}^{\rm free}(0.05~{}{\rm GeV}/c)$ for
neutron-proton ($np$) collisions at $p_{\rm lab}\leq 0.05~{}{\rm GeV}/c$, as
experimental data for lower incident momenta ($p_{\rm lab}$) are unavailable.
For comparison purpose, results from the LBUU calculation without NN
scatterings, i.e., the Vlasov calculation, are also shown in Fig. 1. In both
cases, we use in the initial perturbation the same parameter $\lambda$ $=$
$15~{}{\rm MeV}/c$, which is also used in all the calculations in the present
study, and we find that varying the value of $\lambda$ by $2/3$ almost has no
effects on the value of the GDR width. As shown in the left window of Fig. 1
for the response function $\Delta\langle\hat{Q}_{\rm IVD}\rangle(t)$, the
inclusion of NN scatterings significantly enhances the damping of the
oscillations. The dotted cyan curve in the left window of Fig. 1 represents
the expectation value of the $\hat{Q}_{\rm IVD}$ in the ground state of
[208]Pb from the LBUU calculation with $\sigma_{\rm NN}^{\rm free}$, which is
negligible compared with that in the excited cases. To illustrate more clearly
the effect of collisional damping, we show in the right window of Fig. 1 the
GDR strength function $S(E)$ from the Fourier transformation of the response
function. Note that the Vlasov calculation is carried out for a long evolution
time of $1000~{}{\rm fm}/c$ when the amplitude of the oscillation of
$\Delta\langle\hat{Q}\rangle(t)$ almost vanishes so that the fluctuation in
the calculated strength function from the Fourier transform of
$\Delta\langle\hat{Q}\rangle(t)$ is negligible. We clearly see the large
increase of GDR width after including NN scatterings, namely, the GDR width of
208Pb are $6.5~{}\rm MeV$ and $1.5~{}\rm MeV$ in the LBUU calculations with
and without NN scatterings, respectively.

Figure 2: The GDR width of [208]Pb from LBUU calculations for different values
of $\alpha$ in $\sigma_{\rm NN}^{*}$. The horizontal line represents the RCNP
experimental value of $4.0~{}\rm MeV$. The inset shows the strength function
with $\alpha$ $=$ $1.8$ (solid line) and the shifted one (dash-dotted line) to
match the experimental GDR peak energy. See text for details.

Experimentally, the GDR width of [208]Pb has been well determined to be
$4.0~{}\rm MeV$ from the $\isotope[208]{Pb}(\vec{p},\vec{p}^{\prime})$
reaction carried out at RCNP [37]. Our result from the LBUU calculation with
$\sigma_{\rm NN}^{\rm free}$ thus significantly overestimates the GDR width of
[208]Pb. This is understandable because of the absence of medium effect on the
NN scattering in the calculation. Its inclusion is expected to reduce the NN
cross section, weaken the collisional damping, and result in a smaller GDR
width. The sensitivity of the GDR width to NN scatterings shown in Fig. 1
makes it possible to constrain the medium effect on the NN scattering cross
section.

For $\sigma_{\rm NN}^{*}$, we parameterize it by multiplying the free NN cross
section with a medium-dependent correction factor. Specifically, we choose an
exponential reduction factor as suggested by the $T$-matrix approach in Ref.
[20], i.e.,

$\sigma_{\rm NN}^{*}=\sigma_{\rm NN}^{\rm free}{\rm
exp}\bigg{[}-\alpha\frac{\rho/\rho_{\rm nuc}}{1+(T_{\rm
c.m.}/T_{0})^{2}}\bigg{]}.$ (7)

In the above, $T_{\rm c.m.}$ is the the total kinetic energy of two scattering
test nucleons at the rest frame of the local medium or cell, $\rho_{\rm nuc}$
$=$ $0.16~{}{\rm fm}^{-3}$ is the nuclear normal density, and $T_{0}$ $=$
$0.015~{}{\rm GeV}$. For the parameter $\alpha$, its original value in Ref.
[20] is $0.6$, which is called the Rostock cross section. In the present
study, we treat it as a free parameter to control the strength of medium
effect. Displayed in Fig. 2 is the GDR width $\Gamma$ of [208]Pb obtained with
different values of $\alpha$. As expected, the GDR width decreases with
increasing $\alpha$. To reproduce the experimental value of $\Gamma$ $=$
$4.0~{}{\rm MeV}$ measured at RCNP requires $\alpha$ to be as large as about
$1.8$, which indicates a very large medium reduction of the NN scattering
cross section.

Figure 3: Density dependence of the medium correction at different values for
the total kinetic energy $T_{\rm c.m.}$ of two scattering nucleons using NN
cross sections from Eq. (7) with $\alpha$ $=$ $1.8$, the Rostock cross section
with $\alpha$ $=$ $0.6$, and the FU$4$FP$6$ parameterization.

