# Measuring neutron skin by grazing isobaric collisions

Hao-jie Xu111Corresponding author<EMAIL_ADDRESS>School of Science,
Huzhou University, Huzhou, Zhejiang 313000, China  Hanlin Li222Corresponding
author<EMAIL_ADDRESS>College of Science, Wuhan University of Science and
Technology, Wuhan, Hubei 430065, China  Ying Zhou  School of Physics and
Astronomy, Shanghai Key Laboratory for Particle Physics and Cosmology, and Key
Laboratory for Particle Astrophysics and Cosmology (MOE), Shanghai Jiao Tong
University, Shanghai 200240, China  Xiaobao Wang  School of Science, Huzhou
University, Huzhou, Zhejiang 313000, China  Jie Zhao  Department of Physics
and Astronomy, Purdue University, West Lafayette, Indiana 47907, USA  Lie-Wen
Chen333Corresponding author<EMAIL_ADDRESS>School of Physics and
Astronomy, Shanghai Key Laboratory for Particle Physics and Cosmology, and Key
Laboratory for Particle Astrophysics and Cosmology (MOE), Shanghai Jiao Tong
University, Shanghai 200240, China  Fuqiang Wang444Corresponding author:
<EMAIL_ADDRESS>School of Science, Huzhou University, Huzhou, Zhejiang
313000, China  Department of Physics and Astronomy, Purdue University, West
Lafayette, Indiana 47907, USA

###### Abstract

Neutron skin thickness ($\Delta r_{\rm np}$) of nuclei and the inferred
nuclear symmetry energy are of critical importance to nuclear physics and
astrophysics. It is traditionally measured by nuclear processes with
significant theoretical uncertainties. We recently proposed an indirect
measurement of the $\Delta r_{\rm np}$ by charged hadron multiplicities in
central isobaric collisions at relativistic energies, which are sensitive to
nuclear densities. In this Letter, we propose a direct measurement of the
$\Delta r_{\rm np}$ by using net-charge multiplicities in ultra-peripheral
(grazing) collisions of those isobars, under the assumption that they are
simple superimposition of nucleon-nucleon interactions. We illustrate this
novel approach by the trento and urqmd models.

Nuclei are bound states of protons and neutrons by the overall attractive
nuclear force. The nuclear strong force is isospin symmetric, however, because
of Coulomb interactions, heavy nuclei usually need more neutrons than protons
to remain stable. The root-mean-square radius of neutron distribution in a
heavy nucleus is thus larger than that of the proton distribution. The
difference is referred to as the neutron skin thickness, $\Delta r_{\rm
np}\equiv r_{n}-r_{p}$ Brown (2000). With more neutrons comes the penalty
symmetry energy associated with the asymmetry between the proton and neutron
numbers. By measuring the $\Delta r_{\rm np}$, one gains valuable information
about the nuclear symmetry energy. Of particular interest is the symmetry
energy density slope parameters $L$ at the nuclear saturation density
$\rho_{0}$ Chen et al. (2005); Roca-Maza et al. (2011); Tsang et al. (2012);
Horowitz et al. (2014) and $L_{c}$ at the critical density $\rho_{c}\approx
0.11$ fm-3 Zhang and Chen (2013).

The $\Delta r_{\rm np}$ has traditionally been measured by low-energy electron
and hadron scatterings off nuclei Frois and Papanicolas (1987); Lapikas
(1993); Roca-Maza et al. (2011); Tsang et al. (2012); Tarbert et al. (2014).
Because of the inevitable uncertainties in modeling the strong interaction of
the scattering processes in quantum chromodynamics (QCD) Ray et al. (1992),
large uncertainties on the $L$ and $L_{c}$ persist. Void of the strong
interaction uncertainties, parity-violating scattering processes with
electrons Donnelly et al. (1989); Horowitz et al. (2001) and neutrinos Akimov
et al. (2017), sensitive to the neutral current, have been measured. The
latest result from the Lead Radius Experiment (PREX-II) on the 208Pb, $\Delta
r_{\rm np}$ = $0.283\pm 0.071$ fm Abrahamyan et al. (2012); Adhikari et al.
(2021), still has a large statistical uncertainty. This leads to
$L_{c}=71.5\pm 22.6$ and $L=105\pm 37$ MeVReed et al. (2021), compared to
$L=75\pm 25$ MeVCentelles et al. (2009) from traditional scattering
experiments, where the uncertainty is dominated by statistical one in the
former and the systematic one in the latter. Efforts combining the PREX-II and
astro-nuclear observations may improve the $L_{c}$ and $L$ constraints Yue et
al. (2021).

