# Nuclear deformation effects in photoproduction of $\rho$ mesons in
ultraperipheral isobaric collisions

Shuo Lin<EMAIL_ADDRESS>Department of Modern Physics, University of
Science and Technology of China, Anhui 230026, China  Jin-Yu Hu
<EMAIL_ADDRESS>Department of Modern Physics, University of
Science and Technology of China, Anhui 230026, China  Hao-Jie Xu
<EMAIL_ADDRESS>School of Science,Huzhou University,Huzhou, Zhejiang,
313000, China  Strong-Coupling Physics International Research Laboratory
(SPiRL), Huzhou University, Huzhou, Zhejiang 313000, China  Shi Pu
<EMAIL_ADDRESS>Department of Modern Physics, University of Science and
Technology of China, Anhui 230026, China  Southern Center for Nuclear-Science
Theory (SCNT), Institute of Modern Physics, Chinese Academy of Sciences,
Huizhou 516000, Guangdong Province, China  Qun Wang<EMAIL_ADDRESS>Department of Modern Physics, University of Science and Technology of China,
Anhui 230026, China  School of Mechanics and Physics, Anhui University of
Science and Technology, Huainan,Anhui 232001, China

###### Abstract

We have investigated the $\rho^{0}$ meson photoproduction in ultraperipheral
isobaric collisions between ${}_{44}^{96}\textrm{Ru}+_{44}^{96}\textrm{Ru}$
and ${}_{40}^{96}\textrm{Zr}+_{40}^{96}\textrm{Zr}$ at $\sqrt{s_{NN}}=200$
GeV, employing the dipole model with the equivalent photon approximation. By
implementing the Woods-Saxon distribution to represent the nuclear mass
density, which is derived from density functional theory with an inclusion of
nuclear deformation effects, we have calculated the transverse momentum
$q_{T}$ spectra in isobaric collisions. We observe the characteristic dip
behavior in these spectra, indicative of diffraction phenomena in high-energy
physics. We notice that the deformation effects cause a nearly linear increase
with $q_{T}^{2}$ for $q_{T}^{2}\lesssim 0.015$ $\textrm{GeV}^{2}$, aligning
with experimental observations. We offer a simple explanation for the observed
behavior in these spectra by introducing the effective width of the nuclei in
the thickness function. We also extend our discussion on the $\rho^{0}$ meson
photoproduction with the targets ${}^{63}\textrm{Cu}$,${}^{197}\textrm{Au}$,
and ${}^{238}\textrm{U}$.

##  I Introduction

Relativistic heavy-ion collisions (HIC) provide a platform for exploring novel
phenomena under extreme conditions, such as quantum transport phenomena
associated with chiral anomaly (Kharzeev:2007jp, ; Fukushima:2008xe, ;
Huang:2013iia, ; Pu:2014cwa, ; Chen:2016xtg, ; Hidaka:2017auj, ) (also see
recent reviews (Gao:2020vbh, ; Gao:2020pfu, ; Kharzeev:2015znc, ;
Hidaka:2022dmn, )) in ultra-strong electromagnetic fields (Skokov:2009qp, ;
Bzdak:2011yy, ; Deng:2012pc, ; Roy:2015coa, ), and nonlinear effects of
quantum electrodynamics (QED) (Hattori:2012je, ; Hattori:2012ny, ;
Hattori:2020htm, ; STAR:2019wlg, ; Hattori:2022uzp, ; Adler:1971wn, ;
Schwinger:1951nm, ; Copinger:2018ftr, ; Copinger:2020nyx, ; Copinger:2022jgg,
). Furthermore, such strong electromagnetic fields in HIC, regarded as a huge
flux of quasi-real photons, offer a new opportunity to investigate photon-
photon and photon-nuclear interactions, which are significantly enhanced by
proton numbers of colliding nuclei.

Dilepton photoproduction, one of the key processes in photon-photon
interactions, has been extensively investigated in ultraperipheral collisions
(UPC) (ATLAS:2018pfw, ; STAR:2018ldd, ; STAR:2019wlg, ; ALICE:2022hvk, ) using
the equivalent photon approximation (EPA) (Klein:2016yzr, ; Bertulani:1987tz,
; Bertulani:2005ru, ; Baltz:2007kq, ). However it has been found that EPA
cannot describe the data of dilepton photoproduction in UPC accurately,
primarily due to its omission of the impact parameter and the transverse
momentum dependence of initial photons. Then the EPA has been extended to
generalized EPA or QED in the background field approach (Vidovic:1992ik, ;
Hencken:1994my, ; Hencken:2004td, ; Zha:2018tlq, ; Zha:2018ywo, ;
Brandenburg:2020ozx, ; Brandenburg:2021lnj, ; Li:2019sin, ; Wang:2022ihj, )
based on the factorization theorem (Klein:2018fmp, ; Klein:2020jom, ;
Li:2019yzy, ; Xiao:2020ddm, ) and the QED model incorporating a wave-packet
description of nuclei (Wang:2021kxm, ; Wang:2022gkd, ; Lin:2022flv, ). To
account for transverse momentum broadening effects, higher-order contributions
have been considered, such as Coulomb corrections (Ivanov:1998ka, ;
Eichmann:1998eh, ; Segev:1997yz, ; Baltz:1998zb, ; Baltz:2001dp, ;
Baltz:2003dy, ; Baltz:2009jk, ; Zha:2021jhf, ; Klein:2020jom, ; Sun:2020ygb, )
arising from multiple re-scattering within intense electromagnetic fields and
the Sudakov effect pertaining to the soft photon radiation in the final state
(Li:2019sin, ; Li:2019yzy, ; Klein:2018fmp, ; Klein:2020jom, ; Shao:2023zge, ;
Shao:2022stc, ).