Although an early study on the balance energy, at which the nucleon direct
flow in HICs vanishes, favors a small medium reduction of the NN cross section
[1], more recent studies based on the analysis of the collective flow and
nuclear stopping data [6, 29] as well as the nucleon induced reaction cross
section [30] require a large medium reduction. For comparisons, we also
calculate the GDR width from the LBUU method with two different $\sigma_{\rm
NN}^{*}$, namely, the Fuchs cross section [23], which is obtained from the in-
medium Dirac-Brueckner T matrix, and the FU$4$FP$6$ parameterization, which is
preferred by the nucleon induced reaction cross section [30]. The value of GDR
width of [208]Pb calculated using the FU$4$FP$6$ parameterization is
$4.32~{}\rm{MeV}$, which is consistent with the experimental data. On the
other hand, the values obtained with the Fuchs cross section and the Rostock
cross section with $\alpha$ $=$ $0.6$ based on microscopic calculations are
$5.39~{}\rm{MeV}$ and $5.59~{}\rm{MeV}$, respectively, which both overestimate
the experimental value. In Fig. 3, we show the density dependence of the
medium correction $\sigma_{\rm NN}^{*}/\sigma_{\rm NN}^{\rm free}$ at three
different $T_{\rm c.m.}$ values for the NN cross section in Eq. (7) with
$\alpha$ $=$ $1.8$, the Rostock cross section with $\alpha$ $=$ $0.6$, and the
FU$4$FP$6$ parameterization with the isospin asymmetry $\delta$ set to be
$0.21$ as in [208]Pb. The $T_{\rm c.m.}^{\rm F}$ $\approx$ $0.073~{}\rm{GeV}$
in this figure represents the $T_{\rm c.m.}$ of two nucleons at the Fermi
surface of normal nuclear matter density $\rho_{\rm nuc}$. It is seen that
both the $\alpha$ $=$ $1.8$ case and the FU$4$FP$6$ parameterization, which
can describe the data of the GDR width of [208]Pb, show similar medium
reductions, which are very large compared with that from the Rostock cross
section.

In the inset of Fig. 2, we further show the strength function of the iso-
vector excitation of [208]Pb from the LBUU calculation uisng the cross section
in Eq. (7) with $\alpha$ $=$ $1.8$ together with the experimental measurements
at RCNP [37]. The LBUU result shifted to match the GDR peak energy is also
included for comparison. As can be seen, our calculation nicely reproduces the
shape of the experimental strength function, but overestimates the peak energy
by about $1.4$ MeV. A better agreement for the peak value could be achieved by
varying the mean-field potential or nuclear EOS, which is known to
significantly affect the peak energy of nuclear GDR [54]. Further tests show
that the mean-field potential or nuclear EOS only weakly affects the obtained
GDR width. As to the quantum corrections to the BUU equation [40], it is not
expected to modify our results qualitatively as their effect on the damping of
collective motion of heavy nucleus [55] is insignificant compared with that of
NN scatterings.

_Conclusions._ We have used the LH method to solve the BUU transport equation
with the binary collisions in the collision term treated via the stochastic
approach. With the use of a sufficiently large number of test particles, the
present LBUU method treats the Pauli blocking in the collision term of BUU
equation with very high precision and thus significantly increases the
accuracy in solving the BUU equation. From the accurately calculated GDR width
of [208]Pb, we have found that it depends strongly on the magnitude of the in-
medium NN cross section $\sigma_{\rm NN}^{*}$, and the experimentally measured
GDR width of [208]Pb from the $\isotope[208]{Pb}(\vec{p},\vec{p}^{\prime})$
reaction at RCNP can only be reproduced with a NN cross section that is
significantly reduced in nuclear medium. The large medium reduction of
$\sigma_{\rm NN}^{*}$ raises challenges to microscopic calculations based on
realistic NN interactions. Also, the effects of such a large medium reduction
of $\sigma_{\rm NN}^{*}$ on the widths of other modes of giant resonances in
nuclei and on the dynamics of HICs need to be studied as it may significantly
affect the extracted information on the properties of nuclear matter at
various densities.

_Acknowledgements._ We thank Bao-An Li and Jun Su for useful discussions, and
Meisen Gao, Jie Pu, Chen Zhong and Ying Zhou for the maintenance of the GPU
severs. This work was partially supported by the National Natural Science
Foundation of China under Contracts No. $11905302$, No. $11947214$, No.
$11890714$, No. $11625521$ and No. $11421505$, the Key Research Program of
Frontier Sciences of the CAS under Grant No. QYZDJ-SSW-SLH$002$, the Strategic
Priority Research Program of the CAS under Grants No. XDB$16$ and No.
XDB$34000000$, the Major State Basic Research Development Program (973
Program) in China under Contract No. $2015$CB$856904$, the US Department of
Energy under Contract No. DE-SC$0015266$, and the Welch Foundation under Grant
No. A-$1358$.

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