Recently, we have proposed to use the ratio of charged hadron multiplicities
($N_{\rm ch}$) in most central collisions of isobars
(${}^{96}_{44}$Ru+${}^{96}_{44}$Ru and ${}^{96}_{40}$Zr+${}^{96}_{40}$Zr) at
the Relativistic Heavy Ion Collider (RHIC) to help constrain the $\Delta
r_{\rm np}$ Li et al. (2020). This exploits the sensitivity of particle
production to nucleon density distribution, which differs slightly between Ru
and Zr because of their different $\Delta r_{\rm np}$ values. Note that
$N_{\rm ch}$ is isospin insensitive; it is essentially the same among $pp$,
$pn$, and $nn$ interactions at high energies (see Table 1). Our idea in Ref.
Li et al. (2020) probes the nucleon density and, in turn, the neutron’s
knowing the precisely measured proton’s; it does not directly probe the
neutron density.

The nucleon density difference is, of course, also present in peripheral
collisions but there, the $N_{\rm ch}$ sensitivity to $\Delta r_{\rm np}$ is
weak (see Fig. 4 of Ref. Li et al. (2018) where the isobar difference is only
2%). However, those ultra-peripheral collisions, where the nuclei are only
grazing each other, must have very different mixture of participant protons
and neutrons, and therefore likely yield significantly different net-charge
numbers ($\Delta Q$). This is obvious especially for the case of full
acceptance with exact charge conservation, in which a proton releases one net-
charge number, whereas the contribution from neutrons is none. This difference
should also manifest in limited-acceptance midrapidity detectors. By measuring
$\Delta Q$ in those grazing collisions, one could in principle determine the
difference in the numbers of protons and neutrons participating in those
collisions, which is directly sensitive to $\Delta r_{\rm np}$.

To make our idea more concrete, we list in Table 1 the $\Delta Q$ (and $N_{\rm
ch}$ as well as the net-proton number $\Delta B_{p}$) in minimum bias $pp$,
$pn$, and $nn$ interactions at $\sqrt{s}=200$ GeV, simulated by the pythia
event generator Sjostrand et al. (2006, 2008) (version 8.240). We used Monash
tune Skands et al. (2014) which was based on the LHC data (we will discuss
this particular point at end of the paper). The acceptance cuts for $\Delta Q$
are $|\eta|<1$ and $0.2<p_{T}<2$ GeV/$c$, and we have excluded the
(anti-)protons with $p_{T}<0.4$ GeV/$c$ (where the protons are strongly
contaminated by background in experiments Abelev et al. (2009)). The $N_{\rm
ch}$ is from $|\eta|<0.5$ and $0.2<p_{T}<2$ GeV/$c$ (as usually performed in
experiment). The $\Delta Q$ differs significantly between $pp$ and $nn$
interactions, whereas $\Delta B_{p}$ is almost the same. The charge difference
in the initial baryons that have transported to midrapidity is transferred
nearly entirely to mesons. This difference will imprint in peripheral nucleus-
nucleus (AA) collisions, the magnitude of which depends on $\Delta r_{\rm
np}$. If a peripheral AA collision is a simple superimposition of nucleon-
nucleon (NN) interactions, then one can predict $\Delta Q$ in AA by those in
NN collisions as

$\Delta Q_{AA}\propto q_{AA}\Delta Q_{pp}+(1-q_{AA})\Delta Q_{nn}\,,$ (1)

where $q_{AA}$ is the fraction of protons among the participant nucleons. The
proportionality (normalization) factor is simply the number of NN collisions.
We have taken $\Delta Q_{pn}=(\Delta Q_{pp}+\Delta Q_{nn})/2$ which is a good
assumption as seen in Table 1– we have also verified this in the Hijing model
Wang and Gyulassy (1991); Wang (1997) and the urqmd model Bass et al. (1998);
Bleicher et al. (1999). One can have a similar equation for $\Delta B_{p}$,
but since $\Delta B_{p}$’s are almost the same among $pp$, $pn$ and $nn$
interactions it is of little use for our purpose.