Vector meson photoproduction, a form of photon-nucleus interaction in high-
energy collisions, provide rich information for parton structures inside
nuclei and nucleons (Guo:2021ibg, ; Guo:2023qgu, ; Sun:2021pyw, ;
Koempel:2011rc, ; Brodsky:1994kf, ; Hatta:2019lxo, ; Mantysaari:2022ffw, ;
Mantysaari:2023qsq, ; Arslandok:2023utm, ; Achenbach:2023pba, ). Compared to
$ep$ or $eA$ collisions, UPC provide a strong photon flux, thereby offer an
exceptional opportunity to investigate photon-nucleus processes. New data have
been obtained in recent experiments (ALICE:2019tqa, ; ALICE:2020ugp, ;
ALICE:2021gpt, ; STAR:2019wlg, ; STAR:2021wwq, ) for the study of gluon
distributions (Zhou:2022twz, ; Brandenburg:2022jgr, ; Hagiwara:2021xkf, ;
Zha:2020cst, ). Recently, the high-energy version of the double-slit
interference phenomenon, a milestone in UPC physics, was observed in the
process $\gamma A\rightarrow\rho^{0}A$ and $\rho^{0}\rightarrow\pi^{+}\pi^{-}$
(STAR:2022wfe, ). Such an interference phenomenon was studied using the color
glass condensate effective theory (Xing:2020hwh, ) and the vector meson
dominance model (Zha:2020cst, ). Meanwhile, the asymmetry of $\pi^{+}\pi^{-}$
from $\rho^{0}$ decay was discussed in Refs. (Hagiwara:2020juc, ;
Hagiwara:2021xkf, ; Xing:2020hwh, ), and the $\cos(4\phi)$ azimuthal asymmetry
was highlighted as a possible signal for the elliptic gluon distribution
(Hagiwara:2021xkf, ).

While many studies have been conducted on ultraperipheral Au+Au or Pb+Pb
collisions, photon-photon and photon-nuclear interactions in ultraperipheral
isobaric collisions have not yet been systematically discussed. Unlike
spherical nuclei, Ru and Zr nuclei have deformation in charge and density
distributions. In a recent work by some of us (Lin:2022flv, ), the
distributions of the transverse momentum, invariant mass, and azimuthal angle
for dileptons in photoproduction processes of isobaric collisions have been
calculated. The main result of the study suggests that nuclear structure can
significantly influence these spectra. A subsequent study about the nuclear
deformation effect on dilepton photoproduction is presented in Ref.
(Luo:2023syp, ). The excitation of Ru and Zr nuclei through photon-nuclear
interaction and neutron emission is discussed in Ref. (Zhao:2022dac, ).

Table 1: Parameters in the Woods-Saxon distributions for Ru and Zr in (a)
deformed case and (b) spherical case, followed by the procedure given in Refs.
(Xu:2021qjw, ; Xu:2021vpn, ). The quoted values for $R$ and $a$ are in fm.

(a) with deformation | $R$ | $a$ | $\beta_{2}$ | $\beta_{3}$
---|---|---|---|---
Ru | 5.093 fm | 0.471 fm |  0.16 | 0
Zr | 5.021 fm | 0.517 fm | 0 |  0.20

(b) without deformation $R$ $a$ $\beta_{2}$ $\beta_{3}$ Ru 5.093 fm 0.487 fm 0
0 Zr 5.022 fm 0.538 fm 0 0

In this work, we investigate the photoproduction of $\rho^{0}$ mesons in
ultraperipheral isobaric collisions using the dipole model with EPA. We
introduce deformation parameters into the mass density distribution inspired
by state-of-the-art calculations in density functional theory. We will
calculate transverse momentum spectra of $\rho^{0}$ mesons by photoproduction
in ultraperipheral isobaric Ru+Ru and Zr+Zr collisions at $\sqrt{s_{NN}}=$200
GeV.

The paper is organized as follows. In Sec. II and III, we briefly introduce
the dipole model and phenomenological models and parameters for numerical
calculations, respectively. The numerical results on transverse momentum
spectra in isobaric collisions are presented in Sec. IV. We will discuss
possible application to photon-nuclear interaction at the Electron Ion
Collider (EIC) in Sec. V. A summary of the results is given in Sec. VI.
Throughout this work, we adopt the convention for the metric tensor
$g_{\mu\nu}=\textrm{diag}\\{+,-,-,-\\}$.

##  II Theoretical framework

Before delving into the theoretical framework, we first outline the strategy
of this work. For a comprehensive analysis, one might follow the theoretical
framework established in Refs. (Xing:2020hwh, ; Zha:2020cst, ), which include
the impact parameter and the transverse momentum dependence of the initial
photons. This approach aligns with very recent studies on vector meson
photoproduction in heavy ion collisions(Mantysaari:2023prg, ), as well as the
investigation of fluctuating nucleon substructure in $J/\psi$ photoproduction
in Ref. (Mantysaari:2022sux, ). However, the differential cross sections for
these processes are exceedingly complex, and it presents a challenge to
discern how nuclear deformation effects influence the final differential cross
section. To streamline our discussion, we implement the EPA, thereby
neglecting the impact parameter and transverse momentum dependence of the
initial photons in our current study. We will demonstrate that even with the
EPA, it is possible to capture the essential physics of nuclear deformation
effects in vector meson photoproduction in isobaric collisions.

The cross section for the vector mesons production in the ultraperipheral
heavy ion collisions can be written in the EPA (e.g. also see the review paper
(Bertulani:2005ru, )). In EPA, the vector meson production can be understood
as the photon flux generated by one nucleus collides with another nucleus to
create vector mesons. If we focus on the case in which the photon flux is
generated by the nucleus $A_{1}$, then the cross section can be simply written
as,

$\displaystyle\sigma$ $\displaystyle=$ $\displaystyle\int
d\omega\frac{n_{1}(\omega)}{\omega}\sigma_{\gamma
A_{2}\rightarrow\rho^{0}A_{2}}(\omega),$ (1)

where $\sigma_{\gamma A\rightarrow\rho^{0}A}$ is the cross section for the
sub-progress $\gamma A\rightarrow\rho A$ and $n_{1}(\omega)$ is the photon
flux with energy $\omega$ generated by nucleus $A_{1}$ and can be further
estimated by integrating of the equivalent photon flux per unit area over the
impact parameter.

$\displaystyle n(\omega)$ $\displaystyle=$
$\displaystyle\frac{2Z^{2}\alpha}{\pi\beta^{2}}\left\\{\xi
K_{0}(\xi)K_{1}(\xi)-\frac{\xi^{2}}{2}\left[K_{0}^{2}(\xi)-K_{1}^{2}(\xi)\right]\right\\}.$

with $\xi=2\omega R_{A}/\gamma\beta$, $\alpha,\gamma,Z$ being the fine-
structure constant, Lorentz factor of the nucleus, and the nuclear charge
number, respectively. $K_{i}(\xi)$ are Bessel function of order $i$.

In realistic nucleus-nucleus collisions, both nucleus $A_{1}$ and $A_{2}$ can
be sources of photon flux and then scatter with the alternative nucleus.
Therefore, the cross section in Eq. (1) should be extended as,

$\displaystyle\sigma$ $\displaystyle=$ $\displaystyle\int
dy[n_{1}(y)\sigma_{\gamma A_{2}\rightarrow\rho
A_{2}}(y)+n_{2}(-y)\sigma_{\gamma A_{1}\rightarrow\rho A_{1}}(-y)],$

where $y=\ln(2\omega/m_{V})$ is the momentum rapidity of vector mesons with
$m_{V}$ being the mass of vector meson. We emphasize that there should be
interference terms in the square of amplitude in principle, e.g. see Refs.
(Xing:2020hwh, ; Zha:2020cst, ; Mantysaari:2023prg, ; STAR:2022wfe, ) for the
discussion on the quantum interference in relativistic heavy ion collisions.
In EPA, these interference terms are dropped for simplicity. As mentioned, we
concentrate on the effects caused by the nuclear structure in the current work
instead of other effects. We will comment on the possible contributions from
these interference terms at the end of the next section.

Since we also focus on the effects in the low transverse momentum $q_{T}$
region of vector mesons, i.e. $q_{T}\lesssim 0.13$ GeV, the coherent process
dominates. The coherent differential cross section for vector meson production
in $\gamma A$ can described by the dipole model (Brodsky:1994kf, ;
Kowalski:2003hm, ; Kowalski:2006hc, ; Ryskin:1992ui, ),

$\displaystyle\frac{d\sigma^{\gamma A\rightarrow\rho^{0}A}}{dq_{T}^{2}}$
$\displaystyle=$ $\displaystyle\frac{1}{16\pi}\left|\mathcal{A}\right|^{2},$
(4)

The scattering amplitude $\mathcal{A}$ can be conventionally expressed as the
convolution of the dipole scattering amplitude
$N(\mathbf{r}_{T},\mathbf{b}_{T})$, and the vector meson wave function
$\Psi^{V\rightarrow q\bar{q}*}(r_{\bot},z)$ and photon splitting functions
$\Psi^{\gamma\rightarrow q\bar{q}}(r_{\bot},z)$, in position space,