Table 1: The midrapidity charged hardon multiplicity $N_{\rm ch}$ ($|\eta|<0.5$), the net-charge $\Delta Q$, and net-proton $\Delta B_{p}$ within $|\eta|<1$ and $0.2<p_{T}<2$ GeV/$c$ (excluding protons and antiprotons from $p_{T}<0.4$ GeV/$c$) in minimum bias $pp$, $pn$, and $nn$ interactions at $\sqrt{s}=200$ GeV by pythia (version 8.240, Monash tune Skands et al. (2014)). pythia | $N_{\rm ch}$ | $\Delta Q$ | $\Delta B_{p}$
---|---|---|---
p+p | $3.487$ | $0.0793$ | $0.0158$
p+n | $3.485$ | $0.0255$ | $0.0144$
n+n | $3.483$ | $-0.0279$ | $0.0130$

Now consider the isobar collisions of Ru+Ru and Zr+Zr at $\sqrt{s_{{}_{\rm
NN}}}=200$ GeV. The $\Delta Q$ ratio in Ru+Ru over Zr+Zr collisions, under the
superimposition assumption, is

$R_{\Delta Q}\equiv\frac{\Delta Q_{\rm RuRu}}{\Delta Q_{\rm
ZrZr}}=\frac{q_{\rm RuRu}+\alpha/(1-\alpha)}{q_{\rm
ZrZr}+\alpha/(1-\alpha)}\,,$ (2)

where $\alpha\equiv\Delta Q_{nn}/\Delta Q_{pp}$ is the $\Delta Q$ ratio in
$nn$ to $pp$ interactions; pythia gives $\alpha\simeq-0.352$. The overall
$q_{\rm RuRu}$ and $q_{\rm ZrZr}$ values for the whole nuclei are $44/96$ and
$40/96$, respectively; they would give $R_{\Delta Q}\simeq 1.267$. Of course,
the simple superimposition assumption breaks down in non-peripheral collisions
because of nuclear effects. However, the assumption should be good for grazing
AA collisions, where one expects Eq. (2) to be valid. The general idea to
probe $\Delta r_{\rm np}$ by $R_{\Delta Q}$ is that a sizable $\Delta r_{\rm
np}$ will make the $q_{AA}$ decrease dramatically with increasing impact
parameter ($b$) in those grazing collisions. $\Delta r_{\rm np}$ of 96Zr is
significantly larger than that of 96Ru, so the $R_{\Delta Q}$ ratio amplifies
the $\Delta r_{\rm np}$ sensitivity. The $\Delta r_{\rm np}$ of both nuclei
are controlled by the $L_{c}$ parameter, thus a measurement of $R_{\Delta Q}$
can determine its value.

Following our previous work Li et al. (2020); Xu et al. (2018), we examine
four sets of 96Ru and 96Zr nuclear densities from energy density functional
theory (DFT). One is the standard Skyrme-Hartree-Fock (SHF) model (see, e.g.,
Ref. Chabanat et al. (1997)) using the well-known interaction set SLy4
Chabanat et al. (1998); Wang et al. (2016). The other is the extended SHF
(eSHF) model Chamel et al. (2009); Zhang and Chen (2016) with three sets of
interaction parameters, denoted as Lc47, Lc20 and Lc70, corresponding to
$L_{c}=47.3$, 20 and 70 MeV Zhang and Chen (2014), respectively, all with the
symmetry energy $E_{\rm sym}(\rho_{c})=26.65$ MeV Zhang and Chen (2013). The
Lc47 set is the best fit to data on the nuclear masses and electric dipole
polarizibility in 208Pb Zhang and Chen (2013), and the other two are to
explore the effects of the symmetry energy (and neutron skin) variations.