$\displaystyle\mathcal{A}$ $\displaystyle=$ $\displaystyle 2i\int
d^{2}\mathbf{b}_{T}e^{i\mathbf{\Delta}_{T}\cdot\mathbf{b}_{T}}\int\frac{d^{2}\mathbf{r}_{T}}{4\pi}\intop_{0}^{1}dz$
$\displaystyle\times\Psi^{\gamma\rightarrow
q\bar{q}}(\mathbf{r}_{T},z)N(\mathbf{r}_{T},\mathbf{b}_{T})\Psi^{V\rightarrow
q\bar{q}*}(\mathbf{r}_{T},z),$

where two dimensional $\mathbf{r}_{T}$ is the transverse size of the
$q\overline{q}$ dipole, two dimensional $\mathbf{b}_{T}$ is the impact
parameter of the dipole relative to the target center, $-\mathbf{\Delta}_{T}$
is the nucleus recoil transverse momentum and $z=k^{+}/p^{+}$, with
$k^{+}$being the light cone momentum of the quark in the $q\overline{q}$ pair
and $p^{+}$the light cone momentum of the photon. In the EPA, the transverse
momentum of photon is negligible, thus, the transverse momentum of the vector
meson $q_{T}$ should equal to $\Delta_{T}$. Here, the dipole and nucleus
scattering amplitude
$N(\mathbf{r}_{T},\mathbf{b}_{T})=1-\frac{1}{N_{c}}\textrm{Tr}\left[U(\mathbf{b}_{T}+\mathbf{r}_{T}/2)U^{\dagger}(\mathbf{b}_{T}-\mathbf{r}_{T}/2)\right]$
where $N_{c}$ is number of colors and
$U(\mathbf{x}_{T})=\mathcal{P}\exp\left[ig\int_{-\infty}^{+\infty}dx^{+}A^{-}(x^{+},\mathbf{x}_{T})\right]$
is the Wilson line (Gelis:2010nm, ; Berges:2020fwq, ) with $\mathcal{P}$
standing for the path ordering.

Figure 1: Transverse momentum $q_{T}$ spectra for $\rho^{0}$ meson (a) with
and (b) without deformation effects. The parameters for with and without
deformation effects correspond to the set of parameters (a) and (b) in Table
1, respectively. The yellow and gray areas in both subfigures represent the
results for Ru+Ru and Zr+Zr collisions, respectively, with error estimates
included from the numerical calculations.

The vector meson wave function $\Psi^{V\rightarrow q\bar{q}*}$ is
nonperturbative and cannot be derived from the first principle calculations.
In the current study, we follow the textbook (Kovchegov:2012mbw, ) and
parameterize vector meson wave function $\Psi^{V\rightarrow q\bar{q}*}$
combined with photon splitting function as

$\displaystyle\Psi^{\gamma\rightarrow q\bar{q}}\Psi^{V\rightarrow q\bar{q}*}$
$\displaystyle=$
$\displaystyle\frac{N_{c}ee_{q}}{\pi}\left\\{m_{q}^{2}K_{0}(|\mathbf{r}_{T}|\varepsilon_{f})\Phi^{*}(|\mathbf{r}_{T}|,z)\right.$
$\displaystyle\left.+\frac{\partial
K_{0}(|\mathbf{r}_{T}|\varepsilon_{f})}{|\partial\mathbf{r}_{T}|}\frac{\partial\Phi^{*}(|\mathbf{r}_{T}|,z)}{\partial|\mathbf{r}_{T}|}\left[z^{2}+(1-z)^{2}\right]\right\\},$
(6)

where $e_{q}$ is the charge of quarks, the parameter $\varepsilon_{f}$ in our
case reduces to $\varepsilon_{f}\approx m_{q}$ and
$\Phi^{*}(|\mathbf{r}_{T}|,z)$ is the scalar part of the vector meson wave
function,

$\displaystyle\Phi^{*}(|\mathbf{r}_{T}|,z)$ $\displaystyle=$
$\displaystyle\beta
z(1-z)\exp\left(-\frac{|\mathbf{r}_{T}|^{2}}{R_{T}^{2}}\right),$ (7)

with $\beta=4.47,R_{T}^{2}=21.9\textrm{GeV}{}^{-2}$ for $\rho$ meson.

Figure 2: The ratios of transverse momentum spectra in isobaric collisions.
The blue solid and green dashed lines represent the results with and without
nuclear deformation effects, respectively. The brown dash-dotted and red
dashed lines correspond to the results from body-body and tip-tip
configurations of Ru+Ru collisions, respectively. The parameters for the mass
density distribution are taken from those listed in Table 1.

Figure 3: Thickness functions of Ru nucleus in the transverse plane for
different configurations:(a) the result computed using the set of parameters
labeled as (b) in Table 1; (b) and (c) represent the results of the body-body
and tip-tip configurations, respectively, calculated using the set of
parameters labeled as (a) in Table 1.  Figure 4: Thickness functions of Ru
nucleus as the function of the distance from the center with different
configuration.The blue solid line corresponds to the spherical configuration,
the red dashed line to the body-body configuration aligned with the y-axis,
the brown dash-dotted line to the body-body configuration along the x-axis,
and the green dashed line to the tip-tip configuration.

##  III Models and parameters

In the previous section, we briefly introduce the cross section for vector
meson productions in the UPCs in EPA. We notice that dipole scattering
amplitude $N(\mathbf{r}_{T},\mathbf{b}_{T})$ is non-perturbtaive. In the
current work, we adopt the conventional parameterization for
$N(\mathbf{r}_{T},\mathbf{b}_{T})$ which are widely used in Refs.
(Brodsky:1994kf, ; Kowalski:2003hm, ; Kowalski:2006hc, ; Ryskin:1992ui, ). The
dipole amplitude are connected to the thickness function
$T_{A}(\mathbf{b}_{T})$ by,

$\displaystyle N(\mathbf{r}_{T},\mathbf{b}_{T})$ $\displaystyle=$
$\displaystyle 1-\exp\left[-2\pi
B_{p}AT_{A}(\mathbf{b}_{T})\mathcal{N}(\mathbf{r}_{T})\right],$ (8)

where $A$ is the number of nucleon in the nuclei $A_{1,2}$, $B_{p}$ is a
constant and $\mathcal{N}(r_{\bot})$ is the dipole-nucleon scattering
amplitude. Following the Ref. (Xing:2020hwh, ), $B_{p}$ is chosen as
$4\textrm{ GeV}{}^{-2}$ $\mathcal{N}(r_{\bot})$ is given by modified IP
Saturation model (Lappi:2010dd, ; Lappi:2013am, ),
$\mathcal{N}(r_{\bot})=1-\exp\left[-r_{\bot}^{2}G(x_{g},\mathbf{r}_{T})\right]$,
where $x_{g}$ is the momentum fraction of gluons and $G(x_{g},\mathbf{r}_{T})$
is the gluon distribution function. For convenience, one usually assumes that
$G(x_{g},\mathbf{r}_{T})$ is approximately independent on $\mathbf{r}_{T}$.
Then, $G(x_{g},\mathbf{r}_{T})$ can be further parameterized as
$G(x_{g})=\frac{1}{4}(x_{0}/x_{g})^{\lambda_{\textrm{GBW}}}$ in the Golec-
Biernat and Wüsthoff (GBW) model (Golec-Biernat:1998zce, ; Golec-
Biernat:1999qor, ). The parameters $x_{0}=3\times 10^{-4}$ and
$\lambda_{\textrm{GBW}}=0.29$ are determined by fitting to HERA data
(Kowalski:2006hc, ).

The nuclear thickness function $T_{A}(\mathbf{b}_{T})$ is integrated nuclear
mass density distribution $\rho(z,\mathbf{b}_{T})$ over the beam direction
$z$,

$T_{A}(\mathbf{b}_{T})=\int dz\rho(z,\mathbf{b}_{T}).$ (9)

The mass density distribution is often taken as the Woods-Saxon distribution
in spherical coordinates $(r,\theta,\phi)$,

$\rho(r,\theta,\phi)=\frac{\rho_{0}}{1+\exp\\{[r-R_{0}(\theta,\phi)]/a\\}},$
(10)

where $a$ is the skin diffuseness. To describe the nucleus has the
deformation, the $R_{0}(\theta,\phi)$ can be expanded by the spherical Bessel
function $Y_{l,m}(\theta,\phi)$. Assuming the nuclei are the axisymmetric, one
can get,

$R_{0}(\theta)=R\left[1+\beta_{2}Y_{2,0}(\theta)+\beta_{3}Y_{3,0}(\theta)+...\right],$
(11)

where $\beta_{i}$ are parameters and $R$ is the radius of nucleus.

In the current study, we aim to learn the effects of nuclear deformation in
the isobaric collisions. The flow observables measured in relativistic
isobaric collisions imply a large quadrupole deformation for Ru ($\beta_{{\rm
2,Ru}}=0.16$) and a large octupole deformation for Zr ($\beta_{3,Zr}=0.20$)
(STAR:2021mii, ; Zhang:2021kxj, ). These deformations cannot be well described
by the density functional theory (DFT) calculations, although some efforts
have been made recently (Rong:2022qez, ). We thus follow the procedure given
in Refs. (Xu:2021qjw, ; Xu:2021vpn, ) to obtain the WS parameters $R$ and $a$
for the given $\beta_{i}$, where the first and second radial moments are
matched to the corresponding nucleus calculated by DFT. The corresponding
parameters in the spherical and deformed cases are listed in Tab. 1. The
differential cross sections in this study are computed by the ZMCintegral
package (Zhang_2020cpc_zw, ; Wu:2019tsf, ) (also see Refs. (Zhang:2019uor, ;
Zhang:2022lje, ) for other applications of the package).

##  IV transverse momentum spectra with deformation effects

We depict the differential cross section $d\sigma/dq_{T}^{2}$, which
demonstrates a decreasing trend as a function of $q_{T}^{2}$ , with and
without nuclear deformation effects, in Figs. 1 (a) and (b), respectively. A
notable feature is the distinct minimum in the cross section at
$q_{T}^{2}\simeq 0.024$ $\textrm{GeV}^{2}$, signifying diffraction patterns
characteristic of nuclear collisions. This signature is consistent with
diffraction phenomena extensively discussed in high-energy physics literature
(Kovchegov:2012mbw, ; Good:1960ba, ; Goulianos:1982vk, ). Furthermore, the
deformation effects are observed to elevate the local minima of the cross
section.