With a given nuclear density, one can calculate the $q_{AA}$ parameter in
Ru+Ru and Zr+Zr collisions as a function of $b$ (and $N_{\rm ch}$). We use the
trento model Moreland et al. (2015) to do that. In trento particle production
is related to the reduced thickness, $N_{\rm ch}\propto
T_{R}(p;T_{A},T_{B})\equiv[(T_{A}^{p}+T_{B}^{p})/2]^{1/p}$ Bernhard et al.
(2016); Moreland et al. (2015). We use the parameter $p=0$ (i.e., $N_{\rm
ch}\propto\sqrt{T_{A}T_{B}}$), a gamma fluctuation parameter $k=1.4$, and a
Gaussian nucleon size of $0.6$ fm, which were found to well describe the
multiplicity data in heavy ion collisions Bernhard et al. (2016); Moreland et
al. (2015). The $q_{\rm RuRu}$ and $q_{\rm ZrZr}$ from trento calculations are
shown in Fig. 1. They are sensitive to $\Delta r_{\rm np}$ and thus the
$L_{c}$ parameter, and decrease with decreasing $N_{\rm ch}$ in peripheral
collisions. Both these features indicate the effect of $\Delta r_{\rm np}$ on
$R_{\Delta Q}$. The sensitivity is larger in Zr+Zr than Ru+Ru collisions
because of the larger $\Delta r_{\rm np}$ of Zr.

Figure 1: (Color online). The proton fractions $q_{AA}$ among participant
nucleons as function of charged hadron multiplicity $N_{\rm ch}$ calculated by
trento with the 96Ru and 96Zr nuclear densities from eSHF (Lc20, Lc47, Lc70)
and SHF (SLy4). The dotted and dashed lines indicate the overall values of
44/96 and 40/96 of the entire Ru and Zr nuclei, respectively.  Figure 2:
(Color online). $R_{\Delta Q}$ as function of $N_{\rm ch}$ by Eq. (2) with
$\alpha=-0.352$ from pythia and the $q_{AA}$ from trento in Fig. 1 with four
DFT densities (solid curves). The dashed (dash-dotted) curve uses
$\alpha=-0.315$ ($-0.389$) with the Lc47 density. The dotted line indicates
the overall value of 1.267.

With those ratios and the $\alpha$ value, the $R_{\Delta Q}$ can be obtained
by Eq. (2) directly. Figure 2 shows the $R_{\Delta Q}$ as a function of
$N_{\rm ch}$ using $\alpha=-0.352$ from pythia (Table 1, the dashed curve
using another $\alpha$ value will be discussed later). The differences among
the nuclear densities are obvious at small $N_{\rm ch}$. Although the
calculated $R_{\Delta Q}$ is unlikely correct at large $N_{\rm ch}$ because of
the invalid assumption of simple NN superimposition for those non-peripheral
AA collisions, the $R_{\Delta Q}$ for grazing collisions with small $N_{\rm
ch}$ should be robust and can be used to determine the $\Delta r_{\rm np}$ and
thus the $L_{c}$ parameter. We note that the $\alpha$ value is obtained from
minimum bias NN interactions. However, a grazing AA collision with a
particular $N_{\rm ch}$ value selects biased underlying NN interactions,
especially when $N_{\rm ch}$ is small. We have verified that $\Delta Q_{pp}$
and $\Delta Q_{nn}$ do indeed depend on $N_{\rm ch}$ of NN collisions,
however, their ratio $\alpha$ is insensitive to $N_{\rm ch}$. Using a constant
$\alpha$ in Eq. (2) is, therefore, justified.

Figure 3: (Color online). Effect of nuclear quadrupole deformity on the
$R_{\Delta Q}$, shown by its relative deviation from the spherical case.

The DFT densities we used have assumed spherical nuclei without considering
the possible deformation in their calculations. Their experimental
measurements of the quadrupole deformation parameter are also uncertain,
ranging from $\beta_{2}=0.053$ to $0.158$ for Ru and from $0.08$ to $0.217$
for Zr Deng et al. (2016). To test the sensitivity of our results to nuclear
deformation, we follow Ref. Xu et al. (2021) using the Woods-Saxon
parametrization
$\rho(r,\theta)\propto[1+\exp([r-R(1+\beta_{2}Y_{2}^{0}(\theta))]/a)]^{-1}$
with $\beta_{2}=0.16$ fixed and the $R$ and $a$ reproducing the first and
second radial moments of the DFT-calculated spherical proton and neutron
densities of SHF/SLy4. The changes in $R_{\Delta Q}$ from the spherical case
are shown in Fig. 3 for the three combinations of deformed/spherical Ru and Zr
nuclei. The effect is only a couple of percent, leading possibly to less than
5 MeV uncertainty in the extracted $L_{c}$ parameter. This small effect is not
a surprise because the $R_{\Delta Q}$ is sensitive only to the relative
proton/neutron composition on the nuclear surface, which is not sensitive to
nuclear deformation.