Rather than analyzing the transverse momentum $q_{T}$ spectra for Ru and Zr
independently, the ratio of the differential cross sections for these isotopes
can yield additional insights into nuclear structure. In our preceding
investigations of dilepton photon production in isobaric collisions
(Lin:2022flv, ), we have found that the distributions of charge and mass
density for Ru and Zr can cause the ratios of transverse momentum, invariant
mass, and azimuthal angle distributions for these nuclei to be less than the
fourth power of their charge number ratio, i.e.,
$(Z_{\mathrm{Ru}}/Z_{\mathrm{Zr}})^{4}=(44/40)^{4}$. However, in the present
study, upon disregarding the differences in mass distribution between Ru and
Zr, the differential cross section ratio is anticipated to scale with the
square of the charge number ratio for Ru and Zr, i.e.
$(Z_{\mathrm{Ru}}/Z_{\mathrm{Zr}})^{2}=(44/40)^{2}$.

Surprisingly, we observe a dramatic effect from nuclear deformation present in
Fig. 2. As depicted by the green dashed lines in Fig. 2, we find that the
ratio closely approaches $(44/40)^{2}$ and is nearly independent of
$q_{T}^{2}$ in the absence of nuclear deformation effects. When such effects
are included, as depicted by the blue solid line in Fig. 2, the ratio exhibits
an approximate linear increase with $q_{T}^{2}$ in the low $q_{T}^{2}$ region,
specifically for $q_{T}^{2}\lesssim 0.015$ $\textrm{GeV}^{2}$. This behavior
is consistent with the experimental data (JieZhao:2023, ). When
$q_{T}^{2}\gtrsim 0.015\textrm{ GeV}^{2}$, we find that the numerical errors
increase significantly when considering the effects of nuclear deformation in
Fig. 1 (a). Consequently, to maintain accuracy, we limit our discussion to the
ratios of transverse momentum spectra for $q_{T}^{2}\lesssim 0.015\textrm{
GeV}^{2}$.

To illustrate the impact of nuclear deformation effects on the ratio of
transverse momentum spectra, we analyze two extreme configurations for Ru in
Fig. 2: the tip-tip and body-body Ru+Ru collisions. Intriguingly, the cross
section ratio for tip-tip Ru+Ru collisions relative to averaged Zr+Zr
collisions increases significantly with $q_{T}^{2}$, whereas the ratio for
body-body Ru+Ru collisions decreases with $q_{T}^{2}$.

Now, we offer a simple intuitive explanation for the phenomena observed in
nuclear collisions. The effects of nuclear deformation are manifested as
variations in the shape profiles of the colliding nuclei. These variations can
be quantitatively analyzed using the thickness function, as defined in Eq.
(9). This function is typically parameterized by a Gaussian distribution in
terms of $\mathbf{b}_{T}$, given by

$T_{A}(\mathbf{b}_{T})\sim\exp\left(-\frac{\mathbf{b}_{T}^{2}}{w_{T}^{2}}\right),$
(12)

where $w_{T}$ is a parameter that can be interpreted as the effective width of
the nucleus. To gain a better understanding, we plot the thickness functions
of Ru nucleus in Figs. 3 and 4. Rather than fitting the thickness function to
Gaussian-type functions as in Eq. (12), we estimate $w_{T}$ using

$w_{T}^{2}=\frac{\int
d^{2}\mathbf{b}_{T}T_{A}(\mathbf{b}_{T})\mathbf{b}_{T}^{2}}{\int
d^{2}\mathbf{b}_{T}T_{A}(\mathbf{b}_{T})},$ (13)

The results for various nuclear configurations are summarized in Table 2.

Table 2: Estimation the effective width of nucleus $w_{T}$ for various configurations by using Eq. (13).  | Ru (body) | Ru (tip) | Ru (spherical) | Zr (spherical)
---|---|---|---|---
$w_{T}$ |  $3.628$ fm |  $3.372$ fm |  $3.544$ fm |  $3.571$ fm

We note that the mass density in the tip configuration is more concentrated
than in the spherical and body configurations, particularly in the central
region. Therefore, the effective width $w_{T,\textrm{tip}}^{\mathrm{Ru}}$ is
smaller than $w_{T,\textrm{spherical}}^{\mathrm{Ru}}$. In the case of the
body-body collision configuration, we encounter a divergence in the effective
parameters along the $x$ and $y$ axes, with the averaged
$w_{T,\textrm{body}}^{\mathrm{Ru}}$ , closely mirroring the parameter of the
spherical configuration, suggesting
$w_{T,\textrm{spherical}}^{\mathrm{Ru}}<w_{T,\textrm{body}}^{\mathrm{Ru}}$.
Overall, we observe the following relation,

$w_{T,\textrm{tip}}^{\mathrm{Ru}}<w_{T,\textrm{spherical}}^{\mathrm{Ru}}\approx
w_{T,\textrm{spherical}}^{\mathrm{Zr}}<w_{T,\textrm{body}}^{\mathrm{Ru}}.$
(14)

Consider two rugby balls, representing deformed nuclei, colliding. The contact
area and interaction dynamics change significantly depending on whether the
nuclei hit each other at their ends (tip-tip) or sides (body-body). In tip-tip
collisions, the nuclei interact over a smaller area with higher mass
densities. Conversely, in body-body collisions, the larger contact area
results in a lower mass density distribution.