Next we examine our idea using a dynamical model. We use the urqmd (Ultra
relativistic Quantum Molecular Dynamics, v3.4) Bass et al. (1998); Bleicher et
al. (1999) as it have been widely used to study the conserved charge number
and its fluctuations in heavy ion collisions Adamczyk et al. (2014); Xu et al.
(2016). We simulate events within $b\in[7,20]$ fm since we focus only on
peripheral collisions. The same acceptance cuts have been applied as performed
in pythia simulations. Figure 4 shows $R_{\Delta Q}$ as a function of $N_{\rm
ch}$. Similar splittings are found at low $N_{\rm ch}$ as in Fig. 2. urqmd
simulations of NN interactions indicate $\alpha=-0.344$. Using this $\alpha$
value, the predicted curves by Eq. (2) are superimposed in Fig. 4. The curves
can fairly well describe the urqmd data. This indicates that the grazing
collisions in urqmd with $N_{\rm ch}\lesssim 10$ are indeed simple
superimposition of NN interactions. This is not surprising as only a few
nucleons participate in such a grazing AA collision so any nuclear effect
would be negligible. At higher $N_{\rm ch}$ the urqmd data points deviate from
the curves, presumably because those collisions are not simple NN
superimpositions any more. It may also be viewed as that the effective
$\alpha$ in central AA collisions, because of nuclear effects, is very
different from the one calculated using single NN interactions.

Figure 4: (Color online). $R_{\Delta Q}$ for $b\in[7,20]$fm as function of
$N_{\rm ch}$ ($|\eta|<0.5$) simulated by urqmd with DFT nuclear densities from
eSHF (Lc20, Lc47, Lc70). $\Delta Q$ is calculated from $|\eta|<1$ and
$0.2<p_{T}<2$ GeV/$c$ excluding the (anti-)protons with $p_{T}<0.4$ GeV/$c$.
The open red squares show a calculation for the Lc47 case including
$\Lambda$-hyperon decays. The curves are Eq. (2) with $\alpha=-0.344$ from
urqmd NN interactions and the $q_{AA}$ from trento in Fig. 1.

The large differences among the nuclear densities are mostly due to the
negative value of $\alpha$ as mentioned before. We note that a larger $p_{T}$
cut on (anti)proton makes $\alpha$ more negative, e.g., $\alpha=-0.416$ with
$p_{T}>0.8{\rm GeV}/c$ on (anti)proton, which may make the $R_{\Delta Q}$ more
sensitive to $\Delta r_{\rm np}$. However, excluding any “net-charge” will
weaken the correlation between the initial protons from the incoming nuclei
and the final-state net-charge observable, which would introduce stronger
model dependence. We also note that the finite negative $\alpha$ value is not
merely because we exclude the (anti)proton with $p_{T}<0.4{\rm GeV}/c$; we
have checked that without any $p_{T}$ cut, the $\alpha$ value is $-0.370$ at
mid-rapidity ($|\eta|<1.0$). In our urqmd simulations, the hyperon decays are
not included. The hyperon decays can introduce net-charge because of the
combined effect of the finite acceptance (which does not always cover both
daughter particles from the decay) and the asymmetry of those decays. We have
calculated the effect of $\Lambda$ decays for the Lc47 case, and found that
the effect is negligible (see the open red squares in Fig.4).

We have simulated $\sim 4.5\times 10^{8}$ urqmd events for each isobar system
in Fig. 4. They give a statistical uncertainty on $L_{c}$ of roughly 9 MeV
using data points at $N_{\rm ch}\lesssim 10$. It should be noted that our
proposal requires measurements of ultra-peripheral collisions which are often
difficult because of trigger inefficiencies in experiment, increasing in
severity with decreasing $N_{\rm ch}$. Over $2\times 10^{9}$ minimum bias
events have been collected at the RHIC for each isobar system; they would
likely give similar statistics to our simulated urqmd sample, given the
typical experimental trigger inefficiency. Thus, the statistical uncertainty
on the extracted $L_{c}$ using isobar data is expected to be $\sim$10 MeV.
Since our $R_{\Delta Q}$ observable is a ratio of two isobar systems, many
experimental systematic effects cancel and would unlikely contribute a
significant part to the overall experimental uncertainty. We note that the
electromagnetic interactions could affect the proposed method. However, in
experiment, one can avoid electromagnetic interactions by triggering on
minimum bias hadronic interactions only.

Our idea relies on the state of the art DFT calculation and the assumption of
simple NN superimposition for grazing isobar collisions. The theorectical
uncertainty from the former may be gauged by the two parameter sets of SLy4
and Lc47 shown in Fig. 2, corresponding to similar $L_{c}$ values. Their
difference would give an uncertainty of $\sim$6 MeV. This is rather modest
compared to the substantial uncertainties from QCD modeling of traditional
scattering observables. Another theoretical uncertainty on density
distributions comes from nuclear deformation; this is insignificant as
indicated by Fig. 3.

The assumption of simple NN superimposition should be rather robust for
grazing heavy ion collisions over a wide range of relativistic energies. We
have focused on $\sqrt{s_{{}_{\rm NN}}}=200$ GeV in this Letter because the
isobar collision data recently taken at RHIC were at this energy Koch et al.
(2017). We note that isospin-sensitive observables, such as the
$\pi^{+}/\pi^{-}$ ratio, have been proposed long ago to probe high-density
symmetry energy in low energy nuclear collisions (not particularly in grazing
collisions) Li (2002). In our study of grazing AA collisions, the NN
information imprint in $\Delta Q$ is carried over by the $\alpha$ parameter.
The $\alpha$ value we used in this Letter comes from a particular tune of
pythia and urqmd. It is used only to illustrate our proposed method; its value
should be taken with caution. For example, pythia6 (version 6.416) tuned to
the RHIC data Zhang et al. (2018) would give $\alpha=-0.315$, the result of
which with the Lc47 density is shown by the dashed curve in Fig. 2. Hijing
gives $\alpha=-0.389$ which is shown by the dashed-dot curve in Fig. 2 for
Lc47. This would result in an uncertainty on $L_{c}$ comparable to the range
of the parameter sets in Fig. 2 (which was constrained from traditional
scattering measurements). The determination of the $\alpha$ parameter is
therefore essential. The $\Delta Q$ data in $pp$ interactions are abundant,
that in $nn$ is experimentally not available. Although $nn$ interactions are
hard to perform, their information may be indirectly accessed by proton-
deuteron and/or deuteron-deuteron collisions, yet to be conducted. In
addition, one may gain valuable information on the $\alpha$ parameter from
existing p-Au and d-Au data at RHIC. Provided that the $\alpha$ parameter can
be determined relatively precisely, our proposal will likely result in a
$L_{c}$ value with an overall uncertainty of 10-15 MeV. This would be a
substantial complement to traditional methods with the present overall
$\sim$20 MeV uncertainty Reed et al. (2021).

To summarize, we demonstrate that the net-charge ratio $R_{\Delta Q}$ in
isobar ${}^{96}_{44}$Ru+${}^{96}_{44}$Ru over
${}^{96}_{40}$Zr+${}^{96}_{40}$Zr collisions can be predicted from those in
$pp$ and $nn$ interactions under the assumption that nucleus-nucleus AA
collisions are a simple superimposition of NN interactions. We show using the
urqmd model that this assumption is valid for ultra-peripheral (grazing)
collisions with $N_{\rm ch}\lesssim 10$ within the midrapidity range of
$|\eta|<0.5$. The predicted $R_{\Delta Q}$ depends on the proton and neutron
densities, particularly on the neutron skin thickness $\Delta r_{\rm np}$. The
$R_{\Delta Q}$ measurement in grazing isobar collisions, together with DFT
calculations of nuclear densities can, therefore, determine the $\Delta r_{\rm
np}$ and, in turn, the symmetry energy density slope parameter $L_{c}$.

We have previously proposed Li et al. (2020) an indirect observable, the
$N_{\rm ch}$ ratio in central isobar collisions, to probe the $\Delta r_{\rm
np}$. This observable is isospin insensitive and is based on the fact that
particle production in relativistic heavy ion collisions depends on the total
nuclear (proton+neutron) density which has some sensitivity to the $\Delta
r_{\rm np}$. The $R_{\Delta Q}$ observable proposed here is isospin sensitive,
directly related to the $\Delta r_{\rm np}$. It is based on the simple
superimposition of NN interactions for those grazing AA collisions, and
depends on the difference (proton $-$ neutron) density which is highly
sensitive to the $\Delta r_{\rm np}$. Combining these two observables will
yield stringent constrains on the $\Delta r_{\rm np}$ and, thus, the $L_{c}$
parameter.