To be more specific, we can consider the the averaged dipole amplitude in Eq.
(8),

$N(\mathbf{r}_{T},\mathbf{b}_{T})\simeq 2\pi
B_{p}AT_{A}(\mathbf{b}_{T})\mathcal{N}(\mathbf{r}_{T}).$ (15)

The scattering amplitude $\mathcal{A}$ can then be evaluated as,

$\displaystyle\mathcal{A}$ $\displaystyle\sim$ $\displaystyle
A\int\frac{d^{2}\mathbf{r}_{T}}{4\pi}\intop_{0}^{1}dzB_{p}\mathcal{N}(\mathbf{r}_{T})$
(16) $\displaystyle\times\Psi^{\gamma\rightarrow
q\bar{q}}(\mathbf{r}_{T},z)\Psi^{V\rightarrow q\bar{q}*}(\mathbf{r}_{T},z)$
$\displaystyle\times\int
d^{2}\mathbf{b}_{T}e^{i\mathbf{q}_{T}\cdot\mathbf{b}_{T}}\exp\left(-\frac{\mathbf{b}_{T}^{2}}{w_{T}^{2}}\right),$

where, as mentioned in EPA, we have set $\boldsymbol{\Delta}_{T}\simeq q_{T}$.
It is straightforward to find the $q_{T}^{2}$ dependence of
$\mathcal{A}(q_{T}^{2})$ can be expressed as,

$\mathcal{A}(q_{T}^{2})\propto e^{-\frac{1}{4}q_{T}^{2}w_{T}^{2}},$ (17)

Therefore, the ratio of differential cross section for two types of nucleus is

$\displaystyle\frac{d\sigma_{\mathrm{Ru}}/dq_{T}^{2}}{d\sigma_{\mathrm{Zr}}/dq_{T}^{2}}$
$\displaystyle\propto e^{\delta w_{T}q_{T}^{2}},$ (18)

with

$\delta
w_{T}\equiv-\frac{1}{2}\left[(w_{T}^{\mathrm{Ru}})^{2}-(w_{T}^{\mathrm{Zr}})^{2}\right].$
(19)

Now, we can apply the Eq. (18) to interpret Fig. 2. First, since
$w_{T,\textrm{spherical}}^{\mathrm{Ru}}\approx
w_{T,\textrm{spherical}}^{\mathrm{Zr}}$ as shown in Eq. (14) and the ratio for
two spherical nuclei should be nearly constant, aligning with the green lines
in Fig. 2. In realistic Zr + Zr collisions, we find that octupole deformation
effects of the Zr nucleus have an insignificant numerical impact on the
differential cross section. Thus, $\delta w_{T}$ can be estimated using
$w_{T,\textrm{spherical}}^{\mathrm{Zr}}$ in Table 2 for simplicity. According
to the inequality (14), it is straightforward to assert that the ratio of
cross section of the tip-tip Ru + Ru collisions over the averaged Zr+Zr
collisions increases with $q_{T}^{2}$. We obtain the slope parameter $\delta
w_{T}$ from Eq. (18) is approximately $17\textrm{ GeV}^{-2}$, which is roughly
consistent with the slope represented by the red dashed line in Fig. 2,
obtained from linear fitting. Conversely, the body-body Ru+Ru collisions
exhibit opposite behavior for the ratio of the differential cross section. We
find the slope parameter $\delta w_{T}\simeq-5\textrm{ GeV}^{-2}$. This value
is in close agreement with the slope of the brown dash-dotted line depicted in
Fig. 2, as obtained by linear fitting. The results for the realistic Ru + Ru
collisions with the various configurations, corresponding to the blue solid
line in Fig. 2, should fall within the region bounded by those from the tip-
tip and body-body collisions. As a remark, we find that the ratio of the
differential cross section for realistic Ru + Ru and Zr + Zr collisions are
strongly related to the nuclear structure.

Before end of this section, we would like to comment on Eq. (18). We emphasize
that for realistic Ru + Ru collisions, it is insufficient to estimate the
ratio of cross section using effective width $w_{T}$. The effective width
$w_{T}$ does not retain sufficient detail regarding the diverse configurations
of the quadrupole deformed Ru nucleus, distinct from the octupole deformation
of the Zr nucleus mentioned above. In the comparison, the results computed
using Eq. (LABEL:eq:cross_section_02) represents the average over the
differential cross section for various nuclear configurations. This averaging
process can be thought of as a configuration-by-configuration mean.
Importantly, this approach preserves the information of various nuclear
configuration in the differential cross section. For a more rigorous
assessment of $w_{T}$, one should incorporate an appropriate weight function
into Eq. (13), which relates to the differential cross section calculation in
Eq. (LABEL:eq:cross_section_02). Nevertheless, such an analysis falls beyond
the scope of the present work and is reserved for future studies.