## Acknowledgments

HX thanks Dr. Qiang Zhao and Dr. Wenbin Zhao for useful discussions. FW thanks
Dr. Torbjörn Sjöstrand and Dr. Christian Bierlich for helpful information on
Pythia. This work is supported in part by the National Natural Science
Foundation of China (Grant Nos. 11905059, 11947410, 12035006, 12047568,
12075085, 11625521, U1732138, 11605054, 11505056), the Ministry of Science and
Technology of China (Grant No. 2020YFE0202001), National SKA Program of China
No. 2020SKA0120300, the Zhejiang Provincial Natural Science Foundation of
China (Grant No. LY21A050001), the Natural Science Foundation of Hubei
Province (Grant No. 2019CFB563), and the U.S. Department of Energy (Grant No.
DE-SC0012910).

## References

  * Brown (2000) B. A. Brown, Phys. Rev. Lett. 85, 5296 (2000).
  * Chen et al. (2005) L.-W. Chen, C. M. Ko, and B.-A. Li, Phys. Rev. C72, 064309 (2005), eprint nucl-th/0509009.
  * Roca-Maza et al. (2011) X. Roca-Maza, M. Centelles, X. Vinas, and M. Warda, Phys. Rev. Lett. 106, 252501 (2011), eprint 1103.1762.
  * Tsang et al. (2012) M. B. Tsang et al., Phys. Rev. C86, 015803 (2012), eprint 1204.0466.
  * Horowitz et al. (2014) C. J. Horowitz, E. F. Brown, Y. Kim, W. G. Lynch, R. Michaels, A. Ono, J. Piekarewicz, M. B. Tsang, and H. H. Wolter, J. Phys. G41, 093001 (2014), eprint 1401.5839.
  * Zhang and Chen (2013) Z. Zhang and L.-W. Chen, Phys. Lett. B726, 234 (2013), eprint 1302.5327.
  * Frois and Papanicolas (1987) B. Frois and C. N. Papanicolas, Ann. Rev. Nucl. Part. Sci. 37, 133 (1987).
  * Lapikas (1993) L. Lapikas, Nucl. Phys. A553, 297c (1993).
  * Tarbert et al. (2014) C. M. Tarbert et al., Phys. Rev. Lett. 112, 242502 (2014), eprint 1311.0168.
  * Ray et al. (1992) L. Ray, G. W. Hoffmann, and W. R. Coker, Phys. Rept. 212, 223 (1992).
  * Donnelly et al. (1989) T. W. Donnelly, J. Dubach, and I. Sick, Nucl. Phys. A503, 589 (1989).
  * Horowitz et al. (2001) C. J. Horowitz, S. J. Pollock, P. A. Souder, and R. Michaels, Phys. Rev. C63, 025501 (2001), eprint nucl-th/9912038.
  * Akimov et al. (2017) D. Akimov et al. (COHERENT), Science 357, 1123 (2017), eprint 1708.01294.
  * Abrahamyan et al. (2012) S. Abrahamyan et al., Phys. Rev. Lett. 108, 112502 (2012), eprint 1201.2568.
  * Adhikari et al. (2021) D. Adhikari et al. (PREX), Phys. Rev. Lett. 126, 172502 (2021), eprint 2102.10767.
  * Reed et al. (2021) B. T. Reed, F. J. Fattoyev, C. J. Horowitz, and J. Piekarewicz, Phys. Rev. Lett. 126, 172503 (2021), eprint 2101.03193.
  * Centelles et al. (2009) M. Centelles, X. Roca-Maza, X. Vinas, and M. Warda, Phys. Rev. Lett. 102, 122502 (2009), eprint 0806.2886.
  * Yue et al. (2021) T.-G. Yue, L.-W. Chen, Z. Zhang, and Y. Zhou (2021), eprint 2102.05267.
  * Li et al. (2020) H. Li, H.-j. Xu, Y. Zhou, X. Wang, J. Zhao, L.-W. Chen, and F. Wang, Phys. Rev. Lett. 125, 222301 (2020), eprint 1910.06170.