Table 3: Parameters in the Woods-Saxon distributions (10) for ${}^{63}\textrm{Cu}$,${}^{197}\textrm{Au}$, and ${}^{238}\textrm{U}$, taken from Refs. (Loizides:2014vua, ).  | $R$ | $a$ | $\qquad\beta_{2}\qquad$ | $\qquad\beta_{3}\qquad$
---|---|---|---|---
${}^{63}\textrm{Cu}$ |  4.200 fm |  0.596 fm | 0 | 0
${}^{197}\textrm{Au}$ | 6.380 fm | 0.535 fm | 0 | 0
${}^{238}\textrm{U}$ | 6.831 fm | 0.598 fm | 0.239 | 0
Figure 5: Differential cross section for the process
$\gamma+A\rightarrow\rho^{0}+A$ in EIC as a function of $q_{T}^{2}$. The gluon
momentum fraction $x_{g}$ is fixed at $1.7\times 10^{-3}$.The yellow,gray and
blue areas corresponds to collisions with ${}^{238}\textrm{U}$
,${}^{197}\textrm{Au}$ and ${}^{63}\textrm{Cu}$ nucleus, respectively, with
error estimates included from the numerical calculations.

##  V Possible application to photon-nuclear interaction in EIC

We now extend our discussion on nuclear structure to the EIC physics. Rather
than using Eq. (LABEL:eq:cross_section_02), we focus on the cross section as
described in Eq. (4). We adhere to the same framework to compute the photon-
nuclear interactions with several nuclear targets proposed for the EIC, namely
${}^{63}\textrm{Cu}$,${}^{197}\textrm{Au}$, and ${}^{238}\textrm{U}$. The
nuclear mass density is using the standard Woods-Saxon distribution as defined
in Eq. (10), with the parameters $R,a,\beta_{2}$ are taken from Ref.
(Loizides:2014vua, ) and listed in Table 3.

The differential cross section (4) for the targets
${}^{63}\textrm{Cu}$,${}^{197}\textrm{Au}$, and ${}^{238}\textrm{U}$, with the
typical gluon momentum fraction $x_{g}=1.7\times 10^{-3}$ (Mantysaari:2023qsq,
), is present in Fig. 5. We note a substantial difference in the transverse
momentum spectra among the various nuclear targets. In the region where
$q_{T}^{2}\lesssim 0.006\textrm{ GeV}^{2}$, $\frac{d\sigma_{{\rm
U}}}{dq_{T}^{2}}>\frac{d\sigma_{{\rm Au}}}{dq_{T}^{2}}>\frac{d\sigma_{{\rm
Cu}}}{dq_{T}^{2}}$, which is primarily attributed to the differences in the
number of nucleons.

The characteristic dip behavior, associated with diffraction phenomena in
high-energy physics, is observed for all three types of nuclei, consistent
with expectations. We note that the larger the nuclear radius $R$, the more
rapidly the dip is reached. We also expect that a larger nuclear radius $R$
correlates with a greater effective width $w_{T}$ in the thickness function in
Eq. (12). The decay rate of the differential cross section for the three types
of nuclei before encountering their first dip aligns with our analysis of the
effective width.

##  VI Summary and discussion

We have investigated the $\rho^{0}$ meson photoproduction in ultraperipheral
isobaric collisions between Ru+Ru and Zr+Zr at $\sqrt{s_{NN}}=200$ GeV,
utilizing the dipole model with EPA. In this study, the dipole amplitude is
linked to the thickness functions of the nuclei, which in turn are derived by
integrating the nuclear mass density distributions over beam direction. The
nuclear mass density used here is inspired by DFT calculations and
incorporates the effects of nuclear deformation.

We calculated the transverse momentum spectra in Ru+Ru and Zr+Zr collisions
and observed the characteristic dip behavior in these spectra, indicative of
diffraction phenomena in high-energy physics. We further analyzed the ratio of
the transverse momentum spectra for the two types of nuclei. Our findings
suggest that in the absence of deformation effects, the ratio closely
approaches $(44/40)^{2}$ , whereas the inclusion of deformation effects can
result in an approximate linear increase with $q_{T}^{2}$, for
$q_{T}^{2}\lesssim 0.015$ $\textrm{GeV}^{2}$. This pattern aligns with the
trends observed in experimental data.

We introduce the effective width $w_{T}$ of nuclei within the thickness
function framework and derive the dependence of the differential cross section
on $w_{T}$. By analyzing the variation of $w_{T}$ in tip-tip and body-body
configurations of Ru + Ru collisions, we offer a straightforward and intuitive
explanation for the observed behavior in the ratio of transverse momentum
spectra between the two types of nuclei.

We also studied the $\rho^{0}$ meson photoproduction with the targets
${}^{63}\textrm{Cu}$,${}^{197}\textrm{Au}$, and ${}^{238}\textrm{U}$ proposed
in EIC. We find a significant dependence on the nuclear structure in the
transverse momentum spectra. Our results indicate that it is possible to study
the nuclear structure in the future high energy collisions.

###### Acknowledgements.

We would like to thank Xiao-feng Wang, Cheng Zhang, Du-xin Zheng for helpful
discussion. This work is supported in part by the National Key Research and
Development Program of China under Contract No. 2022YFA1605500, by the Chinese
Academy of Sciences (CAS) under Grants No. YSBR-088 and by National Nature
Science Foundation of China (NSFC) under Grants No. 12075235, No. 12135011,
No. 12275082, No. 12035006, and No. 12075085.

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