  * Li et al. (2018) H. Li, H.-j. Xu, J. Zhao, Z.-W. Lin, H. Zhang, X. Wang, C. Shen, and F. Wang, Phys. Rev. C98, 054907 (2018), eprint 1808.06711.
  * Sjostrand et al. (2006) T. Sjostrand, S. Mrenna, and P. Z. Skands, JHEP 05, 026 (2006), eprint hep-ph/0603175.
  * Sjostrand et al. (2008) T. Sjostrand, S. Mrenna, and P. Z. Skands, Comput. Phys. Commun. 178, 852 (2008), eprint 0710.3820.
  * Skands et al. (2014) P. Skands, S. Carrazza, and J. Rojo, Eur. Phys. J. C 74, 3024 (2014), eprint 1404.5630.
  * Abelev et al. (2009) B. I. Abelev et al. (STAR), Phys. Rev. C79, 034909 (2009), eprint 0808.2041.
  * Wang and Gyulassy (1991) X.-N. Wang and M. Gyulassy, Phys. Rev. D44, 3501 (1991).
  * Wang (1997) X.-N. Wang, Phys. Rept. 280, 287 (1997), eprint hep-ph/9605214.
  * Bass et al. (1998) S. A. Bass et al., Prog. Part. Nucl. Phys. 41, 255 (1998), eprint nucl-th/9803035.
  * Bleicher et al. (1999) M. Bleicher et al., J. Phys. G25, 1859 (1999), eprint hep-ph/9909407.
  * Xu et al. (2018) H.-J. Xu, X. Wang, H. Li, J. Zhao, Z.-W. Lin, C. Shen, and F. Wang, Phys. Rev. Lett. 121, 022301 (2018), eprint 1710.03086.
  * Chabanat et al. (1997) E. Chabanat, J. Meyer, P. Bonche, R. Schaeffer, and P. Haensel, Nucl. Phys. A627, 710 (1997).
  * Chabanat et al. (1998) E. Chabanat, P. Bonche, P. Haensel, J. Meyer, and R. Schaeffer, Nucl. Phys. A635, 231 (1998), [Erratum: Nucl. Phys. A643,441(1998)].
  * Wang et al. (2016) X. B. Wang, J. L. Friar, and A. C. Hayes, Phys. Rev. C94, 034314 (2016), eprint 1607.02149.
  * Chamel et al. (2009) N. Chamel, S. Goriely, and J. M. Pearson, Phys. Rev. C80, 065804 (2009), eprint 0911.3346.
  * Zhang and Chen (2016) Z. Zhang and L.-W. Chen, Phys. Rev. C94, 064326 (2016), eprint 1510.06459.
  * Zhang and Chen (2014) Z. Zhang and L.-W. Chen, Phys. Rev. C90, 064317 (2014), eprint 1407.8054.
  * Moreland et al. (2015) J. S. Moreland, J. E. Bernhard, and S. A. Bass, Phys.Rev. C92, 011901 (2015), eprint 1412.4708.
  * Bernhard et al. (2016) J. E. Bernhard, J. S. Moreland, S. A. Bass, J. Liu, and U. Heinz, Phys. Rev. C94, 024907 (2016), eprint 1605.03954.
  * Deng et al. (2016) W.-T. Deng, X.-G. Huang, G.-L. Ma, and G. Wang, Phys. Rev. C94, 041901 (2016), eprint 1607.04697.
  * Xu et al. (2021) H.-j. Xu, H. Li, X. Wang, C. Shen, and F. Wang, Phys. Lett. B 819, 136453 (2021), eprint 2103.05595.
  * Adamczyk et al. (2014) L. Adamczyk et al. (STAR), Phys. Rev. Lett. 112, 032302 (2014), eprint 1309.5681.
  * Xu et al. (2016) J. Xu, S. Yu, F. Liu, and X. Luo, Phys. Rev. C 94, 024901 (2016), eprint 1606.03900.
  * Koch et al. (2017) V. Koch, S. Schlichting, V. Skokov, P. Sorensen, J. Thomas, S. Voloshin, G. Wang, and H.-U. Yee, Chin. Phys. C 41, 072001 (2017), eprint 1608.00982.
  * Li (2002) B.-A. Li, Phys. Rev. Lett. 88, 192701 (2002), eprint nucl-th/0205002.
  * Zhang et al. (2018) S. Zhang, L. Zhou, Y. Zhang, M. Zhang, C. Li, M. Shao, Y. Sun, and Z. Tang, Nucl. Sci. Tech. 29, 136 (2018), eprint 1803.05767.